Antiwindup design with guaranteed regions of stability: an LMI-based approach

This note addresses the design of antiwindup gains for obtaining larger regions of stability for linear systems with saturating inputs. Considering that a linear dynamic output feedback has been designed to stabilize the linear system (without saturation), a method is proposed for designing an antiwindup gain that maximizes an estimate of the basin of attraction of the closed-loop system. It is shown that the closed-loop system obtained from the controller plus the antiwindup gain can be modeled by a linear system with a deadzone nonlinearity. A modified sector condition is then used to obtain stability conditions based on quadratic Lyapunov functions. Differently from previous works these conditions are directly in linear matrix inequality form. Some numerical examples illustrate the effectiveness of the proposed design technique when compared with the previous ones.     

Anti-windup strategies for discrete-time switched systems subject to input saturation

This paper deals with the design of anti-windup compensator for discrete-time switched systems subject to input saturation. The cases of static and dynamic anti-windup controllers are addressed aiming at maximising the estimate of the basin of attraction of the origin for the closed-loop system. Two aspects of the switching law are taken into account during the design: either it is arbitrary or it is a part of the complete control law. Theoretical conditions allowing to synthesise the anti-windup compensator are mainly described through linear matrix inequalities. Computational oriented conditions are then provided to solve convex optimisation problems that are able to give a constructive solution.     

Anti-windup design of uncertain discrete-time Markovian jump systems with partially known transition probabilities and saturating actuator

This paper carries out a study on the design of anti-windup gains for uncertain discrete-time Markovian jump systems subject to both actuator saturation and partially known transition probabilities. The parameter uncertainties appearing in both the state and input matrices are assumed to be time-varying and norm-bounded. Under the assumption that a set of linear dynamic output feedback controllers have been designed to stabilise the Markovian jump system in the absence of actuator saturation, anti-windup compensation gains are designed for maximising the domain of attraction of the closed-loop system with actuator saturation. Then, by solving a convex optimisation problem with constraints of a set of linear matrix inequalities, the anti-windup compensation gains are obtained. A simulation example is provided to illustrate the effectiveness of the proposed technique.     

基于扩张状态观测器的动态抗饱和补偿器设计方法

针对一类受外界扰动以及执行器饱和影响的不确定非线性系统,提出一种基于扩张状态观测器的动态抗饱和补偿器设计方法.首先通过将系统的不确定项以及外部扰动作为扩张状态,设计线性扩张状态观测器(ESO)对系统的总扰动进行估计;然后,在控制器中引入对扩张状态的估计值,对系统的总扰动进行补偿,设计了动态抗饱和补偿器,将控制器、观测器以及动态抗饱和补偿器的参数求解问题转化为基于LMI不等式组约束的优化问题,确保系统具有尽可能大的收敛域;最后通过数值仿真验证所提出设计方法的有效性.     

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.     

Anti-windup compensation for systems with sensor saturation: a study of architecture and structure

This article proposes a linear dynamic anti-windup strategy for the alleviation of performance and stability problems in systems which are linear apart from saturating sensors. Unlike anti-windup compensation for systems subject to actuator saturation, there is no agreed architecture for applying anti-windup to systems with sensor saturation and therefore attention is devoted to the discussion of various candidate configurations which can be interpreted as a particular choice of a static non-linear map. The main results of the article give existence conditions for compensators which yield global exponential stability and finite ?₂ gain of the overall non-linear closed-loop system. The existence conditions are different to those which have appeared hitherto in the literature.     

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.     

Stabilization of nonlinear time-delay systems with input saturation via anti-windup fuzzy design

This investigation considers stability analysis and control design for nonlinear time-delay systems subject to input saturation. An anti-windup fuzzy control approach, based on fuzzy modeling of nonlinear systems, is developed to deal with the problems of stabilization of the closed-loop system and enlargement of the domain of attraction. To facilitate the designing work, the nonlinearity of saturation is first characterized by sector conditions, which provide a basis for analysis and synthesis of the anti-windup fuzzy control scheme. Then, the Lyapunov–Krasovskii delay-independent and delay-dependent functional approaches are applied to establish sufficient conditions that ensure convergence of all admissible initial states within the domain of attraction. These conditions are formulated as a convex optimization problem with constraints provided by a set of linear matrix inequalities. Finally, numeric examples are given to validate the proposed method.     

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.     

单输入线性系统的改进型动态抗饱和设计

本文基于状态重置的改进型动态抗饱和补偿方案, 研究了具有单输入的线性饱和系统的抗饱和控制问题. 相比较于传统的动态抗饱和补偿方案, 当执行器不饱和时, 改进的动态抗饱和补偿方案把动态抗饱和补偿器的状态重置为零. 所以当执行器不饱和时, 改进的动态抗饱和补偿器将不会对控制器进行补偿. 进一步的, 提出了一个时间依赖的Lyapunov函数来分析闭环系统的稳定性, 并以LMIs的形式给出了闭环系统的控制综合条件. 最后, 通过压电纳米运动平台验证了所提出的改进型动态抗饱和补偿方案的有效性。     

一类执行器幅值与速率受限的线性系统静态抗饱和综合

讨论一类线性约束系统的静态抗饱和综合问题, 将执行器动力学特性引入增广系统, 从而化原系统为仅有幅值饱和的高阶增广系统. 提出基于线性矩阵不等式(LMI)的优化算法, 求得的静态抗饱和增益同时保证闭环系统的局部稳定性与最小化的L2增益. 仿真算例验证了方法的有效性.     

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.     

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.     

A Switching Anti-windup Design Using Multiple Lyapunov Functions

This technical note proposes a switching anti-windup design, which aims to enlarge the domain of attraction of the closed-loop system. Multiple anti-windup gains along with an index function that orchestrates the switching among these anti-windup gains are designed based on the min function of multiple quadratic Lyapunov functions. In comparison with the design of a single anti-windup gain which maximizes a contractively invariant level set of a single quadratic Lyapunov function as a way to enlarge the domain of attraction, the use of multiple Lyapunov functions and switching in the proposed design allows the union of the level sets of the multiple Lyapunov functions, each of which is not necessarily contractively invariant, to be contractively invariant and within the domain of attraction. As a result, the resulting domain of attraction is expected to be significantly larger than the one resulting from a single anti-windup gain and a single Lyapunov function. Indeed, simulation results demonstrate such a significant improvement.      

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.     

A new stability analysis and controller design method for discrete-time linear systems with saturation nonlinearities

The problems of stability analysis and controllers design for discrete-time linear systems subject to state saturation nonlinearities are investigated in this paper. Both full state saturation and partial state saturation are considered. It is well known to all that the controller design problem under state saturation is very difficult and complex to deal with. In order to overcome the difficulty, a new and tractable system is constructed, and it can be proved that the constructed system is with the same domain of attraction as the original system. With the aid of this property, to estimate the domain of attraction of the original system, an LMI-based method is presented for estimating the domain of attraction of the origin for the new constructed system under state saturation. Further, two optimization algorithms are developed for constructing dynamic output-feedback controllers and state feedback controllers, respectively, which guarantee that the domain of attraction of the origin for the closed-loop system is as ’large’ as possible. An example is provided to demonstrate the effectiveness of the new method.     

Stability and stabilization of discrete-time periodic linear systems with actuator saturation

This paper is concerned with the problems of stability and stabilization for discrete-time periodic linear systems subject to input saturation. Both local results and global results are obtained. For local stability and stabilization, the so-called periodic invariant set is used to estimate the domain of attraction. The conditions for periodic invariance of an ellipsoid can be expressed as linear matrix inequalities (LMIs) which can be used for both enlarging the domain of attraction with a given controller and synthesizing controllers. The periodic enhancement technique is introduced to reduce the conservatism in the methods. As a by-product, less conservative results for controller analysis and design for discrete-time time-invariant systems with input saturation are obtained. For global stability, by utilizing the special properties of the saturation function, a saturation dependent periodic Lyapunov function is constructed to derive sufficient conditions for guaranteeing the global stability of the system. The corresponding conditions are expressed in the form of LMIs and can be efficiently solved. Several numerical and practical examples are given to illustrate the theoretical results proposed in the paper.     

Anti-windup controller design using linear parameter-varying control methods

In this paper, we seek to provide a systematic anti-windup control synthesis approach for systems with actuator saturation within a linear parameter-varying (LPV) design framework. The closed-loop induced L2 gain control problem is considered. Different from conventional two-step anti-windup design approaches, the proposed scheme directly utilizes saturation indicator parameters to schedule accordingly the parameter-varying controller. Hence, the synthesis conditions are formulated in terms of linear matrix inequalities (LMIs) that can be solved very efficiently. The resulting gain-scheduled controller is non-linear in general and would lead to graceful performance degradation in the presence of actuator saturation non-linearities and linear performance recovery. An aircraft longitudinal dynamics control problem with two input saturation non-linearities is used to demonstrate the effectiveness of the proposed LPV anti-windup scheme.     

Incorporating anti-windup gain in improving stability of actuator constrained linear multiple state delays systems

This paper proposes the design of anti-windup compensator gain for improving stability of actuator input constrained linear multiple state delays systems. The system state delays are classified into mixed delay-dependent/delay-independent analysis and described by delay-differential equations. The real scalar delays are assumed to be fixed and unknown, but with known coefficient matrices. It is shown that the closed-loop system containing the controller plus the anti-windup gain can be modeled as a linear system with dead-zone nonlinearity. The formulation of anti-windup compensator gain is based on convex optimization using linear matrix inequalities (LMI) that ensure closed-loop asymptotic stability of the system while accounting upper-bound delays. The devised LMIs based on Lyapunov-Krasovskii functionals prove significantly less conservative in giving higher upper bounds delays in the formulation of anti-windup gain besides ensuring closed-loop asymptotic stability.     

Analysis and design of discrete-time linear systems with nested actuator saturations

This paper is concerned with the analysis and design of discrete-time linear systems subject to nested saturation functions. By utilizing a new compact convex hull representation of the saturation nonlinearity, a linear matrix inequalities (LMIs) based condition is obtained for testing the local and global stability of the considered nonlinear system. The estimation of the domain of attraction and the design of feedback gains such that the estimation of the domain of attraction for the resulting closed-loop system is maximized are then converted into some LMIs based optimization problems. Compared with the existing results on the same problems, the proposed solutions are less conservative as more slack variables are introduced into the conditions. A couple of numerical examples are worked out to validate the effectiveness of the proposed approach.