Minimal-order observer and output-feedback stabilization control design of stochastic nonlinear systems

1IntroductionTheproblemofestimatingtheparametersofmultiplesinusoidsinnoisehasre-ceivedconsiderableattentioninthepastthirtyyears,andalotofalgorithmshavebeenestablishedtosolvetheproblem.Amongallofthealgorithms,themaximumlikelihood(ML)estimatorisaprominentone[1],andseveralalgorithms,suchasANP[2]andIMP[3,4]arerelatedtoML.ThedrawbackoftheMLestimatorisitshighcomputationalcomplexity,sothealternatingprojection(AP)algorithm[5]wasdevelopedtomakeitsrealtimerealizationpossible.Butthefundamentaldefic…     

Global output-feedback stabilization for a class of stochastic non-minimum-phase nonlinear systems

In this paper, the problem of output-feedback stabilization is investigated for the first time for a class of stochastic nonlinear systems whose zero dynamics may be unstable. Under the assumption that the inverse dynamics of the system is stochastic input-to-state stabilizable, a stabilizing output-feedback controller is constructively designed by the integrator backstepping method together with a new reduced-order observer design and the technique of changing supply functions. It is shown that, under small-gain type conditions for small signals, the resulting closed-loop system is globally asymptotically stable in probability. The obtained results extend the existing methodology from deterministic systems to stochastic systems. An example is given to demonstrate the main features and effectiveness of the proposed output-feedback control scheme.     

一类具有未知控制方向非线性系统的输出反馈自适应控制

研究了一类控制方向未知非线性系统的输出反馈自适应镇定问题. 首先, 通过一线性状态变换, 将未知控制系数集中起来, 从而将原系统变换为适于控制设计的新系统. 然后, 分别引入状态观测器和参数估计器, 并应用积分反推和调节函数方法, 给出了输出反馈稳定控制律的构造性设计过程. 可以证明，所设计的控制器确保原系统状态渐近收敛到原点, 而其它闭环系统状态有界. 仿真结论验证了所提出方法的有效性.     

一类开关切换系统的输出反馈镇定

对一类子系统为线性的开关切换系统进行观测器设计，并给出系统动态输出反馈可镇定的充分条件。在所设计的观测器基础上，以系统状态的观测值为依据设计各子控制器和切换方案，使整个闭环系统是指数渐近稳定的。对于切换时间间隔受限制的情况，给出了相应的动态输出反馈可镇定条件和镇定控制方案。实例仿真结果表明，所给出的设计方案是可行的。     

Dynamic high-gain scaling: State and output feedback with application to systems with ISS appended dynamics driven by all States

We propose a dynamic high-gain scaling technique and solutions to coupled Lyapunov equations leading to results on state-feedback, output-feedback, and input-to-state stable (ISS) appended dynamics with nonzero gains from all states and input. The observer and controller designs have a dual architecture and utilize a single dynamic scaling. A novel procedure for designing the dynamics of the high-gain parameter is introduced based on choosing a Lyapunov function whose derivative is negative if either the high-gain parameter or its derivative is large enough (compared to functions of the states). The system is allowed to contain uncertain terms dependent on all states and uncertain appended ISS dynamics with nonlinear gains from all system states and input. In contrast, previous results require uncertainties to be bounded by a function of the output and require the appended dynamics to be ISS with respect to the output, i.e., require the gains from other states and the input to be zero. The generated control laws have an algebraically simple structure and the associated Lyapunov functions have a simple quadratic form with a scaling. The design is based on the solution of two pairs of coupled Lyapunov equations for which a constructive procedure is provided. The proposed observer/controller structure provides a globally asymptotically stabilizing output-feedback solution for the benchmark open problem proposed in our earlier work with the provision that a magnitude bound on the unknown parameter be given.     

随机非线性系统的输出反馈控制

针对满足线性增长条件的一类随机非线性系统, 本文研究了输出反馈镇定问题. 然而不同于现有的所有文献, 由于线性增长条件中含有不可量测的状态, 引入了一个待定的高增益观测器. 利用反推设计技术, 构造性地给出了一个输出反馈控制器的设计, 通过适当地选取高增益参数, 保证了闭环系统的零解是概率意义下全局渐近稳定的, 输出几乎处处调节于零.     

具有时变时滞的不确定非完整系统的输出反馈镇定

针对一类含有状态时变时滞的不确定非完整系统, 提出一种输出反馈镇定控制算法. 通过应用不连续的输入-状态变换和缩放变换, 将原始研究系统转换为更利于反馈控制器设计的新系统. 基于此系统设计状态反馈控制律, 通过构造状态观测器、利用必然等价原理给出理想的输出反馈镇定控制器. 分析表明, 所设计的控制器能够使得闭环系统的状态渐近趋于零. 最后通过仿真实例表明了所提出控制策略的有效性.

     

Adaptive observer-based control for a class of nonlinear stochastic systems

In this paper, the adaptive state estimation and state-feedback stabilization problems for a class of nonlinear stochastic systems with unknown constant parameters are studied. The sequential design methods are proposed to construct the adaptive controllers. Adaptive state and parameter estimators are designed by using a stochastic Lyapunov method and the separation theory of the design for the state-feedback gain and observer gain, which guarantees that the closed-loop system is asymptotically stable in the mean-square sense. Sufficient conditions for the existence of parameters estimator are given in terms of linear matrix inequalities. Finally, the numerical examples are provided to illustrate the feasibility of the proposed theoretical results.     

一类高阶次随机非线性系统的输出反馈镇定

针对一类高阶次随机非线性系统,研究其输出反馈镇定问题．通过选择有效的观测器和李雅普诺夫函数,所设计的光滑输出反馈控制器保证了闭环系统的平衡点是依概率全局渐近稳定的,输出几乎处处调节到零．数值仿真验证了控制方案的有效性．     

Continuous output-feedback stabilization for a class of stochastic high-order nonlinear systems

This paper is concerned with the global stabilization via output-feedback for a class of high-order stochastic nonlinear systems with unmeasurable states dependent growth and uncertain control coefficients. Indeed, there have been abundant deterministic results which recently inspired the intense investigation for their stochastic analogous. However, because of the possibility of non-unique solutions to the systems, there lack basic concepts and theorems for the problem under investigation. First of all, two stochastic stability concepts are generalized to allow the stochastic systems with more than one solution, and a key theorem is given to provide the sufficient conditions for the stochastic stabilities in a weaker sense. Then, by introducing the suitable reduced order observer and appropriate control Lyapunov functions, and by using the method of adding a power integrator, a continuous (nonsmooth) output-feedback controller is successfully designed, which guarantees that the closed-loop system is globally asymptotically stable in probability.     

Tracking control for a velocity-sensorless VTOL aircraft with delayed outputs

This paper presents a nonlinear output-feedback control method to force an underactuated vertical take-off and landing (VTOL) aircraft with delayed measurement outputs to asymptotically track a given reference trajectory. The control development is based on a delayed-output observer. The proposed observer consists of two-step observation algorithms reconstructing the system state with multi-delayed time instants. Conditions are given ensuring global exponential convergence to zero of the observation error for the given the delays in the measurements. Moreover, two low-dimensional controllers are designed separately to stabilize the two decomposed flight dynamics, and make the overall closed-loop system stable. The merits of method include its simple implementation and interesting application. Numerical simulations illustrate the effectiveness of the proposed control method.     

Local separation principle for non-linear systems

In this paper, we consider the problem of locally stabilizing non-linear systems by output feedback control. To be precise, the separation principle for non-linear systems are studied, which allows the design of the control law and the observer to be carried out independently. While the previous results assumed the existence of Lyapunov function for the observation error, we assume the existence of a state observer which asymptotically estimates the true state. As a matter of fact, it turns out that the latter condition is necessary for the former condition, but the converse is not true. The separation principle with the reduced order observer is also presented. Under the assumption that a non-linear system has asymptotically stable zero dynamics and is locally detectable, it is shown that the asymptotic output tracking by output feedback control can be achieved.     

On convergence of active disturbance rejection control for a class of uncertain stochastic nonlinear systems

In this paper, the practical mean-square convergence of active disturbance rejection control for a class of uncertain stochastic nonlinear systems modelled by the Itô-type stochastic differential equations with vast stochastic uncertainties is developed. We first design an extended state observer (ESO) to estimate both the unmeasured states and the stochastic total disturbance which includes unknown internal system dynamics, external stochastic disturbance without known statistical characteristics, unknown stochastic inverse dynamics, and uncertainty caused by the deviation of control parameter from its nominal value. The stochastic total disturbance is then cancelled (compensated) in the feedback loop. An ESO-based output-feedback control is finally designed analogously as for the system without uncertainties. The practical mean-square reference tracking and practical mean-square stability of the resulting closed-loop system are achieved. The numerical experiments are carried out to illustrate the effectiveness of the proposed approach.     

一类非线性不确定性系统基于状态观测器的输出反馈镇定控制设计

The output-feedback stabilization control problem is investigated for a class of nonlinear uncertain systems. Based on the multivariable analog of circle criterion, an observer is designed to estimate the system states and hence the dynamical equations that the estimation error satisfies are derived first. Then, by using integral backstepping approach together with completing square technique, the output-feedback stabilization control is constructively designed such that the closedloop system is asymptotically stable. Finally, an example is given to illustrate the main results of this paper.     

Output-feedback control of a class of stochastic nonlinear systems with power growth conditions

This paper investigates the problem of output-feedback stabilization for a class of stochastic nonlinear systems in which the nonlinear terms depend on unmeasurable states besides measurable input. We extend linear growth conditions to power growth conditions and reduce the control effort. By using backstepping technique, choosing a high-gain parameter, an output-feedback controller is designed to ensure the closed-loop system globally asymptotically stable in probability. The efficiency of the output-feedback controller is demonstrated by a simulation example.     

Output-feedback control of a class of high-order stochastic nonlinear systems

This article investigates the problem of output-feedback stabilisation for a class of high-order stochastic non-linear systems in which the diffusion terms depend on unmeasurable states besides the output. By introducing a new rescaling transformation, adopting an effective observer and choosing the appropriate Lyapunov function, an output-feedback controller is constructed to ensure that the equilibrium at the origin of the closed-loop system is globally asymptotically stable in probability, the output can be regulated to the origin almost surely, and the problem of inverse optimal stabilisation in probability is solved. The efficiency of the output-feedback controller is demonstrated by several simulation examples.     

Output-feedback stabilization of a class of uncertain non-minimum-phase nonlinear systems

The problem of global output-feedback stabilization for a class of nonlinear systems whose zero dynamics are not necessarily stable is addressed in this paper. It is shown that, using a novel observer design tool together with standard backstepping and small-gain techniques, it is possible to design a stabilizing output feedback controller which ensures robustness with respect to dynamic uncertainties. The proposed stabilization method generalizes existing tools in several directions. As an example, the method is applied to the stabilization of the benchmark translational oscillator/rotational actuator (TORA) using only measurements of the rotational position.     

Robust stabilization of uncertain systems by periodic feedback

An uncertain control system described by a family parametrized by an unknown parameter that takes values in a known constraint set is considered. The robust stabilization problem is defined as finding a dynamic output controller that globally stabilizes the uncertain system; that is, no matter what the parameter value chosen by ‘Nature’, the closed-loop system is globally asymptotically stable. It is shown that in the class of dynamic periodic controllers the solution to this problem exists under reasonable assumptions given for a general family of abstract non-linear models. Those assumptions, in the case of a family of linear finite-dimensional models, are equivalent to suitable stabilizability and detectability assumptions. For this case, insensitivity of the design with respect to small errors in the data is proved.     

Adaptive output-feedback stabilization of non-local hyperbolic PDEs

We address the problem of adaptive output-feedback stabilization of general first-order hyperbolic partial integro-differential equations (PIDE). Such systems are also referred to as PDEs with non-local (in space) terms. We apply control at one boundary, take measurements on the other boundary, and allow the system’s functional coefficients to be unknown. To deal with the absence of both full-state measurement and parameter knowledge, we introduce a pre-transformation (which happens to be based on backstepping) of the system into an observer canonical form. In that form, the problem of adaptive observer design becomes tractable. Both the parameter estimator and the control law employ only the input and output signals (and their histories over one unit of time). Prior to presenting the adaptive design, we present the non-adaptive/baseline controller, which is novel in its own right and facilitates the understanding of the more complex, adaptive system. The parameter estimator is of the gradient type, based on a parametric model in the form of an integral equation relating delayed values of the input and output. For the closed-loop system we establish boundedness of all signals, pointwise in space and time, and convergence of the PDE state to zero pointwise in space. We illustrate our result with a simulation.     

Adaptive output-feedback control for a class of uncertain stochastic non-linear systems with time delays

In this paper, we investigate the adaptive output-feedback stabilisation for a class of stochastic non-linear systems with time-varying time delays. First, we give some sufficient conditions to ensure the existence and uniqueness of the solution process for stochastic non-linear systems with time delays, and introduce a new stability notion and the related criterion. Then, for a class of stochastic non-linear systems with time-varying time delays, uncertain parameters in both drift and diffusion terms, and general constant virtual control coefficients, we present a systematic design procedure for a memoryless adaptive output-feedback control law by using the backstepping method. It is shown that under the control law based on a memoryless observer, the closed-loop equilibrium of interest is globally stable in probability, and moreover, the solution process can be regulated to the origin almost surely.