Brief paperFeedback control of nonlinear stochastic systems for targeting a specified stationary probability density☆
Introduction
The control of stochastic systems for targeting specified probabilities or statistics has been studied for several decades. However, many well-established regulatory control techniques focus only on two quantities, i.e., mean and variance or covariance, as key design targets (Åström, 1970, Åström and Wittenmark, 1980, Goodwin and Sin, 1984, Lu and Skelton, 1998, Skelton et al., 1998, Wojtkiewicz and Bergman, 2001). Since the output of a nonlinear stochastic system is usually non-Gaussian, mean and variance or covariance are not enough to characterize the output process. Thus, in recent years, there has been a growing interest in the feedback control of nonlinear stochastic systems for targeting a probability density function (PDF). Karny (1996) proposed a randomized controller to minimize the mismatch between the joint PDF of a finite process realization and the target one. The controller depends on the solution of a functional equation, which is simpler than the general dynamic programming equation but still very complicated to obtain the solution. Guo and Wang (2005) proposed a pseudo proportional-integral-derivative (PID) tracking control strategy for general nonlinear stochastic systems, based on a linear B-spline model for the output PDFs. Yang, Guo, and Wang (2009) proposed a constrained proportional-integral (PI) tracking control for output probability distributions, based on two-step neural networks (NNs). Following the square-root B-spline NN approximation to the measured output PDF, the problem was transferred into the tracking of dynamic weights. Although the PID and PI tracking controls in Guo and Wang (2005) and Yang et al. (2009) are easy to implement and user-friendly, the B-spline approximation at each sample time is still of a high computational load. Wang (2003) used a nonlinear AutoRegressive and Moving Average with eXogenous (ARMAX) model with an arbitrary and bounded random input to represent general nonlinear stochastic dynamical systems and developed an optimal control to minimize the difference between the conditional output PDF and the target PDF. Guo, Wang, and Wang (2008) proposed an optimal control strategy for nonlinear ARMAX systems with channel time-delay and non-Gaussian noise to make the output PDF follow the target PDF and use a multi-step-ahead nonlinear cumulative cost function to improve the tracking performance. Crespo and Sun (2003) proposed a discontinuous non-linear feedback law to minimize the error between the output PDF and the target one for a one-dimensional stochastic continuous-time process model. Forbes et al., 2003a, Forbes et al., 2003b and Forbes, Guay, and Forbes (2004a) developed regulatory control synthesis techniques for shaping the PDF of stochastic processes. The regulatory control synthesis techniques include a 9-step algorithm, integrating PDF-shaping into a comprehensive control design procedure.
It should be noted that all the design techniques proposed in Crespo and Sun (2003), Forbes et al., 2003a, Forbes et al., 2003b, Forbes et al., 2004a, Guo and Wang (2005), Guo et al. (2008), Karny (1996), Wang (2003) and Yang et al. (2009) are approximate. The disadvantages of these design techniques are: (1) there is no guarantee that the output PDFs of the system will be exactly the same as a given one; (2) the explicit expressions of the feedback controllers are difficult to obtain and the computational loads are very high; (3) only low-dimensional stochastic process models or some typical systems were presented in the references.
An ideal approach to control the design of nonlinear stochastic systems for targeting a specified PDF is based on the exact solution of the systems. The output of a nonlinear system to Gaussian white noise is a diffusion process and the transition PDF of the output is governed by the Fokker–Planck–Kolmogorov (FPK) equation. The exact transient solutions of the FPK equation have been obtained only for very special one-dimensional nonlinear stochastic systems. The exact stationary solutions of the FPK equation, however, have been obtained for a variety of nonlinear stochastic systems (Lin and Cai, 1995, Zhu, 2006). Among them, five classes of exact stationary solutions have been obtained for the dissipated MDOF Hamiltonian systems (Zhu et al., 1990, Zhu and Yang, 1996). All these exact stationary solutions can be treated as the basis for designing feedback control of nonlinear stochastic systems to target a specified SPDF. However, so far only the exact stationary solution of a one-dimensional nonlinear stochastic system has been used for designing the feedback control of a system to target a specified SPDF (Forbes, Guay, & Forbes, 2004b).
In the present paper, an innovative procedure for designing the feedback control of MDOF nonlinear stochastic systems to target a specified SPDF is proposed, based on the technique for obtaining the exact stationary solutions of the dissipated MDOF Hamiltonian systems. This paper is organized as follows: In Section 2, the technique for obtaining the five classes of exact stationary solutions of the dissipated MDOF Hamiltonian systems is briefly reviewed. Then, the design procedure for the feedback control of MDOF nonlinear stochastic systems to target a specified SPDF is proposed in Section 3. In Section 4, a method for proving that the transient output of the controlled system approaches the target SPDF as is presented. In Section 5, an example is worked out to illustrate the design procedure and its effectiveness. Finally, conclusions and suggestions for future studies are given in Section 6.
Section snippets
Exact stationary solution
Consider a MDOF nonlinear stochastic system expressed by the following dissipated Hamilton equations where and are generalized displacements and generalized momenta, respectively; , ; is a Hamiltonian; are coefficients of damping; are amplitudes of random excitations; are Gaussian white noises in the sense of Stratonovich with correlation functions
Design of control for targeting a specified SPDF
Consider a controlled MDOF nonlinear stochastic system. Suppose that it can be expressed as the following controlled, dissipated Hamiltonian system where are feedback control forces to be designed and the other notations are the same as those in Eq. (1). A derivation of Eq. (6) as that from Eq. (1) to Eq. (4) leads to the following Itô stochastic differential equations
Output of the controlled system
Substituting and determined in the last section into Eq. (7) leads to the following Itô stochastic differential equations of the controlled system It is readily proved by using the technique described in Section 2 that system (15) has the same exact stationary solution as the target SPDF in Eq. (8). is a vector diffusion process with the elliptic differential generator
Example
Consider a 4-DOF non-linear stochastic system governed by the following Itô stochastic differential equations where are constants, and are independent Wiener processes with intensities , respectively. The Hamiltonian and first integrals of the Hamiltonian system associated with the uncontrolled system (17) are
Concluding remarks
In the present paper the technique for obtaining the exact stationary solutions of MDOF nonlinear stochastic systems has been applied for the first time to design feedback control of MDOF nonlinear stochastic systems for targeting a pre-specified SPDF. The proposed control procedure has a number of advantages: it is simple yet yields an exactly analytical control law; the control can be designed offline and no online measurement is required; the physical implications of the feedback control are
C.X. Zhu is currently a Ph.D. candidate at the School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University. She received her B.S. degree from the same university in 2007. She has worked in the Department of Mechanics in Zhejiang University for two years since 2008. Her research interests include stochastic optimal control and process control.
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C.X. Zhu is currently a Ph.D. candidate at the School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University. She received her B.S. degree from the same university in 2007. She has worked in the Department of Mechanics in Zhejiang University for two years since 2008. Her research interests include stochastic optimal control and process control.
W.Q. Zhu is a professor of Zhejiang University and a member of Chinese Academy of Sciences. He graduated from Northwestern Polytechnical University in 1961 and the Graduate School of the same university in 1964. In the last two decades, he visited University of Wisconsin-Madison, MIT, Florida Atlantic University and State University of New York at Buffalo, in USA, University Blaise Pascal in France, Kyoto University in Japan, etc. His research interest is in nonlinear stochastic dynamics and control, and he has proposed and developed a Hamiltonian theoretical framework of nonlinear stochastic dynamics and control.
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The work reported in this paper is supported by the National Natural Science Foundation of China under Grant No. 10772159, 11072212 and No. 10932009, the Zhejiang Natural Science Foundation of China under Grant No. Y7080070 and the Research Fund of Northwestern Polytechnical University under Grant No. JC200802 and JC200937. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor George Yin under the direction of Editor Ian R. Petersen.