I would like to make an optimization for PID control system with three parameters Kp,Ki and Kd. If I choose the difference range of Kp,Ki and Kd. A different results will be got. I am not sure how to choose the range of Parameters.
You must to analyze the paramaters sensitivity factors and starting from these one to built the parameter sensitivity matrix (S). Then you can compute the covariance matrix and to regard the correlation existing between these three parameters (extra-diagonals terms).These ones can be also computed directly for the matrix S using the scalar product between each two columna of this matrix.
Theoretically, if the correlation factors is close to the unity, it is necessary to add suplimentar constraints or informations for the optimization problem in order to have uniqueness of solution or more stability of problem solution.
So, in a more general way you must to search the conditions for each the determinant of the (Stransposed*S) is minimum.
For more details see the book of Beck and Arnold with the title "Parameter Estimation...." (Edition of 1970) or other documents corresponding to statistical estimation of parameters.
In a real and easy practice you can keep constant one of the parameter and to identify only the reamining two ones. After you can add the third one and to re-identify the all in the same time. This strategy can be useful for some problems and can permits to converge to global solution.
from parametric study: stability set of first kind, stability set of second kind, and stability set of third kind
Ranges should be chosen with a certain relation to physical reality in mind. From a practical point of view, choose the ranges as broad as practical realisation possibilities permit. And if the obtained optimization result will yield a physically non-feasible special combination of parameters just try to get as near as possible to that, or specialize your boundary conditions more specifically by defining combined/related boundaries.
It is standard result if you do not define real constraints, restrictions. Then you have to use any optimisation method with restrictions (Newton, gradient m., etc). You can see article: Fikar- Kostur "Optimal proces control", 2012 13th International Carpathian Control Conference (ICCC). Other way is formulation objective function with more criteria. You can remove boundaries into any criteria. In this case you can use current optimisation method.
I would formulate the problem as an optimal control problem and solve it as unconstrained problem i.e.
f(k)= e Q_1 e^T + u Q_2 u^T
Where the weight matrices:
Q_1 : penalty for having an error
Q_2 : penalty for changing the control input - can be used to control the system damping . There might also be some system limitations i.e. how fast can you change the input to the system etc.
The dynamic behavior of the control system can then be controlled by manipulating the weight factors - the procedure are normally applied on MIMO system using a state space formulation
Regards
Benny Endelt
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