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In pole assignment, if the plant has a near pole zero cancellation then in general the control energy required is relatively large. Likewise in adaptive pole-assignment based on the Certainty Equivalence Principle, when the estimated plant has a near pole zero cancellation, the control energy could be excessive. One method to cope with this is to freeze the controller (suspend the application of the Certainty Equivalence principle) for the period of time when plant estimates have "near" pole zero cancellations. In practice, there could be difficulty in setting appropriate thresholds for "nearness".

The central tendency adaptive pole-assignment theory of this section leads to practical methods to assess whether or not there is a "near" pole zero cancellation and a method to select or construct a controller to use when there is such a cancellation. Also search procedures are noted for the more computationally intensive task of seeking the Mode[ukPAl Fk-i] where ukPA denotes a pole assignment control.

Pole-Assignment

For the plant (1.1) consider a pole assignment control scheme

EukPA = -Fyk, AE + BF = H (2.1)

where

E(q-!) = 1 + eiq-1 +...+emq-m , F(q-J) = fiq -1 +...+fnq-n

CH 5 POLE ASSIGNMENT CONTROL

The zeros o f H(q_1) are the assigned poles. The specific form of the controller is such that its free parameters are equal in number to the plant parameters. Also, the controller has a built in delay for implementation purposes.

The polynomial equation (2.1b) can be rewritten as an algebraic equation in terms of the Sylvester matrix Sab as

Sab[<}>]= h ’ <t‘T = [e t fT]> hT= [ lhTi

e x = [ei e2 ... em], f x = [fi f2 ... fnL hx = [hi h2 ... hn+m]* m + 1 —> n —» Sab = 1 ai 1 a2 ai 0 0 0 bi 0 . b2 bi . ‘ T n+m+1 _ I (2.2)

This relationship has the dual form

S

J = h, e1 = [? b^],

A solution o f (2.2) for <j) involves Sa b' 1* which is known to exist if and only if the plant has no pole zero cancellations, or that [3,page 142]

qnA(q_1), qmB(q_1) are coprime (2.4)

Also, under (2.4) it is known that Sef' 1exists, or equivalently that qnF(q_1 ), qmE (q 'i) are coprime.

CH 5 POLE ASSIGNMENT CONTROL

Persistence of Excitation

Should the plant disturbances (external or internal) include a white noise term wk, then the controller selection (2.1) with qnF(q*1 ), qmE(q*1 ) coprime, ensures that the plant states Xkx = [yk-1 ...yk-n Uk-1 ...Uk-m] are persistently exciting, see [4,5]. The controller states Xk are identical to the plant states by design. [In pole assignment where the dimension of (j) is less than that of 9, E has dimension (m-1) which is the minimum permissible, then in the absence of external excitation, one of the closed-loop system states Uk-i is a linear combination of the others, being constrained by the controller relationship, and excitation of plant states is not achieved.]

A Posteriori Densities

The evaluation of f(ukPA! Fk-i) to achieve Mode[ukPAl Fk-i] appears to be too formidable for practical implementation. Consider now the density f(4>| Fk-i). A differentiation of (2.2a) with respect to 0T leads after a number of steps (details are given in the appendix) to the Jacobian, under (2.4), and assuming det(SAß) * 0

Where Sab is obtained from Sab by deleting the first row and column. Likewise for Sef- Nowobserve

(2.5)

I det JI = det Sef det Se f (2.6)

det Sab det Sab

CH 5 POLE ASSIGNMENT CONTROL

mean 0km and covariance Pk. Then using standard arguments

f[<j>(0) I Fk-l] = Kldet J-1(8)lexp (-j lie-efcmilpk1) (2.7)

Here k is some normalizing constant so that the integral o f f[<j)(0) I Fk-i] over all 0

is unity. Also, in case det J(0) is not properly defined, we set f=0.

Note that if a controller were chosen with the dimension o f $ less than that o f 0, then evaluation o f f(<j) I Fk-i) could require integration out o f an auxiliary variable. This would render the central tendency adaptive controllers o f this section impractical for implementation.

Central Tendency Adaptive Pole Assignment:

The control UkPA o f (2.1) can be re-expressed in terms o f the controller states xkCPA as

UkPA = - ^ X k CPA, XkCPAx = [ uk- lPA ...Uk-mPA yk-1 yk-nl (2.8) Thus, a central tendency control can be defined, using obvious notation, as

UkCT = - Mode[ <j) I Fk-i]Txkc c r (2.9a)

Mode[ <}> I Fk-i] maximizes f[4>(0) I Fk-i] (2.9b)

However, there is practical difficulty in implementing (2.9b) over all 0, since for each 0 investigated, the evaluation <{>(0) from (2.2) is required. A compromise is to perform the maximization only over the set o f 0 for which (j>(0) is o f necessity evaluated, namely § i , Ö2, ... ^k, or a subset of these e.g. §k, ök-l> Ök-M for some M. Let the optimizing <}) be denoted 4>k* and the associated as §k*. Now

CH 5 POLE ASSIGNMENT CONTROL

UkCT = - <j)k*TXkCCT (2.10)

Properties.

In maximizing f((}> | Fk_i), the definition (2.7) tells us that there is a "maximization" of det(SAß) balanced by a "minimization" of Il0-0kmllpk1. Thus "near" pole-zero cancellations associated with §k* are avoided without departing "far" from 0km.

A typical scenario for the application of the pseudo mode control law (2.10) for the case when qnA (q -l), qmB(qT ) are coprime is as follows. In the absence of a near pole-zero cancellation associated with §k and with ök converging to 0, then one expects that the controller at each k will be selected according to the Certainty Equivalence Principle. Should there be a "near" pole-zero cancellation associated with §k, then ök will not maximize f[<j)(0i) I Fk_i] over i=l,2,...,k, but rather an estimate 0i for i<k will. One expects that on average ukCT will give an improved control signal, but not necessarily at all k. Should there be a "near" pole-zero cancellation associated with the plant, then if ä k is closer to 0 than previous estimates, even though there is a "near" pole-zero cancellation associated with §k, then we would expect that the approximate pseudo mode control will be the Certainty Equivalence control. Precise convergence properties are given in the next section, and simulations are given in Section 4.

Cautious Adaptive Pole Assignment.

In implementing standard adaptive pole-assignment based on certainty equivalence, ill-conditioning occurs when d etS j^ ic denoted Ak is small, since the control involves a factor Ak_1. The Cautious Control methodology suggests that such a factor is replaced by Ak[Ak2 + where fyk is an estimate of the variance in

CH 5 POLE ASSIGNMENT CONTROL

uncertainty o f Ak. Thus in obvious notion

ukCC = Ak[Ak2 + Here we take a A k 3Ak

aV k aßk

Where Pk is the covariance in uncertainty o f parameter estimates 0k, and

d A k

ö V

AkSMk'x