Rate of convergence in model reference adaptive control

A new adaptive scheme for continuous-time model reference adaptive control systems is proposed. It is shown that this scheme allows us to increase the exponential rate of parameter convergence to an arbitrary, desired level subject to a sufficiently rich reference input and a sufficiently large adaptive gain, while retaining global stability of the overall system. For simplicity and brevity the results are presented for plants with known high-frequency gain kp.     

Extended Lyapunov adaptation law for model reference systems

An extension of the adaptation methods presented previously by Ten Cate are presented. Proof of asymptotic stability of the error is effected using Lyapunov theory. This method completes the triplet of a group of previously published adaptation rules for a variation suggested by Shahein et al. using a plant-model parameter misalignment function.     

Synthesis of discrete model reference adaptive systems

The synthesis of discrete model reference adaptive systems is discussed from the hyperstability point of view. The results included here are an extension of previous results obtained by the author for the continuous case [1].     

Stable adaptive control for a class of nonlinear systems using amodified Lyapunov function

Investigates the adaptive control design for a class of nonlinear systems using Lyapunov's stability theory. The proposed method is developed based on a novel Lyapunov function, which removes the possible controller singularity problem in some of the existing adaptive control schemes using feedback linearization techniques. The resulting closed-loop system is proven to be globally stable, and the output tracking error converges to an adjustable neighborhood of zero     

Tracking and parameter identification for model reference adaptive control

We provide barrier Lyapunov functions for model reference adaptive control algorithms, allowing us to prove robustness in the input‐to‐state stability framework and to compute rates of exponential convergence of the tracking and parameter identification errors to zero. Our results ensure identification of all entries of the unknown weight and control effectiveness matrices. We provide easily checked sufficient conditions for our relaxed persistency of excitation conditions to hold. Our illustrative numerical example demonstrates the performance of the control methods.     

Bifurcation in model reference adaptive control systems

We study the effect of unmodeled dynamics on the performance of a simple model reference adaptive control mechanism. Exact stability boundaries are computed and different routes to chaotic behavior and global instability are described. Typically, we observed a region of period doubling and Hopf bifurcations similar to those observed in the Ziegler-Nichols approach to PID control design. The analysis applies to higher order and certain nonlinear systems as well.     

On parameter convergence in adaptive control

It is well known that the parameter error as well as the model-plant mismatch error in a model reference adaptive scheme tends exponentially to zero iff a certain sufficient richness condition holds for signals inside the time-varying plant control loop. In this paper we give conditions on the reference signal (the exogenous input to the adaptive loop) — namely, that it have as many spectral lines as there are unknown parameters, in order to guarantee parameter convergence.     

Comments on "Improved signal synthesis techniques for model reference adaptive control systems"

In this correspondence a limitation of the synthesis technique of the above paper is pointed out. It is further shown that this technique can be extended to the solution of the adaptive observation problem for linear single variable systems subject to this limitation.     

Liapunov redesign of model reference adaptive control systems

The model reference adaptive control system has proved very popular on account of a ready-made, but heuristically based, rule for synthesizing the adaptive loops-the so-called "M.I.T. rule." A theoretical analysis of loops so designed is generally very difficult, but analyses of quite simple systems do show that instability is possible for certain system inputs. An alternative synthesis based on Liapunov's second method is suggested here, and is applied to the redesign of adaptive loops considered by some other authors who have all used the M.I.T, rule. Derivatives of model-system error are sometimes required, but may be avoided in gain adjustment schemes if the system transfer function is "positive real," using a lemma due to Kalman. This paper amplifies and extends the work of Butchart and Shackcloth reported at the IFAC (Teddington) Symposium, September, 1965.     

Comparative studies of model reference adaptive control systems

A brief but up-to-date survey on existing methods of designing a class of model reference adaptive control systems is given in this paper. A comparison of the merits of the various design rules is also made with particular attention to the M.I.T. rule and the Lyapunov synthesis technique. Subsequently a systematic performance comparison of the various designs, with deterministic as well as stochastic inputs, is presented using the computer simulation studies of two simple gain adjustment schemes. From the dimensionless characteristic graphs obtained the Lyapunov schemes are found to exhibit superior performance over other designs. These graphs also uncover the strange adaptive response of those designs based on gradient methods in that the performance index may increase or decrease with increasing system gain parameters.     

Stable model reference adaptive fuzzy control of a class ofnonlinear systems

In this paper, we propose a new adaptive fuzzy control scheme called model reference adaptive fuzzy control (MRAFC). The MRAFC scheme employs a reference model to provide closed-loop performance feedback for generating or modifying a fuzzy controller's knowledge base. The MRAFC scheme grew from ideas in conventional model reference adaptive control (MRAC). The MRAFC scheme is developed to perform adaptive feedback linearization to a class of nonlinear systems. A class of fuzzy controllers, which can be expressed in an explicit form, is used as the primary controller. Based on Lyapunov's second method, we have developed MRAFC schemes and derived fuzzy rule adaptive laws. Hence, not only the stability of the system can be assured but also the performance, such as the issues of robustness and parameter convergence, of the MRAFC system can be analyzed explicitly. We showed that in the case of no modeling error, the state error converges to zero asymptotically. In the case that persistent excitation is satisfied, we showed that the MRAFC system is asymptotically stable. By considering the periodic signal as reference input signal, we showed that the square wave can make the MRAFC system be persistently excited. The feasibility of applying these techniques has been demonstrated by considering the control of an inverted pendulum in following a reference model response     

Lyapunov design for convergence rate improvement in adaptive control

This note deals with the improvement of the convergence rate and robust properties in an adaptive error model. A new adaptive control algorithm is presented which includes both the estimate of the prediction error to drive the parameter adaptation and the proportional-integral or integral-relay adjustment law. To prevent unbounded growth of feedback gains, a least-squares gain update is used. The adaptive controller is constructed using measurements of state variables only. The design is based on the Lyapunov function method and a new general form of the Lyapunov function was found. As a result, exponential convergence of the Lyapunov function in the presence of persistent excitation, is obtained.     

Improved performance model reference adaptive control withparameter mismatch compensation

An adaptive controller with improved performance characteristics is introduced. The proposed controller extends results in this area since it achieves performance improvement of the zero-state output error in the presence of some uncertainty on the high-frequency gain of the plant     

A parameter estimation perspective of continuous time model reference adaptive control

The problem of adaptive control of continuous time deterministic dynamic systems is re-examined. It is shown that the convergence proofs for these algorithms may be decomposed into “modules” dealing with estimation and control, yielding a “key technical lemma” analogous to that used successfully in the study of discrete time systems. The extra freedom provided by the modular structure is used to formulate existing algorithms in a common framework and to derive several new algorithms. It is also shown how least squares, as opposed to gradient, estimation can be used in continuous time adaptive control.     

On the convergence of a model reference adaptive control algorithm with unknown high frequency gain

A model reference adaptive control algorithm is shown not to be exponentially convergent when the high frequency gain of the plant is unknown. The implication is that such an algorithm will be lacking in some way in robustness.     

Exponential convergence of a model reference adaptive controller for plants with known high frequency gain

A model reference adaptive controller proposed by Morse is studied. It is shown to be exponentially convergent when the high frequency gain is known and when the reference inputs satisfy a persistence of excitation assumption, which allows the inputs to be other than a linear combination of sinusoids.     

A condition for decentralization in model reference adaptive systems

It is shown that when the interconnection pattern within a system conforms to certain prescribed symmetry requirements, decentralized adaptive algorithms for identification and control can be developed within a decomposition-decentralization framework. This is useful in the presence of constraints on the available information structure of the system.     

A hyperstability criterion 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.     

A model reference adaptive control scheme for pure-feedbacknonlinear systems

A model reference adaptive control (MRAC) scheme is presented for nonlinear systems in a pure-feedback canonical form with unknown parameters. The present of parameter uncertainty in the system causes imperfect linearization, i.e. it introduces nonlinear additive terms in the transformed coordinates. Under some mild technical assumptions, global convergence of the output error is established for all initial estimates of the parameter vector lying in an open neighborhood of the true parameters in the parameter space     

Exponential stability and stabilization of uncertain linear time-varying systems using parameter dependent Lyapunov function

In this paper, the problem of exponential stability and stabilization for a class of uncertain linear time-varying systems is considered. The system matrix belongs to a polytope and the time-varying parameter as well as its time derivative are bounded. Based on a time-varying version of Lyapunov stability theorem, new sufficient conditions for the exponential stability and stabilization via parameter dependent state feedback controllers (i.e., a gain scheduling controllers) are given. Using parameter dependent Lyapunov function, the conditions are formulated in terms of two linear matrix inequalities without introducing extra useless decision variables and hence are simply verified. The results are illustrated by numerical examples.