# ADAPTIVE CONTROL PDF

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Direct and Indirect Adaptive Control 8. Model Reference Adaptive Control Adaptive Pole Placement Control. This tutorial paper looks back at almost 50 years of adaptive control trying to establish how much more we need to secure for the industrial community an. Nonlinear and Adaptive Control with Applications . grounding for analysis and design of adaptive control systems and so these form the.

As a result, direct adaptive control is restricted to certain classes of plant models. In general, not every plant can be expressed in a parameterized form involving only the controller parameters, which is also a suitable form for online estimation.

As we show in Chapter 5, a class of plant models that is suitable for direct adaptive control for a particular control objective consists of all SISO LTI plant models that are minimum phase; i.

In general, the ability to parameterize the plant model with respect to the desired controller parameters is what gives us the choice to use the direct adaptive control approach. Note that Figures 1. This identical-in-structure interpretation is often used in the literature of adaptive control to argue that the separation of adaptive control into direct and indirect is artificial and is used simply for historical reasons. 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 suffers from a problem known as the stabilizability problem explained as follows: As shown in Figure 1. Such calculations are possible, provided that the estimated plant is controllable and observable or at least stabilizable and detectable.

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. As we explain in Chapter 6, solutions to this stabilizability problem are possible at the expense of additional complexity.

The principle behind the design of direct and indirect adaptive control shown in Figures 1. The form of the control law is the same as the one used in the case of known plant parameters. In the case of indirect adaptive control the unknown controller parameters are calculated at each time t using the estimated plant parameters generated by the online estimator, whereas in the direct adaptive control case the controller parameters are generated directly by the online estimator.

In both cases the estimated parameters are treated as the true parameters for control design purposes. This design approach is called certainty equivalence CE and can be used to generate a wide class of adaptive control schemes by combining different online parameter estimators with different control laws. In some approaches, the control law is modified to include nonlinear terms, and this approach deviates somewhat from the CE approach.

The principal philosophy, however, that as the estimated parameters converge to the unknown constant parameters the control law converges to that used in the known parameter case, remains the same. Gain scheduling structure. 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 in the following subsections. Since some of these approaches are relatively recent and research is still going on, we will not discuss them further in the rest of the book.

Transitions between different operating points that lead to significant parameter changes may be handled by interpolation or by increasing the number of operating points. The two elements that are essential in implementing this approach are a lookup table to store the values of Kj and the plant measurements that correlate well with the changes in the operating points. The approach is called gain scheduling and is illustrated in Figure 1.

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, 6 Chapter 1. Introduction Figure 1.

Multiple models adaptive control with switching. One of the disadvantages of gain scheduling is that the adjustment mechanism of the controller gains is precomputed offline and, therefore, provides no feedback to compensate for incorrect schedules.

Large unpredictable changes in the plant parameters, however, due to failures or other effects 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 [3,6] and other systems [7, , ].

While gain scheduling falls into the generic definition of adaptive control, we do not classify it as adaptive control in this book due to the lack of online parameter estimation which could track unpredictable changes in the plant parameters. These schemes are based on search methods in the controller parameter space [8] 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 [22, 23]. In some approaches, after a satisfactory controller is found it can be tuned locally using online parameter estimation for better performance [].

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. Without going into specific details, the general structure of this multiple model adaptive control with switching, as it is often called, is shown in Figure 1. Why Adaptive Control 7 In Figure 1.

This by itself could be a difficult task in some practical situations where the plant parameters are unknown or change in an unpredictable manner. Furthermore, since there is an infinite number of plants within any given bound of parametric uncertainty, finding controllers to cover all possible parametric uncertainties may also be challenging.

In other approaches [22, 23], it is assumed that the set of controllers with the property that at least one of them is stabilizing is available. This is achieved by the use of a switching logic that differs in detail from one approach to another. While these methods provide another set of tools for dealing with plants with unknown parameters, they cannot replace the identifier-based adaptive control schemes where no assumptions are made about the location of the plant parameters.

One advantage, however, is that once the switching is over, the closed-loop system is LTI, and it is much easier to analyze its robustness and performance properties. This LTI nature of the closed-loop system, at least between switches, allows the use of the well-established and powerful robust control tools for LTI systems [29] for controller design. These approaches are still at their infancy and it is not clear how they affect performance, as switching may generate bad transients with adverse effects on performance.

Switching may also increase the controller bandwidth and lead to instability in the presence of high-frequency unmodeled dynamics. Guided by data that do not carry sufficient information about the plant model, the wrong controllers could be switched on over periods of time, leading to internal excitation and bad transients before the switching process settles to the right controller.

Some of these issues may also exist in classes of identifier-based adaptive control, as such phenomena are independent of the approach used. The following simple examples illustrate situations 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 linear control law can meet the control objective. The conclusion is that in the Chapter 1. Introduction 8 absence of an upper bound for the plant parameter no linear controller could stabilize the plant and drive the state to zero. The switching schemes described in Section 1. As we will establish in later chapters, 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. Another example where an adaptive control law may have properties superior to those of the traditional linear schemes is the following. It is clear that by increasing the value of the controller gain k, we can make the steady-state value of jc as small as we like. This will lead to a high gain controller, however, which is undesirable especially in the presence of high-frequency unmodeled dynamics.

In principle, however, we cannot guarantee that x will be driven to zero for any finite control gain in the presence of nonzero disturbance d. The adaptive control approach is to estimate online the disturbance d and cancel its effect via feedback.

Therefore, in addition to stability, adaptive control techniques could be used to improve performance in a wide variety of situations where linear techniques would fail to meet the performance characteristics. This by no means implies that adaptive control is the most 1. A Brief History 9 appropriate approach to use in every control problem. The purpose of this book is to teach the reader not only the advantages of adaptive control but also its limitations.

Adaptive control involves learning, and learning requires data which carry sufficient information about the unknown parameters. For such information to be available in the measured data, the plant has to be excited, and this may lead to transients which, depending on the problem under consideration, may not be desirable.

Furthermore, in many applications there is sufficient information about the parameters, and online learning is not required. In such cases, linear robust control techniques may be more appropriate. The adaptive control tools studied in this book complement the numerous control tools already available in the area of control systems, and it is up to the knowledge and intuition of the practicing engineer to determine which tool to use for which application.

The theory, analysis, and design approaches presented in this book will help the practicing engineer to decide whether adaptive control is the approach to use for the problem under consideration.

Starting in the early s, the design of autopilots for high-performance aircraft motivated intense research activity in adaptive control. Highperformance aircraft undergo drastic changes in their dynamics when they move from one operating point to another, which cannot be handled by constant-gain feedback control.

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. In the second approach, referred to as direct adaptive control, the plant 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 Figure 1. The parameter estimate 0 0 specifies an estimated plant model characterized by G 0 0 , which for control design purposes is treated as the "true" plant model and is used to calculate the controller parameter or gain vector Oc by solving a certain algebraic equation, 0t.

The form of the control law 1. Adaptive Control: Identifier-Based 3 Figure 1. Indirect adaptive control structure. Figure 1. Direct adaptive control structure. Therefore, the main problem in indirect adaptive control is to choose the class of control laws C We will study this problem in great detail in Chapters 5 and 6.

The estimate 9c t is then used in the control law without intermediate calculations. The choice of the class of control 4 Chapter 1. Introduction laws C 9C and parameter estimators that generate 9c t so that the closed-loop plant meets the performance requirements is the fundamental problem in direct adaptive control. As a result, direct adaptive control is restricted to certain classes of plant models.

In general, not every plant can be expressed in a parameterized form involving only the controller parameters, which is also a suitable form for online estimation.

As we show in Chapter 5, a class of plant models that is suitable for direct adaptive control for a particular control objective consists of all SISO LTI plant models that are minimum phase; i.

Non-Identifier-Based 5 Figure 1. Gain scheduling structure. Non-Identifier-Based Another class of schemes that fit the generic structure given in Figure 1. 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 in the following subsections.

Since some of these approaches are relatively recent and research is still going on, we will not discuss them further in the rest of the book. Transitions between different operating points that lead to significant parameter changes may be handled by interpolation or by increasing the number of operating points.

The two elements that are essential in implementing this approach are a lookup table to store the values of Kj and the plant measurements that correlate well with the changes in the operating points. The approach is called gain scheduling and is illustrated in Figure 1. 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, 6 Chapter 1. Introduction Figure 1. Multiple models adaptive control with switching. One of the disadvantages of gain scheduling is that the adjustment mechanism of the controller gains is precomputed offline and, therefore, provides no feedback to compensate for incorrect schedules.

Large unpredictable changes in the plant parameters, however, due to failures or other effects 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 [3,6] and other systems [7, , ]. While gain scheduling falls into the generic definition of adaptive control, we do not classify it as adaptive control in this book due to the lack of online parameter estimation which could track unpredictable changes in the plant parameters.

These schemes are based on search methods in the controller parameter space [8] 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 [22, 23]. In some approaches, after a satisfactory controller is found it can be tuned locally using online parameter estimation for better performance []. 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.

Without going into specific details, the general structure of this multiple model adaptive control with switching, as it is often called, is shown in Figure 1.

Why Adaptive Control 7 In Figure 1. This by itself could be a difficult task in some practical situations where the plant parameters are unknown or change in an unpredictable manner.

Furthermore, since there is an infinite number of plants within any given bound of parametric uncertainty, finding controllers to cover all possible parametric uncertainties may also be challenging. In other approaches [22, 23], it is assumed that the set of controllers with the property that at least one of them is stabilizing is available.

This is achieved by the use of a switching logic that differs in detail from one approach to another. While these methods provide another set of tools for dealing with plants with unknown parameters, they cannot replace the identifier-based adaptive control schemes where no assumptions are made about the location of the plant parameters. One advantage, however, is that once the switching is over, the closed-loop system is LTI, and it is much easier to analyze its robustness and performance properties.

This LTI nature of the closed-loop system, at least between switches, allows the use of the well-established and powerful robust control tools for LTI systems [29] for controller design. These approaches are still at their infancy and it is not clear how they affect performance, as switching may generate bad transients with adverse effects on performance.

Switching may also increase the controller bandwidth and lead to instability in the presence of high-frequency unmodeled dynamics. Guided by data that do not carry sufficient information about the plant model, the wrong controllers could be switched on over periods of time, leading to internal excitation and bad transients before the switching process settles to the right controller.

Some of these issues may also exist in classes of identifier-based adaptive control, as such phenomena are independent of the approach used.

The following simple examples illustrate situations 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 linear control law can meet the control objective. The conclusion is that in the Chapter 1. Introduction 8 absence of an upper bound for the plant parameter no linear controller could stabilize the plant and drive the state to zero. The switching schemes described in Section 1.

As we will establish in later chapters, 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. Another example where an adaptive control law may have properties superior to those of the traditional linear schemes is the following.

Consider the same example as above but with an external bounded disturbance d: It is clear that by increasing the value of the controller gain k, we can make the steady-state value of jc as small as we like. This will lead to a high gain controller, however, which is undesirable especially in the presence of high-frequency unmodeled dynamics.

In principle, however, we cannot guarantee that x will be driven to zero for any finite control gain in the presence of nonzero disturbance d.

The adaptive control approach is to estimate online the disturbance d and cancel its effect via feedback. The following adaptive control law can be shown to guarantee signal boundedness and convergence of the state x to zero with time: Therefore, in addition to stability, adaptive control techniques could be used to improve performance in a wide variety of situations where linear techniques would fail to meet the performance characteristics.

This by no means implies that adaptive control is the most 1. A Brief History 9 appropriate approach to use in every control problem. The purpose of this book is to teach the reader not only the advantages of adaptive control but also its limitations.

Adaptive control involves learning, and learning requires data which carry sufficient information about the unknown parameters. For such information to be available in the measured data, the plant has to be excited, and this may lead to transients which, depending on the problem under consideration, may not be desirable. Furthermore, in many applications there is sufficient information about the parameters, and online learning is not required. In such cases, linear robust control techniques may be more appropriate.

The adaptive control tools studied in this book complement the numerous control tools already available in the area of control systems, and it is up to the knowledge and intuition of the practicing engineer to determine which tool to use for which application.

The theory, analysis, and design approaches presented in this book will help the practicing engineer to decide whether adaptive control is the approach to use for the problem under consideration. Starting in the early s, the design of autopilots for high-performance aircraft motivated intense research activity in adaptive control. Highperformance aircraft undergo drastic changes in their dynamics when they move from one operating point to another, which cannot be handled by constant-gain feedback control.

A sophisticated controller, such as an adaptive controller, that could learn and accommodate changes in the aircraft dynamics was needed. Model reference adaptive control was suggested by Whitaker and coworkers in [30, 31] to solve the autopilot control problem. Sensitivity methods and the MIT rule were used to design the online estimators or adaptive laws of the various proposed adaptive control schemes. An adaptive pole placement scheme based on the optimal linear quadratic problem was suggested by Kalman in [32].

The work on adaptive flight control was characterized by a "lot of enthusiasm, bad hardware and nonexisting theory" [33]. 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 [34] caused the interest in adaptive control to diminish. The s 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. Developments in dynamic programming [35, 36], dual control [37] and stochastic control in general, and system identification and parameter estimation [38, 39] played a crucial role in the reformulation and redesign of adaptive control.

By , Parks [40] and others found a way of redesigning the MIT rule-based adaptive laws used in the model reference adaptive control MRAC schemes of the s by applying the Lyapunov design approach. Their work, even though applicable to a special class of LTI plants, set the stage for further rigorous stability proofs in adaptive control for more general classes of plant models. The advances in stability theory and the progress in control theory in the s improved the understanding of adaptive control and contributed to a strong renewed interest in the field in the s.

On the other hand, the simultaneous development and progress in computers and electronics that made the implementation of complex controllers, such as 10 Chapter!. Introduction the adaptive ones, feasible contributed to an increased interest in applications of adaptive control.

The s witnessed several breakthrough results in the design of adaptive control. MRAC schemes using the Lyapunov design approach were designed and analyzed in []. The concepts of positivity and hyperstability were used in [45] to develop a wide class o MRAC schemes with well-established stability properties.

At the same time parallel efforts for discrete-time plants in a deterministic and stochastic environment produced several classes of adaptive control schemes with rigorous stability proofs [44,46]. The excitement of the s and the development of a wide class of adaptive control schemes with wellestablished stability properties were accompanied by several successful applications [47— 49].

The successes of the s, however, were soon followed by controversies over the practicality of adaptive control. As early as it was pointed out by Egardt [41] that the adaptive schemes of the s could easily go unstable in the presence of small disturbances.

The nonrobust behavior of adaptive control became very controversial in the early s when more examples of instabilities were published by loannou 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- s, 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 as well as performance error bounds that are of the order of the modeling error. The work on robust adaptive control continued throughout the s and involved the understanding of the various robustness modifications and their unification under a more general framework [41, ].

In discrete time Praly [57, 58] was the first to establish global stability in the presence of unmodeled dynamics using various fixes and the use of a dynamic normalizing signal which was used in Egardt's work to deal with bounded disturbances.

The use of the normalizing signal together with the switching a-modification led to the proof of global stability in the presence of unmodeled dynamics for continuous-time plants in [59]. The solution of the robustness problem in adaptive control led to the solution of the long-standing problem of controlling a linear plant whose parameters are unknown and changing with time.

By the end of the s several breakthrough results were published in the area of adaptive control for linear time-vary ing plants [5, ]. The focus of adaptive control research in the late s to early s was on performance properties and on extending the results of the s 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 [].

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 [66]. In the late s to early s, 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. Difficulties in establishing global convergence results soon arose since in multilayer neural networks the weights appear in a nonlinear fashion, leading to "nonlinear in the parameters" parameterizations for which globally 1.

This led to the consideration of single layer neural networks where the weights can be expressed in ways convenient for estimation parameterizations. These approaches are described briefly in Chapter 8, where numerous references are also provided for further reading. In the mids to early s, several groups of researchers started looking at alternative methods of controlling plants with unknown parameters []. These methods avoid the use of online parameter estimators in general and use search methods, multiple models to characterize parametric uncertainty, switching logic to find the stabilizing controller, etc.

Research in these non-identifier-based adaptive control techniques is still going on, and issues such as robustness and performance are still to be resolved. Adaptive control has a rich literature full of different techniques for design, analysis, performance, and applications. Several survey papers [74, 75] and books and monographs [5,39,41,,49,50,66,] 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.

The purpose of this book is to present the basic design and analysis tools in a tutorial manner, making adaptive control accessible as a subject to less mathematically oriented readers while at the same time preserving much of the mathematical rigor required for stability and robustness analysis. Some of the significant contributions of the book, in addition to its relative simplicity, include the presentation of different approaches and algorithms in a unified, structured manner which helps abolish much of the mystery that existed in adaptive control.

Furthermore, up to now continuous-time adaptive control approaches have been viewed as different from their discrete-time counterparts. In this book we show for the first time that the continuous-time adaptive control schemes can be converted to discrete time by using a simple approximation of the time derivative. This page intentionally left blank Chapter 2 Parametric Models Let us consider the first-order system where jc, u are the scalar state and input, respectively, and a, b are the unknown constants we want to identify online using the measurements of Jt, u.

The first step in the design of online parameter identification PI algorithms is to lump the unknown parameters in a vector and separate them from known signals, transfer functions, and other known parameters in an equation that is convenient for parameter estimation. We refer to 2. The SPM may represent a dynamic, static, linear, or nonlinear system. For this reason we refer to 2. As we will show later, this property is significant in designing online PI algorithms whose global convergence properties can be established analytically.

We can derive 2. In some cases, the unknown parameters cannot be expressed in the form of the linear in the parameters models. In such cases the PI algorithms based on such models cannot be shown to converge globally.

The transfer function W q is a known stable transfer function. In some applications of parameter identification or adaptive control of plants of the form whose state x is available for measurement, the following parametric model may be used: Chapter 2.

Parametric Models 15 where Am is a stable design matrix; A, B are the unknown matrices; and x, u are signal vectors available for measurement. We refer to this class of parametric models as state-space parametric models 55PM.

Another class of state-space models that appear in adaptive control is of the form where B is also unknown but is positive definite, is negative definite, or the sign of each of its elements is known. The PI problem can now be stated as follows: For the SSPM: Given the measurements of x, u, i. The online PI algorithms generate estimates at each time t, by using the past and current measurements of signals. Convergence is achieved asymptotically as time evolves.

For this reason they are referred to as recursive PI algorithms to be distinguished from the nonrecursive ones, in which all the measurements are collected a priori over large intervals of time and are processed offline to generate the estimates of the unknown parameters.

Generating the parametric models 2. Below, we present several examples that demonstrate how to express the unknown parameters in the form of the parametric models presented above.

Example 2. If we assume that the spring is "linear," i. Parametric Models Figure 2. Mass-spring-dashpot system. We can easily express 2. If not, the parametric model associated with 0 is a constant design parameter we can choose arbitrarily, to obtain Using 2. Another possible parametric model is Chapter 2.

Parametric Models 17 Figure 2. Cart with two inverted pendulums. Parametric Models The above equation can be rewritten as Where In order to avoid the use of differentiators, we filter each side with fourth-order stable i e. Parametric Models 19 Let us assume that we know that one of the constant parameters, i. Then we can obtain a model of the system in the B-SPM form as follows: The input u and output y are available for measurement.

We would like to parameterize the system 2.

We rewrite 2. In order to express 2. This is equivalent to filtering each side with the fourth-order stable filter p-.

The shifted model where all signals are available for measurement can be rewritten in the form of the SPM as where Example 2. For identification purposes, the system may be expressed as and put in the form of the DPM where If we want W s to be a design transfer function with a pole, say at A. We then rewrite it as Letting we obtain another DPM. The model may be also expressed as where Example 2. We can also express the above system as an nth-order differential equation given by Lumping all the parameters in the vector 22 Chapter 2.

Problems 1.

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Consider the mass-spring-dashpot system of Figure 2. Specify any arbitrary parameters or filters used. Express the unknown parameters in the form of a linear parametric model. Assume that u k , y k , and their past values are available for measurement. Problems 23 4. Assume that only the current and the past four values of the signals u and y, i. Parametric Models 8. The online identification procedure involves the following three steps. Step 1. Step 2. The estimation error is used to drive the adaptive law that generates 6 t online.

A wide class of adaptive laws with different H t and e may be developed using optimization techniques and Lyapunovtype stability arguments. Step 3. This step involves the design of the plant input so that the signal vector 0? For example, for 0? We demonstrate the three design steps using the following example of a scalar plant. Parameter Identification: One-Parameter Case Consider the first-order plant model where a is the only unknown parameter and y and u are the measured output and input of the system, respectively.

Step 1: Parametric Model We write 3. Therefore, 3. This property of ms is used to establish the boundedness of the estimated parameters even when 0 is not guaranteed to be bounded. The use of a lower bound 1 is without loss of generality.

One-Parameter Case 27 Using 3. It should be noted that e cannot be generated using 3. Consequently, 3. The simplest one is obtained by using the SPM 3.

In practice, however, the effect of noise on the measurements of 0 0, especially when 0 0 is close to zero, may lead to erroneous parameter estimates. With this approach, 0 0 is adjusted in a direction that makes s smaller and smaller until a minimum is reached at which e — 0 and updating is terminated.

In this scalar case, which leads to the adaptive law Step 3: Stability and Parameter Convergence The adaptive law should guarantee that the parameter estimate 0 0 and the speed of adaptation 0 are bounded and that the estimation error e gets smaller and smaller with time.

Continuous Time Let us start by using 3. It follows from 3. Another way to analyze 3. We consider the function Then or, using 3. Such a property can be easily established when the input u is bounded see Problem 3.

The PE property of — is guaranteed by choosing the input u appropriately. The condition 3. The PI algorithm for estimating the constant a in the plant 3. The above analysis for the scalar example carries over to the vector case without any significant modifications, as demonstrated in the next section. One important difference, however, is that in the case of a single parameter, convergence of the Lyapunov-like function V to a constant implies that the estimated parameter converges to a constant.

Such a result cannot be established in the case of more than one parameter for the gradient algorithm. Two Parameters Consider the plant model where a, b are unknown constants. We would like to generate online estimates for the parameters a, b. Parametric Model Since y, y are available for measurement, we can express 3. Step 2: For simplicity let us assume that the plant is stable, i.

Exponential stability for the equilibrium point 9e — 0 of 3. We answer the above questions in the following section. Definition 3. Continuous Time Since 07 is always positive semidefinite, the PE condition requires that its integral over any interval of time of length T0 is a positive definite matrix.

A more general definition of sufficiently rich signals and associated properties may be found in [95]. Theorem 3. Consider 3. Then is PE if and only ifu is sufficiently rich of order n. The proof of Theorem 3. We demonstrate the use of Theorem 3.

According to Theorem 3. Ignoring the transient terms that converge to zero exponentially fast, we can show that at steady state where 3. We can verify that for z which implies that is PE.

Let us consider the plant model where b is the only unknown parameter. Let us us Theorem 3. In this case, 34 Chapters. Continuous Time according to the linear independence condition of Theorem 3. In general, for each two unknown parameters we need at least a single nonzero frequency to guarantee PE, provided of course that H s does not lose its linear independence as demonstrated by the above example.

The two-parameter case example presented in section 3. Vector Case 35 where If Z s is Hurwitz, a bilinear model can be obtained as follows: In certain adaptive control systems such as MRAC, the coefficients of P s , Q s are the controller parameters, and parameterizations such as 3. Different choices for H t and s t lead to a wide class of adaptive laws with, sometimes, different convergence properties, as demonstrated in the following sections. Different choices for J 9 lead to different algorithms.

As in the scalar case, we start by defining the estimation model and estimation error for the SPM 3. Some straightforward choices for ns include or 3. Gradient Algorithms Based on the Linear Model 37 where a is a scalar and P is a matrix selected by the designer. The estimation error 3. At each time t, J 9 is a convex function of 0 and therefore has a global minimum. The gradient algorithm 3. Continuous Time We choose the Lyapunov-like function Then along the solution of 3.

In addition, it follows from 3. Now from 3. The proof of parts ii and iii is longer and is presented in the web resource [94]. Comment 3. The only free design parameter is the adaptive gain matrix F. Then and 0 0 — [sin t, cos t]T does not have a limit. Example 3. We want to estimate a, b online. While this adaptive law guarantees properties i and ii of Theorem 3. Continuous Time Example 3. For 3. The online estimate of the amplitude and phase can be computed using 3. The estimated disturbance can then be generated and used by the controller to cancel the effect of the actual disturbance d.

The gradient algorithm with integral cost function guarantees that iii! Furthermore, for r — yl, the rate of convergence increases with y. The proof is presented in the web resource [94]. Simulations demonstrate that the gradient algorithm based on the integral cost gives better convergence properties than the gradient algorithm based on the instantaneous cost. The gradient algorithm based on the integral cost has similarities with the least-squares LS algorithms to be developed in the next section.

The basic idea behind LS is fitting a mathematical model to a sequence of observed data by minimizing the sum of the squares of the difference between the observed and computed data. In doing so, any noise or inaccuracies in the observed data are expected to have less effect on the accuracy of the mathematical model.

The LS method has been widely used in parameter estimation both in recursive and nonrecursive forms mainly for discrete-time systems [46, 47, 77, 97, 98]. Least-Squares Algorithms 43 In practice, dn may be due to sensor noise or external sources, etc.

We examine the following estimation problem: Given the measurements of z r , 0 t for 0 by using the measurements of z r , 0 r at some T r for 0 w. Since J 9 is a convex function over K at each time t, its minimum satisfies which gives the LS estimate provided of course that the inverse exists.

For example, when? Let us now extend this problem to the linear model 3. As in section 3. This cost function is a generalization of 3. If ft — 0, the algorithm becomes the pure LS algorithm discussed and analyzed in section 3. In this case 3. Least-Squares Algorithms 45 Theorem 3. If — is PE, then the recursive LS algorithm with forgetting factor 3.

The proof is given in [56] and in the web resource [94J. Since the adaptive law 3. In this case, 3. The pure LS algorithm 3. From 3. The proofs of iii and iv are included in the proofs of Theorems 3. The pure LS algorithm guarantees that the parameters converge to some constant 9 without having to put any restriction on the regressor 0. Convergence of the estimated parameters to constant values is a unique property of the pure LS algorithm. One of the drawbacks of the pure LS algorithm is that the covariance matrix P may become arbitrarily small and slow down adaptation in some directions.

This is due to the fact that which implies that P ' may grow without bound, which in turn implies that P may reduce towards zero. This is the so-called covariance wind-up problem. Another drawback of the pure LS algorithm is that parameter convergence cannot be guaranteed to be exponential. Then we can show by solving the differential equation via integration that 3. In this case, we can show that by solving the differential equations above. In this case p t converges to a constant and no covariance wind-up problem arises.

In fact, the pure LS algorithm with covariance resetting can be viewed as a gradient algorithm with time-vary ing adaptive gain P, and its properties are very similar to those of a gradient algorithm analyzed in the previous section. They are summarized by Theorem 3.

In this case, P 0 may grow without bound. In order to avoid this phe- 48 Chapters. Continuous Time nomenon, the following modified LS algorithm with forgetting factor is used: Substituting for z in 3.

## Adaptive control

The time derivative V of V along the solution of 3. Continuous Time Comment 3. Therefore, the adaptive laws for 9, p, may be written as Theorem 3.

The adaptive law 3. The proof of iii is long and is presented in the web resource [94]. The proof of iv is included in the proof of Theorem 3. The reader is referred to [56, ] for further reading on adaptive laws with unknown high-frequency gain.

If such a priori information is available, we want to constrain the online estimation to be within the set where the unknown parameters are located. For this purpose we modify the gradient algorithms based on the unconstrained minimization of certain costs using the gradient projection method presented in section A. The gradient algorithm with projection is computed by applying the gradient method to the following minimization problem with constraints: Assume that S is given by where g: Parameter Projection 53 Theorem 3.

The gradient adaptive laws of section 3. The adaptive laws 3. The gradient adaptive law 3. Hence, for the Lyapunov-like function V used in section 3. Defining the sets for projection as and applying the projection algorithm 3. In most applications, we may have such a priori information. We define and use 3. Robust Parameter Identificati'n 3. The following examples are used to show how 3. Equation 3. Since a parameter estimator for a, b developed based on 3.

We express the plant as which we can rewrite in the form of the parametric model 3. We can express 3. The principal question that arises is how the stability properties of the adaptive laws that are developed for parametric models with no modeling errors are affected when applied to the actual parametric models with uncertainties. The following example demonstrates that the adaptive laws of the previous sections that are developed using parametric models that are free of modeling errors cannot guarantee the same properties in the presence of modeling errors.

In this book we show for the first time that the continuous-time adaptive control schemes can be converted to discrete time by using a simple approximation of the time derivative. This page intentionally left blank Chapter 2 Parametric Models Let us consider the first-order system where jc, u are the scalar state and input, respectively, and a, b are the unknown constants we want to identify online using the measurements of Jt, u.

The first step in the design of online parameter identification PI algorithms is to lump the unknown parameters in a vector and separate them from known signals, transfer functions, and other known parameters in an equation that is convenient for parameter estimation. We refer to 2. The SPM may represent a dynamic, static, linear, or nonlinear system. For this reason we refer to 2. As we will show later, this property is significant in designing online PI algorithms whose global convergence properties can be established analytically.

We can derive 2. In some cases, the unknown parameters cannot be expressed in the form of the linear in the parameters models. In such cases the PI algorithms based on such models cannot be shown to converge globally. The transfer function W q is a known stable transfer function. In some applications of parameter identification or adaptive control of plants of the form whose state x is available for measurement, the following parametric model may be used: Chapter 2.

Parametric Models 15 where Am is a stable design matrix; A, B are the unknown matrices; and x, u are signal vectors available for measurement. We refer to this class of parametric models as state-space parametric models 55PM.

Another class of state-space models that appear in adaptive control is of the form where B is also unknown but is positive definite, is negative definite, or the sign of each of its elements is known.

The online PI algorithms generate estimates at each time t, by using the past and current measurements of signals. Convergence is achieved asymptotically as time evolves. For this reason they are referred to as recursive PI algorithms to be distinguished from the nonrecursive ones, in which all the measurements are collected a priori over large intervals of time and are processed offline to generate the estimates of the unknown parameters.

Generating the parametric models 2. Below, we present several examples that demonstrate how to express the unknown parameters in the form of the parametric models presented above. Example 2. If we assume that the spring is "linear," i.

Parametric Models Figure 2. Mass-spring-dashpot system. We can easily express 2. Let us assume that only x, the displacement of the mass, is available for measurement. In this case, in order to express 2. Another possible parametric model is Chapter 2. Parametric Models 17 Figure 2. Cart with two inverted pendulums. Parametric Models 19 Let us assume that we know that one of the constant parameters, i. The input u and output y are available for measurement. We would like to parameterize the system 2.

We rewrite 2. In order to express 2. This is equivalent to filtering each side with the fourth-order stable filter p-. The shifted model where all signals are available for measurement can be rewritten in the form of the SPM as where Example 2. For identification purposes, the system may be expressed as and put in the form of the DPM where If we want W s to be a design transfer function with a pole, say at A.

We then rewrite it as Letting we obtain another DPM.

## Adaptive Control Tutorial (Advances in Design and Control)

The model may be also expressed as where Example 2. Problems 1. Consider the mass-spring-dashpot system of Figure 2. Specify any arbitrary parameters or filters used. Express the unknown parameters in the form of a linear parametric model. Assume that u k , y k , and their past values are available for measurement. Problems 23 4. Assume that only the current and the past four values of the signals u and y, i. Parametric Models 8. Assume that M is known.

Chapter 3 3. The online identification procedure involves the following three steps.

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Step 1. Step 2. The estimation error is used to drive the adaptive law that generates 6 t online. A wide class of adaptive laws with different H t and e may be developed using optimization techniques and Lyapunovtype stability arguments.

Step 3. This step involves the design of the plant input so that the signal vector 0? For example, for 0? We demonstrate the three design steps using the following example of a scalar plant. Parameter Identification: Continuous Time 3. Step 1: Parametric Model We write 3. Therefore, 3. Step 2: Parameter Identification Algorithm This step involves the development of an estimation model and an estimation error used to drive the adaptive law that generates the parameter estimates.

Note that the reverse is not true, i. This property of ms is used to establish the boundedness of the estimated parameters even when 0 is not guaranteed to be bounded. The use of a lower bound 1 is without loss of generality. Example: One-Parameter Case 27 Using 3. It should be noted that e cannot be generated using 3. Consequently, 3. The simplest one is obtained by using the SPM 3.

## Adaptive Control Tutorial (Advances in Design and Control)

In practice, however, the effect of noise on the measurements of 0 0, especially when 0 0 is close to zero, may lead to erroneous parameter estimates. With this approach, 0 0 is adjusted in a direction that makes s smaller and smaller until a minimum is reached at which e — 0 and updating is terminated. In this scalar case, which leads to the adaptive law Step 3: Stability and Parameter Convergence The adaptive law should guarantee that the parameter estimate 0 0 and the speed of adaptation 0 are bounded and that the estimation error e gets smaller and smaller with time.

Parameter Identification: Continuous Time Let us start by using 3. It follows from 3. Another way to analyze 3.We choose and the projection operator becomes which may be written in the compact form Chapter 4. Some of the significant contributions of the book, in addition to its relative simplicity, include the presentation of different approaches and algorithms in a unified, structured manner which helps abolish much of the mystery that existed in adaptive control.

In other approaches [22, 23], it is assumed that the set of controllers with the property that at least one of them is stabilizing is available. In principle, however, we cannot guarantee that x will be driven to zero for any finite control gain in the presence of nonzero disturbance d.

In the first approach, referred to as indirect adaptive control, the plant parameters are estimated online and used to calculate the controller parameters.

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