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

Abstract

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 $H(\omega_n,f_n)$ 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 $i\omega_n$ of the reference signals, then all reference signals in $H(\omega_n,f_n)$ can be asymptotically tracked in the presence of disturbances if and only if \[ \bigl(H_K(i\omega_n)^{-1}[1-H_d(n)]f_n^{-1}\bigr)_{n \in I} \in \ell^2. \] Here $H_K(i\omega_n)$, $n \in I$, is the transfer function of the stabilized plant evaluated at $i\omega_n$, and $(H_d(n))_{n \in I}$ is a sequence of disturbancecoefficients for the stabilized plant. Moreover, the sequence $(f_n)_{n\in I}$ consists of weights for the Fourier coefficients of the reference signals. We give four examples to illustrate the theory.