A Recurrent Neural Network for Solving Sylvester Equation With Time-Varying Coefficients

Transcription

1 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL 13, NO 5, SEPTEMBER A Recurrent Neural Network for Solving Sylvester Equation With Time-Varying Coefficients Yunong Zhang, Danchi Jiang, Jun Wang, Senior Member, IEEE Abstract This paper presents a recurrent neural network for solving the Sylvester equation with time-varying coefficient matrices The recurrent neural network with implicit dynamics is deliberately developed in the way that its trajectory is guaranteed to converge exponentially to the time-varying solution of a given Sylvester equation Theoretical results of convergence sensitivity analysis are presented to show the desirable properties of the recurrent neural network Simulation results of time-varying matrix inversion on-line nonlinear output regulation via pole assignment for the ball beam system the inverted pendulum on a cart system are also included to demonstrate the effectiveness performance of the proposed neural network Index Terms Global exponential convergence, matrix inversion, nonlinear output regulation, recurrent neural network, time-varying Sylvester equation I INTRODUCTION THE linear matrix equation, known as the Sylvester equation, is closely related to the analysis synthesis of dynamic systems, eg, in the design of feedback control system via pole assignment In recent years, many studies have been reported on real-time solutions of algebraic equations including matrix inversion Sylvester equation [1] [13] Generally speaking, the methods reported in these references are related to the gradient descent method in optimization, which can be summarized as follows: first, construct a cost function such that its minimal point is the solution to the equation; then, a recurrent neural network is developed to evolve along a descent direction of this cost function until a minimum of the cost function is reached A typical descent direction is defined by the negative gradient However, if the coefficients of the equation are time-varying, then a gradient-based method may not work well Because of the effects of the time-varying coefficients, the negative gradient direction can no longer guarantee the decrease of the cost function Usually a neural network of much faster convergence in comparison to the time-varying coefficients is required for a real-time solution if the gradient-based method adopted The shortcomings of applying such a method to time-varying cases are two-fold: the much faster convergence is usually at cost of Manuscript received March 12, 2001; revised October 4, 2001 This work was supported by the Hong Kong Research Grants Council under Grant CUHK4150/97E Y Zhang J Wang are with the Department of Automation Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong ( ynzhang@acaecuhkeduhk; jwang@acae cuhkeduhk) D Jiang is with Delano Technology Corporation, Toronto, ON L3R 0E8, Canada Publisher Item Identifier S (02) the precision or with stringent restrictions on design parameters, such method is not applicable to the case the coefficients vary quickly or the case of large-scale complex control systems Implicit dynamical systems arise naturally in electronic circuits In most cases, the Kirchoff s laws of a circuit usually define an implicit dynamical system In this paper, a novel recurrent neural network with implicit dynamics is proposed for solving the problem of the time-varying Sylvester equation (including time-varying matrix inversion as a special case) This implicit dynamics are delicately introduced in place of the gradient dynamics such that the computation error can be made decreasing globally exponentially A sensitivity analysis is included to show the proposed method can actually perform very well even in the cases that the recurrent neural network is not precisely constructed Both theoretical simulation results are given to demonstrate the effectiveness efficiency of the proposed neural-network approach The remainder of this paper is organized into six sections In Section II, the problem formulation some preliminaries are discussed In Section III, the recurrent neural network with implicit dynamics is proposed its convergence sensitivity properties are analyzed Section IV presents the online inversion of the time-varying matrices as a special case of solving Sylvester equation in real time In Sections V VI, the proposed neural networks are, respectively, applied to nonlinear output regulation to control the ball beam system the inverted pendulum on a cart system The last section concludes the paper with final remarks II PROBLEM FORMULATION AND PRELIMINARIES Consider the following time-varying Sylvester equation: are timevarying smooth coefficient matrices which, together with their derivatives, are assumed to be known or can be estimated accurately, is the unknown matrix to be obtained The objective of this paper is to solve the time-varying Sylvester equation (1) in real time To lay a basis for the discussion, the following regularity condition is presented 1) Regularity Condition: There exists a positive real number such that (1) (2) /02$ IEEE

2 1054 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL 13, NO 5, SEPTEMBER 2002 denotes the identity matrices in, respectively, the symbol denote the Kronecker product; ie, is a large matrix made by replacing the th entry of with the matrix The following results can be obtained if the regularity condition holds: 1) ; 2) There exists a unique solution to the Sylvester equation (1) The first result can be clearly seen from (2) by simple manipulations of Kronecker product, that is, the left-h sides of the two equations are actually identical It implies that the matrix is nonsingular Because the matrix equation (1) is equivalent to [14] denotes a column vector obtained by stacking all column vectors of together, the second result is proved For the detailed matrix calculus involved in (1) (3), see the Appendix of the paper Lemma: ([14, Th 445]) Equation (1) satisfies the regularity condition if only if the bounded matrices have no common eigenvalue at any time (ie, ) Furthermore, the minimal difference between eigenvalues of are uniformly greater than or equal to III A RECURRENT NEURAL NETWORK A Model Description In the literature, conventional gradient-based neural network approaches [12], [13] have been developed to compute time-invariant Sylvester equation But for the time-varying case like (1), much faster convergence rate of the gradient-based network is usually required in comparison to the time scale of time-varying coefficients The gradient-based neural network often yield relatively large computational errors The proposed recurrent neural network can be viewed as an extension of a gradient-based neural network to time-varying cases First, let us define a matrix-valued function associated to the time-varying Sylvester equation (1) (3) Fig 1 Block diagram of the recurrent neural network model which produces the following implicit dynamic equation of neural network: is an activation state matrix corresponding to the theoretical solution of (1), the constant is a positive design parameter used to scale the convergence rate of the proposed network Unlike classical Hopfield-like recurrent neural network described by an explicit set of differential equations, the proposed network is given by an implicit dynamic equation Based on the results about the regularity condition in Section II, implicit dynamics in (5) can be uniquely converted into an explicit dynamical system in the form of The implicit dynamical equation (5) can be reformulated as, which is a compact form of the following group of equations: (5) (6) (4) Second, to guarantee the convergence of the function, its time derivative should be made negative definite For exponential convergence of, the following descent direction is used: The block diagram of the recurrent neural network realizing the implicit dynamics in (6) is depicted in Fig 1 B Convergence Analysis The following result on global exponential convergence is important Theorem 1: Given matrix functions of (1), if the regularity condition (2) is satisfied, then the state matrix of the implicit dynamical

3 ZHANG et al: AN RNN FOR SOLVING SYLVESTER EQUATION 1055 system (5) starting from any initial state converges exponentially to the theoretical solution of the time-varying Sylvester equation (1) Proof: Let denoting the difference between the neurally computed solution the theoretical solution of (1) It can be directly checked that is the solution to the ensuing dynamics with the initial state disturbance on electronic components in an additive form, ie, Let denote the solution obtained from the perturbed neural system (9) Since (7), the energy function of (4) is as follows: are, respectively, the theoretical solutions to Sylvester equation (1) the perturbed Sylvester equation Define the computing error Assume are unknown smooth matrix functions with uniformly bounded norm (ie, ) such that the regularity condition still holds for the perturbed pair with a smaller positive constant instead of Taking matrix norm operations, in view of the results in Theorem 1, we get is referred to as the Frobenius norm hereafter Then (10) Hence, On the other h, by the regularity condition (2), it follows that holds Thus, for any initial state there As seen from (10), the first term approaches zero exponentially for any initial state, while the second term is the residual error dependent on the variations It follows from the regularity condition (3) that Thus, without loss of generality, assuming, we show that is upper bounded as (8) The proof on global exponential convergence is, thus, complete C Sensitivity Analysis In the realization of the neural-network model (5), there may be some errors involved Such errors may come from truncating error in digital realization or high-order residue of circuit elements (eg, diodes) in analog realization In this section, we show that these errors cannot derail the proposed neural network, even though they may result in some deviation in the solution The coefficient matrices of neural-network model (5) are assumably perturbed due to parameter variation /or Combining (10) (11), we have (11) (12) Based on these above analysis, there hold the following sensitivity properties

4 1056 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL 13, NO 5, SEPTEMBER 2002 Case 2) If are sufficiently small uniformly distributed with the ratio (ie, assuming with ), defining for simplicity, the residual error is, thus Fig 2 The determination of the perturbation bounds The proof on sensitivity properties is, thus, complete Theorem 2: Consider the recurrent neural network (5) with implementation error terms, ie, the perturbed model (9) 1) If are uniformly bounded, respectively, by in, the computation error is also bounded as in (12), provided that the regularity condition still holds with instead of for any perturbation pair 2) The perturbation bounds of are 3) The residual errors of are bounded by, respectively, for two special but practical cases, ie, the case of being sufficiently small, the case of being uniformly distributed with the ratio of to, respectively Proof: Part 1) has been shown in detail following from (9) (12) As for Part 2), since due to the inequality, as illustrated in Fig 2, the lower bound of the minimal difference between eigenvalues of equals ; ie, Part 3) shows that much tighter bounds of the residual error can be achieved for some special practical cases Case 1) If are sufficiently small while has been perturbed seriously, the residual error IV INVERSION OF TIME-VARYING MATRICES Consider the real-time computation of the inverse of a time-varying matrix, which is a special case of solving the time-varying Sylvester equation with The theoretical solution to is the time-varying inverse, which can be solved by using the proposed neural-network model (5) in the following special form: For illustration, we choose (13) so that the exact solution to the time-varying equation can be given as (14) The recurrent neural network (13) is, thus, in the following form: (15) As illustrated in Fig 3, starting from any initial states romly selected in [ 2,2], the neural network converges to the theoretical solution (14) rapidly the exponential convergence rate of (15) is set to The convergence can be expedited effectively by increasing For example, if, the convergence time without an appreciable error is less than 5 s

5 ZHANG et al: AN RNN FOR SOLVING SYLVESTER EQUATION 1057 Fig 3 The online inversion of a time-varying matrix by model (15) Fig 5 The comparison of the computing error, curves correspond to model (15) curves correspond to model (16) Fig 4 The performance of a traditional matrix-inversion neural network (16), the dash-dotted lines denote elements of the theoretical solution to A(t)X(t) =I For comparison, a traditional matrix-inversion neural network based on gradient descent method is shown [5], [6] (16) its performance is illustrated in Fig 4 with same design parameters (ie, ) Fig 5 illustrates the errors of the two neural networks (15) (16) with different initial states It is seen from Figs 3 5 that the solution neurally computed by (13) converges exponentially to (with the maximal steady-state error less than ), as the residual error between the solution neurally computed by (16) is much larger (ie, the maximal steady-state error equals 10) depends inversely on To show the sensitivity characteristics of the proposed neural network (15), consider the smooth parameter variation of a higher frequency sinusoidal form Fig 6 The norm of the computing error corresponding to different ie, is selected as 005, 01, 015, respectively Theoretically, for, the steady-state error of is upper bounded, respectively, by 0104, 0215, according to Part 3) of Theorem 2, while loose bounds can be calculated, respectively, as 2902, 2985, 3078 according to (11) Before determining the bounds, related constants have been estimated as As seen from Fig 6, the simulation results confirm the sensitivity results presented in Section IV, the curve is the norm of computation error in the case of, the curves are corresponding to, respectively V CONTROL OF THE BALL AND BEAM SYSTEM This section presents the simulation results on the application of the recurrent neural network for control of the ball beam system via on-line pole assignment

6 1058 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL 13, NO 5, SEPTEMBER 2002 Consider a nonlinear plant an exosystem given by (17) (18) An error measure is defined The objective of output regulation is to implement a feedback control law such that the closed-loop system is stable the error between the output of the system (17) output of the exosystem (18) asymptotically approaches zero As shown in [16] [17], the key condition for the solution of this problem is the existence of a zero-error manifold for the plant, which can be rephrased as the existence of sufficiently smooth functions such that, (19) Equation (19) is known as the regulator equation or Isidori Byrnes equation The th-order power-series approximation solution to (19) is described below [15] For some positive integer, let, such that, sufficiently approximate the exact solution to the regulator equation (19) for any in an open neighborhood of the origin The functions can be obtained as degree- polynomials in by solving recursive sets of linear matrix equations generated by substituting appropriate polynomial expressions into (19) Afterwards, the control law synthesis needs to compute such that has desired eigenvalues with negative real parts for all, the feedback control is thus constructed as (20) Fig 7 The ball beam system [13], [18] The feedback gain can, thus, be solved by the proposed recurrent neural network (22) (23) is activation state matrices corresponding to To demonstrate the applicability performance of the nonlinear output regulation system based on neural networks, we will apply the proposed model to two nonminimum-phase benchmark systems, respectively; ie, the ball beam system the inverted pendulum on a cart system As illustrated in Fig 7, in the well-known ball--beam system, a beam is made to rotate in a vertical plane by applying a torque at the center of rotation a ball is free to roll on the beam The motion equation is repeated below [19] are the angle, position, the moment of inertia of the beam, respectively, are the mass, moment of inertia, radius of the ball, respectively, m/s is the acceleration of gravity Defining performing the input transformation, the state-space representation of the system is (21) To stabilize the Jacobian linearization pair in real time, the following time-varying Sylvester equations are considered is an instrumental variable matrix, the matrices are design parameters The uniqueness nonsingularity properties of are guaranteed, if is selected such that is observable, in addition to the regularity condition For details, see [12], the design parameters are selected as kg, m, kgm, kgm,, thus, The objective is to design a state-feedback control law such that the ball position asymptotically tracks the reference input Thus, the exogenous system is formulated as

7 ZHANG et al: AN RNN FOR SOLVING SYLVESTER EQUATION 1059 Fig 8 Adaptive tuning of output neurons z the error measure A third-order approximate solution (the subscript (3) omitted for simplicity) to the regulator equation (19) is given as A straightforward derivation gives Fig 9 Feedback gain matrix K(t) obtained from the neural network To illustrate the operating characteristics of the control system based on neural networks, let The design parameters of the proposed neural network can be set to, respectively Fig 8 depicts the adaptive tuning of output neurons Fig 9 depicts the dynamic output regulation feedback gain obtained from the neural network, correspondingly, the trajectories of the resulting closed-loop poles over time are illustrated in Fig 10 with solid lines Fig 11 shows the tracking performance of the proposed neural-network based controller for the ball beam system As seen from Fig 10 simulation data, the closed-loop poles are kept sufficiently close to the desired locations with the relative error less than absolute error less than 00067, as the trajectories of the closed-loop poles by the gradient-based networks shown in [13] are illustrated in Fig 10 with dashed lines to be time-varying in the ranges of [-20023,-19977], [-30533,-29506], [-40564,-39575] even with the design parameter As shown in Fig 11, the maximal steady-state tracking error of the proposed neurocontrol system is, which, under the same th-order approximation solution, is mainly determined by amplitude frequency of the exogenous signal to be tracked

8 1060 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL 13, NO 5, SEPTEMBER 2002 Fig 10 Trajectories of the closed-loop poles over time, solid lines correspond to the proposed neural network dashed lines correspond to the gradient-based model Fig 12 The inverted pendulum on a cart system Fig 11 Tracking performance of the ball beam system the parameters are given by kg, kg, m, kg/s, m/s The objective is to design a state-feedback control law such that the position of the cart asymptotically tracks the reference input while keeping the closed-loop system stable Thus, the exogenous system error measure can be reformulated similar to the ball beam system except for The third-order approximate solution to the regulator equation (19) is given by VI CONTROL OF THE INVERTED PENDULUM ON A CART SYSTEM The inverted pendulum on a cart illustrated in Fig 12, is another nonlinear nonminimum-phase system that can be found at many universities control labs The motion of the system is described as [20], A straightforward computation of (20) gives is the mass of the cart, is the mass of the block on the pendulum, the length of the pendulum, the acceleration of gravity, the coefficient of viscous friction for the motion of the cart, is the angle the pendulum makes with the vertical, the position of the cart, is the applied force Defining, the state space representation of the system is, we have the equation shown at the bottom of the next page Compared with the ball--beam example, it is much harder (if not impossible) to derive the analytic form of nonlinear output regulation feedback gain explicitly since in this case are much complicated Let, the design parameters of the proposed neural network can be selected as [ ] 100, respectively The operating characteristics of neural network the resulting closed-loop system are shown in Figs Fig 13 depicts the dynamic output regulation feedback gain adaptively tuned in response to the variation of

9 ZHANG et al: AN RNN FOR SOLVING SYLVESTER EQUATION 1061 Fig 15 Tracking performance of the inverted pendulum on a cart system Fig 13 Feedback gain K(t) obtained from the neural network mance of the inverted pendulum on a cart system For comparison, to achieve similar results in the same order precision, the corresponding design parameter of gradient-based neural networks [13] is required to be more than It is worth pointing out that, in general, the stability of a linear time-varying system cannot be ensured by only imposing that the closed-loop eigenvalues are stable at each time This is only true for slowly varying systems Fig 14 Trajectories of closed-loop poles over time Fig show that the closed-loop poles are kept sufficiently close to the desired locations with relative errors less than, that the maximal steady-state tracking error is less than The results show that the proposed recurrent neural networks perform the stabilization satisfactorily subsequently lead to a superior tracking perfor- VII CONCLUDING REMARKS A recurrent neural network with implicit dynamics is presented in this paper for solving time-varying Sylvester equation in real time The state trajectory is guaranteed to converge exponentially to the theoretical solution to the given Sylvester equation Analytical results on convergence sensitivity are discussed to show the exponential convergence of error robustness to perturbations Simulation results on time-varying matrix inverse, control of ball beam system inverted pendulum on a cart system are included to demonstrate effectiveness efficiency of the proposed neural network Compared to the gradient-based methods used in time-varying cases, the proposed neural network is guaranteed to be globally exponentially convergent to the exact solution of the time-varying Sylvester equations Further directions may aim at the design analysis of discrete-time neural networks the development of numerical algorithms electronic circuits for solving time-varying Sylvester equations APPENDIX The Appendix summarizes the matrix calculus involved in (1) (3) In Section II, a2 mathematical tool like the Kronecker product is utilized for the study of matrix equations

10 1062 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL 13, NO 5, SEPTEMBER 2002 The Kronecker product of is denoted by matrix Note that product include is defined to be the block The basic properties of the Kronecker By applying the above Kronecker product rules, we have which equals the left-h side of the regularity condition (2) Hence, if the regularity condition (2) holds for a positive real number, then which implies that the is nonsingular With defined by matrix, we associate the vector Since is nonsingular, the solution to (3) is guaranteed to be existent unique In summary, there exists a unique solution to the time-varying Sylvester equation (1), provided that the regularity condition (2) holds REFERENCES [1] J Jang, S Lee, S Shin, An optimization network for matrix inversion, in Neural Information Processing Computation New York: Amer Inst Physics, 1988, pp [2] F L Luo B Zheng, Neural network approach to computing matrix inversion, Appl Math Comput, vol 47, pp , 1992 [3] A Cichocki R Unbehauen, Neural network for solving systems of linear equations related problems, IEEE Trans Circuits Syst, vol 39, pp , 1992 [4], Neural networks for solving systems of linear equations Part II: Minimax least absolute value problems, IEEE Trans Circuits Syst, vol 39, pp , 1992 [5] J Wang, Recurrent neural networks for solving linear matrix equations, Comput Math Applicat, vol 26, pp 23 34, 1993 [6], A recurrent neural network for real-time matrix inversion, Appl Math Comput, vol 55, pp , 1993 [7], Recurrent neural networks for computing pseudoinverses of rank-deficient matrices, SIAM J Sci Comput, vol 18, pp , 1997 [8] J Wang G Wu, A multilayer recurrent neural network for solving continuous-time algebraic Riccati equations, Neural Networks, vol 11, pp , 1998 [9] J Song Y Yam, Complex recurrent neural network for computing the inverse pseudo-inverse of the complex matrix, Appl Math Comput, vol 93, pp , 1998 [10] Y Xia, J Wang, D L Hung, Recurrent neural networks for solving linear inequalities equations, IEEE Trans Circuits Syst I, vol 46, pp , 1999 [11] Y Xia J Wang, A recurrent neural network for solving linear projection equations, Neural Networks, vol 13, pp , 2000 [12] J Wang G Wu, Recurrent neural networks for synthesizing linear control systems via pole assignment, Int J Syst Sci, vol 26, no 12, pp , 1995 [13] Y Zhang J Wang, Recurrent neural networks for nonlinear output regulation, Automatica, vol 37, pp , 2001 [14] R A Horn C R Johnson, Topics in Matrix Analysis Cambridge, UK: Cambridge Univ Press, 1991, pp [15] J Huang W J Rugh, An approximation method for the nonlinear servomechanism problem, IEEE Trans Automat Contr, vol 37, pp , 1992 [16] A Isidori C I Byrnes, Output regulation of nonlinear systems, IEEE Trans Automat Contr, vol 35, pp , 1990 [17] J Huang W J Rugh, Stabilization on zero-error manifolds the nonlinear servomechanism problem, IEEE Trans Automat Contr, vol 37, pp , 1992 [18] S P Bhattacharyya E de Souza, Pole assignment via Sylvester s equation, Syst Contr Lett, vol 1, no 4, pp , 1982 [19] J Hauser, S Satry, P Kokotovic, Nonlinear control via approximate input-output linearization: The ball beam example, IEEE Trans Automat Contr, vol 37, pp , 1992 [20] R Gurumoorthy S R Sers, Controlling nonminimum phase nonlinear systems The inverted pendulum on a cart example, in Proc Amer Contr Conf, San Francisco, CA, 1993, pp As stated in [14], let be unknown, given, the matrix equation is equivalent to the system of By using the Kronecker product the above notation, the time-varying Sylvester equation (1) can be transformed to the vector form (3); ie, Yunong Zhang received the BS MS degrees in automatic control engineering from Huazhong University of Science Technology, Wuhan, China, South China University of Technology, Guangzhou, China, in , respectively He is now pursuing the PhD degree in the Department of Automation Computer-Aided Engineering, the Chinese University of Hong Kong, Hong Kong His research interests include nonlinear systems, robotics, neural networks, signal processing

11 ZHANG et al: AN RNN FOR SOLVING SYLVESTER EQUATION 1063 Danchi Jiang received the BS degree in mathematics from Wuhan University, Wuhan, China, the MS degree in control systems applications from East China Normal University, Shanghai, China, the PhD degree in systems engineering from the Australian National University, Canberra, Australia This work was done while he was a Postdoctoral Research Fellow at the Chinese University of Hong Kong He is currently with Dalano Technology Corporation, Markham, ON, Canada His present research interests include engineering software development, neural computing, system optimization, control system design Jun Wang (S 89 M 90 SM 93) received the BS degree in electrical engineering the MS degree in systems engineering from Dalian University of Technology, China He received the PhD degree in systems engineering from Case Western Reserve University, Clevel, OH He is currently a Professor of Automation Computer-Aided Engineering, the Chinese University of Hong Kong, Hong Kong Prior to going to Hong Kong, he was an Associate Professor at the University of North Dakota, Gr Forks His current research interests include neural networks their engineering applications Dr Wang is an Associate Editor of the IEEE TRANSACTIONS ON NEURAL NETWORKS IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS