Gramian and difference parametric Lyapunov equation based control of discrete-time linear time-varying systems with applications to tracking of nonholonomic systems

This paper investigates the control and observation problems for discrete-time linear time-varying systems. By combining finite-horizon controllability/reconstructibility Gramians with difference parametric Lyapunov equations, new state-feedback and observer design methods are developed. Under suitable finite-horizon Gramian conditions, explicit feedback and observer gains are constructed such that the resulting closed-loop systems are asymptotically stable or uniformly exponentially stable. In addition, boundedness properties of the associated matrix sequences and gains are established. To further reveal the structure of the proposed designs, two classical alternative gain formulas are revisited and further characterised, and precise relationships between the original gains and the alternative ones are derived. These results provide additional insight into the role of the finite-horizon parameters in controller and observer synthesis. The state-feedback and observer designs are then combined to address observer-based output-feedback stabilisation. Sufficient conditions are derived to guarantee the uniform exponential stability of the resulting closed-loop system. Finally, the proposed theory is applied to the trajectory tracking problem of a class of nonholonomic systems.