I am interested to know, why is control set in control systems theory, usually assumed convex, more specifically, a convex cone, a convex hull or a convex set? Please let me know if you've gotten the answer.
Thank you.
This because if the control set is convex, it is possible to use convex optimization techniques in order to get the global minimum or maximum. Engineering problems that can be casted in this framework are very often solvable with efficient numerical solutions. An example is the least-squares solution in systems identification. If the linear system is convex, then the coefficient matrix is positive semidefinite and the global minima (only a point in the case of a strictly positive definite matrix) can be found in a reliable way, using the pseudoinverse. Convexity is important also in robust control techiques and H2 optimal control theory.
For example, if the control set isn't convex in the robust identification and control framework, could be not possible to get the controller that minimizes the desired cost function (for example the Hinf norm for the robust stability) but another one (corresponding to a local minimum) would result from the optimization problem. That obviously would be a problem in controller synthesis.
In conclusion, it's important to know if the control set has this nice property called convexity, because this naturally lead to efficient and reliable numerical solution, that can be solved easily. If that isn't the case, other techniques can be applied but in general getting the global minimum isn't easy neither fast
This because if the control set is convex, it is possible to use convex optimization techniques in order to get the global minimum or maximum. Engineering problems that can be casted in this framework are very often solvable with efficient numerical solutions. An example is the least-squares solution in systems identification. If the linear system is convex, then the coefficient matrix is positive semidefinite and the global minima (only a point in the case of a strictly positive definite matrix) can be found in a reliable way, using the pseudoinverse. Convexity is important also in robust control techiques and H2 optimal control theory.
For example, if the control set isn't convex in the robust identification and control framework, could be not possible to get the controller that minimizes the desired cost function (for example the Hinf norm for the robust stability) but another one (corresponding to a local minimum) would result from the optimization problem. That obviously would be a problem in controller synthesis.
In conclusion, it's important to know if the control set has this nice property called convexity, because this naturally lead to efficient and reliable numerical solution, that can be solved easily. If that isn't the case, other techniques can be applied but in general getting the global minimum isn't easy neither fast
Marco Cattaruzza , Thank you friend. You are correct, but there is a point to share with you. As I know, convex optimization is valid for minimizing convex cost functions like a quadratic regulator problem as an example; J=int(x^2+u^2).dt, which is quadratic hence convex in u.
I know that, if cost is convex, then you can find efficiently, the global minimum and not worry to get troubled or trapped in a local minimum. But my question is the control set itself u ∈ U (not the cost function J) where U is a convex set (convex cone, or convex hull). I cannot imagine in my mind, why the control set U should be convex, even though probably cost is nonconvex in some cases. Thank you again. Saeb.
You are right, but you have to keep in mind that, in general, convex problems are the ones with objective function convex and the functions defining the constaints convex too (i.e. U, in our case, has to be convex). I hope that I helped you.
I suggest you to look into some books:
https://la.epfl.ch/wp-content/uploads/2018/08/ic-32_lectures-17-28.pdf
https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf
These might help to clarify your doubts
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