A new look at the ∞-horizon linear-quadratic tracking problem

The ∞-horizon tracking problem is considered from the point of view of the linear-quadratic optimal control framework. It is well known that this problem does not have a solution in the strict sense because in general the cost is unbounded. However, for applications where the reference signal is generated by an asymptotically stable system, the problem is well posed and enjoys a bounded cost. In other cases where the control interval [T-t₀] is large, the design framework may still provide a suitable, implementable controller. Computationally, one term in the solution is found by solving an algebraic Riccati equation; and the second term involves an auxiliary function ν(t) found by solving a differential equation backward in time to determine ν(0) which is then used in the actual control run. The main contribution of this article is the development of a linear system of equations for ν(0) when T→∞. A simplification occurs for the scalar control of systems in the standard phase canonic (controllable) form. Two examples are included to illustrate the results