Ning Bin i dr.
Pristup igre nulte-sume za H∞ robusno reguliranje jedinstveno perturbiranih bilinearnih kvadratnih sustava
ISSN 1330-3651 (Print), ISSN 1848-6339 (Online)
DOI: 10.17559/TV-20161112172344
A ZERO-SUM GAME APPROACH FOR H∞ ROBUST CONTROL OF SINGULARLY
PERTURBED BILINEAR QUADRATIC SYSTEMS
Ning Bin, Cheng-ke Zhang, Huai-nian Zhu, Ming Cao, Feng Hu
Original scientific paper
A zero-sum game approach for H∞ robust control of continuous-time singularly perturbed bilinear quadratic systems with an additive disturbance input is
presented. By regarding the stochastic disturbance (or the uncertainty) as "the nature player", the H∞ robust control problem is transformed into a twoperson zero-sum dynamic game model. By utilizing the singular perturbation decomposition method to solve the composite saddle-point equilibrium
strategy of the system, the H∞ robust control strategy of the original singularly perturbed bilinear quadratic systems is obtained. A numerical example of a
chemical reactor model is considered to verify the efficiency of the proposed algorithm.
Keywords: bilinear quadratic system; H∞ robust control; singularly perturbed; zero-sum game theory
Pristup igre nulte-sume za H∞ robusno reguliranje jedinstveno perturbiranih bilinearnih kvadratnih sustava
Izvorni znanstveni članak
U radu se opisuje pristup igre nulte sume za H∞ robusno reguliranje trajnih jedinstveno perturbiranih bilinearnih kvadratnih sustava s dodatnim unosom
smetnji. Smatrajući stohastičke smetnje (ili nesigurnost) kao "igrača prirode", problem H∞ robusnog reguliranja pretvara se u model dinamičke igre nultesume za dvije osobe. Primjenom metode dekompozicije singularne perturbacije za rješavanje složene strategije ravnoteže točke opterećenja toga sustava,
dobiva se strategija H∞ robusnog reguliranja originalnih jedinstveno uznemirenih bilinearnih kvadratnih sustava. Provjera učinkovitosti predloženog
algoritma daje se na numeričkom primjeru modela kemijskog reaktora.
Ključne riječi: bilinearni kvadratni sustav; H∞ robusno reguliranje; jedinstveno perturbiran; teorija igre nulte sume
1
Introduction
Robust control is a branch of control theory that
explicitly deals with uncertainty in its approach to
controller design. The established game theory can be
used to solve a robust control problem. The idea is to
regard the control designer as one player, and the
stochastic disturbance (or the uncertainty) as "the nature
player". Thus a robust control problem is converted into a
two-player game problem, that is when anticipating the
nature player’s various disturbance, how the controller
will design his strategy to optimize his goal, and at the
same time to realize the equilibrium with the nature
player. Then by solving the saddle-point equilibrium
strategy or the Nash equilibrium strategy, the robust
control strategy of various performance indexes can be
further obtained.
The approach of game theory has achieved great
success in the robust control of linear systems. David J.
N. Limebeer et al. studied a H∞ control problem for linear
time-varying systems using a game theoretic approach
[1], Ihnseok Rhee and Jason L. Speyer considered a
finite-time interval disturbance attenuation problem for a
time-varying system with uncertainty in the initial
conditions of state based on a LQ game theoretic
formulation where the control plays against adversaries
composed of the process and measurement disturbances
and initial conditions [2]. T. Basar showed that the
discrete-time disturbance rejection problem, formulated in
finite and infinite horizons, and under perfect state
measurements, can be solved by making direct use of
some results on linear-quadratic zero-sum dynamic games
[3]. Dan Shen and Jose B. Cruz converted H∞ optimal
control problems with linear quadratic objective functions
to a regular optimal regulator problem by improving a
game theory based approach [4]. Huai-nian Zhu,
Tehnički vjesnik 24, 3(2017), 777-782
Chengke-Zhang et al. presented a Nash game approach to
obtain a class of stochastic H2/H∞ control for continuoustime Markov jump linear systems [5]. Hiroaki Mukaidani
presented that the H2/H∞ robust control problem for linear
stochastic system governed by Itô differential equation
could be formulated as a Stackelberg differential game
where the leader minimizes an H2 criterion while the
follower deals with the H∞ constraint [6]. Hai-ying Zhou,
Huai-nian Zhu et al. discussed linear quadratic stochastic
zero-sum differential games for discrete-time Markov
jump systems, and constructed the explicit expressions of
the optimal strategies [7]. Tian-liang Zhang, Yu-hong
Wang et al. reviewed newly development in H2/H∞
control of stochastic linear systems with multiplicative
noise based on Nash game approach [8].
However, game theories for singularly perturbed
bilinear systems are seldom discussed, while singularly
perturbed bilinear systems are a quite proper and essential
description tool in describing many practical systems such
as neutron level control problem in a fission reactor, dcmotor, induction motor drives [9], and in financial
engineering problems, Black-Scholes Option Pricing
Model, M. Aoki’s two sector macroeconomic growth
model, P. Chander and F. Tokao’s non-linear input-output
model can all be extended to singularly perturbed bilinear
models in [10÷12].
H∞ robust control of singularly perturbed bilinear
quadratic systems is studied in this paper. By regarding
the stochastic disturbance (or the uncertainty) as "the
nature player", the H∞ robust control problem is
transformed into a two-person zero-sum dynamic game
model. Utilizing the singular perturbation decomposition
method to solve the composite saddle-point equilibrium
strategy of the system, we obtain the H∞ robust control
777
A zero-sum game approach for H∞ robust control of singularly perturbed bilinear quadratic systems
strategy of the original singularly perturbed bilinear
quadratic systems.
2
We introduce some necessary notations. Let Rn×1
denote the n-dimensional vector space and let the norm of
a vector x = [x1x2…xn]T be denoted by
x = (x ∗ x )
 n
xi
=

 i =1
∑



is defined by xTQx.
x = f ( x) + g ( x)u
y = h( x )
(1)
n
with f(0) = 0, h(0) = 0 and x ∈ R is a state vector,
p
u ∈ R m is an input vector, and y ∈ R is a measurable
vector. Then the definition of finite L2-gain is as follows:
Definition 1 [13]: Let γ ≥ 0. System (1) is said to have L2gain less than or equal to g if
2
2
T
y (t ) dt ≤ ∫ γ 2 u (t ) dt
0
(2)
or to have weighted L2-gain not larger than g if
T
∫0
2
y (t ) dt ≤
Q
T
∫0 γ
2
2
u (t ) dt
R
(3)
for all T ≥ 0 and all u ∈L 2 (0 ,T ) , with positive definite
matrices Q, R and y (t ) = h(ϕ (t ,0 , x0 ,u )) denoting the
output of (1) resulting from u for initial state x(0) = 0.
The system has L2-gain < γ if there exists some 0 ≤ γ~ < γ
~
such that (3) holds for γ .
3
Problem statement
Consider the H∞ robust control strategy for the
following time-invariant singularly perturbed bilinear
system:
 x1 (t ) 
εx (t ) =
 2 
 x (t )   M s   
 x (t )  

A 1  +  B +   1  
  u (t ) + Ew(t ) (4a)

M
x
t
x
t
(
)
(
)


f
 2  
 
 2  
 x (t ) 
z (t ) = C  1  + Du (t )
 x2 (t )
778
B 
B =  1,
 B2 
E 
E =  1,
 E2 
x2 (t ) ∈ R n2 are respectively slow and fast
j =1
Consider the following nonlinear system
T
A12 
,
A22 
with n1 + n2 = n, u ∈ R m is a control vector, w ∈ R l
denotes the disturbance input, z (t ) ∈ R q is the penalty
function to be used in the cost function, the small singular
perturbation parameter ε > 0 represents small time
constants, inertias, masses, etc. , and Aij, Bi, Ei, Mi, C, D
(i, j = 1, 2) are constant matrices of appropriate
dimensions, with
 x1 (t )   M s   n1
 M sj  n1 + n2  M sj 
let
 x (t )  M   = ∑ x1 j  M  + ∑ x2 j  M 
 fj 
 fj 
 2   f  
1/ 2
2
Q
A
A =  11
 A21
state variables, x(t ) = [x1 (t ), x2 (t )]T ∈ R n are state vectors
 n m

A =  ∑∑ aij2 
 i =1 j =1 


and the weighted 2-norm x
where
x1 (t ) ∈ R n1 ,
1/ 2
2
The norm of matrix A ∈ R n×m is defined by
∫0
with initial condition
 x1 (0)   x10 
 x (0) =  x 
 2   20 
Preliminaries
1/ 2
Ning Bin et al.
(4b)
j = n1 +1
~
 x (t )   M  
B
~
( x) 
B ( x) =  ~1  = B +  1   s   , then the state Eq.
M
 B2 ( x ) 
 x2 (t )  f  
(4a) can be written as:
 x1 (t ) 
εx (t ) =
 2 
 x (t )  ~
A 1  + B u (t ) + Ew(t )
 x2 (t)
(5)
Letting
~ 
B
 E1 
1
A12 


 
~
A22  , Bε =  ~ε  , Eε =  ε  ,
E

 B2 
 2
ε 
 ε 
ε 
(4a) can be further written as follows:
 A11
 x (t ) 
x(t ) =  1  , Aε =  A21

 x2 (t )
 ε
~
x (t ) = Aε x(t ) + Bε u (t ) + Eε w(t ).
(6)
Thus, the nonlinear robust H∞ control guarantees that
the performance index (7) remains within an upper bound
for a given positive number γ.
∞
{
1
J (u , w) = ∫ z
20
2
−γ 2 w
2
}dt
(7)
The basic game theory idea of robust control design
is to regard the control designer as one player P1, and the
stochastic disturbance (or the uncertainty) as "the nature
player" P2. Thus a robust control problem is converted
into a two-player game problem, that is when anticipating
the nature player P2’s various disturbance, how the
controller P1 will design his strategy to optimize his goal,
and at the same time to realize the equilibrium with the
nature player. Accordingly, the design method of H∞
robust control for system (4) is that: the w*(t, x) tries to
maximize the energy, while the controller or u*(t, x)
simultaneously seeks to minimize it.
Technical Gazette 24, 3(2017), 777-782
Ning Bin i dr.
Pristup igre nulte-sume za H∞ robusno reguliranje jedinstveno perturbiranih bilinearnih kvadratnih sustava
Then the problem is converted to solve the
equilibrium strategy u*(t, x) for the player P1, and the
equilibrium strategy w*(t, x) for the player P2, which
satisfy the following condition:
Thus a two-player zero-sum dynamic game for
players P1 and P2 is constructed.
Decomposition of slow and fast systems
Assumption 1 [13]: The pair (A, B) is completely
controllable and x stays in the controllability domain
defined by
Δ
{
~
~
~
where A0 = A11 − A12 A22 −1 A21 , B01 = B1 − A12 A22 −1 B2 ,
T
J s (u , w) =
=
}
Assumption 2 [13]: The differential Eq. (4) has a
solution defined on [0 ,+∞) for each admissible input
function and x(t ) → 0 as t → ∞ .
The Hamiltonian H(t) corresponding to the system (4)
and performance (7) is:
{
2
−γ
2
w
2
} (
~
+ λ Aε x + Bε u + Eε w
T
)
(8)
u=
∂J ( x)
∂x
1
γ
2
EεT
(9)
(10)
And J(x) satisfies the following HJI equation
T
1
1  ∂J ( x)  ~ −1 ~ T ∂J ( x)
0 = x T C T Cx − 
+
 Bε R1 Bε
2
2  ∂x 
∂x
+
1
2γ
2
EεT
∂J ( x)
∂x
T
 ∂J ( x) 
+
 Aε x
 ∂x 
(11)
under the assumption of DT = 0 and R1 = DTD > 0.
Neglecting the fast modes is equivalent to assuming that
they are infinitely fast, that is letting ε = 0. Without the
fast modes the system (5) reduces to:
~
x1 = A11 x1 + A12 x2 + B1u + E1w
~
0 = A21 x1 + A22 x2 + B2 u + E 2 w
(12a)
(12b)
∫0 (x
T
1
2
∞
2
}dt =
0
∞
)
Q1 x + u T R1u − γ 2 w T w dt =
~
x1s = A0 x1s + B01u s + E0 ws , x1s = x10
(14)
∫ (x1s Q0 x1s + 2 x1s D1u s + 2 x1s D2 ws +
T
T
0
T
)
+ 2u1Ts D3 ws + u1Ts R1s u s − wsT R2 s ws dt
where
−T
−1
−T
−1 ~
Q0 = Q11 + AT21 A22
Q22 A22
A21 , D1 = AT21 A22
Q22 A22
B2 ,
~
−T
−1
−T
−1
D2 = AT21 A22
Q22 A22
E 2 , D3 = B2 T A22
Q22 A22
E2 ,
Q ) is stabilizable
and detectable.
Theorem 1: Under Assumption 3, suppose that the
following algebraic Riccati equation has solution ps
(15)
Then the equilibrium solution of the slow subsystem
can be given by
u s∗ = −(T1 + T2 p s ) x1s
(16a)
ws∗
(16b)
= (T3 + T4 p s ) x1s
Proof: Substituting (14) into (9), we can obtain the
optimal control us of the slow subsystem:
~T
~ T ∂J s ( x)
λ ) (17a)
u s = − R1−1 B01
= − R1−s1 ( D1T x1s + D3 ws + B01
∂x
Substituting (14) into (10), we can obtain the worst
disturbance input ws of the slow subsystem:
ws =
1
γ
2
E0T
∂J s ( x)
= R2−s1 ( D2T x1s + D3T u s + E0T λ )
∂x
(17b)
We get
Assuming that A22 is non-singular, we have
Tehnički vjesnik 24, 3(2017), 777-782
1
2
−γ 2 w
AT p s + p s A − p s Bp s + Q = 0
∂J ( x)
∂x
2
2
∞
Assumption 3: The triplet (A, B,
And worst disturbance input is:
w=
=
∫ {z
1
2
~
−T
−1 ~
−T
−1
R1s = R1 + B2 T A22
Q22 A22
B2 , R2 s = γ 2 − E2 T A22
Q22 A22
E2 .
where λ ∈ R n×1 is the Lagrangian multiplier.
Then the optimal control is given by:
~
− R1−1 BεT
Q12 
, and
Q22 
x 
x 
then x T C T Cx =  1  Q1  1  .
 x2 
 x2 
Substituting the above into (7), we can obtain the
quadratic cost function for the slow subsystem
X c = x ∈ R | ( A, B + {xM }) controllable
n
1
H=
z
2
(13b)
Q
E0 = E1 − A12 A22 −1 E 2 let C T C = Q1 =  11
T
Q12
J (u ∗ , w) ≤ J (u ∗ , w∗ ) ≤ J (u , w∗ ).
4
~
x 2 s = − A22 −1 ( A21 x1s + B2 u s + E 2 ws )
(13a)
u s = −T1 x1s − T2 λ , ws = T3 x1s + T4 λ
779
A zero-sum game approach for H∞ robust control of singularly perturbed bilinear quadratic systems
where
Then the equilibrium strategies of the fast subsystem
are given by
T1 = ( R1−s1 D3 R2−s1 D3T + 1) −1 ( R1−s1 D1T + R1−s1 D3 R2−s1 D2T )
~T
T2 = ( R1−s1 D3 R2−s1 D3T + 1) −1 ( R1−s1 D3 R2−s1 E0T + R1−s1 B01
)
~
u ∗f = − R1−1 B2T p f x2 f
T3 = R2−s1 ( D2T − D3TT1 )
w∗f =
T4 = R2−s1 ( E0T − D3TT2 )
For − λ = Q0 x1s + D1u s + D2 ws + A0T λ , letting λ = p s x1s ,
we have
~
p s ( A0 − B01T1 + E0T3 ) + ( A0T − D1T2 + D2T4 ) p s +
~
+ p s ( E0T4 − B01T2 ) p s + (Q0 − D1T1 + D2T3 ) = 0
(18)
~
Because ( A0 − B01T1 + E0T3 ) T = A0T − D1T2 + D2T4 ,
~
~
let A = A0 − B01T1 + E0T3 , B = B01T2 − E0T4 ,
Q = Q0 − D1T1 + D2T3 .
Then (18) can be written as the following algebraic
Riccati equation:
AT p s + p s A − p s Bp s + Q = 0
In the fast subsystem, we assume that the slow
variables are constant in the boundary layer. Redefining
the fast variables x2f = x2 – x2s, and the fast controls uf = u
– us, wf = w – ws, the fast subsystem is formulated as:
1
A22 x2 f +
x2 f (0) = x20 − x2 s (0).
=
1
2
∞
∫ (x
1
2
∫ {z
∞
2
−γ 2 w
}dt =
0
T
2 f Q22 x 2 f
Composite strategy
The composite strategy pair of the full-order
singularly perturbed system (4) is constructed as follows
[14]:
~
u c = u s∗ + u ∗f = −(T1 + T2 p s ) x1s − R1−1 B2T p f x2 f
wc = ws∗ + w∗f = (T3 + T4 p s ) x1s +
)
(20)
0
~
B22 , Q22 ) are stabilizable and detectable.
Theorem 2: Under Assumption 4, suppose that the
following algebraic Riccati equation has solution pf
γ2
p f x2 f
(23b)
(24a)
(24b)
~
~
−1
[− A21 + B2 (T1 + T2 p s ) −
G1 = −(T1 + T2 p s ) + R1−1 B2T p f A22
− E 2 (T3 + T4 p s )]
~
G2 = − R1−1 B2T p f
E 2T
γ
2
~
−1
p f A22
[− A21 + B2 (T1 + T2 p s ) −
− E 2 (T3 + T4 p s )]
E 2T
γ2
pf
The composite strategy pair constitutes an o(ε) (near)
saddle-point equilibrium of the full-order game. The proof
can be found in [15].
6
by
Numerical example
The bilinear model of a chemical reactor [16] is given
 x1 (t ) 
 x1 (t ) 
εx 2 (t ) = A x2 (t ) + ( B + x1M 1 + x2 M 2 ) u (t ) + Ew(t )
 x (t ) 
z (t ) = C  1  + Du (t )
 x2 (t )
Where
T
p f A22 + A22
pf −

ET 
~
− p f  B2 R1−1 B2T − E 2 22  p f + Q22 = 0

γ 

E 2T
(23a)
with x1 replacing x1s, x2 replacing x2s + x2f, for
~
−1
( A21 x1s + B2 u s + E 2 ws ), we obtain
x2 s = − A22
+ u Tf R1u f − γ 2 wTf w f dt
~
Assumption 4: the triplet ( A22 , B21 , Q22 ) and (A22,
780
5
G4 =
2
(22b)
The proof is similar to that of the slow subsystem.
(19)
ε
Then we can obtain the quadratic cost function for the
fast subsystem
J f (u , w) =
p f x2 f
G3 = (T3 + T4 p s ) −
1~
1
B2 u f + E 2 w f ,
ε
γ2
where
−1
x2 s = − A22
( A21 + G0 ) x1s .
ε
E 2T
(22a)
u c = G1 x1 + G2 x2
wc = G3 x1 + G4 x2
where ps is the solution of the above Riccati equation.
For convenience, let
~
~
G0 = − B21( T1 + T2 p s ) − B22 ( T3 + T4 p s ), then
x 2 f =
Ning Bin et al.
(21)
 − 1 / 8 ,
 3 / 16 5 / 12  ,
B=
A=


 − 50 / 3 − 8 / 3
 0 
Technical Gazette 24, 3(2017), 777-782
Ning Bin i dr.
Pristup igre nulte-sume za H∞ robusno reguliranje jedinstveno perturbiranih bilinearnih kvadratnih sustava
 0
 0
1 
1 0  ,
 − 1
D= ,
M1 =   , M 2 =   , E =   , C = 

 0
1 
 0
0 1
0
1
1
0
 

,
Q=
 x 0 =  0
0
1
 


and x1 and x2 represent the temperature and concentration
of a chemical reaction while u represents the coolant flow
rate around the reactor. We choose γ = 0,5 and ε = 0,001,
and obtain the simulation curves for the optimal control
strategy and the state as follows:
Simulation curves of the composite strageties for U
1.5
U
1
0.5
0
-0.5
0
1
3
times
X1 state
1
4
0.8
8
0.6
6
0.4
0.2
0
5
6
X2 state
10
x2
x1
2
4
2
0
2
times
4
6
0
0
2
times
4
6
Figure 1 Simulation curves of the control strategy and the state
trajectories
7
Conclusions
Many real systems possess the structure of the
singularly perturbed bilinear control systems such as
motor drives, robust control, multi-sector input-output
analysis and option pricing. A game approach for H∞
robust control of continuous-time singularly perturbed
bilinear quadratic systems with an additive disturbance
input is presented in this paper. By regarding the
stochastic disturbance (or the uncertainty) as "the nature
player", the H∞ robust control problem is transformed into
a two-person zero-sum dynamic game model. By utilizing
the singular perturbation decomposition method to solve
the composite saddle-point equilibrium strategy of the
system, the H∞ robust control strategy of the original
singularly perturbed bilinear quadratic systems is
obtained. A numerical example of a chemical reactor
model is considered to verify the efficiency of the
proposed algorithm. The conclusion obtained in this paper
could be applied to deal with many industry engineering
and financial engineering problems.
Acknowledgment
This work is supported by the National Natural
Science Fund of China (71571053), the Natural Science
Fund of Guangdong Province (2014A030310366,
2015A030310218), a Grant from Guangdong Province,
Soft Science Research Project (2016A070705052).
Tehnički vjesnik 24, 3(2017), 777-782
8
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Authors’ addresses
Ning Bin, lecturer
School of Management,
Guangdong University of Technology,
No. 161, Yinglong Road, Tianhe District,
Guangzhou 510520, P. R. China
E-mail: [email protected]
Cheng-ke Zhang, professor
School of Commerce & Economics,
Guangdong University of Technology,
No. 161, Yinglong Road, Tianhe District,
Guangzhou 510520, P. R. China
E-mail: [email protected]
Huai-nian Zhu, lecturer,Corresponding author
School of Commerce & Economics,
Guangdong University of Technology,
No. 161, Yinglong Road, Tianhe District,
Guangzhou 510520, P. R. China
E-mail: [email protected]
Ming Cao, doctor
School of Management,
Guangdong University of Technology,
No. 161, Yinglong Road, Tianhe District,
Guangzhou 510520, P. R. China
E-mail: [email protected]
Feng Hu, lecturer
School of Management,
Guangdong University of Technology,
No. 161, Yinglong Road, Tianhe District,
Guangzhou 510520, P. R. China
E-mail: [email protected]
782
Technical Gazette 24, 3(2017), 777-782