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IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 13, NO. 3, MAY 2002
633
Global Exponential Stability of Recurrent Neural Networks for Synthesizing Linear Feedback Control Systems Via Pole Assignment Yunong Zhang, Student Member, IEEE, and Jun Wang, Senior Member, IEEE
Abstract—Global exponential stability is most desirable stability property of recurrent neural networks. The paper presents new results for recurrent neural networks applied to online computation of feedback gains of linear time-invariant multivariable systems via pole assignment. The theoretical analysis focuses on the global exponential stability, convergence rates, and selection of design parameters. The theoretical results are further substantiated by simulation results conducted for synthesizing linear feedback control systems with different specifications and design requirements. Index Terms—Global exponential stability, pole assignment, recurrent neural networks.
I. INTRODUCTION
A
PROBLEM of major importance in control applications is the synthesis of linear feedback control systems via pole assignment. As known, when all of the state variables of a time-invariant system are completely controllable and measurable, the closed-loop poles of the system can be placed at any desired locations on the complex plane with state feedback through appropriate gains [15]. Since the performance of a feedback control system is mainly determined by its closed-loop poles, pole assignment has been a very effective approach to designing feedback control systems for decades, especially for multivariate systems. Since the mid 1980s, efforts have been directed toward computational aspects of the pole assignment and many numerically reliable algorithms have been proposed (see [1] and the references therein). The numerical algorithms for computing feedback gain matrix are usually based on Kronecker product and Gaussian elimination method or resort to matrix decomposition/transformation approaches. In these methods, the minimal numerical operations are at least proportional to the cube of the system dimension [1], and consequently such algorithms are not efficient enough when applying to large-scale feedback control systems or online solving the time-varying feedback gain in adaptive control systems (e.g., gain scheduling). In view of this, parallel computation schemes have been investigated for pole assignment. Manuscript received October 3, 2000; revised August 9, 2001. This work was supported by the Hong Kong Research Grants Council under Grant CUHK4150/97E. The authors are with the Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong (e-mail: [email protected]; [email protected]). Publisher Item Identifier S 1045-9227(02)04444-2.
As parallel computational models, recurrent neural networks have been developed to solve a wide variety of algebra and optimization problems; e.g., [2]–[7]. The results of these investigations have laid a solid basis for pole assignment using recurrent neural networks. In recent years, recurrent neural networks have been proposed for synthesizing linear control systems through pole assignment; e.g., [8]–[11]. In particular, a couple of recurrent neural networks is proposed in [9] for on-line pole assignment via solving two coupled matrix equations and the recurrent neural networks are proven to be asymptotically stable. This neural-network approach to on-line pole assignment is also applied to online synthesizing Luenberger state estimator as a dual problem in [12] and extended to solving minimum-norm pole assignment problem in [11]. Global exponential stability is one of the most desirable dynamic properties of recurrent neural networks [13], [14]. Being globally exponentially stable, recurrent neural networks can converge to their equilibria most rapidly. Therefore, it makes the neural network approach more efficient for on-line synthesis of feedback control systems via pole assignment. In this paper, new theoretical results are presented for the pole- assignment recurrent neural networks proposed in [9], with the main contributions focusing on the analysis of global exponential stability and convergence rates, and selection of design parameters. Simulation results conducted on various system specifications and design requirements are given to substantiate the theoretical results. The remainder of this paper is organized in six sections. Section II provides the background information for pole assignment via Sylvester equation, and the dynamics of the pole-assignment recurrent neural networks. The new theoretical results on global exponential stability, convergence rates, and selection of design parameters are presented in Sections III–V, respectively. Section VI discusses simulation results to illustrate the operating characteristics and verify the theoretical results. Section VII concludes this paper with final remarks. II. PRELIMINARIES In this section, we first review the pole assignment problem in synthesizing linear time-invariant feedback control systems, then introduce the neural-network approach to pole assignment presented in [9]. Consider a controllable linear time-invariant system as follows:
1045-9227/02$17.00 © 2002 IEEE
(1)
633
Global Exponential Stability of Recurrent Neural Networks for Synthesizing Linear Feedback Control Systems Via Pole Assignment Yunong Zhang, Student Member, IEEE, and Jun Wang, Senior Member, IEEE
Abstract—Global exponential stability is most desirable stability property of recurrent neural networks. The paper presents new results for recurrent neural networks applied to online computation of feedback gains of linear time-invariant multivariable systems via pole assignment. The theoretical analysis focuses on the global exponential stability, convergence rates, and selection of design parameters. The theoretical results are further substantiated by simulation results conducted for synthesizing linear feedback control systems with different specifications and design requirements. Index Terms—Global exponential stability, pole assignment, recurrent neural networks.
I. INTRODUCTION
A
PROBLEM of major importance in control applications is the synthesis of linear feedback control systems via pole assignment. As known, when all of the state variables of a time-invariant system are completely controllable and measurable, the closed-loop poles of the system can be placed at any desired locations on the complex plane with state feedback through appropriate gains [15]. Since the performance of a feedback control system is mainly determined by its closed-loop poles, pole assignment has been a very effective approach to designing feedback control systems for decades, especially for multivariate systems. Since the mid 1980s, efforts have been directed toward computational aspects of the pole assignment and many numerically reliable algorithms have been proposed (see [1] and the references therein). The numerical algorithms for computing feedback gain matrix are usually based on Kronecker product and Gaussian elimination method or resort to matrix decomposition/transformation approaches. In these methods, the minimal numerical operations are at least proportional to the cube of the system dimension [1], and consequently such algorithms are not efficient enough when applying to large-scale feedback control systems or online solving the time-varying feedback gain in adaptive control systems (e.g., gain scheduling). In view of this, parallel computation schemes have been investigated for pole assignment. Manuscript received October 3, 2000; revised August 9, 2001. This work was supported by the Hong Kong Research Grants Council under Grant CUHK4150/97E. The authors are with the Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong (e-mail: [email protected]; [email protected]). Publisher Item Identifier S 1045-9227(02)04444-2.
As parallel computational models, recurrent neural networks have been developed to solve a wide variety of algebra and optimization problems; e.g., [2]–[7]. The results of these investigations have laid a solid basis for pole assignment using recurrent neural networks. In recent years, recurrent neural networks have been proposed for synthesizing linear control systems through pole assignment; e.g., [8]–[11]. In particular, a couple of recurrent neural networks is proposed in [9] for on-line pole assignment via solving two coupled matrix equations and the recurrent neural networks are proven to be asymptotically stable. This neural-network approach to on-line pole assignment is also applied to online synthesizing Luenberger state estimator as a dual problem in [12] and extended to solving minimum-norm pole assignment problem in [11]. Global exponential stability is one of the most desirable dynamic properties of recurrent neural networks [13], [14]. Being globally exponentially stable, recurrent neural networks can converge to their equilibria most rapidly. Therefore, it makes the neural network approach more efficient for on-line synthesis of feedback control systems via pole assignment. In this paper, new theoretical results are presented for the pole- assignment recurrent neural networks proposed in [9], with the main contributions focusing on the analysis of global exponential stability and convergence rates, and selection of design parameters. Simulation results conducted on various system specifications and design requirements are given to substantiate the theoretical results. The remainder of this paper is organized in six sections. Section II provides the background information for pole assignment via Sylvester equation, and the dynamics of the pole-assignment recurrent neural networks. The new theoretical results on global exponential stability, convergence rates, and selection of design parameters are presented in Sections III–V, respectively. Section VI discusses simulation results to illustrate the operating characteristics and verify the theoretical results. Section VII concludes this paper with final remarks. II. PRELIMINARIES In this section, we first review the pole assignment problem in synthesizing linear time-invariant feedback control systems, then introduce the neural-network approach to pole assignment presented in [9]. Consider a controllable linear time-invariant system as follows:
1045-9227/02$17.00 © 2002 IEEE
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