![List of figures ix
2.8 Reset narrow band compensator 43
2.9 Structure of a typical HDD 44
2.10 The Bode plots of the plant and the open loop with the PI controller 45
2.11 Mid-frequency disturbance rejection 45
2.12 Responses to sinusoid disturbance at 800 Hz 46
2.13 Responses to sinusoid disturbance at 800 Hz (with input disturbance,
output disturbance and white noises inserted) 47
3.1 Negative feedback interconnection of H1 and H2 54
3.2 Reset control system 57
3.3 Phase trajectory of an RCS with an unstable baseline system. (Left:
baseline system; right: reset system.) 59
3.4 The Nyquist plots of Hβ(s − ε) for different β 63
3.5 RCS with time-delay 64
3.6 Reset interval for different initial state from the set of after-reset states 75
3.7 Max eigenvalue of eAT t
RPReAt
− P for different r 76
3.8 State response for x(0) = [2 −1 6]T
and r = −0.5 76
3.9 A reset control system 77
4.1 Responses for the uncertainty-free case 108
4.2 Responses for uncertainty γ (t) = 0.2sin(20t) 108
4.3 Responses for constant uncertainty γ (r = −0.5) 109
4.4 The curves of γl and γr over τ. (For any given τ ∈ [0, 1.095],
the RCS is robustly stable respect to time-varying uncertainty
γ (t) ∈ [γl(τ), γr(τ)]) 110
4.5 Initial condition response to x(0) = [−2 2]T
111
4.6 Input up 111
5.1 An RCS with discrete-time triggering condition 123
5.2 Step responses for the baseline system, reset mode, and the RCS with
Rr = 0 124
5.3 Step responses of the reset mode for different Rr ∈ [−1, 1] 124
5.4 Step responses of the RCS for different Rr ∈ [−1, 0] 124
5.5 Step responses for different Rr ( = 4.5Ts) 128
5.6 Step responses of the reset mode for different Rr 128
5.7 Step responses for different (Rr = 0) 128
5.8 Comparison between traditional and proposed reset control 129
5.9 Control input 129](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-14-320.jpg)
![2 Analysis and design of reset control systems
4
3
2
1
0
10−2
10−1
100
w/w0
101
102
ln
coth
υ
−
2
Figure 1.2 The curve of the weighting function ln coth(|v|/2)
However, this requirement often conflicts with the desired phase margin due to
Bode’s gain-phase relation. In order to make this clear, suppose that the open-loop
transfer function L(s) is stable and of minimum phase. The Bode’s gain-phase relation
states that the phase of L( jω) is uniquely determined by its gain. Precisely, the phase
at frequency ω0 (rad/s) is given by [1]
∠ L( jω0) =
1
π
∞
−∞
d ln |L( jω0ev
)|
dv
ln coth
v
2
dv (rad) (1.3)
with v = ln (ω/ω0). The weighting function ln coth (|v|/2) = ln
ω+ω0
ω−ω0
in the integral
is plotted in Figure 1.2. From this figure, one sees that the integration is mainly
contributed by the weighted rate of the gain-decreasing near ω0. Note that
d ln |L( jω0ev
)|
dv
=
1
20
d[20 lg |L( jω)|]
d lg (ω/ω0)
,
thus a rule of thumb states that if the slope of the magnitude curve near ω0 is 20N
dB/decade and this slope does not change much over a decade-wide interval near ω0,
then the phase angle at ω0 can be approximately calculated through [2]
∠ L( jω0) ≈
N
π
∞
−∞
ln coth
v
2
dv =
N
π
·
π2
2
=
πN
2
(rad). (1.4)
The above Bode’s gain-phase relation indicates that one cannot manipulate the gain
and the phase independently and, in order to assure sufficient phase margin, the rate
of the gain-decreasing near the crossover frequency ωc cannot be too fast. In practice,
the slope of the open-loop gain at the gain-crossover frequency should be designed
to be between −20 and −30 dB/decade in order to assure stability. This tradeoff is
explained in Figure 1.3 where the shaded areas represent the bounds defined by the
low frequency, high frequency, and phase specifications, respectively. If one wants to
design a feedback controller to increase low-frequency gain, broaden the bandwidth,
or decrease the high-frequency gain as indicated as the dash line in Figure 1.3, then](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-21-320.jpg)
![Introduction 3
Avoided area
for sufficient
phase margin
Phase (°)
0
−180
Gain (dB)
Avoided area
for large low-
frequency gain
and sufficient
bandwidth
0
Avoided area
for small high-
frequency gain
Figure 1.3 Frequency domain specifications
the phase near the crossover frequency would inevitably decrease and even violate the
phase margin bound. This tradeoff is unavoidable in the framework of linear feedback
control and is called the “cost of feedback” in Reference 3.
The above inherent limitation motivates researchers to adopt nonlinear filters,
which are with smaller phase lag but with similar magnitude slope characteristics,
to relieve the frequency domain limitations imposed by Bode’s gain-phase constraint
explained above. Early efforts along this line include the nonlinear feedback proposed
by J.B. Lewis [4], the nonlinear gain element by R.E. Kalman [5], and other nons-
mooth filters [6, 7]. In recent years, many hybrid control design techniques have been
developed to overcome the limitations of the traditional linear feedback design. Such
techniques include switching control [8], sliding mode control [9], impulsive control
[10], etc.
In this book, we focus on a special kind of hybrid techniques called reset control
whose original motivation is to overcome the inherent limitation imposed by the
Bode’s gain-phase constraint. The study of reset control can be traced several decades
back. The first reset element is the so-called Clegg integrator (CI) proposed by Clegg
in 1958 [11]. The CI is described by the impulsive differential equation [11, 12]
ż = e, ze 0
z+
= 0, ze ≤ 0
(1.5)](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-22-320.jpg)
![4 Analysis and design of reset control systems
which consists of a linear integrator (LI)
ż = e
and a reset mechanism. When the input e and the output z of the integrator have the
same sign, then it evolves according to the LI. On the other hand, if the input and the
output have opposite signs, then the state is reset to zero. The notation z+
denotes
the state of the integrator after the reset. The condition ze ≤ 0 is called the reset
condition which determines when the state of the integrator should be reset. In some
recent literature, the triggering condition ze ≤ 0 is replaced by the zero-crossing of
the input, i.e., e = 0. In this case, the CI is represented by [13]
ż = e, e = 0
z+
= 0, e = 0.
(1.6)
These two models of CI are different and lead to different development of reset control
theory. However, as basic reset elements, these two models have the same output
response to a sinusoidal input. Thus they have the same sinusoidal input describing
function (DF). Figure 1.4 gives the output responses of the CI and the LI to a sinusoidal
input. The DF of the CI is given by [11]
GCI ( jω) = GLI ( jω)
1 + j
4
π
(1.7)
1
0.5
0
−0.5
Input
signal
−1
2
1.5
1
0.5
Output
of
LI
0
0 5 10 15
0 5 10 15
2
1
0
−1
Output
of
CI
−2
0 5 10 15
Output of CI
First-hormonic of the output
Figure 1.4 Output responses of LI and CI to sinusoidal input e = sin t](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-23-320.jpg)
![Introduction 5
where
GLI ( jω) =
1
jω
is the frequency response function of the LI ż = e. The calculation of the sinusoidal
input DFs of reset systems will be discussed in detail in Chapter 2. The formula (1.7)
shows that the DF of the CI equals the frequency response function of the LI scaled
by a complex factor 1 + j(4/π), which means that the reset action does not change
the slope of the logarithmic magnitude-frequency characteristics but causes a phage
lead of
φ = arctan (4/π) ≈ 51.9◦
(1.8)
at all frequencies. This property suggests that reset elements could be used to achieve
required bandwidth with much lower high-frequency gain, which means more rapid
gain-decreasing within the mid-frequency range is allowed, without degrading the
phase margin specification.
Another basic reset element with more flexibility is the so-called first-order reset
element (FORE) which was proposed in Reference 3. The state-space representation
of a zero-crossing type FORE is
ż = λz + e, e = 0
z+
= 0, e = 0
, (λ ≤ 0) (1.9)
whose sinusoidal input DF is
GFORE( jω) = GL( jω) (1 + j(ω)) (1.10)
where
GL( jω) =
1
jω − λ
is the frequency response function of the first-order linear element ż = λz + e and
(ω) =
2
π
1 + eπλ/ω
1 + (λ/ω)2
0.
The Bode plots of the FORE and the corresponding first-order linear element are
compared in Figure 1.5. Different from the CI, a FORE behaves like a linear element
over low-frequency range but like a CI over high-frequency range. This property
provides more useful design flexibility [3]. The DFs of the CI and the FORE have
been applied to many practical systems showing benefits of reset control. See for
instance References 14 and 15.
The benefit of reset control can also be shown in the time domain. In practice,
integral control is usually used to remove steady-state error. But the side effect is that
it may cause large overshoot and prolong the settling time. Theoretically, differential
control can be used to reduce the overshoot caused by integration without degrading
the rise time specification, but in practice, differential control may make the system
sensitive to sensor noise. Replacing the integrator by the CI can help to achieve better
tradeoff between rise time and overshoot. This benefit was recognized in the early](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-24-320.jpg)
![8 Analysis and design of reset control systems
y
y(t)
yI (t – t1)
yL (t)
yI (t – t1)
t2
A1
A1 0
1 + b
–
t1 t
1
O
Figure 1.9 Reducing overshoot through reset
the first reset, the output response of the system with CI is the same as the linear
system. During the interval [t1, t2], there holds that
y(t) = yL(t) −
1
1 + b
A1yI (t − t1).
Note that A1 0, thus the reset action at t1 plays the role of reducing the first peak. If
the underlying linear system T(s) is of the second order with damping ratio ζ ∈ [0, 1),
then the precise value of the overshoot can be calculated as [16, 17]
MR(ζ, b) =
1
1 + b
b − 2ζ exp
ζ
1 − ζ2
(π − arccos ζ) ML (1.11)
where ML is the overshoot of the step response of the linear system T(s) given by
ML = exp
−
πζ
1 − ζ2
.
It is easy to check that if
b ≥ 2ζ exp
ζ
1 − ζ2
(π − arccos ζ) , (1.12)
then
0 ≤ MR(ζ, b) ML.
If b is chosen to be
0 b 2ζ exp
ζ
1 − ζ2
(π − arccos ζ) ,
then
MR(ζ, b) 0,](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-27-320.jpg)
![Introduction 9
i.e., undershoot occurs. The above analysis indicates that if the parameter b is appro-
priately chosen, the replacement of the LI by the CI can reduce or even remove the
overshoot without degrading the rise time specification.
One interesting question is that if the performance achieved by the reset control
can also be achieved by a linear compensator. The following example, which is
constructed in Reference 18, gives a negative answer. Consider a linear control system
depicted in Figure 1.1 with di = 0, do = 0, and n = 0 where the plant P(s) contains
an integrator and C(s) is stabilizing. Suppose that the reference input is the unit step,
i.e., r(t) = 1(t). Define
z(t) =
t
0
e(s)ds.
Then the Laplace transform of z(t) is
Z(s) := L(z(t)) =
1
s
L(e(t)) =
1
s
E(s) =
1
s2
1
1 + P(s)C(s)
.
Thus by the final value theorem of Laplace transform, there holds
∞
0
e(s)ds = lim
s→0
sZ(s) =
1
lim
s→0
sP(s)C(s)
:=
1
Kv
.
Consider the rise time defined by
tr := sup
T
T : y(t) ≤
t
T
, t ∈ [0, T] . (1.13)
There holds
1
Kv
=
∞
0
e(s)ds
≥
tr
0
1 −
s
tr
ds +
∞
tr
e(s)ds
=
tr
2
+
∞
tr
e(s)ds.
If tr 2/Kv, then
∞
tr
e(s)ds 0
which means that the unit step response overshoots. Keep this in mind and consider
the special case that P(s) = 1/s. Suppose that we aim to design a controller such that
the system is stable and satisfies:
1. the steady-state error of the unit ramp response is not greater than 1;
2. the rise time tr of the unit step response is greater than 2 s;
3. the step response does not overshoot.
The requirements (1) and (2) mean that the velocity error constant Kv ≥ 1 and the rise
time tr 2 ≥ 2/Kv.Thus according to the previous analysis, if a linear control is used,](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-28-320.jpg)
![Introduction 11
where xc ∈ Rnc is the controller state, e = r − y is the tracking error, and r is the
reference input, Ac, Bc, Cc, and Dc are constant matrices, ρc is the reset mapping, and
x+
c represents the state of the controller after reset, i.e.,
x+
c (t) = lim
s→t+0
xc(s).
Combining the plant (1.14) and the reset controller (1.15), we have the closed-loop
system
ẋ = Ax + Br, r − Cx = 0
x+
= ρ(x, r), r − Cx = 0
(1.16)
with x = [xT
p , xT
c ]T
and
A =
Ap − BpDcCp BpCc
−BcCp Ac
, B =
BpDc
Bc
C =
Cp 0
, ρ(x, r) =
xp
ρc(xc, r)
.
The system
ẋ = Ax + Br (1.17)
is called the baseline system of the reset system (1.16). For set-point regulation, the
reference input r is a constant input. In this case, a state x̄r is called as an equilibrium
point of reset system (1.16) if
⎧
⎨
⎩
Ax̄r + Br = 0
Cx̄r − r = 0
ρ(x̄r, r) − x̄r = 0.
(1.18)
Any state satisfying the first two equations of (1.18) is an equilibrium point of the
baseline system, and any point satisfying the last equation of (1.18) is an equilibrium
point of the reset mapping. Equation (1.18) actually imposes a constraint on the
reset mapping, i.e., the reset action should not destroy the equilibrium point of the
baseline system. Precisely, suppose that x̄r = [x̄T
pr, x̄T
cr]T
is the equilibrium point of
the baseline system which is to be stabilized.Then the reset mapping should be chosen
such that x̄r is also an equilibrium point of the reset mapping. In the conventional reset
control, the reset mapping is of the form
ρc(xc, r) = x̄cr + Rc(xc − x̄cr), (1.19)
where the matrix Rc is called the reset matrix which is of the form
Rc =
Inc−nρ 0(nc−nρ)×nρ
0nρ×(nc−nρ) 0nρ×nρ
(1.20)
with nρ an integer satisfying 0 ≤ nρ ≤ nc. Im and 0m×n represent the identity matrix
with dimension m and the m × n zero matrix, respectively. This structure means that
whenever the tracking error crosses zero, part of the controller state is reset to its
steady-state value.](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-30-320.jpg)
![Introduction 15
Assumption 1.3. For any ξ0 ∈ Rn
, there is a positive number μ(ξ0) 0 such that
k (ξ0) ≥ μ(ξ0), ∀k.
Note that if Assumption 1.3 holds, then it implicitly implies that deadlock and
beating would not occur and for any initial state ξ0, there is a unique solution ξ(t; ξ0)
to RCS (1.26) which can be represented as
ξ(t; ξ0) = W(t, ξ0)ξ0 (1.33)
where W(t, ξ0) is the state transition matrix given by
W(t, ξ0) = exp (A(t − τk (ξ0)))R exp (A k (ξ0)) · · · R exp (A 0(ξ0)) (1.34)
for t ∈ (τk (ξ0), τk+1(ξ0)].
In general, Assumption 1.3 is difficult to check since it depends on the reset
time instants directly. In practice, the existence of solutions can be assured by time-
regularization. For instance, with time-regularization, the reset controller (1.15)
becomes
⎧
⎨
⎩
ẋc = Acxc + Bce, (e = 0) ∨ (τ ≤ m)
x+
c = ρc(xc, r), (e = 0) ∧ (τ m)
u = Ccxc + Dce
(1.35)
where m is a pre-specified positive number, ∨ and ∧ represent the logical OR and
AND, respectively, and τ is a variable characterizing the time spent since the latest
reset which is governed by the impulsive differential equation
τ̇ = 1, (e = 0) ∨ (τ ≤ m)
τ+
= 0, (e = 0) ∧ (τ m).
(1.36)
The structure of RCS is depicted in Figure 1.13. The closed loop becomes
ξ̇ = Aξ, (ξ /
∈ J c
) ∨ (τ ≤ m)
ξ+
= Rξ, (ξ ∈ J ) ∧ (τ m).
(1.37)
With time-regularization, two successive reset time instants are separated by at least
m units of time. Thus deadlock, beating or Zenoness would not happen and for any
initial state, a unique well-defined solution always exists.
1.2.3 RCSs with discrete-time reset conditions
In practical computer-based implementations, reset controllers have to be discretized.
For instance, the simplest discrete-time counterpart of the zero-crossing condition
e(t) = 0 (1.38)
in reset controller (1.15) is
e(kTs)e((k − 1)Ts) 0 (1.39)](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-34-320.jpg)
![16 Analysis and design of reset control systems
Reset
controller
Triggering
condition
Plant
y
u
e
t
r +
–
Figure 1.13 An RCS with time-regularization
where Ts is the sampling period. In addition, reset actions occur only at sampling time
instants. Thus the discrete-time form of the reset controller (1.15) can be represented
by
⎧
⎪
⎪
⎨
⎪
⎪
⎩
ẋc = Acxc + Bce, t ∈ ((k − 1)Ts, kTs]
xc(kT+
s ) = xc(kTs), ek ek−1 0
xc(kT+
s ) = ρc(xc(kTs), r), ek ek−1 ≤ 0
u = Ccxc + Dce
(1.40)
where ek := e(kTs). A more general discrete-time reset controller considered in this
book is of the form
⎧
⎪
⎪
⎨
⎪
⎪
⎩
ẋc = Acxc + Bce, t ∈ ((k − 1)Ts, kTs]
xc(kT+
) = xc(kT), φ(ek , ek−1) 0
xc(kT+
s ) = ρc(xc(kTs), r), φ(ek , ek−1) ≤ 0
u = Ccxc + Dce.
(1.41)
where
φ : R2
→ R
is called the triggering function which characterizes the reset condition. This form
of triggering condition is capable of describing more general cases. A useful kind of
triggering functions is given by the quadratic form
φ(η) = ηT
Sη, η ∈ R2
(1.42)
where S is a symmetric matrix. For instance, the triggering function with
S =
0 ( /Ts + 1)/2
( /Ts + 1)/2 − /Ts
(1.43)
defines a reset law under which reset time instants are ahead of zero-crossings, where
0 is a nonnegative constant roughly characterizing the time for which the reset
action is ahead of the zero-crossing. Actually, in this case,
φ(ek , ek−1) = ek−1
ek − ek−1
Ts
+ ek
(1.44)](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-35-320.jpg)
![18 Analysis and design of reset control systems
FORE
e ur u
d
y
n
r +
–
C(s) P(s)
Figure 1.15 A FORE RCS
e
v
1
s + b
ur
Figure 1.16 A FORE equivalence
The pole −b of the FORE will be an adjustable parameter in the design procedure.
Denote by ti the ith reset time instant with 0 = t0 t1 t2 · · · tn ≤ ∞.
n might be finite or infinite. In the case when n = ∞, there holds tn = ∞. Suppose
that e(t) = 0 for all t 0. Similar to the discussion regarding the CI in Section 1.1,
the output response of the FORE can be calculated as
ur(t) = e−bt
t
ti
ebs
e(s)ds, t ∈ [ti, ti+1)
= e−bt
t
0
ebs
e(s)ds −
tk ≤t
tk
tk−1
ebs
e(s)ds
= e−bt
t
0
ebs
[e(s) + v(s)] ds
where
v(t) = −
n
k=1
Ak δ(t − tk )
Ak =
tk
tk−1
e−b(tk −s)
e(s)ds.
This means that the FORE can be alternatively represented as a linear system with an
additional input v(t) as depicted in Figure 1.16. Denote by hL(t) and yL(t), respectively,
the impulse and the step responses of the baseline system, i.e., with the FORE being
replaced by 1
s+b
, and by y(t) the step response of the RCS (with d = 0 and n = 0).
Then there holds
y(t) = yL(t) −
n
k=1
Ak hL(t − tk )
= yL(t) −
n
k=1
Ak ẏL(t − tk ).](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-37-320.jpg)
![Introduction 19
For t ∈ [0, t1], we have y(t) = yL(t) and yL(t1) = 0. For t ∈ [t1, t2),
y(t) = yL(t) − A1ẏL(t − t1)
with
A1 =
t1
0
e−b(t1−s)
[1 − yL(s)] ds 0.
From the above formula, if the step response of the baseline system is as depicted in
Figure 1.9, there holds
ẏL(t) 0, t ∈ [0, t̃1],
where t̃1 represents the time instant of the first peak of yL(t). Thus we have
y(t) yL(t), t ∈ [t1, t1 + t̃1].
This intuitively explains that reset tends to reduce the overshoot.
If b = 0, then the FORE degenerates to the CI in which case A1 is the area under
the curve e(t) within the interval [0, t1]. In general, there holds b = 0. Thus, A1 is the
area under the curve e−b(t1−t)
e(t) and the FORE pole plays the role of decreasing A1.
If the pole of the FORE is chosen to be b 0, we have Ak ≈ 0 and hence
y(t) ≈ yL(t), t ≥ 0.
This explains that the replacement of the CI by the FORE provides more flexibility
in the control design.
If the baseline open-loop transfer function is the second-order system
L(s) =
1
s + b
P(s)C(s) =
ω2
n
s(s + 2ζωn)
, (0 ζ 1), (1.47)
then the overshoot of the RCSs is given by
Overshoot = exp
−πζ
1 − ζ2
− , (1.48)
where
=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
R
1 − 4ζ2M + 4ζ2M2
4ζ2
M2
e−ζμ
− 2ζM(1 − 4ζ2
M)e− μ
2ζM
, ζ ≥ 0.5
R
1 − 2ζM + M2
M2
e−ζμ
− M(1 − 2ζM)e− μ
M
, ζ ≤ 0.5
(1.49)
with
R = exp
−
ζ
1 − ζ2
arccos ζ
M =
ωc
b
, μ =
π − arccos ζ
1 − ζ2
.](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-38-320.jpg)
![20 Analysis and design of reset control systems
177
100
80
Overshoot
(%)
60
40
20
0
–20
–40
–60
25 22.5 20 17.5 15 12.5 10 7.5 5 2.5 0 –2.5 –5
176 174
M = 0
M = 0.5
M = 1
M = 2
M = 5
M = 20
–1.84
172 170 166
–ArgL (jwe
) (º)
162 156 147 136 120 104
L ( jwe
)
1 + L ( jwe
)
(dB)
M =
b
we
Figure 1.17 Overshoot with M = ωc/b as parameter
In the above, ωc is the gain-crossover frequency of the open-loop transfer function,
i.e., |L( jωc)| = 1. Denote by GL(s) the closed-loop transfer function, i.e.,
GL(s) =
L(s)
1 + L(s)
=
ω2
n
s2 + 2ζωns + ω2
n
and define D = |GL( jωc)|. The damping ratio ζ can be related to D with
ζ =
1
2D
4D2 − 1
2(D2 − 1)
. (1.50)
Thus the overshoot of the second-order RCS can be represented as a function of
D and M, which is plotted in Figure 1.17 with M the parameter. From this figure,
when b is too small, undershoot occurs which explains the disadvantage of the CI.
Figure 1.17 is obtained from second-order systems but is also a good approxima-
tion for higher-order systems which are dominant by a second-order dynamics. This
is the main tool in the Horowitz’s design procedure.
The Horowitz’s design [3] is depicted in Figure 1.18 where the reset controller
consists of two parts, i.e., the linear compensator C(s) and the FORE with pole −b.
The linear controller C(s) is of the form
C(s) = (s + b)CL(s). (1.51)
Theabovestructureassuresthatthebaselinedynamicsdoesnotchangewhileadjusting
the parameter b, i.e.,
L(s) =
1
s + b
P(s)C(s) = P(s)CL(s).](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-39-320.jpg)
![Introduction 21
r e ur u
d
y
n
C(s) P(s)
v
1
+ + +
–
s + b
Figure 1.18 A FORE RCS equivalence
The Horowitz’s design includes two steps:
● Step 1: Design a linear controller CL(s) such that the baseline system satisfies
both the disturbance rejection and sensor noise suppression at the expense of
violating the gain/phase margin constraint;
● Step 2: Choose a FORE pole −b to reduce the overshoot.
The above design procedure was applied to a tape-speed control system in [14]
where the plant is given by
P(s) =
5.32 × 105
(s2
− 6.95 × 102
s + 2.16 × 105
)
(s + 14.02)(s2 + 7.451 × 101s + 2 × 104)(s2 + 2.03 × 102s + 3.28 × 104)
.
The specifications for this system are as follows:
1. Thedisturbanceattenuationlevelfromd toe withinthefrequencyrangeω ≤ 2 Hz
is not greater than 0.5, i.e.,
1
1 + L( jω)
≤ 0.5, ω ≤ 2 Hz . (1.52)
2. The sensor-noise suppression ratio for ω ≥ 10 Hz is not greater than 0.4, i.e.,
L( jω)
1 + L( jω)
≤ 0.4, ω ≥ 10 Hz . (1.53)
3. The steady-state error for step input is zero and the overshoot is not greater than
20%.
It has been shown that within the framework of linear feedback design, it is difficult to
find a controller which meets all of the above specifications. By using the Horowitz’s
design, a linear controller CL(s) is first designed to meet the first two specifications
with
CL(s) =
11.3(s + 257.1)(s + 3.606)(s2
+ 20.33s + 216.3)
s(s + 220.3)(s + 36.49)(s2 + 14.29s + 62.48)
×
(s2
+ 55.51s + 3679)(s2
+ 75.74s + 1.579 × 104
)
(s2 + 88.25s + 5925)(s2 + 125.7s + 1.579 × 104)
.
The crossover frequency of L(s) = P(s)CL(s) is approximately ωc ≈ 30 rad/s and
−arg L( jωc) ≈ 150◦
. From Figure 1.17, choosing a FORE pole b = 30 (i.e., M = 1)](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-40-320.jpg)
![22 Analysis and design of reset control systems
reduces the overshoot to 20%. Thus the resulting reset controller consists of a FORE
with b = 30 and a linear compensator
C(s) = (s + 30)GL(s). (1.54)
1.3.2 PI + CI reset design
Proportional-plus-integral control with Clegg integrator (PI + CI) reset control
design is a DF-based design method aiming to improve the closed-loop response
under a linear PI controller [19]. The main idea is to reset a percentage of the LI to
retain the benefit of LI such as the capability of eliminating steady-errors, and at the
same time increase the phase margin and the gain-crossing frequency.
The first step of the PI + CI reset controller design is to design a linear PI of the
form
C(s) = Kp
1 +
1
TI s
(1.55)
based on the desired phase margin and crossover frequency, where Kp is the propor-
tional constant and TI is the integral time constant. Then decompose the integral part
as
1
TI s
=
1
Tis
+
1
Tirs
and replace the second integrator in the above decomposition by a CI as depicted in
Figure 1.19. The integral constants satisfy
1
TI
=
1
Ti
+
1
Tir
and the ratio Ti/Tir characterizes the percentage of the integral to be reset. The DF
of the PI + CI compensator is given by
Cr( jω) = Kp
1 +
1
jωTi
+
1
jωTir
1 + j
4
π
. (1.56)
The remaining step is to tune the ratio Ti/Tir according to the final performance
requirement of the RCS.
Kp
1 1 +
+
e
Ti
s
CI
u
1
Tir
Figure 1.19 PI + CI reset controller](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-41-320.jpg)
![Introduction 23
In the case when the plant is second order given by
P(s) =
ω2
n
(s2 + ξωn + jωd)(s2 + ξωn − jωd)
, (1.57)
where ξ (0 ξ 1) is the damping ratio, ωn the undamped natural frequency and
ωd = ωn 1 − ξ2, then the parameters of the PI + CI controller can be determined as
follows.
1. Linear PI tuning. Based on the desired gain-crossover frequency ωg and the
phase margin φm, calculate TI and Kp by
∠P( jωg)C( jωg) = φm
|P( jωg)C( jωg)| = 1.
(1.58)
Or, precisely,
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
φm =
1
2
π + arctan ωgTI − arctan
ωg + ωd
ξωn
− arctan
ωg − ωd
ξωn
Kpω2
n
ωgTI
1 + ω2
gT2
I
[ξ2ω2
n + (ωg + ωd)2][ξ2ω2
n + (ωg − ωd)2]
= 1
(1.59)
2. PI + CI parameter design. Select a feasible crossover frequency ωgr for the
RCS and calculate Tir according to
⎧
⎨
⎩
|P( jωgr)Cr( jωgr)| = 1
1
TI
=
1
Ti
+
1
Tir
.
(1.60)
Simple calculation gives
Tir =
4
π
β(ωgr) − 1/T2
I − ωgr
, (1.61)
where β is given by
β(ωgr) =
ω2
gr
K2
p ω4
n
ξ2
ω2
n + (ωgr + ωd)2
ξ2
ω2
n + (ωgr − ωd)2
. (1.62)
(1.61) gives the range of ωgr one can choose from, i.e.,
β(ωgr) − 1/T2
I ωgr.
The phase margin φrm of the PI + CI RCS is given by
φrm = ∠P( jωgr)Cr( jωgr)
=
1
2
π + arctan [ωgr + 4/(Tirπ)]TI − arctan
ωg + ωd
ξωn
− arctan
ωg − ωd
ξωn
.](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-42-320.jpg)
![24 Analysis and design of reset control systems
Notes
A detailed description of the limitations of linear feedback control can be found in
Reference 1 and the example showing that reset control achieves a set of specifications
which cannot be achieved by linear control comes from Reference 18. The early
developments of reset control including the CI, the first time-domain design around
the CI, and the FORE (the Horiwitz’s design procedure) studied in this chapter are,
respectively, from References 11, 16, and 3. The application example of Horiwitz’s
design procedure to the tape-speed control systems can be found in Reference 14.
See also the References 13, 17, and 15 and the recently published book [20] for
more detailed discussions of traditional reset control analysis. The basic PI + CI reset
control design comes from Reference 19. See also Reference 21 for the application
of PI + CI reset control design to heat exchangers.
The model (1.5) for the CI is from Reference 12. The definitions of solutions
and the well-posedness of RCSs are based on Reference 20. For the theory of general
hybrid systems and impulsive differential equations, reader can refer to References
22, 23, and 10. Other nonlinear techniques aiming to overcome limitations of linear
feedback control can be found in References 24, 8, 9, 7, 4, and 6.
References
[1] M.M. Seron, J.H. Braslavsky, and G.C. Goodwin. Fundamental limitations in
filtering and control. In Communications and Control Engineering. Springer-
Verlag London Ltd., Berlin, Heidelberg, New York, 1997.
[2] J.S. Freudenberg and D.P. Looze. Frequency domain properties of scalar and
multivariable feedback systems. Springer-Verlag New York, Inc., 1988.
[3] I. Horowitz and P. Rosenbaum. Nonlinear design for cost of feedback reduction
in systems with large parameter uncertainty. International Journal of Control,
21(6):977–1001, 1975.
[4] J.B. Lewis. The use of nonlinear feedback to improve the transient response
of a servomechanism. American Institute of Electrical Engineers, Part II:
Transactions of the Applications and Industry, 71(6):449–453, 1953.
[5] E.R.Kalman. Phase-planeanalysisofautomaticcontrolsystemswithnonlinear
gain elements.American Institute of Electrical Engineers, Part II:Applications
and Industry, Transactions of the, 73(6):383–390, 1955.
[6] A.R. Bailey. Stabilisation of control systems by the use of driven limiters.
Electrical Engineers, Proceedings of the Institution of, 113(1):169–174, 1966.
[7] W.C. Foster, D.L. Gieseking, and W.K. Waymeyer. A nonlinear filter for inde-
pendent gain and phase (with applications). Journal of Basic Engineering,
88:457, 1966.
[8] D. Liberzon. Switching in systems and control. In Systems Control:
Foundations Applications, Birkhäuser, Boston, Basel, Berlin, 2003.
[9] C. Edwards and S. Spurgeon. Sliding mode control: theory and applications.
In Systems and Control, CRC Press, Taylor Francis Group, 1998.](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-43-320.jpg)
![Introduction 25
[10] T.Yang. Impulsive control theory. In Lecture Notes in Control and Information
Sciences, vol. 272, Springer-Verlag Berlin, Heidelberg, Springer, 2001.
[11] J.C. Clegg. A nonlinear integrator for servomechanisms. Institute of Electrical
Engineers, Part II:Transactions of theApplications and Industry, 77(1):41–42,
1958.
[12] L. Zaccarian, D. Nesic, and A.R. Teel. First order reset elements and the Clegg
integrator revisited. In American Control Conference, 2005. Proceedings of
the 2005. June 8–10, 2005, vol. 1, pp. 563–568. IEEE, Portland, OR, 2005.
[13] O. Beker. Analysis of reset control systems. Doctoral dissertations, Univer-
sity of Massachusetts Amherst, Available from Proquest. Paper AAI3027178.
http://scholarworks.umass.edu/dissertations/AAI3027178, January 1, 2001.
[14] Y. Zheng, Y. Chait, C.V. Hollot, M. Steinbuch, and M. Norg. Experimental
demonstrationofresetcontroldesign. ControlEngineeringPractice, 8(2):113–
120, 2000.
[15] Y. Chait and C.V. Hollot. On Horowitz’s contributions to reset control.
International Journal of Robust and Nonlinear Control, 12(4):335–355, 2002.
[16] K.R. Krishnan and I.M. Horowitz. Synthesis of a nonlinear feedback system
with significant plant-ignorance for prescribed system tolerances. Inter-
national Journal of Control, 19(4):689–706, 1974.
[17] Q. Chen. Reset control systems: stability, performance and application.
PhD thesis, University of Massachusetts Amherst, Available from Proquest.
Paper AAI9988771. http://scholarworks.umass.edu/dissertations/AAI9988771,
Januray 1, 2000.
[18] O. Beker, C.V
. Hollot, and Y. Chait. Plant with integrator: an example of
reset control overcoming limitations of linear feedback. IEEE Transactions on
Automatic Control, 46(11):1797–1799, 2001.
[19] A. Baños and A. Vidal. Design of PI+CI reset compensators for second order
plants. In IEEE International Symposium on Industrial Electronics, pp. 118–
123. IEEE, 2007.
[20] A. Baños and A. Barreiro. Reset control systems. Springer Science Business
Media, 2011.
[21] A. Vidal and A. Baños. Reset compensation for temperature control: Exper-
imental application on heat exchangers. Chemical Engineering Journal,
159(1–3):170–181, 2010.
[22] R. Goebel, R.G. Sanfelice, and A.R. Teel. Hybrid dynamical systems:
modeling, stability, and robustness. Princeton University Press, 2012.
[23] W.M. Haddad,V
. Chellaboina, and S.G. Nersesov. Impulsive and hybrid dynam-
ical systems: stability, dissipativity, and control. In Princeton Series inApplied
Mathematics, Princeton University Press, Princeton, New Jersey, 2006.
[24] Z. Sun and S.S. Ge. Switched linear systems: control and design. Springer-
Verlag, Berlin, Heidelberg, 2005.](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-44-320.jpg)
![Describing function analysis of reset systems 29
In order to obtain the DF of system (2.1), we study a sinusoid input response
first. Consider the sinusoid excitation input
e(t) = α sin (ωt). (2.6)
The set of the reset time instants {tk } is given by
tk = kπ/ω, k = 0, 1, . . . .
Define
ηk = x(t+
2k ),
ζk = x(t+
2k+1)
and
ψ(t)
=
t
0
e−As
b sin (ωs)ds.
Then the solution of system (2.1) with initial condition x(0+
) = η0 can be represented
by
x(t) =
eA(t−t2k )
ηk + αeAt
[ψ(t) − ψ(t2k )], t ∈ (t2k , t2k+1];
eA(t−t2k+1)
ζk + αeAt
[ψ(t) − ψ(t2k+1)], t ∈ (t2k+1, t2k+2]
(2.7)
where ηk and ζk are determined by the following recursive algorithm
ζk = Re
π
ω A
ηk + αψ(π
ω
) ,
ηk+1 = Re
π
ω A
ζk − αψ(π
ω
) , η0 = x(0+
).
(2.8)
Definition 2.1. The recursion (2.8) is said to converge globally if there exist constant
vectors ζ̄ and η̄ such that
lim
k→+∞
ηk = η̄, (2.9)
lim
k→+∞
ζk = ζ̄ (2.10)
for an arbitrary initial condition η0.
The following lemma gives the necessary and sufficient condition for the global
convergence of the recursion (2.8).
Lemma 2.1. The recursion (2.8) converges globally if and only if ω ∈ R, or
equivalently,
λ(Re
π
ω A
) 1. (2.11)
In this case, we have
η̄
= lim
k→+∞
ηk = −α I + Re
π
ω A
−1
Re
π
ω A
ψ(
π
ω
), (2.12)
ζ̄
= lim
k→+∞
ζk = α I + Re
π
ω A
−1
Re
π
ω A
ψ(
π
ω
). (2.13)](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-48-320.jpg)
![30 Analysis and design of reset control systems
Proof: (Sufficiency) Assume that (2.11) holds, then I + Re
π
ω A
is nonsingular. Let
ξk = ηk + α I + Re
π
ω A
−1
Re
π
ω A
ψ(
π
ω
),
we have
ξk+1 = Re
π
ω A
2
ξk . (2.14)
Since
λ (Re
π
ω A
)2
= λ(Re
π
ω A
)
2
1,
we have
lim
k→+∞
ξk = 0.
Thus (2.12) holds. (2.13) can be proved similarly.
(Necessity) From (2.8) we have
ηk+1 = Re
π
ω A
2
ηk + χ,
where
χ = α Re
π
ω A
2
ψ(
π
ω
) − αRe
π
ω A
ψ(
π
ω
)
is a constant vector. Let η̄ = limk→∞ ηk , then we have
Re
π
ω A
2
η̄ + χ = η̄.
Further, by defining
ξk = ηk − η̄,
we have
lim
t→∞
ξk = 0
for any ξ0 and
ξk+1 = Re
π
ω A
2
ξk .
Thus condition (2.11) holds.
By Lemma 2.1, we have the following result immediately.
Proposition 2.1. Reset system (2.1) with input (2.6) has a globally asymptotically
stable 2π/ω-periodic solution if and only if
λ(Re
π
ω A
) 1. (2.15)
If the above condition holds, then the globally asymptotically stable solution is
given by
x(t) =
eA(t−t2k )
η̄ + αeAt
[ψ(t) − ψ(t2k )], t ∈ (t2k , t2k+1],
eA(t−t2k+1)
ζ̄ + αeAt
[ψ(t) − ψ(t2k+1)], t ∈ (t2k+1, t2k+2]
where η̄ and ζ̄ are defined in (2.12) and (2.13).](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-49-320.jpg)
![Describing function analysis of reset systems 31
Remark 2.2. From Proposition 2.1, the existence of globally asymptotically periodic
solution of an RCS under a sinusoid input with certain frequency ω does not imply
that the base linear system is stable. In addition, condition (2.15) depends on input
frequency ω. But for a linear system without reset (i.e., R = I), condition (2.15)
implies that A must be a stable matrix. In this case, condition (2.15) is equivalent to
Re λ(A) 0 which is independent of ω.
Remark 2.3. From equation (2.14), |λ(Re
π
ω A
)| is related to the convergence rate of the
periodic solution. This can also be seen in the following example and the simulation
result of Section 2.3.
Example 2.1. Consider a reset system with
A =
−1 1
0 1
, b =
0
1
, c =
0 1 , R =
1 0
0 β
.
It is easy to check that
Re
π
ω A
=
e− π
ω 1
2
[e
π
ω − e− π
ω ]
0 βe
π
ω
.
Thus for any fixed input frequency ω 0, the condition (2.15) holds if and only if
|β| e− π
ω .
The output responses of this system for initial condition (5, −1) with β = −0.3 and 0,
respectively, are given in Figure 2.2.
1.5
1
0.5
0
Output
responses
−0.5
−1
−1.5
0 0.5 1 1.5 2
Time (s)
2.5 3 3.5 4
Sinusoidal input
Output response for b = −0.3
Output response for b = 0
Figure 2.2 Output responses for Example 2.1](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-50-320.jpg)
![32 Analysis and design of reset control systems
In practical systems, the actual input contains signals of different frequencies.
Thus the boundedness of solutions under a sinusoid input with arbitrary frequency is
an important property for RCSs. And this property is also the basis for a nonlinear
element to have a DF defined on arbitrary frequency. From Proposition 2.1, the
following result is obvious.
Proposition 2.2. RCS (2.1) has a globally asymptotically stable 2π/ω-periodic
solution under a sinusoid input with arbitrary frequency ω 0 if and only if
|λ(ReAδ
)| 1, ∀δ ∈ R+
. (2.16)
Remark 2.4. From condition (2.16), by letting δ → 0+
, one sees that R must be
such that |λ(R)| ≤ 1. However, A is not necessarily Lyapunov stable. For instance, in
Example 2.1, A is unstable. But if β = 0, then
ReAδ
=
e−Aδ 1
2
[eAδ
− e−Aδ
]
0 0
and condition (2.16) holds.
Remark 2.5. If condition (2.16) holds, then system (2.1) has a sinusoid input DF
defined on arbitrary frequency.
Remark 2.6. If ω /
∈ , then ω2
I + A2
is nonsingular. In this case, it is easy to check
that
ψ(t) = −[ω2
I + A2
]−1
(ωI cos (ωt) + A sin (ωt))e−At
− ωI b.
Thus
ψ(
π
ω
) = [ω2
I + A2
]−1
(I + e− π
ω A
)ωb. (2.17)
2.2 Describing function
2.2.1 General case
In this section, we calculate the DF of system (2.1). According to the analysis of
the previous section, the response y(t) of system (2.1) under a sinusoid input with
frequency ω ∈ R can be represented as
y(t) = yss(t) + yt(t), (2.18)](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-51-320.jpg)
![Describing function analysis of reset systems 33
where yss(t) is the steady-state response which is 2π/ω-periodic and independent of
the initial condition, and yt(t) is the transient response which dies away with time,
i.e.,
lim
t→∞
yt(t) = 0. (2.19)
The sinusoidal DF of system (2.1) is defined by
GR( jω) =
Yss,1( jω)
E( jω)
, (2.20)
where
Yss,1( jω) =
ω
2π
2π
ω
0
yss(t)e−jωt
dt, (2.21)
E( jω) =
ω
2π
2π
ω
0
e(t)e−jωt
dt. (2.22)
In the following, we assume that ω ∈ R . By (2.7), (2.8) and (2.17), when t ∈
(tk , tk+1],
yss(t) = αcT
eAt
(−1)k+1
e−Atk R(ω)ωb + ψ(t) − ψ(tk )
= αcT
eAt
θk (ω) − αcT
−1
(ω)[ωI cos (ωt) + A sin (ωt)]b
where
θk (ω)
= (−1)k+1
e−Atk R(ω)ωb − ψ(tk ) + −1
(ω)ωb
= (−1)k+1
e−Atk [R(ω) − −1
(ω)]ωb.
Thus the fundamental component of the Fourier series of yss(t) can be calculated as
follows
Yss,1( jω) =
ω
2π
2π
ω
0
yss(t)e−jωt
dt
=
αωcT
2π
(I1 + I2) −
αωcT
−1
(ω)
2π
(ωIJ1 + AJ2)b,
where
I1 =
π
ω
0
eAt
θ0(ω)e−jωt
dt = ( jωI − A)−1
(ω)θ0(ω),
I2 =
2π
ω
π
ω
eAt
θ1(ω)e−jωt
dt
= −( jωI − A)−1
(ω)e
π
ω A
θ1(ω),
J1 =
2π
ω
0
e−jωt
cos (ωt)dt =
π
ω
,
J2 =
2π
ω
0
e−jωt
sin (ωt)dt = −j
π
ω
.](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-52-320.jpg)
![34 Analysis and design of reset control systems
Thus we have
Yss,1( jω) = −
jα
2
cT
( jωI − A)−1
(I + jR(ω))b (2.23)
where
R(ω)
= −
2ω2
π
(ω)[R(ω) − −1
(ω)] (2.24)
=
2ω2
π
(I + e
π
ω A
)(I + Re
π
ω A
)−1
(I − R)(ω2
I + A2
)−1
(2.25)
It is very easy to calculate that
E( jω)
=
ω
2π
2π
ω
0
α sin (ωt)e−jωt
dt = −
jα
2
.
The following result follows.
Theorem 2.1. The sinusoidal DF of system (2.1) is given by
GR( jω) = cT
( jωI − A)−1
(I + jR(ω))b (2.26)
which is well-defined on ω ∈ R , where R(ω) is given by (2.25).
Remark 2.7. If the output function is replaced by
y = cT
x + de, (2.27)
then the DF becomes
GR( jω) = cT
( jωI − A)−1
(I + jR(ω))b + d. (2.28)
Remark 2.8. If A has no pure imaginary eigenvalues, i.e., = ∅, and condition
(2.16) holds, then we have R = R+
. That is to say, the DF is well-defined for an
arbitrary positive number ω.
Remark 2.9.
1. When R = I, system (2.1) degenerates to the base linear system. In this case,
by (2.25), it is easy to verify that R(ω) = 0. Hence the DF degenerates to the
frequency response function of the base linear system.
2. If A is a stable matrix, we have
lim
ω→0
e
π
ω A
= 0.
Thus, limω→0 R(ω) = 0. This means that the reset does not have much effect
over the very low-frequency range ω π A .
3. It is easy to check that
R(∞)
= lim
ω→∞
R(ω) =
4
π
(I + R)−1
(I − R).
This indicates that the effect of the reset is almost independent of frequency in a
high-frequency region with ω π A .](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-53-320.jpg)
![Describing function analysis of reset systems 35
Remark 2.10. Though system (2.1) is essentially a nonlinear system, the DF is
independent of the amplitude α of the excitation signal.
Corollary 2.1. If the reset matrix is zero, i.e., R = 0n×0, then the sinusoidal DF of
system (2.1) is given by
GR( jω) = cT
( jωI − A)−1
I + j
2ω2
π
(I + e
π
ω A
)(ω2
I + A2
)−1
b (2.29)
which is well-defined on ω ∈ R+
. In addition, there holds
lim
ω→∞
GR( jω) =
1 + j
4
π
GL( jω) (2.30)
where
GL( jω) = cT
( jωI − A)−1
b
is the frequency response function of the base linear system.
By using Corollary 2.1, the DF of the first-order reset element (FORE)
⎧
⎨
⎩
ż = −bz + e, e = 0
z+
= 0, e = 0
ur = z.
(2.31)
is given by
GFORE( jω) =
1
jω + b
1 + j
2ω2
(1 + e−πb/ω
)
π(ω2 + b2)
. (2.32)
Let b = 0, then we get the DF of the Clegg integrator (CI) as
GCI ( jω) =
1
jω
1 + j
4
π
.
Example 2.2. [2] Consider a second-order plant y(s) = P(s)u(s) with
P(s) =
s + 1
s(s + 0.2)
controlled by a FORE
⎧
⎨
⎩
ẋr = −xr + e, e = 0
z+
= 0, e = 0
u = xr
(2.33)
where e = r − y. The open-loop dynamics is then of the form (2.1) with
A =
⎡
⎣
−0.2 0 2
1 0 0
0 0 −1
⎤
⎦ , b =
⎡
⎣
0
0
1
⎤
⎦ , c = [0.5 0.5 0], R =
⎡
⎣
1 0 0
0 1 0
0 0 0
⎤
⎦ .](https://image.slidesharecdn.com/3418670-250617173835-12912cdf/85/Analysis-And-Design-Of-Reset-Control-Systems-Yuqian-Guo-Lihua-Xie-54-320.jpg)
Analysis And Design Of Reset Control Systems Yuqian Guo Lihua Xie
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- 6. IET CONTROL ENGINEERING SERIES 94 Analysis and Design of Reset Control Systems
- 7. Other volumes in this series: Volume 8 A History of Control Engineering, 1800–1930 S. Bennett Volume 18 Applied Control Theory, 2nd Edition J.R. Leigh Volume 20 Design of Modern Control Systems D.J. Bell, P.A. Cook and N. Munro (Editors) Volume 28 Robots and Automated Manufacture J. Billingsley (Editor) Volume 33 Temperature Measurement and Control J.R. Leigh Volume 34 Singular Perturbation Methodology in Control Systems D.S. Naidu Volume 35 Implementation of Self-tuning Controllers K. Warwick (Editor) Volume 37 Industrial Digital Control Systems, 2nd Edition K. Warwick and D. Rees (Editors) Volume 39 Continuous Time Controller Design R. Balasubramanian Volume 40 Deterministic Control of Uncertain Systems A.S.I. Zinober (Editor) Volume 41 Computer Control of Real-time Processes S. Bennett and G.S. Virk (Editors) Volume 42 Digital Signal Processing: Principles, devices and applications N.B. Jones and J.D.McK. Watson (Editors) Volume 44 Knowledge-based Systems for Industrial Control J. McGhee, M.J. Grimble and A. Mowforth (Editors) Volume 47 A History of Control Engineering, 1930–1956 S. Bennett Volume 49 Polynomial Methods in Optimal Control and Filtering K.J. Hunt (Editor) Volume 50 Programming Industrial Control Systems Using IEC 1131-3 R.W. Lewis Volume 51 Advanced Robotics and Intelligent Machines J.O. Gray and D.G. Caldwell (Editors) Volume 52 Adaptive Prediction and Predictive Control P.P. Kanjilal Volume 53 Neural Network Applications in Control G.W. Irwin, K. Warwick and K.J. Hunt (Editors) Volume 54 Control Engineering Solutions: A practical approach P. Albertos, R. Strietzel and N. Mort (Editors) Volume 55 Genetic Algorithms in Engineering Systems A.M.S. Zalzala and P.J. Fleming (Editors) Volume 56 Symbolic Methods in Control System Analysis and Design N. Munro (Editor) Volume 57 Flight Control Systems R.W. Pratt (Editor) Volume 58 Power-plant Control and Instrumentation: The control of boilers and HRSG systems D. Lindsley Volume 59 Modelling Control Systems Using IEC 61499 R. Lewis Volume 60 People in Control: Human factors in control room design J. Noyes and M. Bransby (Editors) Volume 61 Nonlinear Predictive Control: Theory and practice B. Kouvaritakis and M. Cannon (Editors) Volume 62 Active Sound and Vibration Control M.O. Tokhi and S.M. Veres Volume 63 Stepping Motors, 4th Edition P.P. Acarnley Volume 64 Control Theory, 2nd Edition J.R. Leigh Volume 65 Modelling and Parameter Estimation of Dynamic Systems J.R. Raol, G. Girija and J. Singh Volume 66 Variable Structure Systems: From principles to implementation A. Sabanovic, L. Fridman and S. Spurgeon (Editors) Volume 67 Motion Vision: Design of compact motion sensing solution for autonomous systems J. Kolodko and L. Vlacic Volume 68 Flexible Robot Manipulators: Modelling, simulation and control M.O. Tokhi and A.K.M. Azad (Editors) Volume 69 Advances in Unmanned Marine Vehicles G. Roberts and R. Sutton (Editors) Volume 70 Intelligent Control Systems Using Computational Intelligence Techniques A. Ruano (Editor) Volume 71 Advances in Cognitive Systems S. Nefti and J. Gray (Editors) Volume 72 Control Theory: A guided tour, 3rd Edition J. R. Leigh Volume 73 Adaptive Sampling with Mobile WSN K. Sreenath, M.F. Mysorewala, D.O. Popa and F.L. Lewis Volume 74 Eigenstructure Control Algorithms: Applications to aircraft/rotorcraft handling qualities design S. Srinathkumar Volume 75 Advanced Control for Constrained Processes and Systems F. Garelli, R.J. Mantz and H. De Battista Volume 76 Developments in Control Theory towards Glocal Control L. Qiu, J. Chen, T. Iwasaki and H. Fujioka (Editors) Volume 77 Further Advances in Unmanned Marine Vehicles G.N. Roberts and R. Sutton (Editors) Volume 78 Frequency-Domain Control Design for High-Performance Systems J. O’Brien Volume 81 Optimal Adaptive Control and Differential Games by Reinforcement Learning Principles D. Vrabie, K. Vamvoudakis and F. Lewis Volume 88 Distributed Control and Filtering for Industrial Systems M. Mahmoud Volume 89 Control-based Operating System Design A. Leva et al. Volume 90 Application of Dimensional Analysis in Systems Modelling and Control Design P. Balaguer Volume 91 An Introduction to Fractional Control D. Valério and J. Costa Volume 92 Handbook of Vehicle Suspension Control Systems H. Liu, H. Gao and P. Li Volume 94 Analysis and Design of Reset Control Systems Y. Guo, L. Xie and Y. Wang
- 8. Analysis and Design of Reset Control Systems Yuqian Guo, Lihua Xie and Youyi Wang The Institution of Engineering and Technology
- 9. Published by The Institution of Engineering and Technology, London, United Kingdom The Institution of Engineering and Technology is registered as a Charity in England & Wales (no. 211014) and Scotland (no. SC038698). © The Institution of Engineering and Technology 2016 First published 2015 This publication is copyright under the Berne Convention and the Universal Copyright Convention. All rights reserved. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may be reproduced, stored or transmitted, in any form or by any means, only with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publisher at the undermentioned address: The Institution of Engineering and Technology Michael Faraday House Six Hills Way, Stevenage Herts, SG1 2AY, United Kingdom www.theiet.org While the authors and publisher believe that the information and guidance given in this work are correct, all parties must rely upon their own skill and judgement when making use of them. Neither the authors nor publisher assumes any liability to anyone for any loss or damage caused by any error or omission in the work, whether such an error or omission is the result of negligence or any other cause. Any and all such liability is disclaimed. The moral rights of the authors to be identified as authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. British Library Cataloguing in Publication Data A catalogue record for this product is available from the British Library ISBN 978-1-84919-703-8 (hardback) ISBN 978-1-84919-704-5 (PDF) Typeset in India by MPS Limited Printed in the UK by CPI Group (UK) Ltd, Croydon
- 10. Contents List of figures viii List of table xi Preface xii Acknowledgments xiv 1 Introduction 1 1.1 Motivation of reset control 1 1.2 Basic concepts of RCSs 10 1.2.1 Preliminaries and problem setup 10 1.2.2 Solutions to RCSs 13 1.2.3 RCSs with discrete-time reset conditions 15 1.3 Fundamental theory of traditional reset design 17 1.3.1 Horowitz’s design 17 1.3.2 PI+CI reset design 22 Notes 24 References 24 2 Describing function analysis of reset systems 27 2.1 Sinusoid input response 27 2.2 Describing function 32 2.2.1 General case 32 2.2.2 Gain-balanced FORE 38 2.3 Application to HDD systems 41 2.3.1 Reset narrow band compensator (RNBC) 41 2.3.2 Mid-frequency disturbance compensation 43 2.3.3 Simulation results 46 Notes 48 References 48 3 Stability of reset control systems 51 3.1 Preliminaries 51 3.1.1 Annihilator of matrices 51 3.1.2 Passive systems 52 3.2 Quadratic stability 57 3.3 Stability of RCSs with time-delay 63
- 11. vi Analysis and design of reset control systems 3.4 Reset times-dependent stability 67 3.5 Passivity of RCSs 77 Notes 81 References 82 4 Robust stability of reset control systems 83 4.1 Definitions and assumptions 83 4.2 Quadratic stability 86 4.2.1 RCSs with low-dimensional plants (np ≤ 2) 87 4.2.2 High-dimensional cases 89 4.3 Affine quadratic stability 93 4.4 Robust stability of RCS with time-delay 96 4.5 Examples 106 Notes 112 References 112 5 RCSs with discrete-time reset conditions 115 5.1 Preliminaries and problem setting 116 5.2 Stability analysis 118 5.3 A heuristic design method 122 5.4 Application to track-seeking control of HDD systems 125 5.4.1 System description 125 5.4.2 Baseline controller design 126 5.4.3 Reset mode 127 5.4.4 Stability analysis 127 5.4.5 Simulation results 128 Notes 130 References 130 6 Reset control systems with fixed reset instants 133 6.1 Stability analysis 133 6.1.1 Stability analysis through induced discrete systems 133 6.1.2 Lie-algebraic condition 135 6.2 Moving horizon optimization 137 6.2.1 Trade-off between stability and other performances 140 6.2.2 Observer-based reset control 141 6.3 Optimal reset law design 142 6.3.1 Equivalence between ORL and LQR 144 6.3.2 Solutions to ORL problems 147 6.4 Application to HDD systems 149 6.4.1 Dynamics model of HDD systems 149 6.4.2 Moving horizon optimal reset control 150 6.4.3 Optimal reset control 153 6.5 Application to PZT-positioning stage 160 6.5.1 Modeling of the PZT-positioning stage 160
- 12. Contents vii 6.5.2 Reset control design 161 6.5.3 Experimental results 162 Notes 166 References 167 7 Reset control systems with conic jump sets 169 7.1 Basic idea 169 7.2 L2-gain analysis 172 7.2.1 Passification via reset 174 7.2.2 Finite L2 gain stability 178 Notes 180 References 180 Index 183
- 13. List of figures 1.1 Linear feedback control system 1 1.2 The curve of the weighting function ln coth(|v|/2) 2 1.3 Frequency domain specifications 3 1.4 Output responses of LI and CI to sinusoidal input e = sint 4 1.5 Bode plots of FORE and first-order linear element 6 1.6 The reset control system (RCS) with a CI 7 1.7 CI equivalence 7 1.8 Replacement of the CI with its equivalence 7 1.9 Reducing overshoot through reset 8 1.10 Responses of the RCS to unit ramp and step inputs, respectively 10 1.11 Two different triggering conditions (a) zero-crossing type; (b) sector type 13 1.12 Deadlock 14 1.13 An RCS with time-regularization 16 1.14 Discrete-time reset law defined by the triggering function given in (1.44) 17 1.15 A FORE RCS 18 1.16 A FORE equivalence 18 1.17 Overshoot with M = ωc/b as parameter 20 1.18 A FORE RCS equivalence 21 1.19 PI+CI reset controller 22 2.1 General RCS 28 2.2 Output responses for Example 2.1 31 2.3 The spectrum of the power percentage σk (R, ω) 38 2.4 The phase lead of GFORE in large frequency range: ϕ = ϕ(R, ∞) 40 2.5 Bode plot of GFORE with fr = 0.1 (based on DF) 41 2.6 Bode plot of GFORE with R = −0.5 (based on DF) 42 2.7 Bode diagram of the NBCs with f = 1 × 104 and different ξ1/ξ1 42
- 14. List of figures ix 2.8 Reset narrow band compensator 43 2.9 Structure of a typical HDD 44 2.10 The Bode plots of the plant and the open loop with the PI controller 45 2.11 Mid-frequency disturbance rejection 45 2.12 Responses to sinusoid disturbance at 800 Hz 46 2.13 Responses to sinusoid disturbance at 800 Hz (with input disturbance, output disturbance and white noises inserted) 47 3.1 Negative feedback interconnection of H1 and H2 54 3.2 Reset control system 57 3.3 Phase trajectory of an RCS with an unstable baseline system. (Left: baseline system; right: reset system.) 59 3.4 The Nyquist plots of Hβ(s − ε) for different β 63 3.5 RCS with time-delay 64 3.6 Reset interval for different initial state from the set of after-reset states 75 3.7 Max eigenvalue of eAT t RPReAt − P for different r 76 3.8 State response for x(0) = [2 −1 6]T and r = −0.5 76 3.9 A reset control system 77 4.1 Responses for the uncertainty-free case 108 4.2 Responses for uncertainty γ (t) = 0.2sin(20t) 108 4.3 Responses for constant uncertainty γ (r = −0.5) 109 4.4 The curves of γl and γr over τ. (For any given τ ∈ [0, 1.095], the RCS is robustly stable respect to time-varying uncertainty γ (t) ∈ [γl(τ), γr(τ)]) 110 4.5 Initial condition response to x(0) = [−2 2]T 111 4.6 Input up 111 5.1 An RCS with discrete-time triggering condition 123 5.2 Step responses for the baseline system, reset mode, and the RCS with Rr = 0 124 5.3 Step responses of the reset mode for different Rr ∈ [−1, 1] 124 5.4 Step responses of the RCS for different Rr ∈ [−1, 0] 124 5.5 Step responses for different Rr ( = 4.5Ts) 128 5.6 Step responses of the reset mode for different Rr 128 5.7 Step responses for different (Rr = 0) 128 5.8 Comparison between traditional and proposed reset control 129 5.9 Control input 129
- 15. x Analysis and design of reset control systems 6.1 The model of VCM actuator 149 6.2 Structure of reset controller 150 6.3 Step responses for different µ 151 6.4 Step responses of base linear system, traditional reset control, and the proposed moving horizon optimal reset control 152 6.5 Enlarged view of step responses 153 6.6 Step responses for base linear system and optimal reset control 155 6.7 Step responses for optimal reset control with different t 156 6.8 Step responses for different R/Q 156 6.9 Responses for different step levels 157 6.10 Response and control input for r = 200 µm 157 6.11 Response and control input for r = 50 µm 158 6.12 Output responses for both optimal reset control and CNFC for r = 200 µm 159 6.13 Control inputs for both optimal reset control and CNFC 159 6.14 Piezoelectric (PZT) mircoactuator-positioning stage. (a) PZT-positioning state and (b) the mass-damper-spring model 160 6.15 Frequency responses of the PZT microactuator-positioning stage 161 6.16 Step responses (r = 1 µm) 163 6.17 Time responses for various step levels (r = 2, 3, 4 µm) 164 6.18 Relationship between reset time interval, overshoot, and settling time of the moving horizon optimal RCSs 164 6.19 Time responses to step input r = 1 µm and sinusoidal input disturbance ud = 0.1 sin (200πt)V 165 6.20 Time responses to step input r = 1 µm and sinusoidal sensor noise n = 0.1 sin(200πt)µm 165 6.21 Bode plot for DFs of input disturbance/sensor noise suppression; the excitation level of ud is 0.1V and n is 0.1 µm 166 7.1 The conic subsets F and J 175 7.2 The conic subsets F̃ and J when time-regularization applies 177 7.3 Negative feedback interconnection of reset controller with a plant 178
- 16. List of table 2.1 The responses to disturbance near central frequency f = 800 Hz. (fd: disturbance frequency, A1: error amplitude without RNBC, and A2: error amplitude with RNBC) 48
- 17. Preface Reset control is a special kind of hybrid control methods aiming to overcome inherent limitations of linear feedback design. The first investigation of reset control goes back to J.C. Clegg in 1958 for the reset integrator which is called Clegg integrator (CI) later. The potential application of reset control in relieving Bode’s gain-phase constraints was originally supported by the describing function (DF) analysis. The analysis and design of reset control in the time domain have long been considered to be challenging. The first reset control design procedure was proposed by Horowitz and his co-authors in 1975. However, the reset control had gone through a period of silence before attracting renewed attentions in the last 15 years. Some new analysis techniques and design methods for reset control systems (RCSs) have been proposed and some of them have been applied to improve performance of some practical control systems. This book is devoted to the analysis and design of RCSs with emphasis on RCSs with nonzero reset matrices and their applications to hard disk drive (HDD) servo systems. Except for the basic theory of reset control, some new ideas for analysis and design of reset control are introduced. Chapter 1 gives the motivation and basic concepts of RCSs. Some typical design methods including the Horiwitz’s design and the PI+CI (proportional-plus-integral control with CI) design are also briefly discussed. In Chapter 2, the DF of reset systems is derived and how the reset matrix affects the frequency domain property of a system is analyzed. In this chapter, the DF is applied to the mid-frequency disturbance rejection of HDD servo systems. Chapter 3 collects some of the recent developments on stability analysis of RCSs. These include quadratic stability of RCSs with stable baseline dynamics and reset times-dependent stability for systems with both stable and unstable baseline dynamics. Passivity of RCSs is also discussed in this chapter. Chapter 4 discusses robust stability of RCSs with uncertainties. Quadratic stability and affine quadratic stability are, respectively, investigated for systems with time-varying and constant uncertainties. Robust stability for RCSs with time-delays is also studied in this chapter. From Chapter 5 to Chapter 7, several different reset schemes are developed. Chapter 5 is about RCSs with discrete-time reset conditions, which can be regarded as a discretized version of the traditional reset mechanism. The discretization provides a different perspective on reset control. A heuristic design procedure aiming to achieve perfect transient response is proposed and applied to track-seeking control of HDD systems. Chapter 6 is about optimal reset control design under fixed reset time instants. Both moving horizon optimization and fixed horizon optimization are discussed and the applications of optimal reset law design to HDD systems and a kind of PZT-positioning stage are, respectively, investigated in this
- 18. Preface xiii chapter. Chapter 7 provides the main idea of RCSs with conic jump sets and discusses passivity and finite L2 gain stability under reset control. The main part of the materials in this book (Chapters 2, 4–6) is based on our research over the past decade. The materials in Chapters 1, 3, and 7 are collections of research results by others. Most of the results in Chapter 3 were originally developed for RCS with zero reset matrices, but have been slightly reformulated to adapt to RCSs with nonzero reset matrices in this book. Chapter 7 briefly introduces some recently developed ideas for reset control. Yuqian Guo, Lihua Xie, Youyi Wang 26 April, 2015
- 19. Acknowledgments This work was supported in part by the National Natural Science Foundation of China (61074002, 61473315).
- 20. Chapter 1 Introduction 1.1 Motivation of reset control In linear feedback control design, performance specifications can be given in both the time and the frequency domains. In the frequency domain, specifications are usually given in terms of the gain and phase properties of open-loop transfer functions over certain frequency range. For instance, consider a standard linear feedback system depictedinFigure1.1whereP(s)andC(s)aretheplantandthecontroller, respectively, di, do, and n are the input disturbance, output disturbance, and measurement noise, respectively, r, y, and e are the reference input, output, and tracking error, respectively. In the frequency domain, the output y is related to r, di, do, and n through y = S(s)(do + P(s)di) + (1 − S(s))(r − n) (1.1) where S(s) is the sensitivity function defined by S(s) = 1 1 + L(s) (1.2) with L(s) := P(s)C(s), the open-loop transfer function. In general, low-frequency gain of S( jω) is expected to be small in order to attenuate the disturbances. This in turn requires that the open-loop gain |L( jω)| be large over the low-frequency range. In addition, the open-loop transfer function is required to have sufficient bandwidth for fast response and positioning. On the other hand, in order to reduce the effect of high-frequency noises, the gain of S( jω) over the high-frequency range should be large, which requires a small open-loop gain |L( jω)| over the same frequency range. These specifications require that the transition of the loop gain from low frequency to high frequency should not be too slow. r – e C(s) P(s) di do y n Figure 1.1 Linear feedback control system
- 21. 2 Analysis and design of reset control systems 4 3 2 1 0 10−2 10−1 100 w/w0 101 102 ln coth υ − 2 Figure 1.2 The curve of the weighting function ln coth(|v|/2) However, this requirement often conflicts with the desired phase margin due to Bode’s gain-phase relation. In order to make this clear, suppose that the open-loop transfer function L(s) is stable and of minimum phase. The Bode’s gain-phase relation states that the phase of L( jω) is uniquely determined by its gain. Precisely, the phase at frequency ω0 (rad/s) is given by [1] ∠ L( jω0) = 1 π ∞ −∞ d ln |L( jω0ev )| dv ln coth v 2 dv (rad) (1.3) with v = ln (ω/ω0). The weighting function ln coth (|v|/2) = ln ω+ω0 ω−ω0 in the integral is plotted in Figure 1.2. From this figure, one sees that the integration is mainly contributed by the weighted rate of the gain-decreasing near ω0. Note that d ln |L( jω0ev )| dv = 1 20 d[20 lg |L( jω)|] d lg (ω/ω0) , thus a rule of thumb states that if the slope of the magnitude curve near ω0 is 20N dB/decade and this slope does not change much over a decade-wide interval near ω0, then the phase angle at ω0 can be approximately calculated through [2] ∠ L( jω0) ≈ N π ∞ −∞ ln coth v 2 dv = N π · π2 2 = πN 2 (rad). (1.4) The above Bode’s gain-phase relation indicates that one cannot manipulate the gain and the phase independently and, in order to assure sufficient phase margin, the rate of the gain-decreasing near the crossover frequency ωc cannot be too fast. In practice, the slope of the open-loop gain at the gain-crossover frequency should be designed to be between −20 and −30 dB/decade in order to assure stability. This tradeoff is explained in Figure 1.3 where the shaded areas represent the bounds defined by the low frequency, high frequency, and phase specifications, respectively. If one wants to design a feedback controller to increase low-frequency gain, broaden the bandwidth, or decrease the high-frequency gain as indicated as the dash line in Figure 1.3, then
- 22. Introduction 3 Avoided area for sufficient phase margin Phase (°) 0 −180 Gain (dB) Avoided area for large low- frequency gain and sufficient bandwidth 0 Avoided area for small high- frequency gain Figure 1.3 Frequency domain specifications the phase near the crossover frequency would inevitably decrease and even violate the phase margin bound. This tradeoff is unavoidable in the framework of linear feedback control and is called the “cost of feedback” in Reference 3. The above inherent limitation motivates researchers to adopt nonlinear filters, which are with smaller phase lag but with similar magnitude slope characteristics, to relieve the frequency domain limitations imposed by Bode’s gain-phase constraint explained above. Early efforts along this line include the nonlinear feedback proposed by J.B. Lewis [4], the nonlinear gain element by R.E. Kalman [5], and other nons- mooth filters [6, 7]. In recent years, many hybrid control design techniques have been developed to overcome the limitations of the traditional linear feedback design. Such techniques include switching control [8], sliding mode control [9], impulsive control [10], etc. In this book, we focus on a special kind of hybrid techniques called reset control whose original motivation is to overcome the inherent limitation imposed by the Bode’s gain-phase constraint. The study of reset control can be traced several decades back. The first reset element is the so-called Clegg integrator (CI) proposed by Clegg in 1958 [11]. The CI is described by the impulsive differential equation [11, 12] ż = e, ze 0 z+ = 0, ze ≤ 0 (1.5)
- 23. 4 Analysis and design of reset control systems which consists of a linear integrator (LI) ż = e and a reset mechanism. When the input e and the output z of the integrator have the same sign, then it evolves according to the LI. On the other hand, if the input and the output have opposite signs, then the state is reset to zero. The notation z+ denotes the state of the integrator after the reset. The condition ze ≤ 0 is called the reset condition which determines when the state of the integrator should be reset. In some recent literature, the triggering condition ze ≤ 0 is replaced by the zero-crossing of the input, i.e., e = 0. In this case, the CI is represented by [13] ż = e, e = 0 z+ = 0, e = 0. (1.6) These two models of CI are different and lead to different development of reset control theory. However, as basic reset elements, these two models have the same output response to a sinusoidal input. Thus they have the same sinusoidal input describing function (DF). Figure 1.4 gives the output responses of the CI and the LI to a sinusoidal input. The DF of the CI is given by [11] GCI ( jω) = GLI ( jω) 1 + j 4 π (1.7) 1 0.5 0 −0.5 Input signal −1 2 1.5 1 0.5 Output of LI 0 0 5 10 15 0 5 10 15 2 1 0 −1 Output of CI −2 0 5 10 15 Output of CI First-hormonic of the output Figure 1.4 Output responses of LI and CI to sinusoidal input e = sin t
- 24. Introduction 5 where GLI ( jω) = 1 jω is the frequency response function of the LI ż = e. The calculation of the sinusoidal input DFs of reset systems will be discussed in detail in Chapter 2. The formula (1.7) shows that the DF of the CI equals the frequency response function of the LI scaled by a complex factor 1 + j(4/π), which means that the reset action does not change the slope of the logarithmic magnitude-frequency characteristics but causes a phage lead of φ = arctan (4/π) ≈ 51.9◦ (1.8) at all frequencies. This property suggests that reset elements could be used to achieve required bandwidth with much lower high-frequency gain, which means more rapid gain-decreasing within the mid-frequency range is allowed, without degrading the phase margin specification. Another basic reset element with more flexibility is the so-called first-order reset element (FORE) which was proposed in Reference 3. The state-space representation of a zero-crossing type FORE is ż = λz + e, e = 0 z+ = 0, e = 0 , (λ ≤ 0) (1.9) whose sinusoidal input DF is GFORE( jω) = GL( jω) (1 + j(ω)) (1.10) where GL( jω) = 1 jω − λ is the frequency response function of the first-order linear element ż = λz + e and (ω) = 2 π 1 + eπλ/ω 1 + (λ/ω)2 0. The Bode plots of the FORE and the corresponding first-order linear element are compared in Figure 1.5. Different from the CI, a FORE behaves like a linear element over low-frequency range but like a CI over high-frequency range. This property provides more useful design flexibility [3]. The DFs of the CI and the FORE have been applied to many practical systems showing benefits of reset control. See for instance References 14 and 15. The benefit of reset control can also be shown in the time domain. In practice, integral control is usually used to remove steady-state error. But the side effect is that it may cause large overshoot and prolong the settling time. Theoretically, differential control can be used to reduce the overshoot caused by integration without degrading the rise time specification, but in practice, differential control may make the system sensitive to sensor noise. Replacing the integrator by the CI can help to achieve better tradeoff between rise time and overshoot. This benefit was recognized in the early
- 25. 6 Analysis and design of reset control systems 0 0 −20 −40 −60 Frequency (rad/s) −100 −80 −10 −20 −30 −40 −50 −60 Gain (dB) Phase (°) −70 10−1 100 101 102 103 10−1 100 101 102 103 FORE with λ = −2 Linear element Figure 1.5 Bode plots of FORE and first-order linear element literature regarding reset control and the first quantitative design procedure around the CI was proposed in Reference 16. The control system considered in Reference 16 is depicted in Figure 1.6. In order to explain the effect of the CI, let us first derive an equivalent representation of the CI. Suppose that 0 ≤ t1 t2 · · · tn are the time instants of the zero-crossings of the input of the CI (1.6). Thus the output of the CI z(t) with t0 = 0 is z(t) = t ti e(s)ds, t ∈ [ti, ti+1) = t 0 e(s)ds − tk ≤t tk tk−1 e(s)ds = t 0 e(s) − n k=1 Ak δ(t − tk ) ds where Ak is the signed area Ak = tk tk−1 e(s)ds.
- 26. Introduction 7 C(s) CI b __ s P(s) y u + + – + Figure 1.6 The reset control system (RCS) with a CI u z v 1 __ s Figure 1.7 CI equivalence v u y + + C(s) P(s) + − 1 __ s b __ s Figure 1.8 Replacement of the CI with its equivalence Here δ(t) represents the unit impulse. This indicates that the CI can be equivalently replaced by a LI with an additional input of a sequence of impulses v(t) = − n k=1 Ak δ(t − tk ) as depicted in Figure 1.7. Replacing the CI in Figure 1.6 by the equivalence, we get the structure depicted in Figure 1.8. Suppose that the transfer function from u to y (i.e., the closed-loop transfer function of the system with the CI replaced by the LI) is T(s). Then the transfer function from v to y is 1 1+b T(s). Thus if the step response and the unit impulse response of the linear system T(s) without reset are, respectively, denoted by yL(t) and yI (t), then the output of the system with CI is y(t) = yL(t) − 1 1 + b n k=1 Ak yI (t − tk ). Suppose that the step response of the linear system T(s) exhibits overshoot as depicted in Figure 1.9 and t1 and t2 are the first two reset time instants. Then before
- 27. 8 Analysis and design of reset control systems y y(t) yI (t – t1) yL (t) yI (t – t1) t2 A1 A1 0 1 + b – t1 t 1 O Figure 1.9 Reducing overshoot through reset the first reset, the output response of the system with CI is the same as the linear system. During the interval [t1, t2], there holds that y(t) = yL(t) − 1 1 + b A1yI (t − t1). Note that A1 0, thus the reset action at t1 plays the role of reducing the first peak. If the underlying linear system T(s) is of the second order with damping ratio ζ ∈ [0, 1), then the precise value of the overshoot can be calculated as [16, 17] MR(ζ, b) = 1 1 + b b − 2ζ exp ζ 1 − ζ2 (π − arccos ζ) ML (1.11) where ML is the overshoot of the step response of the linear system T(s) given by ML = exp − πζ 1 − ζ2 . It is easy to check that if b ≥ 2ζ exp ζ 1 − ζ2 (π − arccos ζ) , (1.12) then 0 ≤ MR(ζ, b) ML. If b is chosen to be 0 b 2ζ exp ζ 1 − ζ2 (π − arccos ζ) , then MR(ζ, b) 0,
- 28. Introduction 9 i.e., undershoot occurs. The above analysis indicates that if the parameter b is appro- priately chosen, the replacement of the LI by the CI can reduce or even remove the overshoot without degrading the rise time specification. One interesting question is that if the performance achieved by the reset control can also be achieved by a linear compensator. The following example, which is constructed in Reference 18, gives a negative answer. Consider a linear control system depicted in Figure 1.1 with di = 0, do = 0, and n = 0 where the plant P(s) contains an integrator and C(s) is stabilizing. Suppose that the reference input is the unit step, i.e., r(t) = 1(t). Define z(t) = t 0 e(s)ds. Then the Laplace transform of z(t) is Z(s) := L(z(t)) = 1 s L(e(t)) = 1 s E(s) = 1 s2 1 1 + P(s)C(s) . Thus by the final value theorem of Laplace transform, there holds ∞ 0 e(s)ds = lim s→0 sZ(s) = 1 lim s→0 sP(s)C(s) := 1 Kv . Consider the rise time defined by tr := sup T T : y(t) ≤ t T , t ∈ [0, T] . (1.13) There holds 1 Kv = ∞ 0 e(s)ds ≥ tr 0 1 − s tr ds + ∞ tr e(s)ds = tr 2 + ∞ tr e(s)ds. If tr 2/Kv, then ∞ tr e(s)ds 0 which means that the unit step response overshoots. Keep this in mind and consider the special case that P(s) = 1/s. Suppose that we aim to design a controller such that the system is stable and satisfies: 1. the steady-state error of the unit ramp response is not greater than 1; 2. the rise time tr of the unit step response is greater than 2 s; 3. the step response does not overshoot. The requirements (1) and (2) mean that the velocity error constant Kv ≥ 1 and the rise time tr 2 ≥ 2/Kv.Thus according to the previous analysis, if a linear control is used,
- 29. 10 Analysis and design of reset control systems 1.4 1.5 1 0.5 y = 0.5t 0 0 1 2 3 1.2 1 0.8 0.6 0.4 0.2 Tracking error to a unit ramp input Step response 0 0 5 t t 10 15 Figure 1.10 Responses of the RCS to unit ramp and step inputs, respectively then the step response will exhibit an overshoot. This contradicts the requirement (3). Thus there exists no stabilizing linear controller which meets all these specifications. However, these specifications can be satisfied by replacing C(s) with the FORE (1.9) with λ = −1. See Figure 1.10 which shows that, with this reset controller, the steady- state error of the unit ramp response is 1 and the rise time is approximately 2.42 s. In addition, the unit step response is deadbeat, thus no overshoot occurs. Though reset control has many benefits over linear control, it is worth pointing out that these benefits do not come from any blind resetting but from an appropriate interaction between the reset mechanism and the underlying linear system. Reset- ting might deteriorate the system performance or even destabilize the system if the underlying linear controller is not appropriately designed. 1.2 Basic concepts of RCSs 1.2.1 Preliminaries and problem setup Consider the single-input single-output (SISO) linear time-invariant plant ẋp = Apxp + Bpup y = Cpxp, (1.14) where xp ∈ Rnp , up ∈ R1 , y ∈ R1 , and Ap, Bp, and Cp are constant matrices with proper dimensions. A conventional reset controller can be modeled by the impulsive differential equation ⎧ ⎨ ⎩ ẋc = Acxc + Bce, e = 0 x+ c = ρc(xc, r), e = 0 u = Ccxc + Dce (1.15)
- 30. Introduction 11 where xc ∈ Rnc is the controller state, e = r − y is the tracking error, and r is the reference input, Ac, Bc, Cc, and Dc are constant matrices, ρc is the reset mapping, and x+ c represents the state of the controller after reset, i.e., x+ c (t) = lim s→t+0 xc(s). Combining the plant (1.14) and the reset controller (1.15), we have the closed-loop system ẋ = Ax + Br, r − Cx = 0 x+ = ρ(x, r), r − Cx = 0 (1.16) with x = [xT p , xT c ]T and A = Ap − BpDcCp BpCc −BcCp Ac , B = BpDc Bc C = Cp 0 , ρ(x, r) = xp ρc(xc, r) . The system ẋ = Ax + Br (1.17) is called the baseline system of the reset system (1.16). For set-point regulation, the reference input r is a constant input. In this case, a state x̄r is called as an equilibrium point of reset system (1.16) if ⎧ ⎨ ⎩ Ax̄r + Br = 0 Cx̄r − r = 0 ρ(x̄r, r) − x̄r = 0. (1.18) Any state satisfying the first two equations of (1.18) is an equilibrium point of the baseline system, and any point satisfying the last equation of (1.18) is an equilibrium point of the reset mapping. Equation (1.18) actually imposes a constraint on the reset mapping, i.e., the reset action should not destroy the equilibrium point of the baseline system. Precisely, suppose that x̄r = [x̄T pr, x̄T cr]T is the equilibrium point of the baseline system which is to be stabilized.Then the reset mapping should be chosen such that x̄r is also an equilibrium point of the reset mapping. In the conventional reset control, the reset mapping is of the form ρc(xc, r) = x̄cr + Rc(xc − x̄cr), (1.19) where the matrix Rc is called the reset matrix which is of the form Rc = Inc−nρ 0(nc−nρ)×nρ 0nρ×(nc−nρ) 0nρ×nρ (1.20) with nρ an integer satisfying 0 ≤ nρ ≤ nc. Im and 0m×n represent the identity matrix with dimension m and the m × n zero matrix, respectively. This structure means that whenever the tracking error crosses zero, part of the controller state is reset to its steady-state value.
- 31. 12 Analysis and design of reset control systems Define ξ = x − x̄r, then the closed-loop RCS (1.16) becomes ξ̇ = Aξ, Cξ = 0 ξ+ = Rξ, Cξ = 0 (1.21) with R = Inp 0 0 Rc . In RCSs, the FORE is given by ż = λz + e, e = 0 z+ = zr, e = 0 (1.22) where z is a scalar variable, zr represents the steady-state value of the FORE, and μ is a scalar constant. If λ = 0 and the steady-state error is zero, then zr = 0. Thus a FORE is generally defined as ż = λz + e, e = 0 z+ = 0, e = 0. (1.23) If λ = 0, the FORE degenerates to the CI in which case the steady-state value of the integrator is not necessarily zero. Traditionally, the CI is defined as ż = e, e = 0 z+ = 0, e = 0 (1.24) which should be understood as the case that the equilibrium point has been moved to the origin through linear transformation. Denote J = ker C := {ξ ∈ Rnp+nc | Cξ = 0} (1.25) which is called the jump set of the RCS (1.21). With this definition, the closed-loop RCS can be alternatively represented by ξ̇ = Aξ, ξ / ∈ S ξ+ = Rξ, ξ ∈ J (1.26) where ξ ∈ Rn with n := np + nc. S = J c represents the complementary set of J which is called the flow set. The reset system (1.26) with J defined in (1.25) is called the zero-crossing type reset system since the reset action is triggered by the zero- crossing of the tracking error. The jump set is a hyperplane as depicted in Figure 1.11(a). In general, for single output systems, if Rank C = 1, then we have dim (J ) = dim ( ker C) = n − 1.
- 32. Introduction 13 x2 x2 x1 x(0) x(0) x + (0) x1 I = ker C O O (a) (b) I Figure 1.11 Two different triggering conditions (a) zero-crossing type; (b) sector type The formula (1.26) can be used to model a broader class of RCSs for more general definitions of flow and jump sets. For instance, in Reference 12, FORE is modeled as ż = −λz + e, ez ≥ 0 z+ = 0, ez ≤ 0, (1.27) where the reset action is triggered whenever its input and output have opposite signs. If the control loop includes such a reset element, the resulting reset system will be of the form of (1.26) with conic jump sets, i.e., S = ξ ∈ Rn | ξT Sξ ≥ 0 , (1.28) J = ξ ∈ Rn | ξT Jξ ≥ 0 , (1.29) where S and J are symmetric matrices. In this case, the flow and jump sets are even allowed to be partially overlapped, leading to nonunique solutions. 1.2.2 Solutions to RCSs Consider reset system (1.26) with S = J c , the complementary set of J . For any ξ0 ∈ Rn , a function ξ(t; ξ0) of time t is said to be a solution to RCS (1.26) starting from the initial state ξ0 if there is a countable or finite subset J(ξ0) ⊂ R+ such that: 1. ξ(t; ξ0) is left-continuous in time t for t 0 with ξ(0; ξ0) = ξ0; 2. ξ(t; ξ0) is differentiable in t and satisfies ξ̇(t; ξ0) = Aξ(t; ξ0), ∀t ∈ R+ J(ξ0); 3. For all t ∈ J(ξ0), there hold that ξ(t; ξ0) ∈ J and x(t+ ; ξ0) = Rx(t; ξ0). In the above definition, J(ξ0) is actually the set of reset time instants of the solution ξ(t; ξ0). Since it is required to be countable or finite, one can arrange the
- 33. 14 Analysis and design of reset control systems Ax x Œ ∂ I ∩ Ic IC I Figure 1.12 Deadlock reset time instants in an increasing order and define τk (ξ0) the kth reset time instant of the solution ξ(t; ξ0). The reset interval is defined as k (ξ0) := τk (ξ0) − τk−1(ξ0) with τ0(ξ0) := 0. One important question is that under what conditions, the reset system exists a well-defined solution for any initial condition ξ0. Note that the state-dependent reset mechanism defined above may lead to complex phenomena including deadlock, beating, and Zenoness which destroy the existence of solutions. Deadlock happens when the system can neither evolve continuously nor jump discretely. Figure 1.12 gives a situation of deadlock where ξ ∈ ∂J ∩ J c and the vector Aξ directs toward the inner of J . Here, ∂J represents the boundary of J . Since ξ / ∈ J , jump is impossible. However, the trajectory of the baseline system starting from ξ will go inside the jump set since the vector field points toward the inner of J , which is not allowed for the RCS. In order to avoid deadlock, the RCS should satisfy the following assumption. Assumption 1.1. For any ξ ∈ ∂J ∩ J c , there exists a positive number such that exp (At)ξ ∈ J c , ∀t ∈ (0, ). (1.30) Beating occurs when there are multiple resettings at a single reset time instants, which is not allowed by the definition of solution. The following assumption is imposed to avoid beating. Assumption 1.2. The jump set and its image under the reset mapping are disjoint, i.e., J ∩ R(J ) = ∅. (1.31) For a given RCS, if J ∩ R(J ) = ∅, one can re-define the jump set by removing R(J ) from J , i.e., ˜ J := J R(J ), (1.32) to avoid beating. Zenoness is a phenomenon where there are infinite number of resettings within a compact time interval in which case the solution is only well-defined within a finite time interval. In order to avoid Zenoness, the following assumption is made.
- 34. Introduction 15 Assumption 1.3. For any ξ0 ∈ Rn , there is a positive number μ(ξ0) 0 such that k (ξ0) ≥ μ(ξ0), ∀k. Note that if Assumption 1.3 holds, then it implicitly implies that deadlock and beating would not occur and for any initial state ξ0, there is a unique solution ξ(t; ξ0) to RCS (1.26) which can be represented as ξ(t; ξ0) = W(t, ξ0)ξ0 (1.33) where W(t, ξ0) is the state transition matrix given by W(t, ξ0) = exp (A(t − τk (ξ0)))R exp (A k (ξ0)) · · · R exp (A 0(ξ0)) (1.34) for t ∈ (τk (ξ0), τk+1(ξ0)]. In general, Assumption 1.3 is difficult to check since it depends on the reset time instants directly. In practice, the existence of solutions can be assured by time- regularization. For instance, with time-regularization, the reset controller (1.15) becomes ⎧ ⎨ ⎩ ẋc = Acxc + Bce, (e = 0) ∨ (τ ≤ m) x+ c = ρc(xc, r), (e = 0) ∧ (τ m) u = Ccxc + Dce (1.35) where m is a pre-specified positive number, ∨ and ∧ represent the logical OR and AND, respectively, and τ is a variable characterizing the time spent since the latest reset which is governed by the impulsive differential equation τ̇ = 1, (e = 0) ∨ (τ ≤ m) τ+ = 0, (e = 0) ∧ (τ m). (1.36) The structure of RCS is depicted in Figure 1.13. The closed loop becomes ξ̇ = Aξ, (ξ / ∈ J c ) ∨ (τ ≤ m) ξ+ = Rξ, (ξ ∈ J ) ∧ (τ m). (1.37) With time-regularization, two successive reset time instants are separated by at least m units of time. Thus deadlock, beating or Zenoness would not happen and for any initial state, a unique well-defined solution always exists. 1.2.3 RCSs with discrete-time reset conditions In practical computer-based implementations, reset controllers have to be discretized. For instance, the simplest discrete-time counterpart of the zero-crossing condition e(t) = 0 (1.38) in reset controller (1.15) is e(kTs)e((k − 1)Ts) 0 (1.39)
- 35. 16 Analysis and design of reset control systems Reset controller Triggering condition Plant y u e t r + – Figure 1.13 An RCS with time-regularization where Ts is the sampling period. In addition, reset actions occur only at sampling time instants. Thus the discrete-time form of the reset controller (1.15) can be represented by ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ẋc = Acxc + Bce, t ∈ ((k − 1)Ts, kTs] xc(kT+ s ) = xc(kTs), ek ek−1 0 xc(kT+ s ) = ρc(xc(kTs), r), ek ek−1 ≤ 0 u = Ccxc + Dce (1.40) where ek := e(kTs). A more general discrete-time reset controller considered in this book is of the form ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ẋc = Acxc + Bce, t ∈ ((k − 1)Ts, kTs] xc(kT+ ) = xc(kT), φ(ek , ek−1) 0 xc(kT+ s ) = ρc(xc(kTs), r), φ(ek , ek−1) ≤ 0 u = Ccxc + Dce. (1.41) where φ : R2 → R is called the triggering function which characterizes the reset condition. This form of triggering condition is capable of describing more general cases. A useful kind of triggering functions is given by the quadratic form φ(η) = ηT Sη, η ∈ R2 (1.42) where S is a symmetric matrix. For instance, the triggering function with S = 0 ( /Ts + 1)/2 ( /Ts + 1)/2 − /Ts (1.43) defines a reset law under which reset time instants are ahead of zero-crossings, where 0 is a nonnegative constant roughly characterizing the time for which the reset action is ahead of the zero-crossing. Actually, in this case, φ(ek , ek−1) = ek−1 ek − ek−1 Ts + ek (1.44)
- 36. Introduction 17 e(t) Zero-crossing Reset time instant e(t) t (k – 1) Ts kTs ek – ek – 1 Ts Δ + ek Δ ek ek – 1 Figure 1.14 Discrete-time reset law defined by the triggering function given in (1.44) where ẽ(kTs + ) = ek − ek−1 Ts + ek is a simple linear predictor of the tracking error at t = kTs + , as depicted in Figure 1.14. Another kind of RCSs with discrete-time reset conditions is the RCSs with fixed reset time instants. In this case, the reset controller is of the form ⎧ ⎨ ⎩ ẋc = Acxc + Bce, t ∈ ((k − 1)Ts, kTs) x+ c = ρc(xc, r), t = kTs u = Ccxc + Dce. (1.45) This reset controller can be viewed as a special case of (1.41) where φ is given in (1.42) with S a positive definite matrix. The benefits of introducing discrete-time triggering conditions are multi-fold. First of all, Zeno phenomenon is naturally avoided and for any initial condition, a solution always exists. Second, discrete-time triggering conditions provide more flexibility in control design as explained in Chapters 5 and 6. 1.3 Fundamental theory of traditional reset design 1.3.1 Horowitz’s design Consider a RCS depicted in Figure 1.15, where P(s) is the plant and the controller consists of a linear compensator C(s) cascaded with a FORE. The baseline transfer function of the FORE is 1 s + b , thus the FORE can be represented in state space as ⎧ ⎨ ⎩ ż = −bz + e, e = 0 z+ = 0, e = 0 ur = z. (1.46)
- 37. 18 Analysis and design of reset control systems FORE e ur u d y n r + – C(s) P(s) Figure 1.15 A FORE RCS e v 1 s + b ur Figure 1.16 A FORE equivalence The pole −b of the FORE will be an adjustable parameter in the design procedure. Denote by ti the ith reset time instant with 0 = t0 t1 t2 · · · tn ≤ ∞. n might be finite or infinite. In the case when n = ∞, there holds tn = ∞. Suppose that e(t) = 0 for all t 0. Similar to the discussion regarding the CI in Section 1.1, the output response of the FORE can be calculated as ur(t) = e−bt t ti ebs e(s)ds, t ∈ [ti, ti+1) = e−bt t 0 ebs e(s)ds − tk ≤t tk tk−1 ebs e(s)ds = e−bt t 0 ebs [e(s) + v(s)] ds where v(t) = − n k=1 Ak δ(t − tk ) Ak = tk tk−1 e−b(tk −s) e(s)ds. This means that the FORE can be alternatively represented as a linear system with an additional input v(t) as depicted in Figure 1.16. Denote by hL(t) and yL(t), respectively, the impulse and the step responses of the baseline system, i.e., with the FORE being replaced by 1 s+b , and by y(t) the step response of the RCS (with d = 0 and n = 0). Then there holds y(t) = yL(t) − n k=1 Ak hL(t − tk ) = yL(t) − n k=1 Ak ẏL(t − tk ).
- 38. Introduction 19 For t ∈ [0, t1], we have y(t) = yL(t) and yL(t1) = 0. For t ∈ [t1, t2), y(t) = yL(t) − A1ẏL(t − t1) with A1 = t1 0 e−b(t1−s) [1 − yL(s)] ds 0. From the above formula, if the step response of the baseline system is as depicted in Figure 1.9, there holds ẏL(t) 0, t ∈ [0, t̃1], where t̃1 represents the time instant of the first peak of yL(t). Thus we have y(t) yL(t), t ∈ [t1, t1 + t̃1]. This intuitively explains that reset tends to reduce the overshoot. If b = 0, then the FORE degenerates to the CI in which case A1 is the area under the curve e(t) within the interval [0, t1]. In general, there holds b = 0. Thus, A1 is the area under the curve e−b(t1−t) e(t) and the FORE pole plays the role of decreasing A1. If the pole of the FORE is chosen to be b 0, we have Ak ≈ 0 and hence y(t) ≈ yL(t), t ≥ 0. This explains that the replacement of the CI by the FORE provides more flexibility in the control design. If the baseline open-loop transfer function is the second-order system L(s) = 1 s + b P(s)C(s) = ω2 n s(s + 2ζωn) , (0 ζ 1), (1.47) then the overshoot of the RCSs is given by Overshoot = exp −πζ 1 − ζ2 − , (1.48) where = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ R 1 − 4ζ2M + 4ζ2M2 4ζ2 M2 e−ζμ − 2ζM(1 − 4ζ2 M)e− μ 2ζM , ζ ≥ 0.5 R 1 − 2ζM + M2 M2 e−ζμ − M(1 − 2ζM)e− μ M , ζ ≤ 0.5 (1.49) with R = exp − ζ 1 − ζ2 arccos ζ M = ωc b , μ = π − arccos ζ 1 − ζ2 .
- 39. 20 Analysis and design of reset control systems 177 100 80 Overshoot (%) 60 40 20 0 –20 –40 –60 25 22.5 20 17.5 15 12.5 10 7.5 5 2.5 0 –2.5 –5 176 174 M = 0 M = 0.5 M = 1 M = 2 M = 5 M = 20 –1.84 172 170 166 –ArgL (jwe ) (º) 162 156 147 136 120 104 L ( jwe ) 1 + L ( jwe ) (dB) M = b we Figure 1.17 Overshoot with M = ωc/b as parameter In the above, ωc is the gain-crossover frequency of the open-loop transfer function, i.e., |L( jωc)| = 1. Denote by GL(s) the closed-loop transfer function, i.e., GL(s) = L(s) 1 + L(s) = ω2 n s2 + 2ζωns + ω2 n and define D = |GL( jωc)|. The damping ratio ζ can be related to D with ζ = 1 2D 4D2 − 1 2(D2 − 1) . (1.50) Thus the overshoot of the second-order RCS can be represented as a function of D and M, which is plotted in Figure 1.17 with M the parameter. From this figure, when b is too small, undershoot occurs which explains the disadvantage of the CI. Figure 1.17 is obtained from second-order systems but is also a good approxima- tion for higher-order systems which are dominant by a second-order dynamics. This is the main tool in the Horowitz’s design procedure. The Horowitz’s design [3] is depicted in Figure 1.18 where the reset controller consists of two parts, i.e., the linear compensator C(s) and the FORE with pole −b. The linear controller C(s) is of the form C(s) = (s + b)CL(s). (1.51) Theabovestructureassuresthatthebaselinedynamicsdoesnotchangewhileadjusting the parameter b, i.e., L(s) = 1 s + b P(s)C(s) = P(s)CL(s).
- 40. Introduction 21 r e ur u d y n C(s) P(s) v 1 + + + – s + b Figure 1.18 A FORE RCS equivalence The Horowitz’s design includes two steps: ● Step 1: Design a linear controller CL(s) such that the baseline system satisfies both the disturbance rejection and sensor noise suppression at the expense of violating the gain/phase margin constraint; ● Step 2: Choose a FORE pole −b to reduce the overshoot. The above design procedure was applied to a tape-speed control system in [14] where the plant is given by P(s) = 5.32 × 105 (s2 − 6.95 × 102 s + 2.16 × 105 ) (s + 14.02)(s2 + 7.451 × 101s + 2 × 104)(s2 + 2.03 × 102s + 3.28 × 104) . The specifications for this system are as follows: 1. Thedisturbanceattenuationlevelfromd toe withinthefrequencyrangeω ≤ 2 Hz is not greater than 0.5, i.e., 1 1 + L( jω) ≤ 0.5, ω ≤ 2 Hz . (1.52) 2. The sensor-noise suppression ratio for ω ≥ 10 Hz is not greater than 0.4, i.e., L( jω) 1 + L( jω) ≤ 0.4, ω ≥ 10 Hz . (1.53) 3. The steady-state error for step input is zero and the overshoot is not greater than 20%. It has been shown that within the framework of linear feedback design, it is difficult to find a controller which meets all of the above specifications. By using the Horowitz’s design, a linear controller CL(s) is first designed to meet the first two specifications with CL(s) = 11.3(s + 257.1)(s + 3.606)(s2 + 20.33s + 216.3) s(s + 220.3)(s + 36.49)(s2 + 14.29s + 62.48) × (s2 + 55.51s + 3679)(s2 + 75.74s + 1.579 × 104 ) (s2 + 88.25s + 5925)(s2 + 125.7s + 1.579 × 104) . The crossover frequency of L(s) = P(s)CL(s) is approximately ωc ≈ 30 rad/s and −arg L( jωc) ≈ 150◦ . From Figure 1.17, choosing a FORE pole b = 30 (i.e., M = 1)
- 41. 22 Analysis and design of reset control systems reduces the overshoot to 20%. Thus the resulting reset controller consists of a FORE with b = 30 and a linear compensator C(s) = (s + 30)GL(s). (1.54) 1.3.2 PI + CI reset design Proportional-plus-integral control with Clegg integrator (PI + CI) reset control design is a DF-based design method aiming to improve the closed-loop response under a linear PI controller [19]. The main idea is to reset a percentage of the LI to retain the benefit of LI such as the capability of eliminating steady-errors, and at the same time increase the phase margin and the gain-crossing frequency. The first step of the PI + CI reset controller design is to design a linear PI of the form C(s) = Kp 1 + 1 TI s (1.55) based on the desired phase margin and crossover frequency, where Kp is the propor- tional constant and TI is the integral time constant. Then decompose the integral part as 1 TI s = 1 Tis + 1 Tirs and replace the second integrator in the above decomposition by a CI as depicted in Figure 1.19. The integral constants satisfy 1 TI = 1 Ti + 1 Tir and the ratio Ti/Tir characterizes the percentage of the integral to be reset. The DF of the PI + CI compensator is given by Cr( jω) = Kp 1 + 1 jωTi + 1 jωTir 1 + j 4 π . (1.56) The remaining step is to tune the ratio Ti/Tir according to the final performance requirement of the RCS. Kp 1 1 + + e Ti s CI u 1 Tir Figure 1.19 PI + CI reset controller
- 42. Introduction 23 In the case when the plant is second order given by P(s) = ω2 n (s2 + ξωn + jωd)(s2 + ξωn − jωd) , (1.57) where ξ (0 ξ 1) is the damping ratio, ωn the undamped natural frequency and ωd = ωn 1 − ξ2, then the parameters of the PI + CI controller can be determined as follows. 1. Linear PI tuning. Based on the desired gain-crossover frequency ωg and the phase margin φm, calculate TI and Kp by ∠P( jωg)C( jωg) = φm |P( jωg)C( jωg)| = 1. (1.58) Or, precisely, ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ φm = 1 2 π + arctan ωgTI − arctan ωg + ωd ξωn − arctan ωg − ωd ξωn Kpω2 n ωgTI 1 + ω2 gT2 I [ξ2ω2 n + (ωg + ωd)2][ξ2ω2 n + (ωg − ωd)2] = 1 (1.59) 2. PI + CI parameter design. Select a feasible crossover frequency ωgr for the RCS and calculate Tir according to ⎧ ⎨ ⎩ |P( jωgr)Cr( jωgr)| = 1 1 TI = 1 Ti + 1 Tir . (1.60) Simple calculation gives Tir = 4 π β(ωgr) − 1/T2 I − ωgr , (1.61) where β is given by β(ωgr) = ω2 gr K2 p ω4 n ξ2 ω2 n + (ωgr + ωd)2 ξ2 ω2 n + (ωgr − ωd)2 . (1.62) (1.61) gives the range of ωgr one can choose from, i.e., β(ωgr) − 1/T2 I ωgr. The phase margin φrm of the PI + CI RCS is given by φrm = ∠P( jωgr)Cr( jωgr) = 1 2 π + arctan [ωgr + 4/(Tirπ)]TI − arctan ωg + ωd ξωn − arctan ωg − ωd ξωn .
- 43. 24 Analysis and design of reset control systems Notes A detailed description of the limitations of linear feedback control can be found in Reference 1 and the example showing that reset control achieves a set of specifications which cannot be achieved by linear control comes from Reference 18. The early developments of reset control including the CI, the first time-domain design around the CI, and the FORE (the Horiwitz’s design procedure) studied in this chapter are, respectively, from References 11, 16, and 3. The application example of Horiwitz’s design procedure to the tape-speed control systems can be found in Reference 14. See also the References 13, 17, and 15 and the recently published book [20] for more detailed discussions of traditional reset control analysis. The basic PI + CI reset control design comes from Reference 19. See also Reference 21 for the application of PI + CI reset control design to heat exchangers. The model (1.5) for the CI is from Reference 12. The definitions of solutions and the well-posedness of RCSs are based on Reference 20. For the theory of general hybrid systems and impulsive differential equations, reader can refer to References 22, 23, and 10. Other nonlinear techniques aiming to overcome limitations of linear feedback control can be found in References 24, 8, 9, 7, 4, and 6. References [1] M.M. Seron, J.H. Braslavsky, and G.C. Goodwin. Fundamental limitations in filtering and control. In Communications and Control Engineering. Springer- Verlag London Ltd., Berlin, Heidelberg, New York, 1997. [2] J.S. Freudenberg and D.P. Looze. Frequency domain properties of scalar and multivariable feedback systems. Springer-Verlag New York, Inc., 1988. [3] I. Horowitz and P. Rosenbaum. Nonlinear design for cost of feedback reduction in systems with large parameter uncertainty. International Journal of Control, 21(6):977–1001, 1975. [4] J.B. Lewis. The use of nonlinear feedback to improve the transient response of a servomechanism. American Institute of Electrical Engineers, Part II: Transactions of the Applications and Industry, 71(6):449–453, 1953. [5] E.R.Kalman. Phase-planeanalysisofautomaticcontrolsystemswithnonlinear gain elements.American Institute of Electrical Engineers, Part II:Applications and Industry, Transactions of the, 73(6):383–390, 1955. [6] A.R. Bailey. Stabilisation of control systems by the use of driven limiters. Electrical Engineers, Proceedings of the Institution of, 113(1):169–174, 1966. [7] W.C. Foster, D.L. Gieseking, and W.K. Waymeyer. A nonlinear filter for inde- pendent gain and phase (with applications). Journal of Basic Engineering, 88:457, 1966. [8] D. Liberzon. Switching in systems and control. In Systems Control: Foundations Applications, Birkhäuser, Boston, Basel, Berlin, 2003. [9] C. Edwards and S. Spurgeon. Sliding mode control: theory and applications. In Systems and Control, CRC Press, Taylor Francis Group, 1998.
- 44. Introduction 25 [10] T.Yang. Impulsive control theory. In Lecture Notes in Control and Information Sciences, vol. 272, Springer-Verlag Berlin, Heidelberg, Springer, 2001. [11] J.C. Clegg. A nonlinear integrator for servomechanisms. Institute of Electrical Engineers, Part II:Transactions of theApplications and Industry, 77(1):41–42, 1958. [12] L. Zaccarian, D. Nesic, and A.R. Teel. First order reset elements and the Clegg integrator revisited. In American Control Conference, 2005. Proceedings of the 2005. June 8–10, 2005, vol. 1, pp. 563–568. IEEE, Portland, OR, 2005. [13] O. Beker. Analysis of reset control systems. Doctoral dissertations, Univer- sity of Massachusetts Amherst, Available from Proquest. Paper AAI3027178. http://scholarworks.umass.edu/dissertations/AAI3027178, January 1, 2001. [14] Y. Zheng, Y. Chait, C.V. Hollot, M. Steinbuch, and M. Norg. Experimental demonstrationofresetcontroldesign. ControlEngineeringPractice, 8(2):113– 120, 2000. [15] Y. Chait and C.V. Hollot. On Horowitz’s contributions to reset control. International Journal of Robust and Nonlinear Control, 12(4):335–355, 2002. [16] K.R. Krishnan and I.M. Horowitz. Synthesis of a nonlinear feedback system with significant plant-ignorance for prescribed system tolerances. Inter- national Journal of Control, 19(4):689–706, 1974. [17] Q. Chen. Reset control systems: stability, performance and application. PhD thesis, University of Massachusetts Amherst, Available from Proquest. Paper AAI9988771. http://scholarworks.umass.edu/dissertations/AAI9988771, Januray 1, 2000. [18] O. Beker, C.V . Hollot, and Y. Chait. Plant with integrator: an example of reset control overcoming limitations of linear feedback. IEEE Transactions on Automatic Control, 46(11):1797–1799, 2001. [19] A. Baños and A. Vidal. Design of PI+CI reset compensators for second order plants. In IEEE International Symposium on Industrial Electronics, pp. 118– 123. IEEE, 2007. [20] A. Baños and A. Barreiro. Reset control systems. Springer Science Business Media, 2011. [21] A. Vidal and A. Baños. Reset compensation for temperature control: Exper- imental application on heat exchangers. Chemical Engineering Journal, 159(1–3):170–181, 2010. [22] R. Goebel, R.G. Sanfelice, and A.R. Teel. Hybrid dynamical systems: modeling, stability, and robustness. Princeton University Press, 2012. [23] W.M. Haddad,V . Chellaboina, and S.G. Nersesov. Impulsive and hybrid dynam- ical systems: stability, dissipativity, and control. In Princeton Series inApplied Mathematics, Princeton University Press, Princeton, New Jersey, 2006. [24] Z. Sun and S.S. Ge. Switched linear systems: control and design. Springer- Verlag, Berlin, Heidelberg, 2005.
- 46. Chapter 2 Describing function analysis of reset systems Describing function (DF) is a quasi-linearization of a nonlinear element subject to certain excitation input used to approximately analyze the behavior of nonlinear systems.The sinusoidal DF, which uses sinusoidal inputs as excitation signals, is most widely known. DF is a powerful tool in investigating behaviors of elements with hard nonlinearities including dead zone, backlash, and hysteresis, and has been applied in limit cycle prediction and control design. The basic motivation to study DFs of reset systems is that by far rigorous mathematical tools are not available to analyze the behavior of state-driven reset. This chapter first derives the sinusoidal DF for a general reset system with arbitrary reset matrix, followed by application of the DF in disturbance rejection of hard disk drive (HDD) systems. 2.1 Sinusoid input response Consider the general reset system r : ⎧ ⎨ ⎩ ẋ = Ax + be, e = 0 x(t+ ) = Rx, e = 0 y = cT x, (2.1) where e, y ∈ R1 are the scalar input and output, respectively, x, A, b, R, c are of compatible dimensions. The second equation of (2.1) is called the reset mapping and R is called the reset matrix. x(t+ ) denotes the after-reset state at reset time instant t, i.e., x(t+ ) = lim s→t+0 x(s). System (2.1) can be a reset element or the open loop of a reset control system (RCS). For instance, consider an RCS depicted in Figure 2.1. The plant p and the reset controller rc are, respectively, p : ẋp = Apxp + Bpu y = cT p xp (2.2)
- 47. 28 Analysis and design of reset control systems r e u xp ∑p ∑o ∑rc y + – Figure 2.1 General RCS and rc : ⎧ ⎨ ⎩ ż = Arz + Arpxp + Bre, e = 0 z(t+ ) = Rrz + Rrpxp, e = 0 u = Erz + Erpxp + Eee. (2.3) Combining p and rc, the open-loop system o can be represented in the form of system (2.1) with x = (xT p , zT )T and A = Ap + BpErp BpEr Arp Ar , B = BpEe Br , (2.4) R = I O Rrp Rr , cT = cT p O . (2.5) Remark 2.1. In the early development of reset systems, the reset matrix R is assumed to be zero, i.e., R = 0. See Reference 1 for instance. In this case, all of the states are reset to zero whenever the input crosses zero. From (2.5), the reset matrix R of the open-loop system is never zero, this is because only the state of the controller is allowed to be reset. In this chapter, we assume the reset matrix to be an arbitrary square matrix. For any matrices A and R, we introduce the following notations (if well-defined) for convenience (ω) = ω2 I + A2 , (ω) = I + e π ω A , R(ω) = I + Re π ω A , R(ω) = −1 R (ω)R(ω)−1 (ω), = {ω 0 | ±jω are eigenvalues of A}, R = {ω 0 | λ(Re π ω A ) 1}. The following assertions are obviously true: 1. (ω) (ω 0) is nonsingular if and only if ω / ∈ . 2. If ω ∈ R, then R(ω) is nonsingular.
- 48. Describing function analysis of reset systems 29 In order to obtain the DF of system (2.1), we study a sinusoid input response first. Consider the sinusoid excitation input e(t) = α sin (ωt). (2.6) The set of the reset time instants {tk } is given by tk = kπ/ω, k = 0, 1, . . . . Define ηk = x(t+ 2k ), ζk = x(t+ 2k+1) and ψ(t) = t 0 e−As b sin (ωs)ds. Then the solution of system (2.1) with initial condition x(0+ ) = η0 can be represented by x(t) = eA(t−t2k ) ηk + αeAt [ψ(t) − ψ(t2k )], t ∈ (t2k , t2k+1]; eA(t−t2k+1) ζk + αeAt [ψ(t) − ψ(t2k+1)], t ∈ (t2k+1, t2k+2] (2.7) where ηk and ζk are determined by the following recursive algorithm ζk = Re π ω A ηk + αψ(π ω ) , ηk+1 = Re π ω A ζk − αψ(π ω ) , η0 = x(0+ ). (2.8) Definition 2.1. The recursion (2.8) is said to converge globally if there exist constant vectors ζ̄ and η̄ such that lim k→+∞ ηk = η̄, (2.9) lim k→+∞ ζk = ζ̄ (2.10) for an arbitrary initial condition η0. The following lemma gives the necessary and sufficient condition for the global convergence of the recursion (2.8). Lemma 2.1. The recursion (2.8) converges globally if and only if ω ∈ R, or equivalently, λ(Re π ω A ) 1. (2.11) In this case, we have η̄ = lim k→+∞ ηk = −α I + Re π ω A −1 Re π ω A ψ( π ω ), (2.12) ζ̄ = lim k→+∞ ζk = α I + Re π ω A −1 Re π ω A ψ( π ω ). (2.13)
- 49. 30 Analysis and design of reset control systems Proof: (Sufficiency) Assume that (2.11) holds, then I + Re π ω A is nonsingular. Let ξk = ηk + α I + Re π ω A −1 Re π ω A ψ( π ω ), we have ξk+1 = Re π ω A 2 ξk . (2.14) Since λ (Re π ω A )2 = λ(Re π ω A ) 2 1, we have lim k→+∞ ξk = 0. Thus (2.12) holds. (2.13) can be proved similarly. (Necessity) From (2.8) we have ηk+1 = Re π ω A 2 ηk + χ, where χ = α Re π ω A 2 ψ( π ω ) − αRe π ω A ψ( π ω ) is a constant vector. Let η̄ = limk→∞ ηk , then we have Re π ω A 2 η̄ + χ = η̄. Further, by defining ξk = ηk − η̄, we have lim t→∞ ξk = 0 for any ξ0 and ξk+1 = Re π ω A 2 ξk . Thus condition (2.11) holds. By Lemma 2.1, we have the following result immediately. Proposition 2.1. Reset system (2.1) with input (2.6) has a globally asymptotically stable 2π/ω-periodic solution if and only if λ(Re π ω A ) 1. (2.15) If the above condition holds, then the globally asymptotically stable solution is given by x(t) = eA(t−t2k ) η̄ + αeAt [ψ(t) − ψ(t2k )], t ∈ (t2k , t2k+1], eA(t−t2k+1) ζ̄ + αeAt [ψ(t) − ψ(t2k+1)], t ∈ (t2k+1, t2k+2] where η̄ and ζ̄ are defined in (2.12) and (2.13).
- 50. Describing function analysis of reset systems 31 Remark 2.2. From Proposition 2.1, the existence of globally asymptotically periodic solution of an RCS under a sinusoid input with certain frequency ω does not imply that the base linear system is stable. In addition, condition (2.15) depends on input frequency ω. But for a linear system without reset (i.e., R = I), condition (2.15) implies that A must be a stable matrix. In this case, condition (2.15) is equivalent to Re λ(A) 0 which is independent of ω. Remark 2.3. From equation (2.14), |λ(Re π ω A )| is related to the convergence rate of the periodic solution. This can also be seen in the following example and the simulation result of Section 2.3. Example 2.1. Consider a reset system with A = −1 1 0 1 , b = 0 1 , c = 0 1 , R = 1 0 0 β . It is easy to check that Re π ω A = e− π ω 1 2 [e π ω − e− π ω ] 0 βe π ω . Thus for any fixed input frequency ω 0, the condition (2.15) holds if and only if |β| e− π ω . The output responses of this system for initial condition (5, −1) with β = −0.3 and 0, respectively, are given in Figure 2.2. 1.5 1 0.5 0 Output responses −0.5 −1 −1.5 0 0.5 1 1.5 2 Time (s) 2.5 3 3.5 4 Sinusoidal input Output response for b = −0.3 Output response for b = 0 Figure 2.2 Output responses for Example 2.1
- 51. 32 Analysis and design of reset control systems In practical systems, the actual input contains signals of different frequencies. Thus the boundedness of solutions under a sinusoid input with arbitrary frequency is an important property for RCSs. And this property is also the basis for a nonlinear element to have a DF defined on arbitrary frequency. From Proposition 2.1, the following result is obvious. Proposition 2.2. RCS (2.1) has a globally asymptotically stable 2π/ω-periodic solution under a sinusoid input with arbitrary frequency ω 0 if and only if |λ(ReAδ )| 1, ∀δ ∈ R+ . (2.16) Remark 2.4. From condition (2.16), by letting δ → 0+ , one sees that R must be such that |λ(R)| ≤ 1. However, A is not necessarily Lyapunov stable. For instance, in Example 2.1, A is unstable. But if β = 0, then ReAδ = e−Aδ 1 2 [eAδ − e−Aδ ] 0 0 and condition (2.16) holds. Remark 2.5. If condition (2.16) holds, then system (2.1) has a sinusoid input DF defined on arbitrary frequency. Remark 2.6. If ω / ∈ , then ω2 I + A2 is nonsingular. In this case, it is easy to check that ψ(t) = −[ω2 I + A2 ]−1 (ωI cos (ωt) + A sin (ωt))e−At − ωI b. Thus ψ( π ω ) = [ω2 I + A2 ]−1 (I + e− π ω A )ωb. (2.17) 2.2 Describing function 2.2.1 General case In this section, we calculate the DF of system (2.1). According to the analysis of the previous section, the response y(t) of system (2.1) under a sinusoid input with frequency ω ∈ R can be represented as y(t) = yss(t) + yt(t), (2.18)
- 52. Describing function analysis of reset systems 33 where yss(t) is the steady-state response which is 2π/ω-periodic and independent of the initial condition, and yt(t) is the transient response which dies away with time, i.e., lim t→∞ yt(t) = 0. (2.19) The sinusoidal DF of system (2.1) is defined by GR( jω) = Yss,1( jω) E( jω) , (2.20) where Yss,1( jω) = ω 2π 2π ω 0 yss(t)e−jωt dt, (2.21) E( jω) = ω 2π 2π ω 0 e(t)e−jωt dt. (2.22) In the following, we assume that ω ∈ R . By (2.7), (2.8) and (2.17), when t ∈ (tk , tk+1], yss(t) = αcT eAt (−1)k+1 e−Atk R(ω)ωb + ψ(t) − ψ(tk ) = αcT eAt θk (ω) − αcT −1 (ω)[ωI cos (ωt) + A sin (ωt)]b where θk (ω) = (−1)k+1 e−Atk R(ω)ωb − ψ(tk ) + −1 (ω)ωb = (−1)k+1 e−Atk [R(ω) − −1 (ω)]ωb. Thus the fundamental component of the Fourier series of yss(t) can be calculated as follows Yss,1( jω) = ω 2π 2π ω 0 yss(t)e−jωt dt = αωcT 2π (I1 + I2) − αωcT −1 (ω) 2π (ωIJ1 + AJ2)b, where I1 = π ω 0 eAt θ0(ω)e−jωt dt = ( jωI − A)−1 (ω)θ0(ω), I2 = 2π ω π ω eAt θ1(ω)e−jωt dt = −( jωI − A)−1 (ω)e π ω A θ1(ω), J1 = 2π ω 0 e−jωt cos (ωt)dt = π ω , J2 = 2π ω 0 e−jωt sin (ωt)dt = −j π ω .
- 53. 34 Analysis and design of reset control systems Thus we have Yss,1( jω) = − jα 2 cT ( jωI − A)−1 (I + jR(ω))b (2.23) where R(ω) = − 2ω2 π (ω)[R(ω) − −1 (ω)] (2.24) = 2ω2 π (I + e π ω A )(I + Re π ω A )−1 (I − R)(ω2 I + A2 )−1 (2.25) It is very easy to calculate that E( jω) = ω 2π 2π ω 0 α sin (ωt)e−jωt dt = − jα 2 . The following result follows. Theorem 2.1. The sinusoidal DF of system (2.1) is given by GR( jω) = cT ( jωI − A)−1 (I + jR(ω))b (2.26) which is well-defined on ω ∈ R , where R(ω) is given by (2.25). Remark 2.7. If the output function is replaced by y = cT x + de, (2.27) then the DF becomes GR( jω) = cT ( jωI − A)−1 (I + jR(ω))b + d. (2.28) Remark 2.8. If A has no pure imaginary eigenvalues, i.e., = ∅, and condition (2.16) holds, then we have R = R+ . That is to say, the DF is well-defined for an arbitrary positive number ω. Remark 2.9. 1. When R = I, system (2.1) degenerates to the base linear system. In this case, by (2.25), it is easy to verify that R(ω) = 0. Hence the DF degenerates to the frequency response function of the base linear system. 2. If A is a stable matrix, we have lim ω→0 e π ω A = 0. Thus, limω→0 R(ω) = 0. This means that the reset does not have much effect over the very low-frequency range ω π A . 3. It is easy to check that R(∞) = lim ω→∞ R(ω) = 4 π (I + R)−1 (I − R). This indicates that the effect of the reset is almost independent of frequency in a high-frequency region with ω π A .
- 54. Describing function analysis of reset systems 35 Remark 2.10. Though system (2.1) is essentially a nonlinear system, the DF is independent of the amplitude α of the excitation signal. Corollary 2.1. If the reset matrix is zero, i.e., R = 0n×0, then the sinusoidal DF of system (2.1) is given by GR( jω) = cT ( jωI − A)−1 I + j 2ω2 π (I + e π ω A )(ω2 I + A2 )−1 b (2.29) which is well-defined on ω ∈ R+ . In addition, there holds lim ω→∞ GR( jω) = 1 + j 4 π GL( jω) (2.30) where GL( jω) = cT ( jωI − A)−1 b is the frequency response function of the base linear system. By using Corollary 2.1, the DF of the first-order reset element (FORE) ⎧ ⎨ ⎩ ż = −bz + e, e = 0 z+ = 0, e = 0 ur = z. (2.31) is given by GFORE( jω) = 1 jω + b 1 + j 2ω2 (1 + e−πb/ω ) π(ω2 + b2) . (2.32) Let b = 0, then we get the DF of the Clegg integrator (CI) as GCI ( jω) = 1 jω 1 + j 4 π . Example 2.2. [2] Consider a second-order plant y(s) = P(s)u(s) with P(s) = s + 1 s(s + 0.2) controlled by a FORE ⎧ ⎨ ⎩ ẋr = −xr + e, e = 0 z+ = 0, e = 0 u = xr (2.33) where e = r − y. The open-loop dynamics is then of the form (2.1) with A = ⎡ ⎣ −0.2 0 2 1 0 0 0 0 −1 ⎤ ⎦ , b = ⎡ ⎣ 0 0 1 ⎤ ⎦ , c = [0.5 0.5 0], R = ⎡ ⎣ 1 0 0 0 1 0 0 0 0 ⎤ ⎦ .
- 55. Exploring the Variety of Random Documents with Different Content
- 56. — Miksi olette niin ääneti, ystävät? Markkus, lue meille kappale hartauskirjastasi, joka on laukussasi. Markkus meni hakemaan laukkuaan. Sillävälin kuiskasi marski hiljaa Gröningille: — Tuota kirjaa kuulen kernaasti; en tiedä, onko se luterilainen vai katoolinen, mutta se on koruton ja harras, ja minä uskon, että se on otollinen Jumalalle… Lue, mestari, kappale… Mestari luki. Yksitoikkoisella, laulavalla äänellä hän laski tulemaan sivun ja toisen ja Klaus-herra näytti rauhallisesti kuuntelevan, maatessaan siinä silmä raollaan, hengitys tuskin tuntuvana. Markkus luki, luki edelleen… Vaan hänen lukiessaan nousi Gröning hiljaa, tarttui marskin peitolla lepäävään käteen: Se oli kankea. — Mestari, sinä olet lukenut kylliksi, saneli hän harvakseen, laskien kankean käden vuoteelle. Markkus laski kirjansa alas, koetteli vuoteella makaajan sydäntä, ja toisti hiljaa: — Klaus Fleming on kuollut! — — — Aamun sarastaissa, — huhtikuun 13 päivänä 1597 — ajoi taas yksinäinen ratsastaja samaa mustunutta talvitietä, jota hän edellisenäkin iltana oli kulkenut, vaan tällä kertaa Pohjan pitäjästä Perniöön päin. Matkan teko tuntui nyt vielä paljo raskaammalta kuin eilen. Keli oli tosin yön aikana kovennut ja aamun koetteessa löysi hevonen paremmin tiepaikan, vaan itse ratsastajalla oli taakka raskaampi. Hän oli nyt kuljettamassa sanomaa, ensiksi Perniön
- 57. kuninkaankartanossa odottavalle, levottomalle rouvalle, sitten edelleen läpi maan Turkuun asti, että Viikin vapaaherra, Suomen käskynhaltija, valtakunnan neuvos, marski ja amiraali Klaus Eerikinpoika Fleming oli kuollut. Ja sen sanoman kuljetus oli raskasta ratsastajalle, Kaspar Gröningille. Vaan hän teki tehtävänsä, kertoi sanomansa ja kulon nopeudella lähti se leviämään yli Suomen niemen, syöksähti aateliskartanoihin, pistihe mökkeihin ja muonamiesten majoihin, kaikkialle iskien kuin odottamaton ukkosen salama, kaikkialla herättäen hämmästystä ja monia mietteitä. — Klaus Fleming on kuollut! Noihin sanoihin sisältyi niin paljo, sisältyi levottomuutta ja toivoa, surua ja iloa, epävarmuutta ja helpotuksen tunnetta. Ihmisten oli vaikea kohta oivaltaa tätä sanomaa. Myrsky oli käynyt yli koko maan, jaksamatta kukistaa rautatammea. Kukaan ei tiennyt, että tammi siinä myrskyssä oli saanut vamman, johon se tyyneen tultua itsekseen taittui. Se oli ollut niin voimakas ja kova, se oli yksin hallinnut kaikki ja tuntunut kukistumattomalta. Ja siksi loivat ihmiset, niin ystävät kuin viholliset, tämän sanoman saapuessa katseensa alas ja hokivat itsekseen: — Klaus Fleming on kuollut! Ja näitä sanoja seurasi tavallisesti kysymys: — Mikä kaatoi mahtavan miehen? Taikausko loi siitä kohta omat selityksensä. Pohjanmaan noidat, nuijamiesten vihan perijät, olivat muka kostaneet Nokian ja
- 58. Santavuoren. Herttuan myrkynsekottajat olivat raivanneet pois tuon sitkeän esteen. Vaan se, joka kuolonsanomaa kuljetti, hymähti näille taikatuumille; hän arvasi syyt syvemmät, mutta maailmalle hän ei niitä selittämään käynyt. — Klaus Fleming on kuollut! Mitä tapahtuu nyt? Sitä ei kukaan osannut ennustella. Ihmiset katselivat huolestuneina ja arastellen toisiaan ja puistelivat miettiväisinä päätään. Vaan kuolon sanoman kuljettaja jatkoi matkaansa, saapui Turkuun. Mutta kauaksi hän ei sinne jäänyt, jo viiden päivän perästä oli hän palannut takasin Pohjan pitäjän nimismiehentaloon, vanhan Prinkkalan herran ja muutamain muiden suomalaisten aatelismiesten kanssa noutamaan Turkuun vainajan ruumista. Keli oli nyt aivan lopullaan, kun tämä saattokulkue teki matkaa Turkuun päin. Paksuun, mustaan harsoon kietoutuneena ajoi Ebba Fleming ruumisreen takana syvän surunsa masentamana, Gröning ratsasti ääneti ja miettiväisnä siinä vieressä ja vakavina seurasivat aatelisherrat perästä. Raskasta oli kelitön kulku, raskaita saattajain mielet ja mietteet. Vaan jäykkänä ja kankeana istui vanha Eenokki etummaisen reen ajolaudalla, kyyditen herraansa, kuten niin monasti ennen. Vaan hän tiesi tekevänsä sen nyt viimeisen kerran ja kunnialla ja huolella hän sen tehdä tahtoi, — sittenhän hänellä ei ollutkaan enää virkaa maailmassa. Eräänä lauvantaina huhtikuun lopulla saapui saattokulkue vihdoin Turkuun ja kulki verkalleen melkein lumetonta rantakatua myöten kaupungin läpi linnaan päin. Mustaa lippua kantava airut ratsasti edellä ja julisti hautasaaton saapumisesta kansalle. Kuolinkellot soivat kaupungin ja linnan kirkoissa — ne olivat jo yhtämittaa soineet
- 59. viikon — ja sotaväki, joka kadulla seisoi kunniavahtina, oli vakavaa ja juhlallista. Äänetönnä seisoi väkijoukko. Siellä täällä paljastui pää ruumisreen ohimennessä, vaan yleensä oli vastaanotto kylmää ja tunteetonta. Miettiväisnä Gröning katseli tuota tiheää laumaa, joka rantakadun täytti, — Turkuun oli juuri näinä päivinä meren auettua saapunut ensimmäinen ryhmä saaristolaisia kevätkauppojaan tekemään, olipa jo etäisempiä aluksiakin satamaan saapunut ja toisia siitä valmistautui lähtemään; ja siksi oli sataman partaalla Turun oloihin nähden harvinainen väenpaljous, joka nyt seisoi ääneti ja jäykkänä, mutta kumminkin kunnioittaen, tiepuolissa ruumissaattoa katsellen. Tuo kansa ei ole tässä vietävää vainajata koskaan rakastanut, ajatteli Gröning, — ei, vihannut se häntä on, mutta se kunnioittaa kumminkin sitä rautavoimaa, jonka se ei luullut katkeavankaan… Yhtäkkiä Gröning ikäänkuin havahtui näistä mietteistään. Hän oli tuossa saaristolaisjoukossa ollut vilaukselta tuntevinaan omituisesti tuttavat kasvot … niin siinä ne taas olivat nuo silmät, jotka hänelle veitikkamaisesti tervehdykseksi iskevät. Se on nuori saaristolainen, vieressään talonpoikaistyttö… Kuka voisi se olla…? Gröning katsoo vielä kerran ja tuntee … sehän on Birckholtz, hänen rakas lankonsa, tottatosiaan, se parantumaton veitikka… Gröning nyökäyttää hänelle vastaukseksi päätään ja ilosesti viittaa nuori saaristolainen laivaan päin, jossa juuri nostetaan purjeita. Nyt Gröning ymmärtää asian, nyt hän arvaa jo senkin, kuka tuo tyttö tuossa vieressä on. Ruumisreki vierii verkalleen sataman ohi pitkin lumetonta katua. Vaan kun saapuu seuraava reki, jossa Ebba-rouva istuu, vaaleana ja syvän surun ilmeet kasvoillaan, mutta siltä suorana ja arvokkaana, ja hänen vieressään nuori Juhana Klaunpoika kalvakkana ja vakavana,
- 60. silloin tarttuu talonpoikaispukuun puettu tyttö suonenvedontapaisesti vieressään seisovan nuorukaisen käsivarteen, ikäänkuin turvautuen siihen, ja silmille vedetyn huivin alta näkyy kyyneleitä vierähtävän hienoselle poskelle. Se tunteiden ristiriita, joka niin monesti oli Anna Flemingin mielessä riehunut, on taas saanut hänessä vallan. Tuota vanhusta, joka nyt kylmänä makaa arkussaan, on hän ihaillut, jopa rakastanutkin, tuossa surevassa perheessä on hän lapsuutensa ja nuoruutensa viettänyt ja sen ilot ja surut ovat olleet hänen omiaan. Hän on siinä kärsinyt, vaan se on siltä ollut hänen kotinsa… Tuossa pitäisi hänen nyt olla, lohduttamassa isättömiksi jääneitä ja lieventämässä lesken surua. Vaan sen sijaan hän kiittämättömänä karkurina seisoo siinä kylmäin syrjästäkatsojain riveissä, tuotettuaan tällä paollaan taaskin vereksen lisäsurun omaisilleen… Annassa herää hetkeksi halu juosta esiin siitä väkijoukosta, viskautua tätinsä eteen rekeen ja huudahtaa: tässä olen, rangaiskaa minut…! Vaan hän säikähtää sitä ajatustaan, tarttuu sulhonsa käsivarteen… — Miksi vapiset, tyttöni? kysyy Hieronymo. — Ja kyyneleitä! — tuossa perheessä olet luullakseni jo itkenyt kylliksi. — Olen, vaan he olivat kumminkin kasvattajani, ja huonosti heitä palkitsen. — Huonostiko, — no, pesäero on nyt kerran tapahtunut; päämies on kuollut, ja siitä kuolemasta meidän elämä alkaa. Heitä viimeiset jäähyväiset, Anna, nyt Klaus Flemingin perheelle, ja sitten laivaan…! Laiva, jossa tuo nuori saaristolaispari oli varannut itselleen paikat Ruotsiin matkustaakseen, olikin jo purjehduskunnossa, valmiina lähtemään joensuusta merelle, ja hautasaaton ohimentyä talutti
- 61. Hieronymo morsiamensa sen kannelle. Hetken kuluttua ruvettiin siinä jo köysiä maista irrottamaan ja kääntämään alusta laiturista. Vaan se ei päässyt kumminkaan vielä kääntymään, eräs toinen laiva, lyypekkiläinen kauppalaiva, joka juuri saapui satamaan ja pyrki laituriin, oli sen edessä ahtaalla joella, ja siinä syntyi, laivoja köysin hinattaessa, vielä hetkisen viivykki. Anna Fleming seisoi joenpuoleiseen kaidepuuhun nojauneena ja katseli, kuinka saksalaisessa laivassa miehet köysillä varppailivat. Mutta yhtäkkiä hän näkee tuon laivan kannella tutut kasvot … hän ei voi olla hiljakseen huudahtamatta, sillä sehän on Elina Fincke tuo, joka maston juurella noin pää kumarassa seisoo… Tämä kuulee myös tutun äänen, kohottaa päänsä, tulee lähemmäs ja niin kauan erossa olleet serkukset tervehtivät yhtäkkiä toisiaan ilostunein kasvoin ja vilkkain liikkein sen kapean juovan yli, joka erottaa laivat toisistaan. Tuokion he ääneti seisovat, ikäänkuin tapaillen sanoja ja hämmästyneinä odottamattomasta tapaamisesta, vaan samassa heille molemmille asema selviää. Ja hymy kangistuu heidän huulilleen. — Sinä olet matkalla toteuttamaan unelmaisi onnea, Anna, lausuu Elina miltei katkerasti. Hän on käynyt kasvoiltaan kalvakaksi, hän on tänä yhtenä vuonna paljo vanhennut ja hänen silmissään on raukea pettymyksen ilme. — Niin, minä olen sen saavuttanut, — vihdoinkin. Vaan sinä, — sinä palaat yksin? — Minun unelmani on särkynyt, kaikki on lopussa. — Et saanut häntä Suomeen?
- 62. — Hän ei luottanut itseensä, ei tulevaisuuteensa, hän pakenee kutsumustaan, pakenee minua… — Siis hyljätty — serkku parka! Unelmasi kai oli, Elina, liian suuri. — Niin, minä haaveksin suuruutta, mainetta, tapailin liian korkealle, siksi olen pudonnut alas. Sinä, Anna, haaveksit pikkusta onnea ja taistelit sen puolesta, — olkoon se sulle täyteläinen! Annan silmään kierähtää kyynele, hän ei saa sanaa suustaan; häntä niin säälittää serkkunsa pettymys ja suru, joka hänen koko käytöksestään, jopa äänen värähdyksestäkin, ilmenee, ja hän sille surulle kyllä arvon osaa antaa. Vaan hän ei löydä sanoja tulkitakseen tunteitaan. Hänen tekisi mieli omaa onneaan valaa lääkkeeksi ystävättärensä kirvelevään haavaan, vaan hän tietää, että siihen ei lääke tepsi… Jo on laituriin saatu vapaa kohta, lyypekkiläistä laivaa ruvetaan hinaamaan sinne ja matka serkusten välillä pitenee. Silloin kysyy vielä Anna: — Ja mitä aijot nyt, Elina? — Hämeenlinnaan, vankina elämään, ja muistelemaan unelmaini kaatumista, — ijäksi. Jää hyvästi! Laivat ovat jo etäällä toisistaan, purjeita laskeutuu väliin, orpanukset eivät näe enää toisiaan. Anna vetäytyy kyynel silmässään pois kaidepuun vierestä; serkulla vankeus, hänellä vapaus edessään … liian puolueellinen oli sittenkin kohtalo! Vaan muuttaa sitä ei voi, seurata vain sen johtoa! — Siinä heidän laivaa juuri loitonnetaan laiturista … ja siinä hänen sulhonsa… Hieronymo seisoo
- 63. rannanpuoleiseen kaidepuuhun nojaten keskustelussa jonkun rannalla seisovan tuttavan kanssa … niin, sehän on Gröning, joka linnasta on kiirehtinyt takasin rantaan lankomiestään tapaamaan. Anna kuulee vain viimeiset lauseet siitä heidän hiljaisesta keskustelustaan. — Sinä olit väärällä puolella, lankomies, sitähän sulle aina sanoin, väitti Hieronymo. — Näet nyt! — En tiedä, olinko väärällä, vaan minun täytyi olla sillä puolella. — Miksi täytyi? — Sitä et ehkä sinä ymmärrä. Mutta katsos, minä olen suomalainen… Laiva loittoni rannasta, purjeet alkoivat saada tuulta ja hiljaa lipui alus joen suuta kohden. Kaupunki häipyi näkyvistä saarten taa, joiden rantametsä jo kainosti vihannoi hienolla hiirenkorvalla ja joissa nurmi pälvipaikoissa, nietosten lomissa, teki ensimmäistä nukkaa. Ja vähän ulompana selällä näkyi taas hienoja utuaaltoja liitelevän vedenpinnalla, kun keväinen aurinko lämmitteli talviseltaan kylmiä syvänteitä. Sitä sumukon hilpeää tanssia katseli kauan laivan kaidepuuhun nojautuva tyttö. Hänen mieleensä palasi niin elävästi eräs keväinen veneretki, jolloin kaksi neitosta oli tuon utuaallokon heiluvasta hypystä ennustellut vastaisia vaiheitaan ja suunnelmoinut tulevaisuutensa unelmia ja toiveita. Ja hän tunsi kiitollisuutta sydämmessään, noita kevään kepeitä hengettäriä katsellessaan, niin syvää kiitollisuutta: sillä hänelle ne olivat hänen pienen onnensa taikoneet…
- 64. Vaan etäämpää rannan puolelta näkyivät vielä Turun linnan ja kirkkojen tornit, ja kirkoista kumahteli myötätuuleen kellojen soitto. Siellä hautakelloja soitettiin.
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