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Decentralized Robust Controller for Exponential Stability of Uncertain Systems with Output Feedback
Copyright © IFAC Robust Control Design, Budapest, Hungary, 1997
DECENTRALIZED ROBUST CONTROLLER FOR EXPONENTIAL STABILITY OF UNCERTAIN SYSTEMS WITH OUTPUT FEEDBACK
V.Vesely, D.Rosinova Department ofAutomatic Control Systems, Faculty ofElectrical Engineering and Information Technology, Slovak University of Technology, 812 19 Bratislava, Slovak Republic fox: + +421 7729 734, e-mail: veseZy@j«[email protected]
Abstract. The paper addresses the problem of design of robust decentralized controllers with output feedback for interconnected linear systems with uncertainties. A new class of robust continuous-time and discrete-time controllers is proposed to guarantee exponential stability of uncertain linear systems without matching conditions. The uncertainties possibly nonlinear can appear both in the subsystems and in the interconnections between them. The proposed algorithm is applied to 6th order system model comprising two subsystems with decentralized controller.
Key Words. Decentralized control, Robust stability, Uncertain dynamic systems,
prescribed decay rate. The robust exponential stability of complex systems pays an important role to guarantee robustness and dynamic performance quality of closed-loop system. Both the Lyapunov function method and (Krasovskij, 1959) definition of exponential stability of generalized dynamic systems provide useful tools for a design of static output feedback controller for continuous-time case and static state feedback one for discrete-time case, which guarantees the robust exponential stability of closed-loop system for all admissible uncertainties. In the design procedure of robust decentralized controller no matching conditions of continuoustime or discrete-time linear model are considered in contrast with (Niculescu, 1995) and others.
1. IN1R.ODUCTION The design of controllers for many physical processes faces the influence of uncertainties, which very often cause poor performance and even instability of control systems. Considering these facts the stability robustness is a topic of great practical importance which has attracted a lot of interest for several decades, (Kwakernaak, 1993), (Sezer, 1988). If uncertainties arise in a complex system which consists of N interconnected subsystems, (Chen, 1988), (Ban, 1995), they can be classified corresponding to their position in complex system. Firstly, the internal uncertainties, subsystem and input ones, that arise in particular subsystem. Secondly, the external uncertainties that belong to the part of interconnections among subsystems. Treating the uncertainties matching conditions are frequently used. We assume that the internal and external uncertainties upper bounds are known. The paper studies the robust decentralized control problem, the proposed control design strategy aims at closed-loop system exponential stability with a
The paper is organized as folloM. Section 2 brings the problem statement and some preliminary results for continuous-time systems. The main results concerning robust decentralized stabilization for continuous-time systems are provided in Section 3 and for discrete-time systems in Section 4. In Section 5 the proposed algorithm for decentralized
131
controller design is applied to 6th order system model example, the obtained results are presented.
Assumption 2 Assume that for each subsystem the conditions of Theorem 1 are met.
2. PROBLEM STATEMENTCONTINUOUS-TIME CASE
The system (1) can be rewritten in the following compact form x=(A+M)x+(B+oB)u+h(x) y=Cx
Consider an uncertain complex system Xi = (Ai+OAi)xi+(Bi+oBi)Ui+
t
hfj(xj)
where xT = [xr. ... ,x~] E Rn, uT = [ur. ... ,u~] E Rm yT = [yr. ...,Y~] E RI are the state, control input and output vectors of the complex system respectively, A,M,B,oB, C are block diagonal matrices with block elements A;,M;,B;,oBi,C; respectively and h(x} is a vector function with entries hfj(xj). Nominal system model is
j=ljA
Yi=CiXI, (1) where i E.3 = {1,2, ...,N}, Xi E Rnj,u; E Rnt;,y; E RI are the state, control input and output of the i-th subsystem respectively, A;,Bi,CI are matrices of appropriate dimensions which represent the nominal part of the subsystem, matrices OA i, OBi of appropriate dimensions represent internal uncertainties in each subsystem and they are assumed to be piecewise continuous and bounded on every time, the unknown vector functionshfj(xj) are continuous and . . . sufficiently smooth in X} , piecewise contInuous ID time, so that the system (1) has unique continuous solutions for all initial conditions XI(tO) = XiO E Rn; and all piecewise continuous inputs U I E Rm; . The isolated subsystem nominal model is x;=A;x;+B;u; Yi= C;x;
The decentralized robust control design problem is to find a control law in the form u=Ky=KCx, K=diag(K;), K;
oB;oB; ~ Yb;Ri
(4)
1~!i(Xj)lI:s; ;ilIxJ!
(5)
(10)
(11)
is robustly exponentially stable with a prescribed decay rate. The output feedback matrix K comprises the local feedback matrices K; with local controller parameters. Throughout this paper the following concept of robust exponential stability of complex system will be used. Definition 1 The uncertain closed-loop system (11) is said to be robustly exponentially stable with a decay rate'}.. > 0 if the trivial solution x(t}=O is exponentially stable with a decay rate '}.. for all initial conditions x(to) =Xo E Rn and for all admissible uncertainties given by (3),(4) and (5), i.e. there exist constants c(xo) > 0 and '}.. > 0 such that
IlxCt)1I ~ c(xo)e-~
(12)
Definition 1 is for certain subset of considered systems equivalent to Definition 2. Definition 2 The uncertain closed-loop system (11) is said to be robustly exponentially stable with a decay rate IX > 0 if for all initial conditions x(to) =Xo and for all admissible uncertainties (3), (4), (5) there exists a Lyapunov function for a nominal system (9) V(x): Rn ~R+, V(x) belongs to class Cl, such that for its time derivative along the solution of closed-loop system (11) the following inequalities hold
~) ~ -«V(x), V(x(to»
(6)
where Li is the real symmetric nonnegative definite solution of A;L;+LA;-L;B;BfL;+cfc;+ GfG; = 0
Rm;xJ;
x= [A+M +(B+oB)KC]x+h(x)
where QOi E Rn,xn/ R; E Rm'-I are knOVdl symmetric posttlve definite matrices, ;fj ~ 0, Yai ~ 0, 10i ~ 0 are unknOVdl constants to be determined from conditions of closed-loop stability of system with uncertainties, ij E .3. The necessary and sufficient conditions for output feedback stabilizability of each subsystem are given in (Kucera and Souza. 1995). Theorem 1 (Kucera and Souza, 1995) The nominal i-th subsystem (2) is output feedback stabilizable if and only if i. (A;,B;) is stabilizable and (Ai, Ci) is detectable ii. there exist real matrices K; and G i such that KiC;+B;L;=G;
E
such that the closed-loop uncertain system comprising the plant (8) and decentralized controller (10)
System (1) is assumed to satisfy the following assumptions. Assumption 1 The internal uncertainties in each subsystem and the unknOVdl vector functions which represent the interconnections among the subsystems satisfy the inequalities (3)
(9)
x=Ax+Bu+h(x)
(2)
OA;M; ~ YaiQo;
(8)
with
(7)
= Vo
Ylllxlf ~ Vex) :s; Y211x11 2 ~lr) ~ -'Y311x1f
where Yi, i = 1, 2, 3, are positive constants. 132
(13)
*,
*.
+Qi + CfKfRjK/C i = 0
The relationship between constants in definitions 1 and 2 is )., = c(xo) = Definition 3 The system (8) is called robustly exponential/y stabilizable if there exists K such that the closed-loop system (11) is robustly exponentially stable for all admissible uncertainties (3),(4) and (5).
(18)
where Qi;: Qo; + QIi and Ki E Rm/><1 E matrices of decentralized controller parameters,
are
j
;: {Kj
E
R m/><1; ,Re).,M{Aj + 1-1i + BjKiC j ) 5, O}, iE 3 (19)
Re ).,M(X) denotes maximal real part of eigenvalues
Finally, the problem studied in this paper can be stated as follows. Under assumptions 1,2 find a matrix K such that the closed-loop uncertain system (11) is exponentially stable with a prescribed decay rate a and the bounds "fai, "fbj, i E 3 in (3), (4) have the maximum possible values for given Sij,i:t= j, i,j E 3.
of matrix X. Proof. For the time derivative of V(x) defined by (15) along the solution of (11) ~ obtain
'!
N
hy(xj)}
(20)
Owing to Assumption 1 and (18), the upper bound on (20) is
'! 5,-
("fai+"fbj)~]xj-21IPjllllxjll
t
siillxAI}
piJ=1
where P j is a symmetric positive definite matrix, as a candidate Lyapunov function of the uncontrolled nominal model of i-th isolated subsystem (2). The candidate Lyapunov function of the uncontrolled nominal model of the complex system (9) is given by
where 1
..,
oG j = !f:dOAj+pj.f{:d, uFj ;;:.
oBjKjC j
(21) ~
Jf/;/ +Pj,,"fbi
Using the following inequalities
2ab
(15)
The sufficient conditio:i'J for decentralized output feedback robust exponential stabilizability of complex system (1) with a prescribed decay rate a are given in the following theorem. Theorem 2 The uncertain system (8) is robustly exponentially stabilizable with a prescribed decay rate a by the decentralized output feedback (10) if the following sufficient conditions are satisfied. i. Assumptions 1,2. ii. For the decay rate aj > 0 of the i-th subsystem,
(F + G)T(F + G) 5, (1 + &)FTF + (1 + &-1 )GTG, &> 0
(22) the bound on time derivative of Lyapunov function (21) can be rewritten
dV dt 5, -
2
+(l-&il)Pj"fai+
(16)
aj~a,
the symmetric matrices M j are nonnegative definite for some K j where M j = (aj -a)pj+ Qo;(I-&n)+QIi -~(&il"fai+
-211Pj1BlxjU
+&·;:l"!bj)+(I-&a)CfKfR;KjC j -So;l/ , iE 3
2,;
li+(I-&a)
5,
-
C;K;oB;OB;KjCj
Yl>i
t
xfMjxj
I
(23)
r-l
whereM; is given by (17). Condition ii. of Theorem 2 then implies the robust exponential stability of the system (11), that completes the proof. Q
(17)
and &il,&a ~ 0 are constants, 1j E Rn/XII/ are identity matrices, Qli E R n /XIJ / are ~etric positive pi·l
Q
t Sitllxjll}
~1
definite matrices, Soi =NJ,:;,/Pi)
L jRJ=1
(14)
V/(x/)
r-l
+P/(OAj+oBjKjCj)]xj+2xfpi
In this section ~ consider the problem of decentralized control design which guarantees the robust exponential stability of complex uncertain system (1). Let us choose function
t
(xfPjxj+x[P;Xj)=t {xf[(Aj+BjKjCjlpj+
r-l
3. DECENTRALIZED ROBUST CONTROL DESIGN WITH OUIPUT FEEDBACK
VeX) =
=t
s; + N;.I,
For the next developments the following definition is introduced. Definition 4 Consider linear uncertain system (8) with h(xFO. Assume that the matrix A+BKC is Hurwitz (stable).
).,M(Pj ) denotes the maximal eigen~ue of matrix Pi. Matrices Pi are solutions of the Lyapunov matrix equations (Ai+ I!f:!i+BiKiCi)TPi+Pi(Ai + ~11j+BiKiC/) 133
The robustness stability measure of the considered system is defined as mr= min {OA +oBKC e ®: I<'\.4...oBKQI
A + OA + (B + oB)KC is not Hurwitz} (24) where ® is a set of all possible system uncertainties. Assume that the matrices Qo;, Qli,R; and constants Yai, Yb;, a; > a and ;, for ij e .3 are known. Then the analysis of properties of M; implies that the best way to guarantee nonnegative definiteness of M; is to maximize the robustness stability measure for all uncertain isolated subsystems (hij(xj) = 0 in (I» . It is known that to maximize m ri, i e.3 the maximal eigenvalues 'AM(P;) should be minimized where Pi is a solution to (18). To simplify the task, minimization of 'AM(P;) is replaced by minimization of Tr(P;), Tr(X) denotes the trace of matrix X. To minimire Tr(P t ) the Lagrange function is considered on subsystem level. The Lagrange function for the i-tb subsystem
If each M; is nonnegative definite then stop, else increase ai or the eigenvalue of Qli and go to 2). Algorithms A and B yield the "optimal" values of aj,Qli,Yai andYbj which guarantee the nonnegative definiteness ofM;, i=I, ... ,N.
4. DECENTRALIZED ROBUST CONTROL DESIGN FOR DISCRETE-TIME SYSTEMS Consider the complex linear uncertain system
xt(k+ I) = (Adi+ OAdi)xi(k) + (Bdi + OBdi)Uj(k)+
+
L; =miD max Tr{P i + Wi[(Ai+ ?-fIt + BjKtCj)TP.+ Ki.Pi W;
~i = KjCjWjC; +B:PjWjC; =0
(26)
(28)
(29) where Aci=Ai+ ~;li+BtKiCj To find the solution of matrix equations (26)-(28) for i-tb subsystem we propose an iterative algorithm. Algorithm A I) Choose ~l) such that A ci is (Hurwitz) stable matrix. 2) Calculate p~l) and
w,l)
(32)
OA ~OA di :s; YadiQ ai
oB~oB di :s; YbdiQbi
(33)
the interconnections are upper bounded
IIg ij(Xj)/I :s; ;d!fIixJ
from Lyapunov
(34)
Definitions I and 2 for robust exponential stability of continuous-time systems can be modified to discrete-time systems in obvious way.
matrix equations (27) and (26). 3) Calculate the next approximation of ~2) using (28) ~2) = ~l) _ pj(B;p~l)w,l)C; + ~l)Cjw,l)C;)
gij(Xj), ie .3
where Xt eRn;, U; eR"'; are the state and control vector ofi-th subsystem respectively, Adi e Rn;XII;,Bdi e Rn;x",; are constant matrices. The unknown matrices OA di, oB di of appropriate dimensions represent system deterministic uncertainties which are upper bounded and can be time-varying. Assumption 3 The pair (A di, Bdi) is controllable for each subsystem Assumption 4 The uncertainties OA di, oB di satisfy the inequalities
+Pi(A i + ~;lj+BtKjCt)+Qt+C;K;KiCi]} , i e .3 (25) The necessary conditions for optimal solution are I +AciWj + W;A~ = 0
t
piJ=1
The decentralized control is considered as a local constant state feedback
(30)
where p; > 0, check the stability of A ci, if it is unstable, decrease pj . 4) Calculate p~2) and w,2) for ~2). 5) If Tr(p~2) _ p~l» < 0, then p~l) = p~2), W~l) = w,2), K~l) = ~2) , go to 3) else stop. The proposed procedure for the design of decentralized robust controller that ensures the exponential stability of uncertain system (11) with a prescribed decay rate a is summarized in the next algorithm. AlgorithmB I) Choose ai;?! a, Yai and Ybi, ie .3. 2) Use Algorithm A for i=I,2, ... ,N. 3) Check if the matrices M j , i e.3 are nonnegative definite. The coefficients &.1, &a can be chosen
u;(k) =KdjXi(k), ie .3
(35)
Kdi e Rm;Xll; is a constant matrix of controller parameters. The corresponding uncertain closed-loop system is xt(k+ I) = (A di + BdiKi)xi(k) + (OAdi + OBdiK;)x;(k)+
+
t
jRJ:l
gtj(Xj), ie .3
(36)
The aim of robust control design is to achieve robust exponential stability of the closed-loop system (36) with a prescribed decay rate. The sufficient conditions for robust stability of the system (36) are given in Theorem 3. Theorem 3 The uncertain decentrally controlled closed-loop system (36) is robustly exponentially stable with a 134
Equation (42) is a discrete-time counterpart to the solution of (26)-(28) for state feedback (for the case when Cj=]j). Hence robust decentralized controller can be designed using Algorithm B after corresponding modifications.
decay rate a if the follo\\ing sufficient conditions are satisfied i. Assumptions 3,4. ii. There exist constants &il,&a,&j3 > 0 and positive definite matrices Qj, ie .3 , such that matrices M j are positive semidefinite,
M j = (l-aj)Qi + (ai -a)pj -
1 &il (1
+ &a)').M<.Pi )
5. EXAMPLE - DESIGN OF ROBUST DECENTRALIZED CONTROLLER
[(1 + &i3)YadiQai + (1 + &il)YbdiKfQbiK;]-
-&~l(1+&a1)(N-1)
t
Consider the 6th order system described by
Pj;~
(37)
. [ A 11 A 12 x= A21 A22
fr'ii=l
where 1 > ai ~ a > 0, P j matrix equation
is a solution of Lyapunov
+[ 0.40.4
(1- ai)-l(Adi + BdiKi)TPi(Adi +BdiKj)-Pi = -Qi .
11 21
x+ [ B 1
0 ] u+ 0 B2
]
0.4 12 ]x+ [OB1 0.4 22 0
oB0 2
]
U
(38) Proof. The proof is omitted for brevity, its technique is similar to the proof of Theorem 2.
Y
zi(k+ 1) =AdiZj(k) + BdiUj(k) .
yT
2
=LYi yI1
un e R4 ,
[xi xn
and
(39)
All
Let us assume that the nominal model (39) is stable. Consider now the subsystem with uncertainties and perturbations Z i(k+ 1) =AdiZ i(k) + BdiUi(k) +Ji(Z i)
[Cl0 C0]
xT = e R6, UT = [ui e R4 that is n=6, m=4, 1=4
where The concept of robustness bound (see e.g. (Sezer, 1988» will be used in further development. Consider the nominal isolated subsystem model
=
(40)
=
0.4 11
-6.1 0.23 0.5 ] 0.3 -5 -3.5 [ 1.5 2.3 -5.3
=[
f;(Z i) is unknown deterministic upper bounded vector function
1.5 1.5 1.5 ] 1.1 1.23 1.1 0.6 1 1.5
(41) Maximal bound f.li such that for f;(Z t) conforming to (41) the system (40) remains stable is called robustness bound for (39). The concept of robustness bound for discrete-time systems correspond to robustness stability measure defined for continuous-time system in Definition 4. Similarly to the continuous-time case the analysis of (37) indicates the way to make M; positive semidefinite, the corresponding decentralized control design strategy aims at determining feedback matrices K i such that the robustness bound of the nominal subsystems (39) is maximized. The maximization of f.li implies the minimization of ').M<.Pi ) simultaneously with one-dimensional search of the appropriate ai, Pi being a solution to (38), (Halicka and Rosinova, 1994). Employing the corresponding Lagrange function the outlined strategy provides Pi as a solution to
A 12 =
0.3 0.35 0.4 ] 0.15 -.32 0.3 [ 0.32 0.43 0.22
A2l =
0.23 0.35 0.2 ] 0.2 -.35 0.45 [ 0.4 0.34 0.33
A22 =
-5 .1 0.2 0.23-4.5 [ 2 3.1
[
-1
]
--:8
1 1 1]
0.4 22 = 1.2 1.2 1.2 0.4 0.5 0.2
(1 - ai)-l(A~;Adi -A~PiBdi(B~PiBdi)-l B~;Adi] -Pt +]t=O,
oB l
(42)
and feedback matrix Ki=-(B~PiBdi)-lB~;Adi .
(43)
135
=
0.3 0.4 ] 0.2 0.2 [ 0.3 0.35
The robustness properties of the closed-loop system can be characterized by 0.2 0.2 ] 0.2 0.2 [ 0.3 0.3
oB2 =
0.4 To.4 ~ YaQo, OBTOB ~ YbR, Ya
Q
Cl = C2 = [
~ ~ ~] .
Kl = [-0.1494 -1.512 ] -0.5813 -0.6017
eigenvalues of the first nominal closed-loop subsystem are -5 .6234; -6.4222±2.778li . Subsystem 2 The matrix M2 is positive definite for the following parameters a2 = 4.5, a = 1, Yd2 = I, Yb2 = 1, q2 = 8, Q12 = diag{q2}, Q02 = o.4~o.422' R2 = oB{oB2 , a2, a have the analogical meaning as for the first subsystem. The gain matrix
K2 = [-0.0507 -0.6349 ] -0.3419 -0.2795
0
0 ] R =[RIO] 0 R2 .
Q02'
ACKNOWLEDGMENT This research was supported by grant "Adaptive and robust control of dynamic systems".
REFERENCES Chen, YR. (1988). Decentralized robust control system design for large-scale uncertain systems. Int.J.Control, 47, 1195-1205. Halicka,M.,D.Rosinova (1994). Stability robustness bound estimates of discrete-time systems: Analysis and comparison. Int.J. Control, 60, 297-314. Han,M.C.,YH.Chen (1995). Decentralized control design: uncertain systems with strong interconnections. Int.J.Control, 61, 1363-1385. Kozak,S (1995). Simple and robust PlO controller. Applied Mathematics and Computation, 70, 2-8. Kucera, V.,C.E.DeSouza (1995). A necessary and sufficient condition for output feedback stabilizability. Automatica, 31, 1357-1359. Krasovskij,N.N. (1959). Some tasks about theory of stability motion, Fizmatgiz, Moscow 1959. Kwakernaak,H. (1993). Robust control and Hoc optimization - Tutorial paper. Automatica, 29, 255-273. Niculescu,S.I.,C.E.DeSouza,L.Dugard and J.M.Dion (1995). Robust exponential stability of uncertain systems with time-varying delays. Proceedings of 3rdECC'95, Italy, Sept. 1995, 1802-1807. Sezer,M.E.,D.D.Si1jak (1988). Robust stability of discrete systems.int.J.Controi, 48, 2055-2063.
'
eigenvalues of the second nominal closed-loop subsystem are -4.8084, -5.6354±3.8965i. According to the decentralized design procedure the uncertain closed-loop system (10), (11) with output static feedback controller is exponentially stable with decay rate a = 0.2 and the robustness properties are given by following parameters; Yal = 0.4, Ybl = 0.4, Yd2 = 1, Yb2 = 1. For comparison \W provide the results of centralized design procedure as \Wll. The corresponding matrix M defined by (17) for overall system is positive definite for controller parameters -0.655 -0.4988 0.0918 -0.0945
[Qol
= 1.
The decentralized robust control design that guarantees the exponential stability of closed-loop system with specified decay rate is studied. The algorithm for the decentralized output feedback controller design is proposed based on the sufficient closed-loop stability condition proved in this paper. The proposed control strategy aims at determining robust stabilizing controllers on subsystem level for appropriate nominal subsystem decay rate so that the overall system is robustly exponentially stable. Both and discrete-time case is continuous-time considered. The provided procedure does not require matching conditions. The obtained results are illustrated on example.
RI
-0.1956 -0.8277 [ -0.1551 -0.2012
:=
Yh
6. CONCLUSIONS
Results of decentralized design procedure. Subsystem 1 The matrix M 1 given in (17) is positive definite for the following parameter values al = 2.5, a = 0.2 , al is prescribed for nominal subsystem 1 without uncertainties (interconnections) and a is the resulting decay rate when all uncertainties are incl~ Yal = 0.4, Ybl = 0.4, Q11 =diag{qd, ql = 17, QOl = 0.4[10.4 11 , = oB[oB 1 . The gain matrix computed using Algorithm B
K=
o
= I,
-0.1746 0.134l 0.2264 -0.3265 0.0546 -0.5019 . -0.4810 -0.3236
Robust exponential stability decay rate a = 2, decay rate for the nominal model a 1 := 5. The eigenvalues of nominal closed-loop system are -5.7579± 3.836i, -5.9507±2.8218i -5.0881, -4.829. 136
DECENTRALIZED ROBUST CONTROLLER FOR EXPONENTIAL STABILITY OF UNCERTAIN SYSTEMS WITH OUTPUT FEEDBACK
V.Vesely, D.Rosinova Department ofAutomatic Control Systems, Faculty ofElectrical Engineering and Information Technology, Slovak University of Technology, 812 19 Bratislava, Slovak Republic fox: + +421 7729 734, e-mail: veseZy@j«[email protected]
Abstract. The paper addresses the problem of design of robust decentralized controllers with output feedback for interconnected linear systems with uncertainties. A new class of robust continuous-time and discrete-time controllers is proposed to guarantee exponential stability of uncertain linear systems without matching conditions. The uncertainties possibly nonlinear can appear both in the subsystems and in the interconnections between them. The proposed algorithm is applied to 6th order system model comprising two subsystems with decentralized controller.
Key Words. Decentralized control, Robust stability, Uncertain dynamic systems,
prescribed decay rate. The robust exponential stability of complex systems pays an important role to guarantee robustness and dynamic performance quality of closed-loop system. Both the Lyapunov function method and (Krasovskij, 1959) definition of exponential stability of generalized dynamic systems provide useful tools for a design of static output feedback controller for continuous-time case and static state feedback one for discrete-time case, which guarantees the robust exponential stability of closed-loop system for all admissible uncertainties. In the design procedure of robust decentralized controller no matching conditions of continuoustime or discrete-time linear model are considered in contrast with (Niculescu, 1995) and others.
1. IN1R.ODUCTION The design of controllers for many physical processes faces the influence of uncertainties, which very often cause poor performance and even instability of control systems. Considering these facts the stability robustness is a topic of great practical importance which has attracted a lot of interest for several decades, (Kwakernaak, 1993), (Sezer, 1988). If uncertainties arise in a complex system which consists of N interconnected subsystems, (Chen, 1988), (Ban, 1995), they can be classified corresponding to their position in complex system. Firstly, the internal uncertainties, subsystem and input ones, that arise in particular subsystem. Secondly, the external uncertainties that belong to the part of interconnections among subsystems. Treating the uncertainties matching conditions are frequently used. We assume that the internal and external uncertainties upper bounds are known. The paper studies the robust decentralized control problem, the proposed control design strategy aims at closed-loop system exponential stability with a
The paper is organized as folloM. Section 2 brings the problem statement and some preliminary results for continuous-time systems. The main results concerning robust decentralized stabilization for continuous-time systems are provided in Section 3 and for discrete-time systems in Section 4. In Section 5 the proposed algorithm for decentralized
131
controller design is applied to 6th order system model example, the obtained results are presented.
Assumption 2 Assume that for each subsystem the conditions of Theorem 1 are met.
2. PROBLEM STATEMENTCONTINUOUS-TIME CASE
The system (1) can be rewritten in the following compact form x=(A+M)x+(B+oB)u+h(x) y=Cx
Consider an uncertain complex system Xi = (Ai+OAi)xi+(Bi+oBi)Ui+
t
hfj(xj)
where xT = [xr. ... ,x~] E Rn, uT = [ur. ... ,u~] E Rm yT = [yr. ...,Y~] E RI are the state, control input and output vectors of the complex system respectively, A,M,B,oB, C are block diagonal matrices with block elements A;,M;,B;,oBi,C; respectively and h(x} is a vector function with entries hfj(xj). Nominal system model is
j=ljA
Yi=CiXI, (1) where i E.3 = {1,2, ...,N}, Xi E Rnj,u; E Rnt;,y; E RI are the state, control input and output of the i-th subsystem respectively, A;,Bi,CI are matrices of appropriate dimensions which represent the nominal part of the subsystem, matrices OA i, OBi of appropriate dimensions represent internal uncertainties in each subsystem and they are assumed to be piecewise continuous and bounded on every time, the unknown vector functionshfj(xj) are continuous and . . . sufficiently smooth in X} , piecewise contInuous ID time, so that the system (1) has unique continuous solutions for all initial conditions XI(tO) = XiO E Rn; and all piecewise continuous inputs U I E Rm; . The isolated subsystem nominal model is x;=A;x;+B;u; Yi= C;x;
The decentralized robust control design problem is to find a control law in the form u=Ky=KCx, K=diag(K;), K;
oB;oB; ~ Yb;Ri
(4)
1~!i(Xj)lI:s; ;ilIxJ!
(5)
(10)
(11)
is robustly exponentially stable with a prescribed decay rate. The output feedback matrix K comprises the local feedback matrices K; with local controller parameters. Throughout this paper the following concept of robust exponential stability of complex system will be used. Definition 1 The uncertain closed-loop system (11) is said to be robustly exponentially stable with a decay rate'}.. > 0 if the trivial solution x(t}=O is exponentially stable with a decay rate '}.. for all initial conditions x(to) =Xo E Rn and for all admissible uncertainties given by (3),(4) and (5), i.e. there exist constants c(xo) > 0 and '}.. > 0 such that
IlxCt)1I ~ c(xo)e-~
(12)
Definition 1 is for certain subset of considered systems equivalent to Definition 2. Definition 2 The uncertain closed-loop system (11) is said to be robustly exponentially stable with a decay rate IX > 0 if for all initial conditions x(to) =Xo and for all admissible uncertainties (3), (4), (5) there exists a Lyapunov function for a nominal system (9) V(x): Rn ~R+, V(x) belongs to class Cl, such that for its time derivative along the solution of closed-loop system (11) the following inequalities hold
~) ~ -«V(x), V(x(to»
(6)
where Li is the real symmetric nonnegative definite solution of A;L;+LA;-L;B;BfL;+cfc;+ GfG; = 0
Rm;xJ;
x= [A+M +(B+oB)KC]x+h(x)
where QOi E Rn,xn/ R; E Rm'-I are knOVdl symmetric posttlve definite matrices, ;fj ~ 0, Yai ~ 0, 10i ~ 0 are unknOVdl constants to be determined from conditions of closed-loop stability of system with uncertainties, ij E .3. The necessary and sufficient conditions for output feedback stabilizability of each subsystem are given in (Kucera and Souza. 1995). Theorem 1 (Kucera and Souza, 1995) The nominal i-th subsystem (2) is output feedback stabilizable if and only if i. (A;,B;) is stabilizable and (Ai, Ci) is detectable ii. there exist real matrices K; and G i such that KiC;+B;L;=G;
E
such that the closed-loop uncertain system comprising the plant (8) and decentralized controller (10)
System (1) is assumed to satisfy the following assumptions. Assumption 1 The internal uncertainties in each subsystem and the unknOVdl vector functions which represent the interconnections among the subsystems satisfy the inequalities (3)
(9)
x=Ax+Bu+h(x)
(2)
OA;M; ~ YaiQo;
(8)
with
(7)
= Vo
Ylllxlf ~ Vex) :s; Y211x11 2 ~lr) ~ -'Y311x1f
where Yi, i = 1, 2, 3, are positive constants. 132
(13)
*,
*.
+Qi + CfKfRjK/C i = 0
The relationship between constants in definitions 1 and 2 is )., = c(xo) = Definition 3 The system (8) is called robustly exponential/y stabilizable if there exists K such that the closed-loop system (11) is robustly exponentially stable for all admissible uncertainties (3),(4) and (5).
(18)
where Qi;: Qo; + QIi and Ki E Rm/><1 E
are
j
E
R m/><1; ,Re).,M{Aj + 1-1i + BjKiC j ) 5, O}, iE 3 (19)
Re ).,M(X) denotes maximal real part of eigenvalues
Finally, the problem studied in this paper can be stated as follows. Under assumptions 1,2 find a matrix K such that the closed-loop uncertain system (11) is exponentially stable with a prescribed decay rate a and the bounds "fai, "fbj, i E 3 in (3), (4) have the maximum possible values for given Sij,i:t= j, i,j E 3.
of matrix X. Proof. For the time derivative of V(x) defined by (15) along the solution of (11) ~ obtain
'!
N
hy(xj)}
(20)
Owing to Assumption 1 and (18), the upper bound on (20) is
'! 5,-
("fai+"fbj)~]xj-21IPjllllxjll
t
siillxAI}
piJ=1
where P j is a symmetric positive definite matrix, as a candidate Lyapunov function of the uncontrolled nominal model of i-th isolated subsystem (2). The candidate Lyapunov function of the uncontrolled nominal model of the complex system (9) is given by
where 1
..,
oG j = !f:dOAj+pj.f{:d, uFj ;;:.
oBjKjC j
(21) ~
Jf/;/ +Pj,,"fbi
Using the following inequalities
2ab
(15)
The sufficient conditio:i'J for decentralized output feedback robust exponential stabilizability of complex system (1) with a prescribed decay rate a are given in the following theorem. Theorem 2 The uncertain system (8) is robustly exponentially stabilizable with a prescribed decay rate a by the decentralized output feedback (10) if the following sufficient conditions are satisfied. i. Assumptions 1,2. ii. For the decay rate aj > 0 of the i-th subsystem,
(F + G)T(F + G) 5, (1 + &)FTF + (1 + &-1 )GTG, &> 0
(22) the bound on time derivative of Lyapunov function (21) can be rewritten
dV dt 5, -
2
+(l-&il)Pj"fai+
(16)
aj~a,
the symmetric matrices M j are nonnegative definite for some K j where M j = (aj -a)pj+ Qo;(I-&n)+QIi -~(&il"fai+
-211Pj1BlxjU
+&·;:l"!bj)+(I-&a)CfKfR;KjC j -So;l/ , iE 3
2,;
li+(I-&a)
5,
-
C;K;oB;OB;KjCj
Yl>i
t
xfMjxj
I
(23)
r-l
whereM; is given by (17). Condition ii. of Theorem 2 then implies the robust exponential stability of the system (11), that completes the proof. Q
(17)
and &il,&a ~ 0 are constants, 1j E Rn/XII/ are identity matrices, Qli E R n /XIJ / are ~etric positive pi·l
Q
t Sitllxjll}
~1
definite matrices, Soi =NJ,:;,/Pi)
L jRJ=1
(14)
V/(x/)
r-l
+P/(OAj+oBjKjCj)]xj+2xfpi
In this section ~ consider the problem of decentralized control design which guarantees the robust exponential stability of complex uncertain system (1). Let us choose function
t
(xfPjxj+x[P;Xj)=t {xf[(Aj+BjKjCjlpj+
r-l
3. DECENTRALIZED ROBUST CONTROL DESIGN WITH OUIPUT FEEDBACK
VeX) =
=t
s; + N;.I,
For the next developments the following definition is introduced. Definition 4 Consider linear uncertain system (8) with h(xFO. Assume that the matrix A+BKC is Hurwitz (stable).
).,M(Pj ) denotes the maximal eigen~ue of matrix Pi. Matrices Pi are solutions of the Lyapunov matrix equations (Ai+ I!f:!i+BiKiCi)TPi+Pi(Ai + ~11j+BiKiC/) 133
The robustness stability measure of the considered system is defined as mr= min {OA +oBKC e ®: I<'\.4...oBKQI
A + OA + (B + oB)KC is not Hurwitz} (24) where ® is a set of all possible system uncertainties. Assume that the matrices Qo;, Qli,R; and constants Yai, Yb;, a; > a and ;, for ij e .3 are known. Then the analysis of properties of M; implies that the best way to guarantee nonnegative definiteness of M; is to maximize the robustness stability measure for all uncertain isolated subsystems (hij(xj) = 0 in (I» . It is known that to maximize m ri, i e.3 the maximal eigenvalues 'AM(P;) should be minimized where Pi is a solution to (18). To simplify the task, minimization of 'AM(P;) is replaced by minimization of Tr(P;), Tr(X) denotes the trace of matrix X. To minimire Tr(P t ) the Lagrange function is considered on subsystem level. The Lagrange function for the i-tb subsystem
If each M; is nonnegative definite then stop, else increase ai or the eigenvalue of Qli and go to 2). Algorithms A and B yield the "optimal" values of aj,Qli,Yai andYbj which guarantee the nonnegative definiteness ofM;, i=I, ... ,N.
4. DECENTRALIZED ROBUST CONTROL DESIGN FOR DISCRETE-TIME SYSTEMS Consider the complex linear uncertain system
xt(k+ I) = (Adi+ OAdi)xi(k) + (Bdi + OBdi)Uj(k)+
+
L; =miD max Tr{P i + Wi[(Ai+ ?-fIt + BjKtCj)TP.+ Ki.Pi W;
~i = KjCjWjC; +B:PjWjC; =0
(26)
(28)
(29) where Aci=Ai+ ~;li+BtKiCj To find the solution of matrix equations (26)-(28) for i-tb subsystem we propose an iterative algorithm. Algorithm A I) Choose ~l) such that A ci is (Hurwitz) stable matrix. 2) Calculate p~l) and
w,l)
(32)
OA ~OA di :s; YadiQ ai
oB~oB di :s; YbdiQbi
(33)
the interconnections are upper bounded
IIg ij(Xj)/I :s; ;d!fIixJ
from Lyapunov
(34)
Definitions I and 2 for robust exponential stability of continuous-time systems can be modified to discrete-time systems in obvious way.
matrix equations (27) and (26). 3) Calculate the next approximation of ~2) using (28) ~2) = ~l) _ pj(B;p~l)w,l)C; + ~l)Cjw,l)C;)
gij(Xj), ie .3
where Xt eRn;, U; eR"'; are the state and control vector ofi-th subsystem respectively, Adi e Rn;XII;,Bdi e Rn;x",; are constant matrices. The unknown matrices OA di, oB di of appropriate dimensions represent system deterministic uncertainties which are upper bounded and can be time-varying. Assumption 3 The pair (A di, Bdi) is controllable for each subsystem Assumption 4 The uncertainties OA di, oB di satisfy the inequalities
+Pi(A i + ~;lj+BtKjCt)+Qt+C;K;KiCi]} , i e .3 (25) The necessary conditions for optimal solution are I +AciWj + W;A~ = 0
t
piJ=1
The decentralized control is considered as a local constant state feedback
(30)
where p; > 0, check the stability of A ci, if it is unstable, decrease pj . 4) Calculate p~2) and w,2) for ~2). 5) If Tr(p~2) _ p~l» < 0, then p~l) = p~2), W~l) = w,2), K~l) = ~2) , go to 3) else stop. The proposed procedure for the design of decentralized robust controller that ensures the exponential stability of uncertain system (11) with a prescribed decay rate a is summarized in the next algorithm. AlgorithmB I) Choose ai;?! a, Yai and Ybi, ie .3. 2) Use Algorithm A for i=I,2, ... ,N. 3) Check if the matrices M j , i e.3 are nonnegative definite. The coefficients &.1, &a can be chosen
u;(k) =KdjXi(k), ie .3
(35)
Kdi e Rm;Xll; is a constant matrix of controller parameters. The corresponding uncertain closed-loop system is xt(k+ I) = (A di + BdiKi)xi(k) + (OAdi + OBdiK;)x;(k)+
+
t
jRJ:l
gtj(Xj), ie .3
(36)
The aim of robust control design is to achieve robust exponential stability of the closed-loop system (36) with a prescribed decay rate. The sufficient conditions for robust stability of the system (36) are given in Theorem 3. Theorem 3 The uncertain decentrally controlled closed-loop system (36) is robustly exponentially stable with a 134
Equation (42) is a discrete-time counterpart to the solution of (26)-(28) for state feedback (for the case when Cj=]j). Hence robust decentralized controller can be designed using Algorithm B after corresponding modifications.
decay rate a if the follo\\ing sufficient conditions are satisfied i. Assumptions 3,4. ii. There exist constants &il,&a,&j3 > 0 and positive definite matrices Qj, ie .3 , such that matrices M j are positive semidefinite,
M j = (l-aj)Qi + (ai -a)pj -
1 &il (1
+ &a)').M<.Pi )
5. EXAMPLE - DESIGN OF ROBUST DECENTRALIZED CONTROLLER
[(1 + &i3)YadiQai + (1 + &il)YbdiKfQbiK;]-
-&~l(1+&a1)(N-1)
t
Consider the 6th order system described by
Pj;~
(37)
. [ A 11 A 12 x= A21 A22
fr'ii=l
where 1 > ai ~ a > 0, P j matrix equation
is a solution of Lyapunov
+[ 0.40.4
(1- ai)-l(Adi + BdiKi)TPi(Adi +BdiKj)-Pi = -Qi .
11 21
x+ [ B 1
0 ] u+ 0 B2
]
0.4 12 ]x+ [OB1 0.4 22 0
oB0 2
]
U
(38) Proof. The proof is omitted for brevity, its technique is similar to the proof of Theorem 2.
Y
zi(k+ 1) =AdiZj(k) + BdiUj(k) .
yT
2
=LYi yI1
un e R4 ,
[xi xn
and
(39)
All
Let us assume that the nominal model (39) is stable. Consider now the subsystem with uncertainties and perturbations Z i(k+ 1) =AdiZ i(k) + BdiUi(k) +Ji(Z i)
[Cl0 C0]
xT = e R6, UT = [ui e R4 that is n=6, m=4, 1=4
where The concept of robustness bound (see e.g. (Sezer, 1988» will be used in further development. Consider the nominal isolated subsystem model
=
(40)
=
0.4 11
-6.1 0.23 0.5 ] 0.3 -5 -3.5 [ 1.5 2.3 -5.3
=[
f;(Z i) is unknown deterministic upper bounded vector function
1.5 1.5 1.5 ] 1.1 1.23 1.1 0.6 1 1.5
(41) Maximal bound f.li such that for f;(Z t) conforming to (41) the system (40) remains stable is called robustness bound for (39). The concept of robustness bound for discrete-time systems correspond to robustness stability measure defined for continuous-time system in Definition 4. Similarly to the continuous-time case the analysis of (37) indicates the way to make M; positive semidefinite, the corresponding decentralized control design strategy aims at determining feedback matrices K i such that the robustness bound of the nominal subsystems (39) is maximized. The maximization of f.li implies the minimization of ').M<.Pi ) simultaneously with one-dimensional search of the appropriate ai, Pi being a solution to (38), (Halicka and Rosinova, 1994). Employing the corresponding Lagrange function the outlined strategy provides Pi as a solution to
A 12 =
0.3 0.35 0.4 ] 0.15 -.32 0.3 [ 0.32 0.43 0.22
A2l =
0.23 0.35 0.2 ] 0.2 -.35 0.45 [ 0.4 0.34 0.33
A22 =
-5 .1 0.2 0.23-4.5 [ 2 3.1
[
-1
]
--:8
1 1 1]
0.4 22 = 1.2 1.2 1.2 0.4 0.5 0.2
(1 - ai)-l(A~;Adi -A~PiBdi(B~PiBdi)-l B~;Adi] -Pt +]t=O,
oB l
(42)
and feedback matrix Ki=-(B~PiBdi)-lB~;Adi .
(43)
135
=
0.3 0.4 ] 0.2 0.2 [ 0.3 0.35
The robustness properties of the closed-loop system can be characterized by 0.2 0.2 ] 0.2 0.2 [ 0.3 0.3
oB2 =
0.4 To.4 ~ YaQo, OBTOB ~ YbR, Ya
Q
Cl = C2 = [
~ ~ ~] .
Kl = [-0.1494 -1.512 ] -0.5813 -0.6017
eigenvalues of the first nominal closed-loop subsystem are -5 .6234; -6.4222±2.778li . Subsystem 2 The matrix M2 is positive definite for the following parameters a2 = 4.5, a = 1, Yd2 = I, Yb2 = 1, q2 = 8, Q12 = diag{q2}, Q02 = o.4~o.422' R2 = oB{oB2 , a2, a have the analogical meaning as for the first subsystem. The gain matrix
K2 = [-0.0507 -0.6349 ] -0.3419 -0.2795
0
0 ] R =[RIO] 0 R2 .
Q02'
ACKNOWLEDGMENT This research was supported by grant "Adaptive and robust control of dynamic systems".
REFERENCES Chen, YR. (1988). Decentralized robust control system design for large-scale uncertain systems. Int.J.Control, 47, 1195-1205. Halicka,M.,D.Rosinova (1994). Stability robustness bound estimates of discrete-time systems: Analysis and comparison. Int.J. Control, 60, 297-314. Han,M.C.,YH.Chen (1995). Decentralized control design: uncertain systems with strong interconnections. Int.J.Control, 61, 1363-1385. Kozak,S (1995). Simple and robust PlO controller. Applied Mathematics and Computation, 70, 2-8. Kucera, V.,C.E.DeSouza (1995). A necessary and sufficient condition for output feedback stabilizability. Automatica, 31, 1357-1359. Krasovskij,N.N. (1959). Some tasks about theory of stability motion, Fizmatgiz, Moscow 1959. Kwakernaak,H. (1993). Robust control and Hoc optimization - Tutorial paper. Automatica, 29, 255-273. Niculescu,S.I.,C.E.DeSouza,L.Dugard and J.M.Dion (1995). Robust exponential stability of uncertain systems with time-varying delays. Proceedings of 3rdECC'95, Italy, Sept. 1995, 1802-1807. Sezer,M.E.,D.D.Si1jak (1988). Robust stability of discrete systems.int.J.Controi, 48, 2055-2063.
'
eigenvalues of the second nominal closed-loop subsystem are -4.8084, -5.6354±3.8965i. According to the decentralized design procedure the uncertain closed-loop system (10), (11) with output static feedback controller is exponentially stable with decay rate a = 0.2 and the robustness properties are given by following parameters; Yal = 0.4, Ybl = 0.4, Yd2 = 1, Yb2 = 1. For comparison \W provide the results of centralized design procedure as \Wll. The corresponding matrix M defined by (17) for overall system is positive definite for controller parameters -0.655 -0.4988 0.0918 -0.0945
[Qol
= 1.
The decentralized robust control design that guarantees the exponential stability of closed-loop system with specified decay rate is studied. The algorithm for the decentralized output feedback controller design is proposed based on the sufficient closed-loop stability condition proved in this paper. The proposed control strategy aims at determining robust stabilizing controllers on subsystem level for appropriate nominal subsystem decay rate so that the overall system is robustly exponentially stable. Both and discrete-time case is continuous-time considered. The provided procedure does not require matching conditions. The obtained results are illustrated on example.
RI
-0.1956 -0.8277 [ -0.1551 -0.2012
:=
Yh
6. CONCLUSIONS
Results of decentralized design procedure. Subsystem 1 The matrix M 1 given in (17) is positive definite for the following parameter values al = 2.5, a = 0.2 , al is prescribed for nominal subsystem 1 without uncertainties (interconnections) and a is the resulting decay rate when all uncertainties are incl~ Yal = 0.4, Ybl = 0.4, Q11 =diag{qd, ql = 17, QOl = 0.4[10.4 11 , = oB[oB 1 . The gain matrix computed using Algorithm B
K=
o
= I,
-0.1746 0.134l 0.2264 -0.3265 0.0546 -0.5019 . -0.4810 -0.3236
Robust exponential stability decay rate a = 2, decay rate for the nominal model a 1 := 5. The eigenvalues of nominal closed-loop system are -5.7579± 3.836i, -5.9507±2.8218i -5.0881, -4.829. 136