Input‐to‐state stability of min‐max MPC scheme for nonlinear time‐varying delay systems

This paper studies the robustness problem of the min–max model predictive control (MPC) scheme for constrained nonlinear time‐varying delay systems subject to bounded disturbances. The notion of the input‐to‐state stability (ISS) of nonlinear time‐delay systems is introduced. Then by using the Lyapunov–Krasovskii method, a delay‐dependent sufficient condition is derived to guarantee input‐to‐state practical stability (ISpS) of the closed‐loop system by way of nonlinear matrix inequalities (NLMI). In order to lessen the online computational demand, the non‐convex min‐max optimization problem is then converted to a minimization problem with linear matrix inequality (LMI) constraints and a suboptimal MPC algorithm is provided. Finally, an example of a truck‐trailer is used to illustrate the effectiveness of the proposed results. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Robust MPC for Linear Systems with Structured Time‐Varying Uncertainties and Saturating Actuator

In this paper, a synthesis of model predictive control (MPC) algorithm is presented for uncertain systems subject to structured time‐varying uncertainties and actuator saturation. The system matrices are not exactly known, but are affine functions of a time varying parameter vector. To deal with the nonlinear actuator saturation, a saturated linear feedback control law is expressed into a convex hull of a group of auxiliary linear feedback laws. At each time instant, a state feedback law is designed to ensure the robust stability of the closed‐loop system. The robust MPC controller design problem is formulated into solving a minimization problem of a worst‐case performance index with respect to model uncertainties. The design of controller is then cast into solving a feasibility of linear matrix inequality (LMI) optimization problem. Then, the result is further extended to saturation dependent robust MPC approach by introducing additional variables. A saturation dependent quadratic function is used to reduce the conservatism of controller design. To show the effectiveness, the proposed robust MPC algorithms are applied to a continuous‐time stirred tank reactor (CSTR) process.     

Constrained robust sampled‐data control for nonlinear uncertain systems

In industrial process control, computer control, which makes the closed‐loop system a sampled‐data one containing both continuous‐ and discrete‐time signals, is widely used. In contrast with traditional approximation methods, sampled‐data synthesis, a direct digital controller design procedure without approximation, has received increasing attention during the past few years. However, many of the existing results cannot be applied to sampled‐data control design for the uncertain systems. In this paper, a result of robust asymptotic stability of sampled‐data systems with constraints on the state is presented based on a result on practical stability for these systems. Then the robust sampled‐data control for a class of uncertain nonlinear systems with constraints on the output is developed. The problem is formulated from vehicle steering control with constraint on the side slip angle of body. The result is described by some matrix inequalities which could be solved by an iterative algorithm based on the linear matrix inequality technique. Finally, a numerical example is presented to demonstrate the result. Copyright © 2002 John Wiley & Sons, Ltd.     

Stabilizing static output feedback receding horizon controls for linear discrete time-invariant systems

In this paper, a stabilizing static output feedback receding horizon control (RHC) is proposed for linear discrete time-invariant systems. An inequality condition on the terminal weighting matrices is presented under which the closed-loop stability of static output feedback receding horizon controls is guaranteed. It is shown that the stabilizing static output feedback receding horizon control problem can be represented as a non-linear minimization problem based on linear matrix inequalities (LMIs). An algorithm for solving the presented non-linear minimization problem is proposed. In addition, this result is extended to an arbitrary reference tracking problem that is one of the main features of receding horizon control. Finally, the efficiency of the proposed algorithm is illustrated by numerical examples.     

Robust Exponential Stability and Stabilization of a Class of Nonlinear Stochastic Time‐Delay Systems

This paper presents new exponential stability and delayed‐state‐feedback stabilization criteria for a class of nonlinear uncertain stochastic time‐delay systems. By choosing the delay fraction number as two, applying the Jensen inequality to every sub‐interval of the time delay interval and avoiding using any free weighting matrix, the method proposed can reduce the computational complexity and conservativeness of results. Based on Lyapunov stability theory, exponential stability and delayed‐state‐feedback stabilization conditions of nonlinear uncertain stochastic systems with the state delay are obtained. In the sequence, the delayed‐state‐feedback stabilization problem for a nonlinear uncertain stochastic time‐delay system is investigated and some sufficient conditions are given in the form of nonlinear inequalities. In order to solve the nonlinear problem, a cone complementarity linearization algorithm is offered. Mathematical and/or numerical comparisons between the proposed method and existing ones are demonstrated, which show the effectiveness and less conservativeness of the proposed method.     

GENERALIZED QUADRATIC STABILIZATION FOR DISCRETE‐TIME SINGULAR SYSTEMS WITH TIME‐DELAY AND NONLINEAR PERTURBATION

This paper discusses a generalized quadratic stabilization problem for a class of discrete‐time singular systems with time‐delay and nonlinear perturbation (DSSDP), which the satisfies Lipschitz condition. By means of the S‐procedure approach, necessary and sufficient conditions are presented via a matrix inequality such that the control system is generalized quadratically stabilizable. An explicit expression of the static state feedback controllers is obtained via some free choices of parameters. It is shown in this paper that generalized quadratic stability also implies exponential stability for linear discrete‐time singular systems or more generally, DSSDP. In addition, this new approach for discrete singular systems (DSS) is developed in order to cast the problem as a convex optimization involving linear matrix inequalities (LMIs), such that the controller can stabilize the overall system. This approach provides generalized quadratic stabilization for uncertain DSS and also extends the existing robust stabilization results for non‐singular discrete systems with perturbation. The approach is illustrated here by means of numerical examples.     

Stabilization of continuous‐time homogeneous bilinear systems by constant inputs through nonlinear minimization with applications to the static output feedback stabilization problem

An algorithm to provide constant‐input stabilizing control inputs for multi‐input continuous‐time bilinear systems is proposed in this paper. The algorithm is based on the formal discrete‐time approximation of the system and the unconstrained nonlinear minimization. The key features of the new algorithm are as follows. First, the formal discrete‐time approximation makes the set of stabilizing control inputs star‐shaped centred at the origin, hence the minimization is to be performed only in a neighbourhood of the origin selected by the designer. Second, the algorithm is always capable of finding a solution if one exists, as long as the minimization inside the neighbourhood is successful. Third, by a slight modification, the algorithm permits us to place all the eigenvalues of the system inside a rectangular region in the complex plane, as long as it is feasible. The algorithm is also applicable to the static output feedback stabilization problem of linear time‐invariant systems. Copyright © 2007 John Wiley & Sons, Ltd.     

Sequential LMI approach for the design of a BMI‐based robust observer state feedback controller with nonlinear uncertainties

This paper aims at developing a robust observer–based estimated state feedback control design method for an uncertain dynamical system that can be represented as a linear time‐invariant system connected with an integral quadratic constraint–type nonlinear uncertainty. Traditionally, in existing design methodologies, a convex semidefinite constraint is obtained at the cost of conservatism and unrealistic assumptions. This paper avoids such assumptions and formulates, the design of the robust observer state feedback controller as the feasibility problem of a bilinear matrix inequality (BMI) constraint. Unfortunately, the search for a feasible solution of a BMI constraint is an NP‐hard problem in general. The applicability of a linearization method, such as the variable change method and the congruence transformation, depends on the specific structure of the problem at hand and cannot be generalized. This paper transforms the feasibility analysis of the BMI constraint into an eigenvalue problem and applies the convex‐concave–based sequential linear matrix inequality optimization method to search for a feasible solution. Furthermore, an augmentation of the sequential linear matrix inequality algorithm to improve its numerical stability is presented. In the application part, a vehicle lateral control problem is presented to demonstrate the applicability of the proposed algorithm to a real‐world estimated state feedback control design problem and the necessity of the augmentation for numerical stability.     

Improved robust stability criteria and design of robust stabilizing controller for uncertain linear time‐delay systems

In this paper, an improved linear matrix inequality (LMI)‐based robust delay‐dependent stability test is introduced to ensure a larger upper bound for time‐varying delays affecting the state vector of an uncertain continuous‐time system with norm‐bounded‐type uncertainties. A quasi‐full‐size Lyapunov–Krasovskii functional is chosen and free‐weighting matrix approach is employed. Less restrictive sufficient conditions are derived for robust stability of time‐varying delay systems with norm‐bounded‐type uncertainties. Moreover, the investigation of the stabilization problem with memoryless state‐feedback control is presented such that the stabilizability criteria are obtained in terms of matrix inequalities, which can be solved via utilizing a cone complementarity minimization algorithm. Finally, the problem of output feedback stabilization for square systems is also taken into consideration. The output feedback stabilizability criteria are derived in the form of linear matrix inequalities, which are convex and can be easily solved using interior point algorithms. A plenty of numerical examples are presented indicating that the proposed stability and stabilization methods effectively improve the existing results. Copyright © 2006 John Wiley & Sons, Ltd.     

Simultaneous design of AWC and nonlinear controller for uncertain nonlinear systems under input saturation

This article addresses a novel technique for the simultaneous design of a robust nonlinear controller and static anti‐windup compensator (AWC) for uncertain nonlinear systems under actuator saturation and exogenous bounded input. The system is presumed to have locally Lipschitz nonlinearities, time‐varying uncertainties (appearing both in the linear as well as nonlinear dynamics and both in the state in addition to the output equations), and external norm‐bounded inputs both in the state and the output equations. Several bilinear matrix inequality–based conditions are derived to simultaneously design the robust nonlinear controller and AWC gains for uncertain nonlinear systems by employing the Lyapunov functional, reformulated Lipschitz property, uncertainty bounds, linear parameter‐varying approach, modified local and global sector conditions, iterative linear matrix inequality algorithm, convex optimization procedure, and gain minimization. The proposed multiobjective AWC‐based dynamic robust nonlinear controller guarantees the mitigation of saturation effects, robustness against time‐varying parametric norm‐bounded uncertainties, the asymptotic stability of the closed‐loop nonlinear system under zero external disturbances, and the attenuation of disturbance effects under nonzero external disturbances. The effectiveness of the proposed AWC‐based dynamic robust nonlinear controller synthesis scheme is illustrated by simulation examples.     

Robust stabilization for discrete‐time nonlinear singular systems with mixed time delays

This paper investigates the robust stability of discrete‐time singular systems involving nonlinear disturbance and mixed time delays. The mixed time delays are comprised of both discrete and distributed delays. The interval of discrete time delays can be divided into several subintervals, firstly. Then, in terms of linear matrix inequality (LMI), a suitable state feedback controller is designed for discrete‐time singular systems with nonlinear disturbance and mixed time delays, and the overall closed‐loop system is regular, causal and mean square asymptotically stable. Numerical examples are provided to show the usefulness and effectiveness of the proposed methods, and the results derived from our approaches are less conservative than existing ones. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Observer‐based finite‐time stabilization for discrete‐time switched singular systems with quadratically inner‐bounded nonlinearities

This paper discusses the observer‐based finite‐time stabilization for discrete‐time switched singular systems with quadratically inner‐bounded nonlinear terms. Firstly, based on the Luenberger‐like observer, by using the average dwell time approach, sufficient conditions are proposed to make closed‐loop systems be regular, be causal, as having a unique solution, and be uniformly finite‐time bounded. Then, a new linear matrix inequality sufficient condition for the existence of an observer‐based controller is obtained by using certain matrix decoupling techniques, and the controller is designed. In this paper, the conditions proposed not only give the observer‐based controller design methods but also guarantee the existence and uniqueness of solution for the systems. Since the quadratically inner‐bounded nonlinearities are more general than Lipschitz nonlinearities and one‐sided Lipschitz nonlinearities, compared with previous works, the proposed controller design methods in this paper are also more general than the existing ones. Finally, numerical examples are provided to illustrate the effectiveness of the methods proposed in this paper.     

A STABLE ON‐LINE SELF‐TUNING OPTIMAL PID CONTROLLER FOR A CLASS OF UNKNOWN SYSTEMS

In this paper, a novel self‐tuning method of optimal PID control laws is proposed for both continuous‐time systems and discrete‐time systems. The controlled plant is assumed to be unknown except the system order (or system delay) and the direction of transmitting control input. Through the minimization of PID gains subject to the Lyapunov stability based reaching condition, the tuning of the three PID control gains is transformed to solve the inequality constraint optimization problem. An unknown SISO nonlinear system subject to a unit step input, and the tracking control problem of the piezoelectric actuator (PZA) with unknown dynamics are simulated. The simulation results show that the excellent tracking performance can be achieved.     

Stabilization of nonlinear networked probabilistic interval delay systems with sensor random packet dropout

In this paper, the problem of stabilization for nonlinear networked systems with probabilistic interval delay and sensor random packet dropout is investigated. By employing the information of probabilistic distribution of time‐varying delay and considering random sensor packet dropout with compensation, the nonlinear stochastic delayed system model is established. Based on the obtained model, by choosing an appropriate Lyapunov function and utilizing a new discrete Jensen‐type inequality, sufficient conditions are derived to obtain the relation of the maximum allowable delay bound, delay interval occurrence rate and packet dropout rate to the stochastic stability of nonlinear networked control systems. Two kinds of design procedures for output feedback controller are also presented in terms of solving corresponding linear matrix inequalities. Numerical examples are provided to illustrate the effectiveness and applicability of proposed techniques. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Robust ℋ︁∞ sliding mode control for nonlinear stochastic systems with multiple data packet losses

In this paper, an ??_∞ sliding mode control (SMC) problem is studied for a class of discrete‐time nonlinear stochastic systems with multiple data packet losses. The phenomenon of data packet losses, which is assumed to occur in a random way, is taken into consideration in the process of data transmission through both the state‐feedback loop and the measurement output. The probability for the data packet loss for each individual state variable is governed by a corresponding individual random variable satisfying a certain probabilistic distribution over the interval [0 1]. The discrete‐time system considered is also subject to norm‐bounded parameter uncertainties and external nonlinear disturbances, which enter the system state equation in both matched and unmatched ways. A novel stochastic discrete‐time switching function is proposed to facilitate the sliding mode controller design. Sufficient conditions are derived by means of the linear matrix inequality (LMI) approach. It is shown that the system dynamics in the specified sliding surface is exponentially stable in the mean square with a prescribed ??_∞ noise attenuation level if an LMI with an equality constraint is feasible. A discrete‐time SMC controller is designed capable of guaranteeing the discrete‐time sliding mode reaching condition of the specified sliding surface with probability 1. Finally, a simulation example is given to show the effectiveness of the proposed method. Copyright © 2011 John Wiley & Sons, Ltd.     

基于等分采样的网络控制系统鲁棒故障检测

考虑一类具有未知干扰的时延网络控制系统,对其进行故障检测。通过提高控制器端的采样频率,等分传感器采样周期,将网络时延近似看成控制器端采样周期的整数倍。建立基于离散切换系统的鲁棒H∞故障观测器,将观测器稳定条件归结为一个线性矩阵不等式。通过求解具有线性矩阵不等式约束的最小化问题,对故障检测阈值进行选取,从而增强检测的鲁棒性和灵敏度。仿真示例验证了该方法的有效性。     

Delay‐Dependent Stability Criterion for Discrete‐Time Systems with Time‐Varying Delays

The stability analysis problem is considered for linear discrete‐time systems with time‐varying delays. A novel summation inequality is proposed, which takes the double summation information of the system state into consideration. The inequality relaxes the recently proposed discrete Wirtinger inequality and its improved version. Based on construction of a suitable Lyapunov‐Krasovskii functional and the novel summation inequality, an improved delay‐dependent stability criterion for asymptotic stability of the systems is derived in terms of linear matrix inequalities. Numerical examples are given to demonstrate the advantages of the proposed method.     

A Bounded Real Lemma for Discrete‐Time Markovian Jump Singular Systems

This paper presents a bounded real lemma for discrete‐time Markovian jump singular systems. First, a new necessary and sufficient condition is proposed in terms of a strict linear matrix inequality, which guarantees the stochastic admissibility of the unforced Markovian jump singular systems. Then, a bounded real lemma for discrete‐time Markovian jump singular systems is derived. It is also proven that the bounded real lemma can be described by a strict matrix inequality. Finally, a numerical example is provided to illustrate the effectiveness of the proposed theory. The results are more tractable and reliable in numerical computations than existing conditions.