Closed Form Expressions of Linear Discrete Time System Responses with an Application to PID controllers

General closed—form expressions of linear discrete—time system responses of arbitrary order are presented without proof. The system poles can be real and/or complex, and may be repeated. While these expressions are readily computed from the system in standard forms, they are based on a backward difference formulation, shown to offer some important simplifications and a closer analogy with the continuous case. Expressions are also derived for Lyapunov (Sylvester) equations, whose solution is the corresponding (cross) Grammian matrix, thus allowing evaluation of it without direct reference to the poles of the system. Finally, an example is presented of how these expressions may be utilized to obtain an expression for zero-optimized open-loop PID coefficients.