Fixed‐structure controller design for polytopic systems via LMIs

In this paper, a new approach for fixed‐structure H₂ controller design in terms of solutions to a set of linear matrix inequalities are given. Both discrete‐time and continuous‐time SISO time‐invariant systems are considered. Then the results are extended to systems with polytopic uncertainty. The presented methods are based on an inner convex approximation of the non‐convex set of fixed‐structure H₂ controllers. The designed procedures are initialized either with a stable polynomial or with a stabilizing controller. An iterative procedure for robust controller design is given that converges to a suboptimal solution. The monotonic decreasing of the upper bound on the H₂ norm is established theoretically for both nominal and robust controller design. Copyright © 2014 John Wiley & Sons, Ltd.     

Finite‐time convergence results in robust model predictive control

Robust asymptotic stability (asymptotic attractivity and ?‐δ stability) of equilibrium regions under robust model predictive control (MPC) strategies was extensively studied in the last decades making use of Lyapunov theory in most cases. However, in spite of its potential application benefits, the problem of finite‐time convergence under fixed prediction horizon has not received, with some few exceptions, much attention in the literature. Considering the importance in several applications of having finite‐time convergence results in the context of fixed horizon MPC controllers and the lack of studies on this matter, this work presents a new set‐based robust MPC (RMPC) for which, in addition to traditional stability guarantees, finite‐time convergence to a target set is proved, and moreover, an upper bound on the time necessary to reach that set is provided. It is remarkable that the results apply to general nonlinear systems and only require some weak assumptions on the model, cost function, and target set.     

On L1‐minimization in optimal control and applications to robotics

In this paper, we analyze optimal control problems with control variables appearing linearly in the dynamics. We discuss different cost functionals involving the L^p‐norm of the control. The case p = 0 represents the time‐optimal control, the case p > 1 yields a standard smooth optimal control problem, whereas the case p = 1 leads to a nonsmooth cost functional. Several techniques are developed to deal with the nonsmooth case p = 1. We present a thorough theoretical discussion of the necessary conditions. Two types of numerical methods are developed: either a regularization technique is used or an augmentation approach is applied in which the number of control variables is doubled. We show the precise relations between the L¹‐minimal control and the bang–bang or singular controls in the augmented problem. Using second‐order sufficient conditions (SSC) for bang–bang controls, we obtain SSC for L¹‐minimal controls. The different techniques and results are illustrated with an example of the optimal control for a free‐flying robot which is taken from Sakawa. Copyright © 2006 John Wiley & Sons, Ltd.     

H∞ output tracking control over networked control systems with Markovian jumping parameters

The problem of H_∞ output tracking control over networked control systems (NCSs) with communication limits and environmental disturbances is studied in this paper. A wide range of time‐varying stochastic problem arising in networked tracking control system is reduced to a standard convex optimization problem involving linear matrix inequalities (LMIs). The closed‐loop hybrid NCS is modeled as a Markov jump linear system in which random time delays and packet dropouts are described as two stochastic Markov chains. Gridding approach is introduced to guarantee the finite value of the sequences of transmission delays from sensor to actuator. Sufficient conditions for the stochastic stabilization of the hybrid NCS tracking system are derived by the LMI‐based approach through the computation of the optimal H_∞ performance. The mode‐dependent robust H_∞ output tracking controller is obtained by the optimal iteration method. Numerical examples are given to demonstrate the effectiveness of the proposed robust output tracking controller for NCS. Copyright © 2016 John Wiley & Sons, Ltd.     

Robust H∞ performance analysis for uncertain discrete‐time singular systems with time‐varying delays

This paper considers the problem of robust H∞ performance analysis for uncertain discrete‐time singular systems with time‐varying delays. Firstly, a delay‐dependent stability criterion under the H∞ performance index for the systems is given based on constructing a generalized Lyapunov–Krasovskii function and introducing a new difference inequality. Then, a sufficient condition ensuing the system to be regular, causal as well as stable for all admissible uncertainties is proposed in terms of a set of strict linear matrix inequalities (LMIs). Finally, we provide examples to show the reduced conservatism and effectiveness of the proposed conditions. Copyright © 2014 John Wiley & Sons, Ltd.     

Analysis of optimal controls for a mathematical model of tumour anti‐angiogenesis

Anti‐angiogenic therapy is a novel treatment approach for cancer that aims at preventing a tumour from developing its own blood supply system that it needs for growth. In this paper we consider a mathematical model where the endogenous stimulation term in the dynamics is taken proportional to the number of endothelial cells. This system is an example from a class of mathematical models for anti‐angiogenic treatment that were derived from a biologically validated model by Hahnfeldt, Panigrahy, Folkman and Hlatky. The problem how to schedule a given amount of angiogenic inhibitors to achieve a maximum reduction in the primary cancer volume is considered as an optimal control problem and it is shown that optimal controls are bang‐bang of the type 0a0 with 0 denoting a trajectory corresponding to no treatment and a a trajectory with treatment at maximum dose along that all inhibitors are being exhausted. Copyright © 2007 John Wiley & Sons, Ltd.     

Robust H∞ guaranteed cost control for discrete‐time switched singular systems with time‐varying delay

This paper investigates the problem of control design for a class of uncertain switched singular systems with time‐varying delay. Under mode‐dependent average dwell time and using an appropriate Lyapunov‐Krasovskii functional, the exponential admissibility of the system is analyzed. In order to obtain less conservative conditions, the delay partitioning technique is adopted as well as the improved reciprocally convex approach. By means of the developed admissibility condition, a static output feedback controller is then designed using linear matrix inequality approach. Moreover, by solving an optimization convex problem with constraints, the switched controller is developed to ensure simultaneously the stability of the closed‐loop system and satisfy an optimized upper bound of both the linear quadratic guaranteed cost and the H_∞ norm. Numerical examples are proposed to verify the efficiency and the merits of the method proposed.     

Linear‐quadratic control problems with L1‐control cost

We analyze a class of linear‐quadratic optimal control problems with an additional L¹‐control cost depending on a parameter β. To deal with this nonsmooth problem, we use an augmentation approach known from linear programming in which the number of control variables is doubled. It is shown that if the optimal control for a given is bang‐zero‐bang and the switching function has a stable structure, the solutions are Lipschitz continuous functions of the parameter β. We also show that in this case the optimal controls for β ^* and a with | β ? β ^* | sufficiently small coincide except on a set of measure . Finally, we use the augmentation approach to derive error estimates for Euler discretizations. Copyright © 2014 John Wiley & Sons, Ltd.     

Secondary delay‐partition approach on robust performance analysis for uncertain time‐varying Lurie nonlinear control system

This paper investigates the problem of robust performance analysis for Lurie nonlinear control system with parameter uncertainties and interval time‐varying delays. Based on an augmented Lyapunov‐Krasovskii functional including multiple integral terms, new delay‐dependent robust stability criteria are derived by proposing a novel secondary delay‐partition approach. Moreover, to obtain less conservative stability conditions, an optimized integral inequality is developed by introducing an adjustable parameter ? ₁. Finally, 5 numerical simulation examples are given to illustrate the effectiveness and advantages of the proposed results.     

Robust continuous‐time controller design via structural Youla–Kučera parameterization with application to predictive control

This paper addresses the continuous‐time control of uncertain linear SISO plants and its nominal and robust stability and nominal and robust performance objectives. A specific application of the Youla–Ku?era (Q) parameterization concept leads to a new development of observer‐like controller structures. This method is combined with a nominal design of continuous‐time generalized predictive control suitable for both minimum‐phase and non‐minimum‐phase plants. The subsequent design procedure consists of two steps. Firstly, the nominal stability and nominal performance of the control system are established by using an analytical design methodology, based on a collection of closed‐loop prototype characteristics with definite time‐domain specifications. And secondly, a generic structure of the controller is enhanced by suitable Q‐parameters guaranteeing that the control system has the required robustness properties. The proposed structural (reduced‐order) Q‐parameterization relies on an observer structure of controllers, which can be easily enhanced with certain filters necessary for control robustification. To reduce the complexity of the resulting robust controllers, we suggest using a structural factorization, which allows for simple forms of robustifying (phase‐lag) correctors of low order, easy for implementation, and convenient for optimization and tuning. Two numerical examples are given to illustrate the composed technique and its practical consequences. Copyright © 2004 John Wiley & Sons, Ltd.     

H∞ control of switched linear discrete‐time systems with polytopic uncertainties

In this paper, the problem of designing H_∞ state‐feedback controllers for switched linear discrete‐time systems with polytopic uncertainties is investigated. Two approaches on designing robust and parameter‐dependent H_∞ controllers are proposed and the existence conditions of the desired controllers are derived and formulated in terms of a set of linear matrix inequalities. By solving the corresponding convex optimization problem, the desired controllers are obtained, respectively, and different optimal H_∞ noise‐attenuation level bounds of corresponding closed‐loop systems are given as well. The designed controllers have their own advantages and disadvantages regarding the conservatism and realization complexity. An illustrative example emerging in networked control systems (NCS) and numerical simulations are presented to show the applicability and effectiveness of the obtained theoretic results. Copyright © 2006 John Wiley & Sons, Ltd.     

H∞ state‐feedback controller design for continuous‐time nonhomogeneous Markov jump systems

This paper studies the problem of H_∞ state‐feedback controller design for continuous‐time nonhomogeneous Markov jump systems. The time‐varying transition rate matrix in continuous‐time domain is considered to lie in a convex bounded domain of polytopic type. By constructing a parameter‐dependent Lyapunov function and fully considering the information about the rate of change of time‐varying parameters, a new sufficient condition on the existence of an H_∞ state‐feedback controller is provided in the form of a parameter‐dependent matrix inequality. Moreover, based on the structure characteristics of Lyapunov matrix and transition matrix, the parameter‐dependent matrix inequality is converted into a finite set of linear matrix inequalities, which can be readily solved. Two numerical examples are provided to demonstrate the effectiveness of the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Observer‐based H∞ control for uncertain singular time‐delay systems with actuator saturation

In this paper, the observer‐based H_∞ control problem for uncertain singular time‐delay systems with actuator saturation is concerned. First, a delay‐dependent linear matrix inequality (LMI) condition is obtained which, guarantees that the uncertain singular time‐delay systems with actuator saturation are regular, impulse free, and asymptotically stable with H_∞ performance condition. Then, with this condition, the estimation of stability region and the design method of observer‐based H_∞ controller are given by solving LMIs and convex optimization problem. Finally, some numerical examples are provided to demonstrate the merit of the obtained results. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal trajectories for angular systems on the projective line

We analyze two optimal problems for a class of nonlinear system on the real projective line Popf¹ induced by a class of bilinear control system: the angular system. Two functional costs are considered: time‐optimal and quadratic. According to the Pontryagin Maximum Principle, in the time‐optimal case we show that if the angle system ?Σ satisfies the controllability property, then there exists a minimal time bang‐bang trajectory connecting any two points on ?¹, the noncontrollable case was discussed in closed form in (SIAM J. Control Optim. 2009; 48 (4):2636–2650). On the other hand, in the quadratic cost, the optimal control is a continuous function (Proyecciones J. Math. 2010; 29 (2):145–164). A comparison is also established between the structure of the solutions for the two optimal problems: time‐optimal and quadratic in the controllable and noncontrollable cases. The extremals are obtained from the adjoint system given by the Pontryagin Maximum Principle onto ?¹ via radial projection. An example is given. Copyright © 2011 John Wiley & Sons, Ltd.     

H∞ control for sampled‐data linear systems with two Markov processes

This paper is concerned with the design problem of H_∞ digital switching control for linear continuous systems with Markovian jumping parameters. The controller is digital and monitored by the jumping parameters of the plant. The closed‐loop system is a hybrid one defined on a hybrid time space (composed of a continuous‐time and a discrete‐time) and a sample space. The sample space is specified by two separable continuous‐time discrete‐state Markov processes, one appearing in the open‐loop system, and the other appearing in control action, which is different with the traditional Markovian jumping process. Our attention is focused on designing digital output feedback controllers for the system with two Markovian jumping processes such that both stochastic stability and a prescribed H_∞ performance are achieved. The problem of robust H_∞ control for systems with parameters uncertainties is also studied. It is shown that the sampled‐data control problems for linear Markovian jumping systems with and without parameter uncertainties can be solved in terms of the solutions to a set of intercoupled matrix inequalities. Two numerical examples are given to show the design procedures. Copyright © 2005 John Wiley & Sons, Ltd.     

Robust reliability method and reliability‐based performance optimization for non‐fragile robust control design of dynamic system with bounded parametric uncertainties

An efficient robust reliability method for non‐fragile robust control design of dynamic system with bounded parametric uncertainties is presented systematically, in which the uncertainties existing in the controlled plant and controller realization are taken into account simultaneously in an integrated framework. Reliability‐based design optimization of non‐fragile robust control for parametric uncertain systems is carried out by optimizing the H₂ and H_∞ performances of the closed‐loop system, with the constraints on robust reliabilities. The non‐fragile robust controller obtained by the presented method may possess a coordinated optimum performance satisfying the precondition that the system is robustly reliable with respect to the uncertainties existing in controlled plant and controller. Moreover, the robustness bounds of uncertain parameters can be provided. The presented formulations are within the framework of linear matrix inequality and thus can be carried out conveniently. It is demonstrated by a numerical example that the presented method is effective and feasible. Copyright © 2016 John Wiley & Sons, Ltd.     

Fixed-order scaled ℋ∞ synthesis

Given a stabilizing fixed-order controller, we propose two algorithms which improve its robust stability and robust performance in the framework of the ℋ_∞ control problem with constant scaling. The idea is to formulate the scaled ℋ_∞ control problem as generalized eigenvalue minimization problems involving (non-linear) matrix inequalities, and then to apply co-ordinate descent algorithms which split the problem into successive (quasi)convex minimization problems. These methods can be considered an extension of the standard μ-synthesis method (the D – K iteration) for fixed-order controller design. Our methods are different from the standard D – K -type iterations in that the analytic centres are computed at each step instead of minimizing objective functions. The controllers obtained may not be globally optimal in general, but are guaranteed to be better than the initial controller. Hence, our methods can be used to improve robustness/performance of a given fixed-order stabilizing controller. Illustrative examples are given for a benchmark problem. © 1997 John Wiley & Sons, Ltd.     

Expanded proximate time‐optimal servo control of permanent magnet synchronous motor

This paper extends the existing proximate time‐optimal servomechanism control methodology to the more typical second‐order servo systems with a damping element. A parameterized design of expanded proximate time‐optimal servomechanism control law with a speed‐dependent linear region is presented for rapid and smooth set‐point tracking using a bounded input signal. The control scheme uses the time‐optimal bang‐bang control law to accomplish maximum acceleration or braking whenever appropriate and then smoothly switches into a linear control law to achieve a bumpless settling. The closed‐loop stability is analyzed, and then the control scheme is applied to the position–velocity control loop in a permanent magnet synchronous motor servo system for set‐point position regulation. Numerical simulation has been conducted, followed by experimental verification based on a TMS320F2812 digital signal controller board. The results confirm that the servo system can track a wide range of target references with superior transient performance and steady‐state accuracy. Copyright © 2015 John Wiley & Sons, Ltd.     

Further results on H ∞ control for discrete‐time uncertain singular systems with interval time‐varying delays in state and input

This paper investigates the state feedback robust H _∞ control problem of a class of discrete‐time singular systems with norm‐bounded uncertainties and interval time‐varying delays in state and input. A new bounded real lemma for discrete‐time singular systems with a pair of time‐varying interval state delays is first investigated. Mathematical comparisons of the new bounded real lemma and two existing ones are presented. Then, on the basis of the bounded real lemma proposed here, a sufficient condition in the form of nonlinear matrix inequality, such that the considered state feedback robust H _∞ control problem is solvable, is given. In order to solve the nonlinear matrix inequality, a cone complementarity linearization algorithm is offered. Several numerical examples are presented to show the applicability of the proposed approach. Copyright © 2012 John Wiley & Sons, Ltd.