A multivariable dichotomous optimum-seeking method

Abstract

Before any identification can be performed, it is necessary to determine the plant type. Since methods for doing this are available, it is assumed that this is known at the start of identification. The identification problem then is to determine [FIT Fyfor an!. vector u, where A( u ) is the steady-state operator.It is possible to measure the Jacobian matrices by experimentall>making S changes in the input (letting the system come to the steady-state each time) and recording the resulting outputs. Let si denote a n : Y vector with a one in the i-th position (start the count from zero) and zeros everywhere else. Then the ith column is Since an infinitesimal change cannot be made, a small but finite -, will be used. Thus the identification procedure can be summarized as follows:Step 1: Establish a steady-state with respect to the vector u, and call the output vector observed over a full steady-state period y.Step 2: Establish a steady-state with respect to the vectors ( u + y s i ) with y small for i = O , 1, 2, . . . '%'-I. C a l l the outputs observed over a full period yi, respectively.Step 3: The columns of the matrix Since Fy is specified by the form of the known performance functional, the required identification has been performed. The question arises whether the required information can be obtained without determining the entire matrix aA( u ) / a u .Observe that ( 1 i ) YOW for identification of type-zero plants if A u is chosen to be the vector \\-here RF, is Fy written in reversed order. (The reversal operator R is a square matrix with ones down the secondary diagonal and zeros elsewhere.) Then for small -/ A ( u + YAU) -A(u) -aA(u) R F~ f n. (19) au \Yhen the set of equations (19) are written in reversed order they become R n = BF, (20) where the i, k-th element of the matrix E is B ,al(.\--k-l rkfor 2, k = 0, . . . , -1 ' -1 Thus if the plant has the property that for all i and k , then E = [aA/auIT and the required identification can be performed by making only one variation in the input. Modifications of this simplified procedure can be obtained by first establishing a steady-state with respect to a vector v = Ru. If the plant has the property then using A v = F y will make j the required vector [ a~( u ) / a u ]~~; or if then choosingAv=RFY makes ( 2 3 ) hold. X similar simplification may be possible for type one system., but since in that case A u is an 2 1 ; vector and Fy is a -1 8 vector i t is not apparent how this ma>-be done. R j the required vector. I t ran be readily shown that when A( u ) is linear (it is already assumed to be timeinvariant) that properties (21) and COSCLVSIOSST o recapitulate, it has been shown that the self-optimization problem for unknown and in general nonlinear plants can be solved by an iterative procedure which alternates identification with optimization stages. I t was further pointed out that when a plant is dynamic, care must be taken to insure that the plant is characterized by a unique fixed operator. For type-zero and type-one plant: . . it was shown that by ...