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H∞-Optimal Observer-Based Controller Design for Nonlinear Multivariable Systems
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
203, 573]596 Ž1996.
0398
H `-Optimal Observer-Based Controller Design for Nonlinear Multivariable Systems Feng-Hsiag Hsiao Department of Electrical Engineering, Chang Gung College of Medicine and Technology, 259, Wen-Hwa 1st Road, Kwei-San, Taoyuan Shian, Taiwan 333, Republic of China
and Chin E. Lin, Ciann-Dong Yang, and Chia-Yuang Chang Institute of Aeronautics and Astronautics, National Cheng Kung Uni¨ ersity, Tainan, Taiwan 700, Republic of China Submitted by Harold L. Stalford Received September 8, 1994
We propose a novel algorithm to synthesize an H `-optimal observer-based controller for a nonlinear multivariable system. Based on the parametrization of the observer-based controller, a necessary and sufficient condition to achieve robust stability is developed. Moreover, the Nevanlinna]Pick interpolation theory is employed to test the solvability condition of the robust stabilization problem. Optimal robustness can be achieved via the technique of H `-optimization. The exact solution of the model-matching problem induced by the stability criterion is solved by the method of noniterative computation of the H `-norm. Q 1996 Academic Press, Inc.
I. INTRODUCTION In practical design, there may be times that an engineer faces the problem of controlling a feedback system with nonlinear phenomena such as saturation, relays, and dead zones. In the design of a classical linear control system, stability is measured in terms of such quantities as gain and phase margins, but for nonlinear systems they are not defined. Even though the circle criterion and Popov criterion have been shown to be useful for testing stability of a feedback system with one linear and one 573 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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FIG. 1. The nonlinear multivariable system Ž P, R, S, T, N ..
nonlinear element in a unit feedback control loop, it remains difficult to apply these results directly to a nonlinear system with an unstable plant and an observer-based controller feedback loop Žsee Fig. 1.. Moreover, both criteria are merely sufficient stability criteria. As a controlled plant may be unstable in real systems and the observer-based controller has been widely applied to design a control system for its greater flexibility w1x, an alternative consideration of a necessary and sufficient stability criterion is necessary for the practical case. Based on the parametrization of the observer-based controller, we derive a necessary and sufficient criterion of robust stability for nonlinear multivariable systems no matter whether the controlled plants are stable or not. By means of Nevanlinna]Pick interpolation theory, the necessary and sufficient condition to solve such a problem of robust stabilization is also presented. According to the robust stability criterion and the technique of H `-optimization, we synthesized an observer-based controller not only to stabilize the entire system but also to attain optimal robustness. The exact solution of the model-matching problem induced by the stability criterion is solved by the noniterative computational method of the H `norm Žintroduced in Yang and Yeh w2, 3x.. The feature of our work is that the controlled plant has no stable, square, proper, and minimum-phase constraints. In Section II, some preliminary notations and definitions are given. In Section III, a necessary and sufficient condition for robust stability is derived and we formulate the solvability condition of the robust stabilization problem. In Section IV, the optimal robustness is attained by means of the technique of H `-optimization. An algorithm is summarized to obtain the optimal robust controller in Section V. An example is given in Section VI to illustrate the algorithm. Finally, a conclusion is provided.
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II. PRELIMINARY NOTATIONS AND DEFINITIONS Throughout this paper, we use the following notations:
l i Ž A. A9 A*Ž s . s H Ž j v .4 5 H Ž j v .5 ` ReŽ s . ² f, g : C n=m LŽ s .
the ith eigenvalue of matrix A the transpose of A complex conjugate transpose of AŽ s . maximum singular value of H Ž j v . ' max i Ž l i Ž H *Ž j v . H Ž j v ...1r2 Hardy H ` -norm of H Ž j v . ' supv s H Ž j v .4 the real part of s inner product of f and g ' H0` f 9Ž t . g Ž t . dt space of n = m complex matrices the set of proper stable rational matrices
The results to be presented in the following sections rely heavily on ideas from the concepts of norm. For this reason a number of standard definitions are reviewed. DEFINITION 2.1. The set of real measurable n-vector-valued functions of the real variable t defined on w0, `x is denoted by R n, and the space Ln2 is defined by
½
Ln2 ' f f g R n ,
`
H0
5
f 9 Ž t . f Ž t . dt - ` .
Ž 2.1.
The L2 norm of f g Ln2 , denoted by 5 f 5 2 , is defined by 5 f 52 '
`
žH
0
1r2
f 9 Ž t . f Ž t . dt
/
s ² f , f :1r2 .
Ž 2.2.
Remark 1. The symbol 5 f 5, without subscript, is generally used instead of 5 f 5 2 . DEFINITION 2.2. For a continuous nonlinear memoryless causal operator N: Ln2 ª Lm2 , and for two real numbers a and b such that y` - a F b - `, N is considered to be inside the sector a , b 4 Žsee Fig. 2. if N satisfies the following properties: Ži. N Ž0. s 0 Žii. 5Ž Nx y u x .5 F d 5 x 5 x g Ln2 , where u s Ž a q b .r2 is called the center of the sector and d s Ž b y a .r2 is its radius.
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FIG. 2. The nonlinear element N.
DEFINITION 2.3. If there exists a nonnegative constant k 1 such that 5 y 5 F k1 5 r 5 ,
; r g Ln2
Ž 2.3.
then the system described in Fig. 1 is said to be L2-stable. DEFINITION 2.4. A stable proper rational matrix H Ž s . g LŽ s . is inner if H *Ž j v . H Ž j v . s I ;v , and is outer if H Ž s . has full row rank for all s in the open right-hand plane ŽRHP.. Remark 2 w4x. Every square rational matrix H Ž s . g LŽ s . can be factored as Hi Ž s . Ho Ž s . or H1 o Ž s . H1 i Ž s . in which Hi Ž s . Ž H1 i Ž s .. and Ho Ž s . Ž H1 o Ž s .. are square inner and outer, respectively. If H Ž s . is square inner, then its adjoint matrix H adjŽ s . and its determinant detŽ H Ž s .. are also inner.
III. PROBLEM FORMULATION AND STABILITY ANALYSIS Consider the nonlinear multivariable feedback control system Ž P, R, S, T, N . ŽFig. 1., in which the nonlinearity N is a continuous memoryless causal operator and r belongs to Ln2 . The controlled plant P Žwhich may be unstable . is an LTI Žlinear time-invariant . causal operator. The observer-based controller Ž RŽ s ., SŽ s ., T Ž s .. g LŽ s . is to be deter-
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FIG. 3. The linearized system Ž P, R, S, T, u I ..
mined, and Ty1 Ž s . is assumed to exist. Based on this configuration, our problem is to find an H `-optimal observer-based controller, with which both internal stability and optimal robustness can be attained. We first formulate the observer-based controller Ž RŽ s ., SŽ s ., T Ž s .. and then derive a necessary and sufficient condition for robust stability of the nonlinear multivariable system Ž P, R, S, T, N .. Moreover, the solvability condition of the robust stabilization problem is also proposed. The problem of attaining optimal robustness is discussed in the next section. Here, the linearized system of Ž P, R, S, T, N . Žsee Fig. 3 in which N is replaced by u I . is considered. Let P Ž s ., RŽ s ., SŽ s ., and T Ž s . represent the Laplace transforms of the operators P, R, S, and T, respectively. Suppose that u P Ž s . can be decomposed into the form
u P Ž s . s Ay1 Ž s . B Ž s . s B1 Ž s . Ay1 1 Ž s.
Ž 3.1.
in which AŽ s ., A1Ž s ., B Ž s ., and B1Ž s . constitute any left and right coprime, proper, stable, rational factorizations of u P Ž s .. Select X Ž s ., X 1Ž s ., Y Ž s ., and Y1Ž s . g LŽ s . to satisfy the Bezout identity: yY1 Ž s .
X1Ž s .
yB1 Ž s .
X Ž s.
AŽ s .
BŽ s.
A1 Ž s .
Y Ž s.
s
I
0
0
I
.
Ž 3.2.
I 0
0 . I
Ž 3.3.
Since UV s I implies VU s I, Ž3.2. yields yB1 Ž s .
X Ž s.
yY1 Ž s .
X1Ž s .
A1 Ž s .
Y Ž s.
AŽ s .
BŽ s.
s
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From Ž3.3., one sees that A1 Ž s . Y1 Ž s . s Y Ž s . A Ž s . .
Ž 3.4.
Referring to Fig. 1, we have e Ž t . s r Ž t . y Ž Ty1 R . e Ž t . y Ž Ty1 SPN . e Ž t . which implies e Ž t . s Ž T q R q u SP .
y1
Tr Ž t . y Ž T q R q u SP .
y1
SP Ž N y u I . e Ž t . . Ž 3.5.
From Ž3.1., it is easily seen that
Ž T q R q u SP .
y1
s A1 Ly1
Ž 3.6.
where L Ž s . s Ž T Ž s . q R Ž s . . A1 Ž s . q S Ž s . B1 Ž s . . In order to satisfy the requirement of asymptotic stability of the linearized closed-loop system Ž P, R, S, T, u I ., we must assume that LŽ s . has a stable inverse. The following lemma describes the relationship among RŽ s ., SŽ s ., and T Ž s . which satisfies this requirement. LEMMA 1 w5x. The matrix LŽ s . s ŽT Ž s . q RŽ s .. A1Ž s . q SŽ s . B1Ž s . has a stable in¨ erse Ly1 Ž s . if and only if T Ž s . q R Ž s . s L Ž s . X1Ž s . q M Ž s . B Ž s .
Ž 3.7.
S Ž s . s L Ž s . Y1 Ž s . y M Ž s . A Ž s .
Ž 3.8.
and
for any M Ž s . g LŽ s .. Remark 3. Ža. The problem of determining RŽ s ., SŽ s ., and T Ž s . has been reduced to solving for LŽ s . and M Ž s .. Žb. In order to satisfy the realizability of a controller Ži.e., a proper controller., the choice of LŽ s . and M Ž s . must be subject to the constraint that Ty1 Ž s . RŽ s . and Ty1 Ž s . S Ž s . are proper. In the following, a necessary and sufficient condition of robust stability of the nonlinear multivariable system Ž P, R, S, T, N . is derived. Prior to the study of robust stability, a useful concept is given below.
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LEMMA 2 w1x.
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Let H be a linear stable operator, then 5 H 5 2 s 5 H Ž j v .5 ` .
THEOREM 1. For the nonlinear multi¨ ariable system Ž P, R, S, T, N . shown in Fig. 1, suppose that Ža. N lies inside the sector a , b 4 . Žb. The obser¨ er-based controller Ž RŽ s ., S Ž s ., T Ž s .. g LŽ s . is gi¨ en in Ž3.7. and Ž3.8.. Then the nonlinear system Ž P, R, S, T, N . is L2-stable Ž see Definition 2.3. if and only if there exists a K Ž s . ' Ly1 Ž s . M Ž s . such that
d u
Ž Y Ž s . y A1 Ž s . K Ž s . . B Ž s .
`
J r1 - 1 1
Ž 3.9.
or
d u
Ž I y A1Ž s . Ly1 Ž s . Ž T Ž s . q R Ž s . . .
`
J r1 - 1
Ž 3.10.
is satisfied, in which u s Ž a q b .r2 and d s Ž b y a .r2. Proof. See Appendix. Remark 4. Basically, the circle criterion and Popov criterion are merely sufficient stability criteria; moreover, they cannot be directly applied to the cases where P Ž s . is unstable. Before they are employed to check the stability of a nonlinear system with unstable controlled plant, the system must be transformed to an equivalent system with stable elements. The importance of this theorem is that it directly gives a necessary and sufficient criterion for robust stabilization whether the controlled plant is stable or not. Let FŽ s. '
d u
Ž Y Ž s . y A1 Ž s . K Ž s . . B Ž s . ,
Ž 3.11.
then Ž3.9. is equivalent to FŽ s. 1
`
-1
Ž contractive operator. .
Ž 3.12.
Although the selection of AŽ s ., A1Ž s ., B Ž s ., B1Ž s ., X Ž s . and Y Ž s . is nonunique, the value of r 1 is independent of the choice of AŽ s ., A1Ž s ., B Ž s ., B1Ž s ., X Ž s ., and Y Ž s ..
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The inequality in Ž3.12. is equivalent to w6x F* Ž j v . F Ž j v . - I,
;v G 0
Ž 3.13.
or G Ž j v . ' I y F* Ž j v . F Ž j v . ) 0.
Ž 3.14.
Hence the problem of robust stabilization is converted to choosing a stable rational matrix K Ž s . in the stabilizing observer-based controller Ž RŽ s ., S Ž s ., T Ž s .. to meet the requirement of inequality Ž3.14.. If the Ž s . is analytic in Re s G 0., choose controlled plant P Ž s . is stable Ži.e., Ay1 1 K Ž s . s Ay1 1 Ž s. Y Ž s. y
ž
«u d
E Ž s . By1 Ž s .
/
Ž 3.15.
for any real constant « with absolute value less than one Ži.e., < « < - 1. and EŽ s . is an inner matrix whose numerators contain all the zeros of B Ž s . in RHP, then it is always possible to get a stable matrix K Ž s . to satisfy the robustness constraint Ž3.14.. However, for an unstable controlled plant, such a K Ž s . may or may not exist depending on both the ratio of dru Ži.e., the sector of N . and the characteristics of the controlled plant P Ž s .. Hence not every nonlinear feedback system Ž P, R, S, T, N . can satisfy Theorem 1. This implies that there may be no robust controller Ž RŽ s ., SŽ s ., T Ž s .. existing to stabilize the nonlinear feedback system Ž P, R, S, T, N .. To test the solvability condition for robust stabilization, the Nevanlinna]Pick interpolation theory is hence employed. LEMMA 3 ŽNevanlinna]Pick Interpolation Theory w4, 7x.. Suppose g 1 , g 2 , . . . , gn are distinct points in the open RHP, and F1 , . . . , Fn are complex matrices, all g 1 , g 2 , . . . , gn are of the same order, with 5 Fi 5 - 1 ; i. There exists an F Ž s . g LŽ s . such that 5 F Ž s .5 ` - 1 and F Žg i . s Fi ; i if and only if the Pick matrix D 11 .. Ds . Dn1
??? ???
D1 n .. .
Ž 3.16.
Dn n
is positi¨ e definite, with Di j s
I y F˜i Fj
g ˆi q g j
,
Ž 3.17.
F˜i is the complex conjugate transpose of Fi and g ˆi is the complex conjugate of g i .
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Suppose that the controlled plant P Ž s . has no pole on the j v-axis and all unstable poles are simple. Let A1 Ž s . s A1 i Ž s . A1 o Ž s . ,
Ž 3.18.
B Ž s . s Bo Ž s . Bi Ž s . ,
Ž 3.19.
in which A1 i Ž s . Ž Bi Ž s .. and A1 o Ž s . Ž Bo Ž s .. are the inner and outer matrices of A1Ž s . Ž B Ž s ... From the fact that multiplication Žleft or right. by the inner matrix preserves norms w1, 4x, the inequality in Ž3.9. is equivalent to
d u
Ž Y Ž s . y A1 Ž s . K Ž s . . B o Ž s .
`
-1
Ž 3.20.
which implies
d u
adj Ž Aadj 1 i Ž s . Y Ž s . y A1 i Ž s . A1 Ž s . K Ž s . . B o Ž s .
`
-1
Ž 3.21.
Žsince A1 i Ž s . is square inner, Aadj Ž . w x. 1 i s is also inner 4 and hence
d u
adj Ž Aadj 1 i Ž s . Y Ž s . y A1 i Ž s . A1 i Ž s . A1 o Ž s . K Ž s . . B o Ž s .
s
d u
`
Ž Aadj 1 i Ž s . Y Ž s . y det Ž A1 i Ž s . . A1 o Ž s . K Ž s . . Bo Ž s .
`
-1
Ž 3.22. Žsince Aadj Ž . Ž . Ž Ž .. . 1 i s A1 i s s det A1 i s I . Let F Ž s. '
d u
Ž Aadj 1 i Ž s . Y Ž s . Bo Ž s . y det Ž A1 i Ž s . . A1 o Ž s . K Ž s . Bo Ž s . . Ž 3.23 .
then Ž3.22. is equivalent to F Ž s.
`
- 1.
Ž 3.24.
Suppose that g i , i s 1, . . . , n, are the distinct zeros of detŽ A1 i Ž s .. in RHP Žor equivalently, the unstable poles of the controlled plant P Ž s .., then Fi s F Ž g i . s
d u
Aadj 1 i Ž g i . Y Ž g i . Bo Ž g i . .
Ž 3.25.
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From the above discussion, the solvability condition for robust stabilization can be described as the following corollary: COROLLARY 1. If the controlled plant P Ž s . is stable, there always exists a stable matrix K Ž s . satisfying the robustness constraint Ž3.14. if and only if Fi in Ž3.25. satisfies the constraint 5 Fi 5 - 1 and the matrix D defined by Ž3.16., Ž3.17., and Ž3.25. is positi¨ e definite. Remark 5. Corollary 1 treats only the solvability of K Ž s . for controlled plants of which the unstable poles are simple. Similar results apply even in the case in which the controlled plant has repeated unstable poles w4x. Comment. The solvability condition of the robust stabilization problem can also be verified by attaining the minimum value of r 1 , which is determined by the technique of H ` -optimization Žsee Section 4.. However, the procedure of noniterative computational method of H `-optimization is complicated. Consequently, it may be in practice more efficient and more rapid to test the solvability condition for robust stabilization by employing Corollary 1 rather than using the technique of H `-optimization. IV. ROBUSTNESS OPTIMIZATION IN H `-NORM In this section, following the same procedure as that in our previous work w8x, we formulate the H ` -optimal control design into the problem of optimal robustness. Then a method of noniterative computation of optimal H `-norm is proposed to solve the model-matching problem introduced in Subsection 4.2. 4.1. H `-Optimal Robustness Problem We propose here the complete design procedure to synthesize the optimal controller to satisfy Ž3.9. with optimal robustness. To ensure stability, Ž3.9. must be satisfied. Indeed, the left-hand-side term of Ž3.9. can be considered a measure of robustness. The optimal robustness can be attained by finding the optimal K˜Ž s . such that
d
Ž Y Ž s . y A1Ž s . K˜Ž s . . B Ž s . u
`
s inf
K Ž s.
Jr
d u
Ž Y Ž s . y A1 Ž s . K Ž s . . B Ž s .
`
Ž 4.1.
holds. Thus, by taking back the previous steps in Section 3, we find exactly what T Ž s . q RŽ s . and SŽ s . are. However, it is nearly impossible to characterize uniquely the exact form of the controllers under only the
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consideration of optimal robustness. Nevertheless, to achieve greater flexibility of this controller, we separate T Ž s . and RŽ s . according to the performance specification such as the tracking property. The minimization of Ž4.1. can be considered a 1-block H ` -optimization problem. 4.2. Non-iterati¨ e Computation of Optimal H `-Norm The problem of optimal robustness in Subsection 4.1 leads to the following standard H `-optimization problem: inf T1 Ž s . y T2 Ž s . K Ž s . T3 Ž s .
K Ž s.
`
sr
Ž 4.2.
in which T1 Ž s . s T2 Ž s . s
d u d u
Y Ž s. BŽ s.
Ž 4.2a.
A1 Ž s .
Ž 4.2b.
T3 Ž s . s B Ž s . .
Ž 4.2c.
Here, a general approach is proposed to manage this problem. The noniterative approach presented in Yang and Yeh w2, 3x has been found suitable for our present requirement. The design algorithm is stated as follows: Step a. Reform the stability criterion Ž4.1. to the 1-block H ` -optimization problem: inf T1 Ž s . y T2 Ž s . K Ž s . T3 Ž s .
`
K Ž s.
s inf T1 Ž s . y T2 i Ž s . T2 o Ž s . K Ž s . T3 o Ž s . T3 i Ž s .
`
s inf T1 Ž s . T3Ui Ž s . y T2 i Ž s . T2 o Ž s . K Ž s . T3 o Ž s .
`
K Ž s.
K Ž s.
s inf
K 9Ž s .
T1X Ž s . y T2X Ž s . K 9 Ž s .
s T1X Ž s . y T2X Ž s . K˜9 Ž s .
`
`
Ž 4.3.
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in which T3 Ž s . s T3 o Ž s . T3 i Ž s . : the co-inner]outer factorization of T3 Ž s . Ž 4.3a . T2 Ž s . s T2 i Ž s . T2 o Ž s . : the inner]outer factorization of T2 Ž s . Ž 4.3b . T1X Ž s . s T1 Ž s . T3Ui Ž s . s
d u
Y Ž s . Bo Ž s .
Ž 4.3c.
T2X Ž s . s T2 i Ž s . s A1 i Ž s . K 9 Ž s . s T2 o Ž s . K Ž s . T3 o s
d u
Ž 4.3d.
A1 o Ž s . K Ž s . B o Ž s . .
Ž 4.3e.
Before we apply the noniterative approach to find r , we need first to transform the formulation to the z-domain via the bilinear transformation s s Ž1 q z .rŽ1 y z .. Hence, we have T1X Ž z . y T2X Ž z . K 9 Ž z .
`
s T1X Ž z . y adj Ž T2X Ž z . . s adj Ž
T2X
Ž z. .
T1X
y1
det Ž T2X Ž z . . K 9 Ž z . T2X
Ž z . y det Ž Ž z . . K 9 Ž z .
s F Ž z . y hŽ z . K 9Ž z .
`
`
Ž 4.4.
`
where F Ž z . s adj Ž T2X Ž z . . T1X Ž z . s
d u
adj Ž A1 i Ž z . . Y Ž z . Bo Ž z .
Ž 4.4a.
and h Ž z . s det Ž T2X Ž z . . s det Ž A1 i Ž z . . .
Ž 4.4b.
hŽ z . is expressed in a general form as hŽ z . s
h 0 q h1 z q ??? qh k z k h k q h ky1 z q ??? qh 0 z k
.
Ž 4.5.
The companion matrix Ch of hŽ z . is defined by
Ch s
0 0 .. .
1 0 .. .
0 1 .. .
??? ???
0 0 .. .
0 yh 0rh k
0 yh1rh k
0 yh 2rh k
??? ???
1
yh ky1rh k
. Ž 4.6.
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Step b. Form matrix J and perform the Cholesky decomposition Ik y J *J s Q o QUo
Ž 4.7.
to obtain Q o , in which h0
h1 h0
Js 0
h2 h1 .. .
??? ??? .. . .. .
h ky1 h ky2 .. . h1
hk
h ky1 hk
h ky2 h ky1 .. .
0
h0
??? ??? .. . .. .
h1 h2 .. .
y1
.
h ky1 hk
Ž 4.8 . Step c. Perform the singular value decomposition 1 f s Ž Im m QU0 . F Ž ChT . Ž Im m QUy . 0
s U SV *
Ž 4.9.
Eventually, the minimum norm of Ž4.1. is found as the maximum singular value of f , i.e., r s s Ž f .. At this stage, we have already known the value of r . Thus, by the Nevanlinna]Pick algorithm w9x we can solve for the optimal K˜9Ž s ., which in turn can be used to find the optimal K˜Ž s .. Consequently, both LŽ s . and M Ž s . can then be solved. Moreover, we can solve for T Ž s . q RŽ s . and SŽ s . from Lemma 1.
V. ALGORITHM From the above analysis, our design procedure is summarized in the following algorithm. Problem. Given P Ž s . and N, how can we synthesize a robust H ` -optimal controller and then find the corresponding control parameters RŽ s ., SŽ s ., and T Ž s .? According to the following steps, we solve this problem. Step 1. Find AŽ s ., B Ž s ., A1Ž s ., B1Ž s ., X Ž s ., X 1Ž s ., Y Ž s ., and Y1Ž s . from Ž3.1. as well as Ž3.2. and using Corollary 1 to test the solvability of K Ž s . for the robustness constraint in Ž3.14.. If yes, go to Step 2. Otherwise, it means that no robust observer-based controller can stabilize the system and we stop. Step 2. Solve for T1X Ž s . and T2X Ž s . from Ž4.3c. and Ž4.3d.. Step 3. Obtain T1X Ž z . and T2X Ž z . by the bilinear transformation s s Ž1 q z .rŽ1 y z ..
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Step 4. Use the noniterative approach to find the minimum norm of Ž4.1.. Step 5. Find the optimal K˜9Ž s ., which can in turn be used to find ˜ K Ž s . from Ž4.3e.. Step 6. Choose LŽ s . and M Ž s . to satisfy the factorization K˜Ž s . s y1 Ž . L s M Ž s .. Step 7. Substitute LŽ s . and M Ž s . into Ž3.7. and Ž3.8.; we can then solve for T Ž s . q RŽ s . and SŽ s .. Step 8. We have more freedom to choose Ž RŽ s ., S Ž s ., T Ž s .. to adjust other system performance.
VI. EXAMPLE Let us consider the nonlinear system shown in Fig. 1 with the controlled plant:
P Ž s. s
sy2
y Ž s y 2.
Ž s y 3. Ž s q 2.
Ž s y 3. Ž s q 2. sy2
0
Ž 6.1.
Ž s y 3. Ž s q 2.
and the nonlinear element N s diag n, n4 , in which n satisfies
¡2 e,
nŽ e . s
~
3 1
q
0FeF2 1
e,
3 2 ¢yn Ž ye . ,
Ž 6.2.
e)2 e-0
From Definition 2.2, it is obvious that the nonlinearity N lies inside the sector 12 , 23 4 , that is,
us
7 12
and
ds
1 12
.
Ž 6.3.
Our goal is to synthesize a robust observer-based controller that not only stabilizes the entire feedback system but also attains optimal robustness. Solution. We solve the problem by following step by step the algorithm proposed in the last section.
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Step 1. Performing the coprime factorization Ž3.1. and computing the Bezout identity Ž3.2., we have
A Ž s . s A1 Ž s . s
B Ž s . s B1 Ž s . s
7
Ž s y 3. Ž s q 2. Ž s q 1. Ž s q 6.
0
0
Ž s y 3. Ž s q 2. Ž s q 1. Ž s q 6.
Ž s y 2. Ž s q 1. Ž s q 6.
12
0
,
Ž 6.4.
y Ž s y 2.
Ž s q 1. Ž s q 6. , Ž 6.5. Ž s y 2. Ž s q 1. Ž s q 6.
and s y 20
0
sq1
X Ž s . s X1Ž s . s
s y 20
0
Y Ž s . s Y1 Ž s . s
,
sq1
29 s q 57
29 s q 57
sq1
sq1 29 s q 57
12 7
Ž 6.6.
0
.
Ž 6.7.
sq1
Factorize A1Ž s ., B Ž s . as Ž3.18. and Ž3.19., respectively. Thus sy3 A1 i Ž s . s
sq3 0
A1 o Ž s . s
0 sy3
,
sq3
Ž 6.8.
Ž s q 2. Ž s q 3. Ž s q 1. Ž s q 6.
o
0
Ž s q 2. Ž s q 3. Ž s q 1. Ž s q 6.
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and
Bo Ž s . s
7
Ž s q 2. Ž s q 1. Ž s q 6.
12
y Ž s q 2.
Ž s q 1. Ž s q 6. , Ž s q 2. Ž s q 1. Ž s q 6.
0 sy2
Bi Ž s . s
Ž 6.9.
0
sq2
.
sy2
0
sq2
Substitute equations Ž6.3. ] Ž6.9. into Corollary 1 and then the Pick matrix D is shown to be positive definite. This result means that the robust stabilization problem is solvable. Step 2. From Ž4.3c. as well as Ž4.3d., we have T1X Ž s . s s
d u
Y Ž s . Bo Ž s .
1 Ž s q 2 . Ž 29 s q 57 . 1 0 7 Ž s q 1. 2 Ž s q 6.
0 1
Ž 6.10.
and T2X Ž s . s A1 i Ž s . s
sy3 1 sq3 0
0 1
Ž 6.11.
Step 3. By the bilinear transformation, we obtain T1X Ž z . s
1 Ž 43 y 14 z . Ž 3 y z . Ž 1 y z . 1 0 7 Ž 14 y 10 z . T2X Ž z . s
y1 q 2 z 1 0 2yz
0 , 1
0 . 1
Ž 6.12. Ž 6.13.
Step 4. h Ž z . s det Ž T2X Ž z . . s adj Ž T2X Ž z . . s
1 y 4 z q 4 z2 4 y 4 z q z2
y1 q 2 z 1 0 2yz
0 , 1
,
Ž 6.14. Ž 6.15.
H
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-OPTIMAL
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589
thus 0 1 y 4
Ch s
1
Ž 6.16.
1
F Ž z . s adj Ž T2X Ž z . . T1X Ž z . s
1 Ž 43 y 14 z . Ž 3 y z . Ž 1 y z . Ž y1 q 2 z . 1 0 7 Ž 14 y 10 z . Ž 2 y z .
0 . Ž 6.17. 1
From Ž4.8., we have 1 Js
4
3
y
0
4 1
.
Ž 6.18.
4
Performing the Cholesky decomposition Ž4.7. gives:
'15 Qo s
0
4
'15
3'15
20
20
.
Ž 6.19.
Substituting Eqs. Ž6.16. ] Ž6.19. into Ž4.9. yields the singular value decomposition:
f s U SV * y2.0000 4.0000 s 0 7 0 1
y1.0000 1.9998 0 0
0 0 y2.0000 4.0000
0 0 y1.0000 1.9998
Ž 6.20.
with S s 17 diag 4.9999 4.9999 0.0001 0.00014 . Thus, the maximum singular value of f is
s Žf. s r (
5 7
- 1.
Ž 6.21.
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Step 5. There must exist a diagonal all-pass function matrix H Ž s . g LŽ s . such that H Ž s.
`
T1X Ž s . y T2X Ž s . K 9 Ž s .
s inf
K 9Ž s .
s T1X Ž s . y T2X Ž s . K˜9 Ž s .
`
s
`
5 7
.
Ž 6.22.
As T2X Ž s . has only one unstable zero 3 and T2X Ž s . is all-pass, the optimal all-pass function matrix H Ž s . is Žby the Nevanlinna]Pick algorithm. H Ž s. s
5 7
I2 .
Ž 6.23.
Then the optimal K˜9Ž s . is found: K˜9 Ž s . s
y Ž 5s 2 q 26 s q 28 . Ž s q 3 . 1 2 0 7 Ž s q 1. Ž s q 6.
0 . 1
Ž 6.24.
From Ž4.3e., K˜9 Ž s . s T2 o Ž s . K˜Ž s . T3 o Ž s . s
1 7
A1 o Ž s . K˜Ž s . Bo Ž s . .
Ž 6.25.
Thus, we obtain K˜Ž s . s y
12 Ž 5s 2 q 26 s q 28 . Ž s q 6 . 1 2 0 7 Ž s q 2.
1 . 1
Ž 6.26.
Step 6. Perform the factorization K˜Ž s . s Ly1 Ž s . M Ž s . and then choose LŽ s . s y
7 Ž s q 2.
1 12 Ž 5s 2 q 26 s q 28 . 0
0 1
Ž 6.27.
and M Ž s. s
sq6 1 sq2 0
1 . 1
Ž 6.28.
Step 7. From Ž3.7. and Ž3.8., we have T Ž s. q RŽ s. s
7 Ž s q 1. Ž s q 6.
1 3 Ž s q 2 . Ž 5s q 26 s q 28 . 0 2
0 1
Ž 6.29a.
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-OPTIMAL
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and y5 Ž s q 1 . Ž s q 6 . 1 Ž 5s 2 q 26 s q 28 . 0
SŽ s. s
1 . 1
Ž 6.29b.
Step 8. Here, we have more freedom to choose T Ž s . and RŽ s . in Ž6.29a. to adjust other system performance. For instance, T Ž s . can be chosen such that lim s ª 0 Ž I y B1Ž s . Ly1 Ž s .T Ž s .. s 0 which has a good tracking property Ži.e., this transfer matrix must have zero at s s 0 for tracking step input or a double zero at s s 0 for ramp input.. In this example, in order to track the unit step, we choose
T Ž s. s
Ž s q 1. Ž s q 6.
1 2 5s q 26 s q 28 Ž . 0
1 , 1
Ž 6.30.
and then y Ž s q 1 . Ž s q 6 . Ž 3s y 1 .
y Ž s q 6. Ž s q 1.
3 Ž s q 2 . Ž 5s q 26 s q 28 .
Ž 5s 2 q 26 s q 28 . y Ž s q 1 . Ž s q 6 . Ž 3s y 1 . 3 Ž s q 2 . Ž 5s 2 q 26 s q 28 .
2
RŽ s. s
0
Ž 6.31. Hence the observer-based controller Ž RŽ s ., SŽ s ., T Ž s .. in Ž6.29. ] Ž6.31. not only attains optimal robustness but also achieves the tracking purpose.
VII. CONCLUSION Based on the parametrization of the observer-based controller, a necessary and sufficient condition of robust stabilization is derived for nonlinear multivariable systems no matter whether the controlled plant is stable or not. The solvability of such a robust stabilization problem is also presented by means of Nevanlinna]Pick interpolation theory. According to the criterion of robust stability and the technique of H `-optimization, we have proposed a novel algorithm to synthesize an observer-based controller not only to stabilize the entire system but also to attain optimal robustness.
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APPENDIX If. Equation Ž3.5. can be rewritten as: e Ž t . s A1 Ly1 Tr Ž t . y A1 Ly1 SP Ž N y u I . e Ž t . .
Ž A.1.
From Lemma 1, we have Ly1 S s Y1 y Ly1 MA s Y1 y KA.
Ž A.2.
Substituting ŽA.2. into ŽA.1., we obtain e Ž t . s A1 Ly1 Tr Ž t . y A1 Ž Y1 y KA . P Ž N y u I . e Ž t . s A1 Ly1 Tr Ž t . y uy1 Ž A1Y1 Ay1 y A1 K . B Ž N y u I . e Ž t . s A1 Ly1 Tr Ž t . y uy1 Ž Y y A1 K . B Ž N y u I . e Ž t .
Ž by Ž 3.4. . Ž A.3.
from which e Ž t . F A1 Ly1 Tr Ž t . q uy1 Ž Y y A1 K . B Ž N y u I . e Ž t . F A1 Ly1 T
r Ž t . q uy1 Ž Y y A1 K . B
F A1 Ly1 T
rŽ t. q
d u
Ž Y y A1 K . B
Ž N y u I . eŽ t . eŽ t . .
Ž A.4.
Since A1 Ly1 T and Ž Y y A1 K . B are asymptotically stable, we have Žby Lemma 2. e Ž t . F A1 Ž j v . Ly1 Ž j v . T Ž j v . q
d u
`
rŽ t.
Ž Y Ž j v . y A1 Ž j v . K Ž j v . . B Ž j v .
`
e Ž t . . Ž A.5.
Let A1 Ž j v . Ly1 Ž j v . T Ž j v .
`
s a1 - `
Ž A.6.
and
d u
Ž Y Ž j v . y A1 Ž j v . K Ž j v . . B Ž j v .
`
s a2 - 1
Ž by Ž 3.9. . Ž A.7.
then
Ž 1 y a2 . e Ž t . F a1 r Ž t . « e Ž t . F a3 r Ž t .
Ž A.8.
H
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-OPTIMAL
593
CONTROLLER DESIGN
where a3 '
a1 1 y a2
- `.
Similarly,
˜ Ž t . s PNe Ž t . y Ž t . s Pe s P Ž N y u I . e Ž t . q u Pe Ž t . s Ž I y B1 Ly1 S . P Ž N y u I . e Ž t . q B1 Ly1 Tr Ž t .
Ž from Ž 3.1. , Ž A.1. .
Ž A.9.
Substituting ŽA.2. into ŽA.9., we have y Ž t . s Ž I y B1 Ž Y1 y KA . . P Ž N y u I . e Ž t . q B1 Ly1 Tr Ž t . s Ž I y B1Y1 q B1 KA . P Ž N y u I . e Ž t . q B1 Ly1 Tr Ž t . s Ž XA q B1 KA . P Ž N y u I . e Ž t . q B1 Ly1 Tr Ž t . s uy1 Ž X q B1 K . B Ž N y u I . e Ž t . q B1 Ly1 Tr Ž t .
Ž A.10.
from which y Ž t . F uy1 Ž X q B1 K . B Ž N y u I . e Ž t . q B1 Ly1 Tr Ž t . F uy1 Ž X q B1 K . B F
d u
Ž X q B1 K . B
Ž N y u I . e Ž t . q B1 Ly1 T e Ž t . q B1 Ly1 T
rŽ t. .
rŽ t.
Ž A.11.
Moreover, Ž X q B1 K . B and B1 Ly1 T are asymptotically stable. Then Žby Lemma 2. yŽ t. F
d u
Ž X Ž j v . q B1Ž j v . K Ž j v . . B Ž j v .
q B1 Ž j v . Ly1 Ž j v . T Ž j v .
`
`
eŽ t .
rŽ t. .
Ž A.12.
Let
d u
Ž X Ž j v . q B1Ž j v . K Ž j v . . B Ž j v .
`
s a4 - `
Ž A.13.
and B1 Ž j v . Ly1 Ž j v . T Ž j v .
`
s a5 - `.
Ž A.14.
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Substituting ŽA.8., ŽA.13., and ŽA.14. into ŽA.12., we have y Ž t . F a 4 e Ž t . q a5 r Ž t . F Ž a 3 a 4 q a5 . r Ž t . .
Ž A.15.
It follows that 5 y Ž t .5 F k 1 5 r Ž t .5 in which k 1 s a3 a4 q a5 is a nonnegative finite constant. Only If. By contradiction, suppose that Ž3.10. is false, i.e.,
d
Ž I y A1Ž s . Ly1 Ž s . Ž T Ž s . q R Ž s . . . u
`
s
d u
Ž I y A1 Ly1 Ž T q R . .
G1
Ž A.16.
Žby Lemma 2.. We construct a plant P˜ that is not stabilized by the compensator Ž RŽ s ., S Ž s ., T Ž s ... Rearrange ŽA.3. as
Ž I q uy1 Ž Y y A1 K . B Ž N y u I . . e Ž t . s Ž I q uy1 Ž I y A1 Ly1 Ž T q R . . Ž N y u I . . e Ž t . s A1 Ly1 Tr Ž t .
Ž A.17.
and select constant unitary matrices W and V to achieve the following singular value decomposition, i.e.
s1 0 W I y A1 Ly1 Ž T q R . V s . .. 0
0 s2
??? ..
???
0
.
0 .. . 0
Ž A.18.
sn
in which s 1 G s 2 G ??? G sn are the singular values of w1 y A1 Ly1 ŽT q R .x. By virtue of the facts that the value of 5 A 5 is the largest singular value of A and 5 WA 5 s 5 A 5 for any unitary matrix W, the inequality ŽA.16. thus implies that d s G 1. Ž A.19. u 1 Suppose we define N y u I ' yVWNm Ž ? .
Ž A.20.
where Nm Ž?. is the nonlinear operator within the conic sector yurs 1 , urs 14 which touches the line of slope urs 1 at t s t 1 , i.e., NmŽ eŽ t 1 .. s urs 1 eŽ t 1 . Žsee Fig. 4.. With this assumption, we have Ne Ž t . y u e Ž t . s Nm e Ž t . F
u s1
eŽ t . F d eŽ t .
Ž by Ž A.19. . . Ž A.21.
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-OPTIMAL
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CONTROLLER DESIGN
FIG. 4. The nonlinear operator Nm touching the line with slope urs 1 at t s t1 .
In other words, N is a permissible nonlinear perturbation in our problem. Once the nonlinear operation Nm e touches the line of slope urs 1 at t s t1 ,
I q uy1 Ž I y A1 Ly1 Ž T q R . . Ž N y u I . 4 e Ž t1 .
¡
~
s I y Wy1
¢
s 1ru 0 .. . 0
s e Ž t 1 . y Wy1
s I y Wy1
1 0 .. . 0
1 0 .. . 0
???
0 s 2 ru
..
snru
???
0 .. .
.. ??? 0 s 2rs 1
0 ??? ..
???
0
.
.
¦
¥
WNm e Ž t 1 .
0
0
??? 0 s 2 rs 1
.
0 .. .
§
We Ž t 1 .
0
snrs 1 0 .. . 0
snrs 1
0
W e Ž t1 .
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s Wy1
0 0 .. . 0
0 1 y s 2rs 1
??? ..
???
0
.
0 .. . 0 1 y snrs 1
0
Wy1 e Ž t 1 . . Ž A.22.
This result reflects the fact that I q uy1 Ž I y A1 Ly1 ŽT q R ..Ž N y u I .4 is singular at t s t 1; and from ŽA.17., it is seen that eŽ t 1 . may approach infinity for r / 0. This phenomenon violates the definition of L2-stability. ACKNOWLEDGMENT The authors thank the Defense Technology Coordination Council, Taiwan, R.O.C., for their support of this work under Contract CS-79-0210-D006-24.
REFERENCES 1. M. Vidyasagar, ‘‘Control System Synthesis: A Factorization Approach,’’ MIT Press, Cambridge, MA, 1985. 2. C. D. Yang and F. B. Yeh, A simple algorithm on minimal balanced realization for transfer function matrices, IEEE Trans. Automat. Control, 34 Ž1989., 879]882. 3. C. D. Yang and F. B. Yeh, One-step extension approach to optimal Hankel-norm approximation and H `-optimization problems, in ‘‘Proc. of the 29th IEEE Conference on Decision and Control, Honolulu, Hawaii, 1990,’’ Vol. 5, pp. 2622]2627. 4. M. Vidyasagar and H. Kimura, Robust controllers for uncertain linear multivariable systems, Automatica 22 Ž1986., 85]94. 5. C. H. Lo, ‘‘Necessary and Sufficient Conditions for Robust Stabilization of Observer-Based Compensating Systems,’’ Master’s thesis, Tatung Institute of Technology, Taipei, 1988. 6. P. Delsarte, Y. V. Genin, and Y. Kamp, Schur parametrization of positive definite block-Toeplitz systems, SIAM J. Appl. Math. 36 Ž1979., 35]46. 7. P. Delsarte, Y. V. Genin, and Y. Kamp, The Nevanlinna]Pick problem for matrix-valued functions, SIAM J. Appl. Mat. 36 Ž1979., 47]61. 8. C. E. Lin, F. H. Hsiao, C. D. Yang, and C. Y. Chang, ‘‘Robust Observer-Controller Design in Nonlinear Multivariable Systems: Using Dither as Auxiliary,’’ Defense Technology Coordination Council, CS-79-0210-D006-24, Sep. 1991. 9. B. C. Chang and J. B. Pearson, Optimal disturbance reduction in linear multivariable systems, IEEE Trans. Automat. Control. 29 Ž1984., 880]887. 10. S. Boyd, V. Balakrishnan, and K. Kabamba, A bisection method for computing the H ` of a transfer matrix and related problems, Math. Control Signals Systems 2 Ž1989., 207]219. 11. B. S. Chen, R. W. Liu, J. Murray, and R. Saeks, Feedback system design: The fractional representation approach, IEEE Trans. Automat. Control. 25 Ž1980., 399]412. 12. B. A. Francis, J. W. Helton, and G. Zames, H `-optimal feedback controllers for linear multivariable systems, IEEE Trans. Automat. Control. 29 Ž1984., 888]900.
203, 573]596 Ž1996.
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H `-Optimal Observer-Based Controller Design for Nonlinear Multivariable Systems Feng-Hsiag Hsiao Department of Electrical Engineering, Chang Gung College of Medicine and Technology, 259, Wen-Hwa 1st Road, Kwei-San, Taoyuan Shian, Taiwan 333, Republic of China
and Chin E. Lin, Ciann-Dong Yang, and Chia-Yuang Chang Institute of Aeronautics and Astronautics, National Cheng Kung Uni¨ ersity, Tainan, Taiwan 700, Republic of China Submitted by Harold L. Stalford Received September 8, 1994
We propose a novel algorithm to synthesize an H `-optimal observer-based controller for a nonlinear multivariable system. Based on the parametrization of the observer-based controller, a necessary and sufficient condition to achieve robust stability is developed. Moreover, the Nevanlinna]Pick interpolation theory is employed to test the solvability condition of the robust stabilization problem. Optimal robustness can be achieved via the technique of H `-optimization. The exact solution of the model-matching problem induced by the stability criterion is solved by the method of noniterative computation of the H `-norm. Q 1996 Academic Press, Inc.
I. INTRODUCTION In practical design, there may be times that an engineer faces the problem of controlling a feedback system with nonlinear phenomena such as saturation, relays, and dead zones. In the design of a classical linear control system, stability is measured in terms of such quantities as gain and phase margins, but for nonlinear systems they are not defined. Even though the circle criterion and Popov criterion have been shown to be useful for testing stability of a feedback system with one linear and one 573 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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FIG. 1. The nonlinear multivariable system Ž P, R, S, T, N ..
nonlinear element in a unit feedback control loop, it remains difficult to apply these results directly to a nonlinear system with an unstable plant and an observer-based controller feedback loop Žsee Fig. 1.. Moreover, both criteria are merely sufficient stability criteria. As a controlled plant may be unstable in real systems and the observer-based controller has been widely applied to design a control system for its greater flexibility w1x, an alternative consideration of a necessary and sufficient stability criterion is necessary for the practical case. Based on the parametrization of the observer-based controller, we derive a necessary and sufficient criterion of robust stability for nonlinear multivariable systems no matter whether the controlled plants are stable or not. By means of Nevanlinna]Pick interpolation theory, the necessary and sufficient condition to solve such a problem of robust stabilization is also presented. According to the robust stability criterion and the technique of H `-optimization, we synthesized an observer-based controller not only to stabilize the entire system but also to attain optimal robustness. The exact solution of the model-matching problem induced by the stability criterion is solved by the noniterative computational method of the H `norm Žintroduced in Yang and Yeh w2, 3x.. The feature of our work is that the controlled plant has no stable, square, proper, and minimum-phase constraints. In Section II, some preliminary notations and definitions are given. In Section III, a necessary and sufficient condition for robust stability is derived and we formulate the solvability condition of the robust stabilization problem. In Section IV, the optimal robustness is attained by means of the technique of H `-optimization. An algorithm is summarized to obtain the optimal robust controller in Section V. An example is given in Section VI to illustrate the algorithm. Finally, a conclusion is provided.
H
`
-OPTIMAL
575
CONTROLLER DESIGN
II. PRELIMINARY NOTATIONS AND DEFINITIONS Throughout this paper, we use the following notations:
l i Ž A. A9 A*Ž s . s H Ž j v .4 5 H Ž j v .5 ` ReŽ s . ² f, g : C n=m LŽ s .
the ith eigenvalue of matrix A the transpose of A complex conjugate transpose of AŽ s . maximum singular value of H Ž j v . ' max i Ž l i Ž H *Ž j v . H Ž j v ...1r2 Hardy H ` -norm of H Ž j v . ' supv s H Ž j v .4 the real part of s inner product of f and g ' H0` f 9Ž t . g Ž t . dt space of n = m complex matrices the set of proper stable rational matrices
The results to be presented in the following sections rely heavily on ideas from the concepts of norm. For this reason a number of standard definitions are reviewed. DEFINITION 2.1. The set of real measurable n-vector-valued functions of the real variable t defined on w0, `x is denoted by R n, and the space Ln2 is defined by
½
Ln2 ' f f g R n ,
`
H0
5
f 9 Ž t . f Ž t . dt - ` .
Ž 2.1.
The L2 norm of f g Ln2 , denoted by 5 f 5 2 , is defined by 5 f 52 '
`
žH
0
1r2
f 9 Ž t . f Ž t . dt
/
s ² f , f :1r2 .
Ž 2.2.
Remark 1. The symbol 5 f 5, without subscript, is generally used instead of 5 f 5 2 . DEFINITION 2.2. For a continuous nonlinear memoryless causal operator N: Ln2 ª Lm2 , and for two real numbers a and b such that y` - a F b - `, N is considered to be inside the sector a , b 4 Žsee Fig. 2. if N satisfies the following properties: Ži. N Ž0. s 0 Žii. 5Ž Nx y u x .5 F d 5 x 5 x g Ln2 , where u s Ž a q b .r2 is called the center of the sector and d s Ž b y a .r2 is its radius.
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FIG. 2. The nonlinear element N.
DEFINITION 2.3. If there exists a nonnegative constant k 1 such that 5 y 5 F k1 5 r 5 ,
; r g Ln2
Ž 2.3.
then the system described in Fig. 1 is said to be L2-stable. DEFINITION 2.4. A stable proper rational matrix H Ž s . g LŽ s . is inner if H *Ž j v . H Ž j v . s I ;v , and is outer if H Ž s . has full row rank for all s in the open right-hand plane ŽRHP.. Remark 2 w4x. Every square rational matrix H Ž s . g LŽ s . can be factored as Hi Ž s . Ho Ž s . or H1 o Ž s . H1 i Ž s . in which Hi Ž s . Ž H1 i Ž s .. and Ho Ž s . Ž H1 o Ž s .. are square inner and outer, respectively. If H Ž s . is square inner, then its adjoint matrix H adjŽ s . and its determinant detŽ H Ž s .. are also inner.
III. PROBLEM FORMULATION AND STABILITY ANALYSIS Consider the nonlinear multivariable feedback control system Ž P, R, S, T, N . ŽFig. 1., in which the nonlinearity N is a continuous memoryless causal operator and r belongs to Ln2 . The controlled plant P Žwhich may be unstable . is an LTI Žlinear time-invariant . causal operator. The observer-based controller Ž RŽ s ., SŽ s ., T Ž s .. g LŽ s . is to be deter-
H
`
-OPTIMAL
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CONTROLLER DESIGN
FIG. 3. The linearized system Ž P, R, S, T, u I ..
mined, and Ty1 Ž s . is assumed to exist. Based on this configuration, our problem is to find an H `-optimal observer-based controller, with which both internal stability and optimal robustness can be attained. We first formulate the observer-based controller Ž RŽ s ., SŽ s ., T Ž s .. and then derive a necessary and sufficient condition for robust stability of the nonlinear multivariable system Ž P, R, S, T, N .. Moreover, the solvability condition of the robust stabilization problem is also proposed. The problem of attaining optimal robustness is discussed in the next section. Here, the linearized system of Ž P, R, S, T, N . Žsee Fig. 3 in which N is replaced by u I . is considered. Let P Ž s ., RŽ s ., SŽ s ., and T Ž s . represent the Laplace transforms of the operators P, R, S, and T, respectively. Suppose that u P Ž s . can be decomposed into the form
u P Ž s . s Ay1 Ž s . B Ž s . s B1 Ž s . Ay1 1 Ž s.
Ž 3.1.
in which AŽ s ., A1Ž s ., B Ž s ., and B1Ž s . constitute any left and right coprime, proper, stable, rational factorizations of u P Ž s .. Select X Ž s ., X 1Ž s ., Y Ž s ., and Y1Ž s . g LŽ s . to satisfy the Bezout identity: yY1 Ž s .
X1Ž s .
yB1 Ž s .
X Ž s.
AŽ s .
BŽ s.
A1 Ž s .
Y Ž s.
s
I
0
0
I
.
Ž 3.2.
I 0
0 . I
Ž 3.3.
Since UV s I implies VU s I, Ž3.2. yields yB1 Ž s .
X Ž s.
yY1 Ž s .
X1Ž s .
A1 Ž s .
Y Ž s.
AŽ s .
BŽ s.
s
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From Ž3.3., one sees that A1 Ž s . Y1 Ž s . s Y Ž s . A Ž s . .
Ž 3.4.
Referring to Fig. 1, we have e Ž t . s r Ž t . y Ž Ty1 R . e Ž t . y Ž Ty1 SPN . e Ž t . which implies e Ž t . s Ž T q R q u SP .
y1
Tr Ž t . y Ž T q R q u SP .
y1
SP Ž N y u I . e Ž t . . Ž 3.5.
From Ž3.1., it is easily seen that
Ž T q R q u SP .
y1
s A1 Ly1
Ž 3.6.
where L Ž s . s Ž T Ž s . q R Ž s . . A1 Ž s . q S Ž s . B1 Ž s . . In order to satisfy the requirement of asymptotic stability of the linearized closed-loop system Ž P, R, S, T, u I ., we must assume that LŽ s . has a stable inverse. The following lemma describes the relationship among RŽ s ., SŽ s ., and T Ž s . which satisfies this requirement. LEMMA 1 w5x. The matrix LŽ s . s ŽT Ž s . q RŽ s .. A1Ž s . q SŽ s . B1Ž s . has a stable in¨ erse Ly1 Ž s . if and only if T Ž s . q R Ž s . s L Ž s . X1Ž s . q M Ž s . B Ž s .
Ž 3.7.
S Ž s . s L Ž s . Y1 Ž s . y M Ž s . A Ž s .
Ž 3.8.
and
for any M Ž s . g LŽ s .. Remark 3. Ža. The problem of determining RŽ s ., SŽ s ., and T Ž s . has been reduced to solving for LŽ s . and M Ž s .. Žb. In order to satisfy the realizability of a controller Ži.e., a proper controller., the choice of LŽ s . and M Ž s . must be subject to the constraint that Ty1 Ž s . RŽ s . and Ty1 Ž s . S Ž s . are proper. In the following, a necessary and sufficient condition of robust stability of the nonlinear multivariable system Ž P, R, S, T, N . is derived. Prior to the study of robust stability, a useful concept is given below.
H
LEMMA 2 w1x.
`
-OPTIMAL
CONTROLLER DESIGN
579
Let H be a linear stable operator, then 5 H 5 2 s 5 H Ž j v .5 ` .
THEOREM 1. For the nonlinear multi¨ ariable system Ž P, R, S, T, N . shown in Fig. 1, suppose that Ža. N lies inside the sector a , b 4 . Žb. The obser¨ er-based controller Ž RŽ s ., S Ž s ., T Ž s .. g LŽ s . is gi¨ en in Ž3.7. and Ž3.8.. Then the nonlinear system Ž P, R, S, T, N . is L2-stable Ž see Definition 2.3. if and only if there exists a K Ž s . ' Ly1 Ž s . M Ž s . such that
d u
Ž Y Ž s . y A1 Ž s . K Ž s . . B Ž s .
`
J r1 - 1 1
Ž 3.9.
or
d u
Ž I y A1Ž s . Ly1 Ž s . Ž T Ž s . q R Ž s . . .
`
J r1 - 1
Ž 3.10.
is satisfied, in which u s Ž a q b .r2 and d s Ž b y a .r2. Proof. See Appendix. Remark 4. Basically, the circle criterion and Popov criterion are merely sufficient stability criteria; moreover, they cannot be directly applied to the cases where P Ž s . is unstable. Before they are employed to check the stability of a nonlinear system with unstable controlled plant, the system must be transformed to an equivalent system with stable elements. The importance of this theorem is that it directly gives a necessary and sufficient criterion for robust stabilization whether the controlled plant is stable or not. Let FŽ s. '
d u
Ž Y Ž s . y A1 Ž s . K Ž s . . B Ž s . ,
Ž 3.11.
then Ž3.9. is equivalent to FŽ s. 1
`
-1
Ž contractive operator. .
Ž 3.12.
Although the selection of AŽ s ., A1Ž s ., B Ž s ., B1Ž s ., X Ž s . and Y Ž s . is nonunique, the value of r 1 is independent of the choice of AŽ s ., A1Ž s ., B Ž s ., B1Ž s ., X Ž s ., and Y Ž s ..
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The inequality in Ž3.12. is equivalent to w6x F* Ž j v . F Ž j v . - I,
;v G 0
Ž 3.13.
or G Ž j v . ' I y F* Ž j v . F Ž j v . ) 0.
Ž 3.14.
Hence the problem of robust stabilization is converted to choosing a stable rational matrix K Ž s . in the stabilizing observer-based controller Ž RŽ s ., S Ž s ., T Ž s .. to meet the requirement of inequality Ž3.14.. If the Ž s . is analytic in Re s G 0., choose controlled plant P Ž s . is stable Ži.e., Ay1 1 K Ž s . s Ay1 1 Ž s. Y Ž s. y
ž
«u d
E Ž s . By1 Ž s .
/
Ž 3.15.
for any real constant « with absolute value less than one Ži.e., < « < - 1. and EŽ s . is an inner matrix whose numerators contain all the zeros of B Ž s . in RHP, then it is always possible to get a stable matrix K Ž s . to satisfy the robustness constraint Ž3.14.. However, for an unstable controlled plant, such a K Ž s . may or may not exist depending on both the ratio of dru Ži.e., the sector of N . and the characteristics of the controlled plant P Ž s .. Hence not every nonlinear feedback system Ž P, R, S, T, N . can satisfy Theorem 1. This implies that there may be no robust controller Ž RŽ s ., SŽ s ., T Ž s .. existing to stabilize the nonlinear feedback system Ž P, R, S, T, N .. To test the solvability condition for robust stabilization, the Nevanlinna]Pick interpolation theory is hence employed. LEMMA 3 ŽNevanlinna]Pick Interpolation Theory w4, 7x.. Suppose g 1 , g 2 , . . . , gn are distinct points in the open RHP, and F1 , . . . , Fn are complex matrices, all g 1 , g 2 , . . . , gn are of the same order, with 5 Fi 5 - 1 ; i. There exists an F Ž s . g LŽ s . such that 5 F Ž s .5 ` - 1 and F Žg i . s Fi ; i if and only if the Pick matrix D 11 .. Ds . Dn1
??? ???
D1 n .. .
Ž 3.16.
Dn n
is positi¨ e definite, with Di j s
I y F˜i Fj
g ˆi q g j
,
Ž 3.17.
F˜i is the complex conjugate transpose of Fi and g ˆi is the complex conjugate of g i .
H
`
-OPTIMAL
581
CONTROLLER DESIGN
Suppose that the controlled plant P Ž s . has no pole on the j v-axis and all unstable poles are simple. Let A1 Ž s . s A1 i Ž s . A1 o Ž s . ,
Ž 3.18.
B Ž s . s Bo Ž s . Bi Ž s . ,
Ž 3.19.
in which A1 i Ž s . Ž Bi Ž s .. and A1 o Ž s . Ž Bo Ž s .. are the inner and outer matrices of A1Ž s . Ž B Ž s ... From the fact that multiplication Žleft or right. by the inner matrix preserves norms w1, 4x, the inequality in Ž3.9. is equivalent to
d u
Ž Y Ž s . y A1 Ž s . K Ž s . . B o Ž s .
`
-1
Ž 3.20.
which implies
d u
adj Ž Aadj 1 i Ž s . Y Ž s . y A1 i Ž s . A1 Ž s . K Ž s . . B o Ž s .
`
-1
Ž 3.21.
Žsince A1 i Ž s . is square inner, Aadj Ž . w x. 1 i s is also inner 4 and hence
d u
adj Ž Aadj 1 i Ž s . Y Ž s . y A1 i Ž s . A1 i Ž s . A1 o Ž s . K Ž s . . B o Ž s .
s
d u
`
Ž Aadj 1 i Ž s . Y Ž s . y det Ž A1 i Ž s . . A1 o Ž s . K Ž s . . Bo Ž s .
`
-1
Ž 3.22. Žsince Aadj Ž . Ž . Ž Ž .. . 1 i s A1 i s s det A1 i s I . Let F Ž s. '
d u
Ž Aadj 1 i Ž s . Y Ž s . Bo Ž s . y det Ž A1 i Ž s . . A1 o Ž s . K Ž s . Bo Ž s . . Ž 3.23 .
then Ž3.22. is equivalent to F Ž s.
`
- 1.
Ž 3.24.
Suppose that g i , i s 1, . . . , n, are the distinct zeros of detŽ A1 i Ž s .. in RHP Žor equivalently, the unstable poles of the controlled plant P Ž s .., then Fi s F Ž g i . s
d u
Aadj 1 i Ž g i . Y Ž g i . Bo Ž g i . .
Ž 3.25.
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From the above discussion, the solvability condition for robust stabilization can be described as the following corollary: COROLLARY 1. If the controlled plant P Ž s . is stable, there always exists a stable matrix K Ž s . satisfying the robustness constraint Ž3.14. if and only if Fi in Ž3.25. satisfies the constraint 5 Fi 5 - 1 and the matrix D defined by Ž3.16., Ž3.17., and Ž3.25. is positi¨ e definite. Remark 5. Corollary 1 treats only the solvability of K Ž s . for controlled plants of which the unstable poles are simple. Similar results apply even in the case in which the controlled plant has repeated unstable poles w4x. Comment. The solvability condition of the robust stabilization problem can also be verified by attaining the minimum value of r 1 , which is determined by the technique of H ` -optimization Žsee Section 4.. However, the procedure of noniterative computational method of H `-optimization is complicated. Consequently, it may be in practice more efficient and more rapid to test the solvability condition for robust stabilization by employing Corollary 1 rather than using the technique of H `-optimization. IV. ROBUSTNESS OPTIMIZATION IN H `-NORM In this section, following the same procedure as that in our previous work w8x, we formulate the H ` -optimal control design into the problem of optimal robustness. Then a method of noniterative computation of optimal H `-norm is proposed to solve the model-matching problem introduced in Subsection 4.2. 4.1. H `-Optimal Robustness Problem We propose here the complete design procedure to synthesize the optimal controller to satisfy Ž3.9. with optimal robustness. To ensure stability, Ž3.9. must be satisfied. Indeed, the left-hand-side term of Ž3.9. can be considered a measure of robustness. The optimal robustness can be attained by finding the optimal K˜Ž s . such that
d
Ž Y Ž s . y A1Ž s . K˜Ž s . . B Ž s . u
`
s inf
K Ž s.
Jr
d u
Ž Y Ž s . y A1 Ž s . K Ž s . . B Ž s .
`
Ž 4.1.
holds. Thus, by taking back the previous steps in Section 3, we find exactly what T Ž s . q RŽ s . and SŽ s . are. However, it is nearly impossible to characterize uniquely the exact form of the controllers under only the
H
`
-OPTIMAL
583
CONTROLLER DESIGN
consideration of optimal robustness. Nevertheless, to achieve greater flexibility of this controller, we separate T Ž s . and RŽ s . according to the performance specification such as the tracking property. The minimization of Ž4.1. can be considered a 1-block H ` -optimization problem. 4.2. Non-iterati¨ e Computation of Optimal H `-Norm The problem of optimal robustness in Subsection 4.1 leads to the following standard H `-optimization problem: inf T1 Ž s . y T2 Ž s . K Ž s . T3 Ž s .
K Ž s.
`
sr
Ž 4.2.
in which T1 Ž s . s T2 Ž s . s
d u d u
Y Ž s. BŽ s.
Ž 4.2a.
A1 Ž s .
Ž 4.2b.
T3 Ž s . s B Ž s . .
Ž 4.2c.
Here, a general approach is proposed to manage this problem. The noniterative approach presented in Yang and Yeh w2, 3x has been found suitable for our present requirement. The design algorithm is stated as follows: Step a. Reform the stability criterion Ž4.1. to the 1-block H ` -optimization problem: inf T1 Ž s . y T2 Ž s . K Ž s . T3 Ž s .
`
K Ž s.
s inf T1 Ž s . y T2 i Ž s . T2 o Ž s . K Ž s . T3 o Ž s . T3 i Ž s .
`
s inf T1 Ž s . T3Ui Ž s . y T2 i Ž s . T2 o Ž s . K Ž s . T3 o Ž s .
`
K Ž s.
K Ž s.
s inf
K 9Ž s .
T1X Ž s . y T2X Ž s . K 9 Ž s .
s T1X Ž s . y T2X Ž s . K˜9 Ž s .
`
`
Ž 4.3.
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HSIAO ET AL.
in which T3 Ž s . s T3 o Ž s . T3 i Ž s . : the co-inner]outer factorization of T3 Ž s . Ž 4.3a . T2 Ž s . s T2 i Ž s . T2 o Ž s . : the inner]outer factorization of T2 Ž s . Ž 4.3b . T1X Ž s . s T1 Ž s . T3Ui Ž s . s
d u
Y Ž s . Bo Ž s .
Ž 4.3c.
T2X Ž s . s T2 i Ž s . s A1 i Ž s . K 9 Ž s . s T2 o Ž s . K Ž s . T3 o s
d u
Ž 4.3d.
A1 o Ž s . K Ž s . B o Ž s . .
Ž 4.3e.
Before we apply the noniterative approach to find r , we need first to transform the formulation to the z-domain via the bilinear transformation s s Ž1 q z .rŽ1 y z .. Hence, we have T1X Ž z . y T2X Ž z . K 9 Ž z .
`
s T1X Ž z . y adj Ž T2X Ž z . . s adj Ž
T2X
Ž z. .
T1X
y1
det Ž T2X Ž z . . K 9 Ž z . T2X
Ž z . y det Ž Ž z . . K 9 Ž z .
s F Ž z . y hŽ z . K 9Ž z .
`
`
Ž 4.4.
`
where F Ž z . s adj Ž T2X Ž z . . T1X Ž z . s
d u
adj Ž A1 i Ž z . . Y Ž z . Bo Ž z .
Ž 4.4a.
and h Ž z . s det Ž T2X Ž z . . s det Ž A1 i Ž z . . .
Ž 4.4b.
hŽ z . is expressed in a general form as hŽ z . s
h 0 q h1 z q ??? qh k z k h k q h ky1 z q ??? qh 0 z k
.
Ž 4.5.
The companion matrix Ch of hŽ z . is defined by
Ch s
0 0 .. .
1 0 .. .
0 1 .. .
??? ???
0 0 .. .
0 yh 0rh k
0 yh1rh k
0 yh 2rh k
??? ???
1
yh ky1rh k
. Ž 4.6.
H
`
-OPTIMAL
585
CONTROLLER DESIGN
Step b. Form matrix J and perform the Cholesky decomposition Ik y J *J s Q o QUo
Ž 4.7.
to obtain Q o , in which h0
h1 h0
Js 0
h2 h1 .. .
??? ??? .. . .. .
h ky1 h ky2 .. . h1
hk
h ky1 hk
h ky2 h ky1 .. .
0
h0
??? ??? .. . .. .
h1 h2 .. .
y1
.
h ky1 hk
Ž 4.8 . Step c. Perform the singular value decomposition 1 f s Ž Im m QU0 . F Ž ChT . Ž Im m QUy . 0
s U SV *
Ž 4.9.
Eventually, the minimum norm of Ž4.1. is found as the maximum singular value of f , i.e., r s s Ž f .. At this stage, we have already known the value of r . Thus, by the Nevanlinna]Pick algorithm w9x we can solve for the optimal K˜9Ž s ., which in turn can be used to find the optimal K˜Ž s .. Consequently, both LŽ s . and M Ž s . can then be solved. Moreover, we can solve for T Ž s . q RŽ s . and SŽ s . from Lemma 1.
V. ALGORITHM From the above analysis, our design procedure is summarized in the following algorithm. Problem. Given P Ž s . and N, how can we synthesize a robust H ` -optimal controller and then find the corresponding control parameters RŽ s ., SŽ s ., and T Ž s .? According to the following steps, we solve this problem. Step 1. Find AŽ s ., B Ž s ., A1Ž s ., B1Ž s ., X Ž s ., X 1Ž s ., Y Ž s ., and Y1Ž s . from Ž3.1. as well as Ž3.2. and using Corollary 1 to test the solvability of K Ž s . for the robustness constraint in Ž3.14.. If yes, go to Step 2. Otherwise, it means that no robust observer-based controller can stabilize the system and we stop. Step 2. Solve for T1X Ž s . and T2X Ž s . from Ž4.3c. and Ž4.3d.. Step 3. Obtain T1X Ž z . and T2X Ž z . by the bilinear transformation s s Ž1 q z .rŽ1 y z ..
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Step 4. Use the noniterative approach to find the minimum norm of Ž4.1.. Step 5. Find the optimal K˜9Ž s ., which can in turn be used to find ˜ K Ž s . from Ž4.3e.. Step 6. Choose LŽ s . and M Ž s . to satisfy the factorization K˜Ž s . s y1 Ž . L s M Ž s .. Step 7. Substitute LŽ s . and M Ž s . into Ž3.7. and Ž3.8.; we can then solve for T Ž s . q RŽ s . and SŽ s .. Step 8. We have more freedom to choose Ž RŽ s ., S Ž s ., T Ž s .. to adjust other system performance.
VI. EXAMPLE Let us consider the nonlinear system shown in Fig. 1 with the controlled plant:
P Ž s. s
sy2
y Ž s y 2.
Ž s y 3. Ž s q 2.
Ž s y 3. Ž s q 2. sy2
0
Ž 6.1.
Ž s y 3. Ž s q 2.
and the nonlinear element N s diag n, n4 , in which n satisfies
¡2 e,
nŽ e . s
~
3 1
q
0FeF2 1
e,
3 2 ¢yn Ž ye . ,
Ž 6.2.
e)2 e-0
From Definition 2.2, it is obvious that the nonlinearity N lies inside the sector 12 , 23 4 , that is,
us
7 12
and
ds
1 12
.
Ž 6.3.
Our goal is to synthesize a robust observer-based controller that not only stabilizes the entire feedback system but also attains optimal robustness. Solution. We solve the problem by following step by step the algorithm proposed in the last section.
H
`
-OPTIMAL
587
CONTROLLER DESIGN
Step 1. Performing the coprime factorization Ž3.1. and computing the Bezout identity Ž3.2., we have
A Ž s . s A1 Ž s . s
B Ž s . s B1 Ž s . s
7
Ž s y 3. Ž s q 2. Ž s q 1. Ž s q 6.
0
0
Ž s y 3. Ž s q 2. Ž s q 1. Ž s q 6.
Ž s y 2. Ž s q 1. Ž s q 6.
12
0
,
Ž 6.4.
y Ž s y 2.
Ž s q 1. Ž s q 6. , Ž 6.5. Ž s y 2. Ž s q 1. Ž s q 6.
and s y 20
0
sq1
X Ž s . s X1Ž s . s
s y 20
0
Y Ž s . s Y1 Ž s . s
,
sq1
29 s q 57
29 s q 57
sq1
sq1 29 s q 57
12 7
Ž 6.6.
0
.
Ž 6.7.
sq1
Factorize A1Ž s ., B Ž s . as Ž3.18. and Ž3.19., respectively. Thus sy3 A1 i Ž s . s
sq3 0
A1 o Ž s . s
0 sy3
,
sq3
Ž 6.8.
Ž s q 2. Ž s q 3. Ž s q 1. Ž s q 6.
o
0
Ž s q 2. Ž s q 3. Ž s q 1. Ž s q 6.
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HSIAO ET AL.
and
Bo Ž s . s
7
Ž s q 2. Ž s q 1. Ž s q 6.
12
y Ž s q 2.
Ž s q 1. Ž s q 6. , Ž s q 2. Ž s q 1. Ž s q 6.
0 sy2
Bi Ž s . s
Ž 6.9.
0
sq2
.
sy2
0
sq2
Substitute equations Ž6.3. ] Ž6.9. into Corollary 1 and then the Pick matrix D is shown to be positive definite. This result means that the robust stabilization problem is solvable. Step 2. From Ž4.3c. as well as Ž4.3d., we have T1X Ž s . s s
d u
Y Ž s . Bo Ž s .
1 Ž s q 2 . Ž 29 s q 57 . 1 0 7 Ž s q 1. 2 Ž s q 6.
0 1
Ž 6.10.
and T2X Ž s . s A1 i Ž s . s
sy3 1 sq3 0
0 1
Ž 6.11.
Step 3. By the bilinear transformation, we obtain T1X Ž z . s
1 Ž 43 y 14 z . Ž 3 y z . Ž 1 y z . 1 0 7 Ž 14 y 10 z . T2X Ž z . s
y1 q 2 z 1 0 2yz
0 , 1
0 . 1
Ž 6.12. Ž 6.13.
Step 4. h Ž z . s det Ž T2X Ž z . . s adj Ž T2X Ž z . . s
1 y 4 z q 4 z2 4 y 4 z q z2
y1 q 2 z 1 0 2yz
0 , 1
,
Ž 6.14. Ž 6.15.
H
`
-OPTIMAL
CONTROLLER DESIGN
589
thus 0 1 y 4
Ch s
1
Ž 6.16.
1
F Ž z . s adj Ž T2X Ž z . . T1X Ž z . s
1 Ž 43 y 14 z . Ž 3 y z . Ž 1 y z . Ž y1 q 2 z . 1 0 7 Ž 14 y 10 z . Ž 2 y z .
0 . Ž 6.17. 1
From Ž4.8., we have 1 Js
4
3
y
0
4 1
.
Ž 6.18.
4
Performing the Cholesky decomposition Ž4.7. gives:
'15 Qo s
0
4
'15
3'15
20
20
.
Ž 6.19.
Substituting Eqs. Ž6.16. ] Ž6.19. into Ž4.9. yields the singular value decomposition:
f s U SV * y2.0000 4.0000 s 0 7 0 1
y1.0000 1.9998 0 0
0 0 y2.0000 4.0000
0 0 y1.0000 1.9998
Ž 6.20.
with S s 17 diag 4.9999 4.9999 0.0001 0.00014 . Thus, the maximum singular value of f is
s Žf. s r (
5 7
- 1.
Ž 6.21.
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HSIAO ET AL.
Step 5. There must exist a diagonal all-pass function matrix H Ž s . g LŽ s . such that H Ž s.
`
T1X Ž s . y T2X Ž s . K 9 Ž s .
s inf
K 9Ž s .
s T1X Ž s . y T2X Ž s . K˜9 Ž s .
`
s
`
5 7
.
Ž 6.22.
As T2X Ž s . has only one unstable zero 3 and T2X Ž s . is all-pass, the optimal all-pass function matrix H Ž s . is Žby the Nevanlinna]Pick algorithm. H Ž s. s
5 7
I2 .
Ž 6.23.
Then the optimal K˜9Ž s . is found: K˜9 Ž s . s
y Ž 5s 2 q 26 s q 28 . Ž s q 3 . 1 2 0 7 Ž s q 1. Ž s q 6.
0 . 1
Ž 6.24.
From Ž4.3e., K˜9 Ž s . s T2 o Ž s . K˜Ž s . T3 o Ž s . s
1 7
A1 o Ž s . K˜Ž s . Bo Ž s . .
Ž 6.25.
Thus, we obtain K˜Ž s . s y
12 Ž 5s 2 q 26 s q 28 . Ž s q 6 . 1 2 0 7 Ž s q 2.
1 . 1
Ž 6.26.
Step 6. Perform the factorization K˜Ž s . s Ly1 Ž s . M Ž s . and then choose LŽ s . s y
7 Ž s q 2.
1 12 Ž 5s 2 q 26 s q 28 . 0
0 1
Ž 6.27.
and M Ž s. s
sq6 1 sq2 0
1 . 1
Ž 6.28.
Step 7. From Ž3.7. and Ž3.8., we have T Ž s. q RŽ s. s
7 Ž s q 1. Ž s q 6.
1 3 Ž s q 2 . Ž 5s q 26 s q 28 . 0 2
0 1
Ž 6.29a.
H
`
-OPTIMAL
591
CONTROLLER DESIGN
and y5 Ž s q 1 . Ž s q 6 . 1 Ž 5s 2 q 26 s q 28 . 0
SŽ s. s
1 . 1
Ž 6.29b.
Step 8. Here, we have more freedom to choose T Ž s . and RŽ s . in Ž6.29a. to adjust other system performance. For instance, T Ž s . can be chosen such that lim s ª 0 Ž I y B1Ž s . Ly1 Ž s .T Ž s .. s 0 which has a good tracking property Ži.e., this transfer matrix must have zero at s s 0 for tracking step input or a double zero at s s 0 for ramp input.. In this example, in order to track the unit step, we choose
T Ž s. s
Ž s q 1. Ž s q 6.
1 2 5s q 26 s q 28 Ž . 0
1 , 1
Ž 6.30.
and then y Ž s q 1 . Ž s q 6 . Ž 3s y 1 .
y Ž s q 6. Ž s q 1.
3 Ž s q 2 . Ž 5s q 26 s q 28 .
Ž 5s 2 q 26 s q 28 . y Ž s q 1 . Ž s q 6 . Ž 3s y 1 . 3 Ž s q 2 . Ž 5s 2 q 26 s q 28 .
2
RŽ s. s
0
Ž 6.31. Hence the observer-based controller Ž RŽ s ., SŽ s ., T Ž s .. in Ž6.29. ] Ž6.31. not only attains optimal robustness but also achieves the tracking purpose.
VII. CONCLUSION Based on the parametrization of the observer-based controller, a necessary and sufficient condition of robust stabilization is derived for nonlinear multivariable systems no matter whether the controlled plant is stable or not. The solvability of such a robust stabilization problem is also presented by means of Nevanlinna]Pick interpolation theory. According to the criterion of robust stability and the technique of H `-optimization, we have proposed a novel algorithm to synthesize an observer-based controller not only to stabilize the entire system but also to attain optimal robustness.
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APPENDIX If. Equation Ž3.5. can be rewritten as: e Ž t . s A1 Ly1 Tr Ž t . y A1 Ly1 SP Ž N y u I . e Ž t . .
Ž A.1.
From Lemma 1, we have Ly1 S s Y1 y Ly1 MA s Y1 y KA.
Ž A.2.
Substituting ŽA.2. into ŽA.1., we obtain e Ž t . s A1 Ly1 Tr Ž t . y A1 Ž Y1 y KA . P Ž N y u I . e Ž t . s A1 Ly1 Tr Ž t . y uy1 Ž A1Y1 Ay1 y A1 K . B Ž N y u I . e Ž t . s A1 Ly1 Tr Ž t . y uy1 Ž Y y A1 K . B Ž N y u I . e Ž t .
Ž by Ž 3.4. . Ž A.3.
from which e Ž t . F A1 Ly1 Tr Ž t . q uy1 Ž Y y A1 K . B Ž N y u I . e Ž t . F A1 Ly1 T
r Ž t . q uy1 Ž Y y A1 K . B
F A1 Ly1 T
rŽ t. q
d u
Ž Y y A1 K . B
Ž N y u I . eŽ t . eŽ t . .
Ž A.4.
Since A1 Ly1 T and Ž Y y A1 K . B are asymptotically stable, we have Žby Lemma 2. e Ž t . F A1 Ž j v . Ly1 Ž j v . T Ž j v . q
d u
`
rŽ t.
Ž Y Ž j v . y A1 Ž j v . K Ž j v . . B Ž j v .
`
e Ž t . . Ž A.5.
Let A1 Ž j v . Ly1 Ž j v . T Ž j v .
`
s a1 - `
Ž A.6.
and
d u
Ž Y Ž j v . y A1 Ž j v . K Ž j v . . B Ž j v .
`
s a2 - 1
Ž by Ž 3.9. . Ž A.7.
then
Ž 1 y a2 . e Ž t . F a1 r Ž t . « e Ž t . F a3 r Ž t .
Ž A.8.
H
`
-OPTIMAL
593
CONTROLLER DESIGN
where a3 '
a1 1 y a2
- `.
Similarly,
˜ Ž t . s PNe Ž t . y Ž t . s Pe s P Ž N y u I . e Ž t . q u Pe Ž t . s Ž I y B1 Ly1 S . P Ž N y u I . e Ž t . q B1 Ly1 Tr Ž t .
Ž from Ž 3.1. , Ž A.1. .
Ž A.9.
Substituting ŽA.2. into ŽA.9., we have y Ž t . s Ž I y B1 Ž Y1 y KA . . P Ž N y u I . e Ž t . q B1 Ly1 Tr Ž t . s Ž I y B1Y1 q B1 KA . P Ž N y u I . e Ž t . q B1 Ly1 Tr Ž t . s Ž XA q B1 KA . P Ž N y u I . e Ž t . q B1 Ly1 Tr Ž t . s uy1 Ž X q B1 K . B Ž N y u I . e Ž t . q B1 Ly1 Tr Ž t .
Ž A.10.
from which y Ž t . F uy1 Ž X q B1 K . B Ž N y u I . e Ž t . q B1 Ly1 Tr Ž t . F uy1 Ž X q B1 K . B F
d u
Ž X q B1 K . B
Ž N y u I . e Ž t . q B1 Ly1 T e Ž t . q B1 Ly1 T
rŽ t. .
rŽ t.
Ž A.11.
Moreover, Ž X q B1 K . B and B1 Ly1 T are asymptotically stable. Then Žby Lemma 2. yŽ t. F
d u
Ž X Ž j v . q B1Ž j v . K Ž j v . . B Ž j v .
q B1 Ž j v . Ly1 Ž j v . T Ž j v .
`
`
eŽ t .
rŽ t. .
Ž A.12.
Let
d u
Ž X Ž j v . q B1Ž j v . K Ž j v . . B Ž j v .
`
s a4 - `
Ž A.13.
and B1 Ž j v . Ly1 Ž j v . T Ž j v .
`
s a5 - `.
Ž A.14.
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Substituting ŽA.8., ŽA.13., and ŽA.14. into ŽA.12., we have y Ž t . F a 4 e Ž t . q a5 r Ž t . F Ž a 3 a 4 q a5 . r Ž t . .
Ž A.15.
It follows that 5 y Ž t .5 F k 1 5 r Ž t .5 in which k 1 s a3 a4 q a5 is a nonnegative finite constant. Only If. By contradiction, suppose that Ž3.10. is false, i.e.,
d
Ž I y A1Ž s . Ly1 Ž s . Ž T Ž s . q R Ž s . . . u
`
s
d u
Ž I y A1 Ly1 Ž T q R . .
G1
Ž A.16.
Žby Lemma 2.. We construct a plant P˜ that is not stabilized by the compensator Ž RŽ s ., S Ž s ., T Ž s ... Rearrange ŽA.3. as
Ž I q uy1 Ž Y y A1 K . B Ž N y u I . . e Ž t . s Ž I q uy1 Ž I y A1 Ly1 Ž T q R . . Ž N y u I . . e Ž t . s A1 Ly1 Tr Ž t .
Ž A.17.
and select constant unitary matrices W and V to achieve the following singular value decomposition, i.e.
s1 0 W I y A1 Ly1 Ž T q R . V s . .. 0
0 s2
??? ..
???
0
.
0 .. . 0
Ž A.18.
sn
in which s 1 G s 2 G ??? G sn are the singular values of w1 y A1 Ly1 ŽT q R .x. By virtue of the facts that the value of 5 A 5 is the largest singular value of A and 5 WA 5 s 5 A 5 for any unitary matrix W, the inequality ŽA.16. thus implies that d s G 1. Ž A.19. u 1 Suppose we define N y u I ' yVWNm Ž ? .
Ž A.20.
where Nm Ž?. is the nonlinear operator within the conic sector yurs 1 , urs 14 which touches the line of slope urs 1 at t s t 1 , i.e., NmŽ eŽ t 1 .. s urs 1 eŽ t 1 . Žsee Fig. 4.. With this assumption, we have Ne Ž t . y u e Ž t . s Nm e Ž t . F
u s1
eŽ t . F d eŽ t .
Ž by Ž A.19. . . Ž A.21.
H
`
-OPTIMAL
595
CONTROLLER DESIGN
FIG. 4. The nonlinear operator Nm touching the line with slope urs 1 at t s t1 .
In other words, N is a permissible nonlinear perturbation in our problem. Once the nonlinear operation Nm e touches the line of slope urs 1 at t s t1 ,
I q uy1 Ž I y A1 Ly1 Ž T q R . . Ž N y u I . 4 e Ž t1 .
¡
~
s I y Wy1
¢
s 1ru 0 .. . 0
s e Ž t 1 . y Wy1
s I y Wy1
1 0 .. . 0
1 0 .. . 0
???
0 s 2 ru
..
snru
???
0 .. .
.. ??? 0 s 2rs 1
0 ??? ..
???
0
.
.
¦
¥
WNm e Ž t 1 .
0
0
??? 0 s 2 rs 1
.
0 .. .
§
We Ž t 1 .
0
snrs 1 0 .. . 0
snrs 1
0
W e Ž t1 .
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HSIAO ET AL.
s Wy1
0 0 .. . 0
0 1 y s 2rs 1
??? ..
???
0
.
0 .. . 0 1 y snrs 1
0
Wy1 e Ž t 1 . . Ž A.22.
This result reflects the fact that I q uy1 Ž I y A1 Ly1 ŽT q R ..Ž N y u I .4 is singular at t s t 1; and from ŽA.17., it is seen that eŽ t 1 . may approach infinity for r / 0. This phenomenon violates the definition of L2-stability. ACKNOWLEDGMENT The authors thank the Defense Technology Coordination Council, Taiwan, R.O.C., for their support of this work under Contract CS-79-0210-D006-24.
REFERENCES 1. M. Vidyasagar, ‘‘Control System Synthesis: A Factorization Approach,’’ MIT Press, Cambridge, MA, 1985. 2. C. D. Yang and F. B. Yeh, A simple algorithm on minimal balanced realization for transfer function matrices, IEEE Trans. Automat. Control, 34 Ž1989., 879]882. 3. C. D. Yang and F. B. Yeh, One-step extension approach to optimal Hankel-norm approximation and H `-optimization problems, in ‘‘Proc. of the 29th IEEE Conference on Decision and Control, Honolulu, Hawaii, 1990,’’ Vol. 5, pp. 2622]2627. 4. M. Vidyasagar and H. Kimura, Robust controllers for uncertain linear multivariable systems, Automatica 22 Ž1986., 85]94. 5. C. H. Lo, ‘‘Necessary and Sufficient Conditions for Robust Stabilization of Observer-Based Compensating Systems,’’ Master’s thesis, Tatung Institute of Technology, Taipei, 1988. 6. P. Delsarte, Y. V. Genin, and Y. Kamp, Schur parametrization of positive definite block-Toeplitz systems, SIAM J. Appl. Math. 36 Ž1979., 35]46. 7. P. Delsarte, Y. V. Genin, and Y. Kamp, The Nevanlinna]Pick problem for matrix-valued functions, SIAM J. Appl. Mat. 36 Ž1979., 47]61. 8. C. E. Lin, F. H. Hsiao, C. D. Yang, and C. Y. Chang, ‘‘Robust Observer-Controller Design in Nonlinear Multivariable Systems: Using Dither as Auxiliary,’’ Defense Technology Coordination Council, CS-79-0210-D006-24, Sep. 1991. 9. B. C. Chang and J. B. Pearson, Optimal disturbance reduction in linear multivariable systems, IEEE Trans. Automat. Control. 29 Ž1984., 880]887. 10. S. Boyd, V. Balakrishnan, and K. Kabamba, A bisection method for computing the H ` of a transfer matrix and related problems, Math. Control Signals Systems 2 Ž1989., 207]219. 11. B. S. Chen, R. W. Liu, J. Murray, and R. Saeks, Feedback system design: The fractional representation approach, IEEE Trans. Automat. Control. 25 Ž1980., 399]412. 12. B. A. Francis, J. W. Helton, and G. Zames, H `-optimal feedback controllers for linear multivariable systems, IEEE Trans. Automat. Control. 29 Ž1984., 888]900.