Eero Immonen

On the Internal Model Structure for Infinite-Dimensional Systems: Two Common Controller Types and Repetitive Control

In this paper we shall extend and amplify the recent theory of controllers employing the so-called internal model structure. For exponentially stable closed loop control systems this structure has been shown in another paper to be necessary and sufficient for robust output regulation, also in infinite-dimensional spaces. Here we shall derive conditions under which two controller types occurring frequently in applications have the internal model structure. Under these conditions robust output regulation is achieved—also in infinite-dimensional spaces—if the closed loop system is sufficiently stable. In the case that the closed loop system is only strongly (but not exponentially) stable, it is sometimes possible to obtain conditional robustness. This means that asymptotic tracking/disturbance rejection is not destroyed by small perturbations so long as closed loop stability also persists. Our results allow for infinite-dimensional plants, controllers, and exogenous systems, and as an application of such a general setting we shall consider generalized repetitive control for exponentially stable infinite-dimensional SISO systems.

Output regulation of periodic signals for DPS: an infinite-dimensional signal Generator

In this note, we consider output regulation and disturbance rejection of periodic signals via state feedback in the setting of exponentially stabilizable linear infinite-dimensional systems. We show that if an infinite-dimensional exogenous system is generating periodic reference signals, solvability of the state feedback regulation problem is equivalent to solvability of the so called equations. This result allows us to consider asymptotic tracking of periodic reference signals which only have absolutely summable Fourier coefficients, while in related existing work the reference signals are confined to be infinitely smooth. We also discuss solution of the regulator equations and construct the actual feedback law to achieve output regulation in the single-input-single-output (SISO) case: The output regulation problem is solvable if the transfer function of the stabilized plant does not have zeros at the frequencies i/spl omega//sub n/ of the periodic reference signals and if the sequence ([CR(i/spl omega//sub n/, A+BK)B]/sup -1/ /spl times/(Q/spl phi//sub n/-CR(i/spl omega//sub n/, A+BK)P/spl phi//sub n/)) /sub n/spl isin/z//spl isin/l/sup n2/. A one-dimensional heat equation is used as an illustrative example.

Feedback and Feedforward Output Regulation of Bounded Uniformly Continuous Signals for Infinite-Dimensional Systems

This article presents a state space output regulation theory for linear infinite-dimensional systems (with bounded control, observation, and feedthrough operators) in which the reference and disturbance signals are bounded and uniformly continuous. Output regulation problems are posed and solved for both feedforward and error feedback controllers. Furthermore, under certain assumptions necessary conditions are obtained for those controllers which solve these output regulation problems. One consequence of these necessary conditions is a generalization of the well-known internal model principle. A key feature in our theory is that the exogenous system generating the reference and disturbance signals is infinite-dimensional.

Practical output regulation for bounded linear infinite-dimensional state space systems

We develop the mathematical foundations of practical state space output regulation for bounded infinite-dimensional linear systems. By practical output regulation we mean asymptotic tracking of references and rejection of disturbances with a given accuracy. Our main results are general upper bounds for the norms perturbations to the parameters of the exosystem, the plant and a controller which achieves exact output regulation. These bounds depend explicitly on the desired tracking accuracy @e>0. In this paper, all perturbations are assumed to be bounded, additive and linear. Our results apply for both feedforward and error feedback controllers, and for arbitrary bounded uniformly continuous reference/disturbance signals.

What Periodic Signals Can an Exponentially Stabilizable Linear Feedforward Control System Asymptotically Track

We study asymptotic tracking and rejection of continuous periodic signals in the context of exponentially stabilizable linear infinite-dimensional systems. Our reference signals are in Sobolev-type spaces

On the realization of periodic functions

H(\omega_n,f_n)

A Simple Robust Controller for the Approximate Regulation of Almost Periodic Signals for Linear Systems

and they (as well as the disturbance signals) are generated by an infinite-dimensional exogenous system. We show that there exists a feedforward controller which achieves output regulation if and only if the so-called regulator equations are satisfied and a decomposability condition holds. For SISO systems this result allows us to completely answer the question posed in the title: We show that if the stabilized plant does not have transmission zeros at the frequencies

Some Properties of Infinite-dimensional Systems Capable of Asymptotically Tracking Bounded Uniformly Continuous Signals

i\omega_n

Output regulation for infinite-dimensional linear systems periodic reference signals from Sobolev spaces

of the reference signals, then all reference signals in

Error Feedback Output Regulation of Bounded Uniformly Continuous Signals for Infinite-Dimensional Linear Systems

H(\omega_n,f_n)