Worst-case design in the time domain: The maximum principle and the standardH ∞ problem

This paper presents a time-domain, optimal-control approach to worst-case design, an alternative to frequency-domainH _∞ techniques. The generic linear-quadratic set-up of the “standardH _∞ problem” is discussed. The resultsinclude a characterization of suboptimal values, as well as a parametrization of all suboptimal compensators, interms of two coupled indefinite Riccati equations. Both the usual infinite-horizon, time-invariant case and the finite-horizon, time-varying case, are treated. The latter is beyond the scope of frequency-domain analysis. An earlier version of this work was prepared at the Center for Control Sciences, Brown University, and was supported in part by the Air Force Office of Scientific Research under Contract F49620-86-C-011.     

The 2-block superoptimalAAK problem

This paper presents a solution to a superoptimal version of the 2-blockAAK problem: Given a rational and antistable matrix functionR(s)=[R ₁₁(s)R ₁₂(s)] and a nonnegative integerk, find all superoptimal approximationsQ(s), with no more thank poles in the right-half complex plane, that minimize the supremum, over the imaginary axis, of the singular values of the error functionE(s)=[R ₁₁ (s) R ₁₂ (s)+Q(s)], with respect to lexicographic ordering. Conditions are given for which the superoptimal approximation is unique. In addition, ana priori upper bound on the MacMillan degree of the approximation is provided. The algorithm may be stopped after minimizing a given number of the singular values. This premature termination carries with it a predictable reduction in the MacMillan degree of the approximation. The algorithm only requires standard linear algebraic computations, and is therefore easily implemented. This work has been supported by Engineering and Physical Sciences Research Council Grant GR/J42533.     

State feedback H ∞ control for a class of nonlinear systems using partition of unity method

The system model is nonlinear with respect to all its variables while the output model is linear. The nonlinear system model is firstly converted into an equivalent linear model with error by using partition of unity method. The stability with decay rate and the disturbance attenuation for the nonlinear system are discussed based on the equivalent model. A state feedback H _∞ controller is then proposed in terms of linear matrix inequalities (LMIs). Recommended by Editorial Board member Bin Jiang under the direction of Editor Jae Weon Choi. The authors would like to thank the anonymous reviewers and the editor for their constructive comments based on which this paper has been improved. Dong-Fang Han received the Ph.D. degree in Pure Mathematics from Shantou University in 2008. His research interests include nonlinear control, robust control and time-delay system. Yin-He Wang received the Ph.D. degree in Control Theory and Engineering from Northeast University in 1999. From 2000 to 2002, he was a Post-doctor in the department of automatic control, Northwestern Polytechnic University, China. His research interests include nonlinear systems, adaptive and robust control.     

A bisection method for computing the H∞ norm of a transfer matrix and related problems

We establish a correspondence between the singular values of a transfer matrix evaluated along the imaginary axis and the imaginary eigenvalues of a related Hamiltonian matrix. We give a simple linear algebraic proof, and also a more intuitive explanation based on a certain indefinite quadratic optimal control problem. This result yields a simple bisection algorithm to compute the H_∞ norm of a transfer matrix. The bisection method is far more efficient than algorithms which involve a search over frequencies, and the usual problems associated with such methods (such as determining how fine the search should be) do not arise. The method is readily extended to compute other quantities of system-theoretic interest, for instance, the minimum dissipation of a transfer matrix. A variation of the method can be used to solve the H_∞ Armijo line-search problem with no more computation than is required to compute a single H_∞ norm. Research supported in part by NSF under Grant ECS-85-52465, ONR under Grant N00014-86-K-0112, an IBM faculty development award, and Bell Communications Research.     

A uniqueness result for the isaacs equation corresponding to nonlinear H∞ control

The dynamic programming equation (DPE) corresponding to nonlinear H_∞ control is considered. When the cost grows quadratically in the state, it is well known that there may be an infinite number of viscosity solutions to the DPE. In fact, there may be more than one classical solution when a classical solution exists. For the case of fixed feedback control, it is shown that there exists a unique viscosity solution in the class of solutions meeting a certain growth condition, and a representation in terms of available storage is obtained. For the active control case, where the H_∞ problem is represented by a differential game, a similar representation result is obtained under the assumption of existence of a suboptimal feedback control. This research was partially supported by AFOSR under Grant F49620-95-1-0296 and by ONR under Grant N0014-96-1-0267.     

Mixed <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> filtering for a class of high-speed sampling uncertain systems

The problem of mixed H2/H∞ filtering for polytopic Delta operator systems is investigated. The aim is to design a linear asymptotically stable filter which guarantees that the filtering error system has different performances in different filtering channels. Based on a parameter-dependent Lyapunov function, a new mixed H2/H∞ performance criterion is presented. Upon this performance criterion, a sufficient condition for the full-order mixed H2/H∞ filter is derived in terms of linear matrix inequalities. The filter can be obtained from the solution of a convex optimization problem. The proposed filter design procedure is less conservative than the strategy based on the quadratic stability notion. A numerical example is given to illustrate the feasibility of the proposed approach.     

LMI-based H ∞ synchronization of second-order neutral master-slave systems using delayed output feedback control

The H _∞ synchronization problem of the master and slave structure of a second-order neutral master-slave systems with time-varying delays is presented in this paper. Delay-dependent sufficient conditions for the design of a delayed output-feedback control are given by Lyapunov-Krasovskii method in terms of a linear matrix inequality (LMI). A controller, which guarantees H _∞ synchronization of the master and slave structure using some free weighting matrices, is then developed. A numerical example has been given to show the effectiveness of the method. The simulation results illustrate the effectiveness of the proposed methodology. Recommended by Editorial Board member Bin Jiang under the direction of Editor Jae Weon Choi. This research has been partially funded by the German Research Foundation (DFG) as part of the Collaborative Research Center 637 ‘Autonomous Cooperating Logistic Processes: A Paradigm Shift and its Limitations’ (SFB 637). This work was supported in part by the National Natural Science Foundation of China (60504008), by the Research Fund for the Doctoral Program of Higher Education of China (20070213084), by the Fok Ying Tung Education Foundation (111064). Hamid Reza Karimi born in 1976, received the B.Sc. degree in Power Systems Engineering from Sharif University of Technology in 1998 and M.Sc. and Ph.D. degrees both in Control Systems Engineering from University of Tehran in 2001 and 2005, respectively. From 2006 to 2007, he was a Post-doctoral Research Fellow of the Alexander-von-Humboldt Stiftung with both Technical University of Munich and University of Bremen in Germany. He held positions as Assistant Professor at the Department of Electrical Engineering of the University of Tehran in Iran, Senior Research Fellow in the Centre for Industrial Mathematics of the University of Bremen in Germany and Research Fellow of Juan de la Cierva program at the Department of Electronics, Informatics and Automation of the University of Girona in Spain before he was appointed as an Associate Professor in Control Systems at the Faculty of Technology and Science of the University of Agder in Norway in April 2009. His research interests are in the areas of nonlinear systems, networked control systems, robust filter design and vibration control of flexible structures with an emphasis on applications in engineering. Dr. Karimi was the recipient of the German Academic Awards (DAAD Award) from 2003 to 2005 and was a recipient of the Distinguished Researcher Award from University of Tehran in 2001 and 2005. He received the Distinguished PhD Award of the Iranian President in 2005 and the Iranian Students Book Agency’s Award for Outstanding Doctoral Thesis in 2007. He also received first rank of Juan de la Cierva research program in the field of Electrical, Electronic and Automation Engineering in Spain in 2007. Huijun Gao was born in Heilongjiang Province, China, in 1976. He received the M.S. degree in Electrical Engineering from Shenyang University of Technology, Shengyang, China, in 2001 and the Ph.D. degree in Control Science and Engineering from Harbin Institute of Technology, Harbin, China, in 2005. He was a Research Associate with the Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, from November 2003 to August 2004. From October 2005 to September 2007, he carried out his postdoctoral research with the Department of Electrical and Computer Engineering, University of Alberta, Canada, supported by an Alberta Ingenuity Fellowship and an Honorary Izaak Walton Killam Memorial Postdoctoral Fellowship. Since November 2004, he has been with Harbin Institute of Technology, where he is currently a Professor. His research interests include network-based control, robust control/filter theory, model reduction, time-delay systems, multidimensional systems, and their engineering applications. Dr. Gao is an Associate Editor for the IEEE Transactions on Systems, Man and Cybernetics Part B: Cybernetics, the Journal of Intelligent and Robotics Systems, the Circuits, System and Signal Processing etc. He serves on the Editorial Board of the International Journal of Systems Science, the Journal of the Franklin Institute etc. He was the recipient of the University of Alberta Dorothy J. Killam Memorial Postdoctoral Fellow Prize in 2005 and was a corecipient of the National Natural Science Award of China in 2008. He was a recipient of the National Outstanding Youth Science Fund in 2008 and the National Outstanding Doctoral Thesis Award in 2007. He was an outstanding reviewer for IEEE Transactions on Automatic Control and Automatica in 2008 and 2007 respectively, and an appreciated reviewer for IEEE Transactions on Signal Processing in 2006.     

H∞ State Feedback Delay-dependent Control for Discrete Systems with Multi-time-delay

In this paper, H∞ state feedback control with delay information for discrete systems with multi-time-delay is discussed. Making use of linear matrix inequality (LMI) approach, a time-delay-dependent criterion for a discrete system with multi-time-delay to satisfy H∞ performance indices is induced, and then a strategy for H1 state feedback control with delay values for plant with multi-time-delay is obtained. By solving corresponding LMI, a delay-dependent state feedback controller satisfying H∞ performance indices is designed. Finally, a simulation example demonstrates the validity of the proposed approach.     

<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Control Design Using Dynamic Neural Networks

In this paper, a new approach is investigated for adaptive dynamic neural network-based H _∞ control, which is designed for a class of non-linear systems with unknown uncertainties. Currently, non-linear systems with unknown uncertainties are commonly used to efficiently and accurately express the real practical control process. Therefore, it is of critical importance but a great challenge and still at its early age to design a stable and robust controller for such a process. In the proposed research, dynamic neural networks were constructed to precisely approximate the non-linear system with unknown uncertainties first, a non-linear state feedback H _∞ control law was designed next, then an adaptive weighting adjustment mechanism for dynamic neural networks was developed to achieve H _∞ regulation performance, and last a recurrent neural network was employed as a neuro-solver to efficiently and numerically solve the standard LMI problem so as to obtain the appropriate control gains. Finally, case studies further verify the feasibility and efficiency of the proposed research.     

Gain-scheduled H ∞ filtering of parameter-varying discrete-time systems via parameter-dependent Lyapunov functions

This paper is concerned with the problem of gain-scheduled H _∞ filter design for a class of parameter-varying discrete-time systems. A new LMI-based design approach is proposed by using parameter-dependent Lyapunov functions. Recommended by Editorial Board member Huanshui Zhang under the direction of Editor Jae Weon Choi. This work was supported in part by the National Natural Science Foundation of P. R. China under Grants 60874058, by 973 program No 2009CB320600, but also the National Natural Science Foundation of Province of Zhejiang under Grants Y107056, and in part by a Research Grant from the Australian Research Council. Shaosheng Zhou received the B.S. degree in Applied Mathematics and the M.Sc. and Ph.D. degrees in Electrical Engineering, in January 1992, July 1996 and October 2001, from Qufu Normal University and Southeast University. His research interests include nonlinear control and stochastic systems. Baoyong Zhang received the B.S. and M.Sc. degrees in Applied Mathematics, in July 2003 and July 2006, all from Qufu Normal University. His research interests include and nonlinear systems, robust control and filtering. Wei Xing Zheng received the B.Sc. degree in Applied Mathematics and the M.Sc. and Ph.D. degrees in Electrical Engineering, in January 1982, July 1984 and February 1989, respectively, all from the Southeast University, Nanjing, China. His research interests include signal processing and system identification.     

Delay-dependent robust H ∞ control for uncertain fuzzy Markovian jump systems

This paper is concerned with the problem of delay-dependent robust H _∞ control for uncertain fuzzy Markovian jump systems with time delays. The purpose is to design a mode-dependent state-feedback fuzzy controller such that the closed-loop system is robustly stochastically stable and satisfies an H _∞ performance level. By introducing slack matrix variables, a delay-dependent sufficient condition for the solvability of the problem is proposed in terms of linear matrix inequalities. An illustrative example is finally given to show the applicability and effectiveness of the proposed method. Recommended by Editorial Board member Young Soo Suh under the direction of Editor Jae Weon Choi. This work is supported by the National Science Foundation for Distinguished Young Scholars of P. R. China under Grant 60625303, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20060288021, and the Natural Science Foundation of Jiangsu Province under Grant BK2008047. Yashun Zhang received the B.S. and M.S. degrees in Control Science and Control Engineering from Hefei University of Science and Technology in 2003 and 2006. He is currently a Ph.D. student in Control Science and Control Engineering, Nanjing University of Science and Technology. His research interests include fuzzy control, sliding mode control and nonlinear control. Shengyuan Xu received the Ph.D. degree in Control Science and Control Engineering from Nanjing University of Science and Technology in 1999. His research interests include robust filtering and control, singular systems, time-delay systems and nonlinear systems. Jihui Zhang is a Professor in the School of Automation Engineering of Qingdao University, China. His main areas of interest are discrete event dynamic systems, production planning and control, and operations research.     

Improved Parameter-dependent H∞ Filtering for Polytopic Delta Operator Systems

The problem of H∞ filtering for polytopic Delta operator linear systems is investigated. An improved H∞ performance criterion is presented based on the bounded real lemma. Upon the improved performance criterion, a sufficient condition for the existence of parameter-dependent H∞ filtering is derived in terms of linear matrix inequalities. The designed filter can be obtained from the solution of a convex optimization problem. The filter design makes full use of the parameter-dependent approach, which leads to a less conservative result than conventional design methods. A numerical example is given to illustrate the effectiveness of the proposed approach.     

New approaches to robust l2?l∞ and H∞ filtering for uncertain discrete-time systemsfiltering for uncertain discrete-time systems

The problems of robust l ₂−l _∞ and H _∞ filtering for discrete-time systems with parameter uncertainty residing in a polytope are investigated in this paper. The filtering strategies are based on new robust performance criteria derived from a new result of parameter-dependent Lyapunov stability condition, which exhibit less conservativeness than previous results in the quadratic framework. The designed filters guaranteeing a prescribed l ₂−l _∞ or H _∞ noise attenuation level can be obtained from the solution of convex optimization problems, which can be solved via efficient interior point methods. Numerical examples have shown that the filter design procedures proposed in this paper are much less conservative than earlier results.     

Neural Net-Based H ∞ Control for a Class of Nonlinear Systems

A novel neural net-based approach for H _∞ control design of a class of nonlinear continuous-time systems is presented. In the proposed frameworks, the nonlinear system models are approximated by multilayer neural networks. The neural networks are piecewisely interpolated to generate a linear differential inclusion models by which a linear state feedback H _∞ control law can be constructed. It is shown that finding the permissible control gain matrices can be transformed to a standard linear matrix inequality problem and solved using the available computer software. This revised version was published online in June 2006 with corrections to the Cover Date.     

Robust adaptive control of a class of nonlinear uncertain systems

A smooth robust dynamic feedback controller is constructed, and the problem of robust H∞ almost disturbance attenuation with internal stability is solved for high-order nonlinear systems with parameter uncertainties. Finally, illustrative example and simulation results demonstrate the effectiveness of the proposed method.     

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 LMI solution to the robust synthesis problem for multi-rate sampled-data systems

In this paper we address the asynchronous multi-rate sampled-data H_∞ synthesis problem. Necessary and sufficient conditions are given for the existence of a controller achieving the desired performance, and the problem is shown to be equivalent to a convex optimization problem expressed in the form of linear operator inequalities. In the case where the sample and hold rates are synchronous, these operator inequalities reduce to linear matrix inequalities, for which standard numerical software is available.     

Relaxed stability condition and state feedback H ∞ controller design for T-S fuzzy systems

In this paper, a fuzzy Lyapunov approach is presented for stability analysis and state feedback H _∞ controller design for T-S fuzzy systems. A new stability condition is obtained by relaxing the ones derived in previous papers. Then, a set of LMI-based sufficient conditions which can guarantee the existence of state feedback H _∞ controller for T-S fuzzy systems is proposed. In comparison with the existing literature, the proposed approach not only provides more relaxed stability conditions but also ensures better H _∞ performance. The effectiveness of the proposed approach is shown through two numerical examples. Recommended by Editor Young-Hoon Joo. Xiao-Heng Chang received the B.E. and M.S. degrees from Liaoning Technical University, China, in 1998 and 2004, respectively, and the Ph.D. degree from Northeastern University, China, in 2007. He is currently a Lecturer in the School of Information Science and Engineering, Bohai University, China. His research interests include fuzzy control and robust control as well as their applications. Guang-Hong Yang received the B.S. and M.S. degrees in Northeast University of Technology, China, in 1983 and 1986, respectively, and the Ph.D. degree in Control Engineering from Northeastern University, China (formerly, Northeast University of Technology), in 1994. He was a Lecturer/Associate Professor with Northeastern University from 1986 to 1995. He joined the Nanyang Technological University in 1996 as a Postdoctoral Fellow. From 2001 to 2005, he was a Research Scientist/Senior Research Scientist with the National University of Singapore. He is currently a Professor at the College of Information Science and Engineering, Northeastern University. His current research interests include fault-tolerant control, fault detection and isolation, nonfragile control systems design, and robust control. Dr. Yang is an Associate Editor for the International Journal of Control, Automation, and Systems (IJCAS), and an Associate Editor of the Conference Editorial Board of the IEEE Control Systems Society.