A Theorem on the relation among the controllable abundance, the solution set of the control laws, and the performance of the control systems

In my paper arXiv1705.08064(On Controllable Abundance Of Saturated-input Linear Discrete Systems), the following theorem on the relation among the controllable abundance, the solution set of the control laws, and the performance of the control systems is gotten and proven.

Theorem 1. For the linear discrete time-invariant systems $\varSigma(A(\alpha),B(\alpha))$ with the technic parameter $\alpha$ , if the controllable region of the system $\varSigma(A(\alpha_{1}),B(\alpha_{1}))$ is the sub-set of that of the system $\varSigma(A(\alpha_{2}),B(\alpha_{2}))$ , that is, the control abundance of the system $\varSigma(A(\alpha_{1}),B(\alpha_{1}))$ is bigger than that of the system $\varSigma(A(\alpha_{2}),B(\alpha_{2}))$ , the solution set of the control laws controlling (stabling) any initial state $x_{0}$ to the origin of the state space for the system $\varSigma(A(\alpha_{1}),B(\alpha_{1}))$ is also a sub-set of that for the system $\varSigma(A(\alpha_{2}),B(\alpha_{2}))$ .

According to the above Theorem, the control laws of the system $\varSigma(A(\alpha_{2}),B(\alpha_{2}))$ are with more abundant and more choices than that of the system $\varSigma(A(\alpha_{1}),B(\alpha_{1}))$ , and the control performance and robustness of the control law designing for the system $\varSigma(A(\alpha_{2}),B(\alpha_{2}))$ can be better than that for the system $\varSigma(A(\alpha_{1}),B(\alpha_{1}))$ . Therefore, optimizing the controllable abundance of the controlled systems by the technic parameter $\alpha$ can promote the control performance and robustness of the systems $\varSigma(A(\alpha),B(\alpha))$ .

The above Theorem and conclusions can be generilized to the linear continuous time-invariant systems.

The above results will undoubtedly bring new dawn to optimal system dynamics and control design.