and control design for polytopic continuous‐time Markov jump linear systems with uncertain transition rates

This paper investigates the problems of and state feedback control design for continuous‐time Markov jump linear systems. The matrices of each operation mode are supposed to be uncertain, belonging to a polytope, and the transition rate matrix is considered partly known. By appropriately modeling all the uncertain parameters in terms of a multi‐simplex domain, new design conditions are proposed, whose main advantage with respect to the existing ones is to allow the use of polynomially parameter‐dependent Lyapunov matrices to certify the mean square closed‐loop stability. Synthesis conditions are derived in terms of matrix inequalities with a scalar parameter. The conditions, which become LMIs for fixed values of the scalar, can cope with and state feedback control in both mode‐independent and mode‐dependent cases. Using polynomial Lyapunov matrices of larger degrees and performing a search for the scalar parameter, less conservative results in terms of guaranteed costs can be obtained through LMI relaxations. Numerical examples illustrate the advantages of the proposed conditions when compared with other techniques from the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

New iterative linear matrix inequality based procedure for H2 and H∞ state feedback control of continuous‐time polytopic systems

This article provides new linear matrix inequality (LMI) sufficient conditions for a generalized robust state feedback control synthesis problem for linear continuous‐time polytopic systems. This generalized problem includes the robust stability, H₂ ‐norm, and H_∞ ‐norm problems as special cases. Using a novel general separation result, which separates the state feedback gain from the Lyapunov matrix but with the state feedback gain synthesized from the slack variable, then allows the formulation of LMI sufficient conditions for the generalized problem. Compared to existing parameterized LMI based conditions, where auxiliary scalar parameters are introduced in order to include the quadratic stability conditions (ie, assuming a constant Lyapunov matrix) as a special case, the proposed new conditions are true LMIs and contain as a particular case the optimal quadratic stability solution. Utilizing any initial solution derived by the quadratic or some existing methods as a starting solution, we propose an algorithm based on an iterative procedure, which is recursively feasible in each update, to compute a sequence of nonincreasing upper bounds for the H₂ ‐norm and H_∞ ‐norm. In addition, if no feasible initial solution can be found for some uncertain systems using any existing methods, another algorithm is presented that offers the possibility of obtaining a robust stabilizing gain. Numerical examples from the literature demonstrate that our algorithms can provide less conservative results than existing methods, and they can also find feasible solutions where all other methods fail.     

线性不确定系统的H∞状态反馈鲁棒重复控制

现有的重复控制设计不能同时优化低通滤波器的参数和重复控制器的参数.我们在设计重复控制系统以控制线性不确定对象时,解决了这个问题.首先,引入状态反馈以保证闭系统的鲁棒稳定性,把重复控制器设计问题转化为H∞状态反馈增益的设计问题.为获得低通滤波器最大转折频率,进一步将设计问题转化为基于线性矩阵不等式约束的凸优化问题.提出了一种迭代算法,用以计算低通滤波器的最大转折频率和H∞状态反馈增益.在保证系统鲁棒稳定性的同时,获得最高控制精度的重复控制器和低通滤波器的参数组合.该方法与已有方法比较,它的结果容易验证和求解,因而更适合于实际应用.最后,通过数值实例验证了本文所提方法的有效性.     

Dynamic output‐feedback control for continuous‐time singular Markovian jump systems

This paper considers a dynamic output‐feedback control for continuous‐time singular Markovian jump systems, whereas the existing research studies in literature focused on state‐feedback or static output‐feedback control. While they have only provided the sufficient conditions, this paper successfully obtains the necessary and sufficient condition for the existence of the dynamic output‐feedback control. Furthermore, this condition is expressed with linear matrix inequalities by the so‐called replacement technique. Two numerical examples show the validity of the resulting control.     

Static output feedback control for positive linear continuous‐time systems

This paper studies the problem of designing the static output feedback controller for the positive linear continuous‐time systems. On the basis of a system augmentation approach, a novel characterization on the stable condition of the closed‐loop system is firstly established. Then, a necessary and sufficient condition is given to ensure the existence of the desired static output feedback controller, and an iterative linear matrix inequality algorithm is presented to compute the feedback gain matrix. Finally, a numerical example is provided to illustrate the effectiveness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

Robust output‐feedback control of state‐multiplicative noisy discrete‐time systems

Linear discrete‐time systems with stochastic and deterministic polytopic type uncertainties in their state‐space model are considered. A dynamic output‐feedback controller is obtained via a new approach that allows a derivation of a controller in spite of parameter uncertainty. In the proposed approach, the system is described via a difference equation and an augmented system is then used to obtain the output‐feedback controller parameters. The controller is obtained without assuming a specific structure to the quadratic Lyapunov function, and it is the first time that an output‐feedback controller is obtained for robust state‐multiplicative systems. The controller minimizes the stochastic L₂‐gain of the closed‐loop where a cost function is defined to be the expected value of the standard performance index with respect to the stochastic uncertainty. Two examples are given where the second of which demonstrates the applicability of our theory to a robot manipulator system. Copyright © 2016 John Wiley & Sons, Ltd.     

Output feedback preview control for polytopic uncertain discrete‐time systems with time‐varying delay

This paper considers the preview tracking control problem of polytopic uncertain discrete‐time systems with a time‐varying delay subject to a previewable reference signal. First, a model transformation is employed and a discrete‐time system with a time‐invariant delay and an external disturbance is obtained. The difference operator method can be extended to derive an augmented error system that includes future information on the reference signal. Then, a previewable reference signal is fully utilized through reformulation of the output equation while considering the output feedback. Based on the small gain theorem, a static output feedback controller with preview actions is designed such that the output can asymptotically track the reference signal. Finally, numerical simulation examples also illustrate the superiority of the desired preview controller for the uncertain system in the paper.     

Switched state feedback for uncertain continuous‐time systems with interval delays

The objective of this paper is to design a switched robust control for a class of continuous‐time systems subject to linear fractional uncertainty and interval time delays. The controller is based on state feedback and the LMI‐based stability conditions are derived using an improved Lyapunov–Krasovskii functional. Moreover, the switching rule as well as the state feedback gains are determined from the minimization of a guaranteed cost function. The theoretical results are illustrated with a numerical example. Copyright © 2010 John Wiley & Sons, Ltd.     

Output feedback preview tracking control for discrete-time polytopic time-varying systems

This paper focuses on the problem of static output feedback preview tracking control for discrete-time systems with time-varying parameters subject to a previewable reference signal. We develop a design method of a robust controller with integral and preview actions achieving robust tracking performance. First, an augmented error system including future information on reference signal is constructed by introducing two new related auxiliary variables to the original system state and input. This leads to a tracking problem being transformed into a regulator problem. Then, a previewable reference signal is fully utilised through reformulation of the output equation for the augmented error system while considering a static output feedback. Meanwhile, the static output preview control gains are solved explicitly by the proposed conditions. Finally, a numerical example is given to demonstrate the effectiveness of the proposed method.     

Full‐complexity polytopic robust control invariant sets for uncertain linear discrete‐time systems

This paper presents an algorithm for the computation of full‐complexity polytopic robust control invariant (RCI) sets, and the corresponding linear state‐feedback control law. The proposed scheme can be applied for linear discrete‐time systems subject to additive disturbances and structured norm‐bounded or polytopic uncertainties. Output, initial condition, and performance constraints are considered. Arbitrary complexity of the invariant polytope is allowed to enable less conservative inner/outer approximations to the RCI sets whereas the RCI set is assumed to be symmetric around the origin. The nonlinearities associated with the computation of such an RCI set structure are overcome through the application of Farkas' theorem and a corollary of the elimination lemma to obtain an initial polytopic RCI set, which is guaranteed to exist under certain conditions. A Newton‐like update, which is recursively feasible, is then proposed to yield desirable large/small volume RCI sets.     

Delay‐dependent robust stability analysis for switched time‐delay systems with polytopic uncertainties

In this paper, sufficient conditions are provided for the stability of switched retarded and neutral time‐delay systems with polytopic‐type uncertainties. It is assumed that the delay in the system dynamics is time‐varying and bounded. Parameter‐dependent Lyapunov functionals are employed to obtain criteria for the exponential stability of the system in the form of linear matrix inequality (LMI). Free‐weighting matrices are then provided to express the relationship between the system variables and the terms in the Leibniz–Newton formula. Numerical examples are presented to show the effectiveness of the results. Copyright © 2014 John Wiley & Sons, Ltd.     

On robust state derivative feedback control for uncertain polytopic systems with uncertain sampling period and network-induced delay

This paper addresses the state derivative feedback control problem for uncertain polytopic systems subject to an uncertain sampling period and network-induced delay. The distinctive contribution relies on the direct design of a robust state derivative feedback controller employing an augmented discretized model derived in terms of the state derivative feedback such that network-induced delay and uncertain sampling periods can be incorporated from the original continuous-time state-space representation into the discretized model. Two augmented models are provided to handle longer input time delays, as well as delays less or equal to the sampling period. In this work, all the uncertain parameters are modeled as a polytopic form whose resulting discrete-time model has matrices with polynomial dependence on the uncertain parameters and an additive norm-bounded term featuring the discretization residual error. Moreover, synthesis conditions are derived using a set of linear matrix inequalities (LMI) to solve the stabilization problem for this class of systems under different input time delays. Finally, numerical simulations are carried out to evaluate the effectiveness of the proposed method.     

A new methodology to compute stabilizing control laws for continuous‐time LTV systems

This paper is devoted to the problem of computing control laws for the stabilization of continuous‐time linear time‐varying systems. First, a necessary and sufficient condition to assess the stability of a linear time‐varying system based on the norm of the transition matrix computed over a sequence of successive finite‐time intervals is proposed. A link with a stability condition for an equivalent discrete‐time model is also established. Then, 3 approaches for the computation of stabilizing state‐feedback gains are proposed: a continuous‐time technique, ie, directly derived from the stability condition, not suitable for numerical implementation; a method based on the stabilization of the discrete‐time equivalent model along with a transformation to generate the desired continuous‐time gain; and the computation of stabilizing gains for a set of periodic discrete‐time systems. Finally, by adapting one of the existing methods for the stabilization of periodic discrete‐time systems, an algorithm for the computation of a stabilizing state‐feedback continuous‐time gain is proposed. A numerical example illustrates the validity of the technique.     

Network‐based robust H∞ control of continuous‐time systems with uncertainty

This paper studies the problem of robust H_∞ control for continuous‐time networked control systems (NCSs). A new type of Lyapunov functionals is exploited to derive sufficient conditions for guaranteeing the robust exponential stability and H_∞ performance of the considered system. It is shown that the new result is less conservative than the existing corresponding ones. Meanwhile, a method of eliminating redundant variables to reduce computational complexity is given, which is also applied to design state feedback H_∞ controllers, and the design condition is given in terms of solutions to a set of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness of the proposed methods. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Linear quadratic networked control of uncertain polytopic systems

This paper investigates the problem of designing robust linear quadratic regulators for uncertain polytopic continuous‐time systems over networks subject to delays. The main contribution is to provide a procedure to determine a discrete‐time representation of the weighting matrices associated to the quadratic criterion and an accurate discretized model, in such a way that a robust state feedback gain computed in the discrete‐time domain assures a guaranteed quadratic cost to the closed‐loop continuous‐time system. The obtained discretized model has matrices with polynomial dependence on the uncertain parameters and an additive norm‐bounded term representing the approximation residual error. A strategy based on linear matrix inequality relaxations is proposed to synthesize, in the discrete‐time domain, a digital robust state feedback control law that stabilizes the original continuous‐time system assuring an upper bound to the quadratic cost of the closed‐loop system. The applicability of the proposed design method is illustrated through a numerical experiment. Copyright © 2015 John Wiley & Sons, Ltd.     

Network‐based robust event‐triggered control for continuous‐time uncertain semi‐Markov jump systems

This article investigates the event‐triggered (ET) states feedback robust control problem for a class of continuous‐time networked semi‐Markov jump systems (S‐MJSs). An ET scheme, which depends on semi‐Markov process, is presented to design a suitable controller and save communication resources. To cope with the network transmission delay phenomenon, a time‐delay S‐MJSs model under the ET scheme is introduced to describe this phenomenon. Then, it is assumed that the communication links between event detector and zero‐order holder are imperfect, where the signal quantization and the actuator fault occur simultaneously. The sufficient conditions are derived by means of linear matrix inequalities approach, which guarantees the stochastic stability of the constructed time‐delay S‐MJSs in an optimized performance level. Based on these criteria, the parameters of controller under the ET scheme are readily calculated. Some simulation results with respect to F‐404 aircraft engine system for two kinds of ET parameters are given to validate the proposed method.     

An interval observer for uncertain continuous‐time linear systems

To design robust interval observers for uncertain continuous‐time linear systems, a new set‐integration approach is proposed to compute trajectory tubes for the estimation error. Because this approach, the order‐preserving condition on the dynamics of the estimation error is no longer required. Therefore, synthesis methods can be used to compute observer gains that reduce the impact of the system uncertainties on the accuracy of the estimated state enclosures. The performance of the proposed approach is showcased through illustrative numerical examples.