AP\(\alpha\)I controller for linear systems with nonconstant disturbances (Q1873756)

The authors of this paper address the problem of adaptive stabilization by output feedback of linear time-invariant, lumped, single-input, single-output plants \[ \begin{aligned} \dot x & =Ax+bu+d(t)\\ y & =cx(t) \end{aligned} \] where \(A\) is an \(n\times n\) real matrix, \(x=x(t)\) is an \(n\times 1\) real vector, \(b\) is an \(n\times 1\) real vector, \(c\) is a \(1\times n\) real vector, \(u: \mathbb{R} \to\mathbb{R}\), and \(d(t)\) is a real \(n\times 1\) nonconstant disturbance vector. It is assumed that the plant is unstable minimum phase of degree one. The authors propose an adaptive-proportional-\(\alpha\)-integral controller to stabilize the disturbed system. They compare the performance of their proposed controller to simple adaptive-proportional integral, simple adaptive-proportional-integral-integral, and simple adaptive-proportional-\(\alpha\) controllers by solving the output tracking problem of a step reference. Simulation results presented by the authors show uniformly better performance by their controller.