TWO KINDS OF HARMONIC PROBLEMS IN CONTROL SYSTEMS

Transcription

1 Jrl Syst Sci & Complexity (2009) 22: TWO KINDS OF HARMONIC PROBLEMS IN CONTROL SYSTEMS Zhisheng DUAN Lin HUANG Received: 22 July 2009 c 2009 Springer Science + Business Media, LLC Abstract This paper discusses the harmonic problems in control systems from two aspects: One is the harmonic control among different subsystems, and the other is the harmonic control among multiple inputs. Some intrinsic problems in such systems are discussed. It is pointed out that some subsystems must be unstable to stabilize the whole interconnected system by an example. Especially for discrete-time multi-input systems, a necessary and sufficient condition is presented for the strict decrease of the quadratic optimal performance index with the control input extensions. This shows an essential difference between single-input and multi-input control systems. Finally, some future research directions are discussed in harmonic control of interconnected systems, allocation of multicontrol inputs, fault-tolerant control, and fault-diagnosis. Key words Interconnected system, quadratic optimal control, redundant inputs, Riccati equation, simplectic matrix. 1 Introduction The harmonic problems in control systems can be divided into two aspects: one is the harmonic control problem in large-scale systems composed of multiple subsystems; the other is the harmonic control among multiple inputs. In the last four decades, the large-scale system theory has been extensively studied and applied to electric power systems, economic systems, ecological and social systems. Some interesting results have been established on such basic issues as decentrally fixed modes, decentralized controllers design, diagonal Lyapunov function method and M-matrix method, etc. [1 7]. However, what roles can the interconnections among subsystems play for the stability or performance of large-scale systems? Or how can the subsystems help with each other to achieve some targets? There are hardly interesting results on such problems until recent years. Especially in the past years, the basic idea for guaranteeing the stability of large-scale systems is to stabilize all the subsystems and try to decrease the effects of interconnections or view the interconnections as uncertainties. Under such an idea, the whole large-scale system can still be stable when the interconnections among subsystems are broken. Such a result is very satisfied from the robustness and reliability. Actually, the diagonal Lyapunov function method always gives such kind of results. However, from the viewpoint of using interconnections to enhance stability, such results are unreasonable and unacceptable. A good Zhisheng DUAN Lin HUANG State Key Lab for Turbulence and Complex Systems and Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing , China. duanzs@pku.edu.cn. This research is supported by National Natural Science Foundation of China under Grant Nos and , and the Key Projects of Educational Ministry under Grant No

2 588 ZHISHENG DUAN LIN HUANG idea is to integrate all the subsystems and make them help with each other to realize the effect of > 2. For a whole large-scale system, it is not necessary to make every subsystem stable or optimal. For example, in large-scale electric power systems, it is interesting to avoid the occurrence of big accidents with the cost of allowing the occurrence of small accidents. And in the modern society, the economy always develops fast with the bankruptcy of small enterprises and manufactories. In fact, in the study of large-scale systems, it was also found that some subsystems must be unstable in order to make sure the stability of the whole system in some special interconnected systems. Therefore, it is important for the study of large-scale systems to design controllers such that the whole system is robust stable or optimal with allowing that some subsystems are unstable or with bad performances [8]. On the other hand, it is also interesting for a multi-input control system to achieve better performances by collaboration among multiple control inputs. In modern control theory, since Wonham [9] proved that a multi-input linear system can be transformed equivalently into a single-input system by introducing a state feedback and finding an auxiliary vector in the column space of the input matrix, the essential differences between single-input and multi-input linear systems have not received much attention. It is known that if a single-input linear system is controllable, increasing the columns of the control input matrix does not affect the stabilizability, controllability, and poles assignment of the system under state-feedback. But what is the case when other system performances are considered? what is the role of the increased columns of the input matrix when they are linearly dependent with the original columns? Clearly, these problems are fundamentally important in modern control theory. Research on such problems would be of great importance to understand the effects of redundant control inputs, especially for modern aircraft and robot systems, where redundant actuators are widely used to improve system reliability and performance [10 15]. Modern aircrafts, especially warcrafts, often use redundant effectors, such as canard or thrust vectoring, to improve maneuverability and reliability. New generation aircrafts even innovates the aerodynamic layout, such as the Lockheed-Martin Innovative Control Effector (LM-ICE) aircraft [16]. Since inappropriate control allocation among different types of actuators may reduce the performance of the aircraft, the problem of control allocation has become an important research subject in modern aircraft control [16 18]. Based on the linear quadratic optimal control theory, a necessary and sufficient condition was given for the strict decrease of the quadratic performance index in continuous-time systems [19]. Both theoretic analysis and numerical simulation results show that the system performance could be improved by increasing the number of control inputs. This paper further considers the optimal control problem for discrete-time systems. 2 Interconnections Among Subsystems For a large-scale system composed of multiple subsystems, it is very hard to discuss the effects of interconnections. And there are few results to this field. It is obviously a good way to give some interesting results for the interconnected system composed of two subsystems and get some intuitive idea for general large-scale systems. An interconnected system composed of two subsystems with decentralized control inputs can be described as { ẋ1 = A 1 x 1 + A 12 x 2 + B 1 u 1, (1) ẋ 2 = A 2 x 2 + A 21 x 1 + B 2 u 2, where u 1 = K 1 x 1 and u 2 = K 2 x 2. The state matrix of closed-loop system is ( ) A1 + B A cl = 1 K 1 A 12. A 21 A 2 + B 2 K 2

3 TWO KINDS OF HARMONIC PROBLEMS IN CONTROL SYSTEMS 589 The above control method is the traditional decentralized control method in large-scale systems [1]. The subsystem s state information is used to control the subsystem itself, but one subsystem does not use the other subsystem s information. This decentralized control method is very important in large-scale systems [1]. If take u 1 = K 1 x 1 + K 12 x 2 and u 2 = K 2 x 2 + K 21 x 1, then it becomes the traditional centralized control problem. The fixed mode always appears in the decentralized control issues, which is a natural generalization of the well-known concept of the uncontrollable and unobservable modes in the traditional centralized control problems [1]. In the following, the effects of A 12 and A 21 for the stability of A cl are discussed. First, consider the following example. Example 1 Take A 1 = , A 2 = , B 1 = 0, B 2 = , A 12 = γ, A 21 = K 1 = ( β 0, β 1, β 2 ) and K 2 = ( α 0, α 1, α 2, α 3 ). Computing the determinant of A cl gives det(si A cl ) = + (1 γ)(α 0 β 1 + α 1 β 0 )s + (1 γ)α 0 β 0. By the stability theory of polynomials [3], all coefficients of a monic stable polynomial must be larger than zero. For the above example, the necessary condition for A 1 +B 1 K 1 and A 2 +B 2 K 2 being stable is α i > 0, β j > 0, i = 0, 1, 2, 3, j = 0, 1, 2. According to the determinant of A cl, one can get the following three result: (i) When γ = 1, 0 is at least a second order fixed mode, which is an eigenvalue that can not be changed by any decentralized controllers; (ii) When γ > 1, A 1 +B 1 K 1, A 2 +B 2 K 2 and A cl cannot be stable simultaneously, since for any stable A 1 +B 1 K 1 and A 2 +B 2 K 2 (α i > 0, β i > 0), the constant term in the determinant of A cl is less than zero; that is, in order to stabilize A cl, at least one of A 1 + B 1 K 1 and A 2 + B 2 K 2 must be unstable; (iii) When γ < 1, it is possible that A 1 +B 1 K 1, A 2 +B 2 K 2 and A cl are stable simultaneously. Remark 1 In this example, one can see that the interconnection terms A 12 and A 21 are very simple, but some interesting results appear. For such similar examples, if one chooses higher order subsystems, then 0 may be a higher order fixed mode. From the above discussion, one can see some complexity in decentralized control problems. Compared with centralized control problems, the main difficulty of solving decentralized control comes from the fact that the feedback gain is subject to the structural constraints. Such constraints are of the same nature as the static output ones, which can be viewed as a full state feedback with constraints that select only the measured states. In the last century, Davison and Ozguner [4] pointed out that the fixed modes exist in decentralized control problems, i.e., some closed-loop eigenvalues can not be changed by decentralized controllers. In recent years, a structured characteristic for interconnections is given in [20]. With such a characteristic, the subsystems and the whole system can not be stable simultaneously. And such results can be generalized to discretetime linear systems [21]. The intercrossed feedback control problems were also studied in [20 21], which show the possibility of collaborations in multiple subsystems. The interconnected

4 590 ZHISHENG DUAN LIN HUANG characteristic in [20] can be generalized to large-scale systems, including complex networks. Actually complex dynamical networks can be viewed as special large-scale systems, see [22] and references therein. By now, there are few developments on stabilizability of large-scale systems by decentralized state feedback controllers. Hence, the engineers like to choose decoupled control method. But it is hard to state that what kind of large-scale systems can be decoupled. For example in flight control problems, some coupled factors are determined by their physical environment. When the decouplability is not clear, the decoupled control method may result in accidents. Actually, the linearized mode of the bank-to-turn (BTT) missiles [23 24] is a coupled mode as in (1). It is still an open problem that if the BTT missiles can be controlled by the decoupling method. 3 Collaboration Among Multiple Inputs As stated in Section 1, it is of great importance both in theory and application to improve system performances through collaborations of multiple control inputs. An interesting necessary and sufficient condition was given in [19] for the strict decrease of the quadratic performance index in continuous-time systems with multiple inputs. This paper considers the discrete-time system and establishes the corresponding result. 3.1 Preliminaries Given a discrete-time linear system: { x(k + 1) = Ax(k) + B0 u 0 (k), A R n n, B 0 R n r0, y(k) = Cx(k), C R m n. Take the quadratic performance index N 1 J 0 = (y T (k)y(k) + u T 0 (k)r 0 u 0 (k)), R 0 > 0, N. (3) k=0 Based on the Riccati equation theory of discrete-time systems, one has the following result for quadratic performance optimal control [25 26]. Lemma 3.1 Suppose that (A, B 0 ) is stabilizable, (A, C) is observable and R 0 = R T 0 > 0. Then, the state-feedback controller such that the index (3) achieves minimum is u 0 (k) = (R 0 + B T 0 P 0 B 0 ) 1 B T 0 P 0 Ax(k), where P 0 is the unique positive definite solution of the Riccati equation A T P 0 A P 0 A T P 0 B 0 (R 0 + B T 0 P 0 B 0 ) 1 B T 0 P 0 A + Q = 0, (4) where Q = C T C. Besides, the optimal performance index is J 0 = x(0) T P 0 x(0), where x(0) is the initial condition of (2). By nonsingularity of R 0, the Riccati equation (4) can be rewritten as where = B 0 R 1 0 BT 0 or A T P 0 A P 0 A T P 0 (I + P 0 ) 1 P 0 A + Q = 0, (5) A T P 0 (I + P 0 ) 1 A P 0 + Q = 0. (6) Suppose that A is nonsingular. Then, the simplectic matrix corresponding to the Riccati equation (5) or (6) is ( ) A + G0 A S 0 = T Q A T A T Q A T. (2)

5 TWO KINDS OF HARMONIC PROBLEMS IN CONTROL SYSTEMS 591 Extending the number of the control inputs of system (2), one gets a new system { x(k + 1) = Ax(k) + Bu(k), B = (B0, B 1 ), u(k) = (u T 0 (k), u T 1 (k)) T, B 1 R n r1, y(k) = Cx(k). (7) Compared with system (2), system (7) can be viewed as a new system with redundant control inputs. Take a new quadratic performance index N 1 J = (y T (k)y(k) + u T (k)ru(k)), R > 0, N (8) k=0 for system (7), where R is an arbitrary extension of R 0 with positive definiteness, i.e., ( ) R0 R R = 01 R01 T = R T > 0. (9) R 1 The corresponding Riccati equation is A T P A P A T P B(R + B T P B) 1 B T P A + Q = 0. (10) In the following, the relationship between systems (2) and (7) in the quadratic performance indexes is studied. 3.2 The Effects of the Extended Control Inputs Similarly to [19], the following lemma is useful for discussing the performance index of (7). Lemma 3.2 For any extended matrix R in (9), if B 1 B 0 R0 1 R 01 0, then R can be supposed to be a diagonal extension as ( ) R0 0 R = = R T > 0. (11) 0 R 1 Proof This lemma can be directly obtained by the method of [19]. Remark 2 Clearly, if R(B 1 ) R(B 0 ), where R( ) denotes column space of the corresponding matrix, then B 1 B 0 R0 1 R 01 0 for any matrix R; if R(B 1 ) R(B 0 ), then it is possible that B 1 B 0 R0 1 R 01 = 0. In this case, one can choose R 01 suitably such that B 1 B 0 R0 1 R Therefore, it is not so restrictive to suppose that R is diagonal blocked. By Lemma 3.2, the matrix R in the optimal performance (8) is supposed to be with diagonal characteristic as in (11). Then, the Riccati equation (10) can be rewritten as A T P A P A T P (I + GP ) 1 GP A + Q = 0, (12) where G = + G 1, = B 0 R 1 0 BT 0, and G 1 = B 1 R 1 1 BT 1, or A T P (I + GP ) 1 A P + Q = 0. (13) If A is nonsingular, then the simplectic matrix corresponding to the Riccati equation (12) or (13) is ( ) A + GA S = T Q GA T A T Q A T. Combining the above discussion with the Riccati equation theory [25], one can get the following result.

6 592 ZHISHENG DUAN LIN HUANG Theorem 3.3 Suppose that (A, B 0 ) is stabilizable, (A, C) is observable, and A is nonsingular. Then, one has P 0 P for the Riccati equations (5) and (12). Furthermore, Ker(P 0 P ) {0} holds if and only if the simplectic matrices S 0 and S have a common anti-stable eigenvalue and the corresponding common left eigenvector simultaneously. Proof By G, one has P (I + P ) 1 P (I + GP ) 1. Then, Rewrite (6) as A T P (I + P ) 1 A A T P (I + GP ) 1 A. (14) A T P 0 A G0 P 0 + Q + A T P 0 P 0 A G0 = 0, (15) where A G0 = (I + P 0 ) 1 A. By the Riccati equation theory, A G0 is Schur stable. Subtracting (13) from (15) leads to A T (P 0 P )A G0 (P 0 P ) + A T P A G0 + A T P 0 P 0 A G0 A T P (I + GP ) 1 A = 0. (16) Rewrite the above equation as Note that A T (P 0 P )A G0 (P 0 P ) + A T P A G0 + A T P 0 P 0 A G0 A T P (I + P ) 1 A +A T P (I + P ) 1 A A T P (I + GP ) 1 A = 0. A T P A G0 + A T P 0 P 0 A G0 A T P (I + P ) 1 A = A T (P 0 P )(I + P ) 1 (P 0 P )A G0 and A T P (I + P ) 1 A A T P (I + GP ) 1 A = A T (I + P ) 1 P (I + G 1 P (I + P ) 1 ) 1 G 1 P (I + P ) 1 A, where G 1 = G. Then, (16) can be written as A T (P 0 P )A G0 (P 0 P ) + A T (P 0 P )(I + P ) (P 0 P )A G0 +A T (I + P ) 1 P (I + G 1 P (I + P ) 1 ) 1 G 1 P (I + P ) 1 A = 0. (17) By (P 0 P )(I + P ) (P 0 P ) 0, A T P (I + P ) 1 A A T P (I + GP ) 1 A 0, and the stability of A G0, the Riccati equation (17) implies P 0 P. In the following, the condition for P 0 > P is discussed. If Ker(P 0 P ) {0}, then there exists a vector x 0 such that (P 0 P )x = 0. Multiplying the left-hand and right-hand sides of (17) by A T and A 1 gives (P 0 P ) A T (P 0 P )A 1 + (P 0 P )(I + P ) (P 0 P ) +(I + P 0 )(I + P ) 1 P (I + G 1 P (I + P ) 1 ) 1 G 1 P (I + P ) 1 (I + P 0 ) = 0. (18) By (P 0 P )x = 0, one has (I + P ) 1 (I + P 0 )x = x. On the other hand, based on G 1 = B 1 R1 1 BT 1, multiplying the left-hand and right-hand sides of (18) by x T and x gives x T A T (P 0 P ) = 0, G 1 P x = 0, or B T 1 P x = 0. Then, Ker(P 0 P ) is A 1 invariable. Hence, there exists an eigenvalue λ of A 1 and the corresponding vector x Ker(P 0 P ) such that x T A T = λx T.

7 TWO KINDS OF HARMONIC PROBLEMS IN CONTROL SYSTEMS 593 Obviously, λ and x also satisfy x T A T G = λxt, where A G = (I + GP ) 1 A; that is, A 1 and A 1 G have a common eigenvalue and the corresponding common eigenvector. In fact, λ is an unobservable mode of (A 1, B1 T P ). Combining the simplectic matrices S 0 and S with the Riccati equations, one can get ( P 0, I)S 0 = A T ( P 0, I), (19) ( P, I)S = A T G ( P, I). (20) According to the discussion on the eigenvalue and eigenvector, multiplying the left-hand side of the above equations by x T leads to ( x T P 0, x T )S 0 = λ( x T P 0, x T ), ( x T P, x T )S = λ( x T P, x T ). By x T P 0 = x T P, A T and A T G are anti-stable, λ is a common anti-stable eigenvalue of S 0 and S, and ( x T P 0, x T ) is the corresponding left eigenvector. On the other hand, if S 0 and S have a common anti-stable eigenvalue λ and the corresponding common left eigenvector, such an eigenvector can be supposed to be ( y T, x T ) by the blocks of the simplectic matrices. By the simplectic matrix theory [25] and the assumptions of the theorem, S 0 and S have no eigenvalues on the unit circle, and all the eigenvalues are symmetrical about the unit circle. Furthermore, suppose that the lower Jordan block corresponding to the anti-stable eigenvalues of S 0 and S are Λ 1 and Λ 2, and the matrices composed of the corresponding left eigenvectors and generalized eigenvectors of S 0 and S are ( X 2, X 1 ) and ( Y 2, Y 1 ), respectively. And without loss of generality, suppose that the first row of ( X 2, X 1 ) and ( Y 2, Y 1 ) are ( y T, x T ). Then, and ( X 2, X 1 )S 0 = Λ 1 ( X 2, X 1 ), ( Y 2, Y 1 )S = Λ 1 ( Y 2, Y 1 ) ( y T, x T )S 0 = λ( y T, x T ), ( y T, x T )S = λ( y T, x T ). (21) By the simplectic matrix theory and the assumptions of the theorem, X 1 and Y 1 are nonsingular. Hence, rewrite the above equations as ( X 1 1 X 2, I)S 0 = X 1 1 Λ 1X 1 ( X 1 1 X 2, I), ( Y 1 1 Y 2, I)S = Y 1 1 Λ 1 Y 1 ( Y 1 1 Y 2, I). Comparing the above equations with (19) and (20), one gets X 1 1 Λ 1X 1 = A T, Y 1 1 Λ 2 Y 1 = A T G. (22) Therefore, P 0 = X 1 1 X 2 and P = Y 1 1 Y 2 are solutions to the Riccati equations (6) and (13). Clearly, the first row of X 1 and Y 1, x T is the common eigenvector of A T and A T G corresponding to their common eigenvalue λ. And the first row of X 2 and Y 2, y T, satisfies x T P 0 = x T P = y T. Furthermore, by (21), y T G 1 = 0; that is, y T B 1 = 0. So, x T P B 1 = 0. Combining the result with (22), one knows that λ is an anti-stable unobservable mode of (A 1, B1 T P ), the corresponding left eigenvector x T satisfies x T (P 0 P ) = 0. This leads to the conclusion. Corollary 3.4 Suppose that (A, B 0 ) is stabilizable, (A, C) is observable and A is nonsingular. Then, the simplectic matrices S 0 and S have a common anti-stable eigenvalue and the corresponding common left eigenvector if and only if S 0 has an anti-stable eigenvalue and its left eigenvector (y T, x T ) satisfies y T B 1 = 0. Furthermore, by x T P 0 = y T, one has x T P 0 B 1 = 0.

8 594 ZHISHENG DUAN LIN HUANG Corollary 3.5 Suppose that (A, B 0 ) is stabilizable, (A, C) is observable, A is nonsingular and the extended control input matrix satisfies B 1 = B 0. Then, the simplectic matrices S 0 and S have a common anti-stable eigenvalue and the corresponding left eigenvector if and only if S 0 and A T have a common anti-stable eigenvalue and the corresponding left eigenvector of A T, x T, satisfies x T P 0 B 0 = 0. Remark 3 In fact, according to the simplectic matrix theory, if S 0 and S have common anti-stable eigenvalues and the corresponding left eigenvectors, then they also have common stable eigenvalues and the corresponding right eigenvectors. For convenience of discussing the above corollaries, the anti-stable eigenvalue and left eigenvector are used in Theorem 3.3. But in Ref. [19], the stable eigenvalue and right eigenvector are used for continuous-time linear systems. Compared with the results in [19], the proof of Theorem 3.3 for discrete-time systems is more complicated. The characteristic of P 0 P 0 can be obtained easily in continuoustime systems [19], but it needs more deductions as in the proof of Theorem 3.3 because of the complexity of the discrete-time Riccati equations. Remark 4 If the simplectic matrices S 0 and S have a common anti-stable eigenvalue, they do not necessarily have a common left eigenvector. Theorem 3.3 requires that they have common eigenvalue and eigenvector simultaneously. Example 2 Consider systems (2) and (7) with matrices A = , B 1 = B 0 = 0 0, C = T, R 1 = R 0 = 1. Since B 1 = B 0 and 0.5 is an uncontrollable mode of (A, B 0 ) and (A, [B 0, B 1 ]), 0.5 and 2 are common eigenvalues of S 0 and S. However, S 0 and S do not satisfy the conditions of Corollary 3.5, so they do not have common eigenvectors. And the solutions to Riccati equations (6) and (13) satisfy P 0 > P. Furthermore, we change B 1 into B 1 = (0, 1, 5) T. Then, 2 is a common anti-stable eigenvalue of S 0 and S, and the corresponding left eigenvector of S 0 satisfies the conditions of Corollary 3.4. In this case, P 0 P, but P 0 P. Combining Theorem 3.3 with Lemma 3.1, one gets the following corollary. Corollary 3.6 For systems (2) and (7), the quadratic performance indexes satisfy J J 0. Besides, if the corresponding simplectic matrices do not have common anti-stable eigenvalues and left eigenvectors simultaneously, then J < J 0 for any initial conditions. Remark 5 According to the above discussions, if one repeatedly extends the control input matrices, the quadratic performance index may also be decreased. Remark 6 For simplicity of discussing simplectic matrices, A is supposed to be nonsingular in this paper. If A is singular, then one needs to use generalized eigenvalue method to replace the simpelctic matrix method [25], and the corresponding problem will be much harder to solve. 4 Conclusion This paper discusses some harmonic problems in control systems from two aspects. Although the coupled or decoupled control has been studied several decades, some basic problems are not yet solved, e.g., whether the interconnected systems can be stabilized in a decentralized way. In the aspect of collaborating control in multiple control input systems, the effects of the redundant inputs have not been completely studied. This paper and [19] studied the effects of multiple control inputs in discrete-time and continuous-time linear systems based on the quadratic performance index. Combining the theoretical methods with engineering applications,

9 TWO KINDS OF HARMONIC PROBLEMS IN CONTROL SYSTEMS 595 it is interesting to study the changes of the controller gain and performance indexes with the control input matrix extensions. Together with fault detection and fault-tolerant control [27 29], it is important to study the influences of the fault in different control inputs or the deletion of some control inputs on the system performances. It is also interesting to discuss the necessity and redundancy of the control inputs. Combining the harmonic control in interconnected systems composed of multiple subsystems with collaborating control in multi-input systems, it is of importance both in theoretical study and engineering applications to establish a unified framework in collaborating, redundant, and fault-tolerant control problems. It is a future research direction to develop effective control algorithms such that the systems can achieve good performances and avoid the fast decrease of performance indexes under the occurrence of input faults. References [1] D. D. Siljak, Large-Scale Dynamic Systems, North-Holland, New York, [2] W. B. Gao and W. Huo, Stability, Decentralized Control and Dynamically Hierarchical Control Foundation of Large Scale Systems (in Chinese), Publishing Press of Beijing University of Aeronautics and Astronautics, Beijing, [3] L. Huang, Stability Theory (in Chinese), Publishing Press of Peking University, Beijing, [4] E. J. Davison and U. Ozguner, Characterization of decentralized fixed modes in interconnected systems, Automatica, 1983, 19(2): [5] M. Ikeda, D. D. Siljak, and K. Yasuda, Optimality of decentralized control for large scale systems, Automatica, 1983, 19: [6] Z. C. Shi and W. B. Gao, Stabilization by decentralized control for large scale interconnected systems, Large Scale Systems, 1986, 10: [7] G. H. Yang and S. Y. Zhang, Decentralized control of a class of large scale systems with symmetrically interconnected subsystems, IEEE Trans. Automat. Contr., 1996, 41(5): [8] L. Huang and Z. S. Duan, Complexity in control science, Acta Automatica Sinica, 2003, 29(5): [9] W. M. Wonham, Linear Multivariable Control: A Geometric Approach, Springer-Verlag, New York, [10] M. X. Wang and M. Li, Development of advanced fighter control allocation methods (in Chinese), Aircraft Design, 2006, 3: [11] Z. Y. Zhan and L. Liu, Control allocation for high performance aircraft with multi-control effectors (in Chinese), Flight Dynamics, 2006, 24(1): [12] E. Q. Yang and J. Y. Gao, Research and development on advanced fighter control allocation methods (in Chinese), Flight Dynamics, 2005, 29(3): 1 4. [13] T. Jiang and K. Khorasani, A fault detection, isolation and reconstruction strategy for a satellite s attitude control subsystem with redundant reaction wheels, IEEE International Conference on Systems, Man and Cybernetics, Montreal, Cook Islands, [14] E. Tatlicioglu, D. Braganza, T. C. Burg, et al., Adaptive control of redundant robot manipulators with sub-task objectives, American Control Conference, Seattle, WA, 2008, [15] Z. Cui, X. C. Wang, D. H. Qian, et al., Research on simulation of redundant robot force control, IEEE International Conference on Robotics and Biomimetics, Sanya, China, 2007, [16] J. B. Davidson, F. J. Lallman, and W. T. Bundick, Real-time adaptive control allocation applied to a high performance aircraft, 5th SIAM Conference on Control and Its Application, [17] A. Serrani and M. Bolender, Invited Session: Control of over-actuated systems: Application to guidance and control of aerospace, marine, and terrestrial vehicles, 14th Mediterranean Conference on Control and Automation, Ancona, Italy, [18] L. Zaccarian, On dynamic control allocation for input-redundant control systems, IEEE Conference on Decision and Control, New Orleans, LA, USA, 2007,

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