Adaptive Modeling

4.0 INTRODUCTION

Adaptive plant modeling or plant identification is an important function for all adaptive control systems. In the practical world, the plant to be controlled may be unknown and possibly time variable. In order to apply adaptive inverse control, the plant, if unstable, must first be stabilized with feedback. This may not be easy to do, particularly when the plant dynamics are poorly known. Empirical methods may be needed to accomplish this objective. For present purposes, we assume that the plant is continuous, stable or stabilized, linear and time invariant. A discrete-time adaptive modeling system samples the plant input and output and automatically adjusts its internal parameters to produce a sampled output which is a close match to the samples of the plant output when the samples of the plant input are used as the input to the adaptive model. When the plant and its model produce similar output signals, the adaptive impulse response is a good representation of the plant impulse response. In reality, the discrete-time adaptive model is a model of the samples of the impulse response of the plant, whose z-transform is designated as P(z). The basic idea is illustrated in Fig. 4.1, where all signals and systems are considered to be sampled.

Figure 4.1 Adaptive modeling of a noisy plant.

Discrete-time adaptive modeling and control systems of the type described herein work only with samples of the plant input and output. For adaptive inverse control, the plant appears as a discrete-time system of transfer function P(z). It is P(z) that is modeled and controlled. For this reason, we will henceforth refer to P(z) as the plant.

Plant disturbance and plant output sensor noise will be lumped together for convenience as a single additive disturbance at the plant output, henceforth to be referred to simply as the plant disturbance. It is shown in Fig. 4.1 as a discrete-time additive noise n_kappearing at the plant output. The dynamic output response of the plant is y_k. The overall plant output is z_kgiven by

The discrete time index is k.

The transfer function of the plant is P(z). Its impulse response in vector form is

The components of this vector have values corresponding to the values of the respective impulses of the plant impulse response. The plant input is u_k. The dynamic output response is y_k, the convolution of its input with its impulse response:

Taking z-transforms, this relation becomes

The parameters of the adaptive model in Fig. 4.1 are generally adjusted by an adaptive algorithm to cause the error є_k to be minimized in the mean square sense. The desired response for the adaptive model is z_k.

A common and very useful form of adaptive model or adaptive filter is the tapped delay-line or transversal filter whose tap weighting coefficients are controlled by an adaptive algorithm. This type of adaptive filter is well-known in the literature [2]-[4], [16]. In Fig. 4.1, it converges to develop a transfer function (z) which is an estimate of P(z). The impulse response vector of P(z) is represented by (4.2). The impulse response vector of (z) is represented by (4.5):

There are n weights and each weight is a function of k, adapting iteratively. Other forms of adaptive filter can also be used, and they are well described in the literature [6]-[9]. The various adaptive filters become linear filters when their weights converge or otherwise become fixed. Adaptive filters converge to approximate Wiener solutions when they are adapted to minimize mean square error. Wiener filter theory is useful in predicting asymptotic converged behavior of adaptive filters.

Adaptive models can be generated that are very close representations of unknown plants. At any time k, however, there will be differences between _kand P. These differences will be called mismatch. There are three sources of mismatch.

1. One source of mismatch comes from representing a plant, whose impulse response is really of infinite length, in terms of a model whose impulse response is of finite length.
2. Another source of mismatch is due to inadequacies that may exist in the plant input signal, which may not excite all the important plant modes. This can be cured by using a "persistently exciting" dither signal added to the plant input. Dither helps to make the modeling process sure and solid but has the disadvantage of introducing an additional disturbance into the control system.
3. A third source of mismatch is noise in the model weights due to the adaptive process. Finite amounts of data are used by the adaptive process to determine the model weights. Only if the adaptive process were done infinitely slowly, using an infinite amount of real-time data, would there be no weight noise. Fast adaptation results in noise in the weights of k.

Neglecting all of these limitations for the moment (they will be addressed below), we proceed to examine idealized plant modeling in the presence of plant noise.