On the dual-Kernel, matric convolution integral in Discrete/Continuous control theory; exact, explicit, closed-form expressions for some simple cases

In the generalized-version of modern, MIMO discrete-time control, known as Discrete/Continuous (D/C) Control Theory, the traditional matric convolution integral B in the traditional, exact, discrete-time state-model: x((k +1)T) = Ax(kT)+ Bu(kT) is replaced by a more general convolution-matrix BH that has two independent kernels, which evolve in counterflow directions. The numerical-evaluation of the generalized matric convolution-integral BH is essential in practical applications of D/C-type discrete-time control, but the dual-kernel, counterflow nature of the matric convolution integral BH complicates the application of traditional numerical methods for evaluating convolution integrals. In this paper, symbolic software (MAPLE) is used to develop the exact, closed-form, explicit analytical expressions for the matric convolution-integral BH(T) for a family of simple, time- invariant, low-order examples. Those explicit, closed-form analytical results provide much needed "truth-models" against which numerical-evaluations of BH , using various alternative numerical-integration schemes, can be confidently compared.