Output‐feedback control of a class of stochastic nonlinear systems with linearly bounded unmeasurable states

In this paper, the problems of stochastic disturbance attenuation and asymptotic stabilization via output feedback are investigated for a class of stochastic nonlinear systems with linearly bounded unmeasurable states. For the first problem, under the condition that the stochastic inverse dynamics are generalized stochastic input‐to‐state stable, a linear output‐feedback controller is explicitly constructed to make the closed‐loop system noise‐to‐state stable. For the second problem, under the conditions that the stochastic inverse dynamics are stochastic input‐to‐state stable and the intensity of noise is known to be a unit matrix, a linear output‐feedback controller is explicitly constructed to make the closed‐loop system globally asymptotically stable in probability. Using a feedback domination design method, we construct these two controllers in a unified way. Copyright © 2007 John Wiley & Sons, Ltd.     

Time‐Varying Stabilizers for Stochastic Systems with no Unforced Dynamics

This paper is concerned with the stabilizability of nonlinear stochastic systems with no unforced dynamics. Sufficient conditions allowing to design explicitly time–varying feedback laws which render such systems asymptotically stable in probability are given. The techniques used in this work involve the stochastic Lyapunov analysis combined with the stochastic version of the La Salle invariance principle. The interest of our results is that the systems considered in the present paper cannot in general be stabilized via time‐invariant feedback laws.     

Feedback Stabilization of a Torque Controlled Rigid Robot Corrupted by Noise

Thepurpose of this paper is to study the stabilizability of a classof nonlinear control stochastic differential systems the deterministicpart of which reduces to the equations of a torque controlledrigid robot. In fact, we derive sufficient conditions for theexistence of stabilizing feedback laws which render the equilibriumsolution of the closed-loop system asymptotically stable in probability.     

Bounded smooth state feedback and a global separation principle for non-affine nonlinear systems

This paper develops sufficient conditions for a general nonlinear control system to be locally (resp. globally) asymptotically stabilizable via smooth state feedback. In particular, it is shown that as in the case of affine systems, this is possible if the unforced dynamic system of ∑₁ is Lyapunov stable and appropriate controllability-like rank conditions are satisfied. Our results incorporate a series of well-known stabilization theorems proposed in the literature for affine control systems and extend them to nonaffine nonlinear control systems.     

Nonlinear–nonquadratic optimal and inverse optimal control for stochastic dynamical systems

In this paper, we develop a unified framework to address the problem of optimal nonlinear analysis and feedback control for nonlinear stochastic dynamical systems. Specifically, we provide a simplified and tutorial framework for stochastic optimal control and focus on connections between stochastic Lyapunov theory and stochastic Hamilton–Jacobi–Bellman theory. In particular, we show that asymptotic stability in probability of the closed‐loop nonlinear system is guaranteed by means of a Lyapunov function that can clearly be seen to be the solution to the steady‐state form of the stochastic Hamilton–Jacobi–Bellman equation and, hence, guaranteeing both stochastic stability and optimality. In addition, we develop optimal feedback controllers for affine nonlinear systems using an inverse optimality framework tailored to the stochastic stabilization problem. These results are then used to provide extensions of the nonlinear feedback controllers obtained in the literature that minimize general polynomial and multilinear performance criteria. Copyright © 2017 John Wiley & Sons, Ltd.     

Stochastic differential games and inverse optimal control and stopper policies

In this paper, we consider a two-player stochastic differential game problem over an infinite time horizon where the players invoke controller and stopper strategies on a nonlinear stochastic differential game problem driven by Brownian motion. The optimal strategies for the two players are given explicitly by exploiting connections between stochastic Lyapunov stability theory and stochastic Hamilton–Jacobi–Isaacs theory. In particular, we show that asymptotic stability in probability of the differential game problem is guaranteed by means of a Lyapunov function which can clearly be seen to be the solution to the steady-state form of the stochastic Hamilton–Jacobi–Isaacs equation, and hence, guaranteeing both stochastic stability and optimality of the closed-loop control and stopper policies. In addition, we develop optimal feedback controller and stopper policies for affine nonlinear systems using an inverse optimality framework tailored to the stochastic differential game problem. These results are then used to provide extensions of the linear feedback controller and stopper policies obtained in the literature to nonlinear feedback controllers and stoppers that minimise and maximise general polynomial and multilinear performance criteria.     

On the simultaneous stabilization of stochastic nonlinear systems

In 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.     

State‐feedback stabilization for high‐order stochastic nonlinear systems with stochastic inverse dynamics

For a class of high‐order stochastic nonlinear systems with stochastic inverse dynamics which are neither necessarily feedback linearizable nor affine in the control input, this paper investigates the problem of state‐feedback stabilization for the first time. Under some weaker assumptions, a smooth state‐feedback controller is designed, which ensures that the closed‐loop system has an almost surely unique solution on [0, ∞), the equilibrium at the origin of the closed‐loop system is globally asymptotically stable in probability, and the states can be regulated to the origin almost surely. A simulation example demonstrates the control scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

Finite‐time stabilization of stochastic nonlinear systems with SiISS inverse dynamics

This paper investigates the finite‐time control problem for a class of stochastic nonlinear systems with stochastic integral input‐to‐state stablility (SiISS) inverse dynamics. Motivated by finite‐time stochastic input‐to‐state stability and the concept of SiISS using Lyapunov functions, a novel finite‐time SiISS using Lyapunov functions is introduced firstly. Then, by adopting this novel finite‐time SiISS small‐gain arguments, using the backstepping technique and stochastic finite‐time stability theory, a systematic design and analysis algorithm is proposed. Given the control laws that guarantee global stability in probability or asymptotic stability in probability, our design algorithm presents a state‐feedback controller that can ensure the solution of the closed‐loop system to be finite‐time stable in probability. Finally, a simulation example is given to demonstrate the effectiveness of the proposed control scheme. Copyright © 2017 John Wiley & Sons, Ltd.     

Active disturbance rejection control approach to output‐feedback stabilization of lower triangular nonlinear systems with stochastic uncertainty

In this paper, we apply the active disturbance rejection control approach to output‐feedback stabilization for uncertain lower triangular nonlinear systems with stochastic inverse dynamics and stochastic disturbance. We first design an extended state observer (ESO) to estimate both unmeasured states and stochastic total disturbance that includes unknown system dynamics, unknown stochastic inverse dynamics, external stochastic disturbance, and uncertainty caused by the deviation of control parameter from its nominal value. The stochastic total disturbance is then compensated in the feedback loop. The constant gain and the time‐varying gain are used in ESO design separately. The mean square practical stability for the closed‐loop system with constant gain ESO and the mean square asymptotic stability with time‐varying gain ESO are developed, respectively. Some numerical simulations are presented to demonstrate the effectiveness of the proposed output‐feedback control scheme. Copyright © 2016 John Wiley & Sons, Ltd.     

State‐feedback stabilization for stochastic high‐order nonlinear systems with SISS inverse dynamics

This paper investigates the problem of state‐feedback control for a class of stochastic high‐order nonlinear systems with stochastic inverse dynamics. Under the assumption that the inverse dynamics of the subsystem are stochastic input‐to‐state stable (SISS), by extending through adding a power integrator technique, choosing an appropriate Lyapunov function and using the idea of changing supply function, a smooth state‐feedback controller is explicitly constructed to render the system globally asymptotically stable in probability and the states can be regulated to the origin. A simulation example is provided to show the effectiveness of the proposed scheme. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Globally Stabilizing a Class of Underactuated Mechanical Systems on the Basis of Finite-Time Stabilizing Observer

Globally exponentially stabilizing a class of underactuated mechanical systems (UMS) with nonaffine nonlinear dynamics is investigated in this paper. The considered UMS has a nonaffine nonlinear subsystem that can be globally asymptotically stabilized by saturated feedbacks, but the saturated feedback cannot be analytically expressed in closed-form. This obstacle limits the real-time applications of most controllers presented in literatures. In this paper, a hybrid feedback strategy is presented to globally exponentially stabilize the UMS with nonaffine and strict-feedback canonical forms. The hybrid feedback strategy is characterized by the composition of partial states feedback and partial virtual outputs feedback based on a higher-order finite-time stabilizing observer. The presented hybrid feedback controller can be synthesized by applying Lyapunov stability theory. Some numerical simulations associated with two underactuated nonlinear systems, the Acrobot system and the Inertia-Wheel-Pendulum (IWP) system, are employed to demonstrate the effectiveness of the proposed controller. The presented control strategy can be applied in real time, thus providing a new feasible dynamic model other than the differential flatness systems for synthesizing the mechanical systems of general underactuated legged robots.     

Feedback stabilization of multi‐DOF nonlinear stochastic Markovian jump systems

A feedback control strategy is designed to asymptotically stabilize a multi‐degree‐of‐freedom (DOF) nonlinear stochastic systems undergoing Markovian jumps. First, a class of hybrid nonlinear stochastic systems with Markovian jumps is reduced to a one‐dimensional averaged Itô stochastic differential equation for controlled total energy. Second, the optimal control law is deduced by applying the dynamical programming principle to the ergodic control problem of the averaged systems with an undetermined cost function. Third, the cost function is determined by the demand of stabilizing the averaged systems. A Lyapunov exponent is introduced to analyze approximately the asymptotic stability with probability one of the originally controlled systems. To illustrate the application of the present method, an example of stochastically excited two coupled nonlinear oscillators with Markovian jumps is worked out in detail.     

Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems

In this paper, the problem of decentralized adaptive output-feedback stabilization is investigated for large-scale stochastic nonlinear systems with three types of uncertainties, including parametric uncertainties, nonlinear uncertain interactions and stochastic inverse dynamics. Under the assumption that the inverse dynamics of the subsystems are stochastic input-to-state stable, an adaptive output-feedback controller is constructively designed by the backstepping method. It is shown that under some general conditions, the closed-loop system trajectories are bounded in probability and the outputs can be regulated into a small neighborhood of the origin in probability. In addition, the equilibrium of interest is globally stable in probability and the outputs can be regulated to the origin almost surely when the drift and diffusion vector fields vanish at the origin. The contributions of the work are characterized by the following novel features: (1) even for centralized single-input single-output systems, this paper presents a first result in stochastic, nonlinear, adaptive, output-feedback asymptotic stabilization; (2) the methodology previously developed for deterministic large-scale systems is generalized to stochastic ones. At the same time, novel small-gain conditions for small signals are identified in the setting of stochastic systems design; (3) both drift and diffusion vector fields are allowed to be dependent not only on the measurable outputs but some unmeasurable states; (4) parameter update laws are used to counteract the parametric uncertainty existing in both drift and diffusion vector fields, which may appear nonlinearly; (5) the concept of stochastic input-to-state stability and the method of changing supply functions are adapted, for the first time, to deal with stochastic and nonlinear inverse dynamics in the context of decentralized control.     

Asymptotic stability in probability and stabilization for a class of discrete‐time stochastic systems

This paper investigates asymptotic stability in probability and stabilization designs of discrete‐time stochastic systems with state‐dependent noise perturbations. Our work begins with a lemma on a special discrete‐time stochastic system for which almost all of its sample paths starting from a nonzero initial value will never reach the origin subsequently. This motivates us to deal with the asymptotic stability in probability of discrete‐time stochastic systems. A stochastic Lyapunov theorem on asymptotic stability in probability is proved by means of the convergence theorem of supermartingale. An example is given to show the difference between asymptotic stability in probability and almost surely asymptotic stability. Based on the stochastic Lyapunov theorem, the problem of asymptotic stabilization for discrete‐time stochastic control systems is considered. Some sufficient conditions are proposed and applied for constructing asymptotically stable feedback controllers. Copyright © 2014 John Wiley & Sons, Ltd.     

Further results on global stabilization of discrete nonlinear systems

In this paper we show how recent advances [11,12] in global stabilization of continuous-time nonaffine nonlinear systems with stable free dynamics, can be nontrivially generalized to their discrete-time counterparts, by means of passivity and bounded state feedback. As a consequence, global stabilization results of discrete-time nonlinear systems with triangular structure are established.     

Active disturbance rejection control to MIMO nonlinear systems with stochastic uncertainties: approximate decoupling and output-feedback stabilisation

ABSTRACT

In this paper, we apply the active disturbance rejection control, an emerging control technology, to output-feedback stabilisation for a class of uncertain multi-input multi-output nonlinear systems with vast stochastic uncertainties. Two types of extended state observers (ESO) are designed to estimate both unmeasured states and stochastic total disturbance which includes unknown system dynamics, unknown stochastic inverse dynamics, external stochastic disturbance without requiring the statistical characteristics, uncertain nonlinear interactions between subsystems, and uncertainties caused by the deviation of control parameters from their nominal values. The estimations decouple approximately the system after cancelling stochastic total disturbance in the feedback loop. As a result, we are able to design an ESO-based stabilising output-feedback and prove the practical mean square stability for the closed-loop system with constant gain ESO and the asymptotic mean square stability with time-varying gain ESO, respectively. Some numerical simulations are presented to demonstrate the effectiveness of the proposed output-feedback control scheme.     

Adaptive Stabilization for a Class of Stochastic Nonlinear Systems with Prandtl‐Ishlinskii Hysteresis

This paper deals with a class of stochastic nonlinear systems with unknown hysteresis. A stochastic Lyapunov method is applied for systems in strict‐feedback form driven by unknown Prandtl‐Ishlinskii hysteresis and Wiener noises of unknown covariance. An adaptive controller is obtained which guarantees the global asymptotic stabilization in probability. Simulation results are provided to illustrate the effectiveness of the proposed approach.     

一类上三角随机非线性系统的输出反馈控制

针对一类上三角随机非线性系统的输出反馈控制问题,首先利用反推技术,为其对应的标称系统设计稳定的输出反馈控制器;然后利用低增益齐次占优技术,为整个系统设计输出反馈控制器.所设计的控制器能保证闭环系统的平衡点为依概率全局渐近稳定的,并将低增益齐次占优技术推广到了随机系统,首次解决了一类上三角随机系统的镇定问题.最后通过数值仿真验证了所提出控制方案的有效性.     

Input saturation and global stabilization of nonlinear systems viastate and output feedback

For multi-input multi-output nonlinear systems whose free dynamics are Lyapunov stable, the author shows how the problem of global stabilization via dynamic output feedback can be solved by using the technique of input saturation. The power of this technique is also illustrated by solving the problem of global stabilization via bounded state feedback for affine nonlinear systems with stable unforced dynamics. Analogous results are established for discrete-time nonlinear systems