Reliable Decentralized Stabilization of Linear Systems - UC Davis ...
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 12, DECEMBER 1998
Proof: First of all, notice that ( 1=2 A; B ) is stabilizable if and ^) ^ 1=2 A only if ( 1=2 A; 1=2 B ) is stabilizable. From Lemma A1, (C; is detectable. Notice now that (5) can be written as
X = J^0 J^ + ( 1=2 A0 )X ( 1=2 A)
0 (( 1=2A0 )X ( 1=2B) + J^0 D^ ) 1 (D^ 0D^ + ( 1=2B0 )X ( 1=2B))01 1 (( 1=2B0 )X ( 1=2A) + D^ 0J^)
and the lemma follows from standard results on ARE’s (see [6, p. 263 and Appendix B]). Proof of Proposition 2.3: Consider in (5) Y = Y^ Y^ : From ~) is de~ 1=2 A Lemma A.2, if ( 1=2 A; B ) is stabilizable and (C; ^ tectable, then there exists a unique positive semidefinite solution X ^ is as ^ ) is stable, where K to (5), and it is such that 1=2 (A + B K ^ and Y = Y ^ : If we now set Y = Y ^ we can in (A1) with X = X ^ conclude similarly the existence of a unique semipositive solution X ~ and Y = Y ~ given by (A1) with X = X ~ : After some of (5) and a K algebraic manipulation, we get that ^ (X
0 X~ ) 0 (A + BK^ )((Y^ + X^ ) 0 (Y + X~ ))(A + BK^ ) ~ + X ~ )B )(K ~ 0K ^) ~ 0K ^ )0 (D 0 D + B 0 (Y = (K
and thus ^ 0X ^ ~ ) 0 (A + B K ^ )0 (X (X ^ ^ )0 (Y = (A + B K
0 X~ )(A + BK^ )
0 Y~ )(A + BK^ )
0 K^ )(D0 D + B0 (Y~ + X~ )B)01 (K~ 0 K^ ) ^ 0X ^ ) that X ~ 0: and it follows from stability of 1=2 (A + B K ~ + (K
[12] Y. Ji, H. J. Chizeck, X. Feng, and K. A. Loparo, “Stability and control of discrete-time jump linear systems,” Control Th. and Adv. Tech., vol. 7, pp. 247–270, 1991. [13] A. Laub, “Algebraic aspects of generalized eigenvalue problems for solving Riccati equations,” in Computational and Combinatorial Methods in Systems Theory, C. Byrnes and A. Lindquist, Eds. Amsterdam, The Netherlands: Elsevier, 1986, pp. 213–227. [14] M. Mariton, Jump Linear Systems in Automatic Control. New York: Marcel Dekker, 1990. [15] L. Vandenberghe and S. Boyd, Software for Semidefinite Programming—User’s Guide. Available via anonymous ftp from isl.stanford.edu /pub/boyd/semidef prog, 1994. [16] H. K. Wimmer, “Monotonicity and maximality of solutions of discretetime algebraic Riccati equations,” J. Math. Syst., Estimation and Contr., vol. 2, no. 2, pp. 219–235, 1992.
Reliable Decentralized Stabilization of Linear Systems A. N. G¨unde¸s
Abstract— Reliable stabilization of linear time-invariant multiinput/multi-output plants is considered using a two-channel decentralized controller configuration. Necessary and sufficient conditions are obtained for existence of reliable controllers that maintain stability under the possible failure of either one of the two controllers. All decentralized controllers that achieve reliable stabilization are characterized. Index Terms—Controller design, decentralized control, reliable stabilization.
For a general account on the positive semidefinite partial ordering of maximal solutions of discrete-time ARE’s, see [16]. REFERENCES [1] H. Abou-Kandil, G. Freiling, and G. Jank, “On the solution of discretetime Markovian jump linear quadratic control problems,” Automatica, vol. 31, no. 5, pp. 765–768, 1995. [2] M. Ait-Rami and L. E. Ghaoui, “LMI optimization for nonstandard Riccati equations arising in stochastic control,” IEEE Trans. Automat. Contr., vol. 41, pp. 1666-1671, Nov. 1996. [3] O. L. V. Costa, J. B. R. do Val, and J. C. Geromel, “A convex programming approach to 2 -control of discrete-time Markovian jump linear systems,” Int. J. Contr., vol. 66, pp. 557–579, 1997. [4] O. L. V. Costa and M. D. Fragoso, “Stability results for discrete-time linear systems with Markovian jumping parameters,” J. Math. Analysis and Appl., vol. 179, pp. 154–178, 1993. , “Discrete-time LQ-optimal control problems for infinite Markov [5] jump parameter systems,” IEEE Trans. Automat. Contr., vol. 40, pp. 2076–2088, 1995. [6] M. H. A. Davis and R. B. Vinter, Stochastic Modeling and Control. London, U.K.: Chapman and Hall, 1985. [7] L. El Ghaoui, R. Nikoukhah, and F. Delebecque, LMITOOL: A FrontEnd for LMI Optimization—User’s Guide. Available via anonymous ftp from ftp.ensa.fr /pub/elghaoui/lmitool, 1995. [8] M. D. Fragoso, J. B. Ribeiro do Val, and D. L. Pinto Jr., “Jump linear 1 -control: The discrete-time case,” Contr. Th. and Adv. Tech., vol. 10, pp. 1459–1474, 1995. [9] Z. Gajic and I. Borno, “Lyapunov iterations for optimal control of jump linear systems at steady state,” IEEE Trans. Automat. Contr., vol. 40, no. 11, pp. 1971–1075, 1995. [10] J. C. Geromel, P. L. D. Peres, and S. R. Souza, “ 2 -guaranteed cost control for uncertain discrete-time linear systems,” Int. J. Contr., vol. 57, pp. 853–864, 1993. [11] Y. Ji and H. J. Chizeck, “Controllability, observability and discretetime Markovian jump linear quadratic control,” Int. J. Contr., vol. 48, pp. 481–498, 1988.
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I. INTRODUCTION We consider reliable stabilization of linear time-invariant (LTI), multi-input/multi-output (MIMO) plants under possible sensor or actuator failures using a two-channel decentralized feedback control configuration. The goal is to maintain closed-loop stability when both controllers act together and when either one of the two controllers acts alone. It is assumed that the failure of a controller is recognized and it is taken out of service (i.e., the states in the controller implementation are all set to zero, the initial conditions and outputs of the channel that failed are set to zero for all inputs). Since the introduction of multi-controller systems in [5] and [6], reliable stabilization has been studied for various failure models using full-feedback [2], [8] and decentralized configurations [4], [7]. Conditions for existence of reliable decentralized controllers were given for a class of reliable stabilization problems using genericity arguments in [4]. The reliable stabilization problem considered in this paper is based on the twochannel decentralized configuration and failure model in [7]. The necessary and sufficient conditions here for existence of reliable decentralized controllers include generalizations of the sufficient conditions in [7]. The main results in this paper are the explicit existence conditions for reliable decentralized controllers. Theorem 2 gives an important interpretation of these conditions in terms of the strong stabilizability Manuscript received January 10, 1996; revised February 7, 1997 and August 1, 1997. This work was supported by the NSF under Grant ECS-9257932. The author is with Electrical and Computer Engineering Department, University of California, Davis, CA 95616 USA (e-mail [email protected]). Publisher Item Identifier S 0018-9286(98)09444-6.
0018–9286/98$10.00 1998 IEEE
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 12, DECEMBER 1998
Fig. 1. The decentralized system
Fig. 2. The system
S (P; C1 ):
Fig. 3. The system
S (P; C2 ):
S (P; CD ):
of an associated system and states that strong stabilizability of two of the subblocks of the plant is necessary. Proposition 1 gives a parameterization of all reliable decentralized controllers for stable plants, and Theorem 3 establishes explicit existence conditions when one channel is single-input/single-output (SISO). Theorem 4 gives important sufficient conditions when all channels are MIMO. The following notation is used throughout; due to the input–output approach, the setting can be continuous-time or discrete-time. Notation: Let U denote the region of instability; U contains the extended closed right half-plane (for continuous-time systems) or the complement of the open unit-disk (for discrete-time systems). Let Rp (Rs ) denote proper (strictly proper) rational functions with real coefficients; R denotes proper rational functions with no poles in U ; M(R) denotes the set of matrices with entries in R; M is called R-stable iff M 2 M(R); and M 2 M(R) is called R-stable unimodular iff M 01 2 M(R). A right-coprime factorization (RCF), a left-coprime factorization (LCF), and a bicoprime factorization ~ = ~ 01 N (BCF) of P 2 M(Rp ) are denoted by P = N D01 = D 0 1 ~ ; D; ~ Nbr ; Db ; Nbl ; Gb 2 M(R), D , Nbr Db Nbl + Gb ; N; D; N ~ , and Db are biproper. Let rankP = r ; so 2 U is called a U -zero D (blocking U -zero) of P iff rankP (so ) < r(P (so ) = 0); the poles of P in U are called its U -poles. For M 2 M(R), the norm k 1 k is defined as kM k = sup s2@ U (M (s)); denotes the maximum singular value and @ U denotes the boundary of U .
by (3); a similar description can be obtained for S (P; C2 ). The ~L2 u2 , N ~ D2 ~2 = N ~R2 ~2 = y2 system S (P; C2 ) is described as D in (4); the description for S (P; C1 ) is similar. For j = 1; 2, the transfer function of S (P; Cj ) from (uP ; uCj ) to (yP ; yCj ) is
01 NLj = N~Rj D~ 01 N~Lj Hj = NRj DDj Dj
~CD + N ~C N )P = [D ~C (D
N P + D
~C 1 [N
CD =
C1 0
P12 P22 0
C2
2 Rpn 2n ; 2 Rpn 2n
;
Pjj Cj
2 Rpn 2n
2 Rpn 2n
(1)
no = no1 + no2 , ni = ni1 + ni2 . It is assumed that P and CD have no hidden modes corresponding to eigenvalues in and that (P; CD ) is well-posed. The failure of the j th controller channel is represented by setting Cj = 0; the corresponding j th channel output yCj is also
U
set to zero. When the second (first) channel fails, the system is called
S (P; C1 ) shown in Fig. 2 (S (P; C2 ) shown in Fig. 3).
~ 01 N Using any RCF P = N D01 , any LCF Cj = D c ~c , j = 1; 2, ~ C1; D ~ C 2 ], N ~C 1 ; N ~C 2 ], DP = eP , ~ C = diag[D ~C = diag[N D T uT ]T , u = [uT uT ]T , y = [y T y T ]T , N P = yP , uP = [uP C P 1 P2 C1 C2 P1 P2 T y T ]T , e T T T yC = [yC P = [eP 1 eP 2 ] , the system S (P; CD ) is 1 C2 described in (2) as DD P = NL u, NR P + Gu = y ; S (P; CD ) 01 NL + G from is well-posed, i.e., the transfer-function H = NR DD (uP ; uC ) to (yP ; yC ) is proper, if and only if DD is biproper. The description of S (P; C1 ) as DD1 1 = NL1 u1 , NR1 1 = y1 is given
0
0
In
P yC 1
~ D
0N~
[0
Ino2 ] 0
In
0
NC 2
=
yP yC 1
0
0
0
(2)
P yC 1
yP C 2
(3)
yP C 2
NC 2 ~ C2 D
~ N
I
uP uC
uP uC 1
0
~C 1 N
0
N
yP yC
0 ~ C1 D
0 ]N
=
S
=
0In
I
=
Consider the LTI, MIMO, and two-channel decentralized control system S (P; CD ) shown in Fig. 1. The plant and the decentralized controller are represented by their transfer functions P and CD , respectively P11 P21
uP uC
0 0
D
II. MAIN RESULTS
P =
0
0I
~C ] N
uP uC 2 =
yP : yC 2
(4)
Reliable Stability: The system S (P; CD ) is said to be R-stable iff 2 M(R); similarly, S (P; Cj ) is R-stable iff Hj 2 M(R). The decentralized controller CD is said to be an R-stabilizing controller for P iff CD is proper and S (P; CD ) is R-stable. The pair (C1 ; C2 ) is called a reliable decentralized controller pair iff C1 , C2 are proper and the systems S (P; CD ), S (P; C1 ), S (P; C2 ) are all R-stable. Lemma 1 gives necessary and sufficient conditions for R-stability of S (P; CD ) under normal operation and under the complete failure of one of the controllers. We assume that the coprime factorizations are in canonical forms; the denominator-matrix of any RCF, LCF can H
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 12, DECEMBER 1998
be put into upper (lower) triangular Hermite forms by elementary column (row) operations [9], [1]. Without loss of generality, it is ~ 01 N ~ are given by assumed that the RCF and LCF P = N D01 = D P
= ND
01 =
~ ~ 01 N =D
=
N11
N12
N21
N22
~ 11 D
~ 12 D ~ 22 D
0
0
D21
D22
01
~11 N ~21 N
C 1 D11 + N~C 1 N11 C 2 D21 + N~C 2 N21
~12 N ~22 N
(6) is
R-unimodular,
~ C 2 D22 + N ~C 2 N22 ) (D
and ~ D
C 1 D11 + N~C 1 N11
R-unimodular
is
~ N
C 1 N12 is R-unimodular:
D21
D22
(7) (8)
To obtain a parameterization of all reliable decentralized controller pairs, we use the following characterization of all admissible plants for stability using one controller [1], [3]: Let P 2 Rpn 2n be partitioned as in (1). There exists a decentralized controller CD = diag[C1 ; C2 ] such that S (P; CD ) and S (P; C2 ) are R-stable if and ~ of the form ~ 01 N only if P has an RCF and LCF P = N D01 = D P
=
ND
01 =
~ 01 N ~ = =D
N11
~2 N ~21 V
n
N12
0
N22
N12 U2
I
0
n
0 ~2 ~21 U N
01
N11
~21 N
~ 22 D
01
0
I
D22 N12 V2
~22 N
(9)
~21 2 Rn 2n , , N12 2 Rn 2n , N ~ ~ (N22 ; D22 ) is right-coprime, (D22 ; N22 ) is left-coprime, and ~2 ; V ~2 ; U2 ; V2 2 M(R) satisfy (10); equivalently, P11 0 U ~ 22 P21 2 M(R), P12 D22 U2 P21 2 M(R), P12 D22 2 M(R), D 01 is an RCF and D~ 01 N~22 is an LCF of P22 and U2 where N22 D22 22 satisfies (10)
where
N11
2 Rn 2n
V2
0 ~22 N
U2
~ 22 D
D22 N22
0 ~2 U
~2 V
=
n
0
I
0
:
(10)
n Rn 2n have an I
0
Theorem 1—Stabilizing Controllers: Let P 2 ~ 01 N ~ satisfying (9). The system RCF and LCF P = N D01 = D S (P; C2 ) is R-stable if and only if C2 is given by (11), Q2 2 Rn 2n is such that D~ C2 is biproper [holds for all Q2 2 M(R) when P22 2 M(Rs )]; S (P; CD ) and S (P; C2 ) are both R-stable if and only if C2 and C1 are given by (11) and (12), and Q1 2 Rn 2n is such that D~ C1 is biproper [holds for all Q1 2 M(R) ~21 ) 2 M(Rs )]; S (P; CD ), S (P; C2 ), and when (N11 0 N12 Q2 N S (P; C1 ) are all R-stable (i.e., (C1 ; C2 ) is a reliable decentralized controller pair) if and only if C2 and C1 are given by (11) and (12), where Q1 2 Rn 2n , Q2 2 Rn 2n satisfy condition (13), or
01 (U2 + Q2 D~ 22 )
~22 ) Q2 N
C2 ~2 + D22 Q2 )(V ~2 = (U
(5)
:
~ D
D11
~ 01 N =D C 2 ~C 2
01 = NC 2 D
~C 1 N12 N is R-unimodular C 2 D22 + N~C 2 N22
~ D
C2
= (V2
~ 01 N ~ be any Lemma 1—Decentralized Stability: Let N D01 = D 0 1 ~ 0 1 n 2 n ~ ; let DC NC = NC DC be any LCF, RCF, LCF of P 2 Rp ~ C1; D ~ C = diag[D ~C = diag[N ~ C 2 ], N ~C 1 ; N ~C 2 ], RCF of CD , D NC = diag[NC 1 ; NC 2 ], DC = diag[DC 1 ; DC 2 ]. The system S (P; CD ) is R-stable if and only if DD := (D~ C0D1 + N~0C 1N ) ~ ~ N is R-unimodular. Let the RCF and LCF P = N D = D be as in (5); S (P; CD ) is R-stable if and only if (6) holds. Let 01 ~ 01 N C2 = D system S (P; C2 ) C 2 ~C 2 = NC 2 DC 2 be any LCF, RCF; the 01 ~ 01 N is R-stable if and only if (7) holds. Let C1 = D C 1 ~C 1 = NC 1 DC 1 be any LCF and RCF; S (P; C1 ) is R-stable if and only if (8) holds ~ D
~ C1, D ~ C 2 are biproper equivalently (14), and D
01
D11
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C1
D22
~ 22 D
~ 01 N =D C 1 ~C 1 = (I
0
0
01
N22 Q2 )
Q1 (N11
0
~2 + D22 Q2 )N ~21 Q1 N12 + (U
~21 Q1 N12 (U2 + Q2 D ~ 22 ) +N
(11)
01 Q1
~21 )) N12 Q2 N
is
R-unimodular
is
R-unimodular
(12) (13)
:
(14)
Conditions (13) and (14) lead to the conditions that P must satisfy for the existence of reliable decentralized controller pairs as stated in Theorem 2. Strong R-stabilizability of pseudo-systems related to P are important for existence of reliable decentralized controllers. The following are well known [9]: An LTI system P^ is said to be strongly R-stabilizable iff an R-stable R-stabilizing controller ^ exists for P ^ . In the standard full-feedback system S (P ^; C ^ ^ ), P C is strongly R-stabilizable if and only if it has an even number of U -poles between consecutive pairs of real blocking U -zeros. Let 01 ^ = Np D 01 = D ~ 01 N P p p ~p = Nbr Db Nbl + Gb be any RCF, LCF, 0 1 ~ ^ ^ ~ ^ 2 M(R) if and and BCF of P ; let C = Dc Nc be any LCF; C ~ c is unimodular. Therefore, P ^ is strongly R-stabilizable only if D ~ Np is R~ 2 M(R) exists such that Dp + X if and only if X ~p + N ~p X unimodular; equivalently, X 2 M(R) exists such that D is R-unimodular; equivalently, Xb 2 M(R) exists such that I
+ Xb Gb
0
bl
N
b br b
X N D
is
R-unimodular
:
Lemma 2—Coprime Factorizations and Strong Stabilizability: Let 2 Rpn 2n , partitioned as in (1), have an RCF and LCF P = 01 = D~ 01 N~ of the form (9). Let N12 D01 be an RCF, ND 22 01 ~ 12 be an LCF of P12 ; let D ~ 01 N ~ ~ 01 X Y 12 22 21 be an LCF and X21 Y21 ^ ~ ~ be an RCF of P21 . Define P := P12 (U2 + D22 Q2 )D22 P21 = ~ 22 )P21 . P12 D22 (U2 + Q2 D 0 1 ~ ~21 is a BCF, N12 (U2 + 1) N12 D22 (U2 + D22 Q2 )N 0 1 01 X~ 12 (U~2 + D22 Q2 )N~21 ~ is an RCF, and Y~12 Q2 D22 )X21 Y21 ^ is an LCF of P . ~ 22 has the same 2) P^ is strongly R-stabilizable if and only if det D ~ 22 )X21 , sign at all real blocking U -zeros of N12 (U2 + Q2 D equivalently, det D22 has the same sign at all real blocking U -zeros of X~12 (U~2 + D22Q2 )N~21. Theorem 2—Conditions for Reliable Decentralized Stabilizability: Let P 2 Rpn 2n be partitioned as in (1). 1) If there exists a reliable decentralized controller pair (C1 ; C2 ), then the following four necessary conditions hold: 1) P has ~ 01 N ~ satisfying (9); 2) in (9), an RCF and LCF N D01 = D 0 1 0 1 ~ ~ N12 D22 is an RCF of P12 , D22 N 21 is an LCF of P21 ; 3) P12 , P21 are strongly R-stabilizable; and 4) the sign of det D22 is the same at all real blocking U -zeros of P12 and at all real blocking U -zeros of P21 . ~ 01 N ~ satisfying (9); 2) Let P have an RCF and LCF N D01 = D 0 1 0 1 ~ ~ let N12 D22 be an RCF of P12 , D22 N21 be an LCF of P21 . P
a)
There exist Q1 ; Q2 2 M(R) satisfying (13), or ~2 + equivalently (14), if and only if P^ = P12 (U ~ 22 P21 = P12 D22 (U2 + Q2 D ~ 22 )P21 is D22 Q2 )D strongly R-stabilizable for some Q2 2 M(R).
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b)
Let P22 2 M(Rs ) and let P12 or P21 be strictly proper. There exists a reliable decentralized controller pair (C1 ; C2 ) if and only if P^ is strongly R-stabilizable for some Q2 2 M(R).
Condition 4) of Theorem 2-1) implies 3); the two conditions are equivalent when P12 ; P21 2 M(Rs ). By Theorem 2, if a reliable decentralized controller pair exists, then Q2 2 M(R) exists such ~2 + D22 Q2 )D ~ 22 P21 is strongly R-stabilizable. When that P^ = P12 (U P 2 M(Rs ), strong R stabilizability of P^ becomes necessary and sufficient for existence of reliable decentralized controller pairs. We parameterize all reliable decentralized controller pairs for R-stable plants in Proposition 1. Explicit necessary and sufficient conditions for existence of reliable decentralized controller pairs are stated in Theorem 3 for the important special case when at least one control channel has only one input and one output. Proposition 1—Reliable Decentralized Stabilization for Stable Plants: Let P 2 Rn 2n be R-stable. Then there exists a reliable decentralized controller pair. Furthermore, all reliable decentralized controller pairs (C1 ; C2 ) are parameterized by (15), where Cj is proper if and only if Qj 2 M(R) is such that (I 0 Qj Pjj ) is biproper, which holds for all Qj 2 M(R) when Pjj 2 M(Rs )
f(C1; C2 )jCj = (I 0 Qj Pjj )01 Qj ; Qj 2 M(R); j = 1; 2; (15) I 0 Q1 P12 Q2 P21 is R-unimodularg: By Theorem 2, if reliable decentralized controllers exist, then P 01 is an RCF, has an RCF and LCF satisfying (9), P22 = N22 D22 0 1 0 1 ~ N P12 = N12 D22 is an RCF, and P21 = D 22 ~21 is an LCF. Suppose P22 = 0, or P12 = 0 or P21 = 0; i.e., N22 = 0 or N12 = 0 or ~21 = 0. The pair (0; D22 ) is right-coprime if and only if D22 is N R-unimodular, i.e., P 2 M(R). So when P22 , P12 ; or P21 is zero, reliable decentralized controllers exist if and only if P is R-stable and the parameterization of all reliable decentralized controller pairs is given by (15), where (I 0 Q1 P12 Q2 P21 ) is R-unimodular for all Q1 ; Q2 2 M(R) if P12 or P21 is zero. Theorem 3—Necessary and Sufficient Conditions When P22 Is SISO: Let P 2 Rpn 2n , P11 2 Rpn 2n , P12 2 Rpn 21 , 01 P21 2 Rp12n , P22 2 Rp (i.e., no2 = ni2 = 1). Let P22 = N22 D22 be any coprime factorization. Let P12 or P21 be strictly proper. Let 1 ; 2 ; 1 1 1 ; (arranged in ascending order) denote the distinct real U -zero poles of P22 and let j ; j ; 1 1 1 ; j (arranged in ascending order) denote those distinct real U -poles of P22 for which the sign of N22 (j ) is not equal to the sign of N22 (j +1 ), 1 k `. There exists a reliable decentralized controller pair (C1 ; C2 ) if and only if the four necessary conditions of Theorem 2-1) hold and P22 has an even number of real U -poles in each of the intervals (j ; j +1 ), 1 k ` 0 1, and (j ; 1). Corollary 1—Sufficient Conditions When P22 is SISO: Let P 2 Rpn 2n , P11 2 Rpn 2n , P12 2 Rpn 21 , P21 2 Rp12n ; let 01 2 Rp be any coprime factorization. Let the four P22 = N22 D22 conditions of Theorem 2-1) hold: 1) Let the sign of D22 at the real blocking U -zeros of P12 and P21 be the same as the sign of D22 (1). Reliable decentralized controllers exist if P22 has even number of U -zeros between any pairs of its real U -poles; 2) Reliable decentralized controllers exist if the sign of D22 is the same at all real U -zeros of P22 as the sign of D22 (1). In Corollary 1-1), if P12 or P21 is strictly proper, then the sign of D22 at the real blocking U -zeros of P12 and of P21 being the same as the sign of D22 (1) follows from the necessary conditions 3) and 4) of Theorem 2-1). When P22 2 M(Rs ), the sufficient condition in Corollary 1-2) is equivalent to P22 being strongly R-
= M(Rs ), this condition implies P22 is stabilizable; when P22 2 strongly R-stabilizable. Theorem 4—Conditions for MIMO Channels: Let P 2 Rpn 2n , P11 2 Rpn 2n , P12 2 Rpn 2n , P21 2 Rpn 2n , P22 2 Rpn 2n . Let the four necessary conditions of Theorem 2-1) hold. 1) Let P12 2 M(Rs ) or P21 2 M(Rs ). Let no2 = ni2 > 1; let the sign of det D22 be the same at all common real U zeros of P12 and P21 as the sign of det D22 (1). Reliable decentralized controller pairs exist if rankP12 = ni2 no1 , rankP21 = no2 ni1 . 2) Let P12 2 M(Rs ) or P21 2 M(Rs ). Let P22 2 Rsn 2n ; let rankP22 = no2 = ni2 , rankP12 + rankP21 > no2 = ni2 ; let the sign of det D22 be the same at all real (transmission) U -zeros of P12 and of P21 as the sign of det D22 (1). Reliable decentralized controller pairs exist if the number of 01 ~ between any real (transmission) U -zeros of P22 = D~22 N 22 ~ 22 is even. pair of real blocking U -zeros of D 3) Let P22 2 Rpn 2n , where no2 and ni2 are not both equal to one. Reliable decentralized controller pairs exist if I 2 Rn 2I n P12 2 Rpn 2n has an R-stable left-inverse P12 n 2 n and if P21 2 Rp has an R-stable right-inverse P21 2 Rn 2n . n 2n I 4) Let P21 2 Rp have an R-stable right-inverse P21 2 Rn 2n . Let P11 2 M(Rs ), P12 2 M(Rs ). Let P22 be strongly R-stabilizable. Reliable decentralized controller pairs ~ 2 Rn 2n . ~ 12 = P22 for some L exist if LP I 2 5) Let P12 2 Rpn 2n have an R-stable left-inverse P12 Rn 2n . Let P11 2 M(Rs ), P12 2 M(Rs ). Let P22 be strongly R-stabilizable. Reliable decentralized controller pairs ~ = P22 for some R ~ 2 Rn 2n . exist if P21 R 6) Let P22 2 Rsn 2n . Reliable decentralized controller pairs ~ 12 = P22 for some L ~ 2 Rn 2n and P21 R ~ = P22 exist if LP n 2 n ~ for some R 2 R . Other sufficient conditions for existence of reliable decentralized controllers can be derived from the six general cases in Theorem 4. For example, under the assumptions of case 2), reliable decentralized controllers exist if either P21 has an R-stable right-inverse or P12 has an R-stable left-inverse since rankP21 = no2 or rankP12 = ni2 = no2 implies rankP12 + rankP21 > no2 . III. CONCLUSIONS We considered the design of reliable decentralized controllers that stabilize a given plant P when both controllers act together and when either one of the controllers acts alone. We showed that reliable decentralized controllers exist only if the subblocks P12 and P21 of P are strongly stabilizable. We established necessary and sufficient conditions for existence of reliable decentralized controllers when P22 is SISO and gave sufficient conditions when all subblocks of P are MIMO. We characterized all reliable decentralized controllers in the parameterizations (11), (12). Extensions to decentralized systems with more than two channels would require additional constraints on the plant. APPENDIX A. Proofs The proof of Lemma 1 follows from (2)–(4) using standard arguments. The proof of Theorem 1 follows by Lemma 1 from the assumption that P has an RCF N D01 of the form (9). Proof of Lemma 2: 1) By (10), (D22 ; NC 2 ) is left-coprime. By assumption, ~ 22 ; N ~21 ) is left-coprime, (X21 ; Y21 ) is right-coprime, (D
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~21 2 M(R) exist such that V21 Y21 + and V21 ; U21 ; V~21 ; U U21 X21 = I , D~ 22 V~21 + N~21 U~21 = I , V21 U~21 = U21 V~21 , N~21 Y21 = D~ 22 X21 . Hence V21 + U21 DC 2 N~21 U21 N22 0NC2 N~21 D22 ~ ~ Y 0 2 N~C221X21 D~ C2 +UN~21CN2 V~2221 N~22 = I
~21 ) is left-coprime for all Q2 2 M(R). implies (D22 ; NC 2 N ^ ~ Let P := P12 NC 2 D22 P21 ; since (N12 ; D22 ) is right-coprime 01 NC 2 N~21 ~21 ) is left-coprime, P^ = N12 D22 and (D22 ; NC 2 N 01 = Y~ 01 X~ 12 , there are is a BCF. For P12 = N12 D22 12 V12 ; U12 ; V~12 ; U~12 2 M(R) such that V12 D22 + U12 N12 = I , X~ 12 U~12 + Y~12 V~12 = I , V12 U~12 = U12 V~12 . Hence
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Proof of Proposition 1: By Lemma 1, (7) and (8) hold if 01 N~Cj is an R-stabilizing controller for ~ Cj and only if Cj = D Pjj , j = 1; 2. Therefore, all C1 , C2 are given by (15). By (6), S (P; CD ) is also R-stable if and only if I 0 Q1 P12 Q2 P21 ~C is R-unimodular. The controllers are proper if and only if D is biproper. We give a solution for Qj 2 M(R) such that I 0 Q1 P12 Q2 P21 is R-unimodular and (I 0 Qj Pjj ) is biproper. Choose Q1 ; Q2 2 M(R) strictly proper; let Q1 = 01 Q1 , where 2 IR, j j > kQ1 P12 Q2 P21 k; choosing strictly proper Q1 ; Q2 is sufficient to make (I 0 Qj Pjj ) biproper, and choosing that guarantees k 01 Q1 P12 Q2 P21 k < 1 is sufficient to satisfy (6). This shows existence of reliable decentralized controllers for any R-stable P . The expression for C1 in (15) 0is1 equivalent to (12). ^ 1 (P11 0 P12 Q2 P21 )) Q^ 1 ; by (13), 8 := By (12), C1 = (I 0 Q (I + Q^ 1 P12 Q2 P21 ) is R-unimodular. With Q1 = 801 Q^ 1 , C1 =
(I + Q^ 1 P12 Q2 P21 0 Q^ 1 P11 )01Q^ 1 = (I 0 801 Q^ 1 P11 )01 801Q^ 1 ^ 1 P12 Q2 P21 ) is ~ is equivalent to (15) and 801 = (I 0 801 Q V21 + U21 (DC 2 + N22 V12 NC 2 )N21 U21 N22 U12 ~ ~ ~ R -unimodular if and only if I 0 Q P Q P is R -unimodular. 1 12 2 21 0X12 NC2N21 Y12 01 = D~ 01 N~22 is Proof of Theorem 3: Since P = N D 22 22 22 22 ~ ~ ~ ~22 , D22 , and D~ 22 are used interchangeably. 2 N12 NY~C212 X21 V~12 + N12 (0D~UC212 +N22N~UC12 = I scalar, N22 , N By ~ ~ ~ 2 V21 N22 )U12 Theorem 2, Q1 ; Q2 2 M(R) satisfying (13) and (14) exist if ~C 2 X21 ; Y21 ) is right-coprime, (Y~12 ; and only if P^ = P12~D22 N~C 2 P21 is strongly R-stabilizable for implies (N12 N some Q2 2 R, i.e., D22 has the same sign at all real blocking X~ 12 NC 2 N~21 ) is left-coprime. U 2-2). Since the only U -zeros 0 1 0 1 ~C 2 X21 Y21 = Y~12 X~12 NC 2 N~21 is an RCF and -zeros of Nn122N~1C 2 X21 by Lemma 2) Since N12 N and P21 2 R12n are their blocking U -zeros, of P12 2 R 0 1 0 1 ^ ~ ~ LCF of P , because P22 = N22 D22 = D22 N22 , P21 = s 2 U is a blocking U -zero of P N 12 ~C 2 X21 if and only if it is a 01 N~21 = X21 Y 01 , P12 = N12 D01 = Y~ 01 X~ 12 , P^ is o D~ 22 ~C 2 or of P21 , i.e., 21 22 12 blocking U -zero of P12 , i.e., N12 (so ) = 0, or of N strongly R-stabilizable if and only if det Y21 , equivalently X (s ) = 0. The four conditions of Theorem 2-1) are necessary det Y~12 , det D~ 22 , or det D22 ; has the same sign at all real for21theoexistence of reliable decentralized controllers. The U -poles of U -zeros of N12 N~C2 X21 , equivalently, of X~12 NC2N~21 . P22 are the U -zeros of D22 . By (10), the signs of U2 and N22 are Proof of Theorem 2: ~ 22 . Suppose fj1 ; 1 1 1 ; j` g is empty. the same at all real U -zeros of D ~ 2 2 R exists such that (U2 + Q~ 2 D~ 22 ) has no U -zeros [9]. 1) If reliable decentralized controllers exist, Condition 1) holds Then Q ~ 2 D~ 22 )X21 are those by (9). By Theorem 1, (13) and (14) hold; (13) and (14) Since the only blocking U -zeros of N12 (U2 + Q ~ 22 ; N~21 ) is left- of P12 and P21 , the conditions of Theorem 2-1) are sufficient for (13). imply (N12 ; D22 ) is right-coprime, and (D 01 , If P 2 M(R ), then (V 0 Q~ N~ ) is biproper. If N~ (1) 6= 0, coprime. By (9), Condition 2) holds since P12 = N12 D22 01 N~21 . Conditions 3) and 4) are shown as follows: let Q222 := Q~ 2 +s Q^ 2 , Q^ 2 22 R, Q2^ 2 (221) 6= (V2 0 Q~ 2 N~2222)N~ 01 (1), P21 = D~ 22 22 P12 is strongly R-stabilizable if and only if for any RCF kQ^ 2 k < kD~ 22 (U2 + Q~ 2 D~ 22 )01 k01 . Then D~ C 2 is biproper, i.e., 01 , X~ 2 M(R) exists such that D22 + C2 2 M(Rp ), and N~C 2 is a unit in R. Since P12 2 M(Rs ) or P12 = N12 D22 ~ XN12 is R-unimodular; with X~ = NC 2 N~21 Q1 , (13) implies P21 2 M(Rs ), Q1 satisfying (13) can be chosen strictly proper P12 is strongly R-stabilizable. Similarly, P21 is strongly R- so that D~ C 1 is biproper. Therefore, the conditions of Theorem 2-1) 01 N~21 , X 2 are sufficient for the existence of reliable decentralized controllers. ~ 22 stabilizable if and only if for any LCF P21 = D ~ ~ M(R) exists such that D22 + N21X is R-unimodular; with Suppose fj1 ; 1 1 1 ; j`g is not empty; then N22 has an odd number X = Q1 N12 N~C 2 , (14) implies P21 is strongly R-stabilizable. of zeros in each interval (j ; j +1 ), 1 k `. By (10), ~21 Q1 N12 ) has no U -zeros, N~C 2 (j )N22 (j ) = 1 at the U -zeros j of D~ 22 ; hence, N~C 2 = Since (13) implies det (D22 +NC 2 N det D22 (z12 ) has the same sign as det D22 (z21 ) for all real (U2 + Q2 D~ 22 ) has an odd number of zeros because N~C 2 has even z12 ; z21 2 U , N12 (z12 ) = 0 and N~21 (z21 ) = 0. number of zeros in (j ; j +1 ), 1 k `. Note that j is the 01 NC 2 N~21 is a BCF; first zero of D~ 22 immediately to the left of the real U -zero of N~C 2 in 2) a) By Lemma 2, P^ = N12 D22 ~ 22 immediately to the therefore, P^ is strongly R-stabilizable if and only if (j ; j +1 ) and j +1 is the first zero of D ~ ^ ~ ^ X 2 M(R) exists such that (D22 + NC 2 N21 XN12 ) right of the real U -zero of NC 2 in (j ; j +1 ). If Q2 2 M(R) ~ 22 has the same sign at all real U -zeros of N~C 2 , then is R-unimodular, equivalently (13) holds. exists such that D ~ must have an even number of zeros between j and j +1 D 22 b) By Theorem 1-3), reliable decentralized controllers exist ~C 2 has at least one real U -zero in each of these intervals. Since if and only if Q1 ; Q2 2 M(R) exist satisfying (13), since N ~ C 1 ; D~ C 2 are biproper. It was shown that either P12 (1) = 0 or P21 (1) = 0, the sign of D~ 22 at the U -zero such that D Q1 ; Q2 2 M(R) satisfying (13) exist if and only if of N~C 2 in the last interval (j ; j +1 ) must agree with the sign of P^ is strongly R-stabilizable. Since P22 2 M(Rs ), D~ 22 (1); hence, D~ 22 must have even number of zeros in (j ; 1). (V2 0 Q2 N~22 ) is biproper. If P12 2 M(Rs ) or This proves necessity. For any Q2 2R, the minimum number of U P21 2 M(Rs ), then P^ = P12 D22 N~C 2 P21 2 M(Rs ); zeros of N~C 2 is `, which is the number of intervals where N22 has an ~ 22 . There is Q~ 2 2 R hence, P^ is strongly R-stabilizable if and only if odd number of zeros between real U -zeros of D 0 1 ~ ~ ^ such that ( U + Q ) has exactly ` real U -zeros, with exactly one in D 2 2 22 for any 0a 2 IRnU , (s + a) P is strongly R~ 1 2 M(R) exists such that each of (j ; j +1 ), 1 k `, because (U2 + Q~ 2 D~ 22 ) has an odd stabilizable, equivalently, Q D22 + NC 2 N~21 (s + a)01 Q~ 1 N12 is R-unimodular. Let number of zeros in each of these intervals. If D~ 22 has even number of Q1 := (s + a)01 Q~ 1 2 M(Rs ); then (13) holds and real U -zeros in each of (j ; j +1 ), 1 k ` 0 1, and (j ; 1), ~ 2 D~ 22 )P21 is strongly R-stabilizable. Let then P^ = P12 D22 (U2 + Q D~ C 1 is biproper.
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~ 22 + N~21 Q1 N12 (U2 + Q~ 2 D~ 22 ) is Q1 2 M(R) be such that M1 := D R-unimodular; Q1 can be chosen strictly proper. If N~22 (1) 06=1 0, let Q2 := Q~ 2 + Q^ 2 , Q^ 2 2 R, Q^ 2 (1) 6= (V~2 0 N22 Q~ 2 )(1)N~22 (1), kQ^ 2 k < kD~ 22 M101N~21 Q1 N12 k01 . Then D~ C2 , D~ C1 are biproper, ~ 22 + N~21 Q1 N12 N~C 2 = M1 + i.e., C1 ; C2 are proper. Since D ~ 22 is R-unimodular, (C1 ; C2 ) is a reliable decenN~21 Q1 N12 Q^ 2 D tralized controller pair. Proof of Corollary 1: 1) If P22 has an even number of real U -zeros between consecutive real U -poles, the sign of N22 is the same at all real U ~ 22 . Since fj1 ; 1 1 1 ; j` g is empty, Q2 2 R exists zeros of D ~C 2 has no U -zeros (the only blocking U -zeros of such that N ~ C 2 is biproper, N12 N~C 2 X21 are those of P12 , P21 ) and D ^ ~C 2 P21 is i.e., C2 is proper. For this Q2 , P = P12 D22 N strongly R-stabilizable. Since (s + a)01 P^ is strongly R~ 1 2 M(R) exists such that stabilizable for any 0a 2 IRnU , Q D22 + (U~2 + D22 Q2 )N~21 (s + a)01 Q~ 1 N12 is R-unimodular; ~ C 1 is biproper. Q1 = (s + a)01 Q~ 1 satisfies (13) and D 2) By assumption, the sign of D22 is the same at all real U -zeros of P12 , P21 , and P22 as the sign of D22 (1); hence, P22 is ~ C 2 is Rstrongly R-stabilizable. Let Q2 2 R be such that D ~ unimodular, then D22 has the same sign at all real U -zeros ~C 2 N22 . The sign of D22 at the real U -zeros of N~C 2 is of N ~C 2 P21 is the same as that of D22 (1); hence, P^ = P12 D22 N strongly R-stabilizable. As in the proof of 1), (s + a)01 P^ is also strongly R-stabilizable, since Q1 can be chosen strictly proper, the controllers are proper. Proof of Theorem 4: Let P22 2 Rpn 2n , rankP22 =: r, 3 := diag[1 1 1 1 r ], and 9 := diag[ 1 1 1 1 r ]; there exist Runimodular L 2 Rn 2n , R 2 Rn 2n satisfying (16), where j ; j 2 R, j is biproper, (j ; j ) is coprime, i.e., uj ; vj 2 R exist satisfying vj j + uj j = 1, j = 1; 1 1 1 ; r; j divides j +1 , form). j +1 divides j , j = 1; 1 1 1 ; r 0 1 (see [9], Smith–McMillan 01 = D~22 01 N~22 are given By (16), any RCF and LCF P22 = N22 D22 ~ 2 M(R); let UD := in (17) and (18) for some R-unimodular M; M diag[u1 1 1 1 ur ], VD := diag[v1 1 1 1 vr ]; then (VD 9 + UD 3) = Ir ; U2 ; V2 in (9) are given by (19), where A^ 2 Rn 2n
P22 = L diag[3; 0(n 0r)2(n 0r) ] diag[901 ; I(n 0r) ]R = L diag[901 ; I(n 0r) ] diag[3; 0]R (16)
(N22 ; D22 ) = (L diag[3; 0(n 0r)2(n 0r) ]M; R01 diag[9; I(n 0r) ]M ) (D~ 22 ; N~22 ) = (M~ diag[9; I(n 0r) ]L01 ; ~ diag[3; 0(n 0r)2(n 0r) ]R) M
~ 22 U2 = M 01 diag[UD ; 0(n 0r)2(n 0r) ]L01 + A^D 0 1 V2 = M diag[VD ; I(n 0r) ]R 0 A^N~22 :
(17)
(18)
(19)
Let Q11 2 Rr2r be any upper-triangular matrix whose nondiagonal entries qij 6= 0 are constants, for i; j = 1; 1 1 1 ; r, j > i. For j = 1; 1 1 1 ; r, choose qjj 2 R as follows: Let Z12 , Z21 be the sets of all real U -zeros of P12 and of P21 , respectively; let Z := Z12 Z21 = fz1 ; 1 1 1 ; z` g. Let Zj = fzj1 ; 1 1 1 ; zj` g Z be such that uj (z ) 6= 0 for z 2 Zj . Define qjj 2 R as qjj = ` qjj (1) k=1 (s 0 zjk )(s + a)01 ; 0a 2 IRnU , qjj (1) 2 IRnf0g is such that (vj 0 qjj j )(1) 6= 0; this holds for all qjj (1) when 1 j 2 Rs ; when j 62 Rs , take qjj (1) 6= vj 0 j (1). If Zj is empty, then qjj = qjj (1) 2 IR. With this qjj 2 R, (uj + qjj j ) does not have zeros at any of the real U -zeros of P12 or P21 . If uj = 0, then j is a unit in R and if u1 = 0, then P22 2 M(R).
1) Choose
Q11 Q12 ~ 01 0 Q22 M Q12 2 Rr2(n 0r) , Q22 2 R(n 0r)2(n 0r) has no real blocking U -zeros and no zeros at any real U -zeros of P12 and P21 ; obvious choices for Q22 are any R-unimodular matrix or ~C 2 = (U2 +Q2 D~ 22 ) has the identity In 0r . By construction, N Q2 = 0A^ + M 01
no real blocking U -zeros and no U -zeros coinciding with any real U -zeros of either P12 or P21 . Since rankP12 = ni2 = no2 , 01 if and only if zo 2 U is a U -zero of P12 = N12 D22 rankN12 (zo ) < no2 . Similarly, zo 2 U is a U -zero of P21 = X21 Y2101 if and only if rankX21 (zo ) < no2 . By Lemma 2, P^ is ~ 22 has the same sign strongly R-stabilizable if and only if det D ~C 2 X21 , which are the real at all real blocking U -zeros of N12 N blocking U -zeros of P12 , of P21 and possibly some of the real U -zeros common to P12 and P21 . We prove that no other real blocking U -zeros exist by contradiction: Suppose zo 2 IR \U is ~C 2 X21 (zo ) = 0 but N12 (zo ) 6= 0, X21 (zo ) 6= such that N12 N 0, and zo is not a common zero of P12 and P21 ; since zo may be a zero of one of P12 or P21 , there are two cases: 1) If P12 (zo ) 6= 0, P21 (zo ) 6= 0, then rankN12 (zo ) = ni2 = no2 , rankX21 (zo ) = no2 ; hence, N12 (zo ) 2 IRn 2n has a left^12 and X21 (zo ) 2 IRn 2n has a right-inverse inverse N ^ X21 . Therefore, N12 N~C 2 X21 (zo ) = 0 implies N~C 2 (zo ) = 0, which is a contradiction; and 2) If zo is a zero of either P12 ~C 2 (zo ) = no2 . Either X^ 21 2 IRn 2n or P21 , then rankN ^ 21 = I (if P21 (zo ) 6= 0) or N^12 2 exists such that X21 (zo )X 2 n n ^12 N12 (zo ) = I (if P12 (zo ) 6= 0). exists such that N IR ~ Therefore, N12 NC 2 X21 (zo ) = 0 implies either N12 (zo ) = 0, or X21 (zo ) = 0; again we have a contradiction. Since P12 or P21 is strictly proper, the sign of det D22 is the same at all of these real blocking U -zeros as the sign of det D22 (1); ~C 2 P21 is strongly R-stabilizable. By therefore P^ := P12 D22 N ~ (16) and (19), DC 2 in (11) is biproper, i.e., C2 is proper since det (VD 0 Q11 3)(1) 6= 0. Since P12 2 M(Rs ) or P21 2 M(Rs ), Q1 2 M(R) satisfying (13) can be chosen strictly proper so that C1 is proper. 01) = no2 = ni2 , then rankN22 = 2) If rankP22 = rank(N22 D22 no2 . By (10), det U2 and det N22 have the same sign at all real blocking U -zeros of D22 . By (16), the real blocking U zeros of D22 are those of its smallest invariant factor n . Since the U -zeros of P22 are those of det N22 , the sign of det N22 is the same at all real U -zeros of n . Therefore, q 2 R exists such that (det U2 + q n ) is a unit of R ~C 2 is R-unimodular implies Q2 2 M(R) exists such that N ~C 2 is R-unimodular, the only real [9]. We show that since N ~C 2 X21 are the blocking U -zeros of blocking U -zeros of N12 N P12 , or P21 and possibly some of the real U -zeros of P12 or of P21 : If zo 2 IR \ U is such that P12 (zo ) 6= 0, P21 (zo ) 6= 0, then rankN12 (zo ) + rankX21 (zo ) 0 no2 > 0. There~C 2 X21 (zo ) rankN12 (zo )+rankX21 (zo ) 0 fore, rankN12 N no2 > 0 implies N12 N~C 2 X21 (zo ) 6= 0. Since the signs of det D22 and det D22 (1) are the same at all real U -zeros of P12 and P21 , by Lemma 2, P^ is strongly R-stabilizable. Since P22 2 M(Rs ), the existence of proper controllers follows from Theorem 2-2)-b). 01 has an R-stable left 3) First we show that P12 = N12 D22 inverse if and only if N12 2 Rn 2n has an R-stable left I 2 Rn 2n ; if P I 2 Rn 2n exists such inverse N12 12 I I N12 = D22 . Since (N12 ; D22 ) is that P12 P12 = I , then P12 I + U12 )N12 = I implies (V12 P I + right-coprime, (V12 P12 12 I 2 U12 ) 2 M(R) is a left-inverse of N12 . Conversely, if N12
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I P12 = D01 exists such that N1I2 N12 = I , then N12 22 I P12 = I ; hence, D22 N I 2 M(R) is a leftimplies D22 N12 12 ~21 ~ 01 N inverse of P12 . It can be shown similarly that P21 = D 22 2 n n ~ has an R-stable right-inverse if and only if N21 2 R ~ I 2 Rn 2n . Construct Q11 2 Rr2r has a right-inverse N 21 with qjj chosen as above; since rankP12 (s) = ni2 and rankP21 (s) = no2 for all s 2 U , P12 and P21 have no U -zeros. Therefore, Q11 is chosen so that the nondiagonal entries qij 6= 0 are constants, and qjj 2 R are such that ~C 2 has no real (vj 0 qjj j )(1) 6= 0. To guarantee that N blocking U -zeros, let
Rn 2n
Q11 Q12 ~ 01 M 2 Rn 2n Q21 Q22 Q12 2 Rr2(n 0r) , Q21 2 R(n 0r)2r , Q22 2 R(n 0r)2(n 0r) can be arbitrary if both no2 > 1 and ni2 > 1; if ni2 = 1, let Q12 2 IR12(n 01) be nonzero real; if no2 = 1, let Q21 2 IR(n 01)21 be nonzero I ~I Q ^ ~ real. Let Q1 := N 21 1 N12 2 M(R). Since NC 2 has ^ 1 2 M(R) exists such that no real blocking U -zeros, Q I ~ 22 + N ~I Q ^ ~ ~21 N ~ ^ ~ D 21 1 N12 N12 NC 2 = D22 + Q1 NC 2 is R~ unimodular, i.e., (14) holds. Since DC 2 is biproper, C2 is ^ 1 2 M(R) such that D ^1N ~ 22 + Q ~C 2 proper. There is Q ~ 1 2 M(R) exists such is R-unimodular if and only if Q ~ 1N ~C 2 is R-unimodular; choosing ~ 22 + (s + a)01 Q that D 0 1 ~ Q1 = (s + a) Q1 2 M(Rs ) implies C1 is proper. 4) Let CS be any R-stable R-stabilizing controller for P22 . 01 satisfy Without loss of generality, let the RCF P22 = N22 D22 D22 + CS N22 = I ; hence, N22 D22 = (I 0 N22 CS )N22 . ~2 = Then U2 = CS + T (I 0 N22 CS ), V2 = I 0 T N22 , U CS + D22 T , V~2 = I 0 N22 T , T 2 M(R) satisfy (10). By ~ 01 N ~21 implies N ~21 has a right-inverse assumption, P21 = D 22 I ~ 12 D 01 = P22 = N22 D 01 ~ ~ 12 = LN N21 2 M(R). Also, LP 22 22 ~ 12 = N22 . Let C1 , C2 be given by (11) and implies LN ~ 2 Rn 2n , Q2 = 0T . Then (13) ~I L (12), Q1 = N 21 becomes D22 + CS N22 = I . Since P11 , P12 2 M(Rs ), ~21 ) 2 M(Rs ) implies D ~ C 1 is biproper, i.e., (N11 0 N12 Q2 N C1 2 M(Rp ). Since C2 = CS 2 M(R), (C1 ; C2 ) is a Q2 = 0A^ + M 01
reliable decentralized controller pair. 5) Let CS be as in 4); let D22 + CS N22 = I . By assumption, 01 implies N12 has a left-inverse N12 I 2 M(R). P12 = N12 D22 0 1 ~ 0 1 ~ ~ ~ ~ ~ Also, P21 R = D22 N21 R = P22 = D N22 implies N~21 R~ = N~22 = N22 . Let C1 , C2 be given by (11) and (12), ~ I 2 Rn 2n , Q2 = 0T . The conclusion follows Q1 = RN 12 as in 4). ~ 12 = P22 implies 6) Choose CS as in 4). By assumption, LP ~ 12 = N22 , and P21 R ~ = P22 implies N ~21 R ~ = LN N~22 = N22 . Let Q2 = 0T , Q1 = R~ Q^1 L~ , Q^1 = k rm k0m (CS N22 )m02 CS ; k is any integer such m=2 that k > kCS N22 k and rm are the binomial coefficients. By (13) D22 + CS N22 Q^1 N22 = I 0 CS N22 + k rm k0m (CS N22 )m = (I 0 k01 CS N22 )k is Rm=2 unimodular. Then C1 2 M(Rs ) since Q^1 ; Q1 2 M(Rs ). Since C2 = CS is proper, (C1 ; C2 ) is a reliable decentralized controller pair. REFERENCES [1] A. N. G¨unde¸s and C. A. Desoer, Algebraic Theory of Linear Feedback Systems with Full and Decentralized Compensators, Lecture Notes in Control and Information Sciences, vol. 142. Berlin, Germany: Springer-Verlag, 1990.
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[2] K. D. Minto and R. Ravi, “New results on the multi-controller scheme for the reliable control of linear plants,” in Proc. American Control Conf., 1991, pp. 615–619. [3] C. N. Nett, “Algebraic aspects of linear control system stability,” IEEE Trans. Automat. Contr., vol. 31, pp. 941–949, 1986. ¨ uler, “Reliable decentralized stabiliza¨ [4] K. A. Unyelio˘ glu and A. B. Ozg¨ tion of feedforward and feedback interconnected systems,” IEEE Trans. Automat. Contr., vol. 37, pp. 1119–1132, 1992. ˇ [5] D. D. Siljak, “On reliability of control,” in Proc. 17th IEEE Conf. Decision and Control, 1978, pp. 687–694. , “Reliable control using multiple control systems,” Int. J. Contr., [6] vol. 31, pp. 303–329, 1980. ˇ [7] X. L. Tan, D. D. Siljak, and M. Ikeda, “Reliable stabilization via factorization methods,” IEEE Trans. Automat. Contr., vol. 37, pp. 1786–1791, 1992. [8] M. Vidyasagar and N. Viswanadham, “Algebraic design techniques for reliable stabilization,” IEEE Trans. Automat. Contr., vol. 27, pp. 1085–1095, 1982. [9] M. Vidyasagar, Control System Synthesis: A Factorization Approach. Cambridge, MA: M.I.T. Press, 1985.
Useful Nonlinearities and Global Stabilization of Bifurcations in a Model of Jet Engine Surge and Stall Miroslav Krsti´c, Dan Fontaine, Petar V. Kokotovi´c, and James D. Paduano
Abstract— Compressor stall and surge are complex nonlinear instabilities that reduce the performance and can cause failure of aircraft engines. We design a feedback controller that globally stabilizes a broad range of possible equilibria in a nonlinear compressor model. With a novel backstepping design we retain the system’s useful nonlinearities which would be cancelled in a feedback linearizing design. The design control law is simple and, moreover, it is optimal with respect to a meaningful nonquadratic cost functional. As in a previous bifurcationtheoretic design, we change the character of the bifurcation at the stall inception point from subcritical to supercritical. However, since we do not approach bifurcation control using a normal form but using Lyapunov tools, our controller achieves not only local but also global stability. The controller requires minimal modeling information (bounds on the slope of the stall characteristic and the B -parameter) and simpler sensing (rotating stall is stabilized without measuring its amplitude). Index Terms—Axial flow compressors, backstepping, bifurcation control, jet engines, rotating stall, surge.
I. INTRODUCTION In control engineering the importance of qualitative low-order nonlinear models is twofold. First, they can capture the dominant dynamic phenomena; second, they are testbeds which help refine new nonlinear design methods. One such model, the Moore–Greitzer Manuscript received March 14, 1997. This work was supported in part by the Air Force Office of Scientific Research under Grants F496209610223 and 95-1-0409 and in part by the National Science Foundation under Grant ECS9624386. M. Krsti´c is with the Department of AMES, University of California at San Diego, La Jolla, CA 92093-0411 USA (e-mail: [email protected]). D. Fontaine and P. V. Kokotovi´c are with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106 USA. J. D. Paduano is with the Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139 USA. Publisher Item Identifier S 0018-9286(98)08423-2.
0018–9286/98$10.00 1998 IEEE
Proof: First of all, notice that ( 1=2 A; B ) is stabilizable if and ^) ^ 1=2 A only if ( 1=2 A; 1=2 B ) is stabilizable. From Lemma A1, (C; is detectable. Notice now that (5) can be written as
X = J^0 J^ + ( 1=2 A0 )X ( 1=2 A)
0 (( 1=2A0 )X ( 1=2B) + J^0 D^ ) 1 (D^ 0D^ + ( 1=2B0 )X ( 1=2B))01 1 (( 1=2B0 )X ( 1=2A) + D^ 0J^)
and the lemma follows from standard results on ARE’s (see [6, p. 263 and Appendix B]). Proof of Proposition 2.3: Consider in (5) Y = Y^ Y^ : From ~) is de~ 1=2 A Lemma A.2, if ( 1=2 A; B ) is stabilizable and (C; ^ tectable, then there exists a unique positive semidefinite solution X ^ is as ^ ) is stable, where K to (5), and it is such that 1=2 (A + B K ^ and Y = Y ^ : If we now set Y = Y ^ we can in (A1) with X = X ^ conclude similarly the existence of a unique semipositive solution X ~ and Y = Y ~ given by (A1) with X = X ~ : After some of (5) and a K algebraic manipulation, we get that ^ (X
0 X~ ) 0 (A + BK^ )((Y^ + X^ ) 0 (Y + X~ ))(A + BK^ ) ~ + X ~ )B )(K ~ 0K ^) ~ 0K ^ )0 (D 0 D + B 0 (Y = (K
and thus ^ 0X ^ ~ ) 0 (A + B K ^ )0 (X (X ^ ^ )0 (Y = (A + B K
0 X~ )(A + BK^ )
0 Y~ )(A + BK^ )
0 K^ )(D0 D + B0 (Y~ + X~ )B)01 (K~ 0 K^ ) ^ 0X ^ ) that X ~ 0: and it follows from stability of 1=2 (A + B K ~ + (K
[12] Y. Ji, H. J. Chizeck, X. Feng, and K. A. Loparo, “Stability and control of discrete-time jump linear systems,” Control Th. and Adv. Tech., vol. 7, pp. 247–270, 1991. [13] A. Laub, “Algebraic aspects of generalized eigenvalue problems for solving Riccati equations,” in Computational and Combinatorial Methods in Systems Theory, C. Byrnes and A. Lindquist, Eds. Amsterdam, The Netherlands: Elsevier, 1986, pp. 213–227. [14] M. Mariton, Jump Linear Systems in Automatic Control. New York: Marcel Dekker, 1990. [15] L. Vandenberghe and S. Boyd, Software for Semidefinite Programming—User’s Guide. Available via anonymous ftp from isl.stanford.edu /pub/boyd/semidef prog, 1994. [16] H. K. Wimmer, “Monotonicity and maximality of solutions of discretetime algebraic Riccati equations,” J. Math. Syst., Estimation and Contr., vol. 2, no. 2, pp. 219–235, 1992.
Reliable Decentralized Stabilization of Linear Systems A. N. G¨unde¸s
Abstract— Reliable stabilization of linear time-invariant multiinput/multi-output plants is considered using a two-channel decentralized controller configuration. Necessary and sufficient conditions are obtained for existence of reliable controllers that maintain stability under the possible failure of either one of the two controllers. All decentralized controllers that achieve reliable stabilization are characterized. Index Terms—Controller design, decentralized control, reliable stabilization.
For a general account on the positive semidefinite partial ordering of maximal solutions of discrete-time ARE’s, see [16]. REFERENCES [1] H. Abou-Kandil, G. Freiling, and G. Jank, “On the solution of discretetime Markovian jump linear quadratic control problems,” Automatica, vol. 31, no. 5, pp. 765–768, 1995. [2] M. Ait-Rami and L. E. Ghaoui, “LMI optimization for nonstandard Riccati equations arising in stochastic control,” IEEE Trans. Automat. Contr., vol. 41, pp. 1666-1671, Nov. 1996. [3] O. L. V. Costa, J. B. R. do Val, and J. C. Geromel, “A convex programming approach to 2 -control of discrete-time Markovian jump linear systems,” Int. J. Contr., vol. 66, pp. 557–579, 1997. [4] O. L. V. Costa and M. D. Fragoso, “Stability results for discrete-time linear systems with Markovian jumping parameters,” J. Math. Analysis and Appl., vol. 179, pp. 154–178, 1993. , “Discrete-time LQ-optimal control problems for infinite Markov [5] jump parameter systems,” IEEE Trans. Automat. Contr., vol. 40, pp. 2076–2088, 1995. [6] M. H. A. Davis and R. B. Vinter, Stochastic Modeling and Control. London, U.K.: Chapman and Hall, 1985. [7] L. El Ghaoui, R. Nikoukhah, and F. Delebecque, LMITOOL: A FrontEnd for LMI Optimization—User’s Guide. Available via anonymous ftp from ftp.ensa.fr /pub/elghaoui/lmitool, 1995. [8] M. D. Fragoso, J. B. Ribeiro do Val, and D. L. Pinto Jr., “Jump linear 1 -control: The discrete-time case,” Contr. Th. and Adv. Tech., vol. 10, pp. 1459–1474, 1995. [9] Z. Gajic and I. Borno, “Lyapunov iterations for optimal control of jump linear systems at steady state,” IEEE Trans. Automat. Contr., vol. 40, no. 11, pp. 1971–1075, 1995. [10] J. C. Geromel, P. L. D. Peres, and S. R. Souza, “ 2 -guaranteed cost control for uncertain discrete-time linear systems,” Int. J. Contr., vol. 57, pp. 853–864, 1993. [11] Y. Ji and H. J. Chizeck, “Controllability, observability and discretetime Markovian jump linear quadratic control,” Int. J. Contr., vol. 48, pp. 481–498, 1988.
H
H
H
1733
I. INTRODUCTION We consider reliable stabilization of linear time-invariant (LTI), multi-input/multi-output (MIMO) plants under possible sensor or actuator failures using a two-channel decentralized feedback control configuration. The goal is to maintain closed-loop stability when both controllers act together and when either one of the two controllers acts alone. It is assumed that the failure of a controller is recognized and it is taken out of service (i.e., the states in the controller implementation are all set to zero, the initial conditions and outputs of the channel that failed are set to zero for all inputs). Since the introduction of multi-controller systems in [5] and [6], reliable stabilization has been studied for various failure models using full-feedback [2], [8] and decentralized configurations [4], [7]. Conditions for existence of reliable decentralized controllers were given for a class of reliable stabilization problems using genericity arguments in [4]. The reliable stabilization problem considered in this paper is based on the twochannel decentralized configuration and failure model in [7]. The necessary and sufficient conditions here for existence of reliable decentralized controllers include generalizations of the sufficient conditions in [7]. The main results in this paper are the explicit existence conditions for reliable decentralized controllers. Theorem 2 gives an important interpretation of these conditions in terms of the strong stabilizability Manuscript received January 10, 1996; revised February 7, 1997 and August 1, 1997. This work was supported by the NSF under Grant ECS-9257932. The author is with Electrical and Computer Engineering Department, University of California, Davis, CA 95616 USA (e-mail [email protected]). Publisher Item Identifier S 0018-9286(98)09444-6.
0018–9286/98$10.00 1998 IEEE
1734
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 12, DECEMBER 1998
Fig. 1. The decentralized system
Fig. 2. The system
S (P; C1 ):
Fig. 3. The system
S (P; C2 ):
S (P; CD ):
of an associated system and states that strong stabilizability of two of the subblocks of the plant is necessary. Proposition 1 gives a parameterization of all reliable decentralized controllers for stable plants, and Theorem 3 establishes explicit existence conditions when one channel is single-input/single-output (SISO). Theorem 4 gives important sufficient conditions when all channels are MIMO. The following notation is used throughout; due to the input–output approach, the setting can be continuous-time or discrete-time. Notation: Let U denote the region of instability; U contains the extended closed right half-plane (for continuous-time systems) or the complement of the open unit-disk (for discrete-time systems). Let Rp (Rs ) denote proper (strictly proper) rational functions with real coefficients; R denotes proper rational functions with no poles in U ; M(R) denotes the set of matrices with entries in R; M is called R-stable iff M 2 M(R); and M 2 M(R) is called R-stable unimodular iff M 01 2 M(R). A right-coprime factorization (RCF), a left-coprime factorization (LCF), and a bicoprime factorization ~ = ~ 01 N (BCF) of P 2 M(Rp ) are denoted by P = N D01 = D 0 1 ~ ; D; ~ Nbr ; Db ; Nbl ; Gb 2 M(R), D , Nbr Db Nbl + Gb ; N; D; N ~ , and Db are biproper. Let rankP = r ; so 2 U is called a U -zero D (blocking U -zero) of P iff rankP (so ) < r(P (so ) = 0); the poles of P in U are called its U -poles. For M 2 M(R), the norm k 1 k is defined as kM k = sup s2@ U (M (s)); denotes the maximum singular value and @ U denotes the boundary of U .
by (3); a similar description can be obtained for S (P; C2 ). The ~L2 u2 , N ~ D2 ~2 = N ~R2 ~2 = y2 system S (P; C2 ) is described as D in (4); the description for S (P; C1 ) is similar. For j = 1; 2, the transfer function of S (P; Cj ) from (uP ; uCj ) to (yP ; yCj ) is
01 NLj = N~Rj D~ 01 N~Lj Hj = NRj DDj Dj
~CD + N ~C N )P = [D ~C (D
N P + D
~C 1 [N
CD =
C1 0
P12 P22 0
C2
2 Rpn 2n ; 2 Rpn 2n
;
Pjj Cj
2 Rpn 2n
2 Rpn 2n
(1)
no = no1 + no2 , ni = ni1 + ni2 . It is assumed that P and CD have no hidden modes corresponding to eigenvalues in and that (P; CD ) is well-posed. The failure of the j th controller channel is represented by setting Cj = 0; the corresponding j th channel output yCj is also
U
set to zero. When the second (first) channel fails, the system is called
S (P; C1 ) shown in Fig. 2 (S (P; C2 ) shown in Fig. 3).
~ 01 N Using any RCF P = N D01 , any LCF Cj = D c ~c , j = 1; 2, ~ C1; D ~ C 2 ], N ~C 1 ; N ~C 2 ], DP = eP , ~ C = diag[D ~C = diag[N D T uT ]T , u = [uT uT ]T , y = [y T y T ]T , N P = yP , uP = [uP C P 1 P2 C1 C2 P1 P2 T y T ]T , e T T T yC = [yC P = [eP 1 eP 2 ] , the system S (P; CD ) is 1 C2 described in (2) as DD P = NL u, NR P + Gu = y ; S (P; CD ) 01 NL + G from is well-posed, i.e., the transfer-function H = NR DD (uP ; uC ) to (yP ; yC ) is proper, if and only if DD is biproper. The description of S (P; C1 ) as DD1 1 = NL1 u1 , NR1 1 = y1 is given
0
0
In
P yC 1
~ D
0N~
[0
Ino2 ] 0
In
0
NC 2
=
yP yC 1
0
0
0
(2)
P yC 1
yP C 2
(3)
yP C 2
NC 2 ~ C2 D
~ N
I
uP uC
uP uC 1
0
~C 1 N
0
N
yP yC
0 ~ C1 D
0 ]N
=
S
=
0In
I
=
Consider the LTI, MIMO, and two-channel decentralized control system S (P; CD ) shown in Fig. 1. The plant and the decentralized controller are represented by their transfer functions P and CD , respectively P11 P21
uP uC
0 0
D
II. MAIN RESULTS
P =
0
0I
~C ] N
uP uC 2 =
yP : yC 2
(4)
Reliable Stability: The system S (P; CD ) is said to be R-stable iff 2 M(R); similarly, S (P; Cj ) is R-stable iff Hj 2 M(R). The decentralized controller CD is said to be an R-stabilizing controller for P iff CD is proper and S (P; CD ) is R-stable. The pair (C1 ; C2 ) is called a reliable decentralized controller pair iff C1 , C2 are proper and the systems S (P; CD ), S (P; C1 ), S (P; C2 ) are all R-stable. Lemma 1 gives necessary and sufficient conditions for R-stability of S (P; CD ) under normal operation and under the complete failure of one of the controllers. We assume that the coprime factorizations are in canonical forms; the denominator-matrix of any RCF, LCF can H
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 12, DECEMBER 1998
be put into upper (lower) triangular Hermite forms by elementary column (row) operations [9], [1]. Without loss of generality, it is ~ 01 N ~ are given by assumed that the RCF and LCF P = N D01 = D P
= ND
01 =
~ ~ 01 N =D
=
N11
N12
N21
N22
~ 11 D
~ 12 D ~ 22 D
0
0
D21
D22
01
~11 N ~21 N
C 1 D11 + N~C 1 N11 C 2 D21 + N~C 2 N21
~12 N ~22 N
(6) is
R-unimodular,
~ C 2 D22 + N ~C 2 N22 ) (D
and ~ D
C 1 D11 + N~C 1 N11
R-unimodular
is
~ N
C 1 N12 is R-unimodular:
D21
D22
(7) (8)
To obtain a parameterization of all reliable decentralized controller pairs, we use the following characterization of all admissible plants for stability using one controller [1], [3]: Let P 2 Rpn 2n be partitioned as in (1). There exists a decentralized controller CD = diag[C1 ; C2 ] such that S (P; CD ) and S (P; C2 ) are R-stable if and ~ of the form ~ 01 N only if P has an RCF and LCF P = N D01 = D P
=
ND
01 =
~ 01 N ~ = =D
N11
~2 N ~21 V
n
N12
0
N22
N12 U2
I
0
n
0 ~2 ~21 U N
01
N11
~21 N
~ 22 D
01
0
I
D22 N12 V2
~22 N
(9)
~21 2 Rn 2n , , N12 2 Rn 2n , N ~ ~ (N22 ; D22 ) is right-coprime, (D22 ; N22 ) is left-coprime, and ~2 ; V ~2 ; U2 ; V2 2 M(R) satisfy (10); equivalently, P11 0 U ~ 22 P21 2 M(R), P12 D22 U2 P21 2 M(R), P12 D22 2 M(R), D 01 is an RCF and D~ 01 N~22 is an LCF of P22 and U2 where N22 D22 22 satisfies (10)
where
N11
2 Rn 2n
V2
0 ~22 N
U2
~ 22 D
D22 N22
0 ~2 U
~2 V
=
n
0
I
0
:
(10)
n Rn 2n have an I
0
Theorem 1—Stabilizing Controllers: Let P 2 ~ 01 N ~ satisfying (9). The system RCF and LCF P = N D01 = D S (P; C2 ) is R-stable if and only if C2 is given by (11), Q2 2 Rn 2n is such that D~ C2 is biproper [holds for all Q2 2 M(R) when P22 2 M(Rs )]; S (P; CD ) and S (P; C2 ) are both R-stable if and only if C2 and C1 are given by (11) and (12), and Q1 2 Rn 2n is such that D~ C1 is biproper [holds for all Q1 2 M(R) ~21 ) 2 M(Rs )]; S (P; CD ), S (P; C2 ), and when (N11 0 N12 Q2 N S (P; C1 ) are all R-stable (i.e., (C1 ; C2 ) is a reliable decentralized controller pair) if and only if C2 and C1 are given by (11) and (12), where Q1 2 Rn 2n , Q2 2 Rn 2n satisfy condition (13), or
01 (U2 + Q2 D~ 22 )
~22 ) Q2 N
C2 ~2 + D22 Q2 )(V ~2 = (U
(5)
:
~ D
D11
~ 01 N =D C 2 ~C 2
01 = NC 2 D
~C 1 N12 N is R-unimodular C 2 D22 + N~C 2 N22
~ D
C2
= (V2
~ 01 N ~ be any Lemma 1—Decentralized Stability: Let N D01 = D 0 1 ~ 0 1 n 2 n ~ ; let DC NC = NC DC be any LCF, RCF, LCF of P 2 Rp ~ C1; D ~ C = diag[D ~C = diag[N ~ C 2 ], N ~C 1 ; N ~C 2 ], RCF of CD , D NC = diag[NC 1 ; NC 2 ], DC = diag[DC 1 ; DC 2 ]. The system S (P; CD ) is R-stable if and only if DD := (D~ C0D1 + N~0C 1N ) ~ ~ N is R-unimodular. Let the RCF and LCF P = N D = D be as in (5); S (P; CD ) is R-stable if and only if (6) holds. Let 01 ~ 01 N C2 = D system S (P; C2 ) C 2 ~C 2 = NC 2 DC 2 be any LCF, RCF; the 01 ~ 01 N is R-stable if and only if (7) holds. Let C1 = D C 1 ~C 1 = NC 1 DC 1 be any LCF and RCF; S (P; C1 ) is R-stable if and only if (8) holds ~ D
~ C1, D ~ C 2 are biproper equivalently (14), and D
01
D11
1735
C1
D22
~ 22 D
~ 01 N =D C 1 ~C 1 = (I
0
0
01
N22 Q2 )
Q1 (N11
0
~2 + D22 Q2 )N ~21 Q1 N12 + (U
~21 Q1 N12 (U2 + Q2 D ~ 22 ) +N
(11)
01 Q1
~21 )) N12 Q2 N
is
R-unimodular
is
R-unimodular
(12) (13)
:
(14)
Conditions (13) and (14) lead to the conditions that P must satisfy for the existence of reliable decentralized controller pairs as stated in Theorem 2. Strong R-stabilizability of pseudo-systems related to P are important for existence of reliable decentralized controllers. The following are well known [9]: An LTI system P^ is said to be strongly R-stabilizable iff an R-stable R-stabilizing controller ^ exists for P ^ . In the standard full-feedback system S (P ^; C ^ ^ ), P C is strongly R-stabilizable if and only if it has an even number of U -poles between consecutive pairs of real blocking U -zeros. Let 01 ^ = Np D 01 = D ~ 01 N P p p ~p = Nbr Db Nbl + Gb be any RCF, LCF, 0 1 ~ ^ ^ ~ ^ 2 M(R) if and and BCF of P ; let C = Dc Nc be any LCF; C ~ c is unimodular. Therefore, P ^ is strongly R-stabilizable only if D ~ Np is R~ 2 M(R) exists such that Dp + X if and only if X ~p + N ~p X unimodular; equivalently, X 2 M(R) exists such that D is R-unimodular; equivalently, Xb 2 M(R) exists such that I
+ Xb Gb
0
bl
N
b br b
X N D
is
R-unimodular
:
Lemma 2—Coprime Factorizations and Strong Stabilizability: Let 2 Rpn 2n , partitioned as in (1), have an RCF and LCF P = 01 = D~ 01 N~ of the form (9). Let N12 D01 be an RCF, ND 22 01 ~ 12 be an LCF of P12 ; let D ~ 01 N ~ ~ 01 X Y 12 22 21 be an LCF and X21 Y21 ^ ~ ~ be an RCF of P21 . Define P := P12 (U2 + D22 Q2 )D22 P21 = ~ 22 )P21 . P12 D22 (U2 + Q2 D 0 1 ~ ~21 is a BCF, N12 (U2 + 1) N12 D22 (U2 + D22 Q2 )N 0 1 01 X~ 12 (U~2 + D22 Q2 )N~21 ~ is an RCF, and Y~12 Q2 D22 )X21 Y21 ^ is an LCF of P . ~ 22 has the same 2) P^ is strongly R-stabilizable if and only if det D ~ 22 )X21 , sign at all real blocking U -zeros of N12 (U2 + Q2 D equivalently, det D22 has the same sign at all real blocking U -zeros of X~12 (U~2 + D22Q2 )N~21. Theorem 2—Conditions for Reliable Decentralized Stabilizability: Let P 2 Rpn 2n be partitioned as in (1). 1) If there exists a reliable decentralized controller pair (C1 ; C2 ), then the following four necessary conditions hold: 1) P has ~ 01 N ~ satisfying (9); 2) in (9), an RCF and LCF N D01 = D 0 1 0 1 ~ ~ N12 D22 is an RCF of P12 , D22 N 21 is an LCF of P21 ; 3) P12 , P21 are strongly R-stabilizable; and 4) the sign of det D22 is the same at all real blocking U -zeros of P12 and at all real blocking U -zeros of P21 . ~ 01 N ~ satisfying (9); 2) Let P have an RCF and LCF N D01 = D 0 1 0 1 ~ ~ let N12 D22 be an RCF of P12 , D22 N21 be an LCF of P21 . P
a)
There exist Q1 ; Q2 2 M(R) satisfying (13), or ~2 + equivalently (14), if and only if P^ = P12 (U ~ 22 P21 = P12 D22 (U2 + Q2 D ~ 22 )P21 is D22 Q2 )D strongly R-stabilizable for some Q2 2 M(R).
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b)
Let P22 2 M(Rs ) and let P12 or P21 be strictly proper. There exists a reliable decentralized controller pair (C1 ; C2 ) if and only if P^ is strongly R-stabilizable for some Q2 2 M(R).
Condition 4) of Theorem 2-1) implies 3); the two conditions are equivalent when P12 ; P21 2 M(Rs ). By Theorem 2, if a reliable decentralized controller pair exists, then Q2 2 M(R) exists such ~2 + D22 Q2 )D ~ 22 P21 is strongly R-stabilizable. When that P^ = P12 (U P 2 M(Rs ), strong R stabilizability of P^ becomes necessary and sufficient for existence of reliable decentralized controller pairs. We parameterize all reliable decentralized controller pairs for R-stable plants in Proposition 1. Explicit necessary and sufficient conditions for existence of reliable decentralized controller pairs are stated in Theorem 3 for the important special case when at least one control channel has only one input and one output. Proposition 1—Reliable Decentralized Stabilization for Stable Plants: Let P 2 Rn 2n be R-stable. Then there exists a reliable decentralized controller pair. Furthermore, all reliable decentralized controller pairs (C1 ; C2 ) are parameterized by (15), where Cj is proper if and only if Qj 2 M(R) is such that (I 0 Qj Pjj ) is biproper, which holds for all Qj 2 M(R) when Pjj 2 M(Rs )
f(C1; C2 )jCj = (I 0 Qj Pjj )01 Qj ; Qj 2 M(R); j = 1; 2; (15) I 0 Q1 P12 Q2 P21 is R-unimodularg: By Theorem 2, if reliable decentralized controllers exist, then P 01 is an RCF, has an RCF and LCF satisfying (9), P22 = N22 D22 0 1 0 1 ~ N P12 = N12 D22 is an RCF, and P21 = D 22 ~21 is an LCF. Suppose P22 = 0, or P12 = 0 or P21 = 0; i.e., N22 = 0 or N12 = 0 or ~21 = 0. The pair (0; D22 ) is right-coprime if and only if D22 is N R-unimodular, i.e., P 2 M(R). So when P22 , P12 ; or P21 is zero, reliable decentralized controllers exist if and only if P is R-stable and the parameterization of all reliable decentralized controller pairs is given by (15), where (I 0 Q1 P12 Q2 P21 ) is R-unimodular for all Q1 ; Q2 2 M(R) if P12 or P21 is zero. Theorem 3—Necessary and Sufficient Conditions When P22 Is SISO: Let P 2 Rpn 2n , P11 2 Rpn 2n , P12 2 Rpn 21 , 01 P21 2 Rp12n , P22 2 Rp (i.e., no2 = ni2 = 1). Let P22 = N22 D22 be any coprime factorization. Let P12 or P21 be strictly proper. Let 1 ; 2 ; 1 1 1 ; (arranged in ascending order) denote the distinct real U -zero poles of P22 and let j ; j ; 1 1 1 ; j (arranged in ascending order) denote those distinct real U -poles of P22 for which the sign of N22 (j ) is not equal to the sign of N22 (j +1 ), 1 k `. There exists a reliable decentralized controller pair (C1 ; C2 ) if and only if the four necessary conditions of Theorem 2-1) hold and P22 has an even number of real U -poles in each of the intervals (j ; j +1 ), 1 k ` 0 1, and (j ; 1). Corollary 1—Sufficient Conditions When P22 is SISO: Let P 2 Rpn 2n , P11 2 Rpn 2n , P12 2 Rpn 21 , P21 2 Rp12n ; let 01 2 Rp be any coprime factorization. Let the four P22 = N22 D22 conditions of Theorem 2-1) hold: 1) Let the sign of D22 at the real blocking U -zeros of P12 and P21 be the same as the sign of D22 (1). Reliable decentralized controllers exist if P22 has even number of U -zeros between any pairs of its real U -poles; 2) Reliable decentralized controllers exist if the sign of D22 is the same at all real U -zeros of P22 as the sign of D22 (1). In Corollary 1-1), if P12 or P21 is strictly proper, then the sign of D22 at the real blocking U -zeros of P12 and of P21 being the same as the sign of D22 (1) follows from the necessary conditions 3) and 4) of Theorem 2-1). When P22 2 M(Rs ), the sufficient condition in Corollary 1-2) is equivalent to P22 being strongly R-
= M(Rs ), this condition implies P22 is stabilizable; when P22 2 strongly R-stabilizable. Theorem 4—Conditions for MIMO Channels: Let P 2 Rpn 2n , P11 2 Rpn 2n , P12 2 Rpn 2n , P21 2 Rpn 2n , P22 2 Rpn 2n . Let the four necessary conditions of Theorem 2-1) hold. 1) Let P12 2 M(Rs ) or P21 2 M(Rs ). Let no2 = ni2 > 1; let the sign of det D22 be the same at all common real U zeros of P12 and P21 as the sign of det D22 (1). Reliable decentralized controller pairs exist if rankP12 = ni2 no1 , rankP21 = no2 ni1 . 2) Let P12 2 M(Rs ) or P21 2 M(Rs ). Let P22 2 Rsn 2n ; let rankP22 = no2 = ni2 , rankP12 + rankP21 > no2 = ni2 ; let the sign of det D22 be the same at all real (transmission) U -zeros of P12 and of P21 as the sign of det D22 (1). Reliable decentralized controller pairs exist if the number of 01 ~ between any real (transmission) U -zeros of P22 = D~22 N 22 ~ 22 is even. pair of real blocking U -zeros of D 3) Let P22 2 Rpn 2n , where no2 and ni2 are not both equal to one. Reliable decentralized controller pairs exist if I 2 Rn 2I n P12 2 Rpn 2n has an R-stable left-inverse P12 n 2 n and if P21 2 Rp has an R-stable right-inverse P21 2 Rn 2n . n 2n I 4) Let P21 2 Rp have an R-stable right-inverse P21 2 Rn 2n . Let P11 2 M(Rs ), P12 2 M(Rs ). Let P22 be strongly R-stabilizable. Reliable decentralized controller pairs ~ 2 Rn 2n . ~ 12 = P22 for some L exist if LP I 2 5) Let P12 2 Rpn 2n have an R-stable left-inverse P12 Rn 2n . Let P11 2 M(Rs ), P12 2 M(Rs ). Let P22 be strongly R-stabilizable. Reliable decentralized controller pairs ~ = P22 for some R ~ 2 Rn 2n . exist if P21 R 6) Let P22 2 Rsn 2n . Reliable decentralized controller pairs ~ 12 = P22 for some L ~ 2 Rn 2n and P21 R ~ = P22 exist if LP n 2 n ~ for some R 2 R . Other sufficient conditions for existence of reliable decentralized controllers can be derived from the six general cases in Theorem 4. For example, under the assumptions of case 2), reliable decentralized controllers exist if either P21 has an R-stable right-inverse or P12 has an R-stable left-inverse since rankP21 = no2 or rankP12 = ni2 = no2 implies rankP12 + rankP21 > no2 . III. CONCLUSIONS We considered the design of reliable decentralized controllers that stabilize a given plant P when both controllers act together and when either one of the controllers acts alone. We showed that reliable decentralized controllers exist only if the subblocks P12 and P21 of P are strongly stabilizable. We established necessary and sufficient conditions for existence of reliable decentralized controllers when P22 is SISO and gave sufficient conditions when all subblocks of P are MIMO. We characterized all reliable decentralized controllers in the parameterizations (11), (12). Extensions to decentralized systems with more than two channels would require additional constraints on the plant. APPENDIX A. Proofs The proof of Lemma 1 follows from (2)–(4) using standard arguments. The proof of Theorem 1 follows by Lemma 1 from the assumption that P has an RCF N D01 of the form (9). Proof of Lemma 2: 1) By (10), (D22 ; NC 2 ) is left-coprime. By assumption, ~ 22 ; N ~21 ) is left-coprime, (X21 ; Y21 ) is right-coprime, (D
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 12, DECEMBER 1998
~21 2 M(R) exist such that V21 Y21 + and V21 ; U21 ; V~21 ; U U21 X21 = I , D~ 22 V~21 + N~21 U~21 = I , V21 U~21 = U21 V~21 , N~21 Y21 = D~ 22 X21 . Hence V21 + U21 DC 2 N~21 U21 N22 0NC2 N~21 D22 ~ ~ Y 0 2 N~C221X21 D~ C2 +UN~21CN2 V~2221 N~22 = I
~21 ) is left-coprime for all Q2 2 M(R). implies (D22 ; NC 2 N ^ ~ Let P := P12 NC 2 D22 P21 ; since (N12 ; D22 ) is right-coprime 01 NC 2 N~21 ~21 ) is left-coprime, P^ = N12 D22 and (D22 ; NC 2 N 01 = Y~ 01 X~ 12 , there are is a BCF. For P12 = N12 D22 12 V12 ; U12 ; V~12 ; U~12 2 M(R) such that V12 D22 + U12 N12 = I , X~ 12 U~12 + Y~12 V~12 = I , V12 U~12 = U12 V~12 . Hence
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Proof of Proposition 1: By Lemma 1, (7) and (8) hold if 01 N~Cj is an R-stabilizing controller for ~ Cj and only if Cj = D Pjj , j = 1; 2. Therefore, all C1 , C2 are given by (15). By (6), S (P; CD ) is also R-stable if and only if I 0 Q1 P12 Q2 P21 ~C is R-unimodular. The controllers are proper if and only if D is biproper. We give a solution for Qj 2 M(R) such that I 0 Q1 P12 Q2 P21 is R-unimodular and (I 0 Qj Pjj ) is biproper. Choose Q1 ; Q2 2 M(R) strictly proper; let Q1 = 01 Q1 , where 2 IR, j j > kQ1 P12 Q2 P21 k; choosing strictly proper Q1 ; Q2 is sufficient to make (I 0 Qj Pjj ) biproper, and choosing that guarantees k 01 Q1 P12 Q2 P21 k < 1 is sufficient to satisfy (6). This shows existence of reliable decentralized controllers for any R-stable P . The expression for C1 in (15) 0is1 equivalent to (12). ^ 1 (P11 0 P12 Q2 P21 )) Q^ 1 ; by (13), 8 := By (12), C1 = (I 0 Q (I + Q^ 1 P12 Q2 P21 ) is R-unimodular. With Q1 = 801 Q^ 1 , C1 =
(I + Q^ 1 P12 Q2 P21 0 Q^ 1 P11 )01Q^ 1 = (I 0 801 Q^ 1 P11 )01 801Q^ 1 ^ 1 P12 Q2 P21 ) is ~ is equivalent to (15) and 801 = (I 0 801 Q V21 + U21 (DC 2 + N22 V12 NC 2 )N21 U21 N22 U12 ~ ~ ~ R -unimodular if and only if I 0 Q P Q P is R -unimodular. 1 12 2 21 0X12 NC2N21 Y12 01 = D~ 01 N~22 is Proof of Theorem 3: Since P = N D 22 22 22 22 ~ ~ ~ ~22 , D22 , and D~ 22 are used interchangeably. 2 N12 NY~C212 X21 V~12 + N12 (0D~UC212 +N22N~UC12 = I scalar, N22 , N By ~ ~ ~ 2 V21 N22 )U12 Theorem 2, Q1 ; Q2 2 M(R) satisfying (13) and (14) exist if ~C 2 X21 ; Y21 ) is right-coprime, (Y~12 ; and only if P^ = P12~D22 N~C 2 P21 is strongly R-stabilizable for implies (N12 N some Q2 2 R, i.e., D22 has the same sign at all real blocking X~ 12 NC 2 N~21 ) is left-coprime. U 2-2). Since the only U -zeros 0 1 0 1 ~C 2 X21 Y21 = Y~12 X~12 NC 2 N~21 is an RCF and -zeros of Nn122N~1C 2 X21 by Lemma 2) Since N12 N and P21 2 R12n are their blocking U -zeros, of P12 2 R 0 1 0 1 ^ ~ ~ LCF of P , because P22 = N22 D22 = D22 N22 , P21 = s 2 U is a blocking U -zero of P N 12 ~C 2 X21 if and only if it is a 01 N~21 = X21 Y 01 , P12 = N12 D01 = Y~ 01 X~ 12 , P^ is o D~ 22 ~C 2 or of P21 , i.e., 21 22 12 blocking U -zero of P12 , i.e., N12 (so ) = 0, or of N strongly R-stabilizable if and only if det Y21 , equivalently X (s ) = 0. The four conditions of Theorem 2-1) are necessary det Y~12 , det D~ 22 , or det D22 ; has the same sign at all real for21theoexistence of reliable decentralized controllers. The U -poles of U -zeros of N12 N~C2 X21 , equivalently, of X~12 NC2N~21 . P22 are the U -zeros of D22 . By (10), the signs of U2 and N22 are Proof of Theorem 2: ~ 22 . Suppose fj1 ; 1 1 1 ; j` g is empty. the same at all real U -zeros of D ~ 2 2 R exists such that (U2 + Q~ 2 D~ 22 ) has no U -zeros [9]. 1) If reliable decentralized controllers exist, Condition 1) holds Then Q ~ 2 D~ 22 )X21 are those by (9). By Theorem 1, (13) and (14) hold; (13) and (14) Since the only blocking U -zeros of N12 (U2 + Q ~ 22 ; N~21 ) is left- of P12 and P21 , the conditions of Theorem 2-1) are sufficient for (13). imply (N12 ; D22 ) is right-coprime, and (D 01 , If P 2 M(R ), then (V 0 Q~ N~ ) is biproper. If N~ (1) 6= 0, coprime. By (9), Condition 2) holds since P12 = N12 D22 01 N~21 . Conditions 3) and 4) are shown as follows: let Q222 := Q~ 2 +s Q^ 2 , Q^ 2 22 R, Q2^ 2 (221) 6= (V2 0 Q~ 2 N~2222)N~ 01 (1), P21 = D~ 22 22 P12 is strongly R-stabilizable if and only if for any RCF kQ^ 2 k < kD~ 22 (U2 + Q~ 2 D~ 22 )01 k01 . Then D~ C 2 is biproper, i.e., 01 , X~ 2 M(R) exists such that D22 + C2 2 M(Rp ), and N~C 2 is a unit in R. Since P12 2 M(Rs ) or P12 = N12 D22 ~ XN12 is R-unimodular; with X~ = NC 2 N~21 Q1 , (13) implies P21 2 M(Rs ), Q1 satisfying (13) can be chosen strictly proper P12 is strongly R-stabilizable. Similarly, P21 is strongly R- so that D~ C 1 is biproper. Therefore, the conditions of Theorem 2-1) 01 N~21 , X 2 are sufficient for the existence of reliable decentralized controllers. ~ 22 stabilizable if and only if for any LCF P21 = D ~ ~ M(R) exists such that D22 + N21X is R-unimodular; with Suppose fj1 ; 1 1 1 ; j`g is not empty; then N22 has an odd number X = Q1 N12 N~C 2 , (14) implies P21 is strongly R-stabilizable. of zeros in each interval (j ; j +1 ), 1 k `. By (10), ~21 Q1 N12 ) has no U -zeros, N~C 2 (j )N22 (j ) = 1 at the U -zeros j of D~ 22 ; hence, N~C 2 = Since (13) implies det (D22 +NC 2 N det D22 (z12 ) has the same sign as det D22 (z21 ) for all real (U2 + Q2 D~ 22 ) has an odd number of zeros because N~C 2 has even z12 ; z21 2 U , N12 (z12 ) = 0 and N~21 (z21 ) = 0. number of zeros in (j ; j +1 ), 1 k `. Note that j is the 01 NC 2 N~21 is a BCF; first zero of D~ 22 immediately to the left of the real U -zero of N~C 2 in 2) a) By Lemma 2, P^ = N12 D22 ~ 22 immediately to the therefore, P^ is strongly R-stabilizable if and only if (j ; j +1 ) and j +1 is the first zero of D ~ ^ ~ ^ X 2 M(R) exists such that (D22 + NC 2 N21 XN12 ) right of the real U -zero of NC 2 in (j ; j +1 ). If Q2 2 M(R) ~ 22 has the same sign at all real U -zeros of N~C 2 , then is R-unimodular, equivalently (13) holds. exists such that D ~ must have an even number of zeros between j and j +1 D 22 b) By Theorem 1-3), reliable decentralized controllers exist ~C 2 has at least one real U -zero in each of these intervals. Since if and only if Q1 ; Q2 2 M(R) exist satisfying (13), since N ~ C 1 ; D~ C 2 are biproper. It was shown that either P12 (1) = 0 or P21 (1) = 0, the sign of D~ 22 at the U -zero such that D Q1 ; Q2 2 M(R) satisfying (13) exist if and only if of N~C 2 in the last interval (j ; j +1 ) must agree with the sign of P^ is strongly R-stabilizable. Since P22 2 M(Rs ), D~ 22 (1); hence, D~ 22 must have even number of zeros in (j ; 1). (V2 0 Q2 N~22 ) is biproper. If P12 2 M(Rs ) or This proves necessity. For any Q2 2R, the minimum number of U P21 2 M(Rs ), then P^ = P12 D22 N~C 2 P21 2 M(Rs ); zeros of N~C 2 is `, which is the number of intervals where N22 has an ~ 22 . There is Q~ 2 2 R hence, P^ is strongly R-stabilizable if and only if odd number of zeros between real U -zeros of D 0 1 ~ ~ ^ such that ( U + Q ) has exactly ` real U -zeros, with exactly one in D 2 2 22 for any 0a 2 IRnU , (s + a) P is strongly R~ 1 2 M(R) exists such that each of (j ; j +1 ), 1 k `, because (U2 + Q~ 2 D~ 22 ) has an odd stabilizable, equivalently, Q D22 + NC 2 N~21 (s + a)01 Q~ 1 N12 is R-unimodular. Let number of zeros in each of these intervals. If D~ 22 has even number of Q1 := (s + a)01 Q~ 1 2 M(Rs ); then (13) holds and real U -zeros in each of (j ; j +1 ), 1 k ` 0 1, and (j ; 1), ~ 2 D~ 22 )P21 is strongly R-stabilizable. Let then P^ = P12 D22 (U2 + Q D~ C 1 is biproper.
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~ 22 + N~21 Q1 N12 (U2 + Q~ 2 D~ 22 ) is Q1 2 M(R) be such that M1 := D R-unimodular; Q1 can be chosen strictly proper. If N~22 (1) 06=1 0, let Q2 := Q~ 2 + Q^ 2 , Q^ 2 2 R, Q^ 2 (1) 6= (V~2 0 N22 Q~ 2 )(1)N~22 (1), kQ^ 2 k < kD~ 22 M101N~21 Q1 N12 k01 . Then D~ C2 , D~ C1 are biproper, ~ 22 + N~21 Q1 N12 N~C 2 = M1 + i.e., C1 ; C2 are proper. Since D ~ 22 is R-unimodular, (C1 ; C2 ) is a reliable decenN~21 Q1 N12 Q^ 2 D tralized controller pair. Proof of Corollary 1: 1) If P22 has an even number of real U -zeros between consecutive real U -poles, the sign of N22 is the same at all real U ~ 22 . Since fj1 ; 1 1 1 ; j` g is empty, Q2 2 R exists zeros of D ~C 2 has no U -zeros (the only blocking U -zeros of such that N ~ C 2 is biproper, N12 N~C 2 X21 are those of P12 , P21 ) and D ^ ~C 2 P21 is i.e., C2 is proper. For this Q2 , P = P12 D22 N strongly R-stabilizable. Since (s + a)01 P^ is strongly R~ 1 2 M(R) exists such that stabilizable for any 0a 2 IRnU , Q D22 + (U~2 + D22 Q2 )N~21 (s + a)01 Q~ 1 N12 is R-unimodular; ~ C 1 is biproper. Q1 = (s + a)01 Q~ 1 satisfies (13) and D 2) By assumption, the sign of D22 is the same at all real U -zeros of P12 , P21 , and P22 as the sign of D22 (1); hence, P22 is ~ C 2 is Rstrongly R-stabilizable. Let Q2 2 R be such that D ~ unimodular, then D22 has the same sign at all real U -zeros ~C 2 N22 . The sign of D22 at the real U -zeros of N~C 2 is of N ~C 2 P21 is the same as that of D22 (1); hence, P^ = P12 D22 N strongly R-stabilizable. As in the proof of 1), (s + a)01 P^ is also strongly R-stabilizable, since Q1 can be chosen strictly proper, the controllers are proper. Proof of Theorem 4: Let P22 2 Rpn 2n , rankP22 =: r, 3 := diag[1 1 1 1 r ], and 9 := diag[ 1 1 1 1 r ]; there exist Runimodular L 2 Rn 2n , R 2 Rn 2n satisfying (16), where j ; j 2 R, j is biproper, (j ; j ) is coprime, i.e., uj ; vj 2 R exist satisfying vj j + uj j = 1, j = 1; 1 1 1 ; r; j divides j +1 , form). j +1 divides j , j = 1; 1 1 1 ; r 0 1 (see [9], Smith–McMillan 01 = D~22 01 N~22 are given By (16), any RCF and LCF P22 = N22 D22 ~ 2 M(R); let UD := in (17) and (18) for some R-unimodular M; M diag[u1 1 1 1 ur ], VD := diag[v1 1 1 1 vr ]; then (VD 9 + UD 3) = Ir ; U2 ; V2 in (9) are given by (19), where A^ 2 Rn 2n
P22 = L diag[3; 0(n 0r)2(n 0r) ] diag[901 ; I(n 0r) ]R = L diag[901 ; I(n 0r) ] diag[3; 0]R (16)
(N22 ; D22 ) = (L diag[3; 0(n 0r)2(n 0r) ]M; R01 diag[9; I(n 0r) ]M ) (D~ 22 ; N~22 ) = (M~ diag[9; I(n 0r) ]L01 ; ~ diag[3; 0(n 0r)2(n 0r) ]R) M
~ 22 U2 = M 01 diag[UD ; 0(n 0r)2(n 0r) ]L01 + A^D 0 1 V2 = M diag[VD ; I(n 0r) ]R 0 A^N~22 :
(17)
(18)
(19)
Let Q11 2 Rr2r be any upper-triangular matrix whose nondiagonal entries qij 6= 0 are constants, for i; j = 1; 1 1 1 ; r, j > i. For j = 1; 1 1 1 ; r, choose qjj 2 R as follows: Let Z12 , Z21 be the sets of all real U -zeros of P12 and of P21 , respectively; let Z := Z12 Z21 = fz1 ; 1 1 1 ; z` g. Let Zj = fzj1 ; 1 1 1 ; zj` g Z be such that uj (z ) 6= 0 for z 2 Zj . Define qjj 2 R as qjj = ` qjj (1) k=1 (s 0 zjk )(s + a)01 ; 0a 2 IRnU , qjj (1) 2 IRnf0g is such that (vj 0 qjj j )(1) 6= 0; this holds for all qjj (1) when 1 j 2 Rs ; when j 62 Rs , take qjj (1) 6= vj 0 j (1). If Zj is empty, then qjj = qjj (1) 2 IR. With this qjj 2 R, (uj + qjj j ) does not have zeros at any of the real U -zeros of P12 or P21 . If uj = 0, then j is a unit in R and if u1 = 0, then P22 2 M(R).
1) Choose
Q11 Q12 ~ 01 0 Q22 M Q12 2 Rr2(n 0r) , Q22 2 R(n 0r)2(n 0r) has no real blocking U -zeros and no zeros at any real U -zeros of P12 and P21 ; obvious choices for Q22 are any R-unimodular matrix or ~C 2 = (U2 +Q2 D~ 22 ) has the identity In 0r . By construction, N Q2 = 0A^ + M 01
no real blocking U -zeros and no U -zeros coinciding with any real U -zeros of either P12 or P21 . Since rankP12 = ni2 = no2 , 01 if and only if zo 2 U is a U -zero of P12 = N12 D22 rankN12 (zo ) < no2 . Similarly, zo 2 U is a U -zero of P21 = X21 Y2101 if and only if rankX21 (zo ) < no2 . By Lemma 2, P^ is ~ 22 has the same sign strongly R-stabilizable if and only if det D ~C 2 X21 , which are the real at all real blocking U -zeros of N12 N blocking U -zeros of P12 , of P21 and possibly some of the real U -zeros common to P12 and P21 . We prove that no other real blocking U -zeros exist by contradiction: Suppose zo 2 IR \U is ~C 2 X21 (zo ) = 0 but N12 (zo ) 6= 0, X21 (zo ) 6= such that N12 N 0, and zo is not a common zero of P12 and P21 ; since zo may be a zero of one of P12 or P21 , there are two cases: 1) If P12 (zo ) 6= 0, P21 (zo ) 6= 0, then rankN12 (zo ) = ni2 = no2 , rankX21 (zo ) = no2 ; hence, N12 (zo ) 2 IRn 2n has a left^12 and X21 (zo ) 2 IRn 2n has a right-inverse inverse N ^ X21 . Therefore, N12 N~C 2 X21 (zo ) = 0 implies N~C 2 (zo ) = 0, which is a contradiction; and 2) If zo is a zero of either P12 ~C 2 (zo ) = no2 . Either X^ 21 2 IRn 2n or P21 , then rankN ^ 21 = I (if P21 (zo ) 6= 0) or N^12 2 exists such that X21 (zo )X 2 n n ^12 N12 (zo ) = I (if P12 (zo ) 6= 0). exists such that N IR ~ Therefore, N12 NC 2 X21 (zo ) = 0 implies either N12 (zo ) = 0, or X21 (zo ) = 0; again we have a contradiction. Since P12 or P21 is strictly proper, the sign of det D22 is the same at all of these real blocking U -zeros as the sign of det D22 (1); ~C 2 P21 is strongly R-stabilizable. By therefore P^ := P12 D22 N ~ (16) and (19), DC 2 in (11) is biproper, i.e., C2 is proper since det (VD 0 Q11 3)(1) 6= 0. Since P12 2 M(Rs ) or P21 2 M(Rs ), Q1 2 M(R) satisfying (13) can be chosen strictly proper so that C1 is proper. 01) = no2 = ni2 , then rankN22 = 2) If rankP22 = rank(N22 D22 no2 . By (10), det U2 and det N22 have the same sign at all real blocking U -zeros of D22 . By (16), the real blocking U zeros of D22 are those of its smallest invariant factor n . Since the U -zeros of P22 are those of det N22 , the sign of det N22 is the same at all real U -zeros of n . Therefore, q 2 R exists such that (det U2 + q n ) is a unit of R ~C 2 is R-unimodular implies Q2 2 M(R) exists such that N ~C 2 is R-unimodular, the only real [9]. We show that since N ~C 2 X21 are the blocking U -zeros of blocking U -zeros of N12 N P12 , or P21 and possibly some of the real U -zeros of P12 or of P21 : If zo 2 IR \ U is such that P12 (zo ) 6= 0, P21 (zo ) 6= 0, then rankN12 (zo ) + rankX21 (zo ) 0 no2 > 0. There~C 2 X21 (zo ) rankN12 (zo )+rankX21 (zo ) 0 fore, rankN12 N no2 > 0 implies N12 N~C 2 X21 (zo ) 6= 0. Since the signs of det D22 and det D22 (1) are the same at all real U -zeros of P12 and P21 , by Lemma 2, P^ is strongly R-stabilizable. Since P22 2 M(Rs ), the existence of proper controllers follows from Theorem 2-2)-b). 01 has an R-stable left 3) First we show that P12 = N12 D22 inverse if and only if N12 2 Rn 2n has an R-stable left I 2 Rn 2n ; if P I 2 Rn 2n exists such inverse N12 12 I I N12 = D22 . Since (N12 ; D22 ) is that P12 P12 = I , then P12 I + U12 )N12 = I implies (V12 P I + right-coprime, (V12 P12 12 I 2 U12 ) 2 M(R) is a left-inverse of N12 . Conversely, if N12
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 12, DECEMBER 1998
I P12 = D01 exists such that N1I2 N12 = I , then N12 22 I P12 = I ; hence, D22 N I 2 M(R) is a leftimplies D22 N12 12 ~21 ~ 01 N inverse of P12 . It can be shown similarly that P21 = D 22 2 n n ~ has an R-stable right-inverse if and only if N21 2 R ~ I 2 Rn 2n . Construct Q11 2 Rr2r has a right-inverse N 21 with qjj chosen as above; since rankP12 (s) = ni2 and rankP21 (s) = no2 for all s 2 U , P12 and P21 have no U -zeros. Therefore, Q11 is chosen so that the nondiagonal entries qij 6= 0 are constants, and qjj 2 R are such that ~C 2 has no real (vj 0 qjj j )(1) 6= 0. To guarantee that N blocking U -zeros, let
Rn 2n
Q11 Q12 ~ 01 M 2 Rn 2n Q21 Q22 Q12 2 Rr2(n 0r) , Q21 2 R(n 0r)2r , Q22 2 R(n 0r)2(n 0r) can be arbitrary if both no2 > 1 and ni2 > 1; if ni2 = 1, let Q12 2 IR12(n 01) be nonzero real; if no2 = 1, let Q21 2 IR(n 01)21 be nonzero I ~I Q ^ ~ real. Let Q1 := N 21 1 N12 2 M(R). Since NC 2 has ^ 1 2 M(R) exists such that no real blocking U -zeros, Q I ~ 22 + N ~I Q ^ ~ ~21 N ~ ^ ~ D 21 1 N12 N12 NC 2 = D22 + Q1 NC 2 is R~ unimodular, i.e., (14) holds. Since DC 2 is biproper, C2 is ^ 1 2 M(R) such that D ^1N ~ 22 + Q ~C 2 proper. There is Q ~ 1 2 M(R) exists such is R-unimodular if and only if Q ~ 1N ~C 2 is R-unimodular; choosing ~ 22 + (s + a)01 Q that D 0 1 ~ Q1 = (s + a) Q1 2 M(Rs ) implies C1 is proper. 4) Let CS be any R-stable R-stabilizing controller for P22 . 01 satisfy Without loss of generality, let the RCF P22 = N22 D22 D22 + CS N22 = I ; hence, N22 D22 = (I 0 N22 CS )N22 . ~2 = Then U2 = CS + T (I 0 N22 CS ), V2 = I 0 T N22 , U CS + D22 T , V~2 = I 0 N22 T , T 2 M(R) satisfy (10). By ~ 01 N ~21 implies N ~21 has a right-inverse assumption, P21 = D 22 I ~ 12 D 01 = P22 = N22 D 01 ~ ~ 12 = LN N21 2 M(R). Also, LP 22 22 ~ 12 = N22 . Let C1 , C2 be given by (11) and implies LN ~ 2 Rn 2n , Q2 = 0T . Then (13) ~I L (12), Q1 = N 21 becomes D22 + CS N22 = I . Since P11 , P12 2 M(Rs ), ~21 ) 2 M(Rs ) implies D ~ C 1 is biproper, i.e., (N11 0 N12 Q2 N C1 2 M(Rp ). Since C2 = CS 2 M(R), (C1 ; C2 ) is a Q2 = 0A^ + M 01
reliable decentralized controller pair. 5) Let CS be as in 4); let D22 + CS N22 = I . By assumption, 01 implies N12 has a left-inverse N12 I 2 M(R). P12 = N12 D22 0 1 ~ 0 1 ~ ~ ~ ~ ~ Also, P21 R = D22 N21 R = P22 = D N22 implies N~21 R~ = N~22 = N22 . Let C1 , C2 be given by (11) and (12), ~ I 2 Rn 2n , Q2 = 0T . The conclusion follows Q1 = RN 12 as in 4). ~ 12 = P22 implies 6) Choose CS as in 4). By assumption, LP ~ 12 = N22 , and P21 R ~ = P22 implies N ~21 R ~ = LN N~22 = N22 . Let Q2 = 0T , Q1 = R~ Q^1 L~ , Q^1 = k rm k0m (CS N22 )m02 CS ; k is any integer such m=2 that k > kCS N22 k and rm are the binomial coefficients. By (13) D22 + CS N22 Q^1 N22 = I 0 CS N22 + k rm k0m (CS N22 )m = (I 0 k01 CS N22 )k is Rm=2 unimodular. Then C1 2 M(Rs ) since Q^1 ; Q1 2 M(Rs ). Since C2 = CS is proper, (C1 ; C2 ) is a reliable decentralized controller pair. REFERENCES [1] A. N. G¨unde¸s and C. A. Desoer, Algebraic Theory of Linear Feedback Systems with Full and Decentralized Compensators, Lecture Notes in Control and Information Sciences, vol. 142. Berlin, Germany: Springer-Verlag, 1990.
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[2] K. D. Minto and R. Ravi, “New results on the multi-controller scheme for the reliable control of linear plants,” in Proc. American Control Conf., 1991, pp. 615–619. [3] C. N. Nett, “Algebraic aspects of linear control system stability,” IEEE Trans. Automat. Contr., vol. 31, pp. 941–949, 1986. ¨ uler, “Reliable decentralized stabiliza¨ [4] K. A. Unyelio˘ glu and A. B. Ozg¨ tion of feedforward and feedback interconnected systems,” IEEE Trans. Automat. Contr., vol. 37, pp. 1119–1132, 1992. ˇ [5] D. D. Siljak, “On reliability of control,” in Proc. 17th IEEE Conf. Decision and Control, 1978, pp. 687–694. , “Reliable control using multiple control systems,” Int. J. Contr., [6] vol. 31, pp. 303–329, 1980. ˇ [7] X. L. Tan, D. D. Siljak, and M. Ikeda, “Reliable stabilization via factorization methods,” IEEE Trans. Automat. Contr., vol. 37, pp. 1786–1791, 1992. [8] M. Vidyasagar and N. Viswanadham, “Algebraic design techniques for reliable stabilization,” IEEE Trans. Automat. Contr., vol. 27, pp. 1085–1095, 1982. [9] M. Vidyasagar, Control System Synthesis: A Factorization Approach. Cambridge, MA: M.I.T. Press, 1985.
Useful Nonlinearities and Global Stabilization of Bifurcations in a Model of Jet Engine Surge and Stall Miroslav Krsti´c, Dan Fontaine, Petar V. Kokotovi´c, and James D. Paduano
Abstract— Compressor stall and surge are complex nonlinear instabilities that reduce the performance and can cause failure of aircraft engines. We design a feedback controller that globally stabilizes a broad range of possible equilibria in a nonlinear compressor model. With a novel backstepping design we retain the system’s useful nonlinearities which would be cancelled in a feedback linearizing design. The design control law is simple and, moreover, it is optimal with respect to a meaningful nonquadratic cost functional. As in a previous bifurcationtheoretic design, we change the character of the bifurcation at the stall inception point from subcritical to supercritical. However, since we do not approach bifurcation control using a normal form but using Lyapunov tools, our controller achieves not only local but also global stability. The controller requires minimal modeling information (bounds on the slope of the stall characteristic and the B -parameter) and simpler sensing (rotating stall is stabilized without measuring its amplitude). Index Terms—Axial flow compressors, backstepping, bifurcation control, jet engines, rotating stall, surge.
I. INTRODUCTION In control engineering the importance of qualitative low-order nonlinear models is twofold. First, they can capture the dominant dynamic phenomena; second, they are testbeds which help refine new nonlinear design methods. One such model, the Moore–Greitzer Manuscript received March 14, 1997. This work was supported in part by the Air Force Office of Scientific Research under Grants F496209610223 and 95-1-0409 and in part by the National Science Foundation under Grant ECS9624386. M. Krsti´c is with the Department of AMES, University of California at San Diego, La Jolla, CA 92093-0411 USA (e-mail: [email protected]). D. Fontaine and P. V. Kokotovi´c are with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106 USA. J. D. Paduano is with the Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139 USA. Publisher Item Identifier S 0018-9286(98)08423-2.
0018–9286/98$10.00 1998 IEEE
