Ultimate Boundedness Theorem for Model Reference Adaptive Control Systems

This paper considers the stability problem of the model reference adaptive control systems by means of the properties of hyperstable systems. A theorem concerning the hyperstability of model reference adaptive control systems is presented. This theorem directly gives a structure of the adaption mechanism. The results presented here include all the results obtained by Butchart, Shackcloth, Parks, Winsor, Roy, and Dressler. The hyperstability approach presented in this paper also allows for other solutions to the adaption mechanism and represents a general method for studying this type of adaptive systems. The results are directly applicable to the design of model reference adaptive control systems and they were verified for some particular cases by analogical simulation.

The efficiency of a separately excited DC motor is improved by controlling both armature current and field current simultaneously. The model reference adaptive control system (MRACS) can be useful for this control problem because parameter values and load disturbance (load torque) of controlled objects vary greatly. Two adaptive control systems for efficiency-optimized speed control are proposed on the basis of MRACS theory and the error system method. These are MRACS based on the current ratio method and on the voltage/current ratio method. In these systems, integral action is introduced into MRACS to cope with the step-load disturbance. Although the parameter variation of the voltage/current ratio MRACS method is large, the number of parameters to be adjusted is small as compared with that in the current ratio MRACS method. For each method, fast transient response and good characteristics are obtained for the parameter changes, step-load disturbance, and step desired signal in simulation studies. Efficiency is considerably improved by these methods at a light-load condition.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An interactor matrix introduced by Wolovich & Falb (1976) has an important role in the design of model reference adaptive control systems (MRACS) for a class of multi-input multi-output (MIMO) plants. In the early stage of the research, the interactor was supposed to be diagonal matrix and thus there were no unknown parameters (Goodwin et al., 1980). But, there exist many plants which require non-diagonal interactors (Chan & Goodwin, 1982). And the design of MIMO MRACS with non-diagonal interactors had been discussed, where all elements of the interactor are assumed to be known (Elliott & Wolovich, 1982; Goodwin & Long, 1980). However, this assumption is not adequate for adaptive control systems since the parameters of the interactor depend on the unknown parameters of the plant, i.e., the parameter values and the relative degree of each element of the plant must be used to determine the interactor. Furthermore, even we know all of these information, the structure of the interactor is not determined uniquely. In order to remove the assumption, the MRACS design has been proposed where the degree of diagonal elements and the upper bound of the highest degree of the lower triangular interactor matrix are assumed to be known (Elliott & Wolovich, 1984; Dugard et al., 1984). Under these assumptions, off-diagonal elements of the lower triangular interactor are estimated, and the method seemed suitable for adaptive controller design. However, it is not reasonable to assume the diagonal degrees in MRACS since the determination of the degrees depends on the relative degree and parameter values of each element of a transfer matrix of a given plant. From this view point, an interactor in generic sense was considered under the assumption that the relative degrees of all elements of the transfer function matrix are known (Kase & Tamura, 1990; Mutoh & Ortega, 1993). The method covers almost of all classes of MIMO plants having the same numbers of inputs and outputs generally. But there still exist some rare plants. By the way, there exists an idea of the certainty equivalence principle for the indirect MRACS design, i.e., estimate the unknown parameters of a plant first, then design the controllers on-line, using those estimated parameters. However, the design was seemed very difficult especially for MIMO plants, since large amount of calculation is needed to solve so-called Diophantine equation, beside the derivation of the interactor. In other words, there did not exist a suitable method to solve the Diophantine equation or to derive the interactor matrix. In this chapter, an indirect approach to MIMO MRACS will be shown. For

The objective of this research is to provide a method for design of control systems when the plant involved is highly variable in both structure and parameter values. This typifies many manufacturing processes, which in general are highly sensitive to intrinsic properties of the infeed material, have poorly understood or controlled boundary conditions, and often have highly non-linear characteristics. In addition, the disturbances most frequent in a manufacturing process are best characterized as parameter changes rather than state variable changes. The choice of a parameter adaptive control paradigm is obvious since it has the property of optimizing system response characteristics in the face of wide plant parameter variations. Model Reference Adaptive Systems (MRAS) enjoy an ease of performance specification, and again seem applicable; however, there is a restriction within current theory that the plant be characterized in its structure well enough to have a reference model of identical order. This restriction implies a static plant structure and more a priori information than is often present. During this contract the basic design characteristics of certain MRAS have been studied and problems with reduced order models identified, such as noise and input magnitude sensitivity and instability caused by high frequency learning signals. This has lead to certain solutions and resulting guidelines for design. This is then followed by design studies for a position servomechanism (which serves as the foundation of many automated manufacturing processes). This study expanded the domain from MRAS to self-tuning controllers (STC) and found that both can deal with reduced order models if the initial parameter values are not too distant from the actual values. However, the STC appears to be the more robust. 12 refs., 56 figs.

In this paper a Model Reference Adaptive Control system is presented which is based on qualitative (less precise) models. These qualitative models can represent the underlying physical system accurately even with low order models. Further, multi-valued performance specifications can be expressed explicitly by a qualitative model in the form of a reference model. The Qualitative Model Reference Adaptive Control system utilises a fuzzy qualitative simulation method to represent this reference model. These fuzzy dynamical models are of an analytic form and allow system analysis to be based on fuzzy interval analysis. Therefore, stability of the adaptive loop within Qualitative Model Reference Adaptive Control is based on results of robust parametric stability using interval models.

The control objectives for Model Reference Adaptive Control Systems (MRAC) are specified in terms of a reference model. The control objectives for Stochastic Self-Tuning Regulators (S-STURE) are specified in terms of an ARMA model. These two types of adaptive control systems share a common characteristic (duality). This duality extends the one existing in the linear case with known parameters between modal control (pole-placement) and minimum variance control. Based on the duality properties, a unified presentation of the two techniques is possible; the S-STURE being interpreted as a stochastic MRAC. The underlying concepts and the structure of these adaptive control systems are described. The available theoretical results concerning the analysis of these systems are briefly reviewed. The primary domain of application of these techniques are indicated. An overview of the unsolved problems concludes the paper.

A modified Liapunov design technique for model reference adaptive control system is shown to result in improved system convergence. An adaptive rule, derived on the basis of a new Liapunov function, is compared to the previous rule. A local stability analysis applied to the modified design shows that the error response is more rapidly convergent. Furthermore, system simulations show that the transient response for the adjustable parameters is also improved. A second result presented is a design technique for a class of plants whose parameters cannot be adjusted directly. This design leads to a system with a set of prefilter and feedback adjustable gains as the adaptive parameters and physically realizable linear time-invariant filter networks in both the feedback and prefilter paths. It eliminates the problem of nonunique adaptive laws previously encountered and requires only n - m - 1 derivative networks for its implementation (nth order plant with m zeros); hence, if m = n - 1, no derivative networks are required for implementation. In order to maintain a bounded plant input signal, the zeros of the plant transfer function must be restricted to the open left-half plane.

Model reference adaptive control systems (MRACS) is the one of the useful systems to control the plant containing unknown variable parameters. However, to design MRACS, it has been needed to satisfy model matching condition. In this paper, to improve the response in the case unsatisfying model matching condition, we design an adaptive law containing fractional order integrator in adaptive law, and prove the system's Lyapnov stability.

In spite of the large number of design methods for model reference adaptive control system (MRACS) have been proposed by the researchers of stability theorem, only a few studies of MRACS for multi-input multi-output (MIMO) plant with time-delay have been published to date.In this paper, a design method for discrete MRACS is proposed based on Lyapunov's direct method. The controlled plant is a MIMO linear plant with time-delays. It is shown that the proposed adaptive algorithms gurantee the convergency of plant's output to reference model's response by introducing an auxiliary input vector. For the adaptive algorithms proposed here, only available input-output date of the plant are used and no information on unknown plant parameter is required. Finally, the validity of the proposed adaptive algorithms is demonstrated by numerical examples.

A method is presented for the design of multivariable model reference adaptive control systems by Lyapunov's second method. The method can be applied to a class of systems in which the misaligned elements of the plant matrices cannot be compensated independently.

The problem of designing model reference adaptive control systems for plants with unmodelled dynamics subject to output disturbances is considered. The inadequacies of present model reference adaptive system design philosophies are highlighted, and a new design philosophy is outlined that makes use of the powerful frequency-domain design tools used in classical analysis.

A New Approach to the Design of an Adaptive Control System

This paper presents a method for fuzzy model reference adaptive control system design for a partially known multi-input multi-output (MIMO) linear dynamic plant. Differential or convolutional equations can describe the plant to be controlled, and the fuzzy controller is assumed to be from the MIMO sector-bounded nonstationary nonlinearities class. The task is to obtain robust adaptive stabilization. Some absolute stability theorems and integral evaluations of the state and control vectors are applied in the MIMO fuzzy adaptive controller design procedure for the first time. A method for automatically obtaining a near-optimal nonlinear dynamic reference model is described. A practical step-by-step procedure of the MIMO adaptive fuzzy controller design is proposed, resulting in highly interpretable MIMO fuzzy control rules based on triangular fuzzy sets and strong triangular fuzzy partition. An example of the fuzzy controller design according to the proposed procedure is given.

In this paper, a design of a signal synthesis adaptive controller for a class of linear time-varying uncertain plants using model reference adaptive control techniques is presented. Following a discussion of the general concept, the design of an adaptive controller for unconstrained and constrained control conditions is given. The small ultimate boundness of the state error is considered as the adaptation criterion, which is shown to be satisfied by a Liapunov type stability theorem. To handle the uncertainties that are associated with the plant dynamics and its environment, a min-max concept is employed in the design of the controller. The signal synthesis adaptive approach presented in this paper does not require as many structural assumptions as do most other adaptive approachers, and this is its principal advantage. The state vectors of the plant and model are assumed to be both accessible. Some simulation results are presented which illustrate the effectiveness of the design given here.

In most of the studies of model reference adaptive control, it is assumed that an upper bound on the degree of the controlled system is known. It makes the scope of application of model reference adaptive control too restrictive, since the reasonable upper bound on the degree cannot be specified a priori in many cases. In the present paper, we propose a design method of model reference adaptive control systems for nonlinear systems with unknown degrees and with unknown relative degrees 1 or 2. Even if the degree of the controlled system varies, and even if the relative degree also varies between 1 and 2, the structure of the proposed adaptive controller does not change. It is shown that the resulting control system is uniformly bounded, and that the tracking error converges to an arbitrarily small residual region. Finally, several simulation studies also show the effectiveness of the proposed method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>