Robust H 2 control based on dynamic multipliers

In this article, upper bounds on the worst-case H ₂ performance index relative to structured, feedback perturbations are considered which are based on the minimisation of dual Lagrangean functionals over linearly-parametrised, finite-dimensional classes of dynamic multipliers. It is shown that the minimisation problems in question can be recast as optimisation problems with linear cost functional and Linear matrix inequality (LMI) constraints. An iterative scheme is suggested to generate linearly-parametrised classes of multipliers of increasing dynamic order so that progressively tighter upper bounds can be obtained, as illustrated by two simple numerical examples. Finally, with a view to synthesis procedures based on ‘D–K iterations’ relative to multipliers and controllers, it is shown that the minimisation of the upper-bounds corresponding to given multipliers with respect to linearly-parametrised classes of Youla parameters can also be cast as linear-cost/LMI-constraint problems.     

Robust Wiener filtering with non-parametric spectral uncertainty

In this article, robust least-squares filtering problems are considered for non-parametric multivariate spectral uncertainty defined by the so-called spectral band and generalised-moment constraints. Its major aim is to provide a basis for computing approximate solutions to worst-case, Wiener-filtering optimisation problems involving causal filters and multivariate signals. It hinges upon associating upper and lower bounds on the minimum worst-case performance achievable with causal filters with linear-cost/linear matrix inequality (LC/LMI)-constraint optimisation problems. On the basis of a Lagrangean duality formulation for the worst-case, least-squares performance of a given filter, upper bounds on it are obtained as the optimal values of LC/LMI problems. Then, for linearly parameterised classes of filter transfer functions, a causal filter which optimises such an upper bound on worst-case performance can also be obtained from an LC/LMI optimisation problem. To estimate the amount of conservatism incurred when relying on such upper bounds, optimal, nominal, least-squares performance for a given pair of power spectral densities (for the information and noise signal) is maximised over finite-dimensional, linearly parameterised classes of the latter. Again, such problems are shown to be equivalent to LC/LMI problems and the corresponding optimal values are lower bounds on the minimum worst-case, least-squares error achievable in the original robust filtering problem (say, μ_*). Finally, two simple numerical examples are presented to illustrate how causal filters can be obtained whose worst-case, least-squares performance is quite close to the optimal one (i.e. μ_*).     

Robust performance analysis for systems with structured uncertainty

This paper analyses robust performance measures for linear time-invariant systems with norm-bounded time-varying structured uncertainty. We consider two robust performance measures. One is the worst-case peak value of the error signal in response to the disturbance with a known energy. The other is the worst-case energy of the error signal in response to impulsive disturbance. In both cases, the ‘worst case’ is taken over all admissible uncertainties and disturbances. The notion of robust stability is Q-stability, or the scaled H_∞ norm bound. Our main results provide an upper bound for each of the robust performance measures in terms of a positive definite matrix which satisfies a linear matrix inequality (LMI) together with a scaling matrix. Hence, the best bound in this LMI formulation can be computed by convex programming.     

LMI-based controller synthesis: A unified formulation and solution

This paper proposes a unified approach to linear controller synthesis that employs various LMI conditions to represent control specifications. We define a comprehensive class of LMIs and consider a general synthesis problem described by any LMI of the class. We show a procedure that reduces the synthesis problem, which is a BMI problem, to solving a certain LMI. The derived LMI condition is equivalent to the original BMI condition and also gives a convex parametrization of all the controllers that solve the synthesis problem. The class contains many of widely-known LMIs (for H_∞ norm, H₂ norm, etc.), and hence the solution of this paper unifies design methods that have been proposed depending on each LMI. Further, the class also provides LMIs for multi-objective performance measures, which enable a new formulation of controller design through convex optimization. © 1998 John Wiley & Sons, Ltd.     

Estimating the conservatism of Popov''s criterion for real parametric uncertainties

We consider the problems of robust stability and performance analysis of linear systems subject to parametric uncertainties. The Popov criterion provides lower bounds of stability margins and upper bounds on robust performance. In this paper we propose a method for obtaining upper bounds on stability margin or lower bounds on robust H₂ performance based on the outcome of the lower bound computation for the stability problem or the upper bound computation for the H₂ performance problem. We make several numerical tests.     

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.     

LMI optimization approach to robust H∞ observer design and static output feedback stabilization for discrete‐time nonlinear uncertain systems

A new approach for the design of robust H_∞ observers for a class of Lipschitz nonlinear systems with time‐varying uncertainties is proposed based on linear matrix inequalities (LMIs). The admissible Lipschitz constant of the system and the disturbance attenuation level are maximized simultaneously through convex multiobjective optimization. The resulting H_∞ observer guarantees asymptotic stability of the estimation error dynamics and is robust against nonlinear additive uncertainty and time‐varying parametric uncertainties. Explicit norm‐wise and element‐wise bounds on the tolerable nonlinear uncertainty are derived. Also, a new method for the robust output feedback stabilization with H_∞ performance for a class of uncertain nonlinear systems is proposed. Our solution is based on a noniterative LMI optimization and is less restrictive than the existing solutions. The bounds on the nonlinear uncertainty and multiobjective optimization obtained for the observer are also applicable to the proposed static output feedback stabilizing controller. Copyright © 2008 John Wiley & Sons, Ltd.     

Robust Analysis of Discrete‐Time Lur'e Systems with Slope Restrictions Using Convex Optimization

This paper considers robust stability and robust performance analysis for discrete‐time linear systems subject to nonlinear uncertainty. The uncertainty set is described by memoryless, time‐invariant, sector bounded, and slope restricted nonlinearities. We first give an overview of the absolute stability criterion based on the Lur'e‐Postkinov Lyapunov function, along with a frequency domain condition. Subsequently, we derive sufficient conditions to compute the upper bounds of the worst case H₂ and worst case H∞ performance. For both robust stability testing and robust performance computation, we show that these sufficient conditions can be readily and efficiently determined by performing convex optimization over linear matrix inequalities.     

Robust fuzzy stabilization of dithered chaotic systems using island-based random optimization algorithm

Applying dither to highly nonlinear systems may suppress chaotic phenomena, but dynamic performance, such as convergence rate and disturbance attenuation, is usually not guaranteed. This paper presents a dithered H_∞ robust fuzzy control scheme to stabilize chaotic systems that ensures disturbance attenuation bounds. In the proposed scheme, Takagi-Sugeno (T-S) fuzzy linear models are used to describe the relaxed models of the dithered chaotic system, and fuzzy controllers are designed based on an extension to the concept of parallel distributed compensation (PDC). Sufficient condition for the existence of the H_∞ robust fuzzy controllers is presented in terms of a novel linear matrix inequalities (LMI) form which takes full consideration of modeling error and disturbances, but cannot be solved by the standard procedures. In order to solve the LMI problem and to identify the chaotic systems as T-S fuzzy modes, we propose a compound optimization strategy called the island-based random-walk algorithm (IRA). The algorithm is composed of a set of communicating random-walk optimization procedures concatenated with the down-hill simplex method. The design procedure and validity of the proposed scheme is demonstrated via numerical simulation of the dithered fuzzy control of a chaotic system.     

On control for linear systems with interval time-varying delay

This paper deals with the problem of delay-dependent robust H_∞ control for linear time-delay systems with norm-bounded, and possibly time-varying, uncertainty. The time-delay is assumed to be a time-varying continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available, and no restriction on the derivative of the time-varying delay is needed, which allows the time-delay to be a fast time-varying function. Based on an integral inequality, which is introduced in this paper, and Lyapunov–Krasovskii functional approach, a delay-dependent bounded real lemma (BRL) is first established without using model transformation and bounding techniques on the related cross product terms. Then employing the obtained BRL, a delay-dependent condition for the existence of a state feedback controller, which ensures asymptotic stability and a prescribed H_∞ performance level of the closed-loop systems for all admissible uncertainties, is proposed in terms of a linear matrix inequality (LMI). A numerical example is also given to illustrate the effectiveness of the proposed method.     

Reduction of H∞ state feedback control problems for the MIMO servo systems

We deal with H_∞ state feedback control problem for the multi‐input‐multi‐output (MIMO) servo system and discuss the advantages of the facial reduction (FR) to the resulting linear matrix inequality (LMI) problems. In fact, as far as our usual setting, the dual of the LMI problem is not strictly feasible because the generalized plant has always stable invariant zeros. Thus FR is available to such LMI problems, and we can reduce and simplify the original LMI problem to a smaller‐size LMI problem. As a result, we observe that the numerical performance of the SDP solvers is improved. Also, as a by‐product, we obtain the best performance index of the reduced LMI problem with a closed‐form expression. This helps the H_∞ performance limitation analysis. Another contribution is to reveal that the resulting LMI problem obtained from H_∞ control problem has a finite optimal value, but no optimal solutions under an additional assumption. This is also confirmed in the numerical experiment of this paper. FR also plays an essential role in this analysis.     

Robust stability of nonlinear time‐delay systems with interval time‐varying delay

This paper deals with the problem of obtaining delay‐dependent stability conditions and L₂‐gain analysis for a class of nonlinear time‐delay systems with norm‐bounded and possibly time‐varying uncertainties. No restrictions on the derivative of the time‐varying delay are imposed, though lower and upper bounds of the delay interval are assumed to be known. A Lyapunov–Krasovskii functional approach is proposed to derive novel delay‐dependent stability conditions which are expressed in terms of linear matrix inequalities (LMIs). To reduce conservatism, the work exploits the idea of splitting the delay interval in multiple regions, so that specific conditions can be imposed to a unique functional in the different regions. This improves the computed bounds for certain delay‐dependent integral terms, providing less conservative LMI conditions. Examples are provided to demonstrate the reduced conservatism with respect to the available results in the literature. Copyright © 2010 John Wiley & Sons, Ltd.     

New results on robust H ∞ filtering design for discrete-time piecewise linear delay systems

This paper revisits the problem of robust H _∞ filtering design for a class of discrete-time piecewise linear state-delayed systems. The state delay is assumed to be time-varying and of an interval-like type, which means that both the lower and upper bounds of the time-varying delay are available. The parameter uncertainties are assumed to have a structured linear fractional form. Based on a novel delay-dependent piecewise Lyapunov–Krasovskii functional combined with Finsler's Lemma, a new delay-dependent sufficient condition for robust H _∞ performance analysis is first derived and then the filter synthesis is developed. It is shown that by using a new linearisation technique, a unified framework can be developed so that both the full-order and reduced-order filters can be obtained by solving a set of linear matrix inequalities (LMIs), which are numerically efficient with commercially available software. Finally, a numerical example is provided to illustrate the effectiveness and less conservatism of the proposed approach.     

Low‐order H∞ controller design on the frequency domain by partial optimization

In this paper, an optimization method of low‐order multivariable controllers for H_∞ control is proposed. Starting from a low‐order stabilizing controller, our method gives a sequence of controllers for which the H_∞ norm performance index is monotonically non‐increasing by tuning the numerator coefficient matrices of the low‐order controller. This controller class includes multivariable PID controllers. The proposed method is a descent method where the feasible direction is calculated by solving a linear matrix inequality that represents a sufficient condition for the H_∞ criterion for each frequency. Usefulness is shown by two numerical examples. Copyright © 2009 John Wiley & Sons, Ltd.     

Robust H ∞ model predictive control for uncertain systems using relaxation matrices

In this paper, a robust H _∞ model predictive control (MPC) technique is proposed for time-varying uncertain discrete-time systems in the presence of input constraints and disturbances. We formulate a minimization problem of the upper bound of finite horizon cost function subject to the terminal inequality for an induced l ₂-norm bound. In order to improve system performance, we propose an LMI condition for the terminal inequality by using relaxation matrices. The LMI condition guarantees induced l ₂-norm bounds of the system despite system uncertainty and disturbance. A numerical example shows the effectiveness of the proposed method.     

MODEL SET IDENTIFICATION IN FREQUENCY-DOMAIN AND ITS APPLICATION TO JOINT DESIGN WITH ROBUST CONTROL

In this paper, we propose a new model set identification method for robust control, which determines both nominal models and uncertainty bounds in frequency-domain using periodgrams obtained from experimental data. This method also gives less conservative model sets when we have more experimental data, which is one of the distinguished features compared with the existing model set identification methods. To this end, first, we construct a new noise model set in terms of periodgrams, which consists of hard-bounded (or deterministic) noises but takes account of a low correlation property of noise signals, simultaneously. Then, based on the noise model, we show how to compute the nominal models and the upper bounds of modeling error via convex optimization, which minimize given cost functions. Furthermore, by introducing a weighting function compatible with control performance criterion into the identification cost function, we consider a joint design method of the proposed model set identification and H_∞ control. Numerical examples show the effectiveness of the proposed method.     

LMI OPTIMIZATION APPROACH FOR DELAY‐DEPENDENT H∞ CONTROL OF TIME‐VARYING DELAY SYSTEMS

In this paper, the robust delay‐dependent H_∞ control for a class of uncertain systems with time‐varying delay is considered. An improved state feedback H_∞ control is proposed to minimize the H_∞‐norm bound via the LMI optimization approach. Based on the proposed result, delay‐dependent criteria are obtained without using the model transformation technique or bounded inequalities on cross product terms. The linear matrix inequality (LMI) optimization approach is used to design the robust H_∞ state feedback control. Some numerical examples are given to illustrate the effectiveness of the approach.     

A duality-based approach to the multiobjective H 2/H ∞ optimisation problem

In this article, a duality approach to multiobjective H ₂/H _∞ problems is pursued in which real-rational, para-Hermitian multipliers and real-valued ones are associated to H _∞ and (as usual) H ₂ constraints, respectively. It is shown that the maximisation of a dual functional over all such multipliers yields the optimal value of the original multiobjective H ₂/H _∞ problem. To compute lower bounds on the latter and the corresponding approximate solutions to the original problem, the maximisation of the dual functional over linearly-parameterised, finite-dimensional classes of real-rational multipliers is shown to be equivalent to semi-definite, linear programming problems – once the optimal multipliers in such a class are obtained, the corresponding approximate solutions can be computed from an unconstrained H ₂ problem. Iterative modification of such classes is discussed to obtain increasing sequences of lower bounds on the optimal value of the original problem. This is done on the basis of (locally) increasing directions for the dual functional which go beyond the finite-dimensional class of multipliers considered in a given step. Finally, a numerical example is presented to illustrate the way the presented results can lead to approximate solutions to the multiobjective H ₂/H _∞ problem together with tight estimates of the corresponding deviation from its optimal value.     

Stabilization of the Furuta Pendulum with backlash using H∞‐LMI technique: experimental validation

The rotary inverted pendulum, also named Furuta Pendulum, has been studied extensively for control performance evaluation in under‐actuated mechanisms. The H_∞ control invoking linear matrix inequality (H_∞‐LMI) has been also widely employed for linear control design. This paper deals with the feasibility of the H_∞‐LMI technique to stabilize the rotary inverted pendulum around its unstable equilibrium point when there exists a backlash nonlinearity in the actuator. So, the H_∞‐LMI faces the nonlinear effect in the actuator and the non‐linear pendulum model. Experimental realization of the designed H_∞‐LMI control also shows evidence of the good performance of the controller subject to external perturbation. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society