时间：2022-08-28 08:06:46
Abstract
This paper studies the state feedback stabilization of a kind of nonlinear discrete singular large―scale control systems by using Lyapunov matrix equation， generalized Lyapunov function method and matrix theory. There gives some sufficient conditions for determining the asymptotical stability and instability of the corresponding singular closed―loop large―scale systems while the subsystems are regular， causal and R―controllable. At last， an example is given to show the application of main result.
Key words： Nonlinear discrete singular large―scale system； Control system； Asymptotical stability； Stabilization； State feedback
SUN Shuiling， CHEN Yuanyuan （2012）. Stabilization of a Kind of Nonlinear Discrete Singular Large―Scale Control Systems. Advances in Natural Science， 5（2）， ―0. Available from URL： http：///index.php/ans/article/view/j.ans.1715787020120502.1180
DOI： http：///10.3968/j.ans.1715787020120502.1180
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 is a n ― state vector， is a control input vector， is a n×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 subsystems：
（1）
where and are semi―state vector and vector function， respectively. is a control input vector； Aii，，， they are constant matrices；
denote：
rank （Ei） = ri ≤ ni ， E = Block ― diag （E1， E2， . . . ， Em）， rank （E） = r < n， B = Block ― diag （B1， B2， . . .， Bm）
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 × n constant matrices， B is a
n × m constant matrix， rank （E） = r < n， is a semi―state vector， is a control input vector.
Definition 1 （Yang & Zhang， 2004）： Discrete singular system （2） is said to be regular if
， for some .
Definition 2 （Yang & Zhang， 2004）： The zero solution of discrete singular system （2） is said to be stable if for every e > 0， there exists a δ > 0， such that ||x（k； k0x0）||
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 ≤ k ≤ N ）. Otherwise， it is said to be non―causal.
Now consider the isolated subsystems of systems：
（4）
Assume that all systems of systems （4） are R―controllable， we choose the linear control law
Ui（k） = ―Kixi（k）（i = 1， ...， m） （5）
Then singular closed―loop large―scale systems of systems （1） are given by
（6）
The corresponding closed―loop isolated subsystems are
（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 .
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 ， is a positive semi―definite matrix， then 2uTVv ≤ euTVu + e―1vTVv 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
ATVA ― ETVE = ― ETWE
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））TV（Ex（k））， where V is a positive semi―definite matrix， and rank（ETVE） = rankE =r；
（b）， 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 which satisfies that
（8）
（9）
then when
3Wi ― 2Vi ― 3[δλM（Vi） + （m―1）δλM + （m―1）2δλM]Ii > 0
（i = 1， 2， ...， m） （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 Wi is a positive definite and Vi is a positive semi―definite matrix from Lemma 4， and λM（Vi） denotes the maximum eigenvalue of matrix Vi， ， and Ii is a ni × ni identity matrix.
Proof： systems （7） are regular and causal， as they are asymptotically stable， then given positive definite matrix Wi， Lyapunov matrix equation
have positive semi―definite solution Vi.
Construct quadratic form
as the scalar Lyapunov function of systems （7）.
Let
as the Lyapunov function of systems （1）. We have
By using Lemma 3， choose e = 1， we have
Noticing that
Thus
Noticing that
by using Lemma 5， we know，， therefore .
To prove
By noticing that systems （7） are regular and causal， there exists reversible， matrices Pi， Qi （i = 1，...， m） which satisfy
Let
Where ，is and identity matrix， respectively. Pi， Qi， M，，，，are corresponding dimension constant matrices. Thus the singular closed―loop large―scale systems （6） are equivalent to
Noticing that
we have. holds from .
Noticing that
we have
Noticing that holds from （8）， that is ， so .
Hence， ， that is . 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 Wi，
there exists a positive semi―definite matrix Vi which satisfies
（11）
if there exists a real number which satisfies
（12）
when
（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
（14）
where，
We choose the control law Ui（k） = ―Kixi（k） （i = 1， 2），
and
then
It is easy to test that （8） and （9） are holded， then we know this system （14） is satbilizable 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.
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