Saturation-based switching anti-windup design for linear systems with nested input saturation

This paper proposes a saturation-based switching anti-windup design for the enlargement of the domain of attraction of a linear system subject to nested saturation. A nestedly saturated linear feedback is expressed as a linear combination of a set of auxiliary linear feedbacks, which form a convex hull where the nestedly saturated linear feedback resides. This set of auxiliary linear feedbacks is then partitioned into several subsets. The auxiliary linear feedbacks in each of these subsets form a convex sub-hull of the original convex hull. When the value of the nestedly saturated linear feedback falls into a convex sub-hull, it can be expressed as a linear combination of the subset of all the auxiliary feedbacks that form the convex sub-hull. A separate anti-windup gain is designed for each convex sub-hull by using a common quadratic Lyapunov function and is implemented when the value of the nestedly saturated linear feedback falls into this convex sub-hull. Simulation results indicate that such a saturation-based switching anti-windup design has the ability to significantly enlarge the domain of attraction of the closed-loop system.     

Stability analysis and anti-windup design of switched systems with actuator saturation

The stability analysis and anti-windup design problem is investigated for two linear switched systems with saturating actuators by using the single Lyapunov function approach. Our purpose is to design a switching law and the anti-windup compensation gains such that the maximizing estimation of the domain of attraction is obtained for the closed-loop system in the presence of saturation. Firstly, some sufficient conditions of asymptotic stability are obtained under given anti-windup compensation gains based on the single Lyapunov function method. Then, the anti-windup compensation gains as design variables are presented by solving a convex optimization problem with linear matrix inequality (LMI) constraints. Two numerical examples are given to show the effectiveness of the proposed method.     

A switching anti-windup design based on partitioning of the input space

This paper revisits the problem of enlarging the domain of attraction of a linear system with multiple inputs subject to actuator saturation by designing a switching anti-windup compensator. The closed-loop system consisting of the plant, the controller and the anti-windup compensator is first equivalently formulated as a linear system with input deadzone. We then partition the input space into several regions. In one of these regions, all inputs saturate with the time-derivative of the saturated input being zero. In each of the remaining regions, there is a unique input that does not saturate. The time derivative of the deadzone function associated with the unsaturating input is zero. By utilizing these special properties of the inputs on an existing piecewise Lyapunov function of the augmented state vector containing the deadzone function of inputs, we design a separate anti-windup gain for each region of the input space. The switching from one anti-windup gain to another is activated when the input signals leave one region for another, which can be implemented online since only the measurement of the input signals is required. Simulation results indicate that the proposed approach has the ability to obtain a significantly larger estimate of the domain of attraction than the existing approaches.     

奇异线性系统在执行器饱和受限下不变集条件的改进

本文考虑饱和线性反馈下奇异线性系统扩大吸引域估计的问题.根据每个输入是否饱和,将输入空间分成若干子区域.在每个子区域内部,系统模型中没有显示的部分状态的时间导数可被显式表达.利用含有全部系统状态的二次Lyapunov函数,建立一组双线性矩阵不等式形式的改进的不变集条件.该组条件下,二次Lyapunov函数的水平集可诱导出一个吸引域估计.为得到最大的吸引域估计,构建了以这些双线性矩阵不等式为约束条件的优化问题,并为其求解给出了迭代算法.仿真结果表明本文得到的吸引域估计明显大于现有结果.     

Stability Analysis and Design of Uncertain Discrete‐time Switched Systems with Actuator Saturation Using Antiwindup and Multiple Lyapunov Functions Approach

The stability analysis and anti‐windup design problem is investigated for a class of discrete‐time switched systems with saturating actuators by using the multiple Lyapunov functions approach. Firstly, we suppose that a set of linear dynamic output controllers have been designed to stabilize the switched system without input saturation. Then, we design anti‐windup compensation gains and a switching law in order to enlarge the domain of attraction of the closed‐loop system. Finally, the anti‐windup compensation gains and the estimation of domain of attraction are presented by solving a convex optimization problem with linear matrix inequality (LMI) constraints. A numerical example is given to demonstrate the effectiveness of the proposed design method.     

An asymmetric Lyapunov function for linear systems with asymmetric actuator saturation

This paper proposes an asymmetric Lyapunov function approach to the estimation of the domain of attraction and the domain with a guaranteed regional gain for a linear system subject to asymmetric actuator saturation. Depending on the sign of each of the m inputs, the input space is divided into 2^m regions. In each region, the linear system with asymmetrically saturated inputs can be expressed as a linear system with symmetric dead zone. A quadratic function of the augmented state vector containing the system state and the symmetric dead‐zone function is constructed for each region. From these quadratic functions, an asymmetric Lyapunov function is composed. Furthermore, based on the special properties of the intersections between regions, 2 generalized asymmetric Lyapunov functions are proposed that lead to reduced conservativeness. A set of conditions are established under which the level sets of these asymmetric Lyapunov functions are contractively invariant and are thus estimates of the domain of attraction. Another set of conditions are derived under which the level sets are subsets of the domain with a guaranteed regional gain. Based on these conditions, LMI‐based optimization problems are formulated and solved to obtain the largest level sets as the estimates of the domain of attraction and of the domain with a guaranteed regional gain. Simulation results demonstrate the effectiveness of the proposed approach.     

On IQC approach to the analysis and design of linear systems subject to actuator saturation

This paper establishes IQC-based (Integral Quadratic Constraints) conditions under which an ellipsoid is contractively invariant for a single input linear system under a saturated linear feedback law. Based on these set invariance conditions, the determination of the largest such ellipsoid, for use as an estimate of the domain of attraction, can be formulated and solved as an LMI optimization problem. Such an LMI problem can also be readily adapted for the design of the feedback gain that achieves the largest contractively invariant ellipsoid. While the advantages of the proposed IQC approach remain to be explored, it is shown in this paper that the largest contractively invariant ellipsoid determined by this approach is the same as the one determined by the existing approach based on expressing the saturated linear feedback as a linear differential inclusion (LDI), which is known to lead to non-conservative result in determining the largest contractively invariant ellipsoid for single input systems.     

基于系统输出的嵌套饱和系统抗饱和补偿器设计

针对嵌套输入饱和系统的吸引域扩大问题,本文提出了一种基于系统输出的抗饱和补偿器激发策略,将被控系统输出信号经性能补偿器馈入到抗饱和补偿器激发环节中,形成蕴含系统实时性能信息的抗饱和激发新机制,克服了传统抗饱和激发机制无法直接反映系统性能的缺点.基于上述抗饱和控制新框架,本文建立了抗饱和补偿器及性能补偿器存在的充分条件,并依此构建了优化问题求解最优补偿器增益以实现扩大闭环系统吸引域的目的.仿真结果表明本文方法的有效性.     

Analysis and design for singular discrete linear systems subject to actuator saturation

A method to estimate the domain of attraction for a singular discrete linear system under a saturated linear feedback is established. Simple conditions are derived in terms of an auxiliary feedback matrix for determining if a given ellipsoid is contractively invariant. These conditions are expressed in terms of linear matrix inequalities. The largest contractively invariant ellipsoid can also be determined by solving an optimization problem with linear matrix inequality constraints. This result is extended to the design of feedback gain that results in the largest contractively invariant ellipsoid, which is also a linear matrix inequality optimization problem. A numerical example demonstrates the applicability and effectiveness of the presented method. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Design of multiple anti-windup loops for multiple activations

This paper considers anti-windup design for linear systems subject to actuator saturation.Three anti-windup gains are designed for activations immediately at the occurrence actuator saturation,after the saturation has reached a certain level and in anticipation of the occurrence of saturation,respectively.The design is based on the minimization of L 2 gain from the disturbance to the controlled output of the resulting closed-loop system.Traditional anti-windup scheme involves a single anti-windup loop for immediate activation.A recent innovation is to design a single anti-windup loop for delayed or anticipatory activation,as well as to design two anti-windup gains,one for immediate activation and one for delayed activation.Our design of three anti-windup gains for three different activations is shown through simulation to lead to significant further performance improvement over the previous activation schemes.     

Anti-windup design with guaranteed regions of stability for discrete-time linear systems

The purpose of this paper is to study the determination of stability regions for discrete-time linear systems with saturating controls through anti-windup schemes. Considering that a linear dynamic output feedback has been designed to stabilize the linear discrete-time system (without saturation), a method is proposed for designing an anti-windup gain that maximizes an estimate of the basin of attraction of the closed-loop system in the presence of saturation. It is shown that the closed-loop system obtained from the controller plus the anti-windup gain can be locally modeled by a linear system with a deadzone nonlinearity. Then, based on the use of a new sector condition and quadratic Lyapunov functions, stability conditions in an LMI form are stated. These conditions are then considered in a convex optimization problem in order to compute an anti-windup gain that maximizes an estimate of the basin of attraction of the closed-loop system. Moreover, considering asymptotically stable open-loop systems, it is shown that the conditions can be slightly modified in order to determine an anti-windup gain that ensures global stability. An extension of the proposed results to the case of dynamic anti-windup synthesis is also presented in the paper.     

Absolute stability analysis of discrete-time systems with composite quadratic Lyapunov functions

A generalized sector bounded by piecewise linear functions was introduced in a previous paper for the purpose of reducing conservatism in absolute stability analysis of systems with nonlinearity and/or uncertainty. This paper will further enhance absolute stability analysis by using the composite quadratic Lyapunov function whose level set is the convex hull of a family of ellipsoids. The absolute stability analysis will be approached by characterizing absolutely contractively invariant (ACI) level sets of the composite quadratic Lyapunov functions. This objective will be achieved through three steps. The first step transforms the problem of absolute stability analysis into one of stability analysis for an array of saturated linear systems. The second step establishes stability conditions for linear difference inclusions and then for saturated linear systems. The third step assembles all the conditions of stability for an array of saturated linear systems into a condition of absolute stability. Based on the conditions for absolute stability, optimization problems are formulated for the estimation of the stability region. Numerical examples demonstrate that stability analysis results based on composite quadratic Lyapunov functions improve significantly on what can be achieved with quadratic Lyapunov functions.     

受约束时滞系统的抗饱和补偿器增益设计

Systems that are subject to both time-delay in state and input saturation are considered.We synthesize the anti-windup gain to enlarge the estimation of domain of attraction while guaran-teeing the stability of the closed-loop system. An ellipsoid and a polyhedral set are used to bound the state of the system, which make a new sector condition valid. Other than an iterative algorithm, a direct designing algorithm is derived to compute the anti-windup compensator gain, which reduces the conservatism greatly. We analyze the delay-independent and delay-dependent cases, respectively. Finally, an optimization algorithm in the form of LMIs is constructed to compute the compensator gain which maximizes the estimation of domain of attraction. Numerical examples are presented to demonstrate the effectiveness of our approach.     

The Ellipsoidal Invariant Set of Fractional Order Systems Subject to Actuator Saturation: The Convex Combination Form

The domain of attraction of a class of fractional order systems subject to saturating actuators is investigated in this paper. We show the domain of attraction is the convex hull of a set of ellipsoids. In this paper, the Lyapunov direct approach and fractional order inequality are applied to estimating the domain of attraction for fractional order systems subject to actuator saturation. We demonstrate that the convex hull of ellipsoids can be made invariant for saturating actuators if each ellipsoid with a bounded control of the saturating actuators is invariant. The estimation on the contractively invariant ellipsoid and construction of the continuous feedback law are derived in terms of linear matrix inequalities (LMIs). Two numerical examples illustrate the effectiveness of the developed method.      

一类切换模糊时滞系统的状态反馈控制

针对切换模糊时滞系统,根据平行分布补偿算法,设计了模糊状态反馈控制器.使用切换技术及单Lyapunov函数和多Lyapunov函数方法,给出了这一类切换模糊时滞系统渐近稳定的充分条件及切换律.仿真结果表明了方法的有效性.     

An analysis and design method for linear systems under nested saturation

This paper considers a linear system under nested saturation. Nested saturation arises, for example, when the actuator is subject to magnitude and rate saturation simultaneously. A condition is derived in terms of a set of auxiliary feedback gains for determining if a given ellipsoid is contractively invariant. Moreover, this condition is shown to be equivalent to linear matrix inequalities (LMIs) in the actual and auxiliary feedback gains. As a result, the estimation of the domain of attraction for a given set of feedback gains can be formulated as an optimization problem with LMI constraints. By viewing the feedback gains as extra free parameters, the optimization problem can be used for controller design.     

Stability analysis and antiwindup design of uncertain discrete-time switched linear systems subject to actuator saturation

This paper investigates the stability analysis and antiwindup design problem for a class of discrete-time switched linear systems with time-varying norm-bounded uncertainties and saturating actuators by using the switched Lyapunov function approach.Supposing that a set of linear dynamic output controllers have been designed to stabilize the switched system without considering its input saturation,we design antiwindup compensation gains in order to enlarge the domain of attraction of the closed-loop system in the presence of saturation.Then,in terms of a sector condition,the antiwindup compensation gains which aim to maximize the estimation of domain of attraction of the closed-loop system are presented by solving a convex optimization problem with linear matrix inequality(LMI)constraints.A numerical example is given to demonstrate the effectiveness of the proposed design method.     

An analysis and design method for discrete-time linear systems under nested saturation

This paper considers a discrete-time linear system under nested saturation. Nested saturation arises, for example, in systems with actuators subject to both magnitude and rate saturation. A condition is derived in terms of a set of auxiliary feedback gains for determining if a given ellipsoid is contractively invariant. Moreover, this condition is shown to be equivalent to linear matrix inequalities (LMIs) in the actual and auxiliary feedback gains. As a result, the estimation of the domain of attraction for a given set of feedback gains is then formulated as an optimization problem with LMI constraints. By viewing the feedback gains as extra free parameters, the optimization problem can be used for controller design.     

约束T-S模糊系统设计的模糊Lyapunov方法

通过引入模糊Lyapunov函数,研究一类执行器饱和的离散T-S模糊系统.对系统设计模糊抗积分饱和补偿器,得到系统稳定的充分条件,并扩大了系统的吸引域.这种方法避免了寻求一个满足系统所有模糊规则的公共正定矩阵P.最后,抗积分饱和补偿器增益通过迭代优化算法得到.