Am04 ch5 24oct04-stateand integral

Chapter 5
State and Output Feedback
This chapter describes how feedback can be used shape the local behavior
of a system. Both state and output feedback are discussed. The concepts
of reachability and observability are introduced and it is shown how states
can be estimated from measurements of the input and the output.
5.1 Introduction
The idea of using feedback to shape the dynamic behavior was discussed in
broad terms in Section 1.4. In this chapter we will discuss this in detail for
linear systems. In particular it will be shown that under certain conditions it
is possible to assign the system eigenvalues to arbitrary values by feedback,
allowing us to “design” the dynamics of the system.
The state of a dynamical system is a collection of variables that permits
prediction of the future development of a system. In this chapter we will
explore the idea of controlling a system through feedback of the state. We
will assume that the system to be controlled is described by a linear state
model and has a single input (for simplicity). The feedback control will be
developed step by step using one single idea: the positioning of closed loop
eigenvalues in desired locations. It turns out that the controller has a very
interesting structure that applies to many design methods. This chapter
may therefore be viewed as a prototype of many analytical design methods.
If the state of a system is not available for direct measurement, it is
often possible to determine the state by reasoning about the state through
our knowledge of the dynamics and more limited measurements. This is
done by building an “observer” that uses measurements of the inputs and
outputs of a linear system, along with a model of the system dynamics, to
109

110 CHAPTER 5. STATE AND OUTPUT FEEDBACK
estimate the state.
The details of the analysis and designs in this chapter are carried out for
systems with one input and one output, but it turns out that the structure
of the controller and the forms of the equations are exactly the same for
systems with many inputs and many outputs. There are also many other
design techniques that give controllers with the same structure. A charac-
teristic feature of a controller with state feedback and an observer is that
the complexity of the controller is given by the complexity of the system
to be controlled. Thus the controller actually contains a model of the sys-
tem. This is an example of the internal model principle which says that a
controller should have an internal model of the controlled system.
5.2 Reachability
We begin by disregarding the output measurements and focus on the evolu-
tion of the state which is given by
dx
dt
= Ax + Bu, (5.1)
where x ∈ Rn, u ∈ R, A is an n × n matrix and B an n × 1 matrix. A
fundamental question is if it is possible to ﬁnd control signals so that any
point in the state space can be reached.
First observe that possible equilibria for constant controls are given by
Ax + bu0 = 0
This means that possible equilibria lies in a one (or possibly higher) dimen-
sional subspace. If the matrix A is invertible this subspace is spanned by
A−1B.
Even if possible equilibria lie in a one dimensional subspace it may still
be possible to reach all points in the state space transiently. To explore
this we will ﬁrst give a heuristic argument based on formal calculations with
impulse functions. When the initial state is zero the response of the state
to a unit step in the input is given by
x(t) =
t
0
eA(t−τ)
Bdτ. (5.2)
The derivative of a unit step function is the impulse function δ(t), which
may be regarded as a function which is zero everywhere except at the origin

5.2. REACHABILITY 111
and with the property that
∞
∞
δ(t)dt = 1.
The response of the system to a impulse function is thus the derivative of
(5.2)
dx
dt
= eAt
B.
Similarly we ﬁnd that the response to the derivative of a impulse function
is
d2x
dt2
= AeAt
B.
The input
u(t) = α1δ(t) + α2
˙δ(t) + α¨δ(t) + · · · + αnδ(n−1)
(t)
thus gives the state
x(t) = α1eAt
B + α2AeAt
B + α3A2
eAt
B + · · · + αnAn−1
eAt
B.
Hence, right after the initial time t = 0, denoted t = 0+, we have
x(0+) = α1B + α2AB + α3A2
B + · · · + αnAn−1
B
The right hand is a linear combination of the columns of the matrix
Wr = B AB . . . An−1B . (5.3)
To reach an arbitrary point in the state space we thus require that there are
n linear independent columns of the matrix Wc. The matrix is called the
reachability matrix.
An input consisting of a sum of impulse functions and their derivatives
is a very violent signal. To see that an arbitrary point can be reached with
smoother signals we can also argue as follow. Assuming that the initial
condition is zero, the state of a linear system is given by
x(t) =
t
0
eA(t−τ)
Bu(τ)dτ =
t
0
eAτ
Bu(t − τ)dτ.
It follows from the theory of matrix functions that
eAτ
= Iα0(τ) + Aα1(τ) + . . . + An−1
αn−1(τ)

and we find that
x(t) = B
t
0
α0(τ)u(t − τ) dτ + AB
t
0
α1(τ)u(t − τ) dτ+
. . . + An−1
B
t
0
αn−1(τ)u(t − τ) dτ.
Again we observe that the right hand side is a linear combination of the
columns of the reachability matrix Wr given by (5.3).
We illustrate by two examples.
Example 5.1 (Reachability of the Inverted Pendulum). Consider the inverted
pendulum example introduced in Example 3.5. The nonlinear equations of
motion are given in equation (3.5)
dx
dt
=
x2
sin x1 + u cos x1
y = x1.
Linearizing this system about x = 0, the linearized model becomes
dx
dt
=
0 1
1 0
x +
0
1
u
y = 1 0 x.
(5.4)
The dynamics matrix and the control matrix are
A =
0 1
1 0
, B =
0
1
The reachability matrix is
Wr =
0 1
1 0
. (5.5)
This matrix has full rank and we can conclude that the system is reachable.
This implies that we can move the system from any initial state to any final
state and, in particular, that we can always find an input to bring the system
from an initial state to the equilibrium.
Example 5.2 (System in Reachable Canonical Form). Next we will consider
a system by in reachable canonical form:
dz
dt
=







−a1 −a2 . . . an−1 −an
1 0 0 0
0 1 0 0
...
0 0 1 0







z +







1
0
0
...
0







u = Ãz + ˜Bu

5.2. REACHABILITY 113
To show that Wr is full rank, we show that the inverse of the reachability
matrix exists and is given by
˜W−1
r =





1 a1 a2 . . . an
0 1 a1 . . . an−1
...
0 0 0 . . . 1





(5.6)
To show this we consider the product
˜B Ã ˜B · · · Ãn−1B W−1
r = w0 w1 · · · wn−1
where
w0 = ˜B
w1 = a1
˜B + Ã ˜B
...
wn−1 = an−1B + an−2
ÃB + · · · + Ãn−1
B.
The vectors wk satisfy the relation
wk = ak + ˜wk−1
and iterating this relation we find that
w0 w1 · · · wn−1 =





1 0 0 . . . 0
0 1 0 . . . 0
...
0 0 0 . . . 1





which shows that the matrix (5.6) is indeed the inverse of ˜Wr.
Systems That Are Not Reachable
It is useful of have an intuitive understanding of the mechanisms that make
a system unreachable. An example of such a system is given in Figure 5.1.
The system consists of two identical systems with the same input. The
intuition can also be demonstrated analytically. We demonstrate this by a
simple example.

u
S
S
Figure 5.1: A non-reachable system.
Example 5.3 (Non-reachable System). Assume that the systems in Figure 5.1
are of first order. The complete system is then described by
dx1
dt
= −x1 + u
dx2
dt
= −x2 + u
The reachability matrix is
Wr =
1 −1
1 −1
This matrix is singular and the system is not reachable. One implication of
this is that if x1 and x2 start with the same value, it is never possible to
find an input which causes them to have different values. Similarly, if they
start with different values, no input will be able to drive them both to zero.
Coordinate Changes
It is interesting to investigate how the reachability matrix transforms when
the coordinates are changed. Consider the system in (5.1). Assume that the
coordinates are changed to z = Tx. As shown in the last chapter, that the
dynamics matrix and the control matrix for the transformed system are
Ã = TAT−1
˜B = TB
The reachability matrix for the transformed system then becomes
˜Wr = ˜B Ã ˜B . . . Ãn−1 ˜B =

5.3. STATE FEEDBACK 115
We have
Ã ˜B = TAT−1
TB = TAB
Ã2 ˜B = (TAT−1
)2
TB = TAT−1
TAT−1
TB = TA2
B
...
Ãn ˜B = TAn
B.
The reachability matrix for the transformed system is thus
˜Wr = ˜B Ã ˜B . . . Ãn−1 ˜B = T B AB . . . An−1B = TWr (5.7)
This formula is useful for finding the transformation matrix T that converts
a system into reachable canonical form (using ˜Wr from Example 5.2).
5.3 State Feedback
Consider a system described by the linear differential equation
dx
dt
= Ax + Bu
y = Cx
(5.8)
The output is the variable that we are interested in controlling. To begin
with it is assumed that all components of the state vector are measured.
Since the state at time t contains all information necessary to predict the
future behavior of the system, the most general time invariant control law
is function of the state, i.e.
u(t) = f(x(t))
If the feedback is restricted to be a linear, it can be written as
u = −Kx + Krr (5.9)
where r is the reference value. The negative sign is simply a convention
to indicate that negative feedback is the normal situation. The closed loop
system obtained when the feedback (5.8) is applied to the system (5.9) is
given by
dx
dt
= (A − BK)x + BKrr (5.10)

It will be attempted to determine the feedback gain K so that the closed
loop system has the characteristic polynomial
p(s) = sn
+ p1sn−1
+ . . . + pn−1s + pn (5.11)
This control problem is called the eigenvalue assignment problem or the pole
placement problem (we will define “poles” more formally in a later chapter).
Examples
We will start by considering a few examples that give insight into the nature
of the problem.
Example 5.4 (The Double Integrator). The double integrator is described by
dx
dt
=
0 1
0 0
x +
0
1
u
y = 1 0 x
Introducing the feedback
u = −k1x1 − k2x2 + Krr
the closed loop system becomes
dx
dt
=
0 1
−k1 −k2
x +
0
Kr
r
y = 1 0 x
(5.12)
The closed loop system has the characteristic polynomial
det
s −1
k1 s + k2
= s2
+ k2s + k1
Assume it is desired to have a feedback that gives a closed loop system with
the characteristic polynomial
p(s) = s2
+ 2ζω0s + ω2
0
Comparing this with the characteristic polynomial of the closed loop system
we find find that the feedback gains should be chosen as
k1 = ω2
0, k2 = 2ζω0
To have unit steady state gain the parameter Kr must be equal to k1 =
ω2
0. The control law can thus be written as
u = k1(r − x1) − k2x2 = ω2
0(r − x1) − 2ζ0ω0x2

In the next example we will encounter some difficulties.
Example 5.5 (An Unreachable System). Consider the system
dx
dt
=
0 1
0 0
x +
1
0
u
y = Cx = 1 0 x
with the control law
u = −k1x1 − k2x2 + Krr
The closed loop system is
dx
dt
=
−k1 1 − k2
0 0
x +
Kr
0
r
This system has the characteristic polynomial
det
s + k1 −1 + k2
0 s
= s2
+ k1s = s(s + k1)
This polynomial has zeros at s = 0 and s = −k1. One closed loop eigenvalue
is thus always equal to s = 0 and it is not possible to obtain an arbitrary
characteristic polynomial.
This example shows that the eigenvalue placement problem cannot al-
ways be solved. An analysis of the equation describing the system shows
that the state x2 is not reachable. It is thus clear that some conditions on
the system are required.
The reachable canonical form has the property that the parameters of
the system are the coefficients of the characteristic equation. It is therefore
natural to consider systems on this form when solving the eigenvalue place-
ment problem. In the next example we investigate the case when the system
is in reachable canonical form.
Example 5.6 (System in Reachable Canonical Form). Consider a system in
reachable canonical form, i.e,
dz
dt
= Ãz + ˜Bu =







−a1 −a2 . . . −an−1 −an
1 0 . . . 0 0
0 1 . . . 0 0
...
0 0 . . . 1 0







z +







1
0
0
...
0







u
y = ˜Cz = b1 b2 · · · bn z.
(5.13)

The open loop system has the characteristic polynomial
Dn(s) = det







s + a1 a2 . . . an−1 an
−1 s 0 0
0 −1 0 0
...
0 0 −1 s







.
Expanding the determinant by the last row we find that the following re-
cursive equation for the determinant:
Dn(s) = sDn−1(s) + an.
It follows from this equation that
Dn(s) = sn
+ a1sn−1
+ . . . + an−1s + an
A useful property of the system described by (5.13) is thus that the coef-
ficients of the characteristic polynomial appear in the first row. Since the
all elements of the B-matrix except the first row are zero it follows that the
state feedback only changes the first row of the A-matrix. It is thus straight
forward to see how the closed loop eigenvalues are changed by the feedback.
Introduce the control law
u = − ˜Kz + Krr = −˜k1z1 − ˜k2z2 − . . . − ˜knzn + Krr (5.14)
The closed loop system then becomes
dz
dt
=







−a1 − ˜k1 −a2 − ˜k2 . . . −an−1 − ˜kn−1 −an − ˜kn
1 0 0 0
0 1 0 0
...
0 0 1 0







z +







Kr
0
0
...
0







r
y = b1 b2 · · · bn z
(5.15)
The feedback thus changes the elements of the first row of the A matrix,
which corresponds to the parameters of the characteristic equation. The
closed loop system thus has the characteristic polynomial
sn
+ (al + ˜k1)sn−1
+ (a2 + ˜k2)sn−2
+ . . . + (an−1 + ˜kn−1)s + an + ˜kn

Requiring this polynomial to be equal to the desired closed loop polynomial
(5.11) we find that the controller gains should be chosen as
˜k1 = p1 − a1
˜k2 = p2 − a2
...
˜kn = pn − an
This feedback simply replace the parameters ai in the system (5.15) by pi.
The feedback gain for a system in reachable canonical form is thus
˜K = p1 − a1 p2 − a2 · · · pn − an (5.16)
To have unit steady state gain the parameter Kr should be chosen as
Kr =
an + ˜kn
bn
=
pn
bn
(5.17)
Notice that it is essential to know the precise values of parameters an and
bn in order to obtain the correct steady state gain. The steady state gain is
thus obtained by precise calibration. This is very different from obtaining
the correct steady state value by integral action, which we shall see in later
chapters. We thus find that it is easy to solve the eigenvalue placement
problem when the system has the structure given by (5.13).
The General Case
To solve the problem in the general case, we simply change coordinates so
that the system is in reachable canonical form. Consider the system (5.8).
Change the coordinates by a linear transformation
z = Tx
so that the transformed system is in reachable canonical form (5.13). For
such a system the feedback is given by (5.14) where the coefficients are given
by (5.16). Transforming back to the original coordinates gives the feedback
u = − ˜Kz + Krr = − ˜KTx + Krr
It now remains to find the transformation. To do this we observe that the
reachability matrices have the property
˜Wr = ˜B Ã ˜B . . . Ãn−1 ˜B = T B AB . . . An−1B = TWr

The transformation matrix is thus given by
T = ˜WrW−1
r (5.18)
and the feedback gain can be written as
K = ˜KT = ˜K ˜WrW−1
r (5.19)
Notice that the matrix ˜Wr is given by (5.6). The feedforward gain Kr is
given by equation (5.17).
The results obtained can be summarized as follows.
Theorem 5.1 (Pole-placement by State Feedback). Consider the system given
by equation (5.8)
dx
dt
= Ax + Bu
y = Cx
with one input and one output. If the system is reachable there exits a
feedback
u = −Kx + Krr
that gives a closed loop system with the characteristic polynomial
p(s) = sn
+ p1sn−1
+ . . . + pn−1s + pn.
The feedback gain is given by
K = ˜KT = p1 − a1 p2 − a2 . . . pn − an
˜WrW−1
r
Kr =
pn
an
where ai are the coeﬃcients of the characteristic polynomial of the matrix
A and the matrices Wr and ˜Wr are given by
Wr = B AB . . . An−1
˜Wr =





1 a1 a2 . . . an−1
0 1 a1 . . . an−2
...
0 0 0 . . . 1





−1
Remark 5.1 (A mathematical interpretation). Notice that the eigenvalue
placement problem can be formulated abstractly as the following algebraic
problem. Given an n × n matrix A and an n × 1 matrix B, ﬁnd a 1 × n
matrix K such that the matrix A − BK has prescribed eigenvalues.

5.4. OBSERVABILITY 121
Computing the Feedback Gain
We have thus obtained a solution to the problem and the feedback has been
described by a closed form solution.
For simple problems it is easy to solve the problem by the following
simple procedure: Introduce the elements ki of K as unknown variables.
Compute the characteristic polynomial
det(sI − A + BK).
Equate coefficients of equal powers of s to the coefficients of the desired
characteristic polynomial
p(s) = sn
+ p1sn−1
+ . . . + pn−1 + pn.
This gives a system of linear equations to determine ki. The equations
can always be solved if the system is observable. Example 5.4 is typical
illustrations.
For systems of higher order it is more convenient to use equation (5.19),
this can also be used for numeric computations. However, for large systems
this is not sound numerically, because it involves computation of the charac-
teristic polynomial of a matrix and computations of high powers of matrices.
Both operations lead to loss of numerical accuracy. For this reason there are
other methods that are better numerically. In MATLAB the state feedback
can be computed by the procedures acker or place.
5.4 Observability
In Section 5.3 it was shown that it was possible to find a feedback that gives
desired closed loop eigenvalues provided that the system is reachable and
that all states were measured. It is highly unrealistic to assume that all
states are measured. In this section we will investigate how the state can
be estimated by using the mathematical model and a few measurements. It
will be shown that the computation of the states can be done by dynamical
systems. Such systems will be called observers.
Consider a system described by
dx
dt
= Ax + Bu
y = Cx
(5.20)
where x is the state, u the input, and y the measured output. The problem
of determining the state of the system from its inputs and outputs will be

considered. It will be assumed that there is only one measured signal, i.e.
that the signal y is a scalar and that C is a (row) vector.
Observability
When discussing reachability we neglected the output and focused on the
state. We will now discuss a related problem where we will neglect the input
and instead focus on the output. Consider the system
dx
dt
= Ax
y = Cx
(5.21)
We will now investigate if it is possible to determine the state from observa-
tions of the output. This is clearly a problem of significant practical interest,
because it will tell if the sensors are sufficient.
The output itself gives the projection of the state on vectors that are
rows of the matrix C. The problem can clearly be solved if the matrix C
is invertible. If the matrix is not invertible we can take derivatives of the
output to obtain
dy
dt
= C
dx
dt
= CAx.
From the derivative of the output we thus get the projections of the state
on vectors which are rows of the matrix CA. Proceeding in this way we get








y
dy
dt
d2y
dt2
...
dn−1y
dtn−1








=







C
CA
CA2
...
CAn−1







x (5.22)
We thus find that the state can be determined if the matrix
Wo =







C
CA
CA2
...
CAn−1







(5.23)
has n independent rows. Notice that because of the Cayley-Hamilton equa-
tion it is not worth while to continue and take derivatives of order higher

5.4. OBSERVABILITY 123
Σ
y
S
S
Figure 5.2: A non-observable system.
than n − 1. The matrix Wo is called the observability matrix. A system
is called observable if the observability matrix has full rank. We illustrate
with an example.
Example 5.7 (Observability of the Inverted Pendulum). The linearized model
of inverted pendulum around the upright position is described by (5.4). The
matrices A and C are
A =
0 1
1 0
, C = 1 0
The observability matrix is
Wo =
1 0
0 1
which has full rank. It is thus possible to compute the state from a mea-
surement of the angle.
The calculation can easily be extended to systems with inputs. The state
is then given by a linear combination of inputs and outputs and their higher
derivatives. Diﬀerentiation can give very large errors when there is measure-
ment noise and the method is therefore not very practical particularly when
derivatives of high order appear. A method that works with inputs will be
given the next section.
A Non-Observable System
It is useful to have an understanding of the mechanisms that make a system
unobservable. Such a system is shown in Figure 5.2. Next we will consider

the system in observable canonical form, i.e.
dz
dt
=







−a1 1 0 . . . 0
−a2 0 1 0
...
−an−1 0 0 1
−an 0 0 0







z +







b1
b2
...
bn−1
bn







u
y = 1 0 0 . . . 0 z + Du
A straight forward but tedious calculation shows that the inverse of the
observability matrix has a simple form. It is given by
W−1
o =







1 0 0 . . . 0
a1 1 0 . . . 0
a2 a1 1 . . . 0
...
an−1 an−2 an−3 . . . 1







This matrix is always invertible. The system is composed of two identi-
cal systems whose outputs are added. It seems intuitively clear that it is
not possible to deduce the states from the output. This can also be seen
formally.
Coordinate Changes
It is interesting to investigate how the observability matrix transforms when
the coordinates are changed. Consider the system in equation (5.21). As-
sume that the coordinates are changed to z = Tx. It follows from linear
algebra that the dynamics matrix and the output matrix are given by
Ã = TAT−1
˜C = CT−1
.
The observability matrix for the transformed system then becomes
˜Wo =







˜C
˜C Ã
˜C Ã2
...
˜C Ãn−1








5.5. OBSERVERS 125
We have
˜C Ã = CT−1
TAT−1
= CAT−1
˜C Ã2
= CT−1
(TAT−1
)2
= CT−1
TAT−1
TAT−1
= CA2
T−1
...
˜C Ãn
= CAn
T−1
and we find that the observability matrix for the transformed system has
the property
˜Wo =







˜C
˜C Ã
˜C Ã2
...
˜C Ãn−1







T−1
= WoT−1
(5.24)
This formula is very useful for finding the transformation matrix T.
5.5 Observers
For a system governed by equation (5.20), we can attempt to determine
the state simply by simulating the equations with the correct input. An
estimate of the state is then given by
dˆx
dt
= Aˆx + Bu (5.25)
To find the properties of this estimate, introduce the estimation error
˜x = x − ˆx
It follows from (5.20) and (5.25) that
d˜x
dt
= A˜x
If matrix A has all its eigenvalues in the left half plane, the error ˜x will
thus go to zero. Equation (5.25) is thus a dynamical system whose output
converges to the state of the system (5.20).
The observer given by (5.25) uses only the process input u; the measured
signal does not appear in the equation. It must also be required that the
system is stable. We will therefore attempt to modify the observer so that

the output is used and that it will work for unstable systems. Consider the
following observer
dˆx
dt
= Aˆx + Bu + L(y − Cˆx). (5.26)
This can be considered as a generalization of (5.25). Feedback from the
measured output is provided by adding the term L(y − Cˆx). Notice that
Cˆx = ˆy is the output that is predicted by the observer. To investigate the
observer (5.26), form the error
˜x = x − ˆx
It follows from (5.20) and (5.26) that
d˜x
dt
= (A − LC)˜x
If the matrix L can be chosen in such a way that the matrix A − LC has
eigenvalues with negative real parts, the error ˜x will go to zero. The con-
vergence rate is determined by an appropriate selection of the eigenvalues.
The problem of determining the matrix L such that A − LC has pre-
scribed eigenvalues is very similar to the eigenvalue placement problem that
was solved above. In fact, if we observe that the eigenvalues of the matrix
and its transpose are the same, we ﬁnd that could determine L such that
AT − CT LT has given eigenvalues. First we notice that the problem can be
solved if the matrix
CT AT CT . . . A(n−1)T CT
is invertible. Notice that this matrix is the transpose of the observability
matrix for the system (5.20).
Wo =





C
CA
...
CAn−1





of the system. Assume it is desired that the characteristic polynomial of the
matrix A − LC is
p(s) = sn
+ p1sn−1
+ . . . + pn
It follows from Remark 5.1 of Theorem 5.1 that the solution is given by
LT
= p1 − a1 p2 − a2 . . . pn − an
˜WT
o W−T
o

5.5. OBSERVERS 127
where Wo is the observability matrix and ˜Wo is the observability matrix of
the system of the system
dz
dt
=







−a1 1 0 . . . 0
−a2 0 1 . . . 0
...
−an−1 0 0 . . . 1
−an 0 0 . . . 0







z +







b1
b2
...
bn−1
bn







u
y = 1 0 0 . . . 0
which is the observable canonical form of the system (5.20). Transposing
the formula for L we obtain
L = W−1
o
˜Wo





p1 − a1
p2 − a2
...
pn − an





The result is summarized by the following theorem.
Theorem 5.2 (Observer design by eigenvalue placement). Consider the sys-
tem given by
dx
dt
= Ax + Bu
y = Cx
where output y is a scalar. Assume that the system is observable. The
dynamical system
dˆx
dt
= Aˆx + Bu + L(y − Cˆx)
with L chosen as
L = W−1
o
˜Wo





p1 − a1
p2 − a2
...
pn − an





(5.27)
where the matrices Wo and ˜Wo are given by
Wo =





C
CA
...
CAn−1





, ˜W−1
o =







1 0 0 . . . 0
a1 1 0 . . . 0
a2 a1 1 . . . 0
...
an−1 an−2 an−3 . . . 1








PSfrag replacements
ˆx˙ˆxu −ˆy
y
B Σ
Σ
A
−C
−1
L
Figure 5.3: Block diagram of the observer. Notice that the observer contains
a copy of the process.
Then the observer error ˜x = x − ˆx is governed by a differential equation
having the characteristic polynomial
p(s) = sn
+ p1sn−1
+ . . . + pn
Remark 5.2. The dynamical system (5.26) is called an observer for (the
states of the) system (5.20) because it will generate an approximation of the
states of the system from its inputs and outputs.
Remark 5.3. The theorem can be derived by transforming the system to
observable canonical form and solving the problem for a system in this form.
Remark 5.4. Notice that we have given two observers, one based on pure dif-
ferentiation (5.22) and another described by the differential equation (5.26).
There are also other forms of observers.
Interpretation of the Observer
The observer is a dynamical system whose inputs are process input u and
process output y. The rate of change of the estimate is composed of two
terms. One term Aˆx + Bu is the rate of change computed from the model
with ˆx substituted for x. The other term L(y − ˆy) is proportional to the
difference e = y − ˆy between measured output y and its estimate ˆy = Cˆx.
The estimator gain L is a matrix that tells how the error e is weighted and
distributed among the states. The observer thus combines measurements
with a dynamical model of the system. A block diagram of the observer is
shown in Figure 5.3

5.5. OBSERVERS 129
Duality
Notice the similarity between the problems of finding a state feedback and
finding the observer. The key is that both of these problems are equivalent
to the same algebraic problem. In eigenvalue placement it is attempted to
find L so that A − BL has given eigenvalues. For the observer design it
is instead attempted to find L so that A − LC has given eigenvalues. The
following equivalence can be established between the problems
A ↔ AT
B ↔ CT
K ↔ LT
Wr ↔ WT
o
The similarity between design of state feedback and observers also means
that the same computer code can be used for both problems.
Computing the Observer Gain
The observer gain can be computed in several different ways. For simple
problems it is convenient to introduce the elements of L as unknown param-
eters, determine the characteristic polynomial of the observer det (A − LC)
and identify it with the desired characteristic polynomial. Another alterna-
tive is to use the fact that the observer gain can be obtained by inspection
if the system is in observable canonical form. In the general case the ob-
server gain is then obtained by transformation to the canonical form. There
are also reliable numerical algorithms. They are identical to the algorithms
for computing the state feedback. The procedures are illustrated by a few
examples.
Example 5.8 (The Double Integrator). The double integrator is described by
dx
dt
=
0 1
0 0
x +
0
1
u
y = 1 0
The observability matrix is
Wo =
1 0
0 1

i.e. the identity matrix. The system is thus observable and the problem can
be solved. We have
A − LC =
−l1 1
−l2 0
It has the characteristic polynomial
det A − LC = det
s + l1 −1
−l2 s
= s2
+ l1s + l2
Assume that it is desired to have an observer with the characteristic poly-
nomial
s2
+ p1s + p2 = s2
+ 2ζωs + ω2
The observer gains should be chosen as
l1 = p1 = 2ζω
l2 = p2 = ω2
The observer is then
dˆx
dt
=
0 1
0 0
ˆx +
0
1
u +
l1
l2
(y − ˆx1)
5.6 Output Feedback
In this section we will consider the same system as in the previous sections,
i.e. the nth order system described by
dx
dt
= Ax + Bu
y = Cx
(5.28)
where only the output is measured. As before it will be assumed that u and
y are scalars. It is also assumed that the system is reachable and observable.
In Section 5.3 we had found a feedback
u = −Kx + Krr
for the case that all states could be measured and in Section 5.4 we have
presented developed an observer that can generate estimates of the state ˆx
based on inputs and outputs. In this section we will combine the ideas of
these sections to ﬁnd an feedback which gives desired closed loop eigenvalues
for systems where only outputs are available for feedback.

5.6. OUTPUT FEEDBACK 131
If all states are not measurable, it seems reasonable to try the feedback
u = −Kˆx + Krr (5.29)
where ˆx is the output of an observer of the state (5.26), i.e.
dˆx
dt
= Aˆx + Bu + L(y − Cˆx) (5.30)
Since the system (5.28) and the observer (5.30) both are of order n, the
closed loop system is thus of order 2n. The states of the system are x and
ˆx. The evolution of the states is described by equations (5.28), (5.29)(5.30).
To analyze the closed loop system, the state variable ˆx is replace by
˜x = x − ˆx (5.31)
Subtraction of (5.28) from (5.28) gives
d˜x
dt
= Ax − Aˆx − L(y − Cˆx) = A˜x − LC˜x = (A − LC)˜x
Introducing u from (5.29) into this equation and using (5.31) to eliminate ˆx
gives
dx
dt
= Ax + Bu = Ax − BKˆx + BKrr = Ax − BK(x − ˜x) + BKrr
= (A − BK)x + BK˜x + BKrr
The closed loop system is thus governed by
d
dt
x
˜x
=
A − BK BK
0 A − LC
x
˜x
+
BKr
0
r (5.32)
Since the matrix on the right-hand side is block diagonal, we ﬁnd that the
characteristic polynomial of the closed loop system is
det (sI − A + BK) det (sI − A + LC)
This polynomial is a product of two terms, where the ﬁrst is the charac-
teristic polynomial of the closed loop system obtained with state feedback
and the other is the characteristic polynomial of the observer error. The
feedback (5.29) that was motivated heuristically thus provides a very neat
solution to the eigenvalue placement problem. The result is summarized as
follows.

Theorem 5.3 (Pole placement by output feedback). Consider the system
dx
dt
= Ax + Bu
y = Cx
The controller described by
u = −Kˆx + Krr
dˆx
dt
= Aˆx + Bu + L(y − Cˆx)
gives a closed loop system with the characteristic polynomial
det (sI − A + BK) det (sI − A + LC)
This polynomial can be assigned arbitrary roots if the system is observable
and reachable.
Remark 5.5. Notice that the characteristic polynomial is of order 2n and
that it can naturally be separated into two factors, one det (sI − A + BK)
associated with the state feedback and the other det (sI − A + LC) with the
observer.
Remark 5.6. The controller has a strong intuitive appeal. It can be thought
of as composed of two parts, one state feedback and one observer. The
feedback gain K can be computed as if all state variables can be measured.
The Internal Model Principle
A block diagram of the controller is shown in Figure 5.4. Notice that the
controller contains a dynamical model of the plant. This is called the internal
model principle. Notice that the dynamics of the controller is due to the
observer. The controller can be viewed as a dynamical system with input y
and output u.
dˆx
dt
= (A − BK − LC)ˆx + Ly
u = −Kˆx + Krr
(5.33)
The controller has the transfer function
C(s) = K[sI − A + BK + LC]−1
L (5.34)

5.7. INTEGRAL ACTION 133
PSfrag replacements
r u yx˙x
ˆx
−ˆy
B
B
Σ
Σ
Σ
Σ
R
R
A
A
C
−C
L
−K
Kr
Process
Observer
Figure 5.4: Block diagram of a controller which combines state feedback
with an observer.
5.7 Integral Action
The controller based on state feedback achieves the correct steady state re-
sponse to reference signals by careful calibration of the gain Lr and it lacks
the nice property of integral control. It is then natural to ask why the
the beautiful theory of state feedback and observers does not automatically
give controllers with integral action. This is a consequence of the assump-
tions made when deriving the analytical design method which we will now
investigate.
When using an analytical design method, we postulate criteria and spec-
ifications, and the controller is then a consequence of the assumptions. In
this case the problem is the model (5.8). This model assumes implicitly that
the system is perfectly calibrated in the sense that the output is zero when
the input is zero. In practice it is very difficult to obtain such a model.
Consider, for example, a chemical process control problem where the output
is temperature and the control variable is a large rusty valve. The model
(5.8) then implies that we know exactly how to position the valve to get a
specified outlet temperature—indeed, a highly unrealistic assumption.

Having understood the difficulty it is not too hard to change the model.
By modifying the model to
dx
dt
= Ax + B(u + v)
y = Cx,
(5.35)
where v is an unknown constant, we can can capture the idea that the model
is no longer perfectly calibrated. This model is called a model with an input
disturbance. Another possibility is to use the model
dx
dt
= Ax + Bu
y = Cx + v
where v is an unknown constant. This is a model with an output disturbance.
It will now be shown that a straightforward design of an output feedback for
this system does indeed give integral action. Both disturbance models will
produce controllers with integral action. We will start by investigating the
case of an input disturbance. This is a little more convenient for us because
it fits the control goal of finding a controller that drives the state to zero.
The model with an input disturbance can conveniently be brought into
the framework of state feedback. To do this, we first observe that v is an
unknown constant which can be described by
dv
dt
= 0
To bring the system into the standard format we simply introduce the dis-
turbance v as an extra state variable. The state of the system is thus
z =
x
v
This is also called state augmentation. Using the augmented state the model
(5.35) can be written as
d
dt
x
v
=
A B
0 0
x
v
+
B
0
u
y = C 0
x
v
(5.36)
Notice that the disturbance state is not reachable. If the disturbance can
be measured, the state feedback is then
u = − ˜Kz + Krr = −Kxx − Kvv + Krr (5.37)

5.8. A GENERAL CONTROLLER STRUCTURE 135
The disturbance state v is not reachable. The the effect of the disturbance
on the system can, however, be eliminated by choosing Kv = 1. If the distur-
bance v is known the control law above can be interpreted as a combination
of feedback from the system state and feedforward from a measured distur-
bance. It is not realistic to assume that the disturbance can be measured
and we will instead replace the states by estimates. The feedback law then
becomes
u = −Kxˆz + Krr = −Kx ˆx − ˆv + Krr
This means that feedback is based on estimates of the state and the distur-
bance. There are many other ways to introduce integral action.
5.8 A General Controller Structure
So far reference signals have been introduced simply by adding it to the state
feedback. A more sophisticated way of doing this is shown by the block dia-
gram in Figure 5.5, where the controller consists of three parts: an observer
that computes estimates of the states based on a model and measured pro-
cess inputs and outputs, a state feedback and a trajectory generator that
generates the desired behavior of all states xm and a feedforward signal uff.
The signal uff is such that it generates the desired behavior of the states
when applied to the system, under ideal conditions of no disturbances and
no modeling errors. The controller is said to have two degrees of freedom
because the response to command signals and disturbances are decoupled.
Disturbance responses are governed by the observer and the state feedback
and the response to command signal is governed by the feedforward. To get
some insight into the behavior of the system let us discuss what happens
when the command signal is changed. To fix the ideas let us assume that
the system is in equilibrium with the observer state ˆx equal to the process
state x. When the command signal is changed a feedforward signal uff(t) is
generated. This signal has the property that the process output gives the
desired output xm(t) when the feedforward signal is applied to the system.
The process state changes in response to the feedforward signal. The ob-
server tracks the state perfectly because the initial state was correct. The
estimated state ˆx will be equal to the desired state xm and the feedback
signal L(xm − ˆx) is zero. If there are some disturbances or some modeling
errors the feedback signal will be different from zero and attempt to correct
the situation.
The controller given in Figure 5.5 is a very general structure. There are
many ways to generate the feedforward signal and there are also many dif-

PSfrag replacements
Trajectory &
Feedforward
Generator
r
u
-ˆx
ˆy
xm
ProcessΣ Σ
State
Feedback
Observer
ufb
uff
y
Figure 5.5: Block diagram of a controller based on a structure with two
degrees of freedom. The controller consists of a command signal generator,
state feedback and an observer.
ferent ways to compute the feedback gains L and the gain L of the observer.
The system in Figure 5.5 is an example of the internal model principle
which says that a controller should contain a model of the system to be
controlled and the disturbances action on the system.
Computer Implementation
The controllers obtained so far have been described by ordinary differential
equations. They can be implemented directly using analog computers. Since
most controllers are implemented using digital computers we will briefly
discuss how this can be done.
The computer typically operates periodically, signals from the sensors
are sampled and converted to digital form by the A/D converter, the control
signal is computed, converted to analog form for the actuators, as shown in
Figure 1.3 on page 5. To illustrate the main principles we consider the
controller described by equations (5.29) and (5.30), i.e.
u = −Kˆx + Krr
dˆx
dt
= Aˆx + Bu + K(y − Cˆx)
The first equation which only consists of additions and multiplications can
be implemented directly in a computer. The second equation has to be
approximated. A simple way is to replace the derivative by a difference
ˆx(tk+1) − ˆx(tk)
h
= Aˆx(tk) + Bu(tk) + K(y(tk) − Cˆx(tk))

5.9. EXERCISES 137
where tk are the sampling instants and h = tk+1 −tk is the sampling period.
Rewriting the equation we get
ˆx(tk+1 = ˆx(tk) + h Aˆx(tk) + Bu(tk) + K(y(tk) − Cˆx(tk)) . (5.38)
The calculation of the state only requires addition and multiplication and
can easily be done by a computer. A pseudo code for the program that runs
in the digital computer is
"Control algorithm - main loop
r=adin(ch1) "read setpoint from ch1
y=adin(ch2) "read process variable from ch2
u=C*x+Kr*r "compute control variable
daout(ch1) "set analog output ch1
x=x+h*(A*x+B*u+L*(y-C*x)) "update state estimate
The program runs periodically. Notice that the number of computations
between reading the analog input and setting th analog output has been
minimized. The state is updated after the analog output has been set. The
program has one states x. The choice of sampling period requires some care.
For linear systems the difference approximation can be avoided by ob-
serving that the control signal is constant over the sampling period. An
exact theory for this can be developed. Doing this we get a control law that
is identical to (5.38) but with slightly different coefficients.
There are several practical issues that also must be dealt with. For ex-
ample it is necessary to filter a signal before it is sampled so that the filtered
signal has little frequency content above fs/2 where fs is the sampling fre-
quency. If controllers with integral action are used it is necessary to provide
protection so that the integral does not become too large when the actuator
saturates. Care must also be taken so that parameter changes do not cause
disturbances. Some of these issues are discussed in Chapter 10.
5.9 Exercises
1. Consider a system on reachable canonical form. Show that the inverse of
the reachability matrix is given by
˜W−1
r =





1 a1 a2 . . . an
0 1 a1 . . . an−1
...
0 0 0 . . . 1





(5.39)

Am04 ch5 24oct04-stateand integral

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