Venkataramanan Balakrishnan

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.

Robust constrained model predictive control using linear matrix inequalities

The primary disadvantage of current design techniques for model predictive control (MPC) is their inability to explicitly deal with model uncertainty. In this paper, the authors address the robustness issue in MPC by directly incorporating the description of plant uncertainty in the MPC problem formulation. The plant uncertainty is expressed in the time-domain by allowing the state-space matrices of the discrete-time plant to be arbitrarily time-varying and belonging to a polytope. The existence of a feedback control law minimizing an upper bound on the infinite horizon objective function and satisfying the input and output constraints is reduced to a convex optimization over linear matrix inequalities (LMIs). It is shown that for the plant uncertainty described by the polytope, the feasible receding horizon state feedback control design is robustly stabilizing.

A new CAD method and associated architectures for linear controllers

A computer-aided design (CAD) method and associated architectures are proposed for linear controllers. The design method and architecture are based on recent results that parameterize all controllers that stabilize a given plant. With this architecture, the design of controllers is a convex programming problem that can be solved numerically. Constraints on the closed-loop system, such as asymptotic tracking, decoupling, limits on peak excursions of variables, step response, settling time, and overshoot, as well as frequency-domain inequalities, are readily incorporated in the design. The minimization objective is quite general, with LQG (linear quadratic Gaussian) H/sub infinity / and new l/sub 1/ types as special cases. The constraints and objective are specified in a control specification language which is natural for the control engineer, referring directly to step responses, noise powers, transfer functions, and so on. >

A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its L ∞ -norm

The i-th singular value of a transfer matrix need not be a differentiable function of frequency where its multiplicity is greater than one. We show that near a local maximum, however, the largest singular value has a Lipschitz second derivative, but need not have a third derivative. Using this regularity result, we give a quadratically convergent algorithm for computing the L∞-norm of a transfer matrix.

Robust Kalman filters for linear time-varying systems with stochastic parametric uncertainties

We present a robust recursive Kalman filtering algorithm that addresses estimation problems that arise in linear time-varying systems with stochastic parametric uncertainties. The filter has a one-step predictor-corrector structure and minimizes an upper bound of the mean square estimation error at each step, with the minimization reduced to a convex optimization problem based on linear matrix inequalities. The algorithm is shown to converge when the system is mean square stable and the state space matrices are time invariant. A numerical example consisting of equalizer design for a communication channel demonstrates that our algorithm offers considerable improvement in performance when compared with conventional Kalman filtering techniques.

Semidefinite programming duality and linear time-invariant systems

Several important problems in control theory can be reformulated as semidefinite programming problems, i.e., minimization of a linear objective subject to linear matrix inequality (LMI) constraints. From convex optimization duality theory, conditions for infeasibility of the LMIs, as well as dual optimization problems, can be formulated. These can in turn be reinterpreted in control or system theoretic terms, often yielding new results or new proofs for existing results from control theory. We explore such connections for a few problems associated with linear time-invariant systems.

Synthesis of fixed-structure controllers via numerical optimization

We propose an iterative algorithm for designing linear time-invariant controllers with some prespecified structure. The iterations require the solution of optimization problems based on linear matrix inequalities, in which either the Lyapunov function proving a certain property or the controller to be designed is alternately regarded as the optimization variable (while the other is fixed). A number of structure constraints on the controller (reduced-order, decentralized, etc.) can be addressed using this technique, which also extends to plants with nonlinearities or uncertainties. The algorithm is heuristic in nature, and is not guaranteed to converge globally. However it provides a locally optimal solution which depends on the initialization of the algorithm, and serves as a useful design tool.<<ETX>>

Real-time failure-tolerant control of kinematically redundant manipulators

Considers real-time fault-tolerant control of kinematically redundant manipulators to single locked-joint failures. The fault-tolerance measure used is a worst-case quantity, given by the minimum, over all single joint failures, of the minimum singular value of the post-failure Jacobians. Given any end-effector trajectory, the goal is to continuously follow this trajectory with the manipulator in configurations that maximize the fault-tolerance measure. The computation required to track these optimal configurations with brute-force methods is prohibitive for real-time implementation. We address this issue by presenting algorithms that quickly compute estimates of the worst-case fault-tolerance measure and its gradient. Comparisons show that the performance of the best method is indistinguishable from that of brute-force implementations. An example demonstrating the real-time performance of the algorithm on a commercially available seven degree-of-freedom manipulator is presented.

Linear matrix inequalities in robustness analysis with multipliers

Abstract We show that a number of standard robustness tests can be reinterpreted as special cases of the application of the passivity theorem with the appropriate choice of multipliers. We show how these tests can be performed using convex optimization over linear matrix inequalities.

Algorithms and software for LMI problems in control

A number of important problems from system and control theory can be numerically solved by reformulating them as convex optimization problems with linear matrix inequality (LMI) constraints. While numerous articles have appeared cataloging applications of LMIs to control system analysis and design, there have been few publications in the control literature describing the numerical solution of these optimization problems. The purpose of this article is to provide an overview of the state of the art of numerical algorithms for LMI problems, and of the available software.