Designing SDRE-Based Controller for a Class of Nonlinear Singularly Perturbed Systems

Seyed Mostafa Ghadami, Roya Amjadifard & Hamid Khaloozadeh
International Journal of Robotics and Automation (IJRA), Volume (4) : Issue (1) : 2013 1
Designing SDRE-Based Controller for a Class of Nonlinear
Singularly Perturbed Systems
Seyed Mostafa Ghadami m.gadami@srbiau.ac.ir
Department of Electrical Engineering,
Science and Research Branch,
Islamic Azad University, Tehran, Iran
Roya Amjadifard amjadifard@tmu.ac.ir
Assistant Professor,
Kharazmi University,
Tehran, Iran
Hamid Khaloozadeh h_khaloozadeh@kntu.ac.ir
Professor,
K.N. Toosi University of Technology,
Tehran, Iran
Abstract
Designing a controller for nonlinear systems is difficult to be applied. Thus, it is usually based on
a linearization around their equilibrium points. The state dependent Riccati equation control
approach is an optimization method that has the simplicity of the classical linear quadratic control
method. On the other hand, the singular perturbation theory is used for the decomposition of a
high-order system into two lower-order systems. In this study, the finite-horizon optimization of a
class of nonlinear singularly perturbed systems based on the singular perturbation theory and the
state dependent Riccati equation technique together is addressed. In the proposed method, first,
the Hamiltonian equations are described as a state-dependent Hamiltonian matrix, from which,
the reduced-order subsystems are obtained. Then, these subsystems are converted into outer-
layer, initial layer correction and final layer correction equations, from which, the separated state
dependent Riccati equations are derived. The optimal control law is, then, obtained by computing
the Riccati matrices.
Keywords: Singularly Perturbed Systems, State-Dependent Riccati Equation, Nonlinear Optimal
Control, Finite-Horizon Optimization Problem, Single Link Flexible Joint Robot Manipulator.
1. INTRODUCTION
Designing regulator systems is an important class of optimal control problems in which optimal
control law leads to the Hamilton-Jacobi-Belman (HJB) equation. Various techniques have been
suggested to solve this equation. One of these techniques, which are used for optimizing in
infinite horizon, is based on the state-dependent Riccati equation (SDRE). In this technique,
unlike linearization methods, a description of the system as state-dependent coefficients (SDCs)
and in the form f(x)=A(x)x must be provided. In this representation, A(x) is not unique. Therefore,
the solutions of the SDRE would be dependent on the choice of matrix A(x). With suitable choice
of the matrix, the solution to the equation is optimal; otherwise, the equation has suboptimal
solutions. Bank and Mhana [1] proposed a suitable method for the selection of SDCs. Çimen [2]
provided the condition for the solvability and local asymptotic stability of the SDRE closed-loop
system for a class of nonlinear systems. Khaloozadeh and Abdolahi converted the nonlinear
regulation [3] and tracking [4] problems in the finite-horizon to a state-dependent quasi-Riccati
equation. They also provided an iterative method based on the Piccard theorem, which obtains a
solution at a low convergence rate but good precision. On the other hand, the system discussed
in this study is a class of nonlinear singularly perturbed systems. Naidu and Calise [5] dealt with

the use of the singular perturbation theory and the two time scale (TTS) method in satellite and
interplanetary trajectories, missiles, launch vehicles and hypersonic flight, space robotics. For LTI
singularly perturbed systems, Su et al. [6] and Gajic et al. [7] performed the exact slow-fast
decomposition of the linear quadratic (LQ) singularly perturbed optimal control problem in infinite
horizon by deriving separate Riccati equations. Also, Gajic et al. [8] did the same for the case of
finite horizon. Amjadifard et al. [9, 10] addressed the robust disturbance attenuation of a class of
nonlinear singularly perturbed systems and robust regulation of a class of nonlinear singularly
perturbed systems [11], and also position and velocity control of a flexible joint robot manipulator
via fuzzy controller based on singular perturbation analysis [12]. Fridman [13, 14] dealt with the
infinite horizon nonlinear quadratic optimal control problem for a class of non-standard nonlinear
singularly perturbed systems by invariant manifolds of the Hamiltonian system and its
decomposition into linear-algebraic Riccati equations.
In this study, we extend results of [13, 14] to the finite horizon by slow-fast manifolds of the
Hamiltonian system and its decomposition into SDREs. Our contribution is that, we used the
singular perturbation theory and SDRE method together. In the proposed method, first, the state-
dependent Hamiltonian matrix is derived for the system under study. Then, this matrix is
separated into the reduced-order slow and fast subsystems. Using the singular perturbation
theory, the state equations and SDREs are converted into outer layer, initial layer correction and
final layer correction equations, which are then solved to obtain the optimal control law. The block
diagram of the proposed method is shown in Figure 1.
FIGURE 1: The design procedure stages in the proposed method.
The remainder of this study is organized as follows. Section 2 explains the structure of the
singularly perturbed system for optimization. Section 3 involves in the description of steps of the
design procedure in the proposed method. Section 4 presents the simulation results of the
system used in the proposed method. Finally, the study culminates with indication of remarks in
section 5.
2. PROBLEM FORMULATION
The following nonlinear singularly perturbed system is assumed:
,0)0(,)()( xtxuxBxfxE  (1)
where 1,2=iRx
x
x
tx in
i ,,)(
2
1






 are the states of system, and x=0n is the equilibrium point of the
system (n=n1+n2). This system is full state observable, autonomous, nonlinear in the states, and
affine in the input. Moreover, 1,2=iRB
xxB
xxB
xBRf
xxf
xxf
xf ii n
i
n
i ,,
),(
),(
)(,,
),(
),(
)(
212
211
212
211













are differentiable with respect to x1, x2 for a sufficient number of times. Furthermore, f(0n)=0n,
Optimal
control
law
Description
of the
system as
SDCs
State
dependent
Hamiltonian
matrix
Slow
SDREs
Slow state
equationsSlow
Hamiltonian
matrix
Fast state
equations
Fast
SDREs
Fast
Hamiltonian
matrix

B(x)0nm, xR
n
and 








2212
2111
0
0
nnnn
nnnn
I
I
E

that >0 is a small parameter. Provided these, it
is desired to obtain the optimal control law u(x)R
m
such that for k(x)R
n
, k(0n)=0n and
pointwise positive definite matrix R(x)R
n
R
mm
, the following performance index 𝒥 is minimized.
𝒥           
Ft
t
TT
F dtuxRuxkxktxh
0
(2)
Suppose that k(x), R(x) are differentiable with respect to x1, x2 for a sufficient number of times.
Moreover, tF is chosen such that it is sufficiently large with respect to the dominant time constant
of the slow subsystem, and x(tF) is free.
3. THE PROPOSED METHOD
The singularly perturbed system (1) with performance index (2) is assumed. Defining the co-state
vector 1,2,=iR
xx
xx
x in
i ,,
),(
),(
)(
212
211






 


 the Hamiltonian function is obtained as (3):
).),(),(()),(),(()(
2
1
)()(
2
1
),,( 21221222112111 uxxBxxfuxxBxxfuxRuxkxkux TTTT
  (3)
According to the optimal control theory, the necessary conditions for optimization would be as
follow [2]:
),(,),(),()( 01211211
1
1 txuxxBxxf
H
x T






(4a)
),(,),(),()( 02212212
2
2 txuxxBxxf
H
x T






(4b)
 
  ,|
2
1
)(,
)(
2
1)()()(
1
1
11111
1
Ft
T
F
T
T
T
TTT
x
h
tx
x
xR
u
x
xB
u
x
xf
xk
x
xk
x
H
































































(4c)
 
  ,|
2
1
)(,
)(
2
1)()()(
2
2
22222
2
Ft
T
F
T
T
T
TTT
x
h
tx
x
xR
u
x
xB
u
x
xf
xk
x
xk
x
H































































 
(4d)
.),(),()(0 22121211  xxBxxBuxR
u
H TT



 (4e)
3.1 Description of The System As SDCs (The first step)
A continuous nonlinear matrix-valued function A(x) always exists such that
f(x)=A(x)x (5)
Where A(x):R
n
R
nn
is found by mathematical factorization and is, clearly, non-unique when
n>1. A suitable choice for matrix A(x) is   ,
1
0 |
 

 

d
x
f
xA
xx
where  is a dummy variable that
was introduced in the integration [1]. Then, the relations (4) can be written as:

)(,)()( 0txuxBxxAxE  (6a)
    ft
T
f
T
T
T
TT
T
x
h
txE
x
xR
u
x
xB
u
x
xf
xk
x
xk
x
H
E |
2
1
)(,
)(
2
1)()()(
)( 













































  (6b)
  )(1
xBxRu T
 (6c)
Considering that B(x) and R(x) are nonzero, the optimal control law is proportional to vector .
3.2 Description of The Hamiltonian Matrix As SDCs (The second step)
Assuming that    


1
0 |


d
x
k
xK
xx
is available from k(x)=K(x)x and that Q(x)=K
T
(x)K(x) and
S(x)=B(x)R
-1
(x)B
T
(x), the relations (6) can be rewritten as follow:
,)(,)()( 00 xtxxSxxAxE   (7a)
   xk
x
xK
xxAxxQE
T
i
n
i
i
T








 
)(
[)(
1

             
  ,|
2
1
)(
,]}
)()(
2
1)(
{
1
1
1
11
1
Ft
T
F
m
i
T
i
i
T
m
i
Ti
i
T
T
i
n
i
i
x
h
txE
x
xB
xRxBxBxR
x
xR
xRxB
x
xA
x

































  





 (7b)
Where,
,
)()(
)()(
)(
1
1
1
1

























n
nini
n
ii
i
x
xA
x
xA
x
xA
x
xA
x
xA



(8a)
,
)()(
)()(
)(
1
1
1
1

























n
nini
n
ii
i
x
xK
x
xK
x
xK
x
xK
x
xK



(8b)
,
)()(
)()(
)(
1
1
1
1

























n
nini
n
ii
i
x
xB
x
xB
x
xB
x
xB
x
xB



(8c)

.
)()(
)()(
)(
1
1
1
1

























n
mimi
n
ii
i
x
xR
x
xR
x
xR
x
xR
x
xR



(8d)
Assumption 1: A(x), B(x), Q(x), R(x),
x
xK
x
xB
x
xA





 )(
,
)(
,
)(
and
x
xR

 )(
are bounded in a
neighborhood of  about the region. Then, the expression in the bracket will be ignored because
of being small. This approximation is asymptotically optimal, in that it converges to the optimal
control close to the origin as [2]. Thus, the relations (7) can be written as:
  




















x
xAxQ
xSxA
E
xE
T
)(
)()(


(9)
Remark 1: Suppose that Tsi, TsF are dominant time constants of the slow subsystem for initial and
final layer correction, respectively. In other words,
  islow
si
Jeigreal
T
1
max and
  Fslow
sF
Jeigreal
T
1
max
where, Ji and JF are the Jacobian matrices of Hamiltonian system in
initial and final layer correction and,
   
n
n
F
n
x
ttTF
xx
ttTi
xAxQ
xSxA
J
xAxQ
xSxA
J
0
0
0
)(
)()(
,
)(
)()(
0
0





 

















.
Note that (Tsi+TsF)/2 is the average time constant of the Hamiltonian system and the setting time
is fourfold of one, then a proper selection for tF is
tF > t0+2(Tsi+TsF) (10)
3.3 The Singularly Perturbed SDRE in Finite Horizon
In the proposed method, co-sate vector , can be described as =P(x)x using the sweep method
[3], where,
ji nn
ij
T
RP
xxPxxP
xxPxxP
xP










 ,
),(),(
),(),(
)(
21222121
21212111 
[7] is the unique, non-symmetric,
positive-definite solution of the Riccati matrix equation. By differentiating  with respect to time,
we can write:
xxPxxP )()(   (11)
By substituting (11) in (9) and with rearrangement of one, we have:
   

 










1
0
1
|
2
)(,0)()()()()()()()()(  d
x
h
x
E
txPxQxPxSxPxPxAxAxPxPE xx
T
Fnn
TTT (12)
The relation (12) is called a SDRE for nonlinear singularly perturbed system in finite horizon. It
should be noted that the optimal control law is obtained by computing these Riccati matrices.
The solution conditions for SDRE are that {A(x),B(x)} be stabilizable and {A(x),(Q(x))
1/2
} be
detectable for xR
n
. A sufficient test for the stabilizability condition of {A(x),B(x)} is to check that
the controllability matrix Mc= [B(x),A(x)B(x),…,A
n-1
(x)B(x)] has rank(Mc)=n,x. Similarly, a
sufficient test for detectability of {A(x), (Q(x))
1/2
} is that the observability matrix Mo=[(Q(x))
1/2
,
(Q(x))
1/2
A(x),…, (Q(x))
1/2
A
n-1
(x)] has rank(Mo)=n, x [2]. Furthermore, the closed-loop matrix
A(x)-S(x)P(x) should be pointwise Hurwitz for x. Here,  is any region such that the
Lyapunov function xdxPxxV T








 
1
0
)()(  is locally Lipschitz around the origin [2]. The SDRE in

(12) consist
2
)1)(( 2121  nnnn
differential equations that number of these equations is reduced
by using singular perturbation theory.
3.4 The Separated Hamiltonian Matrices
In the proposed method, by separating the slow and fast variables as
 
,
, 211
1







xx
x
Xs

 
,
, 212
2







xx
x
X f

we can describe the optimization relations (9) in the form of the following
singularly perturbed state-dependent Hamiltonian matrix:
,
),(),(
),(),(
21222121
21122111





















f
s
f
s
X
X
xxHxxH
xxHxxH
X
X



(13)
Where,  
  









 T
jiij
ijij
ij
xxAxxQ
xxSxxA
xxH
),(),(
),(),(
,
2121
2121
21 and I, j=1,2. Thus, we assume that the 2n1
eigenvalues of the system (13) are pointwise small and the remaining 2n2 eigenvalues are
pointwise large, corresponding to the slow and fast responses, respectively. The state and co-
state equations (13) constitute a singularly perturbed, two point boundary value problem
(TPBVP). Hence, the asymptotic solution is obtained as an outer solution in terms of the original
independent variable t, initial layer correction in terms of an initial stretched variable

 0tt 
 ,
and final layer correction in terms of a final stretched variable


ttF 
 [5]. Thus, the
composite solutions can be written as follow:











),(),(),(),(
),(),(),(),(
),(),(),(),(
),(),(),(),(
2222
1111




fFfifof
sFsisos
Fio
Fio
PPtPtP
PPtPtP
xxtxtx
xxtxtx
(14)
where



 0
2
0
10 0,0,
tt
t
tt
tttt FF
F



  . The first terms on the right hand sides of
the above relations represent the outer solution. The second and third terms represent boundary-
layer corrections to the slow manifold near the initial and final times, respectively. Indices o, i and
F correspond to the outer layer, initial, and final correction layers. For any boundary condition on
the slow manifold, states and co-states are given by outer solution. For any boundary condition
out of the slow manifold, the trajectory rapidly approaches the slow manifold according to the fast
manifolds.
We now perform the slow-fast decomposition of the singularly perturbed state-dependent
Hamiltonian matrix, in which H22(x1,x2) must be non-singular for all x1, x2 (in what follows,
dependence upon x1, x2 is not represented, for convenience):


















 




























22
2111
12
21
2222
11
2221
1
22
2222
22
22
2221
1
221211
2222
1
221222
2221
1211
0
0
0
0
nn
nnnn
nn
nn
nnnn
nn
IHH
I
H
HHHH
I
HHI
HH
HH



(15)

Stated differently:
























 
















 









f
s
nn
nnnn
nn
nn
f
s
nnnn
nn
X
X
IHH
I
H
HHHH
X
X
I
HHI
22
2111
12
21
2212
11
2221
1
22
2222
22
22
2221
1
221211
2222
1
221222
0
0
0
0




(16)
New co-sate vector can be described as new=Pnew(xnew)xnew, where ,






f
s
new
x
x
x
 
  ,
,
,







fsf
fss
new
xx
xx


 ,, 21 n
f
n
s RR   and
   
   






fsffsb
fsafss
newnew
xxPxxP
xxPxxP
xP
,,
,,
)(


. Then, the
new slow-fast variables are defined as follow:
  ,
, s
fss
s
s X
xx
x








 (17a)
  ,
, 21
1
22 fs
fsf
f
f XXHH
xx
x






 

 (17b)
Thus, (13) is converted to a new form:
 





 
ff
sfs
HX
HHHHXHHX


22
21
1
221211
1
2212


(18)
Finally, the optimization equations in a singular perturbation model framework with the new
variables are obtained as:
 
     







fssf
fss
HHHHHHHHHHHHHHHH
HHHHH


1221
1
222221
1
2221
1
2222
1
2221
1
22121121
1
22
1221
1
221211


(19)
Moreover, the separated state-dependent Hamiltonian matrices Hs(xs,xf) and H22(x1,x2) are
described in the form of the following:
           
 
  ,)(
),(),(
),(),(
)(,,,,,
11
1111
1111
11
22
2121
2121
22212121
1
2221122111
nn
nn
T
snns
nnsnns
nnfss
O
xxAxxQ
xxSxxA
OxxHxxHxxHxxHxxH




















(20a)
     
 
  .)(
),(),(
),(),(
)(,, 22
2222
2222
22 22
21222122
21222122
222122 nn
nn
T
nn
nnnn
nnfsf O
xxAxxQ
xxSxxA
OxxHxxH 


 










  (20b)

3.5 The slow-fast SDREs (The third step)
In the proposed method, using the singular perturbation theory, the subsystems (19) are
converted into outer-layer and boundary-layer correction subsystems. The separated SDRE
relations are, then derived and solved for obtaining the optimal control law.
Theorem 1: The singularly perturbed system (1) with performance index (2) is assumed. The
slow- fast state equations in the initial layer correction are obtained as follow:
  ),(|,),(),( 011122
*
122
*
11 0
txxxPxxxSxxxAx toosoioosiooso 
(21a)
  
  ),()(|,),(),(),(
),(),(
02
*
022121
*
22
*
12222
*
12122
*
121
22
*
22
*
22
*
12222
*
122
2
0
txtxxxPxxxSPxxxSxxxA
xxPxxxSxxxA
d
dx
otiooioosoiooioo
iooiooioo
i


 (21b)
Also, the slow- fast SDREs in the final layer correction are obtained as follow:
),(|,0),(
),(),(),(
112
*
1
2
*
12
*
12
*
1
11 Ftsonnooso
sooosososooo
T
sooosososo
tPPxxQ
PxxSPPxxAxxAPP
F




(22a)
    ).()(|, 22
*
22222222
*
2222
*
2222 FoFtfFfFofFfFoo
T
oooofF
fF
tPtPPPSPPSPAPSAP
d
dP
F


(22b)
where,  



















 1
0
1
2221
2111
|
2))(())((
))(())((


 d
x
h
x
E
txPtxP
txPtxP
xx
T
FF
F
T
F
. Furthermore, the optimal control
law is as follows:
    ,)),(),(),( 22
*
22
*
122
*
12122
*
1122
*
1
1
iofFoocioo
T
osoioo
T
ioo xxPPxPxxxBxPxxxBxxxRu   (23)
where, Pso and PfF are the unique, symmetric, positive-definite solutions of (22), and
  





 

so
nn
iooioonnfFoc
P
I
xxxHxxxHIPPP 11
22
),(),( 22
*
12122
*
1
1
2222
*
. The solution necessary
conditions of relations (21) and (22) are as follow:
 {Aso(x1o,x
*
2o), Bso(x1o,x
*
2o)} and {A22o(x1o,x
*
2o), B2o(x1o,x
*
2o)} should be stabilizable for
  ., 21
2
*
1
nn
oo RRxx 
 {Aso(x1o,x
*
2o),(Qso(x1o,x
*
2o))
1/2
} and {A22o(x1o,x
*
2o), (Q22o(x1o,x
*
2o))
1/2
} should be detectable for
  ., 21
2
*
1
nn
oo RRxx 
 The outer equations (24) should have solutions (the slow manifolds) as x
*
2o(x1o,P11o),
P
*
21o(x1o,P11o) and P
*
22o(x1o,P11o)
    ,0 2222222212122112121 noooooooooo xPSAxPSPSA  (24a)
    ,0 212121221121221221222222 nnoooooo
T
oooo
T
o QAPPSPAPSPA  (24b)
,0 222222222222222222 nnooooo
T
ooo QPSPPAAP  (24c)
It should be noted that in the above relations, all the elements of the state and Riccati matrices
are dependent on state variables, and have not been represented for simplicity.
Proofs of the theorems are given in appendix.
Remark 2: SDREs in (22) have n1n2 the less differential equations respect to (12).

4. EXAMPLE
Consider a single link flexible joint robot manipulator as it has been introduced in [11]. This link is
directly actuated by a D.C. electrical motor whose rotor is elastically coupled to the link. In this
example, the mathematical model of system is as follows:
uqqkqqI
qqkqmglqI


)(
0)()sin(
2122
2111



(25)
FIGURE 2: Single link flexible joint robot manipulator
In Table 1 there is a complete list of notations of the mathematical model of a single link flexible
joint robot manipulator.
TABLE 1: Notations the mathematical model of a single link flexible joint robot manipulator.
Moreover, parameter values are given in Table 2.
TABLE 2: Parameter values of the single link flexible joint robot manipulator.
Defining 




























2
1
22
1
2
1
13
12
11
1 ,,
x
x
xqx
q
q
q
x
x
x
x 

and =J, state equations are as follow:

























































s
s
xu
xkxkx
x
I
k
x
I
k
x
I
mgl
x
x
x
x
x
x
/0
/0
0
0
0
21211
121111
2
13
2
13
12
11
0
0
3
10
,
1
0
0
0
)sin(




(26)
Notation Description
q1 angular positions of the link
q2 angular positions of the motor
u actuator force (motor torque)
I the arm inertia
J the motor inertia
 the motor viscous friction
mgl the nominal load in the rotor link
K the stiffness coefficient of flexible joint
parameter Value of parameter
I 0.031(Kg.m
2
)
J 0.004(Kg.m
2
)
 0.007
k 7.13
mgl 0.8 (N.m)

It is desired to obtain the optimal control law such that the following performance index 𝒥 is
minimized.
𝒥   
5
0
22
2
2
13
2
12
2
11 dtuxxxx
(27)
In this example, ,1)(,)(,
1
0
0
0
)(,
)sin(
)(
2
13
12
11
21211
121111
2
13











































 xR
x
x
x
x
xkxB
xkxkx
x
I
k
x
I
k
x
I
mgl
x
x
xf

and
h(x(tF))=0. Moreover, f(x), k(x) are differentiable with respect to x for a sufficient number of times
and x=04 is the equilibrium point of the system. Furthermore, t0=0, tF=5, P(x(tF))=044.
Step 1 (Description of the system as SDCs):
To solve the optimization problem, the nonlinear functions f(x), k(x) must first be represented as
SDCs. A suitable choice, considering [1], is as follows:



















  



0
00
)sin(
1000
0100
)(
11
11
1
0
|
kk
I
k
I
k
Ix
xmgld
x
f
xA
xx (28a)















  
1000
0100
0010
0001
)(
1
0
|


d
x
k
xK
xx
(28b)
Step 2 (Description of the Hamiltonian matrix as SDCs):
The separated Hamiltonian matrices can be derived:













































001100
1
00
1
1
1
)sin(
1
00
11
1
0000
)sin(
0
1
1
00
11
000100
),(
22
2
2
2
11
11
22
2
2
2
11
11
222
21
I
kkkk
I
k
Ix
xmglkkk
I
k
I
k
Ix
xmgl
kk
xxHs










(29a)











1
1
),( 2122 xxH (29b)
Step 3.1 (the outer equations):
The relations (24) have solutions as:
1
)(
2
1323122211121211
2
*




 osoosoosooo
o
xPxPxPxxk
x (30a)





























1
1
1
2
23
2
22
2
12
21
*





so
so
so
o
P
kP
k
kP
k
P (30b)
  12
22
*
oP
(30c)
Moreover,   






































 2
1
2
*
1
2
2
*
1
11
11
222
*
1 ),(,
0
1
1
0
),(,
0
)sin(
0
11
100
),({ oosoooso
o
o
ooso xxQxxB
I
k
I
k
Ix
xmgl
kk
xxA




}
100
0
12
21
2
1
12
21
2
1
0
12
21
2
1
12
21
2
1
2
22
2
22
2
22
2
22












































kk
kk
is stabilizable and detectable.
,),({ 2
*
122 ooo xxA
  }1),(,1),( 2
1
2
*
1222
*
12  oooooo xxQxxB is also stabilizable and detectable.
Step 3.2 (the state equations):
According to (21), state variables relations in the initial layer correction are as follow:































s
ooo
osoosoosooo
o
o tx
x
I
k
x
I
k
x
I
mgl
xPxPxPxxk
x
x
/0
0
0
01
121111
2
1323122211121211
13
1
0
3
10
)(,
)sin(
1
)(


 (31a)
0
2
022012
02
2
2
2
1
)(3)(107
)(,1







tPtPk
txx
d
dx soso
ii
i
(31b)
Step 3.3 (the slow-fast SDREs):
The slow- fast SDREs in (22) have 3 the less equations respect to the original SDRE.
Considering (22), the SDRE relations in the final layer correction are as follow:
 
   
33
33_23_13_
23_22_12_
13_12_11_
0)(, 












 Fso
os
T
os
T
os
osos
T
os
ososos
so tP
PPP
PPP
PPP
P





















































































132
23
2
33
122
232322
2
22
22
2223
33
112
232312
11
1133
2
2
2212
2
2121323
11
1123
2
12
22
1213
11
1113
33_
23_
22_
13_
12_
11_
21
1
1
1
2
12
1
)sin(
11
)sin(
1
22
1
)sin(2
so
so
so
so
sososo
sososo
so
so
sososo
o
oso
sosososososo
o
oso
sososo
o
oso
os
os
os
os
os
os
P
P
I
P
kP
kPPP
PkkP
I
P
k
I
P
kP
kPPP
Ix
xmglP
kPPPP
k
I
PP
k
Ix
xmglP
PkkP
I
kP
Ix
xmglP
P
P
P
P
P
P

















(32a)


 1)(,12 222
FfFfFfF
fF
tPPP
d
dP (32b)
Step 3.4 (the optimal control law):
Moreover, the optimal control law is as follow:
ifFosoosoosooo xPxPxPxPxx
k
u 2
2
132312221112212112
)1()(
1
)(
1









(33)
The state equations and SDREs are two-point boundary value problem (TPBVP) and dependent
on state variables, but we have no state values in the whole interval [0,5]. To overcome this
problem we solve the above equations by an iterative procedure [3, 4]. Now, running the
simulation programs, Figures 3, 4 show the angular positions and velocities.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-70
-60
-50
-40
-30
-20
-10
0
10
20
Time(sec)
The angular positions(deg) and first angular velocity(deg/s)
q1
q2
dq1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
-60
-50
-40
-30
-20
-10
0
10
Time(sec)
The angular positions(deg) and first angular velocity(deg/s)
q1
q2
dq1
FIGURE 3: The slow state variables (The angular positions of q1, q2 and angular velocity of 1q ).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-40
-20
0
20
40
60
80
100
Time(sec)
Second angular velocity(deg/s)
dq2
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
0
10
20
30
40
50
60
70
80
90
Time(sec)
Second angular velocity(deg/s)
dq2
FIGURE 4: The fast state variable (angular velocity of 2q ).
Also, Figures 5 and 6 show the Riccati gains.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-20
-15
-10
-5
0
5
10
15
20
Time(sec)
The Riccati gains of P
s
FIGURE 5: The Riccati gains of Ps.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Time(sec)
f
4.985 4.99 4.995 5
0.05
0.1
0.15
0.2
0.25
Time(sec)
f
FIGURE 6: The Riccati gains of Pf.
From Figures 3 and 5, it can be seen that for any initial and final conditions on the slow
manifold, for different values of  , states are given by outer solution. On the other hand,

Figures 4 and 6 show that for any initial and final conditions out of the slow manifold, the
trajectories rapidly approach the slow manifold according to the fast manifolds. Moreover,
Figure 7 shows the optimal control law.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Time(sec)
The optimal control law
FIGURE 7: The optimal control law u.
5. CONCLUSION
With the proposed method in this study, it is seen that the finite-horizon optimization problem of a
class of nonlinear singularly perturbed systems leads to SDREs for slow and fast state variables.
One of the advantages of SDRE method is that knowledge of the Jacobian of the nonlinearity in
the states, similar to HJB equation, is not necessary. Thus, the proposed method has not only
simplicity of the LQ method but also higher flexibility, due to adjustable changes in the Riccati
gains. On the other hand, one of the advantages of the singular perturbation theory is that it
reduces high-order systems into two lower-order subsystems due to the interaction between slow
and fast variables. Note that SDREs in the proposed method have n1n2 the less differential
equations respect to the original SDRE. Thus, the slow-fast SDREs have the simpler computing
than original SDRE and provide good approximations of one.
6. References
[1] S.P Banks and K.J. Mhana. “Optimal Control and Stabilization for Nonlinear Systems.”
IMA Journal of Mathematical Control and Information, vol. 9, pp. 179-196, 1992.
[2] T. Çimen. ”State-Dependent Riccati Equation (SDRE) Control: A Survey,” in Proc. 17th
World Congress; the International Federation of Automatic Control Seoul, Korea, 2008,
pp. 3761-3775.
[3] H. Khaloozadeh and A. Abdolahi. “A New Iterative Procedure for Optimal Nonlinear
Regulation Problem,” in Proc. III International Conference on System Identification and
Control Problems, 2004, pp. 1256-1266.
[4] H. Khaloozadeh and A. Abdolahi. “An Iterative Procedure for Optimal Nonlinear Tracking
Problem,” in Proc. Seventh International Conference on Control, Automation, Robotics
and Vision, 2002, pp. 1508-1512.
[5] D.S. Naidu and A.J. Calise. “Singular Perturbations and Time Scales in Guidance and
Control of Aerospace Systems: A Survey.” Journal of Guidance, Control and Dynamics,
vol. 24, no.6, pp. 1057-1078, Nov.-Dec. 2001.

[6] W.C. Su, Z. Gajic and X. Shen. “The Exact Slow-Fast Decomposition of the Algebraic
Riccati Equation of Singularly Perturbed Systems.” IEEE Transactions on Automatic
Control, vol. 37, no. 9, pp. 1456-1459, Sep. 1992.
[7] Z. Gajic, X. Shen and M. Lim. ”High Accuracy Techniques for Singularly Perturbed
Control Systems-an Overview.” PINSA, vol. 65, no. 2, pp. 117-127, March 1999.
[8] Z. Gajic, S. Koskie and C. Coumarbatch. “On the Singularly Perturbed Matrix Differential
Riccati Equation,” in Proc. CDC-ECC'05, 44th IEEE Conference on Decision and Control
and European Control Conference ECC 2005, Seville, Spain,2005, pp. 14–17.
[9] R. Amjadifard and M.T. H. Beheshti. "Robust disturbance attenuation of a class of
nonlinear singularly perturbed systems." International Journal of Innovative Computing,
Information and Control (IJICIC), vol. 6, pp. 1349-4198, 2010.
[10] R. Amjadifard and M.T.H Beheshti. "Robust stabilization for a class of nonlinear
singularly perturbed systems." Journal of Dynamic Systems, Measurement and Control
(ASME), In Press, 2011.
[11] R. Amjadifard, M.J. Yazdanpanah and M.T.H. Beheshti. "Robust regulation of a class of
nonlinear singularly perturbed systems," IFAC, 2005.
[12] R. Amjadifard, S.E. Khadem and H. Khaloozadeh. "Position and velocity control of a
flexible Joint robot manipulator via fuzzy controller based on singular perturbation
analysis," IEEE International Fuzzy Systems Conference, 2001, pp. 348-351.
[13]E. Fridman. “Exact Slow-Fast Decomposition of the Nonlinear Singularly Perturbed
Optimal Control Problem.” System and Control Letters, vol. 40, pp. 121-131, Jun. 2000.
[14]E. Fridman. “A Descriptor System Approach to Nonlinear Singularly Perturbed Optimal
Control Problem.” Automatica, vol. 37, pp. 543-549, 2001.
Appendix A: The relation between the P(x) and Pnew(xnew)
In order to compute the optimal control law, the relations between the Riccati matrices









),(),(
),(),(
)(
21222121
21212111
xxPxxP
xxPxxP
xP
T

and
   
   






fsffsb
fsafss
newnew
xxPxxP
xxPxxP
xP
,,
,,
)(


must be determined.
Suppose that
   
    










1212
1212
2221
1211
21
1
22
nnnn
nnnn
ll
ll
HH , according to (17), we have:
   
   
 
 
  




















,
,
,)()()(
,)()(
,
1
2112212222
21
1
2112
2
11121121
1
221222
2
11121121
1
211211
221121111211
21
11
TT
f
TT
b
nn
T
a
nn
TT
s
T
f
PlIPlPp
PPlIp
OPllPllPp
OPllPPlIPp
xPlIxPllx





(A1)
Then, for =0, one can write:
    1
2111 ,,
1111
x
xxP
I
x
xxP
I nn
s
fss
nn











 
(A2a)

        2
2122
1
2121
1
2111
21
1
22
,,
0
,,
22121122
x
xxP
I
x
xxP
x
xxP
I
HHx
xxP
I nnnnnn
f
fsf
nn























 
(A2b)
Now, multiplying (A2b) by   22
, nnfsf IxxP  , the following relation is obtained.
    
       2
11
22
0,,,
,
, 2212212121
2111
21
1
22 nfsf
nn
nnfsf xxxPxxPxxxP
xxP
I
HHIxxP 













 
 (A3)
In other words, we have:
  1
)(1 ns Oxx  (A4a)
  11
)(),(),( 2111 nnfss OxxPxxP   (A4b)
  22
)(),(),( 2122 nnfsf OxxPxxP   (A4c)
    12
)(,),( 2121 nnfsc OxxPxxP   (A4d)
Where,       
.
,
,,
2111
21
1
22
11
22 





 

xxP
I
HHIxxPxxP nn
nnfsffsc Also, for =0, we have:
soo xx 1
(A5a)
),(),( 2111 fososoooo xxPxxP  (A5b)
),(),( 2122 fosofoooo xxPxxP  (A5c)
 fosocoooo xxPxxP ,),( 2121  (A5d)
Appendix B: Proof of Theorem 1
a) The optimal control law
According to =P(x)x [3] and (A4), substituting Riccati matrices in (6c), the optimal control law
would result as in (23).
b) The slow manifolds in boundary-layer correction
According to the singular perturbation theory, for =0, the fast variable should be derived with
respect to the slow variable. Substituting =0 in (19), the outer-layer equations are obtained as
follows:
,120| foososso HH     (B1a)
.0 222 2 foon H  (B1b)
Substituting (17b) in (B1b), the following relation is derived:
.0 222212 nfoosoo XHXH  (B2)
In other words, considering (14), we have:
    ,0 2222222212122112121 noooooooooo xPSAxPSPSA  (B3a)
    ,0 212121221121221221222222 nnoooooo
T
oooo
T
o QAPPSPAPSPA 
(B3b)

,0 222222222222222222 nnooooo
T
ooo QPSPPAAP 
(B3c)
For which, x
*
2o(x1o,P11o), P
*
21o(x1o,P11o) and P
*
22o(x1o,P11o) are the solutions. The necessary
conditions for (B3) to be solvable, {A22o(x1o,x
*
2o), B2o(x1o,x
*
2o), (Q22o(x1o,x
*
2o))
1/2
} should be
pointwise stabilizable and detectable for   21
2
*
1 , nn
oo RRxx  [2].
In (B1a), 0| sH for inside and out of the fast manifold, is separated as follows:
         

 0|212121
1
22211221110| ,,,,  xxHxxHxxHxxHHs
 
,,
),(),(
),(),(
100
2121
2121
tttt
xxAxxQ
xxSxxA
T
osos
osos








(B4a)
 
.,
),(),(
),(),(
10
2
*
12
*
1
2
*
12
*
1
FT
oosoooso
oosoooso
tttt
xxAxxQ
xxSxxA











 (B4b)
Substituting (B4) in (B1a), we have:
  ,),(|,),(),(),( 10001112121211 0
tttttxxxxxPxxSxxAx tooosoososo  (B5a)
.),(|,
),(),(),(
),(),(),(
1011
12
*
12
*
112
*
1
12
*
12
*
112
*
1
11
1
FFtso
ooosooo
T
sooooso
ooosooosooooso
osooso
o
tttttPP
xxxPxxAxxxQ
xxxPxxSxxxA
xPxP
x
F




















(B5b)
Thus, assuming that {Aso(x1o,x
*
2o), Bso(x1o,x
*
2o), (Qso(x1o,x
*
2o))
1/2
} is pointwise stabilizable-
detectable for   21
2
*
1 , nn
oo RRxx  [2], with rearrangement of (B5b), the SDRE of the slow
variable is obtained as (22a).
Remark 3: Note that under assumption of above, Pso is unique, symmetric, positive definite
solution of the SDRE (22a) that produces a locally asymptotically stable closed loop solution [2].
Thus the closed-loop matrix As(x1o,x2)-Ss(x1o,x2)Pso is pointwise Hurwitz for (x1o,x2)12.
Here, 12 is any region such that the Lyapunov function is locally Lipschitz around the origin.
c) The fast manifold in initial layer correction
Since the time scale will be changed as

 0tt 
 in the initial layer correction, the time derivative
in this scale will be changed as
dt
d
d
(.)(.)


 in forward time. Considering (4b), we have:
)(|,),(),(),( 0222121212122122
2
0
txxuxxBxxxAxxxA
d
dx
toooo 

(B6)
Substituting (23) in (B6), according to (A4) and (14), the fast state equation in initial layer is
obtained as (21b).
d) The fast manifold in final layer correction
Since the time scale will be changed as


ttF 
 in the final layer correction, the time derivative
in this scale will be changed as
dt
d
d
d (.)(.)


 in backward time:
 
     









fss
f
fs
s
HHHHHHHHHHHHHHHH
d
d
HHHHH
d
d






1221
1
222221
1
2221
1
2222
1
2221
1
22121121
1
22
1221
1
221211

(B7)

Substituting =0 in (B7), we have   120 ns  . Therefore, the final layer correction equation is
obtained as:
 .)(),(|,),( 2102
*
122 FFfffoo
f
txtxxxH
d
d



   (B8)
Now, substituting (20b) and (17b) in (B8), we have:
.),(|,
),(),(
),(),(
222
2
*
1222
*
122
2
*
1222
*
122
FFFtf
ffoo
T
foo
ffoofoo
f
ff
f
f
tttttPP
xPxxAxxxQ
xPxxSxxxA
x
d
dP
d
dx
P
d
dx
F



























 (B9)
Thus, assuming that {A22o(x1o,x
*
2o), B2o(x1o,x
*
2o), (Q22o(x1o,x
*
2o))
1/2
} is stabilizable-detectable for
  21
2
*
1 , nn
oo RRxx  [2], according to (A5) and (14), the SDRE of the fast variable is obtained as
(22b).
Remark 4: Note that under assumption of above, Pf is unique, symmetric, positive definite
solution of the SDRE (22b) that produces a locally asymptotically stable closed loop solution [2].
Thus, the closed-loop matrix A22(x1o,x2)-S22(x1o,x2)P
*
22o is pointwise Hurwitz for (x1o,x2)12.
Here, 12 is any region such that the Lyapunov function is locally Lipschitz around the origin. 

Designing SDRE-Based Controller for a Class of Nonlinear Singularly Perturbed Systems

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