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Stable Adaptive Control and Estimation for Nonlinear Systems: Neural and Fuzzy Approximator Techniques.. For instance, an adaptive controller for the cruise control problem would seek to

Chapter 1

Introduction

1.1 Overview

The goal of a control system is to enhance automation within a system while providing improved performance and robustness For instance, we may develop a cruise control system for an automobile to release drivers from the tedious task of speed regulation while they are on long trips In this case, the output of the plant is the sensed vehicle speed, y, and the input to the plant is the throttle angle, u, as shown in Figure 1.1 Typically, control systems are designed so that the plant output follows some reference input (the driver-specified speed in the case of our cruise control example) while achieving some level of “disturbance rejection.” For the cruise control problem, a disturbance would be a road grade variation or wind Clearly we would want our cruise controller to reduce the effects of such disturbances

on the quality of the speed regulation that is achieved

) Plant

Y

u

Control 4

Figure 1.1 Closed loop control

In the area of “robust control” the focus is on the development of con- trollers that can maintain good performance even if we only have a poor model of the plant or if there are some plant parameter variations In the area, of adaptive control, to reduce the effects of plant parameter variations, robustness is achieved by adjusting (i.e., a,dapting) the controller on-line

Stable Adaptive Control and Estimation for Nonlinear Systems:

Neural and Fuzzy Approximator Techniques.

Jeffrey T Spooner, Manfredi Maggiore, Ra´ul Ord´o˜nez, Kevin M Passino

Copyright  2002 John Wiley & Sons, Inc ISBNs: 0-471-41546-4 (Hardback); 0-471-22113-9 (Electronic)

For instance, an adaptive controller for the cruise control problem would seek to achieve good speed tracking performance even if we do not have a good model of the vehicle and engine dynamics, or if the vehicle dynamics change over time (e.g., via a weight change that results from the addition of cargo, or due to engine degradation over time) At the same time it would try to achieve good disturbance rejection Clearly, the performance of a good cruise controller should not degrade significantly as your automobile ages or if there are reasonable changes in the load the vehicle is carrying

We will use adaptive mechanisms within the control laws when certain parameters within the plant dynamics are unknown An adaptive controller will thus be used to improve the closed-loop system robustness while meet- ing a set of performance objectives If the plant uncertainty cannot be expressed in terms of unknown parameters, one may be able to reformu- late the problem by expressing the uncertainty in terms of a8 fuzzy system, neural network, or some other parameterized nonlinearity The uncertainty then becomes recast in terms of a new set of unknown parameters that may

be adjusted using adaptive techniques

Often, when given the challenge of designing a control system for a par- ticular application, one is provided a model of the plant that contains the dominant dynamic characteristics The engineer responsible for the design

of a control system may then proceed to formulate a control algorithm as- suming that when the model is controlled to within specifications, then the true plant will also be controlled within specifications This approach has been successfully applied to numerous systems More often, however, the controller may need to be adjusted slightly when moving from the design model to the actual implementation due to a mismatch between the model and true system There are also cases when a control system performs well for a particular operating region, but when tested outside that region, performance degrades to unacceptable levels

Y

b, Plant+ A

u Control -4

\ Figure 1.2 Robust control of a# plant with unmodeled dynamics

These issues, among others, are addressed by robust control design When developing a robust control design, the focus is often on maintaining stability even in the presense of unmodeled dynamics or external distur- bances applied to the plant Figure 1.2 shows the situation in which the controller must be designed to operate given any possible plant variation A Unmodeled dynamics are typically associated with every control problem

in which a controller is designed based upon a model This may be due to any one of a number of reasons:

l It may be the case that only a nominal set of parameters are available for the control design If a controller is to be incorporated into a mass- produced product, for example, it may not be practical to measure the exact parameter values for each plant so that a controller can be customized to each particular system

l It may not be cost effective to produce a model that exactly (or even closely) represents the plant’s dynamics It may be possible to spend fewer resources on a robust control design using an incomplete model than developing a high fidelity model so that traditional non-robust techniques may be used

Hence, the approach in robust control is to accept a priori that there will

be model uncertainty, and try to cope with it

The issue of robustness has been studied extensively in the control lit- era:ture When working with linear systems, one may define phase a#nd gain margins which quantify the range of uncertainty a closed-loop system may withstand before becoming unstable In the world of nonlinear control de- sign, we often investigate the stability of a closed-loop system by studying the behavior of a Lyapunov function candidate The Lyapunov function candidate is a mathematical function designed to provide a simplified mea- sure of the control objectives allowing complex nonlinear systems to be analyzed using a scalar differential equation When a controller is designed t,hat drives the Lyapunov function to zero, the control objectives are met If some system uncertainty tends to drive the Lyapunov candidate away from zero, we will often simply add an additional stabilizing term to the control algorithm that dominates the effect of the uncertainty, thereby making the closed-loop system more robust

We will find that by adding a static term in the control law that simply dominates the plant uncertainty, it is often easy to simply stabilize an uncertain plant, however, driving the system error to zero may be difficult

if not impossible Consider the case when the plant is defined by

where x E R is the plant state that we wish to drive to the point x = 1,

u E R is the plant input, and 8 is an unknown constant Since 8 is unknown,

one may not define a static controller that causes 12; = 1 to be a stable equilibrium point In order for 61; = 1 to be a stable equilibrium point, it

is necessary that ? = 0 when x = 1, so U(X) = -0 when z = 1 Since 6’ is unknown, however, we may not define such a controller

In this case, the best that a static nonlinear controller may do is to keep

x bounded in some region around z = 1 If dynamics are included in the nonlinear controller, then it turns out that one may define a control system that does drive x -+ 1 even if B is unknown In this book we will use the approach of adaptive control to help us define such a nonlinear dynamic controller that will stabilize a certain class of nonlinear uncertain systems

An a,daptive controller can be designed so that it estimates some uncertainty within the system, then automatically designs a controller for the estimated plant uncertainty In this way the control system uses information gathered on-line to reduce the model uncertainty, that is, to figure out exactly what the plant is at the current time so that good control can be achieved Considering the system defined by (l.l), an adaptive controller may be defined so that an estimate of 0 is generated, which we will denote 6 If

0 were known, then including a term -8x in the control law would cancel the effects of the uncertainty If 8 + 6 over time, then including the term -6~ in the control law would also cancel the effects of the uncertainty over time This approach is referred to as indirect adaptive control

An indirect approach to adaptive control is made up of an approximator (often referred to as an “identifier” in the adaptive control literature) that

is used to estimate unknown plant parameters and a “certainty equiva- lence” control scheme in which the plant controller is defined (“designed”) aassuming that the parameter estimates a’re their true values The indirect a’daptive approach is shown in Figure 1.3 Here the adjustable approximator

is used to model some component of the system Since the approximation

is used in the control law, it is possible to determine if we have a good estima,te of the plant dynamics If the approximation is good (i.e., we know how t)he plant should behave), then it is easy to meet our control objec- tives If, on the other hand, the plant output moves in the wrong direction, then we ma,y assume that our estimate is incorrect and should be adjusted a,ccordingly

As a1n example of an indirect adaptive controller, consider the cruise control problem where we have an approximator that is used to estimate the vehicle mass and aerodynamic drag Assume that the vehicle dynamics

1 )I-

Figure 1.3 Indirect adaptive control

may be approximated by

where m is the vehicle mass, p is the coefficient of aerodynamic drag, x is the vehicle velocity, and u is the plant input Assume that an approximator has been defined so that estimates of the mass and drag are found such that

& + m and b -+ p Then the control law

u = 62” + r?w(t) may be used so that 2 = v(t) when ti = m and b = p Here v(t) may

be considered a new control input that is defined to drive x to any desired value

Latter in this book, we will learn how to define an approximator for

ti and fi in the above example that allows us to drive x to some desired velocity We will also find that the indirect approach remains stable when k(O) # m and b(O) # p though the initial parameter values may affect the transient performance of the closed-loop system

Yet another approach to adaptive control is shown in Figure 1.4 Here the adjustable approximator acts as a controller The adaptation mecha- nism is then designed to adjust the approximator causing it to match some unknown nonlinear controller that will stabilize the plant and make the closed-loop system achieve its performance objectives

Note tha*t we call this scheme “direct” since there is a direct adjustment

of the parameters of the controller without identifying a model of the plant

1’

Figure 1.4 Direct adaptive control

Direct adaptive control, while a somewhat less popular approach (at least in the neural/fuzzy adaptive control literature), will be considered each time

we consider an indirect scheme in this book Part of the reason we give

a relatively equal treatment to direct adaptive schemes is that in several implementations we have found them to work more effectively than their indirect adaptive counterparts

In this section we outline how neural networks and fuzzy systems can be used as the “approximator” in the adaptive schemes outlined in the previous section Then we discuss the advantages of using neural networks or fuzzy systems as approximators in adaptive systems

Neural networks are parameterized nonlinear functions Their parameters are, for instance, the weights and biases of the network Adjustment of these parameters results in different shaped nonlinearities Typically, the adjustment of the neural network parameters is achieved by a gradient descent approach on an error function that measures the difference between the output of the neural network and the output of the actual system (function) That is, we try to adjust the neural network to serve as an approximator for an unknown function that we only know by how it specifies output values for the given input va,lues (i.e., the training data) Or, viewed a’nother way, we seek to adjust the neural network so that it serves as an

“interpolator” for the input-output da#ta so that if it is presented with input data, it will produce an output that is close to the actual output that the function (system) would create

Due to the wide range of roles tha’t the neural network c&n play in a’daptive schemes we will simply call them “approximators,” and below

we will focus on their properties and advantages It is important to note, however, that neural networks are not unique in their ability to serve as approximators There are conventional approximator structures such as polynomials Moreover, there is the possibility of using a fuzzy system as a#n approximator structure as we discuss next

Historically, fuzzy controllers have stirred a great deal of excitement in some circles since they allow for the simple inclusion of heuristic knowl- edge about how to control a plant rather than requiring exact mathemat- ical models This can sometimes lead to good controller designs in a very short period of time In situations where heuristics do not provide enough information to specify all the parameters of the fuzzy controller a priori, re- searchers have introduced adaptive schemes that use data gathered during the on-line operation of the controller, and special adaptation heuristics, to automatically learn these parameters

Hence, fuzzy systems have served not only their originally intended function of providing an approach to nonadaptive control, but also in adap- tive controllers where, for example, the membership functions are adjusted Fuzzy systems are indeed simply nonlinear functions that are parameter- ized by, for example, membership function parameters In fact, in some situations they are mathematically identical to a certain class of radial ba- sis function neural networks It is then not surprising that we can use fuzzy systems as approximators in the same way that we can use neural networks

It is possible, however, that the fuzzy system can offer an additional ad- vantage in that it may be easier to incorporate heuristic knowledge about how the input-output map for which you are gathering data from should be shaped In some situations this can lead to better convergence properties (simply because it may be easier to initialize the shape of the nonlinearity implemented by the approximator)

In this book we will provide some insights into how to pick an approx- imator (e.g., based on physical considerations); however, the question of which approximator is best to use is still an open research issue In our discussions on approximator properties, when we refer to an “approximator structure,” we mean the nonlinear function that is tuned by the parameters

of the approximator The %ize” of the approximator is some measure of the complexity of the mapping it implements (e.g., for a neural network it might be the total number of parameters used to adjust the network) An- other feature that we will use to distinguish among different approximators

is whether they are “linear in their parameters.” For instance, when only certain parameters in a neural network are adjusted, these may be ones that enter in a linear fashion Clearly, linear in the parameter approximators are a, special case of nonlinear in the parameter approximators and hence they can be more limited in what functions that they can approximate

We will study approximators (neural or fuzzy) that satisfy the “univer- sal approximation property.” If an approximator possesses the universal

8 Introduction

approximation property, then it can approximate any continuous function

on a closed and bounded domain with as much accuracy as desired (how- ever, most often 7 to get an arbitrarily accurate approximation you have to

be willing to increase the size the the approximator structure arbitrarily)

It t urns out that some approximator structures provi de much more efficient parameterized nonlinearities in the sense that to get definite improvement

in approximation accuracy they only have to grow in size in a, linear fashion Other approximator struct ures may have to grow exponentially to achieve small increases in approximation accuracy However, it is important to note that the inclusion of physical domain knowledge may help us to avoid prohibitive increases in the size of the approximator

The “approximation error” is some suitably defined measure (e.g., the maximum distance between the two functions over their domains) of the error between the function you are trying to approximate (e.g., the plant) and the function implemented by the approximator The “ideal approxi- mation error” (also known as the “representation error”) is

error that would result from the best choice of the approxi

the minimum .mator param- eters (i.e., the “ideal parameters”) For a class of neural networks it can

be shown that the ideal approximation error has definite decreases with

an increase in the size of the approximator (i.e., it decreases at a certain rate with a linear increase in

this case you must adjust the

the size of the neural parameter ‘s that enter

network); however, in

in a nonlinear fashion and there are no general guarantees for current algorithms that you will find the ideal parameters Linear in the parameter approximators provide

no such guarantees of reduction of the ideal approximation error; however, when one incorporates physi

cations shows that increases

cal

in

domain knowledge, experience with appli- approximator accuracy can often be found with reasonable increases in the size of the approximator

First, for comparison purposes it is useful to point out that we can broadly think of many conventional adaptive estimation and control approaches for linear systems as techniques that use linear approximation structures for systems with known model order (of course, this is for the state feed- back case and ignores the results for plants where the order is not assumed known) Most often, in these cases, the problems are set up so that the linear approximator (e.g., a linear model with tunable parameters) can perfectly represent the underlying unknown function that it is trying to approximate (e.g., the plant model) However, it may take a certain “per- sistency of excitation” to achieve perfect approximation and conditions for this were derived for adaptive estimation and control

Regardless, thinking along these lines, linear robust adaptive control studies how to tune linear a!pproximators when it is not possible to per-

fectly approximate the unknown function with a linear map In this sense,

it becomes clear why there is such a strong reliance of the methods of on-line approximation based control via neural or fuzzy systems on conventional ro- bust control of linear systems While the universal approximation property guarantees that our approximators can represent the unknown function, for practical reasons we have to limit their size so a finite approximation error arises and must be dealt with; on-line approximation approaches deal with

it in similar (or the same) ways to how it is dealt with in linear robust control

Now, while there is a strong connection to the conventional robust adap- tive control approaches, the on-line approximation based approach allows you to go further since it does not restrict the unknown function to be linear In this way, it provides a logical extension to create nonlinear ro- bust control schemes where there is no need to assume that the plant is a linear parameterization of known nonlinear functions (as in the early work

on a,daptive feedback linearization [192] and the more recently developed systematic approach of adaptive backstepping [115])

It is interesting to note, however, that while there are strong connec- tions to conventional adaptive schemes, there is an additional interesting characteristic of the resulting adaptive systems in that if designed prop- erly they can implement something that is more similar to the way we think of “learning” than conventional adaptive schemes Some on-line ap- proximation based schemes (particularly some that are implemented with approximators that have basis functions with “local support” like radial ba- sis function neural networks and fuzzy systems) achieve local adjustments

to parameters so that only local adjustments to the tuned nonlinearity take place In this case, if designed properly, the controller can be taught one operating condition, then learn a different operating condition, and later return to the first operating condition with a controller that is already properly tuned for that region Another way to think of this is that since

we are tuning nonlinear functions that have an ability to be tuned locally (something a simple linear map cannot do since if you change a parameter

it affects the shape of the map over the whole space) they can remember past tuning to a certain extent

To summarize, in many ways, the advantages of using neural networks or fuzzy systems arise as pr-actical rather than theoretical ben&if,s in the sense that we could avoid their use all together and simply use some conventional a,pproximator structure (e.g., a polynomial approximator structure) The practical benefits of neural networks or fuzzy systems are the following:

l They offer forms of nonlinearities (e.g., the neural network) that are universal a8pproximators (hence more broadly applicable to ma’ny ap- plications) and that offer reduced ideal approximation error for only

a linear increase in the number of parameters

l They offer convenient ways to incorporate heuristics on how to ini- tialize the nonlinearity (e.g., the fuzzy system)

In addition, to help demonstrate the practical nature of the approaches we introduce in this book, there will be an experimental component where we discuss severa’ laboratory implementations of the methods

1.5 Summary

The general control philosophy used within this book may be summarized

as follows:

We use concepts and techniques from robust control theory,

Adaptive approaches are used to compensate for unknown system characteristics, and

When a system uncertainty may be characterized by a function, the problem is reformulated in terms of fuzzy systems or neural networks

to extend the applicability of the adaptive approaches

We will use the traditional controller development and analysis approaches used in robust, adaptive, and nonlinear control, with the mathematical flexibility provided by fuzzy systems and neural networks, to develop a powerful approach to solving many of today’s challenging real-world control problems

Overall, while we understand that many people do not read introduc- tions to books, we tried to make this one useful by giving you a broad view

of the lines of reasoning that we use, and by explaining what benefits the methods may provide to you