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On the other hand, indirect adaptive control aims at estimating the true values of the uncertainties and then uses the uncertainty estimates and their associated estimated model to design the controller. Other deeper subclasses can be identified based on the mathematical nature of the model, eg, linear, nonlinear, continuous, discrete, and hybrid. For the sake of clarity, we have summarized this classification in Figure 1.
In this paper, we will present the surveyed references following the classification given above. We underline again that this paper does not mean to be an exhaustive presentation of all existing adaptive control algorithms, but rather an overview pointing out to monographs, surveys, and recent research papers, in which the readers can find a more detailed presentation of the specific results.
Section 4 contains a few points of comparison between these two classes. Finally, this overview ends with concluding remarks and some general open problems in adaptive control, in Section 5. The model can be the system's uncertain model, used in works related to adaptation to plants' uncertainties see, eg, the work of Ioannou and Sun 1 , or the disturbance uncertain model, used in adaptive disturbance rejection, also known in some literature as adaptive regulation dealing with uncertain disturbance rejection see, eg, the work of Landau et al 2.
As depicted in Figure 1 , we decompose this class into the following two subclasses. In this subclass, we include all the methods that are fully based on a model of the system, in the sense that both the controller and the adaptation filters are based on a model of the system. These methods can be further classified in terms of the approach used to compensate for the model uncertainties, ie, either direct or indirect, end also in terms of the nature of the model and the controller equations.
For instance, the reader is referred to some related books 1 , 2 , 9 , 10 , 24 - 29 , 31 , 55 and related survey papers. The performance of this approach has been discussed in the works of Ioannou et al 58 and van Heusden et al. In the direct adaptive methods, adaptive filters are used to tune the feedback controller for example, via adaptive observers of the disturbance to reject the unknown disturbance, without explicitly estimating its model. Several linear direct adaptive controllers have been proposed in this context see, eg, the works of Marino et al, 74 Marino and Tomei, 75 and Aranovskiy and Freidovich Under the indirect adaptive disturbance rejection methods, we can include numerous algorithms that use the internal model principle to model unknown disturbances and then use adaptive filters to estimate the coefficients of the internal model refer to the works of Bodson and Douglas, 77 Ding, 78 Landau et al, 79 and Serrani 80 as well as to the nice survey papers by Landau et al 15 , We then define a control objective function , such that the control objective is deemed reached if and only if 2.
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This formulation is indeed fairly general. For instance, if one is concerned with a regulation adaptive problem, the objective function Q can be chosen as follows. Similarly, if state or output tracking is the control target, then the objective function can be formulated as 6 where the reference trajectories are solutions of a desired reference model 7.
Of course, to be able to obtain any useful analysis of the nonlinear controllers, some assumptions on the type of nonlinearities had to be formulated. This assumption is important enough to be used as a metric to further classify direct nonlinear controllers in terms of the nature of the plant's parameterization in the unknown parameters.
Indeed, there have been a lot of efforts in dealing with the challenging case of nonlinear uncertainty parameterization see, eg, references 87 - Let us present on a simple example one of these results, namely, the speed gradient—based adaptive method for nonlinear parameterization, which was introduced first in the Russian literature by Fradkov et al, 36 Krasovskii, and Krasovskii and Shendrik. The control objective is to render an uncertain nonlinear system passive, also known as passifying the system, using an adaptive controller of the form 3. Indeed, if the control objective 2 is adaptively achieved, then one can write the inequality 9 which implies passivity of the system.
Since the stability properties of passive systems are well documented see, eg, the work of Byrnes et al , one can then easily claim asymptotic stability of the adaptive feedback system. Finally, asymptotic stabilization can be easily achieved by a simple output feedback from y to v.
Other recent results on nonlinear direct adaptive control can be found, for example, in the works of Wang et al and Cao et al, where the concept of l 1 adaptive control has been extended to nonlinear models. This compensation is done either directly by learning the uncertain part or indirectly by tuning the controller to deal with the uncertainty.
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We have to emphasize that the learning algorithm is solely based on the interaction with the system, and not on the model. Recently, there have been a lot of efforts in this direction of adaptive control. This combination is usually referred to as the dual or modular design for adaptive control. In this line of research in adaptive control, we can cite some related references. The NN is then used to approximate the unknown part of the model. Finally, a controller, based on both the known part and the NN estimate of the unknown part, is determined to realize some desired regulation or tracking performance, see for example.
Next, we consider the reference model 16 where f ref is a known nonlinear smooth function of the desired state trajectories. We assume that f ref is chosen such that the reference trajectories are uniformly bounded in time and orbital, ie, repetitive motion, starting from any desired initial conditions x ref 0. More algorithms and analysis of this type of controllers can be found, for example, in the works of Spooner et al, 34 Wang and Hill, 42 and Lewis et al as well as the references therein.
These algorithms rely on the measurement of the cost function to generate a sequence of desired states that can lead the system to an optimal value for the performance cost function. We present here a simple example of a numerical optimization—based ES control see, eg, chapter 4 in the work of Zhang and Ordonez Consider the following nonlinear dynamics: 18 where is the state vector, u is the control assumed to be a scalar, to simplify the presentation , and f is a known smooth known possibly nonlinear vector function.
We associate with the system modeled by Equation 18 a desired performance cost function Q x , where Q is a scalar smooth function of x. However, the explicit expression of Q as function of x or u is not known. In other words, the only available information is direct measurements of Q and maybe its gradient.
The goal is then to iteratively learn a control input that seeks the minimum of Q. One example of such ES algorithm is given below. Under some assumptions ensuring the existence of a global minimum, which is a stabilizable equilibrium point of 18 , it has been shown that this type of algorithms converges to the minimum of the performance cost function see, eg, chapters 3, 4, and 5 in the work of Zhang and Ordonez We assume that the pair A , B is controllable. In this case, the previous numerical optimization—based ES algorithm reduces as follows.
It has been shown in theorem 4. We cannot possibly present here all of these results; instead, we refer the readers to a few other references for more examples on this topic. However, the DP solutions based on solving the Bellman optimality equation can only be solved efficiently for problems with small state space, action space, and outcome space, ie, this is referred to as the three curses of dimensionality in the work of Powell.
Thus, the controller will be partly based on the model based on B and partly data driven uses RL to compensate for the unknown part A. The LQR optimal control is the controller satisfying However, the above classical solution relies on the full knowledge of model In our case, we assumed that A was an unknown that requires some learning steps. It has been proven in theorem 3. Note that the more challenging case of nonlinear systems has also been studied, for example in, 41 , , , , , , and references therein.
The extension of these results to more general nonlinear PDEs or to nonlinear uncertainty parameterization remains an open problem. Another interesting area in adaptive control is control under input constraints and input bandwidth limitations. As far as state constraints are concerned, the case of strictly feedback form with linear parameterization has been considered recently ; however, an extension to the more general type of nonlinearities remains a challenging problem. Of course, in many real applications, constraints should be enforced both on the actuators and the states.
A relatively new paradigm in adaptive control is the one aiming at a priori performance guarantees, eg, upper bounds on the tracking error imposed a priori by the user.
Adaptive Control : Algorithms, Analysis and Applications
To give the reader a sense of how ES methods work, let us present below a simple ES algorithm. Consider the following general dynamics: 26 where is the state, is the scalar control for simplicity , and is a smooth function. Let us now model this desired performance as a smooth function , which we simply denote J u , since the state vector x is driven by u. To be able to derive some convergence results, we need the following assumptions.
Assumption 1. There exists a smooth function such that Assumption 2. Assumption 3. There exists a maximum , such that Then, based on these assumptions, one can design some simple extremum seekers with proven convergence bounds.
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We can analyze the convergence of the ES algorithm 29 by using the Lyapunov function The derivative of V leads to However, as simple as algorithm 29 might seem, it still requires the knowledge of the gradient of J. This controller is shown see, eg, chapter 3 in the work of Zhang and Ordonez 43 to steer u to the set subject to , which can be made arbitrarily small by the proper tuning of k 1.
Note that the controller requires only measurements of the performance cost, without any need of the system's model. It uses a perturbation signal often sinusoidal to explore the space of control and steers the control variable toward its local optimum, by implicitly following a gradient update.
Tracking and Regulation with Independent Objectives 7. Polynomial Design 7. Time Domain Design 7. Tracking and Regulation with Weighted Input 7. Minimum Variance Tracking and Regulation 7. Design of Minimum Variance Control 7. Generalized Minimum Variance Tracking and Regulation 7. Generalized Predictive Control 7. Controller Equation 7. Closed-Loop Poles 7. Recursive Solutions of the Euclidian Divisions 7.
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Linear Quadratic Control 7. Concluding Remarks 7. Problems 8. Robust Digital Control Design 8. The Robustness Problem 8. The Sensitivity Functions 8. Robust Stability 8. Robustness Margins 8. Model Uncertainties and Robust Stability 8. Robustness Margins and Robust Stability 8. Definition of "Templates" for the Sensitivity Functions Contents note continued: 8. Properties of the Sensitivity Functions 8. Output Sensitivity Function 8.
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Input Sensitivity Function 8. Noise Sensitivity Function 8. Shaping the Sensitivity Functions 8. Other Design Methods 8. Concluding Remarks 8. Problems 9. The Problem 9. The Basic Equations 9. Filtered Recursive Least Squares 9. Filtered Output Error 9. Validation of Models Identified in Closed-Loop 9. Statistical Validation 9.
Pole Closeness Validation 9. Time Domain Validation 9. Comparative Evaluation of the Various Algorithms 9. Simulation Results 9. Concluding Remarks 9. Problems Robust Parameter Estimation The Problem Effect of Disturbances PAA with Dead Zone PAA with Projection Data Normalization The Effect of Data Filtering Alternative Implementation of Data Normalization A Robust Parameter Estimation Scheme Concluding Remarks Direct Adaptive Control Introduction Adaptive Tracking and Regulation with Independent Objectives Basic Design Extensions of the Design Adaptive Tracking and Regulation with Weighted Input Adaptive Minimum Variance Tracking and Regulation The Basic Algorithms Asymptotic Convergence Analysis Martingale Convergence Analysis Robust Direct Adaptive Control Direct Adaptive Control with Bounded Disturbances Direct Adaptive Control with Unmodeled Dynamics An Example Indirect Adaptive Control Adaptive Pole Placement The Basic Algorithm Analysis of the Indirect Adaptive Pole Placement The "Singularity" Problem Contents note continued: Adding External Excitation Robust Indirect Adaptive Control Standard Robust Adaptive Pole Placement Modified Robust Adaptive Pole Placement Adaptive Generalized Predictive Control Adaptive Linear Quadratic Control Adaptive Tracking and Robust Regulation Multimodel Adaptive Control with Switching Principles of Multimodel Adaptive Control with Switching Plant with Uncertainty Multi-Estimator Multi-Controller Supervisor Stability Issues Stability of Adaptive Control with Switching Stability of the Injected System Contents note continued: Application to the Flexible Transmission System Experimental Results Effects of Design Parameters Adaptive Regulation -Rejection of Unknown Disturbances Plant Representation and Controller Design Robustness Considerations Direct Adaptive Regulation Stability Analysis Indirect Adaptive Regulation The Active Vibration Control System Adaptive Feedforward Compensation of Disturbances Basic Equations and Notations Development of the Algorithms Analysis of the Algorithms The Stochastic Case The Case of Non-Perfect Matching Relaxing the Positive Real Condition System Identification Practical Aspects The Digital Control System Selection of the Sampling Frequency Anti-Aliasing Filters Digital Controller Effects of the Digital to Analog Converter Handling Actuator Saturations Anti-Windup Manual to Automatic Bumpless Transfer Effect of the Computational Delay Choice of the Desired Performance The Parameter Adaptation Algorithm Adaptive Control Algorithms Control Strategies Initialization of Adaptive Control Schemes Monitoring of Adaptive Control Systems Passive Hyperstable Systems C.
Passivity -Some Definitions C. Stability of Feedback Interconnected Systems C. Notes Formerly CIP. Description based upon print version of record. Includes bibliographical references pages and index. Also available in print. Electronic reproduction. Access Conditions License restrictions may limit access. Other Form Print version Adaptive control. Dewey Number View online Borrow Buy Freely available Show 0 more links Set up My libraries How do I set up "My libraries"? Not open to the public Held.