JUSTIFICACIÓN DE LAS SEÑALES DE TRANSITO DE LA UNA PUNO.

Some practical aspects of implementing the self tuning predictor will now be considered. Since the recursive estimator is at the heart of the predictor and indeed all self tuners it is appropriate to consider the implementation of the estimator equations (Clarke 1981) . The standard RLS algorithm can present numerical difficulties, particularly using naive programming of the algorithm. Since self tuners in general will be left unattended for long periods of time it is important to emphasise the importance of the estimation algorithm. Furthermore microprocessors are likely to be a practical means of self tuning implementation so that the estimator needs to be reliable even with relatively short word lengths and truncation errors. Indeed numerical issues are of the utmost importance for any reliable implementation of an algorithm.

Theoretically P, the covariance matrix must be a symmetrical and positive semi-definite matrix for stability of the algorithm. However due to truncation errors caused by a finite word length

P may loose this property. The updating equations for P (3.30,3.31) is numerically unsound so that P may become negative definite or skew symmetric,which results in instability in the parameter estimates. The algorithm will then retain its instability until a large control signal is used to make P positive definite again.

In the aerospace industry it was found that the Kalman filter equations suffer from the same difficulties. To circumvent the problem of potential negative definiteness the updating equations have to be cast into numerically stable forms. The two solutions that will be considered depend on the factorization of P into an

upper triangular matrix and its transpose since P is symmetric positive definite. The square root algorithm due to Peterka

(1975) factorizes P into the product

T

k^k (3.142)

where S is upper triangular. Outstanding numerical characteristics and simplicity of the recursive square root approach has lead to its implementation in many practical problems. Kaminski (1971) gives a survey of available techniques. A detailed derivation of the algorithm may be found in Peterka

(1973) who shows that the updating equations can be written as

] si. (3.143)

where

\xl = ^+x(k) Sj^_^sl,x(k) (3.144)

Rearranging and introducing an (n+1) x (n+1), orthogonal matrix

U , where n is the number of parameters being estimated gives

"ic-lT jf(^) I jfik)^ \^k aT 1 _(3.145) Where f(k) = x(k)'^Sy._^ UU^ = I j = (3.146)

By choosing U as the product of n orthogonal matrices U' each of which is used to annul an element of the last column of

I jfik)

lit

gives

U = \H 0\ (3.148)

j- jfjk)

Where H is upper triangular,so that

Sj^ = (3.149)

Æ

From the square root S it is straightforward to calculate the gain matrix K without having to evaluate P.

\^k

The algorithm requires n square root extractions. An alternative to the square root algorithm is the UD algorithm of Thornton and Bierman (1978). The matrix P is expressed as

P = UDU^ • iSlj

Where U is upper triangular with units stored off the diagonal and D is a diagonal matrix. In the self tuning predictor both algorithms were tried. The square root method involves about (4n^

+5n)/2 multiplications and n square root extractions per cycle,

the UD method involves about (3n^ + 3n)/2 multiplications per cycle. Apart from involving less computation the main advantage of the UD algorithm is that the diagonal D matrix corresponds to variances of the individual parameter estimates. Such information is valuable for diagnostic and jacketing software and would involve extra computation with the square root algorithm. Difficulties can also arise from non persistent input data leading to an ill conditioned estimation problem since linear dependent rows arise in the P matrix. This may occur for example when the algorithm is used as an on line identifier for a self tuning regulator whose feedback converges to a constant control law. Control actuator saturation can also result in similar problems since changes in demanded control no longer affect the

process. The instability can be observed in various quantities that are available for supervision purposes, for example the trace of the covariance matrix P and the bursting of parameter estimates. Isermann (1980),Wittenmark and Astrom (1984) discuss in detail configuration aids and aspects of software to supervise parameter adaptive control systems.

M e a s u r e m e n f s P a r a n e t e r s P R O C E SS C A I N S C H E D U L E C O N T R O L L E R

Figure 3.1 Gain Scheduled Adaptive Control Scheme

P a r a n e ters R E F E R E N C E M O D E L P R O C E S S A D J U S T M E N T A L G O R I T H M C O N T R O L L E R

P r o c e s s P a r a n e t e r s C o n t ro Paranet er ser C O N T R O L L E R P R O C E S S P A R A M E T E R E S T I M A TOR C O N T R O L L E R D E S IGN A L G O R I T H M