Shankar P. Bhattacharyya

Robust, fragile, or optimal?

We show by examples that optimum and robust controllers, designed by using the H/sub 2/, H/sub /spl infin//, l/sup 1/, and /spl mu/ formulations, can produce extremely fragile controllers, in the sense that vanishingly small perturbations of the coefficients of the designed controller destabilize the closed-loop control system. The examples show that this fragility usually manifests itself as extremely poor gain and phase margins of the closed-loop system. The calculations given here should raise a cautionary note and draw attention to the larger issue of controller sensitivity which may be important in other nonoptimal design techniques as well.

New results on the synthesis of PID controllers

This paper considers the problem of stabilizing a first-order plant with dead time using a proportional-integral-derivative (PID) controller. Using a version of the Hermite-Biehler theorem that is applicable to quasi-polynomials, the complete set of stabilizing PID parameters is determined for both open-loop stable and unstable plants. The range of admissible proportional gains is first determined in closed form. For each proportional gain in this range, the stabilizing set in the space of the integral and derivative gains is shown to be either a trapezoid, a triangle or a quadrilateral. For the case of an open-loop unstable plant, a necessary and sufficient condition on the time delay is determined for the existence of stabilizing PID controllers.

A generalization of Kharitonov's theorem; Robust stability of interval plants

The robust stability problem is considered for interval plants, in the case of single input (multioutput) or single output (multi-input) systems. A necessary and sufficient condition for the robust stabilization of such plants is developed, using a generalization of V. L. Kharitonovs theorem (1978). The generalization given provides necessary and sufficient conditions for the stability of a family of polynomials delta (s)=Q/sub 1/(s)P/sub 1/(s)+ . . . +Q/sub m/(s)P/sub m/(s), where the Q/sub i/ are fixed and the P/sub i/ are interval polynomials, the coefficients of which are regarded as a point in parameter space which varies within a prescribed box. This generalization, called the box theorem, reduces the question of the stability of the box, in parameter space to the equivalent problem of the stability of a prescribed set of line segments. It is shown that for special classes of polynomials Q/sub i/(s) the set of line segments collapses to a set of points, and this version of the box theorem in turn reduces to Kharitonovs original theorem. >

Observer design for linear systems with unknown inputs

Results from the geometric theory of linear systems are utilized to present a constructive solution to the problem of designing a Luenberger observer to evaluate a given set of linear functions of the state of a linear system subject to unknown or disturbance inputs.

Iterative learning control - A survey and new results

Learning control is an iterative approach to the problem of improving transient behavior for processes that are repetitive in nature. Some results on iterative learning control are presented. A complete review of the literature is given first. Then, a general formulation of the problem is given. Next, a complete analysis of the learning control problem for the case of linear, time-invariant plants and controllers is presented. This analysis offers: insight into the nature of the solution of the learning control problem by deriving sufficient convergence conditions; an approach to learning control for linear systems based on parameter estimation; and an analysis that shows that for finite-horizon problems it is possible to design a learning control algorithm that converges, with memory, in one step. Finally, a time-varying learning controller is given for controlling the trajectory of a nonlinear robot manipulator. A brief simulation example is presented to illustrate the effectiveness of this scheme. 56 refs.

Robust control with structure perturbations

The problem of making a given stabilizing controller robust so that the closed-loop system remains stable for prescribed ranges of variations of a set of physical parameters in the plant. The problem is treated in the state-space and transfer-function domains. In the state-space domain a stability hypersphere is determined in the parameter space using Lyapunov theory. The radius of this hypersphere is iteratively increased by adjusting the controller parameters until the prescribed perturbation ranges are contained in the stability hypersphere. In the transfer-function domain a corresponding stability margin is defined and optimized on the basis of the recently introduced concept of the largest stability hypersphere in the space of coefficients of the closed-loop characteristic polynomial. The design algorithms are illustrated by examples. >

Pole assignment via Sylvester's equation

It is shown that the pole assignment problem can be reduced to solving the linear matrix equations AX − XA = −BG, FX = G successively for X , and then F for almost any choice of G . The result is a new pole assignment procedure and proof of the pole assignment theorem that should play an important role in both theoretical and practical applications.

Controllability, observability and the solution of AX - XB = C

A closed-form finite series representation of the unique solution X of the matrix equation AX − XB=C is developed. Using this representation, the image, kernel, and rank of X are related to the controllable and unobservable subspaces of the (A, C) and (C, B) pairs respectively. Bounds on the rank of X are obtainedin terms of the dimensions of these subspaces. In the case that C has unitary rank, an exact calculation of rank X is made. The generic rank of X with A, B fixed and C generic is evaluated.

Transient response control via characteristic ratio assignment

This note develops an approach to directly control the transient response of linear time-invariant control systems. We begin by considering all-pole transfer functions of order n for which we introduce a set of parameters /spl alpha//sub i/, i=1,...n called the characteristic ratios. We also introduce a generalized time constant /spl tau/. We prove that /spl alpha//sub 1/ and /spl tau/ can be used to characterize the system overshoot to a step input and the speed of response, respectively. By independently adjusting /spl alpha//sub 1/ and /spl tau/ in all-pole systems, arbitrarily small or no overshoot as well as arbitrarily fast speed of response can be achieved. These formulas are used to develop a procedure to design feedback controllers with feedforward or two parameter output feedback type for achieving time response specifications. For a minimum phase plant we show that arbitrary transient response specifications, namely one with independently specified overshoot and specified rise time or speed of response can be exactly attained.

Robust stability under structured and unstructured perturbations

The problem of robust stability for linear time-invariant single-output control systems subject to both structured (parametric) and unstructured (H/sub infinity /) perturbations is studied. A generalization of the small gain theorem which yields necessary and sufficient conditions for robust stability of a linear time-invariant dynamic system under perturbations of mixed type is presented. The solution involves calculating the H/sub infinity /-norm of a finite number of extremal plants. The problem of calculating the exact structured and unstructured stability margins is then constructively solved. A feedback control system containing a linear time-invariant plant which is subject to both structured and unstructured perturbations is considered. The case where the system to be controlled is interval is treated, and a nonconservative, easily verifiable necessary and sufficient condition for robust stability is given. The solution is based on the extremal of a finite number of line segments in the plant parameter property of a finite number of line segments in the plant parameter space along which the points closest to instability are encountered. >