Robust Reliable H∞ Control for Discrete‐Time Systems with Actuator Delays

This article focuses on the robust state feedback reliable H_∞ control problem for discrete‐time systems. Discrete‐time systems with time‐varying delayed control input are formulated. Based on the Lyapunov–Krasovskii method and linear matrix inequality (LMI) approach, delay‐dependent sufficient conditions are developed for synthesizing the state feedback controller for an uncertain discrete‐time system. The parameter uncertainty is assumed to be norm bounded. A design scheme for the state feedback reliable H_∞ controller is proposed in terms of LMIs, which can guarantee the global asymptotic stability and the minimum disturbance attenuation level. Finally, numerical examples are provided to illustrate the effectiveness and reduced conservatism of the proposed methods.     

Robust H∞ control for uncertain discrete‐time stochastic bilinear systems with Markovian switching

This paper is concerned with the problems of robust stochastic stabilization and robust H_∞ control for uncertain discrete‐time stochastic bilinear systems with Markovian switching. The parameter uncertainties are time‐varying norm‐bounded. For the robust stochastic stabilization problem, the purpose is the design of a state feedback controller which ensures the robust stochastic stability of the closed‐loop system irrespective of all admissible parameter uncertainties; while for the robust H_∞ control problem, in addition to the robust stochastic stability requirement, a prescribed level of disturbance attenuation is required to be achieved. Sufficient conditions for the solvability of these problems are obtained in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, explicit expressions of the desired state feedback controllers are also given. An illustrative example is provided to show the effectiveness of the proposed approach. Copyright © 2005 John Wiley & Sons, Ltd.     

New results for the analysis of linear systems with time‐invariant delays

This paper presents a comparison system approach for the analysis of stability and ??_∞ performance of linear time‐invariant systems with unknown delays. The comparison system is developed by replacing the delay elements with certain parameter‐dependent Padé approximations. It is shown using the special properties of the Padé approximation to e^?s that the value sets of these approximations provide outer and inner coverings for that of each delay element and that the robust stability of the outer covering system is a sufficient condition for the stability of the original time delay system. The inner covering system, in turn, is used to provide an upper bound on the degree of conservatism of the delay margin established by the sufficient condition. This upper bound is dependent only upon the Padé approximation order and may be made arbitrarily small. In the single delay case, the delay margin can be calculated explicitly without incurring any additional conservatism. In the general case, this condition can be reduced with some (typically small) conservatism to finite‐dimensional LMIs. Finally, this approach is also extended to the analysis of ??_∞ performance for linear time‐delay systems with an exogenous disturbance. Copyright © 2003 John Wiley & Sons, Ltd.     

Control of discrete‐time periodic linear systems with input saturation via multi‐step periodic invariant sets

This paper studies local control of discrete‐time periodic linear systems subject to input saturation by using the multi‐step periodic invariant set approach. A multi‐step periodic invariant set refers to a set from which all trajectories will enter a periodic invariant set after finite steps, remain there forever, and eventually converge to the origin as time approaches infinity. The problems of (robust) estimation of the domain of attraction, (robust) local stabilization (with bounded uncertainties), and disturbance rejection are considered. Compared with the conventional periodic invariant set approach, which has been used in the literature for local stability analysis and stabilization of discrete‐time periodic linear systems subject to input saturation, this new invariant set approach is capable of significantly reducing the conservatism by introducing additional auxiliary variables in the set invariance conditions. Moreover, the new approach allows to design (robust) stabilizing periodic controller, in the presence of norm bounded uncertainties, whose period is the same as the open‐loop system and is different from the existing periodic enhancement approach by which the period of the controller is multiple times of the period of the open‐loop system. Several numerical examples are worked out to show the effectiveness of the proposed approach. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

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.     

Output‐feedback control of a class of stochastic nonlinear systems with linearly bounded unmeasurable states

In this paper, the problems of stochastic disturbance attenuation and asymptotic stabilization via output feedback are investigated for a class of stochastic nonlinear systems with linearly bounded unmeasurable states. For the first problem, under the condition that the stochastic inverse dynamics are generalized stochastic input‐to‐state stable, a linear output‐feedback controller is explicitly constructed to make the closed‐loop system noise‐to‐state stable. For the second problem, under the conditions that the stochastic inverse dynamics are stochastic input‐to‐state stable and the intensity of noise is known to be a unit matrix, a linear output‐feedback controller is explicitly constructed to make the closed‐loop system globally asymptotically stable in probability. Using a feedback domination design method, we construct these two controllers in a unified way. Copyright © 2007 John Wiley & Sons, Ltd.     

OPTIMAL REJECTION WITH ZERO STEADY‐STATE ERROR OF SINUSOIDAL DISTURBANCES FOR TIME‐DELAY SYSTEMS

This paper studies the problem of optimal rejection with zero steady‐state error of sinusoidal disturbances for linear systems with time‐delay. Based on the internal model principle, a disturbance compensator is constructed to counterbalance the external sinusoidal disturbances, so that the original system can be transformed into an augmented system without disturbances. Then, with the introduction of a sensitivity parameter and expanding power series around it, the optimal disturbance rejection problem can be simplified to the problem of solving an infinite sum of a linear optimal control series without time‐delay or disturbance. The optimal control law for disturbance rejection with zero steady‐state error consists of accurate linear state feedback terms and a time‐delay compensating term, which is an infinite sum of an adjoint vector series. In the presented approach, iteration is required only for the time‐delay compensation series. By intercepting a finite sum of the compensation series, we obtain an approximate physically realizable optimal control law that avoids complex calculation. A numerical simulation shows that the algorithm is effective and easy to implement.     

Delay‐range‐dependent control of nonlinear time‐delay systems under input saturation

This paper describes a delay‐range‐dependent local state feedback controller synthesis approach providing estimation of the region of stability for nonlinear time‐delay systems under input saturation. By employing a Lyapunov–Krasovskii functional, properties of nonlinear functions, local sector condition and Jensen's inequality, a sufficient condition is derived for stabilization of nonlinear systems with interval delays varying within a range. Novel solutions to the delay‐range‐dependent and delay‐dependent stabilization problems for linear and nonlinear time‐delay systems, respectively, subject to input saturation are derived as specific scenarios of the proposed control strategy. Also, a delay‐rate‐independent condition for control of nonlinear systems in the presence of input saturation with unknown delay‐derivative bound information is established. And further, a robust state feedback controller synthesis scheme ensuring L₂ gain reduction from disturbance to output is devised to address the problem of the stabilization of input‐constrained nonlinear time‐delay systems with varying interval lags. The proposed design conditions can be solved using linear matrix inequality tools in connection with conventional cone complementary linearization algorithms. Simulation results for an unstable nonlinear time‐delay network and a large‐scale chemical reactor under input saturation and varying interval time‐delays are analyzed to demonstrate the effectiveness of the proposed methodology. Copyright © 2015 John Wiley & Sons, Ltd.     

Synthesis of Mixed Objective Output Feedback Robust Model Predictive Control

Aiming at the constrained polytopic uncertain system with energy‐bounded disturbance and unmeasurable states, a novel synthesis scheme to design the output feedback robust model predictive control(MPC)is put forward by using mixed H₂/H_∞ design approach. The proposed scheme involves an offline design of a robust state observer using linear matrix inequalities(LMIs)and an online output feedback robust MPC algorithm using the estimated states in which the desired mixed objective robust output feedback controllers are cast into efficiently tractable LMI‐based convex optimization problems. In addition, the closed‐loop stability and the recursive feasibility of the proposed robust MPC are guaranteed through an appropriate reformulation of the estimation error bound (EEB). A numerical example subject to input constraints illustrates the effectiveness of the proposed controller.     

Robust Finite‐Time H∞ Control of a Class of Disturbed Systems using Lmi‐Based Approach

In this paper, the definition of robust finite‐time H_∞ control is presented for a class of disturbed systems. Time‐varying norm‐bounded exogenous disturbance is considered in the system. A state feedback controller is designed, via a Linear Matrix Inequalities (LMIs) approach, which ensures that the closed‐loop system is finite‐time bounded (FTB) and reduces the effect of the disturbance input on the controlled output to a prescribed level. The main result, derived by Lyapunov functions, is a sufficient condition for FTB of disturbed systems and the sufficient condition can be reduced to a feasibility problem involving LMIs. Then a DC motor position control problem is simulated as a demonstration for this study. Simulation results are presented to show the effectiveness of the proposed method as a promising approach for controlling similar disturbed systems.     

A periodic approach for input‐delay problems: Application to network controlled systems affected by polytopic uncertainties

This paper deals with robust stability and stabilization of linear discrete‐time systems subject to uncertainties and network constraints. In network control systems, the control loop is closed over a network, which induces additional dynamics to the original control loop such as delays, sampling, and quantization among many others. This paper focuses on networked induced delays due to unreliable network for which packet losses may occur. An equivalent periodic‐like representation of the resulting system is proposed. This allows first to revisit existing results in this framework and second to take model uncertainties into account by analyzing the closed‐loop model by means of a recent method based on robust control for discrete‐time time‐varying systems. Stability analysis and dynamic state‐feedback stabilization are characterized via new conditions, whose conservatism is extensively discussed. Effectiveness of the proposed methodology is illustrated by numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Global robust output regulation of multivariable systems with nonlinear exosystem

It is well known that multi‐input, multi‐output nature of nonlinear system and generalized exosystem have posed some challenges to output regulation theory. Recently, the global robust output regulation problem for a class of multivariable nonlinear system subject to a linear neutrally stable exosystem has been studied. It has been shown that a linear internal model‐based state feedback control law can lead to the solution of previous problem. In this paper, we will further study the global robust output regulation problem of the system subject to a nonlinear exosystem. By utilizing nonlinear internal model design and decomposing the multi‐input control problem into several single‐input control problems, we will solve the problem by recursive control law design. The theoretical result is applied to the non‐harmonic load torque disturbance rejection problem of a surface‐mounted permanent magnet synchronous motor. Copyright © 2016 John Wiley & Sons, Ltd.     

Stability and control of uncertain ICPT system considering time‐varying delay and stochastic disturbance

This paper studies the problem of robust exponential stability for uncertain inductively coupled power transfer (ICPT) system considering time‐varying delay and stochastic disturbance. Firstly, the model of the system is set up via a time domain method. Secondly, based on the Newton‐Leibnitz formula and stability theory, a new stability analysis of the ICPT system with uncertain parameter and time‐varying delay or stochastic disturbance is presented respectively. The proposed approach can be applied for analyzing other similar systems, and the obtained results can be also used for estimating convergence rate and region of stability. Thirdly, based on the Lyapunov‐Krasovskii functional (LKF) approach and the stochastic stability theory, robust exponential stability criteria in the mean square are derived and the relevant controller is designed. The proposed method can further reduce conservatism. Finally, the correctness and effectiveness of the obtained results are verified by an example and simulations indicate that the larger time delay, the slower attenuation. Simultaneously, the uncertain parameter and the stochastic disturbance have a significant influence on the region of stability. In addition, in respect of reducing conservatism, the effectiveness of the proposed method is demonstrated by comparing with other papers. In addition, the designed controller shows better performance for the ICPT system with the parameter uncertainty, the time‐varying delay, and the stochastic disturbance.     

Finite-time control of linear parameter-varying systems with norm-bounded exogenous disturbance

In this paper, a new approach is presented for finite-time control problems for linear systems subject to time-varying parametric uncertainties and exogenous disturbance. The disturbance is assumed to be time varying and bounded. Sufficient conditions are obtained for the existence of a linear parameter-dependent state feedback gain, which can ensure that the closed-loop system is finite-time bounded (FTB). The conditions can be reduced to feasibility problems involving LMIs. Numerical examples show the validity of the proposed methodology.     

Polynomial filtering for nonlinear stochastic systems with state‐ and disturbance‐dependent noises

This article is concerned with the polynomial filtering problem for a class of nonlinear stochastic systems governed by the Itô differential equation. The system under investigation involves polynomial nonlinearities, unknown‐but‐bounded disturbances, and state‐ and disturbance‐dependent noises ((x,d)‐dependent noises for short). By expanding the polynomial nonlinear functions in Taylor series around the state estimate, a new polynomial filter design method is developed with hope to reduce the conservatism of the existing results. In virtue of stochastic analysis and inequality technique, sufficient conditions in terms of parameter‐dependent linear matrix inequalities (PDLMIs) are derived to guarantee that the estimation error system is input‐to‐state stable in probability. Moreover, the desired polynomial matrix can be obtained by solving the PDLMIs via the sum‐of‐squares approach. The effectiveness and applicability of the proposed method are illustrated by two numerical examples with one concerning the permanent magnet synchronous motor.     

Reduced Order Robust Gain‐Scheduling Control Of The Diesel Apu For Series Hybrid Vehicles

This paper presents a reduced order robust gain‐scheduling approach for the control of the diesel auxiliary power unit (APU) for series hybrid vehicles. The nonlinear plant dynamics are converted into a linear parameter‐varying (LPV) form with parametric uncertainties, in which only partial information of the plant states is available. For this type of LPV system, a new reduced order robust gain‐scheduling synthesis method is proposed based on partial state feedback. The parametric uncertainties are considered using multipliers to reduce the conservatism. The reduced order synthesis problem is solved offline by means of linear matrix inequalities (LMIs), and the synthesis result requires much simpler online computation than the explicit controller formulas do. The synthesis method is applied to the diesel APU controller design, and simulation results are given to demonstrate the controller performance.     

Nonlinear generalization of scaled ℋ︁∞ control: global robustification against nonlinearly bounded uncertainties

This paper proposes a novel approach to the problem of ??₂ disturbance attenuation with global stability for nonlinear uncertain systems by placing great emphasis on seamless integration of linear and nonlinear controllers. This paper develops a new concept of state‐dependent scaling adapted to dynamic uncertainties and nonlinear‐gain bounded uncertainties that do not necessarily have finite linear‐gain, which is a key advance from previous scaling techniques. The proposed formulation of designing global nonlinear controllers is not only a natural extension of linear robust control, but also the approach renders the nonlinear controller identical with the linear control at the equilibrium. This paper particularly focuses on scaled ??^∞ control which is widely accepted as a powerful methodology in linear robust control, and extends it nonlinearly. If the nonlinear system belongs to a generalized class of triangular systems allowing for unmodelled dynamics, the effect of the disturbance can be attenuated to an arbitrarily small level with global asymptotic stability by partial‐state feedback control. A procedure of designing such controllers is described in the form of recursive selection of state‐dependent scaling factors. Copyright © 2004 John Wiley & Sons, Ltd.