In control theory field, the rank of the controllable matrix is for investigating the control ability of the dynamic systems, and that the matrix is full rank means that the systems is the fully state controllable, and then the bigger the rank is, the stronger the control ability of the systems is. Up to now, no precise concept on the control ability is presented.
In fact, the rank of the controllable matrix and the controllablilty are not good measures on the control ability. For example, for the uncontrollable matrix pair (A,B), a little perturbation on the matrix A or B may lead to that the matrix pair is controllable. So, there isn't exist a distinct line between the controllability and uncontrollability concepts and the two concepts are fuzzy. Considered that the modeling error for the system models of the practical dynamic systems, the rank of the controllable matrix and the controllability aren't appropriate for investigating precisely the control ability of the practical dynamic systems.
In the studying on the precise measure on the control ability of the dynamic systems, the volume value of the controllable region under the unit input energy is propose as a new precise measure on that in my paper arXiv1705.08064(On Controllable Abundance Of Saturated-input Linear Discrete Systems). A new theorem on the positive relations among the volume value, the diversity of the control laws (i.e., the size of the solution space of the control laws) in the controller designing, and the performances and robustness of the closed-loop control systems is presented and proven in the paper. That is, the bigger the volume value of the controllable region, the richer the control laws, and then t the closed-loop control systems may be with he better performance and robustness. The volume value is a good measure on the control ability.
How to name the new measure? Which name is appropriate for that, controllable (or controllability) power, controllable (or controllability) abundance, controllable (or controllability) richness, or others? Which colleague gives some helpful suggestions.
In my paper arXiv1705.08064(On Controllable Abundance Of Saturated-input Linear Discrete Systems)，a new measure on the state control ability of the linear discrete systems is presented as an improving and promoting concept of the state controllability in the control theory field. The new measure is not only used to distinguish the controllability or uncontrollability, but also can be used to meter precisely the state control ability of the systems, and maybe is a fundamental concept and index to the dynamic systems.
Based on the new measure, many works in the control field and other related fields can be developed and new results will be with the significant value. Some works related to the controllable abundance can be listed as follows.
1. By computing and optimizing on the controllable abundance, the dynamics and kinematics of the dynamic systems can be improved and promoted, such as, enhancing the control ability of the input variable, extending the controllable region in the state space, and so on.
2. By computing and comparing on the controllable abundance, the input variables with the better control ability are determined from the candidate variables in the modeling of the controlled plants and the control efficient of the control systems can be promoted.
3. Based on the relation theorem among the control abundance, the control ability, and the size and richness of the solution set of the control laws in the controller designing, when the controlled plants are optimized on the controllable abundance, the closed-loop control systems designing by the conventional control methods can be with the better performance and robustness.
4. The controllable abundance can provide the theoretical basis and method 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.
5. The controllable abundance can be used as the objective function (performance index) or the constrained conditions in the optimal control and robust control, and optimizing the control abundance can be promoting the performance and robustness of the closed-loop systems.
6. As the generalization of the controllable abundance concept, the observable abundance can be proposed to meter the observe ability of the dynamic systems and is used to promote the state filtering systems and communication systems.
More meaningful works about the controllable abundance can be proposed and developed, after deeply understanding the new measure and concept.
Dear Ilhan Polat
Thank you for your answers. I always try to find the relevant literatures to my works on the control ability and maybe the relevant literatures was appeared in the earlier period of the state controllabilty studying. Can you tell me where is founded the relevant literatures? Thank you very much.
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