Control Theory
Control theory... deals with the behavior of dynamical systems under the influence of mechanisms that aim to manipulate their behavior. While control systems date back to antiquity, a more formal analysis began with the dynamics analysis of the centrifugal governor, conducted by the physicist James Clerk Maxwell in 1868. A turning point in the application of dynamic control was in the area of flight when the Wright Brothers made their first successful test flights on December 17, 1903. By World War II, control theory was an integral part of fire-control systems, guidance systems, and electronics. Later the 'Race for Space' depended on the development of accurate control systems for spacecraft. Control theory also began to see an increasing use in fields such as economics, sociology, and more recently in the field of biology.
Robust control theory... deals with the control of systems under the influence of uncertainty such as disturbances and unmodeled dynamics.
Our research in robust control concerns the development of new tools for the analysis and synthesis of robust control systems, particularly in the l1 framework. This includes a framework for the analysis and synthesis of
stability and performance robustness in the presence of structured uncertainty described in the l1 norm. Together the synthesis work and the analysis work provide a complete theory and set of tools for treating robustness in the l1 setting. These contributions add to what has become a rich literature in l1 control that began in the original work of Dahleh and Pearson.
Outlined below are some specific theoretical contributions together with selected relevant publications.
A Framework for Robustness under Structured UncertaintyThis work derives nonconservative conditions for the stability and performance robustness of systems in the presence of structured uncertainty in the l1 framework. The conditions are simple and easily computable in terms of the Perron-Frobenius root of a certain nonnegative matrix. The conditions are suitable for robustness analysis of large-scale uncertain dynamical systems.
These results also show that exact conditions for robust performance can be given in terms of conditions for robust stability of a related system.
Some relevant publications:
- M. Khammash and J.B. Pearson, “Robust Disturbance Rejection in l1 Optimal Control Systems,” Systems and Control Letters, 14, pp. 93-101, 1990.
- M. Khammash and J.B. Pearson, “Performance Robustness of Discrete-Time Systems with Structured Uncertainty,” IEEE Transactions on Automatic Control, 36, pp. 398-412, 1991.
- M. Khammash, “Robust Performance: Unknown Disturbances, and Fixed Known Inputs,” IEEE Transactions on Automatic Control, Vol 42, No. 12, pp. 1730-1734, Dec. 1997.
We have developed non-conservative conditions for the performance robustness of time-varying systems in the presence of structured norm-bounded uncertainty. Using these conditions, we obtained a solution to the robustness analysis problem for sampled data systems in the l1 framework. This work on time-varying systems made it possible to explore the merits of time-varying control (as compared to time-invariant control) as it pertains to robust stability and performance of systems with structured uncertainty. Utilizing this machinery, we have shown that time-varying controllers offer no benefit over time-invariant controllers with respect to robustness to structured uncertainty.
Some relevant publications:
- M. Khammash, “Necessary and Sufficient Conditions for the Robustness of Time-Varying Systems with Applications to Sampled-Data Systems,” IEEE Transactions on Automatic Control, Vol. 38, No.1, pp.49-57, January 1993.
- M. Khammash and M. Dahleh, “Time-Varying Control and the Robust Performance of Systems with Structured Norm-Bounded Perturbations,” Automatica, Vol. 28, No. 4, pp. 819-821, July 1992.
Robust Tracking
This research addresses the problem of fixed input tracking and the robustness of this tracking property to uncertainty. The Internal Model Principle has been used to demonstrate tracking robustness when time-invariant uncertainty is present. This has been known for sometime. We have shown that when time-varying uncertainty is considered, the picture changes entirely. No longer can one assure the robustness of tracking under such uncertainty. This work quantifies precisely the role of uncertainty on the robustness of tracking by giving an analytic expression for the achievable tracking error. It has also been extended to solve the robust tracking problem for sampled-data systems.
- M. Khammash, “Robust Steady-State Tracking,” IEEE Transactions on Automatic Control, vol. 40, No. 11, pp. 1872-1880, Nov. 1995.
- L. Zou and M. Khammash, “Robust Steady-State Tracking of Sampled-Data Systems,” proceedings of the 1997 American Control Conference, NM, 1997, pp. 3616-3620.
- L. Zou and M. Khammash, “Robust Steady-State Tracking of Sampled-Data Systems,” Automatica, accepted.
The Scaled-Q Method for Solving the l1 Problem
In this work a new approach for solving MIMO l1 optimal control problems has been developed. We have introduced this approach, which has been referred to as the Scaled-Q approach, in order to alleviate many of the difficulties facing the numerical solution of optimal l1 control problems. In particular, the computations of multivariable zeros and their directions are no longer required.
The Scaled-Q method also avoids the pole-zero cancellation difficulties that existing methods based on zero-interpolation face when attempting to recover the optimal controller from an optimal closed-loop map. Since the Scaled-Q approach is based on solving a regularized auxiliary problem for which the solution is always guaranteed to exist, it can be used regardless of the locations of the system zeros. All solutions are based on finite dimensional linear programming.
Based on the Scaled-Q method we have developed a complete set of efficient software tools for solving the l1 control problem in its most general form.
Some relevant publications:
- M. Khammash, “A New Approach to the Solution of the l1 Control Problem: The Scaled-Q Method,” IEEE Trans. on Automatic Control, vol. 25, No. 2, pp. 180—187, February 2000.
- M. Khammash , “The Scaled-Q Method for Solving l1 Optimization Problems,” proceedings of the 1997 American Control Conference, Albuquerque, NM, 1997, pp. 2997-3001.
Multi-objective Control
We have developed a new approach for solving multi-objective problems. This general approach makes it possible to simultaneously incorporate performance objectives stated in terms of weighted l1, H2, or H-infinity norm constraints while allowing for constraints on the response of a known fixed input. The varying performance criteria are incorporated in the objective function and/or the constraints of an optimization problem that is a convex quadratic programming problem. The underlying theory behind these methods utilizes the Banach-Alaoglu theorem to obtain converging upper and lower bounds to the optimal solution.
Like the solution of the l1 problem using the Scaled-Q approach, the multi-objective methods are numerically tractable and avoid zero interpolation.
Some relevant publications
- M. Salapaka, M. Khammash, and M. Dahleh, “Solution of MIMO H2/ l1 Problem without Zero Interpolation”, SIAM Journal of Optimization and Control, vol. 37, No. 6, pp. 1865—1873, 1999.
- M. Salapaka and M. Khammash, “Multiobjective MIMO Optimal Control Design Using the H2 and l1 Norms,” Automatica, under review.
Synthesis of Globally Optimal Robust Controllers
The analysis conditions for robust performance with structured uncertainty discussed earlier are simple to compute. However the problem of designing controllers that achieve the maximum achievable robustness is difficult in any norm. In the H-infinity setting this problem is non-convex. Reasonable solutions can be obtained using the D-K (or D-G-K) iteration methods, but unfortunately global optimality cannot be guaranteed. In the l1 setting, we have shown that a D-K iteration type method whereby one alternates between solving an l1-optimal control problem and a Perron vector calculation for a nonnegative matrix yields controllers that attempt to maximize robustness. While this iterative method can be effective, it does not guarantee globally optimal solutions and tends to converge prematurely.
To circumvent these problems, we have taken two different approaches to the synthesis problem. In the first approach, we used sensitivity analysis of linear programming to characterize all solutions to the l1-problem for the entire search range of a single d-scaling parameter. This method provides a global solution to the robust synthesis problem in the special case when only two perturbations blocks are present. While this method is quite effective, it does not generalize to the more interesting cases when more than two blocks are present.
Our second approach is based on linear relaxation and provides global optimal solutions for the robust synthesis problem for any number of perturbation blocks. We have shown that linear relaxations of the nonlinear optimization problem give lower bounds for the optimal solution.
The problem structure is then utilized to show that these lower bounds converge to the optimal solution. Since converging upper bounds have also been obtained, the global optimal solution can be found. The only tool needed is linear programming.
Some relevant publications
- M. Khammash and J.B. Pearson, “Analysis and Design for Robust Performance with Structured Uncertainty”, Systems and Control Letters, 20, 179-187, 1993.
- M. A. Dahleh and M. Khammash, “Controller Design in the Presence of Uncertainty,” Automatica, Vol. 29, No. pp. 37-56, January 1993.
- M. Khammash, “Synthesis of Globally Optimal Controllers for Robust Performance with Unstructured Uncertainty,” IEEE Trans. on Auto. Cont., vol. 41, pp. 189-198, Feb. 1996.
- M. Khammash, M. V. Salapaka , T. Vanvoorhis, “Synthesis of globally optimal controllers in l1 using linear relaxation,” proceedings of Conference on Decision and Control, 1998.
- M. Khammash, M. Salapaka, and T. Van Voorhis, “Robust Synthesis in l1: A Globally Optimal Solution,” IEEE Transactions on Automatic Control, accepted.