Robust Input-Output Energy Decoupling for Uncertain Singular Systems

International Journal of Automation and Computing 1 (25) 37-42 Robust Input-Output Energy Decoupling for Uncertain Singular Systems Xin-Zhuang Dong, Qing-Ling Zhang Institute of System Science, Northeastern University, Shenyang, Liaoning 114, PRC Abstract: his paper addresses the robust input-output energy decoupling problem for uncertain singular systems in which all parameter matrices except E exist as time-varying uncertainties. By means of linear matrix inequalities (LMIs), sufficient conditions are derived for the existence of linear state feedback and input transformation control laws, such that the resulting closed-loop uncertain singular system is generalized quadratically stable and the energy of every input controls mainly the energy of a corresponding output, and influences the energy of other outputs as weakly as possible. Keywords: Uncertain singular systems, generalized quadratical stability, input-output energy decoupling, linear matrix inequality (LMI). 1 Introduction During recent years, singular systems have attracted the interest of a number of researchers due to the fact that they appear more naturally in the study of naturally occurring systems than regular systems (state-space systems), e.g. in economic, circuit, and boundary control systems, and chemical processes 1 3]. Many contributions have been made to the study of these systems, and a great number of results on the theory of regular systems have been generalized to the area of singular systems. However, many of the results obtained have been concerned with singular systems in which there is no uncertainty. Recently, some work dealing with the problem of robust stability analysis, and robust control in uncertain singular systems has appeared in the literature 4]. he input-output decoupling problem is a major control problem studied by several researchers for more than two decades. It was first introduced by Morgan 5] as a question of great importance, since it aims to reduce a multi-input, multi-output system to a set of single-input, single-output systems, strongly facilitating in this way a control strategy. he necessary and sufficient conditions for the method of solution were established by 6]. hen a variety of papers appeared on the subject as: group or block decoupling 7] ; decoupling with simultaneous disturbance rejection 8] ; decoupling application in chemical plants 9,1] ; and decoupling in structural analysis 11], etc. An input-output energy decoupling method was first proposed by Mao 12], the chief objective of which was to reduce the relevancy between an inputs and an Manuscript received November 26, 23; revised April 25, 25. Corresponding author. E-mail address: xzdong@amss.ac.cn outputs energy. raditional exact decoupling methods are sensitive to model perturbations. Once perturbations happen, decoupling may be destroyed. he energy decoupling method is an approximate decoupling method. It not only overcomes the drawback described above, but also has a clear physical concept and is easy to understand and as such is likely to be accepted by engineers. In practice, engineers always have ideas in mind as to which output should correspond to which input. Once input and output pairs have been decided, the energy decoupling method can be used to solve the decoupling problem. In this paper, we consider the problem of robust input-output energy decoupling in uncertain singular systems. By means of linear matrix inequalities, sufficient conditions are derived for the existence of state feedback and regular input transformation controllers, such that a closed-loop system is generalized quadratically stable, and the energy of every input controls mainly the energy of a corresponding output, and influences the energy of other outputs as weakly as possible. 2 Problem statement Consider the following class of singular systems in which all parameter matrices except E exist as timevarying uncertainties: Eẋ(t) = Ã(t)x(t) + B(t)u(t) y(t) = Cx(t) + D(t)u(t) (1) x(t) R n is the state, u(t) R m is input, y(t) R m is output, and ranke = r < n, Ã(t), B(t), C(t), D(t) are the time-varying matrices which satisfy the

38 International Journal of Automation and Computing 1 (25) 37-42 following conditions: Ã(t) B(t) A B = + C(t) D(t) C D H1 H 2 ] F(t)E 1 E 2 ] F(t) F(t) I (2) A, B, C, D, H 1, H 2, E 1, E 2 are known constant matrices with appropriate dimensions. he matrix value function F(t) is time-varying Lebesgue integrable, and the uncertainties described above are called allowable uncertainties. Remark 1. In this paper, we do not consider the case in which the derivative matrix E is subjected to norm-bounded uncertainty. Some related papers have been published on robust H 2 state feedback control 13], quadratic robust stabilization 14], and robust stabilization via state feedback 15]. Analysis shows that under certain assumptions, a singular system with normbounded uncertainty in the derivative matrix can be transformed into an equivalent uncertain system in which the derivative matrix is constant. Hence, we only discuss the case in which the derivative matrix E has no uncertainty. A singular system (1) is assumed to not necessarily be regular, but it is assumed that a solution exists for a certain initial value Ex(), a known input u(t) and all allowable uncertainties, e.g. in 16] in which a time-invariant singular system is also not assumed to be regular. he purpose of this paper is to search for state feedback and regular input transformation control laws u(t) = Kx(t) + Gv(t), det G (3) such that a closed-loop system Eẋ(t) = (Ã(t) + B(t)K)x(t) + B(t)Gv(t) y(t) = ( C(t) + D(t)K)x(t) + D(t)Gv(t) (4) performs so that the energy of every input controls mainly the energy of a corresponding output, and influences the energy of other outputs as weakly as possible, for all allowable uncertainties, v(t) R m is new input, and K, G are constant matrices with appropriate dimensions. he process of robust input-output energy decoupling in system (1) can be stated as a search for the matrices K R m n, G R m m such that under a zero initial state, system (4) satisfies y i (t)v i (t)dt α i v i (t) 2 dt τ, v j (t) = (j i), i = 1, 2,, m ŷ i (t) ŷ i (t)dt β i v i (t) 2 dt τ, v j (t) = (j i), i = 1, 2,, m (5) (6) y i (t) is the i-th vector of y(t), v i (t) is the i-th vector of v(t), ŷ i (t) is composed of the other vectors of y(t) except y i (t), and α i, β i are known positive scalars. Since the system we are discussing is an uncertain singular system, our input-output energy decoupling conditions will guarantee the unforced closed-loop system (setting u(t) ) to be regular, stable, and impulsefree simultaneously for all allowable uncertainties, that is, system (4) will also be robustly stable. In order to deal with the problems above more conveniently, we propose the definition of generalized quadratic stability which in fact implies robust stability and is in the form of LMIs. Definition 1. System (1) is called generalized quadratically stable if there exist an invertible matrix X and a positive scalar ε which satisfy the following LMIs EX = X E (7) AX + X A + εh 1 H1 X E1 <. (8) E 1 X εi Denote α = α 1 α 2 α m ] and β = β 1 β 2 β m ]. Definition 2. For system (1), if there exist matrices K, G, such that closed-loop system (4) is generalized quadratically stable and satisfies conditions (5) and (6), for all allowable uncertainties, then system (1) is called input-output energy (α, β K, G) decoupling. Lemma 1. 17] For certain matrices Q, H, E and Q = Q, the inequality Q + HFE + E F H < holds for any F satisfying F F I, if and only if there exists a scalar ε > such that Q + εhh + ε 1 E E <. Lemma 2. 18] Q1 Q he matrix 2 Q > holds if 2 Q 3 and only if or Q 3 >, Q 1 Q 2 Q 1 3 Q 2 > Q 1 >, Q 3 Q 2 Q 1 1 Q 2 >. Lemma 3. If system (1) is generalized quadratically stable, then it is robustly stable. Proof. If system (1) is generalized quadratically stable, then there exist an invertible matrix X and a scalar ε > such that EX = X E AX + X A + εh 1 H 1 + ε 1 X E 1 E 1 X < by Lemma 2. If P = X 1, then the above inequalities are equivalent to the following inequalities E P = P E

X. Z. Dong et al./robust Input-Output Energy Decoupling for Uncertain Singular Systems 39 A P + P A + εp H 1 H 1 P + ε 1 E 1 E 1 < which are equivalent to E P = P E Ã(t) P + P Ã(t) <. By Lemma 1, since F(t) satisfies (2), then system (1) is robustly stable by Proposition 1 in 4]. 3 Main results If system (1) is already generalized quadratically stable, we only need to consider the regular input transformation u(t) = Gv(t) (9) based on which the closed-loop system becomes Eẋ(t) = Ã(t) + B(t)Gv(t) y(t) = C(t) + D(t)Gv(t). (1) heorem 1. he uncertain singular system (1) is input-output energy (α, β, G) decoupling, if there exist invertible matrices X i, Y i and positive scalars ε 1i, ε 2i, i = 1, 2,,m, which satisfy the following LMIs EX i = Xi E (11) Π 1 Π 2 Xi E 1 Π2 Π 3 gi E 2 < (12) E 1 X i E 2 g i ε 1i I EY i = Yi E (13) Γ 1 Bg i Γ 2 Yi E 1 gi B β i gi ˆD i gi E 2 Γ2 ˆD i g i Γ 3 < (14) E 1 Y i E 2 g i ε 2i I Π 1 = AX i + X i A + ε 1i H 1 H 1 Π 2 = Bg i X i C i ε 1i H 1 H 2i Π 3 = 2α i D i g i gi D i + ε 1i H 2i H2i Γ 1 = AY i + Yi A + ε 2i H 1 H1 Γ 2 = Yi Ĉi + ε 2i H 1 Ĥ2i Γ 3 = I + ε 2i Ĥ 2i Ĥ 2i g i is the i-th column of G; C i, D i, H 2i are the i-th rows of C, D, H 2 respectively; and Ĉi, ˆD i, Ĥ2i are composed of the other rows of C, D, H 2 except C i, D i, H 2i respectively. Proof. For every input v i (t), system (1) can be decomposed as, Eẋ(t) = Ã(t)x(t) + B(t)g i v i (t) y(t) = C(t)x(t) + D(t)g i v i (t) and the i-th vector y i (t) of y(t) can be expressed as, y i (t) = (C i +H 2i F(t)E 1 )x(t)+(d i +H 2i F(t)E 2 )g i v i (t) and the other vectors ŷ i (t) can be expressed as ŷ i (t) = (Ĉi+Ĥ2iF(t)E 1 )x(t)+( ˆD i +Ĥ2iF(t)E 2 )g i v i (t) If, V i (t, x) = x(t) E P i x(t) W i (t, x) = x(t) E Q i x(t) invertible matrices P i, Q i satisfy the LMIs hen denote, then E P i = Pi E E Q i = Q i E. (15) C i (t) = C i + H 2i F(t)E 1 Ĉ i (t) = Ĉi + Ĥ2iF(t)E 1 D i (t) = D i + H 2i F(t)E 2 ˆD i (t) = ˆD i + Ĥ2iF(t)E 2 M i (t) = V i (t, x) 2y i (t)v i (t) + 2α i v i (t) 2 = x(t) v i (t) ]L 1 x(t) v i (t) ] N i (t) = Ẇi(t, x) β i v i (t) 2 + ŷ i (t) ŷ i (t) = x(t) v i (t) ]L 2 x(t) v i (t) ] Ã(t) L 1 = P i + Pi Ã(t) P i B(t)g i C i (t) gi B(t) P i C i (t) 2α i 2D i (t)g i Ã(t) L 2 = Q i + Q i Ã(t) ] Q i B(t)g i B(t) Q i β i g i + Ĉi(t) ˆDi (t)g i ] Ĉi(t) ˆDi (t)g i ]. We will consider the conditions under which L 1 <, L 2 <. Notice that L 1 < is the same as Q + HF(t)Ẽ + Ẽ F(t) H < A Q = P i + Pi A gi B P i C i P H = i H 1 H 2i P i Bg i Ci 2α i D i g i gi D i ], Ẽ = E 1 E 2 g i ] From Lemma 1 and Lemma 2, L 1 < is equivalent to 1 2 E1 Pi H 1 2 3 gi E 2 H 2i < E 1 E 2 g i ε 1i I H1 P i H2i ε 1 1i I ]

4 International Journal of Automation and Computing 1 (25) 37-42 for some positive scalar ε 1i, 1 = A P i + P i A 2 = P i Bg i C i 3 = 2α i D i g i g i D i which is equivalent to (12) if we let X i = P 1 i. Similarly, by Lemma 2, L 2 < is equivalent to the following inequality Ã(t)Y i + Yi Ã(t) B(t)gi Yi Ĉ i (t) gi B(t) β i gi ˆD i (t) < Ĉ i (t)y i ˆDi (t)g i I if we let Y i = Q 1 i, and it is the same as ˆQ + ĤF(t)Ê + Ê F(t) Ĥ < AY i + Yi A Bg i Yi Ĉ i ˆQ = gi B β i gi ˆD i Ĉ i Y i ˆDi g i I Ĥ = H 1, Ê = E 1 Y i E 2 g i ] Ĥ 2i which is equivalent to (14) for some positive scalar ε 2i, then it is easy to deduce that (15) is equivalent to (11) and (13). Now we have M i (t), N i (t). If we integrate M i (t), N i (t) from to τ and notice that Ex() =, then can obtain, M i (t)dt = V i (τ, x(τ)) + N i (t)dt = W i (τ, x(τ)) + ( 2y i (t)v i (t) + 2α i v i (t) 2 )dt (ŷ i (t) ŷ i (t) β i v i (t) 2 )dt. Using the non-negative of V i (τ, x(τ)), W i (τ, x(τ)), we have y i (t)v i (t)dt α i v i (t) 2 dt ŷ i (t) ŷ i (t)dt β i v i (t) 2 dt which is the result we want. When system (1) is not generalized quadratically stable, state feedback and the regular input transformation should be considered together. heorem 2. he uncertain singular system (1) is input-output energy (α, β K, G) decoupling, if there exist an invertible matrix X, a matrix Y and positive scalars ε 1i, ε 2i, i = 1, 2,,m, which satisfy the following LMIs, EX = X E (16) Σ 1 Σ 2 Σ 3 Σ 2 Σ 4 gi E 2 < (17) Σ 3 E 2 g i ε 1i I Θ 1 Bg i Θ2 Σ 3 gi B β i gi ˆD i gi E 2 Θ 2 ˆDi g i Θ 3 < (18) Σ 3 E 2 g i ε 2i I Σ 1 = AX + BY + X A + Y B + ε 1i H 1 H 1 Σ 2 = g i B C i X D i Y ε 1i H 2i H 1 Σ 3 = E 1 X + E 2 Y Σ 4 = 2α i D i g i g i D i + ε 1i H 2i H 2i Θ 1 = AX + BY + X A + Y B + ε 2i H 1 H 1 Θ 2 = ĈiX + ˆD i Y + ε 2i Ĥ 2i H 1 Θ 3 = I + ε 2i Ĥ 2i Ĥ 2i and g i, C i, D i, H 2i, Ĉi, ˆD i, Ĥ2i are the same as in heorem 1. Moreover, the state feedback gain can be designed as K = Y X 1. Proof. By applying the sufficient conditions in heorem 1 to the uncertain singular system (4), and allowing X i = Y i = X, i = 1, 2,, m, Y = KX, then we can reach our conclusion. Remark 2. In heorem 2, if K =, it becomes a special case of heorem 1, but not the same as heorem 1. In heorem 1, the matrices X i, Y i, i = 1, 2,, m may be different, but in heorem 2 if K =, a matrix X should satisfy (16) (18) simultaneously, so obviously the conditions in heorem 1 are much looser than in heorem 2 when K =. For a prescribed uncertain singular system (1), based on Definition 1, heorem 1 and heorem 2, we will present an algorithm to solve the problem of inputoutput energy decoupling as follows: Step 1. Solve the LMIs (7) and (8). If there is a solution, go to Step 2, otherwise go to Step 3; Step 2. Solve the LMIs (11), (12), (13), and (14). If there is a solution, then system (1) is input-output energy (α, β, G) decoupling, else end; Step 3. Solve the LMIs (16), (17), and (18). If there is a solution, then system (1) is input-output energy (α, β K, G) decoupling, else end. Remark 3. In this paper, we discuss only the following two cases: the uncertain singular system (1) is already generalized quadratically stable, or the uncertain singular system is generalized quadratically stabilizable via state feedback. his is because we hope

X. Z. Dong et al./robust Input-Output Energy Decoupling for Uncertain Singular Systems 41 that the closed-loop system maintains good inner performance (i.e. regular, stable, and impulse-free) besides the ability of input-output energy decoupling. he problem of input-output energy decoupling in uncertain singular systems that cannot be generalized quadratically stabilized (via state feedback), requires further research in the future. 4 Example Consider the uncertain singular system (1), E = 1 1,A = 1 4 3 1,B = 1 2 1 3 1 1 1 2 1 2 1 C =, D =, H 1 2 1 = 1 1 1.3.5.5.3.5 H 2 =, E.6 1 1 =.1.4.3.2.7 E 2 =.1.2,F(t) = 1 1..2.3 1 Obviously the matrix F(t) satisfies (2). Let α 1 = α 2 = β 1 = β 2 = 1 and X 1 () = X 2 () =. Using the MALAB-LMI oolbox, we can obtain a feasible solution for the robust input-output (α, β K, G) decoupling for system (1) as follows, 1.1137.1147 X =.1147 1.1881 1.381.147 1.7599 1.763.9374.226 Y =.8474.143 1.2522 1.38.6881.1151 K =.777.283.7115.9461.5177 G =.19 1.295 ε 11 = 1.9545, ε 12 =.9898 ε 21 =.6132, ε 22 = 1.2973. Fig.1 shows the simulation curves between the inputs and outputs after each step input for the closedloop system. Fig.1 Step response of the example with state feedback and regular input transformation

42 International Journal of Automation and Computing 1 (25) 37-42 5 Conclusion In this paper, we discuss the problem of robust input-output energy decoupling in uncertain singular systems. By means of LMIs, sufficient conditions are presented for an uncertain singular system to be input-output energy (α, β, G), (α, β K, G) decoupling, while also guaranteeing the closed-loop system to be generalized quadratically stable. An example is given to prove the effectiveness of the conclusions in our paper. References 1] F. L. Lewis, A survey of linear singular systems, Circ., Syst. Sig. Proc., vol. 5, pp. 3 36, 1986. 2] B. Verghese, B. C. Levy,. Kailath, A generalized statespace for singular systems, IEEE rans. Automat. Contr., vol. 26, pp. 811 831, Aug. 1981. 3] R. W. Newcomb, B. Dziurla, Some circuits and system applications of semistate theory, Circ., Syst. Sig. Proc., vol. 8, pp. 235 26, 1989. 4] S. Xu, C. Yang, An algebraic approach to the robust stability analysis and robust stabilization of uncertain singular systems, Int. J. Syst. Sci., vol. 31, pp. 55 61, Jan. 2. 5] B. S. Morgan, he synthesis of linear multivariable systems by state-variable feedback, IEEE rans. Automat. Contr., vol. 9, pp. 44 411, May 1964. 6] P. L. Falb, W. A. Wolovich, Decoupling in the design and synthesis of multivariable control systems, IEEE rans. Automat. Contr., vol. 12, pp. 651 659, May 1967. 7]. G. Koussiouris, Pole assignment while block decoupling a minimal system by state feedback and nonsingular input transformation and the observability of the block decoupling system, Int. J. Control, vol. 32, pp. 443 464, Sep. 198. 8] P. N. Paraskevopoulos, F. N. Koumboulis, G. A. zierakis, Disturbance rejection with simultaneous decoupling of linear time-invariant system, ECC 91, pp. 1784 1788, 1991. 9] P. N. Paraskevopoulos, G. E. Panagiotakis, G. A. Onopas, Decoupling of an absorption column, ECC93, pp. 1132 1136, 1993. 1] P. N. Paraskevopoulos, G. E. Panagiotakis, G. A. Onopas, Decoupling with simultaneous disturbance rejection of a distillation column, Proc. 9th Int. Conf. System Engineering, pp. 14 16, 1993. 11] E. C. Zacharenakis, Input-output decoupling and disturbance rejection problems in structural analysis, Comput. Struct, vol. 55, pp. 441 451, May 1995. 12] W. Mao, J. Chu, Input-output energy decoupling of linear time-invariant system, Control heory and Applications, vol. 19, pp. 146 148, Feb. 22. ( in Chinese) 13] K. akaba, Robust H 2 control of descriptor system with time-varying uncertainty, Int. J. Control, vol. 63, pp. 741 75, Apr. 1998. 14] K. Yasuda and F. Noso, Decentralized quadratic stabilization of interconnected descriptor systems, Proc. 35th IEEE CDC, pp. 4264 4269, 1996. 15] C. Lin, J. L. Wang, G. H. Yang, J. Lam, Robust stabilization via state feedback for descriptor systems with uncertainties in the derivative matrix, Int. J. Control, vol. 73, pp. 47 415, May 2. 16] P. C. Muller, M. Hou, On the observer design for descriptor systems, IEEE rans. Automat. Contr., vol. 38, pp. 1666 1671, Nov. 1993. 17] L. Xie, Output feedback H control of systems with parameter uncertainty, Int. J. Control, vol. 63, pp. 741 75, Mar. 1996. 18] A. Albert, Conditions for positive and nonnegative definiteness in terms of pseudoinverses, SIAM J. Appl. Math., vol. 17, pp. 434 44, Nov. 1969. Xin-Zhuang Dong graduated from the Institute of Information Engineering of People s Liberation Army, China, in 1994. She received the M. S. degree from the Institute of Electronic echnology of People s Liberation Army, in 1998 and the Ph.D. degree from Northeastern University, China, in 24. She is currently a postdoctoral fellow at the Key Laboratory of Systems and Control, CAS. Her research interests include singular and nonlinear systems, especially the control of singular systems such as H control, passive control and dissipative control. Qing-Ling Zhang received the Ph.D. degree from Northeastern University, China, in 1995. He is currently a professor with the Institute of Systems Science, Northeastern University. His research interests include singular systems, fuzzy systems, decentralized control, and H 2 /H control.