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Novel bounded real lemma for discrete-time descriptor systems: Application to H∞ control design
Automatica 48 (2012) 449–453
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Novel bounded real lemma for discrete-time descriptor systems: Application to H∞ control design✩ Mohammed Chadli a,1 , Mohamed Darouach b a
UPJV-MIS (E.A. 4290), University of Picardie Jules Verne, 7 Rue du Moulin Neuf, 80000, Amiens, France
b
CRAN-CNRS (UMR 7039), UHP, Nancy I IUT de Longwy, 186 Rue de Lorraine, 54400 Cosnes et Romain, France
article
info
Article history: Received 12 November 2010 Received in revised form 23 August 2011 Accepted 26 September 2011 Available online 15 December 2011 Keywords: Descriptor systems Linear systems H∞ control Bounded real lemma Strict LMI
abstract This paper concerns the bounded real lemma for discrete-time descriptor systems. A new formulation of the bounded real lemma for these systems is given. It extends the recent results presented in Zhang, Xia, and Shi (2008) and gives necessary and sufficient conditions in strict LMI (linear matrix inequality) which is more suitable for the control design than those presented in Zhang, Xia, and Shi (2008). An application to the H∞ control design is given. A numerical example is presented to show the applicability of our approach. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction and preliminaries Descriptor systems (also known as singular or differential algebraic systems) have attracted particular interest in the literature for their various applications such as economics, robotics, electrical and chemical systems (Dai, 1989; Lewis, 1986; Luenberger, 1979; Mills & Goldenberg, 1989). The analysis and the controller design for such systems have received considerable attention in the past decades (Dai, 1989). Due to its general description, such a model has been employed to design controllers and observers in different areas of research for theoretical development as well as for practical applications (Darouach, 2006; Masubuchi, Kamime, Ohara, & Suda, 1997; Rehm & Allgower, 2004; Varga, 1995; Xia, Zhang, & Boukas, 2008; Yao, Guan, Chen, & Ho, 2006). Interesting results on controllers and observers design are developed for descriptor linear systems using algebraic methods (Darouach, 2006, and references therein) or matrix inequality approaches (Varga, 1995; Xu & Lam, 2004; Xu & Yang, 2000; Yung, 2008; Zhang, Xia, & Shi, 2008). These results are extended to design H∞ controller where necessary and sufficient conditions
✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Mayuresh V. Kothare under the direction of Editor André L. Tits. E-mail addresses: [email protected] (M. Chadli), [email protected] (M. Darouach). 1 Tel.: +33 322827680.
0005-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2011.10.003
are given in nonlinear forms. Uncertain descriptor linear systems are also studied in Ji, Su, and Chu (2007) and Xu and Lam (2004, 2006) and sufficient LMI (linear matrix inequality) conditions are given. On the other hand, the bounded real lemma for discretetime descriptor systems was derived using a non-strict LMI and has the drawback that it cannot be directly solved using classical numerical tools (Hsiung, 1998; Masubuchi et al., 1997; Wang, Yung, & Chang, 1998; Xu & Yang, 2000). Recently strict LMI conditions, which are more tractable and numerically reliable, have been proposed (see Ji et al., 2007; Xu & Lam, 2006; Zhang, Xia, & Shi, 2008, and references therein). Although the results of Zhang, Xia, and Shi (2008) give strict LMI admissibility conditions for H∞ analysis, the result cannot be used directly to design H∞ controllers or observers. Thus, our aim is to give a new formulation to the results of Zhang, Xia, and Shi (2008) and to show how we can apply this formulation in the H∞ control design for discrete-time descriptor systems. This paper proposes strict LMI admissibility conditions to the H∞ control design using existing LMI tools (Boyd, El Ghaoui, Feron, & Balakrishnan, 1994) and the slack variables technique developed for standard systems (Oliveira, Bernussou, & Geromel, 1999). First, necessary and sufficient admissibility conditions are given. From the admissibility conditions obtained, a novel bounded real lemma of discrete-time descriptor systems is given in strict LMI conditions. Then necessary and sufficient conditions to design H∞ controller are given with intermediate free parameters for more relaxation. A numerical example is given to illustrate the proposed approach.
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M. Chadli, M. Darouach / Automatica 48 (2012) 449–453
Notation. Throughout this paper, Rn and Rn×m denote, respectively, the n dimensional Euclidean space and the set of all n × m real matrices. The superscript ‘‘T ’’ denotes matrix transposition, the notation X ≥ Y (respectively, X > Y ) where X and Y are symmetric matrices, means that X − Y is positive semi-definite (respectively, positive definite), the symbol (∗) denotes the transpose elements in the symmetric positions. For a matrix M ∈ Rn×m with rank M = r, let M ⊥ ∈ R(n−r )×n be any matrix such that M ⊥ M = 0 and M ⊥ M ⊥⊤ > 0, note that M ⊥ exists if and only if M has linearly dependent rows. When M is of full row rank matrix we take M ⊥ = 0 by convention. Consider the discrete-time descriptor linear system described by: Ex(t + 1) = Ax(t ) + Bu u(t ) + Bw w(t )
(1a)
y(t ) = Cx(t ) + Dw(t )
(1b)
where x(t ) ∈ R is the state vector, y(t ) ∈ R is the output vector, u(t ) ∈ Rm is the input vector, w(t ) ∈ Rq is the disturbance vector, A ∈ Rn×n , Bu ∈ Rn×m , Bw ∈ Rn×q , C ∈ Rp×n and D ∈ Rp×q . Matrix E may be singular with rankE = r ≤ n. For this system with w(t ) = 0 and u(t ) = 0, we can give the following definitions (Dai, 1989; Rehm & Allgower, 2004; Xu & Lam, 2004). n
Let the pair (E , A) be given, then it is always possible to find I 0
nonsingular matrices M and N such that E = M
M
A1 A3
A2 A4
A⊤ (P − E ⊥⊤ QE ⊥ )A − E ⊤ PE < 0.
0 0
N and A =
N. This decomposition can be obtained via singular
value decomposition of matrix E followed by scaling of the bases. Then, we have the following lemmas (Dai, 1989).
Lemma 2. The pair (E , A) is regular if and only if there exist two nonsingular matrices U and V such that E = U U
¯
A 0
0 I
I 0
0
N
V and A =
V , where N is a nilpotent matrix.
Definition 2. The discrete-time descriptor system (1) is said to be admissible with an H∞ performance γ if it is admissible (for w(t ) = 0) and ∥Gwy (z )∥∞ < γ i.e. ∥y(t )∥2 < γ 2 ∥w(t )∥2 for w(t ) ̸= 0. 2. Main results The following lemma gives necessary and sufficient conditions for system (1), with u(t ) = 0 and w(t ) = 0, to be admissible. Theorem 4. The discrete-time descriptor system (1) or the pair (E , A) is admissible if and only if the following equivalent statements hold. (i) There exists a matrix X = X ⊤ satisfying the following LMI: E ⊤ XE ≥ 0,
A⊤ XA − E ⊤ XE < 0.
(2)
(4)
(iv) There exist matrices P > 0, Q = Q , F and G satisfying the following LMI:
⊤ −E PE + A⊤ F ⊤ + FA (∗)
−F + A⊤ G⊤ < 0. (5) P − E ⊥⊤ QE ⊥ − G − G⊤
(v) These exist matrices P > 0, Q = Q ⊤ , F and G satisfying the following LMI:
−EPE ⊤ + AF ⊤ + FA⊤ (∗)
−F + AG⊤ < 0. P − E ⊥⊤⊥ QE ⊥⊤ − G − G⊤
(6)
Proof. It is proved in Zhang, Xia, and Shi (2008) that statements (2) and (3) are equivalent. Now, we can prove that (3) and (4) are equivalent. This can be obtained from the fact that the pair (E , A) is admissible if and only if the pair (E ⊤ , A⊤ ) is admissible. In fact, we have det(zE − A) = det(zE − A)⊤ = det(zE ⊤ − A⊤ ) and deg(det(zE − A)) = deg(det(zE ⊤ − A⊤ )⊤ ), which shows that the pair (E , A) is regular and causal if and only if (E ⊤ , A⊤ ) is regular and causal. On the other hand, from Lemmas 2 and 3, if the pair (E , A) is regular and causal, there exist two nonsingular matrices U and V such thatE = I 0
U
0 0
V and A = U
and A⊤ = V ⊤
A¯ ⊤ 0
A¯ 0
0 I
0 I
V , then we have E ⊤ = V ⊤
I 0
0 0
U⊤
U ⊤ , in which case the stability of the
pair (E , A) is reduced to that of A¯ or equivalently the stability of A¯ ⊤ which proves the equivalence of inequalities (3) and (4). Consequently, it suffices to prove that condition (5) is equivalent to condition (3). Sufficiency—Condition (3) is satisfied by pre-multiplying (5) by
⊤
I A⊤ and post-multiplying it by I A⊤ . Then, if condition (5) is satisfied, the discrete-time descriptor system (E , A) is admissible. Necessity—Assume that the system (E , A) is admissible, i.e. condition (3) is satisfied. Then there always exist a matrix G such that R − G − G⊤ < 0 with R = P − E ⊥⊤ QE ⊥ . Its follows that
A⊤ RA − E ⊤ PE
If the descriptor system is regular, the transfer function matrix from w(t ) to y(t ) is given by Gwy (z ) = C (zE − A)−1 Bw + D.
satisfying the following
⊤
Lemma 3. Let the pair (E , A) be regular, then it is causal if and only if N = 0.
⊤
A(P − E ⊥⊤⊥ QE ⊥⊤ )A⊤ − EPE ⊤ < 0.
Lemma 1. The pair (E , A) is causal if and only if A4 is nonsingular.
(3)
(iii) There exist matrices P > 0 and Q = Q LMI:
p
Definition 1. • The pair (E , A) is said to be regular if det(zE − A) is not identically zero. • The pair (E , A) is said to be causal if deg(det(zE − A)) = rank E. • The pair (E , A) is said to be stable if |λ(E , A)| < 1 ∀λ(E , A) ∈ {z | det(zE − A) = 0}. • The pair (E , A) is said to be admissible if it is regular, causal and stable.
(ii) There exist matrices P > 0 and Q = Q ⊤ satisfying the following LMI:
(∗)
0 R − G − G⊤
Pre-multiplying (7) by
I
−A
0 I
I 0
< 0.
−A⊤ I
(7) and post-multiplying it by
, we get
⊤ −E PE + A⊤ RA + A⊤ (R − G − G⊤ )A −(R − G − G⊤ )A
(∗) < 0. R − G − G⊤
(8)
Then by choosing F = A⊤ R − A⊤ G, we obtain condition (5). Inequalities (5) and (6) are equivalent. This can be obtained from the fact that the admissibility of the pair (E , A) is equivalent to the admissibility of (E ⊤ , A⊤ ). This completes the proof. Remark 1. Note that for the standard case, i.e. when E = I, we get E ⊥ = 0 and LMI conditions (5) or (6) are reduced to the existence of matrices P > 0, F and G as given in Peaucelle, Arzelier, Bachelier, and Bernussou (2000) or with F = 0 as it is stated in the earlier work of Oliveira et al. (1999). Now, we propose new strict LMI conditions for the discretetime descriptor system (1) to be admissible with an H∞ performance γ .
M. Chadli, M. Darouach / Automatica 48 (2012) 449–453
Theorem 5. The discrete-time descriptor system (1) is admissible with an H∞ performance γ if and only if the following equivalent statements hold. (i) There exist matrices P > 0 and Q = Q ⊤ satisfying the following LMI:
Ω1 (∗)
Ω2 <0 Ω3
(9)
with
where K is a gain matrix of an appropriate dimension. The problem is to design K , such that system (15) is admissible when w(t ) = 0 and ∥Gwy (z )∥∞ < γ i.e. ∥y(t )∥2 < γ 2 ∥w(t )∥2 for w(t ) ̸= 0, where Gwy (z ) = C (zE − A − Bu K )−1 Bw + D. On the other hand, let us consider the following system
¯ (t ) + B¯ w w(t ) E¯ x¯ (t + 1) = Ax
(16a)
y(t ) = C¯ x¯ (t ) + Dw(t )
(16b)
⊤ where x¯ (t ) = x(t )⊤ u(t )⊤ and E 0 A Bu , E¯ = , A¯ = K −I m 0 0 Bw , C¯ = C 0 . B¯ w =
Ω1 = −E ⊤ PE + A⊤ (P − E ⊥⊤ QE ⊥ )A + C ⊤ C Ω2 = C ⊤ D + A⊤ (P − E ⊥⊤ QE ⊥ )Bw ⊥⊤ Ω3 = −γ 2 I + D⊤ D + B⊤ QE ⊥ )Bw . w (P − E (ii) There exist matrices P > 0, Q = Q ⊤ , F and G satisfying the following LMI:
⊤ −E P E + F A + A⊤ F ⊤ (∗)
−F + A⊤ G⊤ P − E ⊥⊤ QE ⊥ − G − G⊤
< 0.
(10)
(iii) There exist matrices P > 0, Q = Q ⊤ , F and G satisfying the following LMI:
−E P E ⊤ + F A⊤ + AF ⊤ (∗)
−F + AG⊤ ⊤⊥⊤ P −E QE ⊤⊥ − G − G⊤
<0
¯ wy (z ) = C¯ (z E¯ − A¯ )−1 B¯ w + D. Then we have the following Let G lemma. Lemma 6. System (15) is admissible for w(t ) = 0 and ∥Gwy (z )∥∞ < γ for w(t ) ̸= 0 if and only if system (16) is admissible for w(t ) = 0 ¯ wy (z )∥∞ < γ for w(t ) ̸= 0. and ∥G Proof. From system (16) we have
z E¯ − A¯ = (11)
A C
A=
Bw , D
E =
P P = 0
E 0
0
γ Ip.q 0 .
0 , Ip
Q Q= 0
, (12)
0
Proof. It is proved in Zhang, Xia, and Shi (2008) that (9) is a necessary and sufficient condition for the discrete-time descriptor system (1) to be admissible with an H∞ performance γ . Thus it suffices to prove that condition (10) is equivalent to (9). Indeed, conditions (9) can be written in the following equivalent form A C
Bw D
⊤
R 0
0 Ip
A C
Bw E ⊤ PE − D 0
0 γ 2 Iq
<0
(13)
with R = P − E ⊥⊤ QE ⊥ . Then (13) can be rewritten as follows
A (P − E ⊤
⊥⊤
QE )A − E P E < 0 ⊥
⊤
(14)
which has the same form as (3). Then the rest of the proof follows from the proof of Theorem 4. Then conditions (9) and (10) are equivalent, and condition (10) is equivalent to (11). This ends the proof. Remark 2. Note that LMI conditions (10)–(11) are more suitable for design problems than condition (9). They can be used easily to design observer or controller with LMI formulation. The following section shows how to design an H∞ controller using existing numerical solvers.
zE − A −K In 0
−Bu Im
−Bu
Im
zE − A − Bu K 0
0 Im
In −K
0 . Im
Then it is easy to see that the regularity and the stability of (15) are equivalent to those of (16). Also rank E¯ = rank E, then the causality of (15) is equivalent to that of (16). On the other hand, we have ¯ wy (z ), which proves the lemma. Gwy (z ) = G Remark 3. The augmented system (16), obtained by using the technique of Chadli, Darouach, and Daafouz (2008), does not change the finite modes or impulsive modes of the original descriptor system (Verghese, Levy, & Kailath, 1981). When the output signal is depending on the input signal: y(t ) = Cx(t ) + By u(t ) + Dw(t ), we can substitute (16b) by y(t ) = [CBy ]¯x(t ) + Dw(t ). Now, by using the results of Lemma 6 and Theorem 5, we can give the following theorem. Theorem 7. The discrete-time descriptor system (16) is admissible with an H∞ performance γ if and only if there exist matrices P > 0, Q = Q ⊤ , F , G and K such that the following equivalent statements hold
⊤ −E¯ P E¯ + F A¯ + A¯ ⊤ F ⊤ −F + A¯ ⊤ G⊤ <0 (∗) P − E¯ ⊥⊤ QE¯ ⊥ − G − G⊤ ¯ ⊤ ¯ ⊤ −E¯ P E¯ ⊤ + F A¯ ⊤ + AF −F + AG (ii) (∗) P − E¯ ⊤⊥⊤ QE¯ ⊤⊥ − G − G⊤ <0
(i)
where P and Q are defined in (12) and
A¯ = 3. H∞ controller design
=
(17)
0
with
451
A¯ C¯
B¯ w , D
E¯ =
E¯ 0
0 γI
(18)
with definitions (17). The discrete-time descriptor linear system (1) with the state feedback control u(t ) = Kx(t ) becomes Ex(t + 1) = (A + Bu K )x(t ) + Bw w(t )
(15a)
y(t ) = Cx(t ) + Dw(t )
(15b)
Remark 4. Note that condition (9) of Zhang, Xia, and Shi (2008) cannot be easily used to design the H∞ controller, which is numerically intractable. The given necessary and sufficient conditions are in the Bilinear Matrix Inequality (BMI) form and introduce
452
M. Chadli, M. Darouach / Automatica 48 (2012) 449–453
supplementary free variables. Note that these given BMI design conditions can be easily solved using existing numerical tools such as the YALMIP software (Löfberg, 2008). Nevertheless, strict LMI formulation with intermediate variable is proposed in the following section. The conditions derived lead to less of conservatism for the design problem as it is proved for standard system (Oliveira et al., 1999; Peaucelle et al., 2000). Also note that, the extension of these conditions to uncertain descriptor systems are very simple when compared with Zhang, Xia, and Shi (2008). These extensions are not given in this note for lack of space.
F4 , F3
G G= 1 0
F F1 = 11 F12
גM , M
G4 G3
G11 G1 = G12
גM M
with = ⊤גIm 0m×(n−m) and M ∈ Rm×m is a nonsingular matrix. Then the bilinear forms of F A¯ and GA¯ become
F1 + F4 C¯ F1 B¯ w + F4 D ¯ FA= F3 C¯ F3 D G1 + G4 C¯ G1 B¯ w + G4 D GA¯ = ¯ G3 C
G3 D
with
F11 Bu − גM F12 Bu − M
G11 Bu − גM G12 Bu − M
F A + גN F1 = 11 F12 A + N
G1 =
G11 A + גN G12 A + N
1 E= 0
where N = MK . Then, we get LMI design conditions for Theorem 7 in variables M , N , F11 , F12 , F3 , F4 , G11 , G12 , G3 and G4 . The controller is then given by K = M −1 N. For dual conditions, the same reasoning can be used with the variable change N = MK ⊤ . Note that this LMI formulation is obtained at the expense of necessary and sufficient conditions. However, compared with existing sufficient LMI design conditions (Ji et al., 2007), our results introduce more of slack free variables, and then more of relaxation for the design problem. The following section gives a numerical example to show this benefit. Remark 5. It should be understood that obtaining LMI formulation to design H∞ state feedback control implies some restrictions since the matrices F and G are not completely free. Another design condition in LMI terms can be obtained also by choosing = ג0n.m . These different choices will end with different LMI conditions whose results do overlap, i.e. no inclusion property can be found. Therefore, their conservatism is different according to the examples treated. 5. Numerical example In this section, a numerical example is given to show the effectiveness of the proposed results. It deals with the H∞ state
2 , 0
2.5 A= 1.7
1 , 0
Bu =
C = 1
1 ,
1.0 , 0.8
Bw =
1 1
D = 0.
Solving the condition of Theorem 7, using the proposed LMI design procedure, with E ⊥ = [0 1], i.e. E ⊥ = [0 1 0 0] and γ = 0.43, we get Q = 68.0752, M = 6.2347 and P =
Now, emphasis is placed upon tractable design conditions instead of the numerically intractable sufficient and necessary conditions proposed in the literature (Xu & Lam, 2004; Xu & Yang, 2000; Yung, 2008; Zhang, Xia, & Shi, 2008). Indeed, to get LMI design conditions (Theorem 7), particular structures of matrices F and G are used as follows: F F = 1 0
4. LMI design procedure
feedback control for the descriptor system (1) with the following data:
1.6208 0.9134 0.3450
N = −5.6927
0.9134 26.6530 3.0067
0.3450 3.0067 , −22.6309
−1.1921
−1.5297 1.3950 6.2347 − 0 . 9735 0 . 9231 0 F1 = , −2.6539 2.8129 6.2347 8.9668 −9.0058 6.2347 0 G1 = 6.4853 −6.6273 . 2.7045 −2.5045 6.2347
−1 Then the controller gain is obtained from K = M N = −0.9131 −0.1912 . Note that with the same γ = 0.43, the result of Ji et al. (2007) fails to find a feasible solution, i.e. to find
a controller gain K . Moreover, it is shown that the result of Ji et al. (2007) leads to a smaller gain compared with that of Xu and Lam (2004). This example shows the benefit of the given results. 6. Conclusion In this paper, a new formulation of the bounded real lemma for discrete-time descriptor linear system is given. Necessary and sufficient admissibility conditions in strict LMI form are derived. Then necessary and sufficient H∞ controller synthesis conditions in strict BMI terms are developed whereas existing results are in nonlinear forms. The proposed new formulation is more adapted to the H∞ control problem design and less conservative, by introducing slack variables, than the existing results. Sufficient LMI conditions for the control problem are also proposed. These conditions are obtained under some constraints and are restrictive compared with the BMI formulation. The conditions derived can be extended to uncertain discrete-time descriptor linear systems. An example is given to illustrate the results obtained. Acknowledgments The authors would like to thank the reviewers for their valuable suggestions. References Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in systems and control theory. Philadelphia, PA: SIAM. Chadli, M., Darouach, M., & Daafouz, J. (2008). Static output stabilisation of singular LPV systems: LMI formulation. In 47th IEEE-CDC. Cancun, 9–11 December (pp. 4793–4796). Dai, L. (1989). Singular control systems. New York: Springer. Darouach, M. (2006). Solution to Sylvester equation associated to linear descriptor systems. Systems and Control Letters, 55(10), 835–838. Hsiung, K. L. (1998). Lyapunov inequality and bounded real lemma for discrete-time descriptor systems. In Proceedings of the 37th IEEE conference on decision and control, Florida, USA (pp. 289–290). Ji, X., Su, H., & Chu, J. (2007). Robust state feedback H∞ control for uncertain linear discrete singular systems. IET Control Theory & Applications, 1, 195–200. Lewis, F. L. (1986). A survey of linear singular systems. Circuits, Systems, and Signal Processing, 5, 3–36.
M. Chadli, M. Darouach / Automatica 48 (2012) 449–453 Luenberger, D. J. (1979). Nonlinear descriptor systems. Journal of Economics Dynamics and Control, 1, 219–242. Masubuchi, I., Kamime, Y., Ohara, A., & Suda, N. (1997). H∞ control for descriptor systems: a matrix inequalities approach. Automatica, 3(4), 669–673. Mills, J. K., & Goldenberg, A. A. (1989). Force and position control of manipulators during constrained motion tasks. IEEE Transactions on Robotics and Automation, 38, 30–46. Oliveira, M. C., Bernussou, J., & Geromel, J. C. (1999). A new discrete-time robust stability condition. Systems & Control Letters, 37, 261–265. Peaucelle, D., Arzelier, D., Bachelier, O., & Bernussou, J. (2000). A new robust D-stability condition for real convex polytopic uncertainty. Systems & Control Letters, 40(1), 21–30. Rehm, A., & Allgower, F. (2004). Causal H∞ Control of discrete-time descriptor systems: an LMI approach in two steps. In Proc. 16th int. symposium on mathematical theory of networks and systems, MTNS-2004, Leuven, Belgium, July 5–9. Varga, A. (1995). On stabilization methods of descriptor systems. Systems & Control Letters, 24, 133–138. Verghese, G. C., Levy, B. C., & Kailath, T. (1981). A generalized state-space for singular systems. IEEE Transactions on Automatic Control, 26, 811–831.
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Automatica journal homepage: www.elsevier.com/locate/automatica
Technical communique
Novel bounded real lemma for discrete-time descriptor systems: Application to H∞ control design✩ Mohammed Chadli a,1 , Mohamed Darouach b a
UPJV-MIS (E.A. 4290), University of Picardie Jules Verne, 7 Rue du Moulin Neuf, 80000, Amiens, France
b
CRAN-CNRS (UMR 7039), UHP, Nancy I IUT de Longwy, 186 Rue de Lorraine, 54400 Cosnes et Romain, France
article
info
Article history: Received 12 November 2010 Received in revised form 23 August 2011 Accepted 26 September 2011 Available online 15 December 2011 Keywords: Descriptor systems Linear systems H∞ control Bounded real lemma Strict LMI
abstract This paper concerns the bounded real lemma for discrete-time descriptor systems. A new formulation of the bounded real lemma for these systems is given. It extends the recent results presented in Zhang, Xia, and Shi (2008) and gives necessary and sufficient conditions in strict LMI (linear matrix inequality) which is more suitable for the control design than those presented in Zhang, Xia, and Shi (2008). An application to the H∞ control design is given. A numerical example is presented to show the applicability of our approach. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction and preliminaries Descriptor systems (also known as singular or differential algebraic systems) have attracted particular interest in the literature for their various applications such as economics, robotics, electrical and chemical systems (Dai, 1989; Lewis, 1986; Luenberger, 1979; Mills & Goldenberg, 1989). The analysis and the controller design for such systems have received considerable attention in the past decades (Dai, 1989). Due to its general description, such a model has been employed to design controllers and observers in different areas of research for theoretical development as well as for practical applications (Darouach, 2006; Masubuchi, Kamime, Ohara, & Suda, 1997; Rehm & Allgower, 2004; Varga, 1995; Xia, Zhang, & Boukas, 2008; Yao, Guan, Chen, & Ho, 2006). Interesting results on controllers and observers design are developed for descriptor linear systems using algebraic methods (Darouach, 2006, and references therein) or matrix inequality approaches (Varga, 1995; Xu & Lam, 2004; Xu & Yang, 2000; Yung, 2008; Zhang, Xia, & Shi, 2008). These results are extended to design H∞ controller where necessary and sufficient conditions
✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Mayuresh V. Kothare under the direction of Editor André L. Tits. E-mail addresses: [email protected] (M. Chadli), [email protected] (M. Darouach). 1 Tel.: +33 322827680.
0005-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2011.10.003
are given in nonlinear forms. Uncertain descriptor linear systems are also studied in Ji, Su, and Chu (2007) and Xu and Lam (2004, 2006) and sufficient LMI (linear matrix inequality) conditions are given. On the other hand, the bounded real lemma for discretetime descriptor systems was derived using a non-strict LMI and has the drawback that it cannot be directly solved using classical numerical tools (Hsiung, 1998; Masubuchi et al., 1997; Wang, Yung, & Chang, 1998; Xu & Yang, 2000). Recently strict LMI conditions, which are more tractable and numerically reliable, have been proposed (see Ji et al., 2007; Xu & Lam, 2006; Zhang, Xia, & Shi, 2008, and references therein). Although the results of Zhang, Xia, and Shi (2008) give strict LMI admissibility conditions for H∞ analysis, the result cannot be used directly to design H∞ controllers or observers. Thus, our aim is to give a new formulation to the results of Zhang, Xia, and Shi (2008) and to show how we can apply this formulation in the H∞ control design for discrete-time descriptor systems. This paper proposes strict LMI admissibility conditions to the H∞ control design using existing LMI tools (Boyd, El Ghaoui, Feron, & Balakrishnan, 1994) and the slack variables technique developed for standard systems (Oliveira, Bernussou, & Geromel, 1999). First, necessary and sufficient admissibility conditions are given. From the admissibility conditions obtained, a novel bounded real lemma of discrete-time descriptor systems is given in strict LMI conditions. Then necessary and sufficient conditions to design H∞ controller are given with intermediate free parameters for more relaxation. A numerical example is given to illustrate the proposed approach.
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Notation. Throughout this paper, Rn and Rn×m denote, respectively, the n dimensional Euclidean space and the set of all n × m real matrices. The superscript ‘‘T ’’ denotes matrix transposition, the notation X ≥ Y (respectively, X > Y ) where X and Y are symmetric matrices, means that X − Y is positive semi-definite (respectively, positive definite), the symbol (∗) denotes the transpose elements in the symmetric positions. For a matrix M ∈ Rn×m with rank M = r, let M ⊥ ∈ R(n−r )×n be any matrix such that M ⊥ M = 0 and M ⊥ M ⊥⊤ > 0, note that M ⊥ exists if and only if M has linearly dependent rows. When M is of full row rank matrix we take M ⊥ = 0 by convention. Consider the discrete-time descriptor linear system described by: Ex(t + 1) = Ax(t ) + Bu u(t ) + Bw w(t )
(1a)
y(t ) = Cx(t ) + Dw(t )
(1b)
where x(t ) ∈ R is the state vector, y(t ) ∈ R is the output vector, u(t ) ∈ Rm is the input vector, w(t ) ∈ Rq is the disturbance vector, A ∈ Rn×n , Bu ∈ Rn×m , Bw ∈ Rn×q , C ∈ Rp×n and D ∈ Rp×q . Matrix E may be singular with rankE = r ≤ n. For this system with w(t ) = 0 and u(t ) = 0, we can give the following definitions (Dai, 1989; Rehm & Allgower, 2004; Xu & Lam, 2004). n
Let the pair (E , A) be given, then it is always possible to find I 0
nonsingular matrices M and N such that E = M
M
A1 A3
A2 A4
A⊤ (P − E ⊥⊤ QE ⊥ )A − E ⊤ PE < 0.
0 0
N and A =
N. This decomposition can be obtained via singular
value decomposition of matrix E followed by scaling of the bases. Then, we have the following lemmas (Dai, 1989).
Lemma 2. The pair (E , A) is regular if and only if there exist two nonsingular matrices U and V such that E = U U
¯
A 0
0 I
I 0
0
N
V and A =
V , where N is a nilpotent matrix.
Definition 2. The discrete-time descriptor system (1) is said to be admissible with an H∞ performance γ if it is admissible (for w(t ) = 0) and ∥Gwy (z )∥∞ < γ i.e. ∥y(t )∥2 < γ 2 ∥w(t )∥2 for w(t ) ̸= 0. 2. Main results The following lemma gives necessary and sufficient conditions for system (1), with u(t ) = 0 and w(t ) = 0, to be admissible. Theorem 4. The discrete-time descriptor system (1) or the pair (E , A) is admissible if and only if the following equivalent statements hold. (i) There exists a matrix X = X ⊤ satisfying the following LMI: E ⊤ XE ≥ 0,
A⊤ XA − E ⊤ XE < 0.
(2)
(4)
(iv) There exist matrices P > 0, Q = Q , F and G satisfying the following LMI:
⊤ −E PE + A⊤ F ⊤ + FA (∗)
−F + A⊤ G⊤ < 0. (5) P − E ⊥⊤ QE ⊥ − G − G⊤
(v) These exist matrices P > 0, Q = Q ⊤ , F and G satisfying the following LMI:
−EPE ⊤ + AF ⊤ + FA⊤ (∗)
−F + AG⊤ < 0. P − E ⊥⊤⊥ QE ⊥⊤ − G − G⊤
(6)
Proof. It is proved in Zhang, Xia, and Shi (2008) that statements (2) and (3) are equivalent. Now, we can prove that (3) and (4) are equivalent. This can be obtained from the fact that the pair (E , A) is admissible if and only if the pair (E ⊤ , A⊤ ) is admissible. In fact, we have det(zE − A) = det(zE − A)⊤ = det(zE ⊤ − A⊤ ) and deg(det(zE − A)) = deg(det(zE ⊤ − A⊤ )⊤ ), which shows that the pair (E , A) is regular and causal if and only if (E ⊤ , A⊤ ) is regular and causal. On the other hand, from Lemmas 2 and 3, if the pair (E , A) is regular and causal, there exist two nonsingular matrices U and V such thatE = I 0
U
0 0
V and A = U
and A⊤ = V ⊤
A¯ ⊤ 0
A¯ 0
0 I
0 I
V , then we have E ⊤ = V ⊤
I 0
0 0
U⊤
U ⊤ , in which case the stability of the
pair (E , A) is reduced to that of A¯ or equivalently the stability of A¯ ⊤ which proves the equivalence of inequalities (3) and (4). Consequently, it suffices to prove that condition (5) is equivalent to condition (3). Sufficiency—Condition (3) is satisfied by pre-multiplying (5) by
⊤
I A⊤ and post-multiplying it by I A⊤ . Then, if condition (5) is satisfied, the discrete-time descriptor system (E , A) is admissible. Necessity—Assume that the system (E , A) is admissible, i.e. condition (3) is satisfied. Then there always exist a matrix G such that R − G − G⊤ < 0 with R = P − E ⊥⊤ QE ⊥ . Its follows that
A⊤ RA − E ⊤ PE
If the descriptor system is regular, the transfer function matrix from w(t ) to y(t ) is given by Gwy (z ) = C (zE − A)−1 Bw + D.
satisfying the following
⊤
Lemma 3. Let the pair (E , A) be regular, then it is causal if and only if N = 0.
⊤
A(P − E ⊥⊤⊥ QE ⊥⊤ )A⊤ − EPE ⊤ < 0.
Lemma 1. The pair (E , A) is causal if and only if A4 is nonsingular.
(3)
(iii) There exist matrices P > 0 and Q = Q LMI:
p
Definition 1. • The pair (E , A) is said to be regular if det(zE − A) is not identically zero. • The pair (E , A) is said to be causal if deg(det(zE − A)) = rank E. • The pair (E , A) is said to be stable if |λ(E , A)| < 1 ∀λ(E , A) ∈ {z | det(zE − A) = 0}. • The pair (E , A) is said to be admissible if it is regular, causal and stable.
(ii) There exist matrices P > 0 and Q = Q ⊤ satisfying the following LMI:
(∗)
0 R − G − G⊤
Pre-multiplying (7) by
I
−A
0 I
I 0
< 0.
−A⊤ I
(7) and post-multiplying it by
, we get
⊤ −E PE + A⊤ RA + A⊤ (R − G − G⊤ )A −(R − G − G⊤ )A
(∗) < 0. R − G − G⊤
(8)
Then by choosing F = A⊤ R − A⊤ G, we obtain condition (5). Inequalities (5) and (6) are equivalent. This can be obtained from the fact that the admissibility of the pair (E , A) is equivalent to the admissibility of (E ⊤ , A⊤ ). This completes the proof. Remark 1. Note that for the standard case, i.e. when E = I, we get E ⊥ = 0 and LMI conditions (5) or (6) are reduced to the existence of matrices P > 0, F and G as given in Peaucelle, Arzelier, Bachelier, and Bernussou (2000) or with F = 0 as it is stated in the earlier work of Oliveira et al. (1999). Now, we propose new strict LMI conditions for the discretetime descriptor system (1) to be admissible with an H∞ performance γ .
M. Chadli, M. Darouach / Automatica 48 (2012) 449–453
Theorem 5. The discrete-time descriptor system (1) is admissible with an H∞ performance γ if and only if the following equivalent statements hold. (i) There exist matrices P > 0 and Q = Q ⊤ satisfying the following LMI:
Ω1 (∗)
Ω2 <0 Ω3
(9)
with
where K is a gain matrix of an appropriate dimension. The problem is to design K , such that system (15) is admissible when w(t ) = 0 and ∥Gwy (z )∥∞ < γ i.e. ∥y(t )∥2 < γ 2 ∥w(t )∥2 for w(t ) ̸= 0, where Gwy (z ) = C (zE − A − Bu K )−1 Bw + D. On the other hand, let us consider the following system
¯ (t ) + B¯ w w(t ) E¯ x¯ (t + 1) = Ax
(16a)
y(t ) = C¯ x¯ (t ) + Dw(t )
(16b)
⊤ where x¯ (t ) = x(t )⊤ u(t )⊤ and E 0 A Bu , E¯ = , A¯ = K −I m 0 0 Bw , C¯ = C 0 . B¯ w =
Ω1 = −E ⊤ PE + A⊤ (P − E ⊥⊤ QE ⊥ )A + C ⊤ C Ω2 = C ⊤ D + A⊤ (P − E ⊥⊤ QE ⊥ )Bw ⊥⊤ Ω3 = −γ 2 I + D⊤ D + B⊤ QE ⊥ )Bw . w (P − E (ii) There exist matrices P > 0, Q = Q ⊤ , F and G satisfying the following LMI:
⊤ −E P E + F A + A⊤ F ⊤ (∗)
−F + A⊤ G⊤ P − E ⊥⊤ QE ⊥ − G − G⊤
< 0.
(10)
(iii) There exist matrices P > 0, Q = Q ⊤ , F and G satisfying the following LMI:
−E P E ⊤ + F A⊤ + AF ⊤ (∗)
−F + AG⊤ ⊤⊥⊤ P −E QE ⊤⊥ − G − G⊤
<0
¯ wy (z ) = C¯ (z E¯ − A¯ )−1 B¯ w + D. Then we have the following Let G lemma. Lemma 6. System (15) is admissible for w(t ) = 0 and ∥Gwy (z )∥∞ < γ for w(t ) ̸= 0 if and only if system (16) is admissible for w(t ) = 0 ¯ wy (z )∥∞ < γ for w(t ) ̸= 0. and ∥G Proof. From system (16) we have
z E¯ − A¯ = (11)
A C
A=
Bw , D
E =
P P = 0
E 0
0
γ Ip.q 0 .
0 , Ip
Q Q= 0
, (12)
0
Proof. It is proved in Zhang, Xia, and Shi (2008) that (9) is a necessary and sufficient condition for the discrete-time descriptor system (1) to be admissible with an H∞ performance γ . Thus it suffices to prove that condition (10) is equivalent to (9). Indeed, conditions (9) can be written in the following equivalent form A C
Bw D
⊤
R 0
0 Ip
A C
Bw E ⊤ PE − D 0
0 γ 2 Iq
<0
(13)
with R = P − E ⊥⊤ QE ⊥ . Then (13) can be rewritten as follows
A (P − E ⊤
⊥⊤
QE )A − E P E < 0 ⊥
⊤
(14)
which has the same form as (3). Then the rest of the proof follows from the proof of Theorem 4. Then conditions (9) and (10) are equivalent, and condition (10) is equivalent to (11). This ends the proof. Remark 2. Note that LMI conditions (10)–(11) are more suitable for design problems than condition (9). They can be used easily to design observer or controller with LMI formulation. The following section shows how to design an H∞ controller using existing numerical solvers.
zE − A −K In 0
−Bu Im
−Bu
Im
zE − A − Bu K 0
0 Im
In −K
0 . Im
Then it is easy to see that the regularity and the stability of (15) are equivalent to those of (16). Also rank E¯ = rank E, then the causality of (15) is equivalent to that of (16). On the other hand, we have ¯ wy (z ), which proves the lemma. Gwy (z ) = G Remark 3. The augmented system (16), obtained by using the technique of Chadli, Darouach, and Daafouz (2008), does not change the finite modes or impulsive modes of the original descriptor system (Verghese, Levy, & Kailath, 1981). When the output signal is depending on the input signal: y(t ) = Cx(t ) + By u(t ) + Dw(t ), we can substitute (16b) by y(t ) = [CBy ]¯x(t ) + Dw(t ). Now, by using the results of Lemma 6 and Theorem 5, we can give the following theorem. Theorem 7. The discrete-time descriptor system (16) is admissible with an H∞ performance γ if and only if there exist matrices P > 0, Q = Q ⊤ , F , G and K such that the following equivalent statements hold
⊤ −E¯ P E¯ + F A¯ + A¯ ⊤ F ⊤ −F + A¯ ⊤ G⊤ <0 (∗) P − E¯ ⊥⊤ QE¯ ⊥ − G − G⊤ ¯ ⊤ ¯ ⊤ −E¯ P E¯ ⊤ + F A¯ ⊤ + AF −F + AG (ii) (∗) P − E¯ ⊤⊥⊤ QE¯ ⊤⊥ − G − G⊤ <0
(i)
where P and Q are defined in (12) and
A¯ = 3. H∞ controller design
=
(17)
0
with
451
A¯ C¯
B¯ w , D
E¯ =
E¯ 0
0 γI
(18)
with definitions (17). The discrete-time descriptor linear system (1) with the state feedback control u(t ) = Kx(t ) becomes Ex(t + 1) = (A + Bu K )x(t ) + Bw w(t )
(15a)
y(t ) = Cx(t ) + Dw(t )
(15b)
Remark 4. Note that condition (9) of Zhang, Xia, and Shi (2008) cannot be easily used to design the H∞ controller, which is numerically intractable. The given necessary and sufficient conditions are in the Bilinear Matrix Inequality (BMI) form and introduce
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M. Chadli, M. Darouach / Automatica 48 (2012) 449–453
supplementary free variables. Note that these given BMI design conditions can be easily solved using existing numerical tools such as the YALMIP software (Löfberg, 2008). Nevertheless, strict LMI formulation with intermediate variable is proposed in the following section. The conditions derived lead to less of conservatism for the design problem as it is proved for standard system (Oliveira et al., 1999; Peaucelle et al., 2000). Also note that, the extension of these conditions to uncertain descriptor systems are very simple when compared with Zhang, Xia, and Shi (2008). These extensions are not given in this note for lack of space.
F4 , F3
G G= 1 0
F F1 = 11 F12
גM , M
G4 G3
G11 G1 = G12
גM M
with = ⊤גIm 0m×(n−m) and M ∈ Rm×m is a nonsingular matrix. Then the bilinear forms of F A¯ and GA¯ become
F1 + F4 C¯ F1 B¯ w + F4 D ¯ FA= F3 C¯ F3 D G1 + G4 C¯ G1 B¯ w + G4 D GA¯ = ¯ G3 C
G3 D
with
F11 Bu − גM F12 Bu − M
G11 Bu − גM G12 Bu − M
F A + גN F1 = 11 F12 A + N
G1 =
G11 A + גN G12 A + N
1 E= 0
where N = MK . Then, we get LMI design conditions for Theorem 7 in variables M , N , F11 , F12 , F3 , F4 , G11 , G12 , G3 and G4 . The controller is then given by K = M −1 N. For dual conditions, the same reasoning can be used with the variable change N = MK ⊤ . Note that this LMI formulation is obtained at the expense of necessary and sufficient conditions. However, compared with existing sufficient LMI design conditions (Ji et al., 2007), our results introduce more of slack free variables, and then more of relaxation for the design problem. The following section gives a numerical example to show this benefit. Remark 5. It should be understood that obtaining LMI formulation to design H∞ state feedback control implies some restrictions since the matrices F and G are not completely free. Another design condition in LMI terms can be obtained also by choosing = ג0n.m . These different choices will end with different LMI conditions whose results do overlap, i.e. no inclusion property can be found. Therefore, their conservatism is different according to the examples treated. 5. Numerical example In this section, a numerical example is given to show the effectiveness of the proposed results. It deals with the H∞ state
2 , 0
2.5 A= 1.7
1 , 0
Bu =
C = 1
1 ,
1.0 , 0.8
Bw =
1 1
D = 0.
Solving the condition of Theorem 7, using the proposed LMI design procedure, with E ⊥ = [0 1], i.e. E ⊥ = [0 1 0 0] and γ = 0.43, we get Q = 68.0752, M = 6.2347 and P =
Now, emphasis is placed upon tractable design conditions instead of the numerically intractable sufficient and necessary conditions proposed in the literature (Xu & Lam, 2004; Xu & Yang, 2000; Yung, 2008; Zhang, Xia, & Shi, 2008). Indeed, to get LMI design conditions (Theorem 7), particular structures of matrices F and G are used as follows: F F = 1 0
4. LMI design procedure
feedback control for the descriptor system (1) with the following data:
1.6208 0.9134 0.3450
N = −5.6927
0.9134 26.6530 3.0067
0.3450 3.0067 , −22.6309
−1.1921
−1.5297 1.3950 6.2347 − 0 . 9735 0 . 9231 0 F1 = , −2.6539 2.8129 6.2347 8.9668 −9.0058 6.2347 0 G1 = 6.4853 −6.6273 . 2.7045 −2.5045 6.2347
−1 Then the controller gain is obtained from K = M N = −0.9131 −0.1912 . Note that with the same γ = 0.43, the result of Ji et al. (2007) fails to find a feasible solution, i.e. to find
a controller gain K . Moreover, it is shown that the result of Ji et al. (2007) leads to a smaller gain compared with that of Xu and Lam (2004). This example shows the benefit of the given results. 6. Conclusion In this paper, a new formulation of the bounded real lemma for discrete-time descriptor linear system is given. Necessary and sufficient admissibility conditions in strict LMI form are derived. Then necessary and sufficient H∞ controller synthesis conditions in strict BMI terms are developed whereas existing results are in nonlinear forms. The proposed new formulation is more adapted to the H∞ control problem design and less conservative, by introducing slack variables, than the existing results. Sufficient LMI conditions for the control problem are also proposed. These conditions are obtained under some constraints and are restrictive compared with the BMI formulation. The conditions derived can be extended to uncertain discrete-time descriptor linear systems. An example is given to illustrate the results obtained. Acknowledgments The authors would like to thank the reviewers for their valuable suggestions. References Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in systems and control theory. Philadelphia, PA: SIAM. Chadli, M., Darouach, M., & Daafouz, J. (2008). Static output stabilisation of singular LPV systems: LMI formulation. In 47th IEEE-CDC. Cancun, 9–11 December (pp. 4793–4796). Dai, L. (1989). Singular control systems. New York: Springer. Darouach, M. (2006). Solution to Sylvester equation associated to linear descriptor systems. Systems and Control Letters, 55(10), 835–838. Hsiung, K. L. (1998). Lyapunov inequality and bounded real lemma for discrete-time descriptor systems. In Proceedings of the 37th IEEE conference on decision and control, Florida, USA (pp. 289–290). Ji, X., Su, H., & Chu, J. (2007). Robust state feedback H∞ control for uncertain linear discrete singular systems. IET Control Theory & Applications, 1, 195–200. Lewis, F. L. (1986). A survey of linear singular systems. Circuits, Systems, and Signal Processing, 5, 3–36.
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