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The problem of system classification can be stated (after Zadeh) as determination, on the basis of observation of inputs and outputs, of one among specified classes of systems to which the system under test belongs. If the system is stochastic and the amount of input-output data is finite the system under test cannot be classified with certainty, i.e. even after the experiment the experimenter cannot state with certainty which one of his hypotheses is true. In the present paper the Bayesian statistics, with subjective interpretation of probability as a measure of belief, is employed to extract the relevant information contained in the data. Given the prior probability distribution (a model of the prior uncertainty) the aposterior probability distribution on the set of hypotheses (conditioned on the data observed) is calculated in order to obtain a rational basis for some kind of decision (e.g. the choice of model structure). The unknown order of a linear system is a special case of this kind. Power and limitations of the theory are demonstrated on examples.

In Bayesian statistics the concept of probability is interpreted as a rational measure of belief which is used to descri be mathemat ically the uncertain re lation between the statistician and the external world. The statistical inference is understood as a correction of prior subject ive probability distribution by objective data. The paper shows that on this Bayesian basis it is possible to build up a consistent theory of system identification. The following problems are considered: one-shot and real time identification, estimation and prediction in closed control loop, redundant and ~identifiable parameters, time-varying parameters and adaptivity.

A self-tuning controller with continuously operating control law updated in a discrete way is presented. The self-tuning is based on realtime identification of a stochastic continuous-process model and LQ-optimum control synthesis. The input and the output are filtered so that the derivatives of the filtered signals are available and their linear relation is estimated. From this relation and from the filter employed a stochastic model of the controlled process is reconstructed and used to update the control law. The controller can be realized purely digitally with high sampling rate for filtering and control and with a lower one for more complex algorithms of identification and control-law updating. The performance of the controller is demonstrated on simulations of a nontrivial case.

Linear finite-memory output predictors updated in real time appear to be a suitable internal representation of the system in a digital self-tuning controller. A new time-domain method of quadratic-optimum control synthesis for systems described by such predictors is presented. The synthesis covers both the servo problem and the regulation problem, including the program control (with preprogrammed command signal) and the feedforward from measurable external disturbances. Unlike the standard Riccati equation, the method leads to algorithms (or explicit formulae in low-order cases) which are numerically robust and therefore suitable for real-time computation using microprocessors with reduced word length.

Self-tuning control using an ARMA, or a theoretically equivalent Delta model to represent the controlled process is considered. If the order ηof the model, the time delay d,and the parameters of the moving-average part (the c—parameters) are chosen a priori, then the remaining parameters of the autoregression and regression part (a—and 6—parameters), which are left for realtime estimation, determine the set of models which can be covered by the preselected model structure. In the paper more than one prior choices of {n, d, c} are considered as uncertain hypotheses. Using the Bayesian approach the probabilities of particular hypotheses, conditioned on the observed data, are determined recursively in real time. Well feasible algorithms are developed for this purpose and demonstrated on simulated examples. It is shown that positional and incremental models can be considered as just special choices of the c—parameters.

Delta model is an alternative form of the standard ARMA model. Although theoretically equivalent it appears numerically more robust and therefore more suitable for real-time computation in adaptive and self-tuning control. In the paper a suitable state- space representation of Delta models is introduced and algorithms for recursive and joint estimation of the main parameters and of the state are developed. For the chosen canonical state representation the problem can be solved exactly within Gaussian distributions. Algorithmic and numerical aspects are emphasized. Simple PASCAL procedures suitable for microprocessor implementation are listed in Appendix.

Simple self-tuning controllers suitable for microprocessor im-plementation have been developed. Special care has been devoted to numerical stability of the algorithms. The controllers described in the paper are believed to cover a rather broad class of industrial control problems if the sampling period is chosen sufficiently long. The rule of thumb is 5 to 7 samples within the significant part of the system step response. If required, the controllers can be equipped with a self-tuning feedforward and are designed both for regulation and for servo applications, including program control.

A new method of LQ optimum control design for ARMAX processes is presented. It operates in time domain and solves the problem for finite control horizon but at the same time makes it possible to determine the stationary control law, to which the solution converges for growing control horizon, in terms of polynomial equations. The solution of the standard Ricatti equation is replaced by sequential factorization and de-composition of certain band matrices. This leads to efficient and numerically stable algorithms. Unlike the polynomial theory, the stationary solution obtained is unique (if it exists) and optimal also for transient and nonstationary parts of the control process. Both regulation and servo problems are considered.

The design of the digital controller frequently follows the aim to obtain the system with the fastest response. By the practical application of this criterion, cases are often encountered where the transient error, caused by an input of typical form, has a very short duration but an inadmissible magnitude. This shortcoming is removed by the numerical method of controller design presented in the paper.