Controllable abundance vs controllable ellipsoid

     In my paper arXiv1705.08064(On Controllable Abundance Of Saturated-input Linear Discrete Systems), the state region $R_{c,T}$ constituted by all possible state $x_{0}$ that can be controlled to the origin of the state space by the unit input variable $\left(\left\Vert u_{t}\right\Vert _{\infty}\leq1,t\in[0,T]\right)$ in the finite time $T$ is defined as the unit controllable region, and its volume is used for metering the control ability of the linear systems, named as the controllable abundance. In fact, in the state-space control theory,the controllable ellipsoid $E_{c,T}$ is defined based on the controllable Grammian matrix and it is proven that the ellipsoid is constituted by all possible state $x_{0}$ that can be controlled to the origin by the unit-energy input variable $\left(\int_{0}^{T}\left\Vert u_{t}\right\Vert _{2}\mathrm{d}t\leq1,t\in[0,T]\right)$ . The ellipsoid and its volume can be used for metering the control ability of the linear systems, also.

     As the two measures on the control ability, the difference between the unit controllable region and the controllable ellipsoid are:

     1. The boundary of the unit controllable region $R_{c,T}$ is constituted by the farthest initial state that can be controlled to the origin by the unit input variable in the finite time $T$ , i.e., the optimal time (the fastest time) that the initial state $x_{0}$ in the boundary of $R_{c,T}$ is controlled to the origin is $T$ . But, the boundary of the controllable ellipsoid $E_{c,T}$ is constituted by the farthest initial state that can be controlled to the origin by the unit-energy input variable in the finite time T, i.e., the control process that the initial state $x_{0}$ in the boundary of $E_{c,T}$ is controlled to the origin in the time T can be with the optimal energy (fuel). Therefore, the controllable abundance is directly related to time optimal control problem and but the controllable ellipsoid is to energy optimal control problem.

    2. It is proven that the size (volume) of controllable region $R_{c,T}$ is proportional to the size (volume) of the solution space of the control law for solving the controller design problem, i.e., the bigger the controllable abundance (the volume of controllable region) is, the bigger the volume of the solution space is. Therefore, the control laws for the systems with the bigger controllable abundance can be with more diversity and richness, and then the closed-loop controlled systems can be with the better performance index and robustness. But for the controllable ellipsoid $E_{c,T}$ , the similar conclusion is not found as yet.

   3. Based on the computation and optimization of the controllable abundance, the theoretical supports and methods can be founded for determine the control horizon, the optimization horizon, the target state, the reference trajectory in the many control methods, such as, optimal control, adaptive control, predictive control, receding horizon optimal control, and so on. But, the similar results for the controllable ellipsoid  is not found as yet.

    According the above preliminary discussion, we can conclude that the controllable abundance as the measure on the control ability is with more direct, more general, and bigger application range than the controllable ellipsoid.