Adaptive H∞ Control of Piecewise‐Linear Systems with Parametric Uncertainties and External Disturbances

In this paper, we study the adaptive H_∞ control problem for piecewise linear systems with parametric uncertainties and external disturbances. Motivated by the vector projection technique and dissipation theory, we propose a piecewise H_∞ controller and its associated projection‐type parameter adaptation law, which are capable of guaranteeing the system stability and minimizing the effect of disturbances. Moreover, the common quadratic Lyapunov function method is used such that all of the synthesis conditions are formulated as linear matrix inequities and thus can be solved efficiently. Finally, a numerical example is provided to illustrate the results.     

Stability and stabilization of piecewise‐affine slab systems subject to Wiener process noise

The main contribution of this paper is to propose a convex formulation of sufficient conditions for both stability analysis and synthesis of stabilizing controllers for stochastic piecewise affine (PWA) systems with multiplicative noise. One of the main difficulties in PWA systems is the fact that the affine terms in the dynamics make it extremely difficult to formulate the synthesis problem as a convex optimization or even convex feasibility program. The presence of multiplicative noise modeled as a Wiener process adds an additional level of difficulty to the analysis and synthesis procedures. Sufficient conditions for stability of stochastic PWA slab systems in the mean square sense are developed first using a stochastic globally quadratic Lyapunov function. Second, PWA state feedback controllers are designed such that the closed‐loop system is stochastically exponentially mean square stable. The conditions for both stability and stabilization are formulated as LMIs, which can then be solved efficiently using currently available software packages. A numerical example shows the effectiveness of the approach. Copyright © 2013 John Wiley & Sons, Ltd.     

基于异步观测器的离散分段仿射系统控制: LMI方法

基于分段二次Lyapunov函数为离散分段仿射系统提出了一种基于观测器的控制律. 考虑的主要问题是原系统当前所在作用域未知, 且无法根据测量输出推断. 通过将凸多面体作用域用椭球体外逼近, 并将矩阵等式约束用奇异值分解技术予以处理, 所提控制方法能够被转化为线性矩阵不等式(LMI)描述, 比现有的只能转化为双线性矩阵不等式的方法更容易求解. 最后, 将所提方法应用到混沌系统控制.     

不确定离散时滞分段系统的广义H2弹性控制

针对一类不确定离散时滞分段系统,研究了广义H2稳定性分析和带有时滞的状态反馈弹性控制器设计问题.通过构造适当的离散分段二次李亚普诺夫函数,利用分段二次李亚普诺夫稳定性理论,给出了对于所有的容许参数不确定性,闭环系统是广义H2稳定的充分条件;在此基础上,基于线性矩阵不等式(LMI)处理方法,提出了带有时滞的状态反馈弹性控制器增益阵的设计方案.仿真实例验证了所提方法的有效性.     

Asymptotic stability and disturbance attenuation properties for a class of networked control systems

In this paper, stability and disturbance attenuation issues for a class of Networked Control Systems (NCSs) under uncertain access delay and packet dropout effects are considered. Our aim is to find conditions on the delay and packet dropout rate, under which the system stability and H∞ disturbance attenuation properties are preserved to a desired level. The basic idea in this paper is to formulate such Networked Control System as a discrete-time switched system. Then the NCSs’ stability and performance problems can be reduced to the corresponding problems for switched systems, which have been studied for decades and for which a number of results are available in the literature. The techniques in this paper are based on recent progress in the discrete-time switched systems and piecewise Lyapunov functions.     

Asymptotic stability and disturbance attenuation properties for a class of networked control systems

In this paper, stability and disturbance attenuation issues for a class of Networked Control Systems (NCSs) under uncertain access delay and packet dropout effects are considered. Our aim is to find conditions on the delay and packet dropout rate, under which the system stability and H∞ disturbance attenuation properties are preserved to a desired level. The basic idea in this paper is to formulate such Networked Control System as a discrete-time switched system. Then the NCSs’ stability and performance problems can be reduced to the corresponding problems for switched systems, which have been studied for decades and for which a number of results are available in the literature. The techniques in this paper are based on recent progress in the discrete-time switched systems and piecewise Lyapunov functions.     

多面体上彷射系统可达性问题研究

研究n维多面体上自治仿射系统的可达性问题. 目的在于得到多面体上最大正不变集和每个极大面的反向可达集(吸引域). 研究最大稳定不变仿射子空间, 并给出不变集和吸引域的性质, 最后通过分割算法确定两者的边界.     

Global Practical Stabilization of Discrete-Time Switched Affine Systems via a General Quadratic Lyapunov Function and a Decentralized Ellipsoid

This paper addresses the problem of global practical stabilization of discrete-time switched affine systems via state-dependent switching rules. Several attempts have been made to solve this problem via different types of a common quadratic Lyapunov function and an ellipsoid. These classical results require either the quadratic Lyapunov function or the employed ellipsoid to be of the centralized type. In some cases, the ellipsoids are defined dependently as the level sets of a decentralized Lyapunov function. In this paper, we extend the existing results by the simultaneous use of a general decentralized Lyapunov function and a decentralized ellipsoid parameterized independently. The proposed conditions provide less conservative results than existing works in the sense of the ultimate invariant set of attraction size. Two different approaches are proposed to extract the ultimate invariant set of attraction with a minimum size, i.e., a purely numerical method and a numerical-analytical one. In the former, both invariant and attractiveness conditions are imposed to extract the final set of matrix inequalities. The latter is established on a principle that the attractiveness of a set implies its invariance. Thus, the stability conditions are derived based on only the attractiveness property as a set of matrix inequalities with a smaller dimension. Illustrative examples are presented to prove the satisfactory operation of the proposed stabilization methods.      

LMI-based robust control of uncertain discrete-time piecewise affine systems

The main contribution of this paper is to present stability synthesis results for discrete-time piecewise affine (PWA) systems with polytopic time-varying uncertainties and for discrete-time PWA systems with norm-bounded uncertainties respectively. The basic idea of the proposed approaches is to construct piecewise-quadratic (PWQ) Lyapunov functions to guarantee the stability of the closed-loop systems. The partition information of the PWA systems is taken into account and each polytopic operating region is outer approximated by an ellipsoid, then sufficient conditions for the robust stabilization are derived and expressed as a set of linear matrix inequalities (LMIs). Two examples are given to illustrate the proposed theoretical results.     

Robust Sliding Mode Control Design for Mismatched Uncertain Systems with a Performance

This paper presents an optimal design of sliding mode controller with a constraint for a general class of uncertain systems. The method proposed here takes into account nonmatching disturbance signals and nonmatching system model uncertainties. Without requiring any transformation or restriction on the input matrix, our approach leads to a straightforward design which can be easily solved by using linear matrix inequality (LMI) technique. By solving the problem of optimal sliding motion design with some quadratic constraints based on LMIs, a mixed approach is presented to reduce the effect of uncertainties and disturbances on the sliding motion. Finally, we give simulation results for controlling a chemical reactor to illustrate the applicability of the proposed scheme for uncertain time‐delay systems or mismatched uncertain systems.     

Robust output‐feedback control of state‐multiplicative noisy discrete‐time systems

Linear discrete‐time systems with stochastic and deterministic polytopic type uncertainties in their state‐space model are considered. A dynamic output‐feedback controller is obtained via a new approach that allows a derivation of a controller in spite of parameter uncertainty. In the proposed approach, the system is described via a difference equation and an augmented system is then used to obtain the output‐feedback controller parameters. The controller is obtained without assuming a specific structure to the quadratic Lyapunov function, and it is the first time that an output‐feedback controller is obtained for robust state‐multiplicative systems. The controller minimizes the stochastic L₂‐gain of the closed‐loop where a cost function is defined to be the expected value of the standard performance index with respect to the stochastic uncertainty. Two examples are given where the second of which demonstrates the applicability of our theory to a robot manipulator system. Copyright © 2016 John Wiley & Sons, Ltd.     

Pre‐filtering and post‐filtering in gain‐scheduled output‐feedback control

Linear matrix inequality (LMI) design conditions for gain‐scheduled output‐feedback control rely on assumptions constraining either system or controller matrices. Throughout the literature, it is common practice to avoid imposing restrictive assumptions on the controller, which may appear undesirable, in favor of state augmentations via pre‐filtering and post‐filtering to construct auxiliary augmented systems that comply with the alternative assumptions on the system matrices. This technique brings in the additional cost of increased state dimensions of the resulting gain‐scheduled output‐feedback controllers. In this paper, we explore the interplay and inherent trade‐offs between state augmentation, controller structure, and performance. We revisit LMI design conditions for quadratic output‐feedback control and demonstrate that state augmentation via pre‐filtering and post‐filtering in order to avoid constraints on the controller matrices is never advantageous even without taking into account the added complexity and propensity for numerical issues associated with state augmentation. As an additional contribution, we extend this observation to recently introduced modified LMI conditions allowing combined – however less restrictive – assumptions on system and controller matrices. Copyright © 2016 John Wiley & Sons, Ltd.     

Adaptive Decentralized NN Control of Nonlinear Interconnected Time‐Delay Systems with Input Saturation

In this paper, a novel decentralized adaptive neural control scheme is proposed for a class of interconnected large‐scale uncertain nonlinear time‐delay systems with input saturation. Radial basis function (RBF) neural networks (NNs) are used to tackle unknown nonlinear functions. Then, the decentralized adaptive NN tracking controller is constructed by combining Lyapunov–Krasovskii functions and the dynamic surface control (DSC) technique, along with the minimal‐learning‐parameters (MLP) algorithm. The stability analysis subject to the effect of input saturation constraints are conducted with the help of an auxiliary design system based on the Lyapunov–Krasovskii method. The proposed controller guarantees uniform ultimate boundedness (UUB) of all of the signals in the closed‐loop large‐scale system, while the tracking errors converge to a small neighborhood around the origin. An advantage of the proposed control scheme lies in the number of adaptive parameters of the whole system being reduced to one and in the solution of the three problems of “computational explosion,” “dimension curse,” and “controller singularity”. Finally, simulation results along with comparisons are presented to demonstrate the advantages, effectiveness, and performance of the proposed scheme.     

Parameterized LMIs for robust and state feedback control of continuous‐time polytopic systems

This paper presents new extended linear matrix inequality (LMI) characterizations for the synthesis of robust and state feedback controllers for continuous‐time linear time‐invariant systems with polytopic uncertainty. Based on a suitable change of variables and the Elimination Lemma, the proposed robust control design techniques are stated as extended LMI conditions parameterized in terms of 2 scalar parameters. One parameter is shown to belong to a bounded domain, thus limiting the scalar search domain. For the other parameter, a bounded subset is provided from numerical experiments. The benefits of the methodology are illustrated through numerical simulations performed on an uncertain model borrowed from the literature. It is shown that the proposed LMI relaxations can provide less conservative results with fewer scalar searches than some existing methods in the literature.     

基于LMI方法的离散分段线性系统静态输出反馈控制

研究了离散分段线性系统静态输出反馈控制问题. 基于分段二次李亚普诺夫函数, 给出了静态输出反馈镇定控制器综合的新的充分的线性矩阵不等式条件. 同时, 通过使用Finsler引理, 引入一组具有特殊结构的松弛变量以减少设计保守性. 与以后方法相比, 所提出的方法具有好的性能, 并且在已有方法失效的情况下仍可用. 文中还将此方法推广到H无穷控制. 最后给出三个例子说明方法的有效性.     

Stability of sampled-data piecewise affine systems: A time-delay approach

This paper addresses the stability analysis of sampled-data piecewise-affine (PWA) systems consisting of a continuous-time plant in feedback connection with a discrete-time emulation of a continuous-time state feedback controller. The sampled-data system is first considered as a continuous-time system with a variable time delay. Conditions under which the trajectories of the sampled-data closed-loop system will converge to an attracting invariant set are then presented. It is also shown that when the sampling period converges to zero, these conditions coincide with sufficient conditions for non-fragility of the stabilizing continuous-time PWA state feedback controller. The results are successfully applied to a helicopter example.     

Non‐Fragile Observer‐Based Control for Uncertain Neutral‐Type Systems via Sliding Mode Technique

This paper is concerned with the problem of observer‐based control for a class of uncertain neutral‐type systems subjected to external disturbance by utilizing sliding mode technique. A novel sliding mode control (SMC) strategy is proposed based on the state estimate and the output. A new sufficient condition of robust asymptotic stability with disturbance attenuation level for the overall systems composed of the original system and error system in the sliding mode is derived in terms of a linear matrix inequality (LMI). Then, a new adaptive controller is designed to guarantee the reachability of the predefined sliding surface in finite‐time. Finally, numerical examples are provided to verify the effectiveness of the proposed method.     

A unified dissipativity approach for stability analysis of piecewise smooth systems

The main objective of this paper is to present a unified dissipativity approach for stability analysis of piecewise smooth (PWS) systems with continuous and discontinuous vector fields. The Filippov definition is considered for the solution of these systems. Using the concept of generalized gradients for nonsmooth functions, sufficient conditions for the stability of a PWS system are formulated based on Lyapunov theory. The importance of the proposed approach is that it does not need any a-priori information about attractive sliding modes on switching surfaces, which is in general difficult to obtain. A section on application of the main results to piecewise affine (PWA) systems followed by a section with extensive examples clearly show the usefulness of the proposed unified methodology. In particular, we present an example with a stable sliding mode where the proposed method works and previously suggested methods fail.     

Adaptive Mittag‐Leffler Stabilization of a Class of Fractional Order Uncertain Nonlinear Systems

This paper deals with the stabilization of a class of commensurate fractional order uncertain nonlinear systems. The fractional order system concerned is of the strict‐feedback form with uncertain nonlinearity. An adaptive control scheme combined with fractional order update laws is proposed by extending classical backstepping control to fractional order backstepping scheme. The asymptotic stability of the closed‐loop system is guaranteed under the construction of fractional Lyapunov functions in the sense of generalized Mittag‐Leffler stability. The fractional order nonlinear system investigated can be stabilized asymptotically globally in presence of arbitrary uncertainty. Finally illustrative examples and numerical simulations are performed to verify the effectiveness of the proposed control scheme.     

Dynamic event‐triggered control for linear time‐invariant systems with ‐gain performance

In this paper, a novel dynamic event‐triggered control scheme is presented for linear time‐invariant systems. Under this control scheme, criteria that guarantee the asymptotic stability and the ‐stability are derived, by which the triggered parameters and the feedback gain can be codesigned. The stability criteria are derived by using Lyapunov‐based analysis tools, and a new Lyapunov‐Krasovskii functional is constructed to further reduce conservatism. Moreover, the projection technique and the mathematical induction are introduced in the stability analysis. Compared with the existing results for static strategies, the proposed dynamic strategy is more flexible and generates fewer events. Finally, simulation examples are provided to demonstrate the effectiveness of this new scheme.