Adaptive Prediction and Control

Before introducing the adaptive prediction algorithm it is instructive to give an overview of adaptive control. The adaptive predictor is in fact a specialisation of a particularly simple adaptive control algorithm. Astrom (1983) presents a survey of Adaptive Control. The definition of adaptive control is a controversial issue but a suitable definition here would be

"An adaptive controller changes or adapts the design of the control algorithm to accommodate changes in the plant or environment"

The starting point for adaptive control is the classical feedback structure , with a process and a controller with adjustable parameters. The adaptive control problem is to find a suitable way of changing the controller parameters in response to changes in process and disturbance dynamics. Initial attempts by researchers to implement adaptive algorithms were stifled (Kalman 1958) by lack of relevant theory and perhaps more importantly inadequate computer hardware. Rapid developments in both theory and computer hardware has led to a great expansion of the field both in academic and industrial circles. The three main schemes

for parameter adaptive control are : gain scheduling ,model reference adaptive control MRAC, and self-tuning.

3.9.1 Gain Scheduling

In this scheme the adaption is based on feedforward measurements of the disturbance or parameter variations,Figure 3.1. The concept of gain scheduling originated as a means of combating large variations in parameters when high performance aircraft are operated within their normal flight envelopes. Mach numbers and dynamic pressures are measured to give an indication of changing process dynamics and used to schedule changes in the regulator parameters. Conceptually gain scheduling poses no problems. The efficiency of the scheme depends on the accuracy of the feedforward signals and prior information about the response of the controlled system. Stability and performance are typically evaluated by simulation with particular attention given to the transition between different operating conditions.

3.9.2 Model Reference Adaptive Control MRAC

The basic scheme of MRAC is shown in Figure 3.2. In MRAC the regulator can be thought of as consisting of two loops. The inner loop is ar ordinary feedback loop composed of the process and the controller. The adaption or outer loop appears as a supplementary feedback loop which feeds back the difference between the model states and states of the adjustable system. The parameters of the controller are adjusted by the outer or adaption loop. The outer loop is thus also a control loop. Finding an adjustment mechanism that brings the difference,to zero and guarantees stability is non trivial and cannot be (Landau 1973) solved using a simple linear feedback from the difference to the controller parameters. A MRAC scheme can often be represented as an equivalent feedback which is nonlinear and time varying.

The adaption mechanism can be equivalently thought of as either modifying the system parameters or as generating an auxiliary

input signal. Initially MRAC schemes were based on local parametric optimization theory. Stability analysis of the resulting schemes was however extremely complicated even for simple cases. These problems motivated the development of adaptive schemes based on stability results of Liapunov and more recently the hyperstability theory of Popov. The main advantage of MRAC is high speed of adaption. It is interesting to note that the parameter identification problem is in fact the "dual" to the MRAC scheme. The recursive determination of a "parametric plant" model using input-output data involves an adjustment of the parameters of the model in order to obtain the desired dynamic characteristics of the plant. Aspects of this duality are discussed in Landau.

3.9.3 Self Tuning Control

Self tuning is another method for automatically tuning controller parameters. Such a scheme is shown in Figure 3.3. The regulator can be thought of as being composed of three parts : a controller, a parameter estimator , and a third part which relates controller parameters to the estimated parameters according to some design algorithm. The feedback controller is in the form of a difference equation which acts upon measured outputs and any feedforward signals to generate the control action. The control design algorithm simply accepts the estimated parameter values and ignores their uncertainties. Such a procedure is called "certainty-equivalence" in stochastic control terminology (Astrom 1970). It essentially implies that the control law can be designed without any consideration for stochastic effects. It is theoretically possible but practically difficult to introduce cautious and dual strategies that take into account parameter uncertainties. Certainty equivalence and the related concepts of neutrality and separability are discussed in Jacobs and Patchell (1973). A certainty equivalence formulation considerably simplifies the algorithm. It may be possible to omit the control design strategy by reformulating the process model equations to yield an "implicit" self-tuner where

the parameter estimator directly produces the coefficients of the controller. Figure 3.3 shows an "explicit" self-tuner where the process model is first identified and then used in the control design stage.

Self tuning was originally proposed by Kalman in 1958 who attempted a hybrid computer implementation using,thermionic valves and multiturn potentiometers. However it was the key paper of Astrom and Wittenmark (1973) which gave the great impetus to the subject.

An algorithm is called self tuning if as the number of input and output samples tends to infinity, the control signal generated becomes that which would be produced by the corresponding feedback law designed on the basis of known process dynamics. Since there are many ways to design a control systems and to estimate parameters there are many varieties of self-tuning regulators.

The proof of the self tuning property assumes time invariant dynamics so that strictly speaking self tuning regulators are not adaptive since adaptive implies that the process dynamics are changing. However slowly time varying processes can easily be handled by self tuners. A detailed account of the various types of self tuners may be found in Harris and Billings (1981). Self tuners have unlike most of modern control theory found rapid application in industry, although most commercial self-tuners are little more than automated Ziegler-Nichols PID tuning aids !