Vibrational and adaptive control of a class of distributed parameter systems described by parabolic partial differential equations

Hong, Keum Shik

Permalink

https://hdl.handle.net/2142/22910 Copy

Description

Title

Vibrational and adaptive control of a class of distributed parameter systems described by parabolic partial differential equations

Author(s)

Hong, Keum Shik

Issue Date

1991

Doctoral Committee Chair(s)

Bentsman, Joseph

Department of Study

Engineering, Mechanical

Discipline

Engineering, Mechanical

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Engineering, Mechanical

Language

eng

Abstract

• Two nonclassical control techniques, vibrational control and direct model reference adaptive control, for a class of distributed parameter systems described by parabolic partial differential equations (PDE's) are discussed.
• Vibrational control is an open loop control technique which proposes a utilization of zero mean parametric excitation to a dynamical system to achieve desired control objectives. The vibrational control problem consists of establishing the existence of parametric vibrations which stabilize an unstable system, and in synthesizing these parametric vibrations. The transient behavior analysis for a vibrationally controlled system is also included in the vibrational control problem. Stability criteria for linear oscillatory parabolic PDE's are discussed first. Vibrational control for nonlinear parabolic PDE's is considered for nonlinearities that give rise to two classes of vibrations; vector additive, and linear multiplicative. Since vibrational control strategy requires no on-line measurements, vibrational stabilization is a powerful alternative in situations when feedback and/or feedforward are difficult or impossible to apply due to the restrictions on sensing and actuation.
• The second half of this work addresses direct adaptive control for parabolic PDE's with constant or spatially-varying coefficients. It is assumed that the distributed measurement and the distributed control are both possible. The adaptation laws are obtained by the Lyapunov redesign method. It is shown that the concept of persistency of excitation in infinite dimensional adaptive systems needs to be investigated in relation to time variable, spatial variable, and boundary conditions. It is demonstrated that even a constant input signal is sufficiently rich in the sense that it guarantees the convergence of parameter errors to zero. Averaging theorems for two-time scale systems which involve a finite dimensional slow system and an infinite dimensional fast system are developed. The exponential stability of the adaptive algorithm, which is critical in finite dimensional adaptive control in terms of tolerating disturbances and unmodeled dynamics, is shown by applying averaging.
• In both vibrational control and direct adaptive control, averaging methods are being used for stability analysis. In vibrational control, which utilizes a qualitative change of global behavior of attractors caused by oscillations, the stability properties of the nonautonomous system is deduced from the stability properties of the averaged autonomous system. In adaptive control the whole closed loop system becomes time-varying by continuous modification of the control laws, and the stability of the closed loop is concluded from the exponential stability of the averaged system. Examples and computer simulations are provided to support the theory in both cases.

Type of Resource

text

Permalink

http://hdl.handle.net/2142/22910

Copyright and License Information

Copyright 1991 Hong, Keum Shik

Owning Collections