- Email: [email protected]
Continuous stabilization controllers for singular bilinear systems: The state feedback case
Automatica 42 (2006) 309 – 314 www.elsevier.com/locate/automatica
Brief paper
Continuous stabilization controllers for singular bilinear systems: The state feedback case夡 Guoping Lua , Daniel W.C. Hob,∗ a College of Electrical Engineering, Nantong University, Jiangsu, 226007, China b Department of Mathematics, City University of Hong Kong, 83, Tat Chee Avenue, Hong Kong
Received 3 November 2003; received in revised form 8 September 2005; accepted 13 September 2005 Available online 5 December 2005
Abstract Global asymptotic stabilization for a class of singular bilinear systems is first studied in this paper. New approaches are developed by means of the LaSalle invariant principle for nonlinear systems. A new set of sufficient condition is first derived via the continuous static state feedback, the feedback not only guarantees the existence of solution but also the global asymptotical stabilization for the closed loop system. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Singular systems; Bilinear; Stabilization; State feedback control
1. Introduction Many real-world systems can be adequately approximated by bilinear models rather than linear models (Mohler, 1991). Control of bilinear systems (BS) has been a topic of recurring interest over the last few decades since this special family of nonlinear systems is of considerable interests in both theory and applications, see for example Chen (1998), Chen and Tsao (2000), Lu, Zheng, and Zhang (2003b), Mohler (1991). Singular bilinear systems (SBS) can be regarded as a generalized BS, or a special class of singular nonlinear systems, and have been addressed in the literature, (see Lewis, Mertzios, & Marszalek, 1991; Zasadzinski, Ali, Rafaralahy, & Magarotto, 2001; Zasadzinski, Magarotto, Rafaralahy, & Ali, 2003). These works present some solid preliminary results on SBS, however, there are still many challenging issues to be addressed. Stabilization is one of the fundamental issues for singular nonlinear systems and has been investigated in Guo and Malabre (2003), Wang, Yung, and Chang (2002), Wu and 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Naomi E. Leonard under the direction of Editor Hassan Khalil. ∗ Corresponding author. Tel.: +852 2788 8652; fax: +852 2788 7446. E-mail addresses: [email protected] (G. Lu), [email protected] (D.W.C. Ho).
0005-1098/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2005.09.010
Mizukami (1994). Local asymptotic stabilization was presented for general singular nonlinear systems in Wang et al. (2002). Robust stabilization design was obtained for a class of singular systems with nonlinear perturbation in Guo and Malabre (2003). Stability and robust stabilization of nonlinear descriptor (singular) systems were considered in Wu and Mizukami (1994). The authors also addressed the stability issue for a class of nonlinear descriptor systems with uncertain perturbations in Lu, Ho, and Yeung (2003a). So far, little works exists for the stabilization issue of SBS. In particular, the global asymptotic stability/stabilization for SBS has seldom been reported in the literature. In addition, one of the fundamental issues for control of nonlinear singular systems is how to guarantee the existence of solutions for the closed loop system and then asymptotic stability. Another fundamental issue is how to get an sufficient stabilization condition which is independent of the system decomposition. These have been seldom reported in the literature. In this paper, we will solve both fundamental issues for a class of SBS. The objective of this paper is to present continuous global asymptotic stabilization controller designs for SBS. It should be mentioned that the sufficient conditions presented in this paper are dependent on the original system matrices and independent of the state transformation on the system matrices of the original SBS. In this paper, the static state feedback is constructed by using the extended Lyapunov stability theory,
310
G. Lu, D.W.C. Ho / Automatica 42 (2006) 309 – 314
LaSalle’s Lemma. Under the given sufficient condition, state feedback controller can be constructed to guarantee the existence of solution for the closed loop systems and the global asymptotic stability simultaneously. The proposed approach extends the results for BS in the literature. It is worth mentioning that the approach developed in this paper is different from those for singular nonlinear systems in Guo and Malabre (2003), Wang et al. (2002), Wu and Mizukami (1994). This paper is organized as follows. Section 2 presents some assumptions and preliminary results for the SBS. Section 3 presents a sufficient condition for continuous static state feedback. Section 4 is the conclusions of the paper. Notation: W T : transpose of matrix W ∈ Rn×m ; W : [max (W T W )]1/2 , i.e. the square root of the maximal eigenvalue of W T W ; X −T : transpose of matrix X −1 ; I (Ir ): identity √ matrix of appropriate dimensions (of Rr×r ); x = x T x, where x = (x1 x2 · · · xn )T ∈ Rn ; Throughout this note, for symmetric matrices X and Y, X Y , respectively, X > Y ): X−Y is positive semi-definite (respectively, positive definite); X Y respectively, X < Y ): X − Y is negative semi-definite (respectively, negative definite). Matrices, if not explicitly stated, are assumed to have compatible dimensions. For convenience, the notation GAS denotes global asymptotic stability, globally asymptotically stable, or global asymptotic stabilization. 2. Preliminaries Consider the following SBS E x˙ = Ax +
m
Bi xui = Ax + B(x)u,
(1)
i=1
where x ∈ Rn and u = [u1 , u2 , . . . , um ]T ∈ Rm are the system state and control input, respectively. The derivative matrix E ∈ Rn×n is singular, we shall assume that 0 < rank E = r < n. A, Bi ∈ Rn×n for 1i m are constant matrices, and B(x) = [B1 x B2 x · · · Bm x]. Assumption 1 (Dai, 1989; Masubuchi, Kamitane, Ohara, & Suda, 1997). (1) The pair (E, A) is regular, that is, det(E −A) is not identically zero. (2) The pair (E, A) is impulse free, that is, deg(det(E −A)) = rank E. From Assumption 1, there exist two invertible matrices M0 and N0 ∈ Rn×n satisfying M0 EN 0 =
Ir 0
0 , 0
M0 AN 0 =
A0 0
0 In−r
,
(2)
where A0 ∈ Rr×r . If there exists an eigenvalue 0 with a positive real part for det(E − A) = 0, then 0 is also an eigenvalue of A0 . Suppose that control input u = u(x) = [u1 (x), u2 (x), . . . , um (x)]T is continuous and u(0) = 0. Apply the state transformation = (T1 T2 )T = N0−1 x, where 1 ∈ Rr
and 2 ∈ Rn−r . Then SBS (1) is equivalent to ˙ 1 = A0 1 + (Ir 0)M0
m
Bk N0 uk (N ),
(3)
k=1
0 = 2 + (0 In−r )M0
m
Bk N0 uk (N ),
(4)
k=1
where M0 m k=1 Bk N0 uk (N ) m M0 Bk N0 |uk (N )| k=1 m = M0 Bk N0 |uk (N )| → 0,
→ 0.
k=1
Then SBS (1) is equivalent to the following simple form: ˙ 1 = A0 1 + (o()) · · · o()T , 0 = 2 + (o()) · · · o()T ,
(5)
where lim→0 o()/ = 0. From the second equation of (5), there exists a positive constant scalar with 0 < < 1 such that 2 / <√ when is sufficiently small, which implies 2 /1 / 1 − 2 when is sufficiently small. From this, we obtain 1/2 o() o() 2 lim = lim 1+ 2 2 = 0, (6) 1 1 →0 1 →0 which implies that the linearized part of system (3) in a neighborhood of equilibrium solution is ˙ 1 = A0 1 . From Theorem 7.3 in Verhulst (1990), we have that the trivial solution of system (3) is unstable. That is, SBS (1) cannot be stabilized via a continuous state feedback. We sum up the above discussion and present the result as follows. Proposition 2. There is no continuous state feedback u = u(x) with u(0)=0 such that the closed loop system is asymptotically stable for SBS (1) if one of the eigenvalues has a positive real part for the pair (E, A). In this paper, we only concentrate on continuous stabilization feedback controller designs. To this end, it follows from Proposition 2 that we have to assume that all eigenvalues for equation det(E − A) = 0 satisfy Re() 0. For simplicity, we make an additional assumption for SBS (1). Assumption 3. The rth order equation det(E − A) = 0 has r distinct eigenvalues k with Re(k ) 0, k = 1, 2, . . . , r. Assumption 3 implies that A0 has r distinct eigenvalues k . Hence there exist an invertible S0 ∈ Rr×r and block diagonal matrix ∈ Rr×r satisfying A0 S0 = S0 ,
+ T 0.
(7)
G. Lu, D.W.C. Ho / Automatica 42 (2006) 309 – 314
Let
P0 = M0T
(S0 S0T )−1 0
0 −In−r
state feedback controller which guarantees the existence of the solution and GAS for the closed loop system.
N0−1 ,
(8) Theorem 6. If Assumptions 1–4 hold, then there exists a continuous static state feedback control which globally asymptotically stabilizes SBS (1).
then from (2) and (8) we have I 0 (S0 S0T )−1 0 N0−1 E T P0 = N0−T r 0 0 0 −In−r (S0 S0T )−1 0 = N0−T N0−1 0. 0 0 Similarly, P0T E = N0−T
(S0 S0T )−1 0
(9)
0 N0−1 = E T P0 0. 0
(10)
In addition, (2)–(8) yield
0 −2In−r
S := {P ∈ R : E P = P E 0 rank(P ) = n, and P T A + AT P 0} = ∅. T
MAN = diag{A1 , In−r },
N0−1
(15)
where A1 ∈ Rr×r . Suppose that matrix P ∈ S satisfies Assumption 4 and then constructing the following form of continuous state feedback controller candidate T u = u(x) = −−1 0 (x)B (x)P x,
(16)
where (11)
Thus, (9)–(11) and rank(P0 )=n imply that the following matrix set is not empty. That is, n×n
Proof. Assumption 1 implies that there exist two nonsingular matrices M, N ∈ Rn×n such that the following standard decomposition holds. MEN = diag{Ir , 0},
P0T A + AT P0 T −1 T T −1 −T (S0 S0 ) A0 + A0 (S0 S0 ) = N0 0 0.
311
(x) = (1 + N −1 x2 )−1 , m T T (0 In−r )MB k N N Bk P N . 0 = 5 1 +
(17)
k=1
T
(12)
In order to construct a continuous GAS control law for system (1), we make further assumption as follows.
We first show that there exist solutions for any given compatible initial conditions for the closed loop system (1) and (16). Decompose SBS (1) into the following form:
Assumption 4. There exists a P ∈ S such that input matrix B(·) satisfies
z˙ 1 = A1 z1 + (Ir 0)MB(N z)u,
(18)
0 = z2 + f (z1 , z2 ),
(19)
qkT P B(qk ) = 0,
where
k = 1, 2, . . . , r,
(13)
where qk is an eigenvector from (k E − A)qk = 0, 1 k r. In the proof of our main result, the following lemma will be used, which can be regarded as an extension of the known Lyapunov stability theorem. Lemma 5 (LaSalle’s Lemma, Vidyasagar, 1993). Consider an rth order nonlinear system z˙ 1 = f (z1 ),
N
−1
x=z=
z1 z2
,
z ∈ Rn , z1 ∈ Rr , z2 ∈ Rn−r ,
f (z1 , z2 ) = (0 In−r )MB(N z)u. Rewrite f (z1 , z2 ) as follows: m
f (z1 , z2 ) = −
−1 0 Bk N zk (z), (0 In−r )M T 1+z z
(14)
where f (·) : Rr → Rr is smooth vector field in Rr . If there exists a Lyapunov function V (z1 ) such that the nonlinear system (14) satisfies V˙ (z1 )0, then any trajectory of system (14) tends to the largest positive invariant set included in set M = {z1 ∈ Rr | V˙ (z1 ) = 0} when t → +∞. 3. Static state feedback We now present a sufficient condition of GAS for SBS (1) by means of continuous state feedback. It should be mentioned that the following theorem is independent to the partition (or state transformation) of the original SBS (1). In addition, the proof of the following theorem presents a design of continuous
(20)
k=1
where k (z) = zT N T BkT P N z. After some manipulations, we have the Jacobian matrix jf (z1 , z2 )/jz2 as follows: m jf (z1 , z2 ) −2 −1 = − 0 (0 In−r )MB k N jz2 (1 + zT z)2 k=1 k (z) 0 0 T k (z) + × zz In−r In−r 1 + zT z zzT 0 T T T + . (21) N (Bk P + P Bk )N In−r 1 + zT z
312
G. Lu, D.W.C. Ho / Automatica 42 (2006) 309 – 314
Noticing the definition of 0 in (17) and |k (z)|N T BkT P Nz2 ,
zzT z2 ,
(22)
z˙ 1 = A1 z1 ,
we have jf (z1 , z2 ) < 1, jz
(23)
2
which implies that Jacobian matrix j(z2 + f (z1 , z2 )) jf (z1 , z2 ) = In−r + jz2 jz2 is nonsingular for any z2 ∈ Based on the Implicit Functions Theorem, there exists a unique continuously differentiable function z2 = (z1 ) with (0) = 0, then it follows from (16) and (17) that the RHS of the dynamics (18) is continuously differentiable on z1 . That is, there exists a solution of z1 from the dynamics (18) for any compatible initial condition, which implies the existence of solution for closed loop system (1) and (16) for any compatible initial conditions. Next we show that the dynamics are globally asymptotically stable. For the given P above, it is easy to show that P can be rewritten as follows (Lu et al., 2003a; Xu, Dooren, Stefan, & Lam, 2002):
P1 P3
0 P4
x T P T B(x) = 0,
z2 = (z1 ) = 0.
(28)
We now show that any solution z1 =z1 (t) satisfying (28) implies z1 = 0. If z1 = z1 (t) satisfies (28), then for any initial condition z1 (0), we have z1 =eA1 t z1 (0). In addition, from (k E−A)qk =0 there exists k ∈ Cr such that k qk = N , (k Ir − A1 )k = 0. (29) 0, Eq. (29) implies that there exist ak ∈ C, 1 k r, such that
Rn−r .
P = MT
In order to show GAS for the closed loop system. We first show GAS of sub-state z1 . To this end, from inequality (27), V˙ = 0 implies that u = 0, that is,
N −1 ,
(24)
where P1 = P1T > 0, P1 ∈ Rr×r , P3 ∈ R(n−r)×r , P4 ∈ R(n−r)×(n−r) .
z1 Considering x = Nz = N (z , then the dynamics (18) 1) are only dependent on sub-state z1 . For this dynamics, choose the following Lyapunov function candidate V = z1T P1 z1 ,
(25)
then V is positive definite, then the derivative of V along the dynamics (18) yields V˙ = 2z1T P1 A1 z1 + 2z1T P1 (Ir 0)MB(x)u.
(26)
z1 (0) =
r
ak k .
(30)
k=1
That is, z1 (t) = eA1 t z1 (0) =
r
ak ek t k .
(31)
k=1
Then (29) and (31) imply r z1 k Nz = N = ak ek t N 0 0 =
r
k=1
a k e k t q k .
(32)
k=1
Dividing the set of the r distinct eigenvalues of the pair (E, A) into the following form: {k : 1 k r} =
s
i ,
i=0
where 0 = {±j k : ik 0, k = 1, 2, . . . , p0 }, i = {− i ± j ik : i > 0, ik 0, k = 1, 2, . . . , pi }, √ j = −1, 1 < 2 < · · · < s , i = 1, 2, . . . , s. Let 1 > 2 > · · · > p0 0, 1 = j 1 and 2 = −j 1 . For convenience, let (x)=x T P T B(x), then by (32), zT N T P T B(N z)= 0 can be written in the following form.
r k t ak e qk k=1
Noticing algebraic equation (19), and after some manipulations, we have V˙ = 2
=2
P 1 z1 P3 z1 + P 4 z2 P1 z1 P3 z1 + P 4 z2
T T
A1 z1 + (Ir 0)MB(x)u 0 A1 z1 + (Ir 0)MB(x)u z2 + (0 In−r )MB(x)u
= x T (P T A + AT P )x T T T − 2−1 0 (x)x P B(x)B (x)P x 0.
(27)
= a12 (q1 )ej2 1 t + a22 (q2 )e−j2 1 t + (t) = 0,
(33)
where (t) is the error term. Since k , 1 k r are distinct eigenvalues, the exponential terms ej2 1 t and e−j2 1 t cannot be linearly represented by the terms e1 t el t , e2 t el t and ek t el t (k, l=3, 4, . . . , r.) in the function (t). Thus zT N T P T B(N z)= 0 implies that a12 (q1 )ej2 1 t + a22 (q2 )e−j2 1 t = [a12 (q1 ) + a22 (q2 )] cos 2 1 t + [a12 (q1 ) − a22 (q2 )] sin 2 1 t = 0.
(34)
G. Lu, D.W.C. Ho / Automatica 42 (2006) 309 – 314
Noticing that cos 2 1 t and sin 2 1 t are linear independent, then from (34), we have a12 (q1 ) + a22 (q2 ) = 0 = a12 (q1 ) − a22 (q2 ), then from Assumption 4, we obtain a1 =a2 =0. With a1 =a2 =0, solution (31) is rewritten as z1 (t) = eA1 t z1 (0) =
r
ak ek t k .
(35)
k=3
We continue to substitute solution (35) into zT N T P T B(N z)=0 and use the above procedure for obtaining z1 (t) = eA1 t z1 (0) =
s
ak ek t k .
(36)
i=1 k ∈ i
Noticing (e 1 t N z1 (t)) = e2 1 t (N z1 (t)), then ⎛ ⎞ s ( + )t ⎝ ak e 1 k qk ⎠ = 0.
(37)
i=1 k ∈ i
If k ∈ 1 , then 1 + k is 0 or pure imaginary value. From the same proof above, we have ak = 0 if k ∈ 1 . Further, ak = 0 for all 1 k r can be obtained by consecutively using this procedure. Hence z1 ≡ 0 is the unique solution from the equations in (28). It follows that z1 ≡ 0 is the unique solution for V˙ = 0. With this result, GAS of sub-state z1 is established by applying Lemma 5. The continuity of function z2 = (z1 ) implies GAS of z2 from the above result on sub-state z1 . Therefore the closed loop system of (1) and (16) is GAS. This completes the proof. Remark 7. The sufficient condition in Theorem 6 is characterized by the system matrices and is independent of any state transformation. By means of the LaSalle invariant principle, Theorem 6 presents a new approach to study state feedback stabilization controller for SBS, which is easy to construct based on the system matrices. The result in this section also extends the main result in Rahn (1996) for nonsingular single input BS into multi-input SBS. In addition, Theorem 6 forms a fundamental framework for the development of stabilization via dynamic output feedback. 4. Conclusions This paper addresses GAS for MIMO SBS. Sufficient condition for the global asymptotic stabilization via continuous static state feedback is first presented. The above results will lead to other important research topics such as robust control
313
problem and dynamic output feedback for SBS. Another interesting topic for future research is to extend the existing global stabilization results to more general class of SBS by means of discontinuous controllers which is now under investigation. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant No. 60474076, NSF Grant BK2003034 from Jiangsu Province and NSF Grant 04KJB510105 from the Jiangsu Provincial Department for Education. This work was also supported by two grants from RGC of HKSAR, China (CityU 101103, 101004). The authors are also grateful to the reviewers for thorough and constructive reviews. References Chen, M. S. (1998). Exponential stabilization of a constrained bilinear system. Automatica, 34(8), 989–992. Chen, M. S., & Tsao, S. T. (2000). Exponential stabilization of a class of unstable bilinear systems. IEEE Transaction on Automatic Control, 45(5), 989–992. Dai, L. (1989). Singular control systems. Berlin, Germany: Springer. Guo, L., & Malabre, M. (2003). Robust H∞ control for descriptor systems with non-linear uncertainties. International Journal of Control, 76(12), 1254–1262. Lewis, F. L., Mertzios, B. G., & Marszalek, W. (1991). Analysis of singular bilinear systems using Walsh functions. IEE Proceedings-Control Theory and Applications, 138(2), 89–92. Lu, G. P., Ho, D. W. C., & Yeung, L. F. (2003a). Generalized quadratic stability for perturbated singular Fsystems. In Proceedings of 42th IEEE conference on decision and control. Maui, Hawaii, USA. Lu, G. P., Zheng, Y. F., & Zhang, C. S. (2003b). Dynamical output feedback stabilization of MIMO bilinear systems with undamped natural response. Asian Journal of Control, 5(2), 251–260. Masubuchi, I., Kamitane, Y., Ohara, A., & Suda, N. (1997). H∞ control for descriptor systems: a matrix inequalities approach. Automatica, 33, 669–673. Mohler, R. R. (1991). Nonlinear systems: V.2 Application to bilinear control. Englewood Cliffs, NJ: Prentice-Hall. Rahn, C. D. (1996). Stabilizability conditions for strictly bilinear systems with purely imaginary spectra. IEEE Transaction on Automatic Control, 41, 1346–1347. Verhulst, F. (1990). Nonlinear differential equations and dynamical systems. Berlin, Heidelberg, Germany: Springer. Vidyasagar, M. (1993). Nonlinear systems analysis. Englewood Cliffs, New Jersey: Prentice-Hall. Wang, H. S., Yung, C. F., & Chang, F. R. (2002). H∞ control for nonlinear descriptor systems. IEEE Transaction on Automatic Control, 47, 1919–1925. Wu, H. S., & Mizukami, K. (1994). Stability and robust stabilization of nonlinear descriptor systems with uncertainties. In Proceedings of the 33rd IEEE conference on decision and control (Vol. 3) (pp. 2772–2777), FL, USA. Xu, S. Y., Dooren, P. V., Stefan, R., & Lam, J. (2002). Robust stability and stabilization for singular systems with state delay and parameter uncertainty. IEEE Transaction on Automatic Control, 47, 1122–1128. Zasadzinski, M., Ali, H. S., Rafaralahy, H., & Magarotto, E. (2001). Disturbance decoupled diagnostic observer for singular bilinear systems. In Proceedings of the American control conference (Vol. 2) (pp. 1455–1460).
314
G. Lu, D.W.C. Ho / Automatica 42 (2006) 309 – 314
Zasadzinski, M., Magarotto, E., Rafaralahy, H., & Ali, H. S. (2003). Residual generator design for singular bilinear systems subjected to unmeasurable disturbances: An LMI approach. Automatica, 39, 703–713. Professor Guoping Lu received the B.S. degree from the Department of Applied Mathematics, Chengdu University of Science and Technology, China, in 1984, and the M.S. and Ph.D. degrees from the Department of Mathematics, East China Normal University, China, in 1989 and 1998, respectively. He is currently a Professor at the College of Electrical Engineering, Nantong University, Jiangsu, China. His current research interests include nonlinear control, robust control and networked control.
Dr. Daniel W. C. Ho received a first class B.Sc., M.Sc. and Ph.D. degrees in mathematics from the University of Salford (UK) in 1980, 1982 and 1986, respectively. From 1985 to 1988, Dr. Ho was a Research Fellow in Industrial Control Unit, University of Strathclyde (Glasgow, Scotland). In 1989, he joined the Department of Mathematics, City University of Hong Kong, where he is currently an Associate Professor. He is an Associate Editor of Asian Journal of Control.His research interests include H-infinity control theory, robust pole assignment problem, adaptive neural wavelet identification, nonlinear control theory.
Brief paper
Continuous stabilization controllers for singular bilinear systems: The state feedback case夡 Guoping Lua , Daniel W.C. Hob,∗ a College of Electrical Engineering, Nantong University, Jiangsu, 226007, China b Department of Mathematics, City University of Hong Kong, 83, Tat Chee Avenue, Hong Kong
Received 3 November 2003; received in revised form 8 September 2005; accepted 13 September 2005 Available online 5 December 2005
Abstract Global asymptotic stabilization for a class of singular bilinear systems is first studied in this paper. New approaches are developed by means of the LaSalle invariant principle for nonlinear systems. A new set of sufficient condition is first derived via the continuous static state feedback, the feedback not only guarantees the existence of solution but also the global asymptotical stabilization for the closed loop system. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Singular systems; Bilinear; Stabilization; State feedback control
1. Introduction Many real-world systems can be adequately approximated by bilinear models rather than linear models (Mohler, 1991). Control of bilinear systems (BS) has been a topic of recurring interest over the last few decades since this special family of nonlinear systems is of considerable interests in both theory and applications, see for example Chen (1998), Chen and Tsao (2000), Lu, Zheng, and Zhang (2003b), Mohler (1991). Singular bilinear systems (SBS) can be regarded as a generalized BS, or a special class of singular nonlinear systems, and have been addressed in the literature, (see Lewis, Mertzios, & Marszalek, 1991; Zasadzinski, Ali, Rafaralahy, & Magarotto, 2001; Zasadzinski, Magarotto, Rafaralahy, & Ali, 2003). These works present some solid preliminary results on SBS, however, there are still many challenging issues to be addressed. Stabilization is one of the fundamental issues for singular nonlinear systems and has been investigated in Guo and Malabre (2003), Wang, Yung, and Chang (2002), Wu and 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Naomi E. Leonard under the direction of Editor Hassan Khalil. ∗ Corresponding author. Tel.: +852 2788 8652; fax: +852 2788 7446. E-mail addresses: [email protected] (G. Lu), [email protected] (D.W.C. Ho).
0005-1098/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2005.09.010
Mizukami (1994). Local asymptotic stabilization was presented for general singular nonlinear systems in Wang et al. (2002). Robust stabilization design was obtained for a class of singular systems with nonlinear perturbation in Guo and Malabre (2003). Stability and robust stabilization of nonlinear descriptor (singular) systems were considered in Wu and Mizukami (1994). The authors also addressed the stability issue for a class of nonlinear descriptor systems with uncertain perturbations in Lu, Ho, and Yeung (2003a). So far, little works exists for the stabilization issue of SBS. In particular, the global asymptotic stability/stabilization for SBS has seldom been reported in the literature. In addition, one of the fundamental issues for control of nonlinear singular systems is how to guarantee the existence of solutions for the closed loop system and then asymptotic stability. Another fundamental issue is how to get an sufficient stabilization condition which is independent of the system decomposition. These have been seldom reported in the literature. In this paper, we will solve both fundamental issues for a class of SBS. The objective of this paper is to present continuous global asymptotic stabilization controller designs for SBS. It should be mentioned that the sufficient conditions presented in this paper are dependent on the original system matrices and independent of the state transformation on the system matrices of the original SBS. In this paper, the static state feedback is constructed by using the extended Lyapunov stability theory,
310
G. Lu, D.W.C. Ho / Automatica 42 (2006) 309 – 314
LaSalle’s Lemma. Under the given sufficient condition, state feedback controller can be constructed to guarantee the existence of solution for the closed loop systems and the global asymptotic stability simultaneously. The proposed approach extends the results for BS in the literature. It is worth mentioning that the approach developed in this paper is different from those for singular nonlinear systems in Guo and Malabre (2003), Wang et al. (2002), Wu and Mizukami (1994). This paper is organized as follows. Section 2 presents some assumptions and preliminary results for the SBS. Section 3 presents a sufficient condition for continuous static state feedback. Section 4 is the conclusions of the paper. Notation: W T : transpose of matrix W ∈ Rn×m ; W : [max (W T W )]1/2 , i.e. the square root of the maximal eigenvalue of W T W ; X −T : transpose of matrix X −1 ; I (Ir ): identity √ matrix of appropriate dimensions (of Rr×r ); x = x T x, where x = (x1 x2 · · · xn )T ∈ Rn ; Throughout this note, for symmetric matrices X and Y, X Y , respectively, X > Y ): X−Y is positive semi-definite (respectively, positive definite); X Y respectively, X < Y ): X − Y is negative semi-definite (respectively, negative definite). Matrices, if not explicitly stated, are assumed to have compatible dimensions. For convenience, the notation GAS denotes global asymptotic stability, globally asymptotically stable, or global asymptotic stabilization. 2. Preliminaries Consider the following SBS E x˙ = Ax +
m
Bi xui = Ax + B(x)u,
(1)
i=1
where x ∈ Rn and u = [u1 , u2 , . . . , um ]T ∈ Rm are the system state and control input, respectively. The derivative matrix E ∈ Rn×n is singular, we shall assume that 0 < rank E = r < n. A, Bi ∈ Rn×n for 1i m are constant matrices, and B(x) = [B1 x B2 x · · · Bm x]. Assumption 1 (Dai, 1989; Masubuchi, Kamitane, Ohara, & Suda, 1997). (1) The pair (E, A) is regular, that is, det(E −A) is not identically zero. (2) The pair (E, A) is impulse free, that is, deg(det(E −A)) = rank E. From Assumption 1, there exist two invertible matrices M0 and N0 ∈ Rn×n satisfying M0 EN 0 =
Ir 0
0 , 0
M0 AN 0 =
A0 0
0 In−r
,
(2)
where A0 ∈ Rr×r . If there exists an eigenvalue 0 with a positive real part for det(E − A) = 0, then 0 is also an eigenvalue of A0 . Suppose that control input u = u(x) = [u1 (x), u2 (x), . . . , um (x)]T is continuous and u(0) = 0. Apply the state transformation = (T1 T2 )T = N0−1 x, where 1 ∈ Rr
and 2 ∈ Rn−r . Then SBS (1) is equivalent to ˙ 1 = A0 1 + (Ir 0)M0
m
Bk N0 uk (N ),
(3)
k=1
0 = 2 + (0 In−r )M0
m
Bk N0 uk (N ),
(4)
k=1
where M0 m k=1 Bk N0 uk (N ) m M0 Bk N0 |uk (N )| k=1 m = M0 Bk N0 |uk (N )| → 0,
→ 0.
k=1
Then SBS (1) is equivalent to the following simple form: ˙ 1 = A0 1 + (o()) · · · o()T , 0 = 2 + (o()) · · · o()T ,
(5)
where lim→0 o()/ = 0. From the second equation of (5), there exists a positive constant scalar with 0 < < 1 such that 2 / <√ when is sufficiently small, which implies 2 /1 / 1 − 2 when is sufficiently small. From this, we obtain 1/2 o() o() 2 lim = lim 1+ 2 2 = 0, (6) 1 1 →0 1 →0 which implies that the linearized part of system (3) in a neighborhood of equilibrium solution is ˙ 1 = A0 1 . From Theorem 7.3 in Verhulst (1990), we have that the trivial solution of system (3) is unstable. That is, SBS (1) cannot be stabilized via a continuous state feedback. We sum up the above discussion and present the result as follows. Proposition 2. There is no continuous state feedback u = u(x) with u(0)=0 such that the closed loop system is asymptotically stable for SBS (1) if one of the eigenvalues has a positive real part for the pair (E, A). In this paper, we only concentrate on continuous stabilization feedback controller designs. To this end, it follows from Proposition 2 that we have to assume that all eigenvalues for equation det(E − A) = 0 satisfy Re() 0. For simplicity, we make an additional assumption for SBS (1). Assumption 3. The rth order equation det(E − A) = 0 has r distinct eigenvalues k with Re(k ) 0, k = 1, 2, . . . , r. Assumption 3 implies that A0 has r distinct eigenvalues k . Hence there exist an invertible S0 ∈ Rr×r and block diagonal matrix ∈ Rr×r satisfying A0 S0 = S0 ,
+ T 0.
(7)
G. Lu, D.W.C. Ho / Automatica 42 (2006) 309 – 314
Let
P0 = M0T
(S0 S0T )−1 0
0 −In−r
state feedback controller which guarantees the existence of the solution and GAS for the closed loop system.
N0−1 ,
(8) Theorem 6. If Assumptions 1–4 hold, then there exists a continuous static state feedback control which globally asymptotically stabilizes SBS (1).
then from (2) and (8) we have I 0 (S0 S0T )−1 0 N0−1 E T P0 = N0−T r 0 0 0 −In−r (S0 S0T )−1 0 = N0−T N0−1 0. 0 0 Similarly, P0T E = N0−T
(S0 S0T )−1 0
(9)
0 N0−1 = E T P0 0. 0
(10)
In addition, (2)–(8) yield
0 −2In−r
S := {P ∈ R : E P = P E 0 rank(P ) = n, and P T A + AT P 0} = ∅. T
MAN = diag{A1 , In−r },
N0−1
(15)
where A1 ∈ Rr×r . Suppose that matrix P ∈ S satisfies Assumption 4 and then constructing the following form of continuous state feedback controller candidate T u = u(x) = −−1 0 (x)B (x)P x,
(16)
where (11)
Thus, (9)–(11) and rank(P0 )=n imply that the following matrix set is not empty. That is, n×n
Proof. Assumption 1 implies that there exist two nonsingular matrices M, N ∈ Rn×n such that the following standard decomposition holds. MEN = diag{Ir , 0},
P0T A + AT P0 T −1 T T −1 −T (S0 S0 ) A0 + A0 (S0 S0 ) = N0 0 0.
311
(x) = (1 + N −1 x2 )−1 , m T T (0 In−r )MB k N N Bk P N . 0 = 5 1 +
(17)
k=1
T
(12)
In order to construct a continuous GAS control law for system (1), we make further assumption as follows.
We first show that there exist solutions for any given compatible initial conditions for the closed loop system (1) and (16). Decompose SBS (1) into the following form:
Assumption 4. There exists a P ∈ S such that input matrix B(·) satisfies
z˙ 1 = A1 z1 + (Ir 0)MB(N z)u,
(18)
0 = z2 + f (z1 , z2 ),
(19)
qkT P B(qk ) = 0,
where
k = 1, 2, . . . , r,
(13)
where qk is an eigenvector from (k E − A)qk = 0, 1 k r. In the proof of our main result, the following lemma will be used, which can be regarded as an extension of the known Lyapunov stability theorem. Lemma 5 (LaSalle’s Lemma, Vidyasagar, 1993). Consider an rth order nonlinear system z˙ 1 = f (z1 ),
N
−1
x=z=
z1 z2
,
z ∈ Rn , z1 ∈ Rr , z2 ∈ Rn−r ,
f (z1 , z2 ) = (0 In−r )MB(N z)u. Rewrite f (z1 , z2 ) as follows: m
f (z1 , z2 ) = −
−1 0 Bk N zk (z), (0 In−r )M T 1+z z
(14)
where f (·) : Rr → Rr is smooth vector field in Rr . If there exists a Lyapunov function V (z1 ) such that the nonlinear system (14) satisfies V˙ (z1 )0, then any trajectory of system (14) tends to the largest positive invariant set included in set M = {z1 ∈ Rr | V˙ (z1 ) = 0} when t → +∞. 3. Static state feedback We now present a sufficient condition of GAS for SBS (1) by means of continuous state feedback. It should be mentioned that the following theorem is independent to the partition (or state transformation) of the original SBS (1). In addition, the proof of the following theorem presents a design of continuous
(20)
k=1
where k (z) = zT N T BkT P N z. After some manipulations, we have the Jacobian matrix jf (z1 , z2 )/jz2 as follows: m jf (z1 , z2 ) −2 −1 = − 0 (0 In−r )MB k N jz2 (1 + zT z)2 k=1 k (z) 0 0 T k (z) + × zz In−r In−r 1 + zT z zzT 0 T T T + . (21) N (Bk P + P Bk )N In−r 1 + zT z
312
G. Lu, D.W.C. Ho / Automatica 42 (2006) 309 – 314
Noticing the definition of 0 in (17) and |k (z)|N T BkT P Nz2 ,
zzT z2 ,
(22)
z˙ 1 = A1 z1 ,
we have jf (z1 , z2 ) < 1, jz
(23)
2
which implies that Jacobian matrix j(z2 + f (z1 , z2 )) jf (z1 , z2 ) = In−r + jz2 jz2 is nonsingular for any z2 ∈ Based on the Implicit Functions Theorem, there exists a unique continuously differentiable function z2 = (z1 ) with (0) = 0, then it follows from (16) and (17) that the RHS of the dynamics (18) is continuously differentiable on z1 . That is, there exists a solution of z1 from the dynamics (18) for any compatible initial condition, which implies the existence of solution for closed loop system (1) and (16) for any compatible initial conditions. Next we show that the dynamics are globally asymptotically stable. For the given P above, it is easy to show that P can be rewritten as follows (Lu et al., 2003a; Xu, Dooren, Stefan, & Lam, 2002):
P1 P3
0 P4
x T P T B(x) = 0,
z2 = (z1 ) = 0.
(28)
We now show that any solution z1 =z1 (t) satisfying (28) implies z1 = 0. If z1 = z1 (t) satisfies (28), then for any initial condition z1 (0), we have z1 =eA1 t z1 (0). In addition, from (k E−A)qk =0 there exists k ∈ Cr such that k qk = N , (k Ir − A1 )k = 0. (29) 0, Eq. (29) implies that there exist ak ∈ C, 1 k r, such that
Rn−r .
P = MT
In order to show GAS for the closed loop system. We first show GAS of sub-state z1 . To this end, from inequality (27), V˙ = 0 implies that u = 0, that is,
N −1 ,
(24)
where P1 = P1T > 0, P1 ∈ Rr×r , P3 ∈ R(n−r)×r , P4 ∈ R(n−r)×(n−r) .
z1 Considering x = Nz = N (z , then the dynamics (18) 1) are only dependent on sub-state z1 . For this dynamics, choose the following Lyapunov function candidate V = z1T P1 z1 ,
(25)
then V is positive definite, then the derivative of V along the dynamics (18) yields V˙ = 2z1T P1 A1 z1 + 2z1T P1 (Ir 0)MB(x)u.
(26)
z1 (0) =
r
ak k .
(30)
k=1
That is, z1 (t) = eA1 t z1 (0) =
r
ak ek t k .
(31)
k=1
Then (29) and (31) imply r z1 k Nz = N = ak ek t N 0 0 =
r
k=1
a k e k t q k .
(32)
k=1
Dividing the set of the r distinct eigenvalues of the pair (E, A) into the following form: {k : 1 k r} =
s
i ,
i=0
where 0 = {±j k : ik 0, k = 1, 2, . . . , p0 }, i = {− i ± j ik : i > 0, ik 0, k = 1, 2, . . . , pi }, √ j = −1, 1 < 2 < · · · < s , i = 1, 2, . . . , s. Let 1 > 2 > · · · > p0 0, 1 = j 1 and 2 = −j 1 . For convenience, let (x)=x T P T B(x), then by (32), zT N T P T B(N z)= 0 can be written in the following form.
r k t ak e qk k=1
Noticing algebraic equation (19), and after some manipulations, we have V˙ = 2
=2
P 1 z1 P3 z1 + P 4 z2 P1 z1 P3 z1 + P 4 z2
T T
A1 z1 + (Ir 0)MB(x)u 0 A1 z1 + (Ir 0)MB(x)u z2 + (0 In−r )MB(x)u
= x T (P T A + AT P )x T T T − 2−1 0 (x)x P B(x)B (x)P x 0.
(27)
= a12 (q1 )ej2 1 t + a22 (q2 )e−j2 1 t + (t) = 0,
(33)
where (t) is the error term. Since k , 1 k r are distinct eigenvalues, the exponential terms ej2 1 t and e−j2 1 t cannot be linearly represented by the terms e1 t el t , e2 t el t and ek t el t (k, l=3, 4, . . . , r.) in the function (t). Thus zT N T P T B(N z)= 0 implies that a12 (q1 )ej2 1 t + a22 (q2 )e−j2 1 t = [a12 (q1 ) + a22 (q2 )] cos 2 1 t + [a12 (q1 ) − a22 (q2 )] sin 2 1 t = 0.
(34)
G. Lu, D.W.C. Ho / Automatica 42 (2006) 309 – 314
Noticing that cos 2 1 t and sin 2 1 t are linear independent, then from (34), we have a12 (q1 ) + a22 (q2 ) = 0 = a12 (q1 ) − a22 (q2 ), then from Assumption 4, we obtain a1 =a2 =0. With a1 =a2 =0, solution (31) is rewritten as z1 (t) = eA1 t z1 (0) =
r
ak ek t k .
(35)
k=3
We continue to substitute solution (35) into zT N T P T B(N z)=0 and use the above procedure for obtaining z1 (t) = eA1 t z1 (0) =
s
ak ek t k .
(36)
i=1 k ∈ i
Noticing (e 1 t N z1 (t)) = e2 1 t (N z1 (t)), then ⎛ ⎞ s ( + )t ⎝ ak e 1 k qk ⎠ = 0.
(37)
i=1 k ∈ i
If k ∈ 1 , then 1 + k is 0 or pure imaginary value. From the same proof above, we have ak = 0 if k ∈ 1 . Further, ak = 0 for all 1 k r can be obtained by consecutively using this procedure. Hence z1 ≡ 0 is the unique solution from the equations in (28). It follows that z1 ≡ 0 is the unique solution for V˙ = 0. With this result, GAS of sub-state z1 is established by applying Lemma 5. The continuity of function z2 = (z1 ) implies GAS of z2 from the above result on sub-state z1 . Therefore the closed loop system of (1) and (16) is GAS. This completes the proof. Remark 7. The sufficient condition in Theorem 6 is characterized by the system matrices and is independent of any state transformation. By means of the LaSalle invariant principle, Theorem 6 presents a new approach to study state feedback stabilization controller for SBS, which is easy to construct based on the system matrices. The result in this section also extends the main result in Rahn (1996) for nonsingular single input BS into multi-input SBS. In addition, Theorem 6 forms a fundamental framework for the development of stabilization via dynamic output feedback. 4. Conclusions This paper addresses GAS for MIMO SBS. Sufficient condition for the global asymptotic stabilization via continuous static state feedback is first presented. The above results will lead to other important research topics such as robust control
313
problem and dynamic output feedback for SBS. Another interesting topic for future research is to extend the existing global stabilization results to more general class of SBS by means of discontinuous controllers which is now under investigation. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant No. 60474076, NSF Grant BK2003034 from Jiangsu Province and NSF Grant 04KJB510105 from the Jiangsu Provincial Department for Education. This work was also supported by two grants from RGC of HKSAR, China (CityU 101103, 101004). The authors are also grateful to the reviewers for thorough and constructive reviews. References Chen, M. S. (1998). Exponential stabilization of a constrained bilinear system. Automatica, 34(8), 989–992. Chen, M. S., & Tsao, S. T. (2000). Exponential stabilization of a class of unstable bilinear systems. IEEE Transaction on Automatic Control, 45(5), 989–992. Dai, L. (1989). Singular control systems. Berlin, Germany: Springer. Guo, L., & Malabre, M. (2003). Robust H∞ control for descriptor systems with non-linear uncertainties. International Journal of Control, 76(12), 1254–1262. Lewis, F. L., Mertzios, B. G., & Marszalek, W. (1991). Analysis of singular bilinear systems using Walsh functions. IEE Proceedings-Control Theory and Applications, 138(2), 89–92. Lu, G. P., Ho, D. W. C., & Yeung, L. F. (2003a). Generalized quadratic stability for perturbated singular Fsystems. In Proceedings of 42th IEEE conference on decision and control. Maui, Hawaii, USA. Lu, G. P., Zheng, Y. F., & Zhang, C. S. (2003b). Dynamical output feedback stabilization of MIMO bilinear systems with undamped natural response. Asian Journal of Control, 5(2), 251–260. Masubuchi, I., Kamitane, Y., Ohara, A., & Suda, N. (1997). H∞ control for descriptor systems: a matrix inequalities approach. Automatica, 33, 669–673. Mohler, R. R. (1991). Nonlinear systems: V.2 Application to bilinear control. Englewood Cliffs, NJ: Prentice-Hall. Rahn, C. D. (1996). Stabilizability conditions for strictly bilinear systems with purely imaginary spectra. IEEE Transaction on Automatic Control, 41, 1346–1347. Verhulst, F. (1990). Nonlinear differential equations and dynamical systems. Berlin, Heidelberg, Germany: Springer. Vidyasagar, M. (1993). Nonlinear systems analysis. Englewood Cliffs, New Jersey: Prentice-Hall. Wang, H. S., Yung, C. F., & Chang, F. R. (2002). H∞ control for nonlinear descriptor systems. IEEE Transaction on Automatic Control, 47, 1919–1925. Wu, H. S., & Mizukami, K. (1994). Stability and robust stabilization of nonlinear descriptor systems with uncertainties. In Proceedings of the 33rd IEEE conference on decision and control (Vol. 3) (pp. 2772–2777), FL, USA. Xu, S. Y., Dooren, P. V., Stefan, R., & Lam, J. (2002). Robust stability and stabilization for singular systems with state delay and parameter uncertainty. IEEE Transaction on Automatic Control, 47, 1122–1128. Zasadzinski, M., Ali, H. S., Rafaralahy, H., & Magarotto, E. (2001). Disturbance decoupled diagnostic observer for singular bilinear systems. In Proceedings of the American control conference (Vol. 2) (pp. 1455–1460).
314
G. Lu, D.W.C. Ho / Automatica 42 (2006) 309 – 314
Zasadzinski, M., Magarotto, E., Rafaralahy, H., & Ali, H. S. (2003). Residual generator design for singular bilinear systems subjected to unmeasurable disturbances: An LMI approach. Automatica, 39, 703–713. Professor Guoping Lu received the B.S. degree from the Department of Applied Mathematics, Chengdu University of Science and Technology, China, in 1984, and the M.S. and Ph.D. degrees from the Department of Mathematics, East China Normal University, China, in 1989 and 1998, respectively. He is currently a Professor at the College of Electrical Engineering, Nantong University, Jiangsu, China. His current research interests include nonlinear control, robust control and networked control.
Dr. Daniel W. C. Ho received a first class B.Sc., M.Sc. and Ph.D. degrees in mathematics from the University of Salford (UK) in 1980, 1982 and 1986, respectively. From 1985 to 1988, Dr. Ho was a Research Fellow in Industrial Control Unit, University of Strathclyde (Glasgow, Scotland). In 1989, he joined the Department of Mathematics, City University of Hong Kong, where he is currently an Associate Professor. He is an Associate Editor of Asian Journal of Control.His research interests include H-infinity control theory, robust pole assignment problem, adaptive neural wavelet identification, nonlinear control theory.