Robust stability analysis of uncertain switched linear systems with unstable subsystems

The problem of robust stability for switched linear systems with all the subsystems being unstable is investigated. Unlike the most existing results in which each switching mode in the system is asymptotically stable, the subsystems may be unstable in this paper. A necessary condition of stability for switched linear systems is first obtained with certain hypothesis. Then, under two assumptions, sufficient conditions of exponential stability for both deterministic and uncertain switched linear systems are presented by using the invariant subspace theory and average dwell time method. Moreover, we further develop multiple Lyapunov functions and propose a method for constructing multiple Lyapunov functions for the considered switched linear systems with certain switching law. Several examples are included to show the effectiveness of the theoretical findings.     

不确定系统的鲁棒与随机模型预测控制算法比较研究

近几十年来,不确定系统模型预测控制的理论和应用得到了飞速发展.本文简要地回顾了不确定系统中鲁棒模型预测控制和随机模型预测控制的发展历史,总结了它们的相关应用,并较为细致地分析了线性不确定系统模型预测控制的各种主要算法.通过总结各种算法的通用模型、运作方式、问题规模,以及它们保证递归可行性、稳定性的方法,分析了部分算法可行域间的关系,揭示了各种算法的主要特点、适用场合和未来可发展方向,并通过仿真实例直观地分析了各种算法的性能和可靠性.     

Conditions under which suboptimal nonlinear MPC is inherently robust

We address the inherent robustness properties of nonlinear systems controlled by suboptimal model predictive control (MPC), i.e., when a suboptimal solution of the (generally nonconvex) optimization problem, rather than an element of the optimal solution set, is used for the control. The suboptimal control law is then a set-valued map, and consequently, the closed-loop system is described by a difference inclusion. Under mild assumptions on the system and cost functions, we establish nominal exponential stability of the equilibrium, and with a continuity assumption on the feasible input set, we prove robust exponential stability with respect to small, but otherwise arbitrary, additive process disturbances and state measurement/estimation errors. These results are obtained by showing that the suboptimal cost is a continuous exponential Lyapunov function for an appropriately augmented closed-loop system, written as a difference inclusion, and that recursive feasibility is implied by such (nominal) exponential cost decay. These novel robustness properties for suboptimal MPC are inherited also by optimal nonlinear MPC. We conclude the paper by showing that, in the absence of state constraints, we can replace the terminal constraint with an appropriate terminal cost, and the robustness properties are established on a set that approaches the nominal feasibility set for small disturbances. The somewhat surprising and satisfying conclusion of this study is that suboptimal MPC has the same inherent robustness properties as optimal MPC.     

输入状态稳定的鲁棒预测控制

以有界干扰非线性系统为研究对象,设计一种基于近似可达集的鲁棒预测控制方法。该方法以鲁棒控制不变集作为终端约束集,采用一种简单的三次多项式逼近预测控制的待优化控制律,通过在线优化求解三次多项式的各项系数,并从理论上证明了所设计的鲁棒预测控制律可以使系统输入状态稳定。最后通过仿真实例验证了所提出的鲁棒预测控制方法的可行性和有效性。     

Robust stability analysis of linear systems with parametric uncertainty

This article is concerned with the problem of robust stability analysis of linear systems with uncertain parameters. By constructing an equivalent system with positive uncertain parameters and using the properties of these parameters, a new stability analysis condition is derived. Due to making use of the properties of uncertain parameters, the new proposed method has potential to give less conservative results than the existing approaches. A numerical example is given to illustrate the effectiveness of the proposed method.     

Robust absolute stability of Lurie interval control systems

This paper considers robust absolute stability of Lurie control systems. Particular attention is given to the systems with parameters having uncertain, but bounded values. Such so‐called Lurie interval control systems have wide applications in practice. In this paper, a number of sufficient and necessary conditions are derived by using the theories of Hurwitz matrix, M matrix and partial variable absolute stability. Moreover, several algebraic sufficient and necessary conditions are provided for the robust absolute stability of Lurie interval control systems. These algebraic conditions are easy to be verified and convenient to be used in applications. Three mathematical examples and a practical engineering problem are presented to show the applicability of theoretical results. Numerical simulation results are also given to verify the analytical predictions. Copyright © 2007 John Wiley & Sons, Ltd.     

Notions of exponential robust stochastic stability,ISS and their Lyapunov characterizations

Concepts of exponential global robust stability for stochastic control systems are analysed in terms of Lyapunov functions. The main result of the paper constitutes a generalization of a converse stability theorem due to Khasminskii for stochastic differential equations and establishes that, under certain hypotheses, the origin is robustly exponentially stable in the rth mean, if and only if the system admits a Lyapunov function which is smooth except possibly at the origin. The main result concerning robust asymptotic stability enable us to derive a Lyapunov‐like characterization for the concept of stochastic input‐to‐state stability (ISS). Copyright © 2003 John Wiley & Sons, Ltd.     

Robust stability of positive switched systems with dwell time

This paper studies robust stability of positive switched systems (PSSs) with polytopic uncertainties in both discrete-time and continuous-time contexts. By using multiple linear copositive Lyapunov functions, a sufficient condition for stability of PSSs with dwell time is addressed. Being different from time-invariant multiple linear copositive Lyapunov functions, the Lyapunov functions constructed in this paper are time-varying during the dwell time and time-invariant afterwards. Then, robust stability of PSSs with polytopic uncertainties is solved. All conditions are solvable via linear programming. Finally, illustrative examples are given to demonstrate the validity of the proposed results.     

Economic MPC of nonlinear systems with nonmonotonic Lyapunov functions and its application to HVAC control

This paper proposes a Lyapunov‐based economic model predictive control (MPC) scheme for nonlinear systems with nonmonotonic Lyapunov functions. Relaxed Lyapunov‐based constraints are used in the MPC formulation to improve the economic performance. These constraints will enforce a Lyapunov decrease after every few steps. Recursive feasibility and asymptotical convergence to the steady state can be achieved using Lyapunov‐like stability analysis. The proposed economic MPC can be applied to minimize energy consumption in heating ventilation and air conditioning control of commercial buildings. The Lyapunov‐based constraints in the online MPC problem enable the tracking of the desired set‐point temperature. The performance is demonstrated by a virtual building composed of 2 adjacent zones.     

Attractors and partial stability of nonlinear dynamical systems

This paper represents a study of the key concepts of attracting sets (submanifolds) and partially stable systems aimed at establishing relations between the local properties and the unification of the methodologies of their analysis. The use of recent results of stability theory and techniques of geometric control enables one to generalize Lyapunov-like sufficient conditions and propose simplified solutions for the problems of set attractivity and partial stability.     

Stability analysis of constrained MPC applied to discrete-time nonlinear with CLF system

The model predictive control （MPC） strategy with a control Lyapunov function （CLF） as terminal cost is commonly used for its guaranteed stability. In most of the cases, CLF is locally designed, and the region of attraction is limited, especially when under control constraints. In this article, the stability and the region of attraction of constrained MPC that is applied to the discrete-time nonlinear system are explicitly analyzed. The region of feasibility is proposed to substitute the region of attraction, which greatly reduces the calculation burden of terminal constraints inequalities and guarantees the stability of the MPC algorithm. Also, the timevariant terminal weighted factor is proposed to improve the dynamic performance of the close-loop system. Explicit experiments verify the effectiveness and feasibility of the relative conclusions, which provide practically feasible ways to stabilize the unstable and/or fast-dynamic systems.     

Stability analysis of constrained MPC with CLF applied to discrete-time nonlinear system

The problem of counting the number of spanning trees is an old topic in graph theory with important applications to reliable network design. Usually, it is desirable to put forward a formula of the number of spanning trees for various graphs, which is not only interesting in its own right but also in practice. Since some large graphs can be composed of some existing smaller graphs by using the product of graphs, the number of spanning trees of such large graph is also closely related to that of the corresponding smaller ones. In this article, we establish a formula for the number of spanning trees in the lexicographic product of two graphs, in which one graph is an arbitrary graph G and the other is a complete multipartite graph. The results extend some of the previous work, which is closely related to the number of vertices and Lapalacian eigenvalues of smaller graphs only.     

On stability and stabilization of periodic discrete-time systems with an application to satellite attitude control

An alternative stability analysis theorem for nonlinear periodic discrete-time systems is presented. The developed theorem offers a trade-off between conservatism and complexity of the corresponding stability test. In addition, it yields a tractable stabilizing controller synthesis method for linear periodic discrete-time systems subject to polytopic state and input constraints. It is proven that in this setting, the proposed synthesis method is strictly less conservative than available tractable synthesis methods. The application of the derived method to the satellite attitude control problem results in a large region of attraction.     

Global stability results for systems under sampled‐data control

In this work sufficient conditions for uniform input‐to‐output stability and uniform input‐to‐state stability are presented for finite‐dimensional systems under feedback control with zero‐order hold. The conditions are expressed by means of single and vector Lyapunov functions. Copyright © 2008 John Wiley & Sons, Ltd.     

饱和系统鲁棒控制及其在飞行控制中的应用

给出一种饱和系统鲁棒稳定性的分析和设计方法.通过对不确定系统的H∞优化过程的分析,并根据其反馈控制原理,按全部和部分输入超过限制分别考虑不同指标下的优化计算问题,得到了输入受限制条件下的3种Riccati方程.飞行控制器设计和仿真结果表明,采用该方法可给出满意的控制效果.     

具有脉冲干扰和可变时滞的区间关联大系统的鲁棒指数稳定性

研究了一类具有脉冲干扰和可变时滞区间关联大系统的鲁棒指数稳定性.假设该系统的关联函数满足全局Lipschitz条件,基于矢量Lyapunov函数法和数学归纳法,给出确保该关联系统鲁棒指数稳定的充分条件.最后给出一个数值算例用以说明本文所得到结论的正确性和有效性.     

Robust stability of state-predictive and Smith control systems for plants with a pure delay

Robust stability of the state-predictive and the Smith control systems for plants with a pure delay is studied. First, the Nyquist-type stability condition is derived for robust stability. Then, the condition is transformed and a robust stability criterion is given in terms of the Nyquist plots of the complementary sensitivity function and the error function. Based on this criterion, we present a method to obtain graphically stability margins in the delay-time and the gain. © 1998 John Wiley & Sons, Ltd.     

An improved approach to robust stability analysis and controller synthesis for LPV systems

This paper revisits the problem of robust stability analysis and synthesis for linear parameter varying (LPV) systems. By introducing two new slack variables, new conditions for the polyquadratic stability and for the affine quadratic stability of LPV systems are presented in terms of linear matrix inequalities. These results are then applied for robust controller synthesis. Owing to the extra freedom degree introduced by the real slack variable, this proposed approach could lead to less conservative results than that of from the literature, especially for robust controller synthesis. The effectiveness and superiority are demonstrated by the comparison between this proposed approach and the existing results on a well‐known numerical example. Copyright © 2010 John Wiley & Sons, Ltd.     

Stability and control of nonlinear systems described by retarded functional equations: a review of recent results

This paper reports on recent results in a series of the work of the authors on the stability and nonlinear control for general dynamical systems described by retarded functional differential and difference equations. Both internal and external stability properties are studied. The corresponding Lyapunov and Razuminkhin characterizations for input-to-state and input-to-output stabilities are proposed. Necessary and sufficient Lyapunov-like conditions are derived for robust nonlinear stabilization. In particular, an explicit controller design procedure is developed for a new class of nonlinear time-delay systems. Lastly, sufficient assumptions, including a small-gain condition, are presented for guaranteeing the input-to-output stability of coupled systems comprised of retarded functional differential and difference equations.