Controlling the equilibria of nonlinear stochastic systems based on noisy data
Introduction
Control problems for randomly forced systems attract a great deal of attention from science and engineering communities. The development of a stochastic control theory was initiated in [24], and continued by many investigators (see, for instance, [25], [3], [35], [15], [19] and bibliography therein).
For linear dynamic systems, the theory of control and stabilization has been well developed. Indeed, many fundamental results in linear control theory have appeared under a state-space framework. Basic advances in the Linear-Quadratic-Gaussian problem are presented in [4]. Here, the most famous is the separation principle that combines Kalman filter (linear-quadratic estimator) and a linear-quadratic regulator [34], [18]. Idea of H2 and controllers was an important contribution to the development of the linear theory [14]. Mathematically, the most linear control problems are reduced to the analysis and solution of matrix equations and inequalities [36], [11], [21], [30], [31], [10], [32]. Computational aspects of linear control theory are discussed in [12].
Today, the control of nonlinear stochastic systems is a challenging problem in the modern engineering [33]. Besides control, there is also research about filtering on nonlinear stochastic systems [22], [23]. An interplay of nonlinearity and stochasticity can generate new unexpected dynamic regimes that are not observed in linear systems [20], [2], [17], [26]. It is important to note the essential difference in the mathematical description of stochastic dynamics of linear systems with additive Gaussian noise and nonlinear systems with state- and control-dependent noise. Solutions of the first type of systems are Gaussian processes, and its dynamics are fully described using the first two moments governed by linear deterministic equations. Analysis and control for the second type of systems are much more difficult. Indeed, a comprehensive probabilistic description of the stochastic flows in nonlinear systems is given by the Kolmogorov–Fokker–Planck equation. Analytically, this equation can be solved only in very specific cases, thereby only statistical properties are studied via numerical simulation in general. Even more unfortunately, for solving control problems, this approach is not applicable. In these circumstances, analytical approximations of probabilistic distributions become the only alternative. For this purpose, the quasipotential method is widely used [13], [16], for which a stochastic sensitivity function (SSF) technique was proposed in [28], [6], [8] and successfully applied to the stabilization of stochastic attractors and suppression of chaos [7], [9]. It was shown that the underlying reason, from an engineering point of view, for unacceptable noise-induced large-amplitude oscillations and stochastic transformations, lies in the high sensitivity of initial deterministic attractors to random disturbances.
In control systems, the stochastic sensitivity can be changed by choosing an appropriate regulator that consequently decreases the stochastic sensitivity and suppress the undesirable noise-induced oscillations. This approach was suggested for controlling the stochastic sensitivity of the equilibria of nonlinear systems [5]. In the case of having complete information, a construction of the regulator is reduced to the analysis of linear matrix equations.
Notice that, in practice, the available information on the current system state usually contains random noise or errors [29]. So, a development of the control theory for nonlinear stochastic systems with incomplete information is a very important new subject of research.
The aim of the present paper is to extend the theory of synthesis of the minimal stochastic sensitivity at equilibria to the case of incomplete information when the observations of the system states contain random errors. From a mathematical point of view, the principal difference lies in the fact that in the case of incomplete information, a synthesis of the regulator is reduced to the analysis of quadratic, but not linear, matrix equations. The main contributions of this paper are to provide a solution to the aforementioned important but challenging technical problem of minimizing the sensitivity of nonlinear stochastic systems by means of control using noisy data.
The paper is organized as follows. In Section 2, we briefly present the stochastic sensitivity function technique for the equilibria of a general nonlinear dynamic system. We devote Section 3 to the problem of the synthesis of randomly-forced system equilibria with a required dispersion. In stochastic sensitivity synthesis, we analyze an important general case with noisy observations. Main mathematical results for a general n-dimensional case are summarized in Theorem 1. In Section 4, we derive a solution of the minimization problem of the stochastic sensitivity (Theorem 2). In Section 5, for an important class of two-dimensional nonlinear oscillators, we consider details of this new control theory and give explicit formulas for the attainability sets and parameters of the stabilizing regulator. In Section 6, we apply these theoretical results to the suppression of unwanted noise-induced large-amplitude oscillations for the stochastic van der Pol system. Our general theory is finally applied to the minimization of the stochastic sensitivity in a 3D nonlinear oscillator in Section 7.
Section snippets
Stochastic sensitivity of equilibrium for nonlinear system
Consider a nonlinear stochastic system of the formwhere is a continuously differentiable vector-function, is a δ-correlated white Gaussian noise vector satisfying , and is the identity matrix. Here, is an -matrix-function characterizing the dependence of disturbances on the states, and ε is a scalar parameter of the noise intensity.
It is supposed that the corresponding deterministic system (1) (with therein) has an
Control of stochastic sensitivity by regulator with noisy data
Consider a nonlinear controlled stochastic system,with , and the control input u is synthesized by the feedback regulatorwith a constant -matrix K.
It is assumed that the corresponding deterministic system (6) (with and u=0 therein) has an equilibrium whose stability is undetermined. The synthesis of a required stochastic sensitivity function for the equilibrium of system (6), (7) with full information was studied in [5].
For the
Minimization of stochastic sensitivity
In many practically important engineering tasks, it is required to build a controller which substantially reduces the variation of random states around the equilibria, corresponding to the basic operating mode. For the fixed level of noise, the decrease of this variation means a decrease of the stochastic sensitivity (see (5)). Therefore, we study the following control problem connected with the minimization of the stochastic sensitivity.
Consider a criterion functionHere, the scalar
Stabilizing the equilibrium of a 2D nonlinear stochastic oscillator
Consider a nonlinear stochastic oscillator with control input in the form ofwhere and σ are scalar functions, ξ is white Gaussian noise satisfying , and ε is the noise intensity.
Rewrite Eq. (29) as a system:Assume that the uncontrolled deterministic system (30) (with therein) has an equilibrium . Now consider a feedback,where k1
Synthesis of the minimal stochastic sensitivity for the stochastic van der Pol oscillator
Consider the following stochastic system:where ξ is standard white Gaussian noise. The deterministic uncontrolled system (35) (with therein) is the well-known van der Pol oscillator [1], [27] with saddle-node bifurcation at . For parameters , this system exhibits (see Fig. 1a) a stable equilibrium (black circle) and a stable limit cycle (solid line) separated by an unstable limit cycle (red dashed line).
Under stochastic
Minimization of stochastic sensitivity for a 3D nonlinear stochastic oscillator
Consider a nonlinear stochastic oscillator with control input in the form ofwhere f and σ are scalar functions, is white Gaussian noise satisfying , and ε is the noise intensity. Assume that the deterministic uncontrolled oscillator (37) (with ) has an equilibrium .
Rewrite Eq. (37) asConsider a feedback with noisy observations,where are
8. Conclusion
In this paper, a stabilization problem for nonlinear stochastic system with incomplete information was considered. A new constructive method for stochastic sensitivity synthesis was suggested. Mathematically, this problem of synthesis of feedback regulator was reduced to the solution of the corresponding quadratic matrix equation for regulator׳s parameters. An analysis of the attainability of the assigned stochastic sensitivity was discussed. Explicit formulas of the regulator are given in
Acknowledgement
This research was supported by the Hong Kong Research Grant Council under the GRF Grant CityU11201414, the Government of the Russian Federation (Act 211, Contract no. 02.A03.21.0006) and RFBR (16-08-00388).
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