Stochastic Control Lyapunov Function Research Articles

Design stabilisers for multi-input affine control stochastic systems via stochastic control Lyapunov functions

This paper studies the problem of state feedback stabilisation, in the sense of a weak solution, for a class of multi-input affine control stochastic systems whose drift and diffusion terms are dependent on the control and for which classical methods are not applicable. Based on the generalised stochastic Lyapunov theorem and on the technique of stochastic control Lyapunov functions, sufficient conditions for the existence of a continuous stabilising feedback control are proposed, and a state feedback controller can be developed to guarantee the closed-loop system globally asymptotically stable in probability. This work generalises previous results on the stabilisation of single-input affine control stochastic systems. The obtained results are illustrated by a numerical example.

On the simultaneous stabilization of stochastic nonlinear systems

SummaryIn this paper, the problem of simultaneous stabilization in probability by state feedback is investigated for a class of stochastic nonlinear systems whose drift and diffusion terms are dependent on the control and for which classical methods are not applicable. Under the assumption that a collection of stochastic control Lyapunov functions (SCLFs) is known and based on the generalized stochastic Lyapunov theorem, we derive sufficient conditions for the simultaneous stabilization in probability by a continuous state feedback controller that we explicitly compute. We also derive a necessary condition when the system coefficients satisfy some regularity conditions. This work generalizes previous results on the simultaneous stabilization of stochastic nonlinear systems. The obtained results are illustrated by a numerical example.

Continuous simultaneous stabilization of single‐input nonlinear stochastic systems

AbstractThis paper investigates the simultaneous stabilization of a collection of continuous single‐input non‐linear stochastic systems, with coefficients that are not necessarily locally Lipschitz. A sufficient condition for the existence of a continuous simultaneously stabilizing feedback control is proposed — it is based on the generalized stochastic Lyapunov theorem and on the technique of stochastic control Lyapunov functions. This condition is also necessary, provided that the system's coefficients satisfy some regularity conditions. Moreover, the proposed feedback can be chosen to be bounded under the assumption that appropriate control Lyapunov functions are known. All the proposed simultaneously stabilizing state feedback controllers are explicitly constructed. Finally, two simulation examples are provided to demonstrate the effectiveness of the proposed approach.

Universal Feedback Controllers and Inverse Optimality for Nonlinear Stochastic Systems

Abstract In this paper, we develop a constructive finite time stabilizing feedback control law for stochastic dynamical systems driven by Wiener processes based on the existence of a stochastic control Lyapunov function. In addition, we present necessary and sufficient conditions for continuity of such controllers. Moreover, using stochastic control Lyapunov functions, we construct a universal inverse optimal feedback control law for nonlinear stochastic dynamical systems that possess guaranteed gain and sector margins. An illustrative numerical example involving the control of thermoacoustic instabilities in combustion processes is presented to demonstrate the efficacy of the proposed framework.

Inverse optimal control for asymptotic trajectory tracking of discrete‐time stochastic nonlinear systems in block controllable form

SummaryThis paper concerns an inverse optimal control–based trajectory tracking of discrete‐time stochastic nonlinear systems. It is assumed that the nonlinear system can be transformed to the so called nonlinear block controllable form. Additionally, the synthesized control law minimizes a cost functional, which is posteriori determined. In contrast to the optimal control technique, this scheme avoids to solve the stochastic Hamilton‐Jacobi‐Bellman equation, which is not an easy task. Based on a discrete‐time stochastic control Lyapunov function, the proposed optimal controller is analyzed. The proposed approach is applied successfully to the two degrees‐of‐freedom helicopter with uncertainties in real time.

Global Asymptotic Stabilization of Nonlinear Deterministic Systems Using Wiener Processes

The objective of this study is to present a noise-assisted stabilization method and a constructive method for designing a stabilizing controller. We investigate the stabilization of a deterministic nonlinear affine system by using a controller with a Wiener process. Our approach focuses on the global asymptotic stabilization of nonholonomic systems. The proposed method for designing a stabilizing controller is based on the notion of a stochastic control Lyapunov function. We show that if there exists a stochastic control Lyapunov function with small control property, the designed controller globally asymptotically stabilizes a given system with probability one.

Linearly Solvable Stochastic Control Lyapunov Functions

This paper presents a new method for synthesizing stochastic control Lyapunov functions for a class of nonlinear stochastic control systems. The technique relies on a transformation of the classical nonlinear Hamilton-Jacobi-Bellman partial differential equation to a linear partial differential equation for a class of problems with a particular constraint on the stochastic forcing. This linear partial differential equation can then be relaxed to a linear differential inclusion, allowing for relaxed solutions to be generated using sum of squares programming. The resulting relaxed solutions are in fact viscosity super/subsolutions, and by the maximum principle are pointwise upper and lower bounds to the underlying value function, even for coarse polynomial approximations. Furthermore, the pointwise upper bound is shown to be a stochastic control Lyapunov function, yielding a method for generating nonlinear controllers with pointwise bounded distance from the optimal cost when using the optimal controller. These approximate solutions may be computed with non-increasing error via a hierarchy of semidefinite optimization problems. Finally, this paper develops a-priori bounds on trajectory suboptimality when using these approximate value functions, as well as demonstrates that these methods, and bounds, can be applied to a more general class of nonlinear systems not obeying the constraint on stochastic forcing. Simulated examples illustrate the methodology.

Stochastic Asymptotic Stabilizers for Deterministic Input-Affine Systems Based on Stochastic Control Lyapunov Functions

In this paper, a stochastic asymptotic stabilization method is proposed for deterministic input-affine control systems, which are randomized by including Gaussian white noises in control inputs. The sufficient condition is derived for the diffucion coefficients so that there exist stochastic control Lyapunov functions for the systems. To illustrate the usefulness of the sufficient condition, the authors propose the stochastic continuous feedback law, which makes the origin of the Brockett integrator become globally asymptotically stable in probability.

Adaptive Neural Network Output-feedback Stabilization for a Class of Stochastic Nonlinear Strict-feedback Systems

Neural network(NN) control scheme is first introduced into a class of stochastic nonlinear strict-feedback systems.Based on the well known backstepping method and the technique of nonlinear observer design,a suitable stochastic control Lyapunov function is then proposed to construct an adaptive neural network output-feedback controller.Under some conditions,it is shown that the equilibrium of the closed-loop system is stable in probability.A simulation example is given to illustrate the effectiveness of the proposed control scheme.

Stochastic nonlinear stabilization — II: Inverse optimality

After considering the stabilization of a specific class of stochastic nonlinear systems in a companion paper, in this second part, we address the classical question of when is a stabilizing (in probability) controller optimal and show that for every system with a stochastic control Lyapunov function it is possible to construct a controller which is optimal with respect to a meaningful cost functional. Then we return to the problem from Part I and design an optimal backstepping controller whose cost functional includes penalty on control effort and which has an infinite gain margin.