Sensor blending for direct adaptive control of non-minimum phase linear infinite-dimensional systems in Hilbert space

2017 ◽  
Author(s):  
Mark J. Balas ◽  
Susan A. Frost
Keyword(s):  
Hilbert Space ◽  
Adaptive Control ◽  
Minimum Phase ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Direct Adaptive Control

Adaptive Tracking Control for Linear Infinite Dimensional Systems

2016 ◽  
Author(s):  
Mark J. Balas ◽  
Susan A. Frost
Keyword(s):  
Hilbert Space ◽  
Adaptive Control ◽  
Finite Basis ◽  
Bounded Operators ◽  
Linear Diffusion ◽  
Adaptive Tracking Control ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Direct Adaptive Control ◽  
Diffusion Systems

Tracking an ensemble of basic signals is often required of control systems in general. Here we are given a linear continuous-time infinite-dimensional plant on a Hilbert space and a space of tracking signals generated by a finite basis, and we show that there exists a stabilizing direct adaptive control law that will stabilize the plant and cause it to asymptotically track any member of this collection of signals. The plant is described by a closed, densely defined linear operator that generates a continuous semigroup of bounded operators on the Hilbert space of states. There is no state or parameter estimation used in this adaptive approach. Our results are illustrated by adaptive control of general linear diffusion systems.

Augmentation of Fixed Gain Controlled Infinite Dimensional Systems With Direct Adaptive Control

2020 ◽  
Author(s):  
Mark J. Balas
Keyword(s):  
Hilbert Space ◽  
Adaptive Control ◽  
Adaptive Optics ◽  
Open Loop ◽  
Loop System ◽  
Linear Diffusion ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Transmission Zeros ◽  
Direct Adaptive Control

Abstract Linear infinite dimensional systems are described by a closed, densely defined linear operator that generates a continuous semigroup of bounded operators on a general Hilbert space of states and are controlled via a finite number of actuators and sensors. Many distributed applications are included in this formulation, such as large flexible aerospace structures, adaptive optics, diffusion reactions, smart electric power grids, and quantum information systems. In this paper, we focus on infinite dimensional linear systems for which a fixed gain linear infinite or finite dimensional controller is already in place. We augment this controller with a direct adaptive controller that will maintain stability of the full closed loop system even when the fixed gain controller fails to do so. We prove that the transmission zeros of the combined system are the original open loop transmission zeros, and the point spectrum of the controller alone. Therefore, the combined plant plus controller is Almost Strictly Dissipative (ASD) if and only if the original open loop system is minimum phase, and the fixed gain controller alone is exponentially stable. This result is true whether the fixed gain controller is finite or infinite dimensional. In particular this guarantees that a controller for an infinite dimensional plant based on a reduced -order approximation can be stabilized by augmentation with direct adaptive control to mitigate risks. These results are illustrated by application to direct adaptive control of general linear diffusion systems on a Hilbert space that are described by self-adjoint operators with compact resolvent.

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