Stabilization Of Stochastic Nonlinear Systems Research Articles

Feedback stabilization and adaptive stabilization of stochastic nonlinear systems by the control Lyapunov function

Our aims of this paper are twofold: On one hand, we study the asymptotic stability in probability of stochastic differential system, when both the drift and diffusion terms are affine in the control. We derive sufficient conditions for the existence of control Lyapunov functions (CLFs) leading to the existence of stabilizing feedback laws which are smooth, except possibly at the equilibrium state. On the other hand, we consider the previous systems with an unknown constant parameters in the drift and introduce the concept of an adaptive CLF for stochastic system and use the stochastic version of Florchinger's control law to design an adaptive controller. In this framework, the problem of adaptive stabilization of nonlinear stochastic system is reduced to the problem of non-adaptive stabilization of a modified system.

State feedback stabilization of stochastic nonlinear systems with SiISS inverse dynamics

AbstractThis paper further considers a more general class of stochastic nonlinear systems with stochastic integral input‐to‐state stability (SiISS) inverse dynamics and drift and diffusion terms depending upon the other states besides stochastic inverse dynamics and the first state. By skillfully choosing the designed functions and the update laws of parameters, and using the important mathematical tools established in IEEE Trans. Automat. Contr. 2010; 55(2):304–320, a unifying framework of state feedback controller is proposed to guarantee that all the signals of the closed‐loop system are bounded almost surely and the states can be regulated to zero almost surely. A simulation example demonstrates the effectiveness of the control scheme. Copyright © 2011 John Wiley & Sons, Ltd.

Contribution to the stabilization of stochastic nonlinear systems with time delays

AbstractIn this paper, the problem of feedback stabilization of stochastic differential delay systems is considered. The systems under study are also nonlinear and nonaffine. By using a LaSalle-type theorem for stochastic systems, general conditions for stabilizing the closed-loop system with delays are obtained. In addition, stabilizing state feedback control laws are proposed. An example illustrates the proposed approach.

Sampled-data exponential stabilization of a class of non-linear stochastic systems

In this paper, the problem of sampled-data exponential stabilization of non-linear stochastic systems with interval parameter uncertainties is considered. The parameter uncertainties are characterized by a matrix bound which is natural in real applications and the non-linearity is assumed to satisfy the boundedness condition. By using the input delay approach, the stochastic sampled-data system is transformed into a stochastic continuous time-delay system. State-feedback and output-feedback controller designs for the non-linear stochastic systems are proposed based on the exponential stability condition such that the system is robustly exponentially stable. Finally, an illustrative example is provided to show the effectiveness of the proposed approach.

Output-feedback stabilization for stochastic nonlinear systems whose linearizations are not stabilizable

This brief paper investigates the problem of output-feedback stabilization for a class of high-order stochastic nonlinear systems which are neither necessarily feedback linearizable nor affine in the control input. Based on the ideas of the homogeneous systems theory and the adding a power integrator technique, an output-feedback controller is constructed to ensure that the equilibrium at the origin of the closed-loop system is globally asymptotically stable (GAS) in probability. The efficiency of the output-feedback controller is demonstrated by a simulation example.

Robust PDF control with guaranteed stability for non-linear stochastic systems under modelling errors

The authors consider the robust probability density function (PDF) control problem for non-linear stochastic systems. The controlled systems are described by using non-Gaussian non-linear difference equations with the un-modelled uncertainty and modelling error, as well as time delays. The non-linear multi-step-ahead predictive cost function is adopted and the relationship between the PDFs of the input and output via the uncertain mapping is established. With these formulations, the cost function can be bounded by a non-linear functional of the control input and the known bounds of the errors. Design algorithms are presented for the robust sub-optimal controller and further for the stabilisation controllers. Simulations are given to demonstrate the effectiveness of the proposed control algorithm.

Inverse optimal stabilization for stochastic nonlinear systems whose linearizations are not stabilizable

After considering the stabilization of a class of high-order stochastic nonlinear systems which are neither necessarily feedback linearizable nor affine in the control input, in this brief paper, we further address the problem of state-feedback inverse optimal stabilization in probability, i.e., our redesigned stabilizing backstepping controller is also optimal with respect to meaningful cost functionals.

Output feedback stabilization for stochastic nonlinear systems in observer canonical form with stable zero-dynamics

We study the problem of output feedback stabilization for stochastic nonlinear systems. We consider a class of stochastic nonlinear systems in observer canonical form with stable zero-dynamics. We introduce a sequence of state transformations that transform the system into a lower triangular structure that is amenable for integrator backstepping design. Then we design the output feedback controller and prove that the closed-loop system is bounded in probability. Under an infinite-horizon risk-neutral cost criterion, the controller designed can guarantee an arbitrarily small long term average cost. Furthermore, when the disturbance vector field vanishes at the origin, the closed-loop system is asymptotically stable in the large, and the risk-neutral cost is guaranteed to be zero. With special care, the controller preserves the equilibrium of the nonlinear system.

Stabilization of stochastic nonlinear systems driven by noise of unknown covariance

This paper poses and solves a new problem of stochastic (nonlinear) disturbance attenuation where the task is to make the system solution bounded by a monotone function of the supremum of the covariance of the noise. This is a natural stochastic counterpart of the problem of input-to-state stabilization in the sense of Sontag (1989). Our development starts with a set of new global stochastic Lyapunov theorems. For an exemplary class of stochastic strict-feedback systems with vanishing nonlinearities, where the equilibrium is preserved in the presence of noise, we develop an adaptive stabilization scheme (based on tuning functions) that requires no a priori knowledge of a bound on the covariance. Next, we introduce a control Lyapunov function formula for stochastic disturbance attenuation. Finally, we address optimality and solve a differential game problem with the control and the noise covariance as opposing players; for strict-feedback systems the resulting Isaacs equation has a closed-form solution.

Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance

We address the class of stochastic output-feedback nonlinear systems driven by noise whose covariance is time varying and bounded but the bound is not known a priori. This problem is analogous to deterministic disturbance attenuation problems. We first design a controller which guarantees that the solutions converge (in probability) to a residual set proportional to the unknown bound on the covariance. Then, for the case of a vanishing noise vector field, we design an adaptive controller which, besides global stability in probability, guarantees regulation of the state of the plant to zero with probability one.

Robustness of exponential stability of nonlinear stochastic systems

Robustness of exponential stability of nonlinear stochastic systems

Adaptive Stabilization of Stochastic Nonlinear Systems

The paper poses and solves the problem of adaptive stabilization of an equilibrium solution for stochastic nonlinear systems. We consider the class of strict-feedback systems driven by noise whose covariance is unknown (with no bound available to the designer) and possibly time varying. Our solution employs a refined form of backstepping with a Lyapunov function quartic in state error variables and quadratic in parameter error. The paper includes a set of new global stochastic Lyapunov and LaSalle-type theorems.

Adaptive Stabilization of Nonlinear Stochastic Systems

The purpose of this paper is to study the problem of asymptotic stabilization in probability of nonlinear stochastic differential systems with unknown parameters. With this aim, we introduce the concept of an adaptive control Lyapunov function for stochastic systems and we use the stochastic version of Artstein's theorem to design an adaptive stabilizer. In this framework the problem of adaptive stabilization of a nonlinear stochastic system is reduced to the problem of asymptotic stabilization in probability of a modified system. The design of an adaptive control Lyapunov function is illustrated by the example of adaptively quadratically stabilizable in probability stochastic differential systems.

Feedback stabilization of nonlinear stochastic systems by locally bounded controls

This paper deals with output regulation and stabilization of nonlinear control stochastic systems using Lyapunov-based methods. Sufficient conditions for global output regulation and stabilization using locally bounded state feedback are obtained. The approach used here is motivated by the work of Zhang and Hirschorn [9] in the deterministic case, that we generalize when the equation is corrupted by a multiplicative noise

Guaranteed-cost stabilization of non-linear stochastic systems

The control of uncertain non-linear discrete-time systems having stochastic cone-bounded non-linearities is considered. First, a quadratic performance bound and a guaranteed-cost optimal state feedback controller are derived. Then, an auxiliary system is introduced. It is shown that the quadratic optimal control for this auxiliary system is the same as the guaranteed-cost control for the original system, and, therefore, the existence of the infinite-horizon guaranteed-cost controller can be based on the stabilizability and observability properties of the auxiliary system. Finally, the stochastic boundedness of the controlled uncertain system is proved based on the properties of the auxiliary system. © 1998 John Wiley & Sons, Ltd.

The stability of stochastically perturbed orbital motions

When investigating the orbital stability of non-linear stochastic systems, two forms of first-approximation systems (with noise of types I and II) are considered. The P-stability of first-approximation systems is defined. A necessary and sufficient condition for P-stability is that the Lyapunov matrix differential equation should possess a periodic solution. An equivalent form is proposed for this criterion, using which one can reduce the problem of stability for stochastic systems to determining the spectral radius of a certain positive operator. When that is done, lower (upper) bounds for the spectral radius yield necessary (sufficient) conditions for stability. The possibilities of obtaining constructive estimates are demonstrated for a system with one type II noise. A parametric stability criterion, which is a stochastic analogue of the well-known Poincaré criterion, is given for a two-dimensional system (the spectral radius is found in explicit form).

Stochastic stability of non-linear systems

The investigation of motion stability of a non-linear system can be simplified by focusing on a crucial variable, such as the amplitude or the total energy of the dominant mode. For a randomly excited system, this can be accomplished using the stochastic averaging or quasi-conservative averaging procedure. It is shown that the stochastic differential equation so obtained is generally associated with singular boundaries, in the sense that either the diffusion coefficient vanishes, or the drift coefficient becomes unbounded. The condition for a dynamical system to be asymptotically stable in probability can be determined from the behaviors of sample functions at or near the two boundaries. Criteria are established to characterize various sample behaviors near each type of singular boundary. A procedure is also developed to obtain the conditions for the asymptotic stability of the statistical moments. Examples are given for illustration.

Exponential stabilization of non-linear stochastic systems

We consider the stabilization of non-linear systems whose parameters are subjected to “white noise”. For stochastic systems with non-linear feedback, we derive sufficient (frequency-domain) conditions of exponential stabilization by a controller that uses information about the system output (incomplete system state information). The problem of the stabilization of linear stochastic systems has been studied in some detail /1–3/. Yet for non-linear stochastic systems we only have general theorems that reduce the stabilization problem to finding a stochastic Lyapunov function /4, 5/. In this paper, we derive sufficient conditions of exponential stabilization by methods of the theory of absolute stochastic stability. The advantages of these methods are well-known: the specific Lyapunov function is not required, and its existence in the class of functions “quadratic form plus integrals over non-linearities” is easily checked /6/. The latest results of this theory for stochastic systems /7/ make it possible to solve the stabilization problem for a wide class of non-linear systems with parametric disturbances.

Application of Yakubovich criterion for stability of nonlinear stochastic systems

Frequency domain criteria for stability of a class of multiloop nonlinear stochastic systems are presented in this paper. The Yakubovich criterion is used in the investigation of asymptotic stability with probability one. The obtained results are illustrated by the example.

On the stability of non-linear stochastic systems

Abstract The results give stochastic analogues for third-, fourth-, fifth- and sixth-order nonlinear deterministic systems. The construction of a V-function when exponentially stable w. p. 1, satisfies an exponential inequality along the process, ℒ V(x)≤−D V(x). Here we modify Kushner's method to construct a V-function which satisfies such an inequality.