A sum of squares approach to backstepping controller synthesis for piecewise affine and polynomial systems

This paper develops a backstepping controller synthesis methodology for piecewise polynomial (PWP) systems in strict form. The main contribution of the paper is to formulate sufficient conditions for controller design for PWP systems in strict form as a sum of squares feasibility problem under the assumption that an initial control Lyapunov function exists to start the iterative backstepping procedure. This problem can then be translated into a convex SDP problem and solved by available software packages. The controller synthesis problem for PWP systems in strict feedback form is divided into two cases. The first case consists of the construction of a sum of squares polynomial control Lyapunov function for PWP systems with discontinuous vector fields. The second case addresses the construction of a PWP control Lyapunov function for PWP systems with continuous vector fields. One major advantage of the proposed method is the fact that it can handle systems with discontinuous vector fields and sliding modes. The new synthesis method is applied to several numerical examples. One of these examples offers the first convex optimization solution to piecewise affine (PWA) control of a benchmark circuit system addressed before in the literature using non‐convex PWA control solutions. Copyright © 2013 John Wiley & Sons, Ltd.     

Passivity and passification of stochastic impulsive memristor‐based piecewise linear system with mixed delays

This paper is concerned with the problem of passivity analysis and passification for a class of stochastic impulsive memristor‐based piecewise linear (PWL) systems with mixed delays and nonlinearity disturbances. Based on the PWL memristor, a PWL system is set up. And some novel sufficient conditions are derived to ensure the passivity/passification performance, such that, for all admissible stochastic disturbances and nonlinearity, the closed‐loop stochastic impulsive memristor‐based PWL system is passive in the sense of expectation. Copyright © 2013 John Wiley & Sons, Ltd.     

Passive stochastic feedback stability-Part II. Applications

In Part I of this paper it was shown that the concept of passivity, used extensively in deterministic feedback stability, could be extended to stochastic feedback systems described through Markov theory to produce stochastic feedback stability theorems. To apply the theory developed in Part I, one is required to do a passivity analysis of the sub-systems of the feedback system. In this paper, six particular types of stochastic systems (sub-systems) are considered and a passivity analysis of each is performed. These systems have the appearance of standardnth-order linear differential systems with a random process entering at some point. The randomness is either white noise, impulsive noise or a Markov jump process. Once the passivity analysis has been made, feedback stability theorems can be derived for many feedback systems containing one or more of these six systems as sub-systems. Two such feedback systems are considered in the paper.     

Passivity Analysis and Synthesis for a Class of Discrete‐Time Switched Stochastic Systems with Time‐Varying Delay

This paper deals with the problems of passivity and passification for a class of discrete‐time switched stochastic systems with time‐varying delay. Based on the average dwell time approach, the piecewise Lyapunov function technique, and the free‐weighting matrix method, a new Lyapunov functional is proposed and sufficient conditions for mean‐square exponential stability and stochastic passivity are developed under average dwell time switching. Moreover, an estimate of state decay can be calculated in terms of linear matrix inequalities (LMIs). Then, the solvability condition for passification is established and the corresponding controller is designed. Two numerical examples are given to show the effectiveness of the proposed methods.     

Safety and Reachability of Piecewise Linear Hybrid Dynamical Systems Based on Discrete Abstractions

In this paper, a novel methodology for analysis of piecewise linear hybrid systems based on discrete abstractions of the continuous dynamics is presented. An important characteristic of the approach is that the available control inputs are taken into consideration in order to simplify the continuous dynamics. Control specifications such as safety and reachability specifications are formulated in terms of partitions of the state space of the system. The approach provides a convenient general framework not only for analysis, but also for controller synthesis of hybrid systems. The research contributions of this paper impact the areas of analysis, verification, and synthesis of piecewise linear hybrid systems.     

Approaches to robust ℋ︁∞ static output feedback control of discrete‐time piecewise‐affine systems with norm‐bounded uncertainties

This paper investigates the problem of robust ??_∞ static output feedback controller design for a class of discrete‐time piecewise‐affine systems with norm‐bounded time‐varying parametric uncertainties. The objective is to design a piecewise‐linear static output feedback controller guaranteeing the asymptotic stability of the resulting closed‐loop system with a prescribed ??_∞ disturbance attenuation level. Based on a piecewise Lyapunov function combined with S‐procedure, Projection lemma, and some matrix inequality convexifying techniques, several novel approaches to the static output feedback controller analysis and synthesis are developed for the underlying piecewise‐affine systems. It is shown that the controller gains can be obtained by solving a set of strict linear matrix inequalities (LMIs) or a family of LMIs parameterized by one or two scalar variables, which are numerically efficient with commercially available software. Finally, three simulation examples are provided to illustrate the effectiveness of the proposed approaches. Copyright © 2010 John Wiley & Sons, Ltd.     

Passivity-based consensus for linear multi-agent systems under switching topologies

This paper studies the passivity-based consensus analysis and synthesis problems for a class of stochastic multi-agent systems with switching topologies. Based on Lyapunov methods, stochastic theory, and graph theory, new different storage Lyapunov functions are proposed to derive sufficient conditions on mean-square exponential consensus and stochastic passivity for multi-agent systems under two different switching cases, respectively. By designing passive time-varying consensus protocols, the solvability conditions for the passivity-based consensus protocol synthesis problem, i.e., passification, are derived based on linearization techniques. Numerical simulations are provided to illustrate the effectiveness of the proposed methods.     

Stability Analysis and $H_{infty}$ Controller Design of Discrete-Time Fuzzy Large-Scale Systems Based on Piecewise Lyapunov Functions

This paper is concerned with stability analysis and $H_{infty}$ decentralized control of discrete-time fuzzy large-scale systems based on piecewise Lyapunov functions. The fuzzy large-scale systems consist of $J$ interconnected discrete-time Takagi–Sugeno (T–S) fuzzy subsystems, and the stability analysis is based on Lyapunov functions that are piecewise quadratic. It is shown that the stability of the discrete-time fuzzy large-scale systems can be established if a piecewise quadratic Lyapunov function can be constructed, and moreover, the function can be obtained by solving a set of linear matrix inequalities (LMIs) that are numerically feasible. The $H_{infty}$ controllers are also designed by solving a set of LMIs based on these powerful piecewise quadratic Lyapunov functions. It is demonstrated via numerical examples that the stability and controller synthesis results based on the piecewise quadratic Lyapunov functions are less conservative than those based on the common quadratic Lyapunov functions.      

T-S-K模糊控制系统的等价优化方法及其在控制器设计中的作用

针对T-S-K模糊控制系统分区线性化综合方法，本文提出了基于模糊规则分解与合并的等价优化方法，并且从理论上论证了这种优化方法的两种作用，从而可以有效地优化不确定项的上界估计值,改善分区线性化综合方法的应用效果．     

Equilibrium-independent passivity: A new definition and numerical certification

We extend the traditional notion of passivity to a forced system whose equilibrium is dependent on the control input by defining equilibrium-independent passivity, a system property characterized by a dissipation inequality centered at an arbitrary equilibrium point. We provide a necessary input/output condition which can be tested for systems of arbitrary dimension and sufficient conditions to certify this property for scalar systems. An example from network stability analysis is presented which demonstrates the utility of this new definition. We then proceed to numerical certification of equilibrium-independent passivity using sum-of-squares programming. Finally, through numerical examples we show that equilibrium-independent passivity is less restrictive than incremental passivity.     

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.     

PERFORMANCE ANALYSIS OF CONTROL SYSTEMS WITH INPUT CONSTRAINTS VIA PIECEWISE QUADRATIC STORAGE FUNCTIONS

Input constraints exist in almost all real‐world systems and frequently cause unexpected undesirable phenomena, such as performance degradation or instability. Although many studies have focused on this problem, precise performance analysis of whole control systems with constraints is still difficult. In this paper, such control systems are represented as piecewise affine systems, and global and local L₂ incremental gain analysis methods via piecewise quadratic storage functions are presented.     

A unified approach for stability analysis of a class oftime-varying nonlinear systems

It is shown that an integral with negative feedback from its output terminal through a strictly passive system is still strictly passive. By using successive negative feedback around integrals, the conditions of strict passivity and a unified approach for constructing strictly passive systems are obtained for time-varying nonlinear systems of different orders. A new concept of “dissipation factor” is defined in this paper. The stability of a class of time-varying nonlinear systems is examined by using this passivity analysis     

方块脉冲函数用于非线性系统的分析以及最优控制的综合

本文用方块脉冲函数方法讨论了解非线性微分方程系统的收敛性和稳定性.在分析和综 合非线性系统最优控制规律中,得到了分段恒定解答的递推算法,算法证明简单,除对基于二 次型性能指标的线性时变系统有效外,也可用于求解线性最优控制系统的Riccati方程.     

Tracking and synchronisation for a class of PWA systems

In this paper, the tracking problem for a class of discontinuous piecewise affine (PWA) systems is addressed. We propose an observer-based output-feedback control design, consisting of a feedforward, a piecewise affine feedback law and a model-based observer, solving the tracking problem. These synthesis results can also be employed to tackle the master-slave synchronisation problem for PWA systems. It is shown that for certain classes of PWA systems the design is characterised in terms of linear matrix inequalities. The results are illustrated by application to mechanical systems with discontinuous friction characteristics.     

Uniform ultimate boundedness analysis and synthesis for linear systems with dead‐zone in the actuators

The presence of the dead‐zone nonlinearity is common in many physical systems and in general degrades the control systems performance or stability. This note considers an interconnection between a continuous time multiple‐input linear system and a dead‐zone nonlinearity associated with a state feedback. The issue of analyzing the property of uniform ultimate boundedness (UUB) of a system and the issue of designing a control under UUB constraint are studied. Firstly, the dead‐zone is replaced via a linear transformation introducing the saturation. Secondly, we improve the size of the UUB area that can be found in the literature, by expressing the saturation in terms of parametrized piecewise affine bounds. This allows also a link between standard approaches such ones using piecewise affine models or ones considering global cone bounded sector conditions. In the framework of analysis and synthesis, sufficient conditions are obtained by linear matrix inequalities. Illustrative examples are given to spotlight the efficiency of our approach. Copyright © 2014 John Wiley & Sons, Ltd.     

A Piecewise Envelope Approach to H ∞ Control of Nonlinear Systems

This paper provides a computable approach for the H _∞ control synthesis of nonlinear systems using the piecewise envelope method. Firstly, the computation method of piecewise envelope is presented such that the required piecewise linear differential inclusions (PWLDIs) can be obtained by solving a linear optimization problem. Then, we propose the H _∞ control design method for PWLDIs using piecewise Lyapunov function theory, all the synthesis conditions are formulated as the feasibility of linear matrix inequalities, hence computationally tractable. Because of the inclusion relationship, the H _∞ controllers designed for PWLDIs are also effective for original nonlinear systems. The validity of the proposed approach is tested by application to control a wheeled mobile robot.     

Incremental passivity and output tracking of switched nonlinear systems

This article studies incremental passivity and the output tracking problem for switched nonlinear systems. The concept of incremental passivity for switched systems is given using the proposed weak-storage functions. Furthermore, conditions for a switched nonlinear system to be incrementally passive are obtained without the requirement of the incremental passivity conditions for subsystems. It is shown that once the incremental passivity is assured, the output tracking problem for switched systems is solvable via the designed controllers, even though the problem for none of subsystems is solvable.     

无源性、稳定性和最优性

本文考察仿射非线性控制系统的无源性、稳定性的最优性，以及这三者之间的关系，为分析稳定性，首先研究非线性控制系统和其线性近似的无源性之间的关系。然后指出，就仿射非线性控制系统而言，无源性、稳定性和最优性在某种意义下是反馈等价的。最后讨论了带量测输出的非线性系统的无源性、稳定性和最优性三者之间的关系。