1 chapter 1 introduction to adaptive control – –adaptive control: identifier-based –...

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Chapter 1Chapter 1

Introduction to Adaptive Control

– Adaptive Control: Identifier-Based – Adaptive Control: Non–Identifier-Based– Gain Scheduling – Why Adaptive Control– A Brief History

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Introduction

• Adapt means to "change (oneself) so that one's behavior will conform to new or changed circumstances."

• The words adaptive systems and adaptive control have been used as early as 1950.

• We use the following specific definition of adaptive control: Adaptive control is the combination of a parameter estimator, which generates parameter estimates online, with a control law in order to control classes of plants whose parameters are completely unknown and/or could change with time in an unpredictable manner.

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Introduction

• The choice of the parameter estimator, the choice of the control law, and the way they are combined leads to different classes of adaptive control schemes.

• Adaptive control as defined above has also been referred to as identifier-based adaptive control in order to distinguish it from other approaches referred to as non-identifier-based, where similar control problems are solved without the use of an online parameter estimator.

• The design of autopilots for high-performance aircraft was one of the primary motivations for active research in adaptive control in the early 1950s.

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Introduction

• The controller structure consists of a feedback loop and a controller with adjustable gains, as shown in following Figure.

General adaptive control structure for aircraft control.

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Adaptive Control: Identifier-Based

The class of adaptive control schemes studied in this course is characterized by the combination of an online parameter estimator, with a control law. The way the parameter estimator, also referred to as adaptive law, is combined with the control law gives rise to two different approaches:

1- In the first approach, referred to as indirect adaptive control, the plant parameters are estimated online and used to calculatethe controller parameters. In other words, at each time t, the estimated plant is formed and treated as if it is the true plant in calculating the controller parameters. This approach has also been referred to as explicit adaptive control, because the controller design is based on an explicit plant model.

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2- In the second approach, referred to as direct adaptive control, theplant model is parameterized in terms of the desired controller parameters, which are then estimated directly without intermediate calculations involving plant parameter estimates. This approach has also been referred to as implicit adaptive control because the design is based on the estimation of an implicit plant model.

The basic structure of indirect adaptive control is shown in following Figure. The plant model G(*) is parameterized with respect to some unknown parameter vector *.

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Indirect adaptive control structure.

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Direct adaptive control structure.

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In general, direct adaptive control is applicable to SISO linear plants which are minimum phase, since for this class of plants the parameterization of the plant with respect to the controller parameters for some controller structures is possible. Indirect adaptive control can be applied to a wider class of plants with different controller structures, but it suffersfrom a problem known as the stabilizability problem explained as follows: The controller parameters are calculated at each time t based on the estimated plant. Such calculations are possible, provided that the estimated plant is controllable and observable or at least stabilizable and detectable.

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Since these properties cannot be guaranteed by the online estimator in general, the calculation of the controller parameters may not be possible at some points in time, or it may lead to unacceptable large controller gains. So, solutions to this stabilizability problem are possible at the expense of additional complexity. Efforts to relax the minimum-phase assumption in direct adaptive control and resolve the stabilizability problem in indirect adaptive control led to adaptive control schemes where both the controller and plant parameters are estimated online, leading to combined direct/indirect schemes that are usually more complex .

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Adaptive Control: Non-Identifier-Based

Another class of schemes that do not involve online parameter estimators is referred to as non-identifier-based adaptive control schemes. In this class of schemes, the online parameter estimator is replaced with search methods for finding the controller parameters in the space of possible parameters, or it involves switching between different fixed controllers, assuming that at least one is stabilizing or uses multiple fixed models for the plant covering all possible parametric uncertainties or consists of a combination of these methods. We briefly describe the main features, advantages, and limitations of these non-identifier-based adaptive control schemes. Some of these approaches are relatively recent and research is still going on.

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The gain scheduler consists of a lookup table and the appropriate logic for detecting the operating point and choosing the corresponding value of control gains from the lookup table. With this approach, plant parameter variations can be compensated by changing the controller gains as functions of the input, output, and auxiliary measurements. The advantage of gain scheduling is that the controller gains can be changed as quickly as the auxiliary measurements respond to parameter changes. Frequent and rapid changes of the controller gains, however, may lead to instability; therefore, there is a limit to how often and how fast the controller gains can be changed.

• Gain Scheduling

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Adaptive Control: Non-Identifier-Based • Gain Scheduling

Gain scheduling structure.

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Adaptive Control: Non-Identifier-Based • Gain Scheduling

One of the disadvantages of gain scheduling is that the adjustment mechanism of the controller gains is precomputed offline and provides no feedback to compensate for incorrect schedules. A careful design of the controllers at each operating point to meet certain robustness and performance measures can accommodate some uncertainties in the values of the plant parameters. However large unpredictable changes in the plant parameters, may lead to deterioration of performance or even to complete failure. Despite its limitations, gain scheduling is a popular method for handling parameter variations in flight control and other systems. While gain scheduling falls into the generic definition of adaptive control, we do not classify it as adaptive control due to the lack of online parameter estimation which could track unpredictable changes in the plant parameters.

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• Multiple Models• Search Methods, and • Switching Schemes

A class of non-identifier-based adaptive control schemes emerged over the years which do not explicitly rely on online parameter estimation. These schemes are based on search methods in the controller parameter space until the stabilizing controller is found or the search method is restricted to a finite set of controllers, one of which is assumed to be stabilizing. In some approaches, after a satisfactory controller is found it can be tuned locally using online parameter estimation for better performance.

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• Multiple Models• Search Methods, and • Switching Schemes

Since the plant parameters are unknown, the parameter space is parameterized with respect to a set of plant models which is used to design a finite set of controllers so that each plant model from the set can be stabilized by at least one controller from the controller set. A switching approach is then developed so that the stabilizing controller is selected online based on the I/O data measurements. Without going into specific details, the general structure of this multiple model adaptive control with switching, as it is often called, is shown in next Figure.

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Multiple models adaptive control with switching

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Why Adaptive Control

The choice of adaptive control as a solution to a particular control problem involves understanding of the plant properties as well as of the performance requirements. The following simple example illustrates situation where adaptive control is superior to linear control.

Consider the scalar plant

where u is the control input and x the scalar state of the plant. The parameter a is unknown. We want to choose the input u so that the state x is bounded and driven to zero with time. If a is a known parameter, then the following linear control law can meet the control objective.

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Why Adaptive Control

0x a s t

In the absence of an upper bound for the plant parameter no linear controller could stabilize the plant and drive the state to zero. As we will establish later , the adaptive control law

guarantees that all signals are bounded and x converges to zero no

matter what the value of the parameter a is. This simple example demonstrates that adaptive control is a potential approach to use in situations where linear controllers cannot handle the parametric uncertainty.

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A Brief History

• Early 1950s, the design of autopilots for high-performance aircraft motivated intense research activity in adaptive control.

• 1958, 1961, Model reference adaptive control was suggested by Whitaker and coworkers in to solve the autopilot control problem.

• 1958, An adaptive pole placement scheme based on the optimal linear quadratic problem was suggested by Kalman.

• The lack of stability proofs and the lack of understanding of the properties of the proposed adaptive control schemes coupled with a disaster in a flight test caused the interest in adaptive control to diminish.

• The 1960s became the most important period for the development of control theory and adaptive control in particular. State-space techniques and stability theory based on Lyapunov were introduced.

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A Brief History• Developments in dynamic programming, dual control and

stochastic control, and system identification and parameter estimation played a crucial role in the reformulation and redesign of adaptive control.

• By 1966, Parks and others found a way of redesigning the MIT rule-based adaptive laws used in the model reference adaptive control (MRAC) schemes using the Lyapunov design approach.

• The advances in stability theory and the progress in control theory in the 1960s improved the understanding of adaptive control and contributed to a strong renewed interest in the field in the 1970s.

• On the other hand, the simultaneous development and progress in computers and electronics that made the implementation of complex controllers, such as the adaptive ones, feasible contributed to an increased interest in applications of adaptive control.

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A Brief History• The 1970s, several breakthrough results in the design of adaptive

control.

• The concepts of positivity were used to develop a wide class of MRAC schemes with well-established stability properties.

• At the same time several classes of adaptive control schemes produced for discrete-time plants.

• The excitement of the 1970s and the development of a wide class of adaptive control schemes with well established stability properties were accompanied by several successful applications.

• The successes of the 1970s, however, were soon followed by controversies over the practicality of adaptive control.

• As early as 1979 it was pointed out by Egardt that the adaptive schemes of the 1970s could easily go unstable in the presence of small disturbances.

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A Brief History• 1980s, The nonrobust behavior of adaptive control became very

controversial when more examples of instabilities were published by loannou et al. and Rohrs et al.

• Rohrs's example of instability stimulated a lot of interest, and the objective of many researchers was directed towards understanding the mechanism of instabilities and finding ways to counteract them.

• By the mid- 1980s, several new redesigns and modifications were proposed and analyzed, leading to a body of work known as robust adaptive control.

• An adaptive controller is defined to be robust if it guarantees signal boundedness in the presence of "reasonable" classes of unmodeled dynamics and bounded disturbances

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A Brief History• The work on robust adaptive control continued throughout the

1980s and involved the understanding of the various robustness modifications and their unification under a more general framework.

• In discrete time Praly was the first to establish global stability in the presence of unmodeled dynamics.

• By the end of the 1980s several results were published in the area of adaptive control for linear time-varying plants.

• The focus of adaptive control research in the late 1980s to early 1990s was on performance properties and on extending the results of the 1980s to certain classes of nonlinear plants with unknow parameters.

• These efforts led to new classes of adaptive schemes, motivated from nonlinear system theory as well as to adaptive control schemes with improved transient and steady-state performance.

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A Brief History• New concepts such as adaptive backstepping, nonlinear damping,

and tuning functions are used to address the more complex problem of dealing with parametric uncertainty in classes of nonlinear systems .

• In the late 1980s to early 1990s, the use of neural networks as universal approximators of unknown nonlinear functions led to the use of online parameter estimators to "train" or update the weights of the neural networks.

• Adaptive control has a rich literature full of different techniques for design, analysis, performance, and applications. Several survey papers and books and thesis have already been published.

• Despite the vast literature on the subject, there is still a general feeling that adaptive control is a collection of unrelated technical tools and tricks.

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THE END