Low-order H/sub /spl infin// controller design for an active suspension system via LMIs

In this paper we consider the mixed H/sub 2//H/sub /spl infin//-control problem for the class of discrete-time linear systems with parameters subject to Markovian jump linear systems. It is assumed that both the state variable and the jump variable are available to the controller. The transition probability matrix may not be exactly known, but belongs to an appropriate convex set. For this controlled discrete-time Markovian jump linear system, the problem of interest can be stated in the following way: find a robust (with respect to the uncertainty on the transition Markov probability matrix) mean-square stabilizing state and jump feedback controller that minimizes an upper bound for the H/sub /spl infin//-norm, under the restriction that the H/sub /spl infin//-norm is less than a prespecified value /spl delta/. The problem of the determination of the smallest H/sub /spl infin//-norm is also addressed. We present an approximate version of these problems via linear matrix inequality optimization.

H/sub /spl infin// and adaptive nonlinear control technique is provided for the active suspension system which is divided into two parts: the main car body part and the hydraulic actuator part. H/sub /spl infin// control design is used for the body part where the frequency response property is important for the ride quality of the automobile, and adaptive nonlinear control design is worked out for the actuator part which has high nonlinearity and parametric uncertainty. These two design methods are combined by a modified version of backstepping.

This paper deals with H/sub /spl infin////spl mu/ DIA control of the magnetic suspension system. H/sub /spl infin// DIA control is an H/sub /spl infin// control problem which treats a mixed Disturbance and an Initial-state uncertainty Attenuation (DIA) and supplies H/sub /spl infin// controls with good transients. H/sub /spl infin// DIA controller has a good initial response property, however its robust performance might be improved. We propose H/sub /spl infin////spl mu/ DIA Control which is to find a multi-objective controller to achieve both the H/sub /spl infin// DIA condition for good initial responses/transient responses and the structured singular value /spl mu/ condition for robust performance. We apply this proposed approach to the magnetic suspension system, and design a robust controller which has both good properties. Finally experimental results show effectiveness of the proposed control system design framework.

In this note, we propose new receding horizon H/sub /spl infin// control (RHHC) schemes for linear input-constrained discrete time-invariant systems with disturbances. The proposed control schemes are based on the dynamic game problem of a finite-horizon cost function with a fixed finite terminal weighting matrix and a one-horizon cost function with time-varying finite terminal weighting matrices, respectively. We show that the resulting RHHCs guarantee closed-loop stability in the absence of disturbances and H/sub /spl infin// norm bound for 2-norm bounded disturbances. We also show that the proposed schemes can easily be implemented via linear matrix inequality optimization. We illustrate the effectiveness of the proposed schemes through simulations.

A new class of reduced-order controllers is obtained for the H/sub /spl infin// problem. The reduced-order controller does not compromise the performance attained by the full-order controller. An algorithm for deriving the reduced-order H/sub /spl infin// controller is presented in the case of continuous time. The reduction in order is related to unstable invariant zeros of the subsystem from disturbance inputs to measurement outputs. In the case where the subsystem has no infinite zeros, the resulting order of the H/sub /spl infin// controller is lower than that of the existing reduced-order H/sub /spl infin// controller designs which are based on reduced-order observer design. Furthermore, the mechanism of the controller order reduction is analyzed on the basis of the two-Riccati equation approach.

The matrix rank approximation has shown high effectiveness in the matrix rank minimization (MRM) problem, which aims to recover the underlying low-rank structure from the observed matrix by imposing the rank constraint. The nuclear norm, serving as a convex surrogate of matrix rank, is employed in the MRM problem by shrinking singular values of the observed entry. However, this substitution treats each singular value equally, which is virtually $ℓ_1$-norm penalty of the singular value vector. Theoretically, the rank function of the matrix can be considered as $ℓ_0$-norm of its singular values. Consequently, minimizing the nuclear norm frequently results in biased solutions in various applications. In this article, we first propose a novel nonconvex rank approximation, named tight and flexible rank (TFR) approximation, to describe rank function effectively. Specifically, the TFR approximation can more tightly approach the rank function and exhibit greater flexibility in handling diverse singular values, as compared to existing nonconvex rank approximations. Furthermore, we apply TFR approximation to matrix completion and develop a solving algorithm with guaranteed convergence based on the framework of proximal alternating minimization. Extensive experiments reveal that the proposed matrix completion model with TFR approximation outperforms several existing state-of-the-art convex and nonconvex methods.

A fault detection filter is designed by minimising the response to confounding noises whilst keeping the fault response constant. A linear quadratic optimisation problem, related to /spl Hscr//sub 2/ and /spl Hscr//sub /spl infin// optimal control is used. The Q matrix of the associated Riccati equation is indefinite in contrast to the indefinite R matrix of /spl Hscr//sub /spl infin// control. We extend linear optimal control theory to solve the fault detection problem in this framework.

In this paper, a mixed H/sub 2//H/sub /spl infin// decentralized variable structure control for car-like mobile robot (CLMR) was developed. Under no uncertain condition, a linear discrete-time dynamic model for steering and translation subsystem was individually achieved by the recursive least-squares parameter estimation. Then the H/sub 2/-norm of output error and weighted control input was minimized to obtain a control such that smaller energy consumption with bounded tracking error was assured. In addition, the H/sub /spl infin//-norm of weighted sensitivity between output disturbance (caused by the interactions among subsystems, modeling error, and external load) and output error was minimized to attenuate the effect of output disturbance. For further improving system performance, a switching control for every subsystem was designed. The experiment of CLMR for tracking an S trajectory was carried out to evaluate the usefulness of the proposed control system.

Solving optimization problems with sparse or low-rank optimal solutions has been an important topic since the recent emergence of compressed sensing and its matrix extensions such as the matrix rank minimization and robust principal component analysis problems. Compressed sensing enables one to recover a signal or image with fewer observations than the “length” of the signal or image, and thus provides potential breakthroughs in applications where data acquisition is costly. However, the potential impact of compressed sensing cannot be realized without efficient optimization algorithms that can handle extremely large-scale and dense data from real applications. Although the convex relaxations of these problems can be reformulated as either linear programming, second-order cone programming or semidefinite programming problems, the standard methods for solving these relaxations are not applicable because the problems are usually of huge size and contain dense data. In this dissertation, we give efficient algorithms for solving these “sparse” optimization problems and analyze the convergence and iteration complexity properties of these algorithms. Chapter 2 presents algorithms for solving the linearly constrained matrix rank minimization problem. The tightest convex relaxation of this problem is the linearly constrained nuclear norm minimization. Although the latter can be cast and solved as a semidefinite programming problem, such an approach is computationally expensive when the matrices are large. In Chapter 2, we propose fixed-point and Bregman iterative algorithms for solving the nuclear norm minimization problem and prove convergence of the first of these algorithms. By using a homotopy approach together with an approximate singular value decomposition procedure, we get a very fast, robust and powerful algorithm, which we call FPCA (Fixed Point Continuation with Approximate SVD), that can solve very large matrix rank minimization problems. Our numerical results on randomly generated and real matrix completion problems demonstrate that this algorithm is much faster and provides much better recoverability than semidefinite programming solvers such as SDPT3. For example, our algorithm can recover 1000 × 1000 matrices of rank 50 with a relative error of 10−5 in about 3 minutes by sampling only 20 percent of the elements. We know of no other method that achieves as good recoverability. Numerical experiments on online recommendation, DNA microarray data set and image inpainting problems demonstrate the effectiveness of our algorithms. In Chapter 3, we study the convergence/recoverability properties of the fixed-point continuation algorithm and its variants for matrix rank minimization. Heuristics for determining the rank of the matrix when its true rank is not known are also proposed. Some of these algorithms are closely related to greedy algorithms in compressed sensing. Numerical results for these algorithms for solving linearly constrained matrix rank minimization problems are reported. Chapters 4 and 5 considers alternating direction type methods for solving composite convex optimization problems. We present in Chapter 4 alternating linearization algorithms that are based on an alternating direction augmented Lagrangian approach for minimizing the sum of two convex functions. Our basic methods require at most O(1/e) iterations to obtain an e-optimal solution, while our accelerated (i.e., fast) versions require at most O(1/ 3 ) iterations, with little change in the computational effort required at each iteration. For more general problem, i.e., minimizing the sum of K convex functions, we propose multiple-splitting algorithms for solving them. We propose both basic and accelerated algorithms with O(1/e) and O(1/ 3 ) iteration complexity bounds for obtaining an e-optimal solution. To the best of our knowledge, the complexity results presented in these two chapters are the first ones of this type that have been given for splitting and alternating direction type methods. Numerical results on various applications in sparse and low-rank optimization, including compressed sensing, matrix completion, image deblurring, robust principal component analysis, are reported to demonstrate the efficiency of our methods.

This brief is concerned with the reduced-order H/sub /spl infin// filtering problem for Markovian jumping linear systems with time delay, where the jumping parameters are modeled by a discrete-time Markov process. Sufficient conditions for the existence of the reduced-order H/sub /spl infin// filter are proposed in terms of linear matrix inequalities (LMI) and a coupling nonconvex matrix rank constraint. In particular, the sufficient conditions for the existence of the zeroth-order (or static) H/sub /spl infin// filter can be expressed in terms of a set of strict LMIs. The explicit parameterization of the desired filter is also given. Finally, a numerical example is given to illustrate the proposed approach.

There exist a direct and an indirect methods in designing reduced-order H/sub /spl infin// controller. The former method is difficult and there exists no effective way presently. The latter method has been used conventionally. However, the resultant controller may be conservative. We propose a method of designing a less conservative reduced-order controller by using genetic algorithm (GA). Since it is difficult to search reduced-order controller directly, we search for the full-order controller whose reduced-order controller has approximate performance of the optimal full-order H/sub /spl infin// controller. Then, it is reduced by the reduction technique. In order to search the desired full-order controller effectively, Glover's parameterization is applied. Furthermore, we propose the coding which searches only optimal full-order controllers, since the desired full-order controller is expected to be one of numerous optimal full-order controllers. In order to show the effectiveness of the proposed method, an numerical example is given.

This paper presents a multi-stage design approach that uses a multiobjective genetic algorithm (MOGA) as the framework for optimization and multiobjective preference articulation. An H/sub /spl infin// loop-shaping technique is used to design controllers based on a linear state-space model of a gas turbine engine (GTE). A non-linear model is then used to assess performance of the controller in meeting various stability, design and performance requirements. The computational load of applying MOGA to the design of a control system for the Spey engine for the H/sub /spl infin// strategy is very high. Alternative model approximations, response surface models, are used in order to speed up the design process. Regression analysis is applied to fit linear models to this data for various control responses. To assist the design process, a neural network is trained to classify possible designs to avoid unstable solutions. These simple models are used to design the controller within the framework of a MOGA. The final designs are checked using the original non-linear model. Good results indicate the viability of this approach for application to complex designs involving expensive computational models.

This paper describes an approach to the reduction of controllers for the normalized coprime factor robustness problem as well as the normalized H/sub /spl infin// problem. It is shown that a relative error approximation of a coprime factor representation of any suboptimal controller leads to a stability guarantee and an upper bound on the performance degradation when the reduced order controller is implemented. When the approximation is performed on the controller generator, guaranteed a priori stability and performance bounds are obtained in terms of the synthesis Riccati equation solutions of the normalized H/sub /spl infin// control problems.

The optimal control design of the ground-vehicle active suspension system is presented. The active suspension system is to improve the vehicle ride comfort by isolating vibrations induced by the road profile and vehicle velocity. The vehicle suspension system is approached by a quarter car model. Dynamic equations of the system are derived by applying Newton’s second law. The control law of the active suspension system is designed using linear quadratic regulator (LQR) method. Performance evaluation is done by benchmarking the active suspension system to a passive suspension system. Both suspension systems are simulated in computer. The simulation results show that the active suspension system significantly improves the vehicle ride comfort of the passive suspension system by reducing 50.37% RMS of vertical displacement, 45.29% RMS of vertical velocity, and 1.77% RMS of vertical acceleration.

H/sup /spl infin//-output feedback control is applied to the control of automobile active suspensions based on a dynamic model of the full vehicle. The output feedback control is desirable from the viewpoint of implementation in the sense that it reduces the number of measurements drastically compared with state feedback used currently in suspension control. In this paper, the authors show that a linearized model can be block-decoupled by a similarity transformation, which reduces the controller complexity significantly. The uncertainties of the vehicle model are properly taken into account in the derivation of the H/sup /spl infin// controller. The controller is actually implemented in a commercial car, and the performance is evaluated both by simulations and experiments. The performance obtained proves to be quite satisfactory.