Disturbance decoupling and robustness of stability

Abstract

The authors consider the finite-dimensional linear time-invariant system described by \[ \left\{\begin{aligned}\dot x(t) &= Ax(t) + Bu(t) + Ed(t)\\ y(t) &= Cx(t)\\ z(t) &= Dx(t) \end{aligned}\right. \] The equations represent a plant with state \(x\in\mathcal X\), control input \(u\in\mathcal U\), measurements \(y\in\mathcal Y\), and controlled outputs \(z\in\mathcal Z\) subjected to the disturbances \(d\in\mathcal D\), where \(\mathcal X, \mathcal U, \mathcal D, \mathcal Y, \mathcal Z\) are finite-dimensional vector spaces. It is assumed that the pair \((A, B)\) is stabilizable and the pair \((C, A)\) is detectable. The disturbance decoupling problem with robust stability is introduced as an extension of the classical disturbance decoupling problem to deal with the uncertainties in the plant description. It is shown that this problem is equivalent to an \(H_\infty\) optimization problem with equality constraints which can be solved by standard techniques when the transfer matrix from the control input to controlled output is left invertible or the transfer matrix from the disturbance input to measured output is right invertible. By formulating the disturbance decoupling and robust stability requirements in terms of subspace-valued functions, an upper bound on the achievable robustness of stability is derived that applies in the general case. Although this is the best possible bound that can be obtained by a finite dimensional pointwise geometrical analysis it is shown that the actual robustness of stability is in general smaller and the pointwise geometrical analysis does not give sharp bounds. Nevertheless, this result is useful to identify structural properties of the plant that cause the disturbance decoupling requirement to be restrictive over the robustness of stability. An example of a second-order plant with damping ratio subject to the unknown distrurbance is considered.