Design of Reduced-Order, H2 Optimal Controllers Using a Homotopy Algorithm

The minimal dimension of a linear-quadratic-gaussian (LQG) compensator is usually equal to the dimension of the design plant. This deficiency can lead to implementation problems when considering control-design for high-order systems such as flexible structures and has led to the development of methodologies for the design of optimal (or near optimal) controllers whose dimension is less than that of the design plant. This paper develops a new homotopy algorithm for the design of reduced-order, H2 optimal controllers. The algorithm has been implemented in MATLAB and the results are illustrated using a benchmark, non-colocated flexible structure control problem.

[1]  R. Skelton,et al.  Controller reduction using canonical interactions , 1987, 26th IEEE Conference on Decision and Control.

[2]  S. Richter.  A Homotopy Algorithm for Solving the Optimal Projection Equations for Fixed-Order Dynamic Compensation: Existence, Convergence and Global Optimality , 1987, 1987 American Control Conference.

[3]  R. Decarlo,et al.  A homotopy-method for eigenvalue assignment using decentralized state feedback , 1984, 1982 21st IEEE Conference on Decision and Control.

[4]  D. Bernstein,et al.  Homotopy Methods for Solving the Optimal Projection Equations for the H2 Reduced Order Model Problem , 1992 .

[5]  S. Richter,et al.  Continuation methods: Theory and applications , 1983 .

[6]  F. J. Gould,et al.  Homotopy methods and global convergence , 1983 .

[7]  R. T. Haftka,et al.  Tracing the Efficient Curve for Multi-objective Control-Structure Optimization , 1991 .

[8]  J. H. Avila.  The Feasibility of Continuation Methods for Nonlinear Equations , 1974 .

[9]  Anthony J. Calise,et al.  Fixed-order dynamic compensation for multivariable linear systems , 1988 .

[10]  Richard W. Longman,et al.  A homotopy approach to the feedback stabilization of linear systems , 1987 .

[11]  R. Kellogg,et al.  Pathways to solutions, fixed points, and equilibria , 1983 .

[12]  Jerry Sun.  How to perform differentiations on matrices , 1990 .

[13]  Arthur E. Bryson,et al.  Attitude Control of a Flexible Spacecraft , 1978 .

[14]  J. Junkins,et al.  A Sequential Linear Optimization Approach for Controller Design , 1985 .

[15]  Emmanuel G. Collins,et al.  Construction of low authority, nearly nonminimal LQG compensators for reduced-order control design , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[16]  M. Athans,et al.  Optimal limited state variable feedback controllers for linear systems , 1971 .

[17]  Emmanuel G. Collins,et al.  An input normal form homotopy for the L2 optimal model order reduction problem , 1994, IEEE Trans. Autom. Control..

[18]  James D. Turner,et al.  Optimal distributed control of a flexible spacecraft during a large-angle maneuver , 1984 .

[19]  P. Makila,et al.  Computational methods for parametric LQ problems--A survey , 1987 .

[20]  M. Mercadal.  HOMOTOPY APPROACH TO OPTIMAL, LINEAR QUADRATIC, FIXED ARCHITECTURE COMPENSATION , 1991 .

[21]  Beresford N. Parlett,et al.  Reduction to Tridiagonal Form and Minimal Realizations , 1992, SIAM J. Matrix Anal. Appl..

[22]  R. Skelton,et al.  A note on balanced controller reduction , 1984 .

[23]  R. H. Cannon,et al.  Experiments in control of flexible structures with noncolocated sensors and actuators , 1984 .

[24]  David C. Hyland,et al.  On direct versus indirect methods for reduced-order controller design , 1990 .

[25]  M. A. Athans,et al.  The role and use of the stochastic linear-quadratic-Gaussian problem in control system design , 1971 .

[26]  J. Yorke,et al.  The homotopy continuation method: numerically implementable topological procedures , 1978 .

[27]  B. Anderson,et al.  Controller Reduction: Concepts and Approaches , 1987, 1987 American Control Conference.

[28]  J. R. Newsom,et al.  Reduced-Order Optimal Feedback Control Law Synthesis for Flutter Suppression , 1982 .

[29]  Raymond A. DeCarlo,et al.  Continuation methods: Theory and applications , 1983, IEEE Transactions on Systems, Man, and Cybernetics.

[30]  E.G. Collins,et al.  Robust decentralized control laws for the ACES structure , 1991, IEEE Control Systems.

[31]  V. Mukhopadhyay.  Stability robustness improvement using constrained optimization techniques , 1987 .

[32]  D. Denery.  An Identification Algorithm That Is Insensitive to Initial Parameter Estimates , 1971 .

[33]  Thomas Kailath,et al.  Linear Systems , 1980 .

[34]  J. Junkins,et al.  Robust nonlinear least squares estimation using the Chow-Yorke homotopy method , 1984 .

[35]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[36]  D. Bernstein,et al.  The optimal projection equations for fixed-order dynamic compensation , 1984 .

[37]  George A. Geist.  Reduction of a general matrix to tridiagonal form using a hypercube multiprocessor , 1991 .

[38]  Udo Kuhn,et al.  Fresh look into the design and computation of optimal output feedback controls for linear multivariable systems , 1987 .

[39]  L. Watson.  Numerical linear algebra aspects of globally convergent homotopy methods , 1986 .

[40]  Huibert Kwakernaak,et al.  Linear Optimal Control Systems , 1972 .

[42]  R. Decarlo,et al.  A continuation algorithm for eigenvalue assignment by decentralized constant-output feedback , 1985 .

[43]  Raymond A. DeCarlo,et al.  Feedback gain optimization in decentralized eigenvalue assignment , 1986, Autom..

[44]  Yi Liu,et al.  Coprime factorization controller reduction with bezout identity induced frequency weighting , 1990, Autom..

[45]  L. Watson,et al.  Tracing structural optima as a function of available resources by a homotopy method , 1988 .

[46]  Emmanuel G. Collins,et al.  High performance, accelerometer-based control of the Mini-MAST structure at Langley Research Center , 1991 .

[47]  Arthur E. Bryson,et al.  Design of low-order compensators using parameter optimization , 1985, Autom..