SOLVING ALGEBRAIC RICCATI EQUATION FOR SINGULAR SYSTEM BASED ON MATRIX SIGN FUNCTION

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1 International Journal of Innovative Computing, Information and Control ICIC International c 2013 ISSN Volume 9, Number 7, July 2013 pp SOLVING ALGEBRAIC RICCATI EQUATION FOR SINGULAR SYSTEM BASED ON MATRIX SIGN FUNCTION Chih-Cheng Huang 1, Jason Sheng-Hong Tsai 1,, Shu-Mei Guo 2, Yeong-Jeu Sun 3 and Leang-San Shieh 4 1 Department of Electrical Engineering 2 Department of Computer Science and Information Engineering National Cheng Kung University No 1, University Road, Tainan City 701, Taiwan n @mailnckuedutw; Corresponding authors: { shtsai; guosm }@mailnckuedutw 3 Department of Electronic Engineering I-Shou University No1, Sec 1, Syuecheng Rd, Dashu District, Kaohsiung City 84001, Taiwan yjsun@isuedutw 4 Department of Electrical and Computer Engineering University of Houston N 308 Engineering Building 1 Houston, Texas , USA lshieh@uhedu Received April 2012; revised August 2012 Abstract The objective of this paper is to propose a constructive methodology for determining the appropriate weighting matrices {Q, R}, which guarantees the solvability of the generalized algebraic Riccati equation and for solving the generalized Riccati equation via the matrix sign function for the stabilizable singular system A decomposition technique is developed to decompose the singular system into a controllable reduced-order regular subsystem and a non-dynamic subsystem As a result, the well-developed analysis and synthesis methodologies developed for a regular system can be applied to the reducedorder regular subsystem Finally, we transform the results obtained for the reduced-order regular subsystem back to those for the original singular system Illustrative examples are presented to show the effectiveness and accuracy of the proposed methodology Keywords: Riccati equation, Singular system, Matrix sign function 1 Introduction Singular systems are often encountered in many fields of science and engineering systems, including circuits, economic systems, boundary control systems and chemical processes 1 Over the past decades, much effort has been invested in the analysis, synthesis and applications of singular systems due to the fact that singular systems appear more nature to represent the real systems than the regular systems (state-space systems) 1-5 The real singular systems usually consist of the non-dynamic subsystems and the dynamic subsystems, which are mathematically governed by the mixed representation of algebraic and differential equations The complex nature of the singular systems often encounters many difficulties in finding the analytical and numerical solutions to such systems, particularly when there is a need for their control Over the past decades, the theory and design of linear quadratic regulator (LQR) for optimal control of the regular systems have been well-developed and successfully applied to many practical design problems 6-10 Instead of tuning the controllers to satisfy the desirable classical control specifications for regular systems, the optimal controller can be easily designed by tuning the weighting matrices {Q, R} in the algebraic Riccati equation, 2771

2 2772 C-C HUANG, J S-H TSAI, S-M GUO, Y-J SUN AND L-S SHIEH for which many analytical and numerical solutions are available The methodologies to find specific weighting matrices {Q, R} for optimal control of regular systems have been well-developed in the literature but not for singular systems, which is an open problem to be solved The motivation of this paper is to propose a constructive methodology for determining the appropriate weighting matrices {Q, R}, which guarantees the solvability of the generalized algebraic Riccati equation and for solving the Riccati equation via the matrix sign function method for the singular systems A decomposition technique is developed to decompose the singular system into a reduced-order regular subsystem and a non-dynamic subsystem As a result, the well-known analysis and synthesis methodologies developed for a regular system can be applied to the reduced-order regular subsystem Finally, we transform the results obtained for the reduced-order regular subsystem back to those for the original singular system The computationally fast and numerically stable matrix sign function method is used to obtain the solution of the generalized algebraic Riccati equation for optimal control of the linear continuous-time singular system Consider the stabilizable 1 n-th order generalized linear, time-invariant system characterized by Eẋ(t) Ax(t) + Bu(t), (1) where x(t) R n is the states, u R m is the control, E R, A R and B R n m are real constant matrices, and E is possibly singular In recent studies, the algebraic Riccati equation (ARE) for the regular system has been generalized to the ARE 18,19 with the nonsingular matrix E in (1) The generalized Riccati equation 19 is given by A T P E + E T P A E T P BR 1 B T P E + Q O, (2) where Q R, R R m m and P R are real constant matrices It should remark that the generalized Riccati Equation (2) might have no solution, even if the selected Q and R are positive-definite matrices, and E is a singular matrix For instance, let Iκ O As O Bs E, A, B, O E f O I n κ B f n m Qs 0 Q, R m m > O, 0 Q f Ps 0 and P, where I κ denotes the κ κ identity matrix and E f is in the Jordan 0 P f canonical form From (2), we have A T s P s O Ps A s O + O P f E f O Ef T P P sb s R 1 Bs T P s P s B s R 1 Bf T P fe f f Ef T P fb f R 1 Bs T P s Ef T P fb f R 1 Bs T P f E f Qs O Ok O +, O Q f O O n k which implies A T s P s + P s A s + P s B s R 1 B T s P s + Q s O κ, (3) P s B s R 1 B T f P f E f O κ (n κ), (4) E T f P f B f R 1 B T s P s O (n κ) κ, (5)

3 SOLVING ALGEBRAIC RICCATI EQUATION FOR SINGULAR SYSTEM 2773 P f E f + E T f P f + E T f P f B f R 1 B T s P f E f + Q f O (n κ) (6) For P s > 0 and any non-null matrices B f and B s, (4) yields P f E f O (n κ), which induces, for example, O 3, (7) where denotes free variables Similarly, (5) gives Ef T P f O (n κ), which induces, for example, O 3 (8) As a result, the pairs of (7) and (8) indicate that P f is a null matrix, where the last-rightbottom element denotes as a free variable This also implies that P is not a positivedefinite matrix In general, the respective E f and P f can be given by and For example, let E fi E f block diagonal {E f1, E f2,, E fl } P f block diagonal {P f1, P f2,, P fl }, E f j where 1 i < j < k l, which gives P fi, P f j , and E fk, and P fk where denotes free variables The triple (4)-(6) also gives Q f O From the above illustrative examples, we can conclude that P and Q are not positive-definite matrices Therefore, even if the selected Q and R are positive-definite matrices, and E is a singular matrix, the generalized Riccati Equation (2) might have no solution By utilizing the neural network approaches but without explicitly providing a constructive way for determining the weighting matrices {Q, R}, various solution methods for the generalized Riccati equation in (2) can be found in This paper proposes a constructive method to determine the weighting matrices {Q, R} for the solution of the generalized Riccati equation in (2) for singular systems via the computationally fast and numerically stable matrix sign function method 2 Problem Formulation and Main Result Consider the controllable linear continuous-time singular system E r ẋ(t) A r x(t) + B r u(t), (10) where x(t) R n is the states, u(t) R m is the control, E r R is a singular matrix, and A r R and B r R n m are real constant matrices The singular system is assumed to,, (9a) (9b)

4 2774 C-C HUANG, J S-H TSAI, S-M GUO, Y-J SUN AND L-S SHIEH be controllable at finite and impulsive modes The singular system can be transformed into the slow and fast subsystem models 24, such as (Appendix A) where ˆx ˆxs ˆx f n 1, Ê Iκ O O Ê ˆx(t) ˆx(t) + ˆBu(t), (11) Êf,  Âs O O I n κ, ˆB ˆB s ˆB f the Os denote null matrices with appropriate sizes, Ê f is in the Jordan canonical form with d blocks of sizes u 1, u 2,, u d, and d i1 u i column (row) number of Êf Lemma 21 Given the linear controllable continuous-time singular system (10), the generalized algebraic Riccati equation for the steady-state linear quadratic regulator is n m A T r P r E r + E T r P r A r E T r P r B r R 1 r P r E r + Q r O n (12) Proof: For the finite-time linear quadratic regulator (LQR) problem, let the quadratic cost function for the singular system (10) be chosen as Tf min J c 1 x T (t)q r x(t) + u T (t)r r u(t) dt, (13) u(t) 2 0 where Q r O, R r > O, and T f is the final time Here, the Pontryagin s maximum principle 9 is used to solve this optimization problem Define a Hamiltonian as H(t) 1 ( x T (t)q r x(t) + u T (t)r r u(t) ) + λ T (A r x(t) + B r u(t)), 2 where λ(t) R n 1 is an un-determined multiplier function The state and costate equations are respectively given as H(t) λ(t) A rx(t) + B r u(t) E r ẋ(t), H(t) x(t) Q rx(t) + A T r λ(t) Er T λ(t), and the stationary condition is H(t) u(t) R ru(t) + Br T λ(t) O Solving the last equation yields the optimal control law in terms of the costate equation as u(t) Rr 1 Br T λ(t) (14) Substituting (14) into (10) yields E r ẋ A r x(t) B r Rr 1 Br T λ(t), which can be combined with the costate equation to give the homogeneous Hamiltonian system as Er ẋ(t) Ar B r Rr 1 Br T x(t) E r λ(t) Q r A T r λ(t) The coefficient matrix in (15) is called the Hamiltonian matrix Let which implies E T r λ(t) E T r P r (t)e r x(t) and λ(t) P r (t)e r x(t), (15) u(t) R 1 r B T r P r E r x(t), (16),

5 SOLVING ALGEBRAIC RICCATI EQUATION FOR SINGULAR SYSTEM 2775 with an unknown n n auxiliary matrix function P r (t) To find the auxiliary function P r (t), we differentiate the costate equation in (16) and use the state equation in (10) with the control law in (16) to get Er T λ(t) Er T P r (t)e r x(t) + Er T P r (t)e r ẋ(t) Er T P r (t)e r x(t) + Er T P r (t)a r x(t) B r R 1 Br T (17) P r (t)e r x(t) Now, from the costate equation, for all t, we have P r (t)e r x(t) Q r + A T r P r (t)e r + Er T P r (t)a r Er T P r (t)b r R 1 Br T P r (t)e r x(t), E T r Er T P r (t)e r Q r + A T r P r (t)e r + Er T P r (t)a r Er T P r (t)b r Rr 1 Br T P r (t)e r (18) The P r (t) in (18) is a null matrix in steady state Hence, we have A T r P r E r + E T r P r A r E T r P r B r R 1 r P r E r + Q r O n (19) This is the generalized algebraic Riccati equation used to determine the steady-state linear quadratic regulator for the linear continuous-time singular system (10) This completes the proof Lemma 22 Let ˆP f and Êf be two matrices, where Êf is a singular matrix of the single Jordan canonical form The following semi-positive definite matrix ˆP f O(n 1) (n 1) O (n 1) 1 O 1 (n 1) 1 1 satisfies the constraints ˆP f Êf O and ÊT f ˆP f O, where the denotes a free variable Proof: Let Ê f From the constraint ˆP f Êf O, we have ˆP f O n (n 1) n 1 Similarly, let Ê T f and by the constraint ÊT f ˆP f O, we have Hence, from above results we have ˆP f O(n 1) n 1 n ˆP f O(n 1) (n 1) O (n 1) 1 O 1 (n 1) 1 1 r r (20)

6 2776 C-C HUANG, J S-H TSAI, S-M GUO, Y-J SUN AND L-S SHIEH This completes the proof Remark 21 Let Êf be a null matrix The matrix ˆP f would satisfy the constraints ˆP f Êf O and ÊT f ˆP f O, where s denote free variables Theorem 21 Given the singular system in (10), which is assumed to be controllable at finite and impulsive modes and can be decomposed into a reduced-order regular subsystem and a non-dynamic subsystem by the approach shown in Appendix A Then, consider the generalized algebraic Riccati equation for the steady-state linear quadratic regulator, which is optimal in the sense of the quadratic cost function (13) for the controllable linear continuous-time singular system in (10), as The solution P r of (21) is given by A T r P r E r + E T r P r A r E T r P r B r R 1 r P r E r + Q r O (21) P r ( ((αe r + βa r )MW V ) 1)T ˆP ((αer + βa r )MW V ) 1, (22) where ˆPs O ˆP, (23) O O n κ ˆP s in (23) is a solution of the following Riccati equation:  s ˆPs + ˆP s  s ˆP s ˆBs ˆR 1 ˆPs + ˆQ s O κ, (24) ˆQ s and ˆR in (24) are both selected positive-definite matrices, and {M, W, V } in (22) are constant matrices and {α, β} in (22) are real constants (see Appendix A) The resulting weighting matrices in the original cost function in (13) become where and Q r ( (MV ) 1)T ˆQ (MV ) 1, (25) ˆQs O ˆQ, (26) O O n k R r ˆR (27) The solution of the Riccati equation ˆP s in (24) guarantees the stability of the reduced-order regular subsystem in (A17) as well as the stability of the singular system without having the impulsive mode in (10) Proof: Let ˆQ ˆQ s O O ˆQ, where ˆQ s R κ κ and ˆR R m m are positive-definite f matrices From (12), we have ÂT ˆP s s O O ˆP + ˆP ˆR 1 s  s O ˆP s ˆBs ˆBT ˆP ˆR 1 s s ˆPs ˆBs ˆBT ˆP f f Ê f f Ê f O ˆP + ÊT f f Ê T ˆP ˆR 1 f f ˆBf ˆBT ˆP s s Ê T ˆP ˆR 1 f f ˆBf ˆBT ˆP s f Ê f + ˆQ s O Ok O O ˆQ, f O O n k

7 which gives SOLVING ALGEBRAIC RICCATI EQUATION FOR SINGULAR SYSTEM 2777  T ˆP s s + ˆP s  s + ˆP ˆR 1 s ˆBs ˆBT s s + ˆQ s O κ, (28) ˆR 1 ˆP s ˆBs ˆBT f f Ê f O κ (n κ), (29) Ê T ˆP ˆR 1 f f ˆBf ˆBT s s O (n κ) κ, (30) ˆP f Ê f + ˆP ÊT f f + ˆP ˆR 1 ÊT f f ˆBf ˆBT s f Ê f + ˆQ f O (n κ) (31) From (29), (30), and Lemma 22, we have the Êf and ˆP f shown in (9a) and (9b) For simplicity in analysis, we let the free variable be zero, which yields ˆPs O ˆP (32) O O n κ By (29), (30), and (21), we have ˆQ f O n κ, (33) which gives ˆQs O ˆQ O O n κ In addition, from (16) and Appendices A and B, we have the linear quadratic regulator u(t) ˆR 1 ˆBT ˆP ʈx(t) ˆR 1 (V 1 T B) ˆP (V 1 ĒV )V 1 x(t) ˆR 1 ((W V ) 1 T B) ˆP ((W V ) 1 ẼV )V 1 M 1 x(t) ˆR 1 ((MW V ) 1 B n ) T ˆP ((MW V ) 1 E n MV )V 1 M 1 x(t) ˆR 1 (((αe r + βa r )MW V ) 1 B r ) T ˆP (((αer + βa r )MW V ) 1 E r MV ) V 1 M 1 x(t) ˆR 1 Br T (((αe r + βa r )MW V ) 1 ) T ˆP ((αer + βa r )MW V ) 1 E r x(t) ˆR 1 Br T P r E r x(t), which yields P r ( ((αe r + βa r )MW V ) 1)T ˆP ((αer + βa r )MW V ) 1, where {M, W, V } are constant matrices and {α, β} are real constants (see Appendix A) Furthermore, from (13), we have where min J c 1 u(t) 2 Tf 0 x T (t)q r x(t) + u T (t)r r u(t) dt, ˆx T (t) ˆQˆx(t) x T (t)(v 1 ) T ˆQV 1 x(t) x T (t)(m 1 ) T (V 1 ) T ˆQV 1 M 1 x(t) x T (t)(m 1 ) T (V 1 ) T ˆQV 1 M 1 x(t) x T (t) ( (MV ) 1)T ˆQ (MV ) 1 x(t) x T (t)q r x(t), which gives Q r ( (MV ) 1)T ˆQ(MV ) 1,

8 2778 C-C HUANG, J S-H TSAI, S-M GUO, Y-J SUN AND L-S SHIEH where V and M are matrices (see Appendix A) Notice that rank(q r ) rank( ˆQ) rank( ˆQ s ) and R r ˆR The proof of the claim that the solution of the generalized Riccati equation guarantees the stability of the reduced-order regular subsystem can be found in literature 9,10 Since the singular system can be transformed into a reduced-order regular subsystem and a non-dynamic subsystem, the stability of the reduced-order regular subsystem assures the stability of the singular system without having impulsive mode Besides, the Q r developed in (25) is not an arbitrary matrix This completes the proof 3 Illustrative Examples To show the effectiveness and accuracy of the proposed methodology, a pure mathematical model is utilized in Example 31 and a practical system is adapted in Example 32, where the sub-matrix Ē2 in (A11) in Example 31 has the Jordan-type eigenstructure and the sub-matrix Ē2 in Example 32 is a null matrix Example 31 Consider the controllable linear continuous-time singular system 24 described in (10) with E r , A r I 6, Br T Taking α 0 and β 1, then E n E r, A n A r and B n B r, which induces 0E n +A n I 6 By definition of the standard form, {E r, A r } is in the standard form Because E n is singular, ie, E n includes some zero eigenvalues, we can utilize the bilinear transform to find the similarity transformation matrix M of E n is necessary Taking ρ 05 and using the algorithm described in Appendix A, we have Ẽ (E n ρi 6 )(E n + ρi 6 ) 1 ) sign (Ẽ + ) sign (Ẽ ), sign (Ẽ,, ,

9 M SOLVING ALGEBRAIC RICCATI EQUATION FOR SINGULAR SYSTEM 2779 ( )) ( )) ind sign (Ẽ + ind sign (Ẽ From (A10), we obtain M 1 E n M Ē1 O O Ē2 M 1 I3 O A n M, M 1 B n O I , BT 1 B T 2 T (34) From (A12), we have 1 0 W E O O 1 (I β n κ αe 2 ) , (35) which gives W 1 E 1 1 O O β(i n κ αe 2 ) , (36) W 1 Ē n , W 1 Ā n , T W 1 Bn Based on (A14) and the fact that Ēf is in the Jordan form, we have Ê f V I 6, (37), Â s ,

10 2780 C-C HUANG, J S-H TSAI, S-M GUO, Y-J SUN AND L-S SHIEH ˆB s , ˆBf Solving the algebraic Riccati equation Âs ˆP s + ˆP s  s ˆP ˆR 1 s ˆBs ˆPs + ˆQ s O 3, where ˆQ s 10 5 I 3 and ˆR I 2, yields ˆP s , where some fractional bits are truncated at here By (22)-(27) and (36)-(37), we have ˆP ˆPS O 3 O 3 O 3 P r ( ((αe r + βa r )MW V ) 1)T ˆP ((αer + βa r )MW V ) , ˆQs O ˆQ 3 I3 O , O 3 O 3 O 3 O Q r ((MV ) 1 ) T ˆQ(MV ) , R r ˆR I 2 Substituting the computed P r and Q r into (21) yields A T r P r E r + Er T P r A r Er T P r B r Rr 1 P r E r + Q r O6, which shows that the solution is quite satisfactory Example 32 Consider the controllable linear continuous-time singular circuit system (Figure 1) 1, where R 1, 000Ω, inductance L 1H, capacitances C F, C 2 02F, and voltage u(t) is the control input The state vector is x(t) v c1 (t) v c2 (t) i 2 (t) i 1 (t) T, where the vc1 (t), v c2 (t), i 2 (t), and i 1 (t) are voltages of capacitors and amperages of the currents flowing over them,

11 SOLVING ALGEBRAIC RICCATI EQUATION FOR SINGULAR SYSTEM 2781 respectively equation where E r Figure 1 The singular circuit system According to Kirchoff s second law, we may establish the following state Taking α 0 and β 1 to have E n E r ẋ(t) A r x(t) + B r u(t), , A r , B r , A n I 4, B T n , which induces 0E n + A n I 4 By definition of the standard form, {E r, A r } is in the standard form Because E n is singular, ie, E n includes some zero eigenvalues, we can utilize the bilinear transform to find the similarity transformation matrix M of E n Taking ρ 005 and using the algorithm described in Appendix A, we have Ẽ (E n ρi 6 )(E n + ρi 6 ) 1 ) sign (Ẽ ) sign (Ẽ + ) sign (Ẽ ,,,,

12 2782 C-C HUANG, J S-H TSAI, S-M GUO, Y-J SUN AND L-S SHIEH M ( )) ( )) ind sign (Ẽ + ind sign (Ẽ (38) From (A10) and (A12), we have M 1 Ē1 O E n M O Ē , M 1 I3 O T A n M, M 1 B n BT 1 B 2 T , O I 1 W E O O 1 (I β n κ αe 2 ) , (39) which implies W 1 E 1 1 O O β(i n κ αe 2 ) , (40) W 1 Ē n , W 1 Ā n , W 1 Bn T Based on (A14) and the fact that Ēf is null, we have Ê f 0,  s V I 4, (41), ˆBs 1 0 0, ˆBf 1 Solving the algebraic Riccati equation Âs ˆP s + ˆP s  s ˆP ˆR 1 s ˆBs ˆPs + ˆQ s O 3, where ˆQ s 10 5 I 3 and ˆR I 1, yields ˆP s where some fractional bits are truncated at here By (22)-(27) and (40)-(41), we have ˆP ˆPs O 3 1 O 1 3 O 1 P r ( ((αe r + βa r )MW V ) 1)T ˆP ((αer + βa r )MW V ) 1

13 SOLVING ALGEBRAIC RICCATI EQUATION FOR SINGULAR SYSTEM ˆQ ˆQs O 3 1 O 1 3 O 1 Q r ( (MV ) 1)T ˆQ (MV ) 1 I3 O 3 1 O 1 3 O 1 R r ˆR I 1 Substituting the computed P r and Q r into (21) yields , 10 5, 105, A T r P r E r + Er T P r A r Er T P r B r Rr 1 P r E r + Q r O4, which shows that the solution is very satisfactory Here, we would like to point out that no toolbox in MATLAB can be used to solve this problem 4 Conclusion This paper shows that even if the selected Q and R are positive-definite matrices, and E is a singular matrix, the generalized Riccati Equation (2) might have no solution Therefore, this paper aims to propose a constructive methodology for determining the appropriate weighting matrices {Q, R}, which guarantees the solvability of the generalized algebraic Riccati equation for the controllable linear continuous-time singular system based on the matrix sign function method A decomposition technique is developed to decompose the singular system into a reduced-order regular subsystem and a non-dynamic subsystem, so that the singular problem can be converted into a standard regular problem As a result, the computationally fast and numerically stable matrix sign function method 25 can be utilized to solve the generalized algebraic Riccati equation for the singular system Finally, we transform the obtained results obtained for the regular system back to those for the original singular system The developed design methodology for finding the LQR can be extended to find the optimal tracker for singular systems Illustrative examples are presented to show the effectiveness and accuracy of the proposed methodology Acknowledgement This work was supported by the National Science Council of Taiwan under contracts NSC E MY3, NSC E MY3, and NS C E REFERENCES 1 L Dai, Singular Control Systems, Springer-Verlag, Berlin, Germany, S Xu and C Yang, An algebraic approach to the robust stability analysis and robust stabilization of uncertain singular systems, International Journal of Systems Science, vol31, pp55-61, 2000

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15 SOLVING ALGEBRAIC RICCATI EQUATION FOR SINGULAR SYSTEM L S Shieh, Y T Tsay and R E Yates, Some properties of matrix-sign functions derived from continued fractions, IEE Proc of Control Theory Applications, vol130, no3, pp , F R Ganmacher, The Theory of Matrices II, Chelsea, New York, R Nikoukhah, A S Willsky and B C Levy, Boundary-value descriptor systems: Well-posedness, reachability and observability, International Journal Control, vol46, no5, pp , S L Campbell, Singular Systems of Differential Equations I, Pitman, New York, J S H Tsai, L S Shieh, J L Zhang and N P Coleman, Digital redesign of pseudo-continuous-time suboptimal regulators for large-scale discrete systems, Control-Theory and Advanced Technology, vol5, no1, pp37-65, L S Shieh, Y T Tsay, S W Lin and N P Coleman, Block-diagonalization and blocktriangularization of a matrix via the matrix sign function, International Journal of Systems Science, vol15, no11, pp , T Kailath, Linear Systems, Prentice-Hall, Englewood Chiffs, New Jersey, 1983 Appendix A: Singular System Descriptions 24 A1 Preliminaries for decomposition of singular systems Consider the linear continuous-time singular system as follows E r ẋ(t) A r x(t) + B r u(t), (A1) where x(t) R n is the state vector and u(t) R m is the input These constant matrices E r, A r and B r all have appropriate dimensions, and E r is a singular matrix The matrix sign function of a square matrix A C with Re(σ(A)) 0 is defined 26 as follows sign(a) 2sign + (A) I n, (A2) where I n is an n n identity matrix and sign + (A) 1 2πi (λi n A) 1 dλ, (A3) c + where c + is a simple closed contour in right-half plane of λ and encloses all right-half plane eigenvalues of A For another thing, the matrix sign function 27,28 is also defined as sign(a) A( A 2 ) 1 A 1 ( A 2 ), (A4) where the matrix A 2 denotes the principal square root of A 2 Preserving the eigenvectors of the original matrix is a main feature of the matrix sign function This property is useful for engineering problem to study the eigenstructures of matrices and develop applications The singular matrix E r can be modified by using bilinear transformation Ẽ r (E r ρi n )(E r + ρi n ) 1, (A5) where ρ is the radius of a circle with center at the origin so that the circle only contains zero eigenvalues and no eigenvalues of E r located on the circle Therefore, the eigenvalues within the circle are mapped into the left-half plane of the complex s-plane, and the others are mapped into the right-half plane of the complex s-plane A2 The regular pencil and the standard pencil Definition A21 29 Let E r and A r be two square constant matrices if det(se r A r ) 0, for all s, then (se r A r ) is called a regular pencil Definition A22 30 Let (se n A n ) be a regular pencil If there exists scalars α and β such that αe n + βa n I n, then (se n A n ) is called a standard pencil Note that for any regular pencil, (se r A r ) can be easily transformed into a standard one by multiplying (αe r + βa r ) 1 to E r and A r respectively, where α and β are scalars such

16 2786 C-C HUANG, J S-H TSAI, S-M GUO, Y-J SUN AND L-S SHIEH that det(αe r + βa r ) 0 Hence, the matrix coefficients of a standard pencil (se n A n ) becomes E n (αe r + βa r ) 1 E r, (A6) A n (αe r + βa r ) 1 A r (A7) The modified system retains its state vector x(t) and the matrices (E n, A n ) have the following properties Lemma A ) E n A n A n E n, which means that E n and A n commute each other 2) E n and A n have the same eigenspaces The above properties enable us to decompose a singular system into a reduced-order regular subsystem (slow subsystem) and a nondynamic subsystem (fast subsystem) A3 Decomposition of singular systems Consider the continuous-time singular system (A1) It is well known that the singular system can be decomposed into slow and fast subsystem From (A6) and (A7), the regular pencil (se r A r ) can be transformed into a standard one (se n A n ) Note that since E r is a singular matrix, which has at least one zero eigenvalue, β cannot be equal zero Hence, multiply (A1) by (αe r + βa r ) 1 can get the following equation E n ẋ(t) A n x(t) + B n u(t), (A8) where E n (αe r + βa r ) 1 E r, A n (αe r + βa r ) 1 A r and B n (αe r + βa r ) 1 B r Because αe n +βa n I n, the pencil (se n A n ) is a standard one which has the properties mentioned in Lemma A21 In order to decompose system (A8), we convert state space x(t) into x(t) by x(t) M x(t) (A9) where the constant matrix M is a block modal matrix of E n and determined by means of the extended matrix sign function The M matrix of state space transformation is given as follows: Step 1: Find sign(ẽn) using the extended matrix sign function with an adequate ρ, where Ẽ n (E n ρi n )(E n + ρi n ) 1 Step 2: Find sign + (Ẽn) 1 2 I n + sign(ẽn) and sign (Ẽn) 1 2 I n sign(ẽn) Step 3: Construct the matrix M ind(sign + (Ẽn))ind(sign (Ẽn)), (A10) where ind( ) represents the collection of the linearly independent column vectors of ( ) Substituting (A9) into (A10) and multiplying M 1 on the left, the equation can be rewritten as M 1 E n M x(t) M 1 A n M x(t) + M 1 B n u(t) 1 β (I n αe n ) x(t) + M 1 B n u(t) ie, Ē1 O O Ē2 x(t) ( 1 Ik αē1) β O 1 β O ( In κ αē2) x(t) + B1 B 2 u(t), (A11) where x(t) x T s (t), x T f (t)t, and M 1 E n M block diagonal { Ē 1, Ē2} Ē1 is invertible with rank(ē1) deg{det(se r A r )}, BT 1, B T T 2 M 1 B n and Ē2 is a nilpotent matrix

17 SOLVING ALGEBRAIC RICCATI EQUATION FOR SINGULAR SYSTEM 2787 with dimension (n κ) (n κ) Notice that since det(i n κ αe 2 ) 1, it is invertible Let W E 1 O O 1 (I, (A12) β n κ αe 2 ) which implies W 1 E 1 1 O O β ( ) 1 I n κ αe 2 (A13) Simplifying (A11) by multiplying W 1 on both sides, one has Iκ O O β ( ) 1 x(t) 1 β (Ē 1 1 αi κ ) O x(t) I n κ αē2 Ē 2 O I n κ Ē B 1 β ( 1 u(t), I n κ αē2) B2 Iκ O A x(t) s O Bs x(t) + u(t), (A14) O Ēf O I n κ B f where Ēf β ( ) 1 (Ē 1 I n κ αē2) Ē 2, A s 1 β 1 αi κ, Bs Ē 1 1 B 1, Bf β(i n κ αē2) 1 B2 Let x(t) V ˆx(t), (A15) where ˆx(t) ˆx Ts (t), ˆx Tf T (t) x T s (t), (U 1 x T T f (t)) and Iκ O V (A16) O U U is a modal matrix of Ē f with dimension (n κ) (n κ) such that U 1 Ē f U is in the Jordan canonical form The function JORDAN in MATLAB can be utilized to compute the generalized eigenvectors and the Jordan canonical form of a Jordan matrix Substituting (A15) into (A14) and multiplying it by V 1, we obtain Ik O Âs O ˆx(t) ˆx(t) + ˆB s u(t), (A17) O Êf O I (n κ) ˆB f where Êf U 1 Ē f U,  s Ās, ˆBs B s and ˆB f U 1 Bf Notice that Êf is in the Jordan block form with d blocks of sizes u 1, u 2,, u d, where d i1 u i column (row) number of Êf Appendix B: Solving Riccati Equation via Matrix Sign Function 32 The Riccati equation for the controllable continuous-time system (Âs, ˆB s ) with weighting matrices ˆQ s (> O) and ˆR(> O) is given by  s ˆPs + ˆP s  s ˆP s ˆBs ˆR 1 ˆPs + ˆQ s O (B1a) The steady state solution of this Riccati equation, ˆPs (> O) with ( ˆQ s, Âs) detectable, can be easily computed using the properties of the matrix sign function 25,33 and the

18 2788 C-C HUANG, J S-H TSAI, S-M GUO, Y-J SUN AND L-S SHIEH eigenvalue-eigenvector approach 34 Consider the Hamiltonian associated with the given system  H s ˆB T s ˆRB s ˆQ (B1b) s ÂT s The following algorithm can be utilized to obtain the solution ˆP s, H k+1 1 Hk + H 1 2 k, H0 H and lim H k sign(h) k Let sign + (H) 1 2 I 2n + sign(h) Construct a block modal matrix X as X ind ( sign + (H) ), ind ( I 2n sign + (H) ) (B2a) (B2b) X11 X 12, (B3a) X 21 X 22 where ind( ) represents the collection of the linearly independent column vectors of ( ) Then, we have ˆP s X 22 (X 12 ) 1 (B3b) To alleviate the problems of computing H 1 k, the Hamiltonnain can transformed into a symmetric form as follows 25 Ĥ ĴH On I n ˆQs  T s H I n O n  s ˆB (B4a) T s ˆRB s and Then, the algorithm in (B2) becomes Ĥ k+1 Ĥk ( ) lim ĴĤk sign(h) k 1 ĴĤk, Ĥ0 ĴH (B4b) The computation of the inverse of the symmetric matrix Ĥk is much simpler than computing the inverse of H k The Riccati solution ˆP s is again given by (B3)