以線性矩陣不等式為基礎之多角形描述系統強韌性分析與
設計
計畫類別： 個別型計畫 計畫編號： NSC91-2213-E-151-002- 執行期間： 91 年 08 月 01 日至 92 年 07 月 31 日 執行單位： 國立高雄應用科技大學電機工程系 計畫主持人： 方俊雄 計畫參與人員： 郭景湖 劉永勝 賴政德 楊景貿 報告類型： 精簡報告 處理方式： 本計畫可公開查詢中 華 民 國 92 年 10 月 14 日
行政院國家科學委員會專題研究計畫成果報告
以線性矩陣不等式為基礎之多角形描述系統強韌性分析與設計
Robustness Analysis and Design for Polytopic Descriptor Systems
--An LMI Approach
計畫編號：NSC 91-2213-E-151-002
執行期限：91 年 8 月 1 日至 92 年 7 月 31 日主持人：方俊雄教授 國立高雄應用科技大學 電機系
參與計畫人員：郭景湖 劉永勝 賴政德 楊景貿
一、中文摘要 本計畫針對多角形描述系統，探討其強韌可容許性分析與設計問題。使用線 性矩陣不等式的技巧，得到多角形描述系統強韌可容許性的充分條件，並應用此 條件來設計強韌靜態輸出回授控制器。 關鍵詞：多角形描述系統、強韌穩定性、強韌控制器、線性矩陣不等式。 AbstractThe robust admissibility analysis and design problems of polytopic descriptor systems are investigated in this project. The problems are solved by using linear matrix inequality (LMI). Sufficient LMI conditions for robust admissibility of polytopic descriptor systems are proposed. The conditions are then applied to synthesize a robust static output feedback controller.
Keyword: polytopic descriptor systems, robust stability, LMI.
二、緣由與目的 近年來在控制系統的分析與合成中，線性矩陣不等式(LMI)的技術，變成一 種非常重要的工具，這是因為 LMI 具有良好的運算特性，一旦問題轉化成 LMI 型式，數值的計算有很好的收斂性 [1，2]。另一個重要原因是 LMI 在表達理論 的結果，具有較簡潔的型式，非常方便理論的推衍。描述系統的分析，使用 LMI 技術，已有相當多的研究成果[3，4，5，6]。 最近 LMI 的技術發展有了新的趨勢，不等式條件中引入更多或更複雜型式 的調節變數(slack variables) [7， 8]，藉以放寬條件對變數的限制，或者使用參
14]。描述系統的分析，截至目前為止，未見任何文獻引用此新技術。本計畫的 目的是要將前述 LMI 最新技術的發展引進描述系統，解決多角形描述系統強韌 穩定度分析及相關問題。
三、結果與討論
Consider the continuous-time and the discrete-time descriptor systems described by
( )
( )
( )
,( )
Ex t& =Ax t +Bu t y=Cx t (1)Ex k
(
+ =1)
Ax k( )
+Bu k( )
, y=Cx k( )
(2)whereE A, ∈Rn n× ,B∈Rn m× , and C∈Rp n× . The matrix E can be singular. Assume
( )
rank E ≤n. In the literature, the system (1)/(2) is said to be admissible if it is regular, impulse-free/cause, and Hurwitz /Schur stable. Assume the matrix A is not precisely known, but belongs to a convex bounded set Ω
1 1 : , 0, 1 N N i i i i i i A A δ A δ δ = = Ω ≡ = ≥ =
∑
∑
. ( 3 )In (3), Ai, i=1, 2,...,N represent the vertices of Ω and are known in advance. The problem considered in this project is to determine if the system (1)/(2)is admissible for all A∈Ω. If not, we will find a static output feedback control law u= −Fy such that (1)/(2) is admissible for all A∈Ω . Both continuous-time and discrete-time cases are studied in this project.
The notation used is fairly standard. M ≥0 means that matrix M is positive semidefinite and M <0negative definite. In is the identify matrix of dimension n. With the unit simplex set represented by
( 1 2 ) 1 : , ,..., , 0, 1 N N i i i δ δ δ δ δ δ δ = ∆ ≡ = ≥ =
∑
. ( 4 )the set Ω in (3) can be simply rewritten as
{
A( )
δ δ ∈∆:}
. 1. Continuous-time casesIn the subsection, sufficient conditions for robust admissibility and static output feedback stabilization of the system (1) with polytopic type uncertainties (3) will be derived in terms of LMIs characterizations. For the development of main results, some preliminary lemmas are introduced and derived.
Lemma 1: For a fixed descriptor system Ex t&
( )
=Ax t( )
, it is admissible if and only if there exists a matrix Y such that0 T T E Y =Y E≥ ( 5 ) 0 T T A Y+Y A< . ( 6 )
For some special cases, Lemma 1 can be applied to deal with robust admissibility of system (1), the following lemma shows the special cases.
Lemma 2: If there exists a matrix P such that all vertex systems satisfying
0 T T E P=P E≥ ( 7 ) 0, 1,.., T T i i A P+P A < i= N ( 8 ) then the system (1) is admissible for all A∈Ω.
The results of Lemma 2 may be too conservative since the same P is required for all vertex systems. It is natural to ask if there exists a parameter- dependent matrix
( )
1 0, 1,.., N i i i P δ δ P i N = =∑
> = ( 9 ) such that( )
( )
( ) ( ) ( ) ( )
0 0 T T T T E P P E A P P A δ δ δ δ δ δ = ≥ + < ∀ ∈ ∆δ . ( 1 0 )The following theorem provides a possible way to construct such matrix.
Theorem 1: If there existPi, Xii, i=1,..,N,Xik, i=1,..,N−1, k = +i 1,..,N such that
0, 1,.., T T i i E P =P E≥ i= N ( 1 1 ) , 1,.., T T i i i i ii A P+P A <X i= N ( 1 2 ) T T T T T i k k i i k k i ik ik A P +A P+P A +P A ≤X +X 1,.., 1, 1,.., i= N− k= +i N ( 1 3 ) 11 12 1 12 22 2 1 2 0 n T n T T n n nn X X X X X X X X X ≤ L L M M O M L ( 1 4 )
then the system (1) is robustly admissible
In the next, we will find a static output feedback control law u= −Fy such that (1) is admissible for all A∈Ω. This is called the problem of static output feedback stabilization.
Lemma 3: The problem of static output feedback stabilization of system (1) is solvable if there exists a parameter- dependent matrix P
( )
δ and F such that( ) ( )
T 0 T E P δ =P δ E≥ ( 1 5 )( )
(
)
T( ) ( ) ( )
T(
)
A δ −BFC P δ +P δ A δ −BFC 0 < . ( 1 6 )Theorem 2: If there exist Pi, N, Xii, i=1,..,N, Xik, i=1,..,N−1k= +i 1,..,N and a nonsingular matrix M such that
0, 1,.., T T i i E P =P E≥ i= N ( 1 7 ) T T T T T i i i i ii A P+P A −C N B −BNC<X 1,.., i= N ( 1 8 ) 2 2 , T T T T T T T i k k i i k k i T ik ik A P A P P A P A C N B BNC X X + + + − − ≤ + 1,.., 1, 1,.., i= N− k= +i N ( 1 9 ) 11 12 1 12 22 2 1 2 0 n T n T T n n nn X X X X X X X X X ≤ L L M M O M L ( 2 0 ) , 1,.., T i P B=BM i= N ( 2 1 )
then the static output feedback gain 1
F=M N− is the solution our to problem.
Remark 1: Without loss of generality, one can assume M is nonsingular. If M is singular, there exists a sufficiently small ε >0such that
p
Mε =M+εI
is nonsingular. Corresponding to the Mε, a new matrix Piε defined by
, 1,..,
i i
Pε = +P εI i= N still satisfies (17)-(21).
Remark 2: For the case of state feedback (i.e.C=In), then the inequalities in Theorem
2 can be simplified as 0, 1,.., T T i i E P =P E≥ i= N ( 2 2 ) T T T T i i i i ii A P+P A −N B −BN<X 1,.., i= N ( 2 3 ) 2 2 , T T T T T T i k k i i k k i T ik ik A P A P P A P A N B BN X X + + + − − ≤ + 1,.., 1, 1,.., i= N− k= +i N ( 2 4 ) 11 12 1 12 22 2 1 2 0 n T n T T n n nn X X X X X X X X X ≤ L L M M O M L ( 2 5 ) , 1,.., T i P B=BM i= N . ( 2 6 ) 2. Discrete-time cases
In this subsection, we present results for discrete-time cases. Before presenting the main result, an admissibility condition for a fixed discrete-time descriptor system is introduced.
Lemma 4: For a fixed discrete-time system Ex k
(
+ =1)
Ax k( )
, it is admissible if and only if there exist a matrix TY =Y and two matrices S, G such that
0 T EYE ≥ ( 2 7 ) 0. T T T T T T T AS S A EYE AG S G A S Y G G + − − < − − − ( 2 8 )
Next lemma is derived for robust admissibility of system (2), based on Lemma 4. Lemma 5: If there exist a matrix P=PT and two matrices S and G such that all
vertex systems satisfying
0 T EPE ≥ ( 2 9 ) 0 T T T T i i i T T T i A S S A EPE A G S G A S P G G + − − < − − − ( 3 0 )
then the system (2) is admissible for all A∈Ω.
The results of Lemma 5 may be too conservative since the same solutions P, G, S are required for all vertex systems. It is natural to ask if there exist the parameter-dependent matricesP
( )
δ , G( )
δ and S( )
δ satisfying( )
T 0 EP δ E ≥ ( 3 1 )( ) ( ) ( ) ( )
( )
( ) ( )
( )
T T T T T A S S A EP E G A S δ δ δ δ δ δ δ δ + − − ( ) ( ) ( )
( ) ( ) ( )
0 T T A G S P G G δ δ δ δ δ δ − < − − ( 3 2 )for all δ ∈∆. The following theorem provides a possible way to construct such matrix.
Theorem 3: If there exist Si, Gi, Xii, T i i P=P , i=1,..,N and Xik, i=1,..,N−1, 1,.., k= +i N such that 0, 1,.., T i EP E ≥ i= N ( 3 3 ) T T T T i i i i i i i i ii T T T i i i i i i A S S A EP E A G S X G A S P G G + − − < − − − 1,.., i= N ( 3 4 ) T T T T T T i k k i i k k i i k T T T T i k k i i k A S A S S A S A EP E EP E G A G A S S + + + − − + − − T T T i k k i i k ik ik T T i k i k i k A G A G S S X X P P G G G G + − − ≤ + + − − − − 1,.., 1, 1,.., i= N− k= +i N ( 3 5 )
1 2 T T n n nn X X X M M O M L
then system (2) is robustly admissible.
In the next, we will find a static output feedback control law u= −Fy such that (2) is admissible for all A∈Ω.
Lemma 6: The problem of static output feedback stabilization of system (2) is solvable if there exist the parameter- dependent matricesP
( )
δ , G( )
δ , S( )
δ , and F such that( )
T 0 EP δ E ≥ ( 3 7 ) ( )(
)
( ) ( )(
( ))
( ) ( )(
( ))
( ) T T T T T A BFC S S A BFC EP E G A BFC S δ δ δ δ δ δ δ δ − + − − − − ( )
(
)
( ) ( )
( )
( )
( )
0 T T A BFC G S P G G δ δ δ δ δ δ − − < − − . ( 3 8 )Based on Lemma 6, next theorem gives an LMI condition to solve the problem.
Theorem 4: If there exist T i i
P =P , N , Si, Gi, Xii, i=1,..,N, Xik, i=1,..,N ,
1,..,
k= +i N and a nonsingular matrix M such that
0, 1,.., T i EP E ≥ i= N ( 3 9 )
(
)
(
)
T T T T i i i i i T T T i i i A S S A BNC BNC EP E G A BNC S + − − − − − 1 , 1,.., T i i i ii T i i A G BNC S X i N P G G − − < = − − ( 4 0 )(
)
(
)
2 2 2 T T T T i k k i i k k i T T T i k T T T T T i k k i i k A S A S S A S A BNC BNC EP E EP E G A G A BNC S S + + + − − − − + − − − 2 T T T i k k i i k ik ik T T i k i k i k A G A G BNC S S X X P P G G G G + − − − ≤ + + − − − − 1,.., 1, 1,.., i= N− k= +i N ( 4 1 )11 12 1 12 22 2 1 2 0 n T n T T n n nn X X X X X X X X X ≤ L L M M O M L ( 4 2 ) , 1,.., i CS =MC i= N ( 4 3 ) , 1,.., i CG =MC i= N ( 4 4 )
then the static output feedback gain 1
F=NM− is the solution to our problem.
Remark 3: Without loss of generality, one can assume M is nonsingular. If M is singular, there exists a sufficiently small ε >0such that
p
Mε =M+εI
is nonsingular. Corresponding to the Mε, two new matrices Siε andGiε, defined by
, 1,.., i i Sε = +S εI i= N , 1,.., i i Gε =G +εI i= N still satisfy (39)-(44).
Remark 4: For the case of state feedback (i.e. C=In),Gi, Si, and M become the same variables. Let Gi=Si =M =H, then the inequalities in Theorem 4 can be simplified as 0, 1,.., T i EP E ≥ i= N
( )
( )
T T T T i i i T T T i A H H A BN BN EP E H A BN H + − − − − − ( 4 5 ) , 1,.., T i ii T i A H BN H X i N P H H − − < = − − ( 4 6 ) 11 12 1 12 22 2 1 2 0. n T n T T n n nn X X X X X X X X X ≤ L L M M O M L ( 4 7 ) 四、計劃成果自評 本計劃已依原來之計畫書之進度完成 100%，目前相關成果已整理兩篇論文 出版在下列研討會論文集：2002 Chinese Automatic Control Conference：Stabilization of Uncertain Linear
Systems via Static Output Feedback Control.
2003 American Control Conference：LMI Conditions for Static Output Feedback
Control of Descriptor Systems.
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