Pierre T. Kabamba

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.

Worst-case analysis and design of sampled-data control systems

Sampled-data systems with continuous-time inputs and outputs, where some state variables evolve in continuous time and others evolve in discrete time, are considered. Methods are presented for computing and optimizing the L/sub 2/-induced norm of such systems, considered as operators relating square integral signals. These worst-case analysis and design results are counterparts of the H/sub infinity /-norm analysis and synthesis for purely continuous- or discrete-time systems. The analysis shows that the L/sub 2/-induced norm of a sampled-data system is smaller than gamma if certain discrete-time descriptor systems have no eigenvalue of magnitude one, and if a certain discrete-time linear time-invariant plant depending on gamma has H/sub infinity / norm smaller than gamma . The synthesis shows that the optimal attenuation problem for a four-block continuous-time plant with a digital controller can be solved with a gamma -iteration on a certain discrete-time four-block plant which depends on gamma . >

Planar, time-optimal, rest-to-rest slewing maneuvers of flexible spacecraft

The control problem of time-optimal, rest-to-rest slewing of a flexible spacecraft through a large angle is considered. The flexible spacecraft is modeled as a linear, elastic, undamped, nongyroscopic system suitable for analysis of planar rotational maneuvers. Minimum-time open-loop planar maneuvers are studied. The control histories are found to be bang-bang with multiple switches in each control variable. The optimal control history is shown to have an important time symmetry property. The switching times, final time, and costates at midmaneuver satisfy a system of nonlinear algebraic equations that can be solved using a homotopy method. An upper bound on attitude error due to control spillover is obtained. This helps to determine, a priori, the number of vibrational modes that need to be actively suppressed at the final time such that a prespecified pointing accuracy is guaranteed after the maneuver has been completed. A time-optimal slewing example is discussed to demonstrate the applicability of the results.

Simultaneous controller design for linear time-invariant systems

The use of generalized sampled-data hold functions (GSHF) in the problem of simultaneous controller design for linear time-invariant plants is discussed. This problem can be stated as follows: given plants P/sub 1/, P/sub 2/, . . ., P/sub N/, find a controller C which achieves not only simultaneous stability, but also simultaneous optimal performance in the N given systems. By this, it is meant that C must optimize an overall cost function reflecting the closed-loop performance of each plant when it is regulated by C. The problem is solved in three aspects: simultaneous stabilization, simultaneous optimal quadratic performance, and simultaneous pole assignment in combination with simultaneous intersampling performance. >

Adaptive Sinusoidal Disturbance Rejection in Linear Discrete-Time Systems—Part I: Theory

An adaptive regulation approach against disturbances consisting of linear combinations of sinusoids with unknown and/or time varying amplitudes, frequencies, and phases for SISO LTI discrete-time systems is considered. The new regulation approach proposed is based on constructing a set of stabilizing controllers using the Youla parametrization of stabilizing controllers and adjusting the Youla parameter to achieve asymptotic disturbance rejection. Three adaptive regulator design algorithms are presented and their convergence properties analyzed. Conditions under which the on-line algorithms yield an asymptotic controller that achieves regulation are presented. Conditions both for the case where the disturbance input properties are constant but unknown and for the case where they are unknown and time varying are given. In the case of error feedback, the on-line controller construction amounts to an adaptive implementation of the Internal Model Principle. The performance of the adaptation algorithms is illustrated through a simulation example. A companion paper [4] describes the implementation and evaluation of the algorithms for the problem of noise cancellation in an acoustic duct.

Worst case analysis and design of sampled data control systems

Considered are sampled data systems with continuous-time inputs and outputs, where some state variables evolve in continuous-time and others evolve in discrete-time. A method is presented for computing and optimizing the L/sub 2/ induced norm of such systems considered as operators relating square integrable signals.<<ETX>>

Control of semiconductor manufacturing equipment: real-time feedback control of a reactive ion etcher

This paper describes the development of real-time control technology for the improvement of manufacturing characteristics of reactive ion etchers. A general control strategy is presented. The principal ideas are to sense key plasma parameters, develop a dynamic input-output model for the subsystem connecting the equipment inputs to the key plasma variables, and design and implement a multivariable control system to control these variables. Experimental results show that this approach to closed-loop control leads to a much more stable etch rate in the presence of a variety of disturbances as compared to current industrial practice. >

Balanced gains and their significance for L^{2} model reduction

The singular values do not give enough information for model reduction in the L2sense. Other invariants are introduced which help describe the contribution of each state to the L2magnitude of the impulse response. This leads to another suboptimal balanced truncation criterion.

Monodromy eigenvalue assignment in linear periodic systems

This paper addresses the problem of assigning the eigenvalues of the monodromy matrix of a linear continuous-time periodic system by periodic state feedback. A controllability criterion is given which slightly strengthens known results. An explicit expression for a periodic feedback gain which performs the pole assignment is given. When the system is controllable over one period, the whole monodromy matrix is assignable by periodic feedback.

An LQR/LQG theory for systems with saturating actuators

An extension of the LQR/LQG methodology to systems with saturating actuators, referred to as SLQR/SLQG, where S stands for saturating, is obtained. The development is based on the method of stochastic linearization, whereby the saturation is replaced by a gain, calculated as a function of the variance of the signal at its input. Using the stochastically linearized system and the Lagrange multiplier technique, solutions of the SLQR/SLQG problems are derived. These solutions are given by standard Riccati and Lyapunov equations coupled with two transcendental equations, which characterize both the variance of the signal at the saturation input and the Lagrange multiplier associated with the constrained minimization problem. It is shown that, under standard stabilizability and detectability conditions, these equations have a unique solution, which can be found by a simple bisection algorithm. When the level of saturation tends to infinity, these equations reduce to their standard LQR/LQG counterparts. In addition, the paper investigates the properties of closed-loop systems with SLQR/SLQG controllers and saturating actuators. In this regard, it is shown that SLQR/SLQG controllers ensure semi-global stability by appropriate choice of a parameter in the performance criterion. Finally, the paper illustrates the techniques developed by a ship roll damping problem.