# Stabilization of a kind of nonlinear discrete singular large-scale control systems.

INTRODUCTION

With the development of modern control theory and the permeation into other application area, one kind of systems with extensive form has appeared which form follows as:

EX (t) = f(X(t), t, u(t))

Where X(t) [member of] [R.sup.n] is a n--state vector, u(t) [member of] [R.sup.m] is a m--control input vector, E is a n x n matrix, it is usually singular. This kind of systems generally is called as the singular control systems. It appeared large in many areas such as the economy management, the electronic network, robot, bioengineering, aerospace industry and navigation and so forth. Singular large-scale control systems have a more practical background. The actual production process can be described preferably by singular large-scale control systems, particularly by discrete singular large-scale control systems. The causality of discrete singular systems makes related results complicated and challenging for us. At present, the research results of the problem above are seldom. The asymptotical stability (Sun & Peng, 2009; Sun & Chen, 2004) and stabilization (Yang & Zhang, 2004; Sun & Chen, 2011) of discrete linear singular large-scale systems has been considered by Lyapunov function method. This paper consider the state feedback stabilization of a kind of nonlinear discrete singular large-scale control systems by introduce weighted sum Lyapunov function method, and give its interconnecting parameters regions of stability.

DEFINITIONS AND PROBLEM FORMULATION

Consider the nonlinear discrete singular large-scale control systems with m subsystems:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are semi-state vector and vector function, respectively. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a control input vector; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], they are constant matrices;

[m.summation over (i=1)] [n.sub.i] = n, [m.summation over (i=1)] [r.sub.i] = r, [m.summation over (i=1)] [m.sub.i] = l,

denote:x(k) = [[x.sup.T.sub.1](k), [x.sup.T.sub.2](k), ..., [x.sup.T.sub.m](k)] [member of] [R.sup.n]

rank ([E.sub.i]) = [r.sub.i] [less than or equal to] [n.sub.i], E = Block - diag ([E.sub.1], [E.sub.2], ..., [E.sub.m]), rank (E) = r < n, B = Block - diag ([B.sub.1], [B.sub.2], ..., [B.sub.m])

Now we give some concepts about discrete singular system:

Ex(k + 1) = Ax(k) (2)

and discrete singular control system:

Ex(k + 1) = Ax(k) + Bu(k)(k = 1, ..., N) (3)

where E and A are n x n constant matrices, B is a n x m constant matrix, rank (E) = r < n, x(k) [member of] [R.sup.n] is a semi-state vector, u(k) [member of] [R.sup.m] is a control input vector.

Definition 1 (Yang & Zhang, 2004): Discrete singular system (2) is said to be regular if

det(zE - A) [not equal to] 0, for some z [member of] C.

Definition 2 (Yang & Zhang, 2004): The zero solution of discrete singular system (2) is said to be stable if for every [epsilon] > 0, there exists a [delta] > 0, such that [parallel]x(k; [k.sub.0][x.sub.0])[parallel] < e, for all k [greater than or equal to] [k.sub.0], whenever the arbitrary initial consistency value x([k.sub.0]) = [x.sub.0] which satisfies [parallel][x.sub.0][parallel] < [delta].

Definition 3 (Yang & Zhang, 2004): Discrete singular control system (3) is said to be causal if x(k) can be uniquely determined by x(0) and control input vectors u(0), u(1), u(k) for any k(0 [less than or equal to] k [less than or equal to] N). Otherwise, it is said to be non-causal.

Now consider the isolated subsystems of systems:

[E.sub.i][x.sub.i](k + 1) = [A.sub.i][x.sub.i](k) + [B.sub.i][U.sub.i](k)(i = 1, 2, ..., m) (4)

Assume that all systems of systems (4) are R-controllable, we choose the linear control law

[U.sub.i](k) = -[K.sub.i][x.sub.i](k)(i = 1, ..., m) (5)

Then singular closed-loop large-scale systems of systems (1) are given by

[E.sub.i][x.sub.i](k + 1) = ([A.sub.ii] - [B.sub.i][K.sub.i])[x.sub.i](k) + [m.summation over (j=1, j[not equal to] i)] [A.sub.ij][x.sub.j](k) + [f.sub.i](x(k), k) (i = 1, ..., m) (6)

The corresponding closed-loop isolated subsystems are

[E.sub.i][x.sub.i](k + 1) = ([A.sub.ii] - [B.sub.i][K.sub.i])[x.sub.i](k)(i = 1, ..., m) (7)

In order to investigate the stabilization of discrete singular large-scale control systems (1), we give the following lemmas:

Lemma 1 (Yang & Zhang, 2004): The system (3) is said to be R-controllable if

rank [zE - A B] = n

for some z [member of] C.

Lemma 2 (Yang & Zhang, 2004): Discrete singular control system (3) is said to be causal if and only if

deg {det (zE - A)} = rank (E)

Lemma 3 (Yang & Zhang, 2004): Assume that u, v [member of] [R.sup.n], V [member of] [R.sup.nxn] is a positive semi-definite matrix, then 2[u.sup.T]Vv [less than or equal to] [eu.sup.T]Vu + [e.sup.-1][v.sup.T]Vv holds for all e > 0.

Lemma 4 (Wo, 2004): Assume that the system (2) is regular and causal, then it is asymptotically stable if and only if given positive definite matrix W, there exists a positive semi-definite matrix V which satisfies

[A.sup.T]VA - [E.sup.T]VE = - [E.sup.T]WE

Lemma 5 (Wo, 2004): Assume that the system (2) is regular, causal, and there exists a function v(Ex) which satisfies the following conditions, then the sub-equilibrium state of systems (2) Ex = 0 is asymptotically stable.

(a) v(Ex) = [(Ex(k)).sup.T] V(Ex(k)), where V is a positive semi-definite matrix, and rank([E.sup.T]VE) = rank E = r;

(b) [DELTA]v(Ex(k)) [less than or equal to] -[(Ex(k)).sup.T] W(Ex(k)), here W is a positive definite matrix.

MAIN RESULTS

Theorem 1: Assume that all isolated subsystems (4) of systems (1) are R-controllable, all closed-loop isolated subsystems (7) are regular, causal and asymptotically stable, and there exist real numbers [mu] > 0, [delta] > 0 which satisfies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

then when

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

the zero solution of the singular closed-loop large-scale systems (6) are asymptotically stable, the discrete singular large-scale control systems (1) are stabilizable. The interconnecting parameter region of stability is given by (10). Here [W.sub.i] is a positive definite and [V.sub.i] is a positive semi-definite matrix from Lemma 4, and [[lambda].sub.M]([V.sub.i]) denotes the maximum eigenvalue of matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [I.sub.i] is a [n.sub.i] x [n.sub.i] identity matrix.

Proof: systems (7) are regular and causal, as they are asymptotically stable, then given positive definite matrix [W.sub.i], Lyapunov matrix equation

[([A.sub.ii] - [B.sub.i][K.sub.i]).sup.T] [V.sub.i]([A.sub.ii] - [B.sub.i][K.sub.i]) - [E.sub.i.sup.T][V.sub.i][E.sub.i] = -[E.sub.i.sup.T][W.sub.i][E.sub.i]

have positive semi-definite solution [V.sub.i].

[v.sub.i][[E.sub.i][x.sub.i](k)] = [[[E.sub.i][x.sub.i](k)].sup.T] [V.sub.i][[E.sub.i][x.sub.i](k)]

as the scalar Lyapunov function of systems (7).

Let v[Ex(k)] = [m.summation over (i=1)] [v.sub.i][[E.sub.i][x.sub.i](k)]

as the Lyapunov function of systems (1). We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By using Lemma 3, choose e = 1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Noticing that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Noticing that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by using Lemma 5, we know, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

To prove

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By noticing that systems (7) are regular and causal, there exists reversible, matrices [P.sub.i], [Q.sub.i](i = 1, ..., m) which satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Where [I.sup.(1).sub.i], [I.sup.(2).sub.i] is [r.sub.i] x [r.sub.i] and ([n.sub.i] - [r.sub.i]) x ([n.sub.i] - [r.sub.i]) identity matrix, respectively. [P.sub.i], [Q.sub.i], M, [A.sup.(1).sub.ij], [A.sup.(12).sub.ij], [A.sup.(21).sub.ij], [A.sup.(2).sub.ij] are corresponding dimension constant matrices. Thus the singular closed-loop large-scale systems (6) are equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Noticing that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Noticing that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we have

([A.sup.(21).sub.ij][z.sup.(1).sub.j](k) + [A.sup.(2).sub.ij][z.sup.(2).sub.j](k)) = (0 [I.sup.(2).sub.i])[P.sub.i][A.sub.ij][x.sub.j](k).

Noticing that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] holds from (8), that is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The Theorem 1 is proved.

Theorem 2: Assume that all subsystems (4) of system (1) are R-controllable, all closed-loop isolated systems (7) are regular, causal, and given positive definite matrix [W.sub.i], there exists a positive semi-definite matrix [V.sub.i] which satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

if there exists a real number m > 0 which satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

when

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (13)

the zero solution of the discrete singular closed-loop large-scale systems (6) are unstable, the discrete singular large-scale control systems(1) are not stabilizable.

Proving is similar with Theorem 1, here it can be omitted.

EXAMPLE

Consider the following 5-order discrete singular large-scale control system which consists of two sub-systems

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We choose the control law [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is easy to test that (8) and (9) are holded, then we know this system (14) is stabilizable from Theorem 1.

CONCLUSION

In this paper, the state feedback stabilization of a kind of nonlinear discrete singular large-scale control systems is investigated by using generalized Lyapunov function method. According to the bound limit parameter of interconnecting terms, there gives some sufficient conditions for determining the asymptotical stability and unstability of the singular closed-loop large-scale system while the subsystems are regular, causal, and R-controllable.

DOI: 10.3968/j.ans.1715787020120502.1180

REFERENCES

[1] YANG, Dongmei, & ZHANG, Qingling (2004). Singular Systems. Beijing: Science Press.

[2] WO, Songlin, & ZOU, Yun (2003). The Asymptotical Stability and Stabilization for Discrete Singular Large-Scale Systems. Systems Engineering And Electronics, 25(10), 1246-1250.

[3] Sun, Shuiling, & Chen, Yunyun (2011). State Feedback Stabilization of Discrete Singular Large-Scale Control Systems. Studies in Mathematical Sciences, 2(2), 36-42.

[4] Sun, Shuiling, & Peng, Ping (2009). Connective Stability of Singular Linear Time-Invariant Large-Scale Dynamical Systems with Sel-Intraction. Advances in Diffrential Equations and Conntrol Processes, 3(1), 63-76.

[5] Sun, Shuiling, & Chen, Chaotian (2004). Connective Stability of a Family of Nonlinear Discrete Large-Scale Systems. Journal of Lanzhou University of Technology, 31(5), 124-127.

[6] Wo, Songlin (2004). Stability and Decentralized Control for the Singular Large-Scale Systems. Nanjing University of Science and Technology, 35(2), 47-52.

SUN Shuiling [a],*; CHEN Yuanyuan [a]

[a] Guangdong Polytechnic Normal University, Guangzhou, China.

* Corresponding author.

Received 7 February 2012; accepted 22 May 2012