Two-stage H ∞ optimization approach to multirate controller design

A two-stage H _?? optimization methodology is presented for designing multirate controllers achieving comparable H _?? closed-loop performance with that of an optimal fast rate design at a reduced real-time computational load. In the first stage, a fast rate controller is designed by solving an H _?? optimal control problem formulated to address the design specifications; then the fast rate controller is decomposed into low frequency and high frequency parts. In the second stage, the high frequency part of the controller is retained and the low frequency part is redesigned at the slow rate using lifting and H _?? optimization. In the proposed approach, potential problems of aliasing due to the multirate nature of the controller are addressed by taking into consideration closed-loop performance.     

An LMI approach to robust reduced-order H ∞ filter design for polytopic uncertain systems

This paper proposes a method for robust reduced-order H _∞ filter design for polytopic uncertain systems, using linear matrix inequalities (LMIs). Sufficient LMI conditions for both robust full- and reduced-order H _∞ filter design are derived. Convex optimization problems are formulated and solved to obtain optimal H _∞ filters by using the resulting LMI conditions. The resulting conditions do not involve any non-convex rank constraints, and thus the proposed method for H _∞ filter design guarantees global optimum solutions. Numerical examples are presented to show the effectiveness of the proposed method. Recommended by Editorial Board member Huanshui Zhang under the direction of Editor Young Il Lee. This work was supported by the Brain Korea 21 Project and the Basic Research Program of the Korea Science and Engineering Foundation under grant R01-2006-000-11373-0. Hyoun-Chul Choi received the B.S., M.S., and Ph.D. degrees in Control and Instrumentation Engineering from Ajou University, Suwon, Korea, in 1995, 1997, and 2006, respectively. He was a Visiting Researcher at Griffith University, Brisbane, Australia, from 2001 to 2002, and a Postdoctoral researcher at Ajou University, Suwon, Korea, from 2006 to 2007. Since 2008, he has been with ASRI, School of Electrical Engineering and Computer Science, Seoul National University, Seoul, Korea, where he is currently a Postdoctoral Researcher. His research interests include LMI-based control, optimal and robust control, network-based control, and mechatronics. Dongkyoung Chwa received the B.S. and M.S. degrees from the Department of Control and Instrumentation Engineering in 1995 and 1997, respectively, and the Ph.D. degree from the School of Electrical and Computer Engineering in 2001, all from Seoul National University, Seoul, Korea. From 2001 to 2003, he was a Postdoctoral Researcher with Seoul National University. In 2003, he was a Visiting Research Fellow at The University of New South Wales, Australian Defence Force Academy, and was the Honorary Visiting Academic at the University of Melbourne, Melbourne, Australia. In 2004, he was a BK21 Assistant Professor with Seoul National University. Since 2005, he has been an Assistant Professor with the Department of Electrical and Computer Engineering, Ajou University, Suwon, Korea. His research interests are nonlinear, robust, and adaptive control theories and their applications to the robotics, underactuated systems including wheeled mobile robots, underactuated ships, cranes, and guidance and control of flight systems. Suk-Kyo Hong received the B.S., M.S., and Ph.D. degrees in Electrical Engineering from Seoul National University, Seoul, Korea, in 1971, 1973, and 1981, respectively. His major graduate research works were centered on speed control of induction motors. He was an Exchange Professor at Rensselaer Polytechnic Institute, Troy, NY, from 1982 to 1983, and at the Institut National de Recherche en Informatique et en Automatique, France, from 1988 to 1989. He has been with the faculty of the Department of Electrical and Computer Engineering, Ajou University, Suwon, Korea, since 1976, and was a Visiting Professor at Griffith University, Australia, in 2001 and 2002. His current research interests include robust robot control, microprocessor applications, factory automation, and computer integrated manufacturing.     

An application of H∞ design to model-following

The present work shows the application of H_∞ optimal control to the imperfect model-following problems where exact model-following fails to be implemented. The significance of this work lies in the improvement of the ill-conditioning problem in the H_∞ algorithm for the strictly proper plant. This is achieved by perturbing the plant in a suitable way. The optimal results can be recovered asymptotically under certain conditions for the scalar case. Such an approach provides a way to solve the problem when the existing H_∞ algorithm fails. This paper also highlights the degree constraints of choosing an ideal model. The model-following controller is illustrated by practical examples and compared with H₂ results.     

The Davidon method of solution of the algebraic matrix Riccati equation†

An algorithm is presented for solving the algebraic matrix Riccati equation using the Fletcher-Powell reformulation of Davidon's method of function minimization. The function to be minimized, as well as its gradient vector required by the minimization process, are evaluated in closed-form, thereby preserving the simplicity and stability properties of the minimization procedure. The quadratic convergence of the algorithm is not dependent on the initial choice of the approximate solution.     

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.     

An innovation approach to optimize a Kalman filter with an H ∞ error bound by secant method

Designing a Kalman filter with a constraint on the H _∞ norm of the estimation error was first developed by Bernstein and Haddad in 1989. The main result is a sufficient condition for characterizing the Kalman filter. In this paper, similar to the standard Kalman filter, the properties of orthogonal principles are also shown to be preserved. Furthermore, the uniqueness, as opposed to an H _∞ filter, of the filter is implied by the orthogonal principles. An innovative approach to obtaining the minimum energy with a constraint on the H _∞ norm of the estimation error is proposed since the original work of Bernstein and Haddad does not, in general, reach the minimum energy of the estimation error. By means of the Secant method, the energy of the estimation error can be reduced as much as possible, under the condition that the H _∞ error bound is still satisfied.     

H ∞ approach to T-S fuzzy controller for limiting reconstruction errors

In this article, the reconstruction error between a real system to be controlled and its Takagi–Sugeno (T-S) fuzzy model is considered in the context of control system design. Accordingly, we propose an H _∞ approach to an adaptive controller that consists of two parts: one is obtained by solving certain linear matrix inequalities (fixed part) and the other is acquired using a fuzzy approximator in which the related parameters are tuned by an adaptive law (variable part). The proposed controller can guarantee a convergent and uniformly bounded control state while maintaining the stability of all the signals involved.     

A geometric approach to H∞ control of nonlinear Markovian jump systems

This paper first discusses the H_∞ control problem for a class of general nonlinear Markovian jump systems from the viewpoint of geometric control theory. Following with the updating of the Markovian jump mode, the appropriate diffeomorphism can be adopted to transform the system into special structures, which establishes the basis for the geometric control of nonlinear Markovian jump systems. Through discussing the strongly minimum-phase property or the strongly γ-dissipativity of the zero-output dynamics, the H_∞ control can be designed directly without solving the traditional coupled Hamilton–Jacobi inequalities. A numerical example is presented to illustrate the effectiveness of our results.     

A receding–horizon approach to the nonlinear H∞ control problem

The receding–horizon (RH) methodology is extended to the design of a robust controller of H_∞ type for nonlinear systems. Using the nonlinear analogue of the Fake H_∞ algebraic Riccati equation, we derive an inverse optimality result for the RH schemes for which increasing the horizon causes a decrease of the optimal cost function. This inverse optimality result shows that the input–output map of the closed-loop system obtained with the RH control law has a bounded L₂-gain. Robustness properties of the nonlinear H_∞ control law in face of dynamic input uncertainty are considered.     

H∞ approach to monotonically convergent ILC for uncertain time-varying delay systems

This paper deals with iterative learning control (ILC) design for uncertain time-delay systems. Monotonic convergence of the resulting ILC process is studied, and a sufficient condition within an H_∞-based framework is developed. It is shown that under this framework, delay-dependent conditions can be obtained in terms of linear matrix inequalities (LMIs), together with formulas for gain matrices design. A numerical example is provided to illustrate the effectiveness of the robust H_∞-based approach to ILC designed via LMIs.     

Multiple Lyapunov functions approach to observer-based H ∞ control for switched systems

This article investigates the issue of H _∞ control for a class of continuous-time switched Lipschitz nonlinear systems. None of the individual subsystems is assumed to be stabilisable with H _∞ disturbance attenuation. Based on a generalised multiple Lyapunov functions (GMLFs) approach, which removes the nonincreasing requirement at switching points, a sufficient condition for the solvability of the H _∞ control problem under a state estimation-dependent switching law is presented. Observers, controllers and a switching law are simultaneously designed. As an extension, a sufficient condition for exponential stabilisability is also given.     

Analysis and control of general logical networks – An algebraic approach

Since Boolean network is a powerful tool in describing the genetic regulatory networks, accompanying the development of systems biology, the analysis and control of Boolean networks have attracted much attention from biologists, physicists, and systems scientists. From mathematical point of view, the dynamics of a Boolean (control) network is a discrete-time logical dynamic process. This paper surveys a recently developed technique, called the algebraic approach, based on semi-tensor product. The new technique can deal with not only Boolean networks, which allow each node to take two values, but also k-valued networks, which allow each node to take k different values, and mix-valued networks, which allow nodes to take different numbers of values.The paper provides a comprehensive introduction to the new technique, including (1) mathematical background of this new technique – semi-tensor product of matrices and the matrix expression of logic; (2) dynamic models of Boolean networks, and general (multi- or mix-valued) logical networks; (3) the topological structure of Boolean networks and general networks; (4) the basic control problems of Boolean/general control networks, which include the controllability, observability, realization, stability and stabilization, disturbance decoupling, identification and optimization, etc.; (5) some other related applications.     

An H∞ suboptimal robust control approach for systems with uncertainties and data dropouts

This paper studies the design of control systems subject to plant uncertainties and data losses in the channel connecting the plant sensor with the controller. The controller design has two main objectives. The first one is to robustify the control law against plant uncertainties. The other one is to achieve good performance by minimising the variance of the error signal. Data losses are modelled as an independent and identically distributed sequence of Bernoulli random variables. For analysis and design, this random variable is replaced by an additive noise plus gain channel model. To cope with structural uncertainties in the model of the plant, an H_∞ control technique is employed. The controller is synthesised in order to make the closed-loop system robust against structural uncertainties of the nominal model, while achieving optimal performance of the system in the presence of dropouts.     

An LMI approach for H ∞ analysis and control of discrete-time piecewise affine systems

In this paper we investigate some analysis and control problems for discrete-time hybrid systems in the piece-wise affine form. By using arguments from the dissipativity theory for non-linear systems, we show that H X analysis and synthesis problems can be formulated and solved via linear matrix inequalities by taking into account the switching structure of the considered system. In this paper we address the generalized problem of controlling hybrid systems whose switching structure does not depend only on the state but also on the control input.     

A new approach to robust reliable H∞ control for uncertain nonlinear systems

This paper focuses on the optimal robust reliable H_∞ control for a class of uncertain nonlinear systems with actuator faults. A new method of annihilating uncertain matrix is proposed. Based on this approach, a new method of disposing of the phenomenon of uncertain matrices multiplication is provided. In accordance with the method, the optimal robust reliable H_∞ control problem for uncertain nonlinear systems is settled by employing state feedback, in terms of linear matrix inequality (LMI). Finally, two illustrative examples are given to show the feasibility and validity of the proposed method.     

Unified approach to H∞-optimization,Hankel approximation and balanced realization problems

A unified approach is proposed here to solve three problems simultaneously, namely H ∞ -optimization, Hankel approximation and minimal balanced realization for scalar systems. It is shown that all these problems can be reduced to the same standard form wherein merely the singular-value decomposition (SVD) of an easily constructed square matrix is required to establish the final solutions. The formulation is solely in terms of the coefficients of a transfer function and the resulting algorithm can be performed in a very computationally efficient way.