Keith Glover

State-space solutions to standard H/sub 2/ and H/sub infinity / control problems

Simple state-space formulas are derived for all controllers solving the following standard H/sub infinity / problem: For a given number gamma >0, find all controllers such that the H/sub infinity / norm of the closed-loop transfer function is (strictly) less than gamma . It is known that a controller exists if and only if the unique stabilizing solutions to two algebraic Riccati equations are positive definite and the spectral radius of their product is less than gamma /sup 2/. Under these conditions, a parameterization of all controllers solving the problem is given as a linear fractional transformation (LFT) on a contractive, stable, free parameter. The state dimension of the coefficient matrix for the LFT, constructed using the two Riccati solutions, equals that of the plant and has a separation structure reminiscent of classical LQG (i.e. H/sub 2/) theory. This paper is intended to be of tutorial value, so a standard H/sub 2/ solution is developed in parallel. >

State-space formulae for all stabilizing controllers that satisfy and H ∞ norm bound and relations to risk sensitivity

Abstract Given a linear system, all stabilizing controllers such that a specified closed-loop transfer function has H ∞ norm less than a given scalar, are parametrized. This characterization involves the solution to two algebraic Riccati equations, each with the same order as the system, and further gives gives feasible controllers also with this order. The relationship to the risk-sensitive LQG stochastic control problem is established, giving an equivalence between robust and stochastic control.

A loop-shaping design procedure using H/sub infinity / synthesis

A design procedure is introduced which incorporates loop shaping methods to obtain performance/robust stability tradeoffs, and a particular H/sub infinity / optimization problem to guarantee closed-loop stability and a level of robust stability at all frequencies. Theoretical justification of this technique is given, and the effect of loop shaping on closed-loop behavior is examined. The procedure is illustrated in a controller design for a flexible space platform. >

Robust stabilization of normalized coprime factor plant descriptions with H/sub infinity /-bounded uncertainty

The problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by H/sub infinity / bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system. This problem can be reduced to a Nehari extension problem directly and gives an optimal stability margin. All controllers satisfying a suboptimal stability margin are characterized, and explicit state-space formulas are given. >

Parametrizations of linear dynamical systems: Canonical forms and identifiability

We consider the problem of what parametrizations of linear dynamical systems are appropriate for identification (i.e., so that the identification problem has a unique solution, and all systems of a particular class can be represented). Canonical forms for controllable linear systems under similarity transformation are considered and it is shown that their use in identification may cause numerical difficulties, and an alternate approach is proposed which avoids these difficulties. Then it is assumed that the system matrices are parametrized by some unknown parameters from a priori system knowledge. The identiability of such an arbitrary parametrization is then considered in several situations. Assuming that the system transfer function can be identified asymptotically, conditions are derived for local and global identifiability. Finally, conditions for identifiability from the output spectral density are given for a system driven by unobserved white noise.

Model reduction of multidimensional and uncertain systems

We present model reduction methods with guaranteed error bounds for systems represented by a Linear Fractional Transformation (LFT) on a repeated scalar uncertainty structure. These reduction methods can be interpreted either as doing state order reduction for multi-dimensionalsystems, or as uncertainty simplification in the case of uncertain systems, and are based on finding solutions to a pair of Linear Matrix Inequalities (LMIs). A related necessary and sufficient condition for the exact reducibility of stable uncertain systems is also presented.

Mixed /spl Hscr//sub 2/ and /spl Hscr//sub /spl infin// performance objectives. II. Optimal control

This paper considers the analysis and synthesis of control systems subject to two types of disturbance signals: white signals and signals with bounded power. The resulting control problem involves minimizing a mixed /spl Hscr//sub 2/ and /spl Hscr//sub /spl infin// norm of the system. It is shown that the controller shares a separation property similar to those of pure /spl Hscr//sub 2/ or /spl Hscr//sub /spl infin// controllers. Necessary conditions and sufficient conditions are obtained for the existence of a solution to the mixed problem. Explicit state-space formulas are also given for the optimal controller. >

Realisation and approximation of linear infinite-dimensional systems with error bounds

The class of linear infinite-dimensional systems with finite-dimensional inputs and outputs whose impulse response h satisfies

Mixed /spl Hscr//sub 2/ and /spl Hscr//sub /spl infin// performance objectives. I. Robust performance analysis

h \in L_1 \cap L_2 (0,\infty ;\mathbb{C}^{p \times m} )

The application of scheduled H/sub infinity / controllers to a VSTOL aircraft

and induces a nuclear Hankel operator is said to be of nuclear type. For this class of systems it is shown that balanced or output normal realisations always exist and their truncations converge to the original system in various topologies. Furthermore, explicit