Robust Linear Controller Design: Time Domain. Approach. BOR-SEN CHEN .4ND CHING-CHANG WONG. Abswuct-In this note a new robust linear controller ...
IEEE TRANSACTIONS ON AUTOMATIC CONTROL. VOL. AC-32, NO. 2, FEBRUARY 1987
161
Robust Linear Controller Design: Time Domain Approach BOR-SEN CHEN .4ND CHING-CHANG WONG Abswuct-In this note a new robust linear controller design in the time domain is introduced for multivariable systems with linear or nonlinear time-varying model uncertainties. The Gronwall lemma is employed to investigate the robust stability conditions which are based on the upper norm-bounds of the uncertainties. The parameters of a dynamic controller are selected to satisfy the requirements of robust stability under plant uncertainties.
*
-h
I. INTRODUCTION The main theoretical problems of robust controller design are to find a linear time-invariant feedback controller for the incompletely known plant and to develop design strategies for a linear controller in the presence of severe model uncertainties. Over the last few years, several authors [I][7] have treated this problem from the frequency domain by means of norm or norm-like (the largest singular value) inequalities. In this note, a new robust controller design procedure in the time domain is introduced for the multivariable dynamic systems, with parametrical as well as structural linear or nonlinear time-varying model uncertainties. The Gronwall lemma is employed to investigate the robust stability conditions which are based on the upper norm-bounds of the uncertainties. The parameters of a dynamic controller are selected to satisfy the requirement of robust stability under plant uncertainties. 11. PROBLEM FORMULATION Consider the following multivariable dynamic system with parametrical uncertainties: X=AX+B~+AA(X)+AB(~)
( I ) y=Cx+Du+AC(x)+AD(u),
X
(1 )
wherex E R"I is the state vector, u E R" is the input vector, y E R' is the output vector, and A , B, C, and D are constant matrices. AA(x), AB@), AC(x), and AD@) are nonlinear time-varying parametrical uncertainties with the following known upper norm-bounds. (SeeFig. 1 as an illustration.)
Fig. 1.
Y
U
Fig. 2.
Thus, system ( 5 ) represents the "approximate model" of the plant. Without loss of generality, we will assume that (A, B ) is controllable and ( C , A ) is observable. If the plant uncertainties are unknown and cannot be expressed as parametrical uncertainties in (J) but can be described by the input-output form as a strqctural model evor with an upper bound (see Fig. 2), then X=Ax+Bu
(Ir) y = C x + D u + A h
* u,
x(0) =x0
where the operator * denotes the convolution while the structural model uncertainty Ah is linear time-varying but bounded by the following inequality:
In this note the dynamic controller has the following structure [ 111: X,= A,xr,+ B,y
,=I
where x, E R"2 and A,, B,, K,, and K2 have appropriate dimensions. Thus, our problems are formulated as follows. Problem (I): The first design problem is to choose the parameters A,, jl All = max I A,, 1 (column sum) (4) B,, K , , and K2 in the dynamic controller (8) such that the closed-loop ,=I approximate system of ( 5 ) and (8) is asymptotically stable and at the same time the closed-loop system with uncertainties of (1) and (8) is also where x;, i = 1, 2, . . ., n denotes the element of the vector x and A,, j , j asymptotically stable, Le., the uncertainties can be tolerated in our design = 1, 2, . . . , n denotes the entries of matrix A . and the controller is a robust controller. This system ( I ) yields an approximate dynamic system as follows: Problem (IZ): The second design problem is to choose the dynamic controller (8) such that the closed-loop approximate system of ( 5 ) and (8) is asymptotically stable and at the same time the closed-loop system with uncertainties of (6) and (8) is also asymptotically stable. and the induced matrix norm corresponding to the vector norm is given as
J
Manuscript received May 30. 1986: revised September 23, 1986. This paper is based on a prior submission of iiovember 6 , 1985. The authors are mith the Department of Electrical Engineering, Tatung Institute of Technology. Taipei, Taiwan, Republic of China. IEEE Log Number 8612206.
In. ROBUSTSTABILIZATION In problem (I), the closed-loop system with nonlinear parametrical uncertainties is described by (1) and (8). Combining (1) and (8): we get
0018-9286/87/0200-0161$01.00
0 1987 IEEE
162
IEEE TRANSACTIONS ON AUTOMATIC NO. AC-32,VOL. CONTROL,
2, FEBRUARY 1987
ACx, and A D u , respectively, and these uncertainties are bounded by the following inequalities:
I I U II
y=[C+DKZ DK,]
[zr]
5
01,
IIUII
IIACII 5
&.
[ I m ( (5 04.
5
62, (1 6)
We can also prove that if we choose parameters A,, B,, K I ,and KZ such that the approximate closed-loop system (12) is asymptotically stable and (15)is satisfied, then the linear parametrical perturbed closed-loop system is also asymptotically stable. For problem (II), combining (6) and (8) leads to the following closedloop system with linear model uncertainty Ah
+[AC(x)+AD(u)].
Let us define
L
A
y=C,f+[Ah
Then the closed-loop system with parametrical uncertainties can be represented as
And the approximate closed feedback system is given by
L
* (K,x,+K,x)].
A
(17)
Then we get the following theorem. Theorem 2: In problem m),assume the linear model uncertainty Ah is nom-bounded by (7) and if we can choose the dynamic controller (8) such that approximate closed-loop system (12) is asymptotically stable and
then closed-loop system (17) with model uncertainty is also asymptotically stable. Proof: See Appendix B. Remark: Suppose the model uncertainty Ah is nonlinear and bounded by the following inequality:
Let us define the transition matrix +(t) of (12) as +(t)=expAr
IlWu(t))ll
(13)
5
r’llu(r)li,
(1 9)
then change (17) into
and suppose
r
I I + ( t ) [ [ Im
exp ( - a t )
t 2 0
1
(14) L
for some constants m > 0, cy > 0. In order to satisfy the requirement of (14), we must choose the parameters A,, B,, K , , and K2 of dynamic controller (8) such that all the eigenvalues of A are distinct and in the 1.h.p. It is also seen that -cy = m e Re ( A , @ ) ) , where Xi(A),i = 1, 2, . . ., n denotes the eigenvalues of A . That is, - cy is the real part of the eigenvalue nearest to the imaginary axis. Using the Gronwall lemma [12], [14], we get the following theorem. Theorem I: In problem (I), suppose the nonlinear parametrical uncertainties are bounded by (2) and if we choose the control parameters of (8) such that the approximateclosed-loop system (12) is asymptotically stable and the following inequality is satisfied:
~ ~ > . ~ [ P I + ( P ~ + P ~ I I B ~ I I~)1( 1I Il [l K ) +~P ~ 1 1 ~ ~ l 1 1
(13
then the nonlinear parametrical perturbed closed-loop system (1 1) is also asymptotically stable (;.e., the nonlinear perturbations AA(x), AB@), AC(x), and AD@) can be tolerated). Proof: See Appendix A. Remarks: 1) In the input-output modeling case (Le., transfer function), the nonlinear uncertainties are considered to be static in the loop [9], [IO]. In this case, the nonlinear uncertainties are in a dynamic model. As a result, the approach under investigation will be more practical in the true control system. 2) From the above analysis, it is seen that robustness margins are given by the eigenvalue closest to the imaginary axis. 3) If A, = B, = K l = 0 and Kr = - Fx, i.e., the dynamic controller (8) becomes the state feedback case u = - Fx, then the robust stability condition (15) is reduced to cy > m[& + L ? ~ 1 1 F l 1 ] . 4) The inequality (15) is only a sufficient condition. That is. even if (15) does not hold, we cannot say that no robust controller exists. 5) When the parametrical uncertainties are linear in problem (I), Le., AA(x), AB(u), AC(x), and U ( U in )( I ) are changed to AAx, ABu,
(20)
y=CX+Ah(u)
and this problem is reduced to problem (I) with AA(x) = AB@) = A C ( x ) = 0, and AD@) = Ah(u). From the above analysis, a controller design procedure is proposed as follows: 1) Check the upper norm-bounds ofthe system uncertainties and adequately give negative eigenvalues X,,i = 1, 2, . . ., n, where n = n, + n2. 2) Choose an adequate matrix U to compute A = U - I diag [X, . . . A,]U and get a. (Since - cy is determined by the real part of the eigenvalue of A nearest to the imaginary axis.) 3) Find the corresponding A , , B,, KI,K2 by the eigenvalue assignment technique and check if the robust stability condition is satisfied. If so, go to step 5 ) . 4) Shift eigenvalues to the left by X ; = A, + Ah,, i = 1, 2, . . ., n, where Ah, are negative numbers. and then go to step 2). 5) Obtain the robust controller from (8). Example: Consider the following linear dynamic system: x=
[ -;-4 [; e] x+
u
Suppose this system has the following nonlinear parametrical perturations:
IEEE TRANSACTIONS ON AUTOMATIC CONTROL.
163
VOL. AC-32. NO. 2, FEBRUARY 1987
How do w'e design a robust controller to stabilize the linear dynamic system (21) under the perturbations such as (22)? Solution: From Theorem 1 , we get
lIM(x)ll
5 1.5llxII.
IlWu)ll
5 0.211ull,
l.e., P I = 1 . 5 , &=0.2, 183~0,B4=0.
From the above design algorithm, we get the eigenvalues of A as - 3, -4, - 6 , - 8 (i.e.> 01 = 3) and the dynamical controller as
.=[ -1
0I 1 x , + [
-; 4
Fig. 3. Responses of the nonlinear perturbed system with robust controller in example.
x
then robust stability inequality (15) is satisfied and dynamic controller (23) is a robust controller which can stabilize perturbed system (22). The simulation result is shown in Fig. 3.
IV. CONCLUSIOK
or
In this note, a new robust controller design in the time domain has been introduced for the multivariable system with linear, nonlinear, or timevarying model uncertainties. We have employed the Gronwall inequality to investigate the robust stability condition. Our design work is simplified to choose the dynamical control parameters to satisfy the requirement of robust stability.
APPEXDIX A PROOF OF THEOREM1
From (lo), we get
APPENDIX B PROOF OF THF0Kt:hi
2
From (17). we get r
1
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-32. NO. 2, FEBRUARY 1987
164 Performing the operator
11.1)
to both sides, we get
=M< m.
REFERENCES
I I
and
[31 ]
We get
]
]
1 ] ]
1.e., ] ]
]
If
] ]
N. R. Sandell, Jr.. “Robust stability of linear dynamic systems with application to singular perturbation theory,” Aufomatica, vol. 15, pp. 467470, July 1979. J.C. Doyle and G.Stein. “Multivariable feedback design: Concepts for a classicall modem synthesis,” IEEE Trans. Automat. Contr., vol. AC-26, pp. 4-16, Feb. 1981. M. J. Chen and C. A. Desoer, “Necessary and sufficient condition for robust stability of linear distributed feedback system,” Int. 3. Contr., vol. 35,no. 2, pp. 255-267, 1982. J. B. Cmz. Jr., I. S . Frwdenberg. and D. P. Looze, “A relationship between sensitivity and stability of multivariable feedback system,” IEEE Trans. Automat. Contr., vol. AC-26, pp. 66-74, Feb. 1981. I. Postlethwaite, J. M. Edmunds, and A. G. J. Macfarlane, “Principal gains and principal phases in the analysis of linear multivariable feedback systems.” IEEE Trans. Automat. Contr., vol. AC-26. pp. 3 2 4 6 , Feb. 1981. H. Kimura, “Robust stabilizability fora class of transfer functions,” IEEE Trans. Automat. Contr.. vol. AC-29, pp. 788-793, Sept. 1984. M. G. Safonov, A. J. h u b , and G. L. Hamnan, “Feedback properties of multivariable systems: The role and use of the return difference matrix,” IEEE Trans. Automat. Contr., vol. AC-26, pp. 47-65. Feb. 1981. I. C. Hsu and A. U. Meyer, ModernControlPrinciples and Applications. New York: McGraw-Hill, 1968. C. A. Desoer and M. Vidyasagar, Feedback Systems:Input-OutputProperties. New York: Academic, 1975. M. Vidyasagar. NonlinearSystemsAnalysis. Englewood Cliffs. NJ:F’renticeHall, 1978. C.T.Chen, LinearSystemTheoryandDesign. New York: CBS College Publishing. 1984. E. Hille, Lectures on OrdinaryDijferentialEquafions. London: AddisonWesley, 1969. R. H. Martin. Jr., Nonlinear Operators and Differential Equationsin Banach Space. New York: Wiley, 1976. E. Kreyszig, Introductory Function AnalysiswithApplications. New York: Wiley, 1978.
then
Since
An Algorithm for Computing the Roots of a Complex Polynomial
Performing the operator IlYll
11.11
to both we sides,
JOHN F. AURAND
get
= IIqI llfll + II[ A h * (K*xr+&x11 II.
Abmuct-An algorithm is proposed for determining all the roots of a polynomial with complex coefficients. A description is given of the methodand an example application is included. The method offers guaranteed convergence, a straightforward structure, and excellent performance (including multiple root extraction).
Integrating both sides, we get
imJlyJJ 5, \IC11JJ*JJ 1, * IlCll im llfll 1, IlAhIl llcll 1- llfll dt+ll[G, lm 1- [If11 dt 5
dl+
5
dt+
5
\\[Ah (X&+&x)IlI
dt
X1111
I. INTRODUCTION
II(Klx,+Kzx)ll dt
IIAhII dr
This note presents an algorithm for determining the roots of a polynomial with complex coeffkients, with guaranteed convergence and dt Manuscript received June 24, 19x6: revised September 24, 1986. The author is with the Department of Electrical Engineering and Computer Engineering. Iowa State University, Ames, IA 5001 I. IEEE Log Number 861 1954.
0018-9286/87/0200-0164$01.00
0 1987 IEEE
