Quite often a designer accepts a determined form of the control algorithm with unknown parameters and the problem consists in finding the values of these parameters optimizing the control quality, i.e. minimizing the per- formance index Q. Thus the choice of the optimal control algorithm is re- stricted to the choice from a class of algorithms determined by the ac- cepted form. The problem of finding the optimal values of parameters in a given form of the control algorithm is called a parametric optimization. For a deterministic plant and full information on this plant, one should as- sume a determined form of the control algorithm if the absolutely optimal algorithm (without a restriction mentioned above) is too difficult to find or to perform. In the case of uncertainties caused by a non-deterministic be- haviour of the plant or an incomplete information on the plant, the parame- ters in the assumed form of the control algorithm may be changed in an adaptation process described in Chap. 11.
Let us denote by a a vector of parameters in the control algorithm, which should be found by minimization of Q. For example, in the linear control algorithm considered in Sect. 5.2, the components of the vector a may be all entries of matrices in the description of this algorithm, or only some of them if the rest are fixed and their values are not to be chosen by a designer. The problem of the static optimization for the control system may be considered as a problem of the optimal control for a static plant, presented in Sect. 4.1. The control system to be optimized may be treated
98 5 Parametric Optimization
as a static plant where Q is an output y and the vector of the parameters a is an input. Of course, it is a discrete control with a long control interval, sufficient for the estimation of Q. To find the optimal value a* it is nec- essary to determine the function Q=Φ(a) and then to minimize this func- tion with respect to a, taking into account constraints concerning a if there are any. To obtain the function Q=Φ(a) one should determine functions of time describing the control process, i.e. functions used in the formula defining Q, e.g. ε(t) or εn if the integral or additive performance index evaluates the control error. This is a problem of the control system analysis mentioned in Sect. 2.6.
In Sects. 5.2, 5.3 and 5.4, the control analysis and the parametric opti- mization for selected cases of linear system will be considered. It will be shown that for linear stationary systems and quadratic performance in- dexes it is possible to determine Q using operational transform of time functions included into the formula for Q (Laplace transform or Z trans- form in a continuous or discrete case, respectively).
For nonlinear systems, in very simple special cases only it is possible to obtain an analytical solution of differential or difference equations describ- ing the control process and consequently, to obtain a formula for Q. Usu- ally, for the fixed value a, only the approximate value of Q may be calcu- lated by applying respective numerical methods. Then a* is determined by using one of successive approximation methods mentioned in Sect. 4.1, i.e. the successive approximations am of the exact result a* are determined in a way analogous to that presented for um in Sect. 4.1. For example, the algorithm analogous to (4.12) has the form
am+1 = am – Kwm
where wm denotes an approximate value of the gradient of Q with respect to a, in m-th step of calculations.
In the formulation and solution of the parametric optimization problem the given assumed form of the control algorithm is used. In practice, this form may be accepted as a result of a designer’s experience or an experi- ence of a human operator controlling real plants. We shall return to this problem in the third part of the book, in the considerations concerning the control in uncertain systems. Let us note that different given a priori forms of the control algorithm with the numerical values of parameters in these forms may be compared by using the performance index Q which may be calculated or obtained as a result of simulations for the known control plant. For the given control plant and two assumed forms of the control
algorithm, this form is better for which the minimum value of Q (i.e. the value Q for a = a*) is smaller. In Sect. 5.5 we shall present several fre- quently used forms of the control algorithm (or forms of a controller) in a closed-loop system: a linear controller (in particular, PID controller) and three nonlinear controllers (including so called fuzzy controller). Let us note that the comparison of these controllers based on Q requires the knowledge of the plant necessary to determine the value Q. So, a general statement that e.g. a fuzzy controller is better than a linear PID controller (or on the contrary) has no sense, even when the controllers with their op- timal parameters a* are compared. The result of comparison depends not only on the controller but also on a form of the plant equation and values of its parameters.
The parametric optimization is applied to different cases of the control systems with different forms of the performance index. This is not always the integral form considered above, especially with the limits of integration 0, ∞, which requires a convergence to zero of a function to be integrated. For example, in the case of two-position controller which will be presented in Sect. 5.5, it is easy to note that ε(t) is a periodic function for t greater than a certain number. Then as a performance index Q we can use an inte- gral of ε2(t) or |ε(t)| for the time interval equal to the period of ε(t). The amplitude of ε(t) may also be used as a performance index in this case.