J.W. Helton

H/sup infinity / control for nonlinear systems with output feedback

The basic question of nonlinear H/sup infinity / control theory is to decide, for a given two-port system, when feedback that makes the full system dissipative and internally stable exists. This problem can also be viewed as a question about circuits, and, after translation, also has a game-theoretic statement. Several necessary conditions for solutions to exist are presented, and sufficient conditions for a certain construction to lead to a solution are given. >

Worst case analysis in the frequency domain: The H ∞ approach to control

This paper describes an approach to frequency domain design using Hxmethods. We outline the main tools available and describe how these methods can be applied to control problems and to some gain optimization problems in circuits. In particular, we present a new method for combining several competing constraints which arise in control (such as tracking error and bandwidth constraints) into a single type of constraint which H^{\infty} methods analyze explicitly. A computer program which implements and teaches this method is on tape for a VAX. It is available on request.

Frequency response algorithms for H/sub infinity / optimization with time domain constraints

A very broad framework for control system design is considered that encompasses frequency-response methodologies for H/sub infinity / optimization that solve various aspects of the control design problem and that are less well known that state-space methods. The focus is on linear programming, Lawsons algorithm, and Trefethens algorithm. A modified Lawsons algorithm is proposed and related to Trefethans method. The modified algorithm is shown to be significantly faster than linear programming and Lawsons algorithm. It is also shown how to extend the modified Lawsons algorithm so as to handle time-domain constraints in addition to frequency-domain specifications, which distinguishes it from other H/sub infinity / optimization methods. Some steps are taken toward dealing with time domain constraints within an H/sub infinity / optimization framework. >

Optimal biped walking with a complete dynamical model

We solve the problem of generating symmetric, periodic minimum energy gaits for a 5-link biped robot moving in the sagittal plane of forward motion. We seek to approximate natural walking motion through the minimization of actuation energy. The model we use has considerably more structure than those previously studied. We deal with a fully nonlinear minimum energy path planning problem on a 14-dimensional state space. Also, a large number of constraints must be considered, including contact and collision effects. Our solution required development of various symbolic, dynamical algorithms relating to multibody systems and use of powerful numerical optimal control software. Solving the minimum energy walking problem including saturation and algebraic constraints amounts to solving a Hamilton-Jacobi-Bellman type equation along the optimal path. We use the path planning software DIRCOL which provides a substantial decrease in computing time required for generating solutions. We discuss numerical optimization and other modeling issues.

Convex Matrix Inequalities Versus Linear Matrix Inequalities

Most linear control problems lead directly to matrix inequalities (MIs). Many of these are badly behaved but a classical core of problems are expressible as linear matrix inequalities (LMIs). In many engineering systems problems convexity has all of the advantages of a LMI. Since LMIs have a structure which is seemingly much more rigid than convex MIs, there is the hope that a convexity based theory will be less restrictive than LMIs. How much more restrictive are LMIs than convex MIs? There are two fundamentally different classes of linear systems problems: dimension free and dimension dependent. A dimension free MI is a MI where the unknowns are g -tuples of matrices and appear in the formulas in a manner which respects matrix multiplication. Most of the classic Mis of control theory are dimension free. Dimension dependent Mis have unknowns which are tuples of numbers. The two classes behave very differently and this survey describes what is known in each case about the relation between convex Mis and LMIs. The proof techniques involve and necessitate new developments in the field of semialgebraic geometry.

Computer simplification of formulas in linear systems theory

Currently, the three most popular commercial computer algebra systems are Mathematica, Maple, and MACSYMA. These systems provide a wide variety of symbolic computation facilities for commutative algebra and contain implementations of powerful algorithms in that domain. The Grobner basis algorithm, for example, is an important tool used in computation with commutative algebras and in solving systems of polynomial equations. On the other hand, most of the computation involved in linear control theory is performed on matrices, which do not commute, and Mathematica, Maple, and MACSYMA are weak in the area of noncommutative operations. The paper reports on applications of a powerful tool, a noncommutative version of the Grobner basis algorithm. The commutative version of this algorithm is implemented in most major computer algebra packages. The noncommutative version is relatively new.

Dissipative control systems synthesis with full state feedback

In this paper we present a general framework for synthesizing state feedback controllers to achieve any desired closed loop dissipative behavior. Special cases include positive real andH∞ controller synthesis. We show that the solution to this dissipative controller synthesis problem is equivalent to the existence of a solution to a partial differential inequality (nonlinear systems case) or an algebraic Riccati inequality (linear systems case). Stability results can be obtained under appropriate detectability assumptions, and a generalization of the strict bounded real lemma is given. We also present an application of the results to a robust stabilization problem.

Minimal energy control of a biped robot with numerical methods and a recursive symbolic dynamic model

The problem of constructing a nonlinear controller for a biped robot optimal with respect to a minimal energy performance criteria is considered. The solution of this difficult, highly nonlinear problem is facilitated by the conjunction of several new developments in numerical optimal control and constrained recursive dynamic models for robotic systems. A 5-link biped model is used with the full dynamics and a uniform distribution of mass at each link. Contacts are modeled as inelastic, and the full dynamics together with the contact and collision forces are calculated efficiently using a recursive symbolic representation of the dynamics. The flexibility and modularity of our dynamics algorithms allows one to construct reduced unconstrained models which do not suffer from integration difficulties. The numerical optimal control software used is powerful and quick enough to handle high dimensional nonlinear systems. The result of our experiment is a walking controller which is optimal with respect to a type of minimum energy performance.

An information state approach to nonlinear J-inner/outer factorization

Proposes a theory for obtaining J-inner/outer factorizations of nonlinear systems based on the previously developed information state framework for output feedback differential games and H/spl infin/ control.<<ETX>>

Sufficient conditions for optimization of matrix functions

Inequalities involving matrix polynomials and associated optimization problems have become very important in engineering. Commonplace in design problems are performance functions /spl Gamma/(X,Y) which are convex in X and convex in Y but which are not jointly convex, and the problem is to minimize the highest eigenvalue of /spl Gamma/. In a previous paper (1997) we derived first order tests for coordinate optimization (the most common approach to these problems) and to first order optimality tests for a true optimum. This article treats second order optimality conditions for optimization of matrix functions. Second order tests are important because optimization of matrix valued /spl Gamma/ based on linearization or coordinate descent will often produce critical points which are not local solutions to the problem. Sufficient conditions, especially if they include second order information, become very valuable in these cases, as they can be used to tell which among the critical points correspond to true local optimal points, and to provide good update directions. Also we introduce and characterize a strong notion of matrix convexity which appears suited to many well behaved engineering problems.