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This thesis for the Master of Science degree by Alexandros John Ampsefidis has been approved for the Department of Electrical Enqineerinq and Computer Science by T. Bialasiewicz Wall William R. Roemish Date
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Ampsefidis, Alexandros John (M.S., Electrical Engineering) Lyapunov Design of a New Model-Reference Adaptive Control System Thesis directed by Associate Professor Jan T. Bialasiewicz In this thesis a new approach to adaptive modelreference control is presented. A design procedure for single-input/single-output systems has been developed and the results are verified by computer simulation. The new adaptive control algorithm is simpler than the already existing approaches. In addition, for implementing the new adaptive mechanism, it is not required that the very strong perfect model-following (PMF) conditions are met. The algorithm presented in this thesis is designed by the use of Lyapunov techniques. It guarantees asymptotic stability, provided that the transfer function of the equivalent error system is strictly positive real. Also, it is shown that this adaptive control algorithm guarantees that the error remains bounded under much less restrictive positivity conditions. Simulation results show that the new adaptive control system is highly robust to the variations in plant dynamics. In particular, the controller can easily stabilize a plant with poles in the right half of the complex plane.
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Furthermore, the performance of the new model-reference adaptive control system is compared to the performance of the adaptive model-following control system. The comparison of results indicates that the new adaptive algorithm is at least as effective as current adaptive designs, which require the satisfaction of much stronger conditions. Finally, the simple and unique approach to adaptive control presented in this thesis shows good promise. for feasible solutions to several more complex problems. The form and content of this abstract are approved. I recommend its publication. Signed. Jan T. Bialasiewicz iv
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LYAPUNOV DESIGN OF A NEW MODEL-REFERENCE by Alexandros John Ampsefidis B.S., University of Colorado, 1988 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Department of Electrical Engineering and Computer Science 1990 .. ............ .. {.
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CONTENTS CHAPTER 1. INTRODUCTION . . . . . . 1 2. DEVELOPMENT OF A NEW MODEL-REFERENCE ADAPTIVE CONTROL SYSTEM . . 2 .1. 2.2. 2.3. Introduction . . . . Model-Reference Intelligent Control System . . . . New Adaptive Control System . . . 2.3.1. Error Equations and the Adaptive Law of the New MRAC System . . 2.3.2. Stability Analysis of the . 5 . 5 . 6 . 9 . 14 New MRAC System . . 17 2 4 Summary 26 3. ADAPTIVE MODEL-FOLLOWING CONTROL SYSTEM 28 3.1. Introduction . 28 3.2. Linear Part of the AMFC System 28 3.3. Adaptive Part of the AMFC System . 33 3.3.1. Basic Principles of Hyperstability Theory . . 38 3.3.2. Conditions for the Hyperstability of the AMFC System 40 4. DYNAMICAL EQUATIONS OF THE PLANT 4.1. Introduction . 4.2. Dynamic Equations of the 45 45 Single Link Manipulator . . 46
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4.3. Dynamic Equation of a D.C. Motor .. 51 4.4. Dynamic Equations of the Plant in the State Space 53 5. SIMULATION RESULTS . . . 55 5.1. Introduction 55 5.2. Numerical Equations of the Plant and the Reference Model 55 5.3. Simulation Results for the New MRAC System 63 5.4. Simulation Results for the AMFC System 5.5. Compqrison of the Two Adaptation Methods Based on the Simulation 7 5 Results 84 6. CONCLUSIONS . . . . . . 86 BIBLIOGRAPHY . . . . . 89 vi
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ACKNOWLEDGEMENTS This study would not have been possible without the cooperation and support of many people to whom I will always be grateful. I especially wish to express my sincere gratitude to my advisor Dr. Jan T. Bialasiewicz for his support and guidance during the preparation of this thesis. I would also like to thank Professor Edward T. Wall for his advice and encouragement, and for his serving on my committee. Furthermore, I would like to thank Dr. William R. Roemish for his encouragement and for serving on my committee. Finally, I would like to thank my family for their financial and moral support which was instrumental in the completion of my studies in the United States.
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CHAPTER 1 INTRODUCTION The purpose of model-reference control is to match the response of a system or plant to that of a reference model. The plant design specifications are included in the model such that the step response of the model will have the specified rise time, overshoot and settling time. A controller can be designed, which uses the model inputs, the model states, and the error between plant and model outputs to generate the appropriate control signals. These control signals, which are the inputs to the plant, are used to drive the outputs of the plant to track the outputs of the model. When the plant parameters are not well-known, it may be desirable to utilize an adaptive control scheme, which adjusts the control law on-line, to reduce the effects of the unknown parameters or plant uncertainties. Therefore, the use of an adaptive controller, if it is properly designed, can result in a satisfactory plant performance with only a limited knowledge of the plant structure and parameters. Model-reference adaptive control systems can be divided into two classes. The first class is composed of the indirect adaptive
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controllers, in which on-line estimates of the plant parameters are used for control law adjustment. The second class, to which the controller designed in this thesis belongs, contains the direct adaptive controllers for which no effort is made to identify the plant parameters In other words, the control law is directly adjusted to minimize the error between the plant and model states. The first direct model-reference adaptive control system was designed by the performance index minimization method proposed by Whitaker [1]. However, this so-called "MIT design rule" could not ensure the stability of the adaptive system. Therefore, subsequent research efforts were directed toward the development of stable adaptive algorithms. One of the first researchers to use the Lyapunov second method to design a stable adaptive controller for single-input/single-output systems was Parks [2]. Direct model-reference adaptive control schemes, designed for multi-input/multi-output systems by using Lyapunov techniques, were developed by Grayson [3], and Winsor and Roy [4]. However, these algorithms require the satisfaction of the Erzberger perfect modelfollowing conditions. In other words, these adaptive controllers function properly only if there exists a certain structural relationship between the plant and the model. 2
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Another adaptive algorithm for multi-input/multioutput continuous systems subject to the perfect modelfollowing conditions was developed by Landau [5]. The stability of this system is ensured by the hyperstability criterion of Popov. Furthermore, in recent years, the concept of the command generator tracker (CGT), which was developed by Broussard [6], has been used to design adaptive controllers, which do not require the satisfaction of the perfect model-following conditions. A paper with this type of controller has been written by Mabius and Kaufman [7]. \ In this thesis a new algorithm for direct modelreference adaptive control of single-input/single-output systems is presented. This algorithm does not require the satisfaction of the PMF conditions. Also, this algorithm, which is designed by using Lyapunov techniques, guarantees asymptotic stability, provided that the transfer function of the equivalent error system is strictly positive real (SPR). In addition, it is shown that the adaptive algorithm guarantees that the error will remain bounded under less restrictive positivity conditions. Furthermore, the new model-reference adaptive control (MRAC) system is extremely simple compared with other control algorithms. Despite its simplicity the new 3
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adaptive controller is at least as effective as the more complex adaptive mechanisms. This fact is demonstrated by comparing the new MRAC system to the adaptive modelfollowing control (AMFC) system. Finally, this thesis is organized in the following manner. The second chapter (Chapter 2) presents the design and stability analysis of the new MRAC system. In Chapter 3 the AMFC system is analyzed. Chapter 4 presents the dynamical equations of the plant (used for the simulation), which is a robotic arm manipulator driven by a d.c. motor. In Chapter 5 the simulation results are presented and the performance of the new MRAC system is compared to the performance of the AMFC system. Chapter 6 presents concluding comments and several proposals on how to conduct further research on the properties of the new MRAC system. 4
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2 OF A NEW MODEL-REFERENCE 2.1 Introduction This chapter presents the development and stability analysis of a new model-reference adaptive control (MRAC) system. The design of the controller is based on Direct Method. The formulation of the design statement permits a suspension of the conditions for perfect model-following (PMF). In other words, it is not required that the structure of the plant resemble that of the model. Therefore, this new adaptive controller should be able to control a wide range of plants. The simplicity and the low order of the adaptive algorithm are distinct features, which represent a significant improvement in adaptive design. In addition, simulation has shown that the new adaptive mechanism.is robust under severe plant parameter changes. Finally, the organization of this chapter is given. Section 2.2 deals with the model-reference intelligent control (MRIC) system [8], which is the foundation of the new adaptive algorithm. Section 2.3, the main contribution of this thesis, describes the development and stability analysis of the new MRAC system. The last part of this chapter,
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section 2.4, gives a presentation of some concluding remarks, which concern the new adaptive algorithm. 2.2 Model-Reference Intelligent Control System The idea of the model-reference intelligent control (MRIC) system was first introduced by J. T. Bialasiewicz and J. C. Proano [8]. The MRIC system, which is shown in Fig. 2.1, consists of three parts. The first part, the Ym + ,+ REFERENCE el ADAP'I'JVE + Yp ' PWIT "' "' I .,. }IODEL lAW Uc U /I' Xe STATE / / ESTUIATOR Fig. 2.1 Model reference intelligent control system. reference model, incorporates the design characteristics of the plant. The purpose of the control system is to constrain the plant step response to track the reference model step response. The second part of the MRIC system consists of the state estimator. The estimator is 6
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described by the differential equation of the reference model with proper feedback. Due to feedback the estimator response is faster than that of the reference model. Therefore, the speed of the adaptation process is improved. The third part of the MRIC system is the adaptive law. This part is designed as a proportional plus integral (PI) controller whose function is to drive the error signal, which is defined as the output of the model minus the output of the plant, to zero. In other words, the adaptive law continuously solves the pole placement problem. To complete the analysis of MRIC system the differential equations, which describe the adaptive system of Fig. 2.1, are considered next. In the following equations the A, B and C are n x n, n x 1, and 1 x n matrices respectively. The plant is described by (2.1) (2.2) where Xp e R n, Y P e I. and u e 1R.. The reference model is described by (2.3) 7
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(2.4) where Xm e lR n, Ym e lR and r e lR. The state estimator equations are (2.5) (2.6) where X8 e lR n and Y e e lR and the error, e2 is defined as (2.7) The adaptive control law is given by (2.8) (2.9) (2.10) where Uc e lll, 8 e lll n and e1 e :JR. Also, the input to the plant may be expressed as u -Uc + I -e T XB + I (2.11) Finally, for convenience it can be assumed, as in reference [8], that Cp = Cm = Ce = C where C = [1, 0, 0 0 ] From simulations it was observed that the application of the MRIC system results is an adaptive system, which is robust under extreme plant parameter variations, even if these variations lead to plant instability.. However, there are two main problems, which 8
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arise when using the MRIC system. The first is that in reference [8] a treatment of the stability of the MRIC system is not given. In other words, it is not formally determined that e1 (see equation (2.10)) will approach zero asymptotically. A further implication of the absence of a formal stability proof is that the designer is not provided with guidelines for the selection of the estimator feedback gains, which would result in a stable system that would force e1 to zero. The second problem, to be observed, is that the adaptive time of the unmodified MRIC system is relatively large. It follows from this that the controller cannot be used to control processes in which the adaptation time is a critical factor. The new MRAC system developed in the following section eliminates both of these problems. 2.3 Hew Model-Reference Adaptive Control System The design of the new MRAC system is based on a modification of the MRIC system. This modification is done by using ideas presented in references [9] and [10]. In order to develop the new adaptive algorithm it is necessary to formulate the new statement of the design problem. The idea of the augmented system, shown in Fig. 2.2, is central to this formulation. The system shown in Fig. 2.2 is a typical feedback control system. The state estimator and the matrix of 9
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+ u PLANT Yp .... .... .... .. ; .. .. +' Uc GAIN Xe STATE / MATRIX ESTJMATOR j Fig. 2.2 Augmented system. gains are located in the feedback path. The state estimator is.simply a filter that supplies the gain controller with the required auxiliary signals. All the necessary concepts have now been defined so that the equation, which defines the behavior of the augmented system can be written. The plant shown in Fig. 2.2 is given by (2.12) (2.13) The state estimator is described by ( 2 .14) 10
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(2.15) and the input to the plant is assumed to be U Uc + I -K( t) Xe + I (2.16) The combination of equation (2.12) with (2.16) results in the following (2.17) The augmented system, with state vector X -[ XpT, X8T 1 T, described by the equations (2.17), (2.14) and (2.13), can be written in the following state space form (2.18) Y -[Cp 0] [ :: l (2.19) Note that Y = Yp. Now, equations (2.18) and (2.19} can be rewritten as follows X -AX+ BI (2.20) Y ex (2.21) with the obvious definitions of the matrices A, B and C. The state vector X is a 2n-vector because it is assumed . that Xp and X8 are n-vectors. Also, the matrices in the 11
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equation for Xp and Xe have appropriate dimensions. The outputs YP, Ye, Y and the inputs r and U belong to the set R. In addition, it is assumed that the plant (described by equation (2.12) and (2.13)) is time invariant and linear with unknown parameters. Since the plant parameters are not known the elements of the gain matrix (refer to Fig. 2.2) are assumed to be time variant. This gain matrix, of dimension 1 x n, is called K(t). Having developed the augmented system equations the design problem can be stated next. The output of the augmented system (which is of order 2n) is required to track the output of a n-th order reference model of the for.m Xm-AmXm + Bmi ym-cmxm (2.22) (2.23) This model represents the desired input-output behavior of the augmented system. Since the augmented system input and the reference model input are the same the tracking problem can be expressed bythe following equation (2.24) Therefore, the purpose of the gain matrix K(t) is to adjust the dynamics of the augmented system so that it performs as a stable reference model. 12
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For the following adaptive system it is assumed that there exists an ideal target system of the form x -A* x + Br which satisfies the equation (2.25) (2.26) (2.27) It is important to note that both the augmented system and the ideal target system have the same input and output matrices. Also, it is assumed that the system described by equations (2.25) and (2.26) has the same order as the augmented system. Moreover, it assumed that the ideal system matrix A* is (2.28) where K is an unknown constant gain . In the stability analysis of the new MRAC system it will be shown that if a gain matrix K exists such that the two positivity constraints are satisfied, the new adaptive control system is asymptotically stable. The reader should note that the value of K is not required to be known, only its existence has to be guaranteed. 13
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Before the stability of the new MRAC system is proven the generalized error equations have to be developed. Also, it is necessary to show how K(t) should be adjusted to drive the generalized state error to zero as t approaches infinity. 2.3.1 Error Equations and the Adaptive Law of the Hew MRAC System The generalized state error is defined as and the output error as (2.29) (2.30) The differential equations for the state error may be obtained by differentiating equation (2.29). That is ex x -X x -A X + A X X ( 2 31) . Substituting the appropriate expression for x and X in (2.31) gives ex A x + Bz A X + A X -AX -BI A (X -X) + (A -A) X (2.32) A ex + (A -A) X where 14
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(A '-A) X[ [ A, l [A, BpK( t) ) Xp LCp Am-LCe LCp Am-LCe xe -I: B,(K -0 K( t)) l [ :: ]-[ B,(K-:(t)) x.l (2.33) --[ :] (K(t) K> x.--B'(K(t) K) x. with (2.34) Therefore, the generalized error state equation (2.32) can be written in the following form: (2.35) Equation (2.35) together with equation (2.30) describes the dynamics of the error system, which is shown in Fig. 2. 3. Also this system, with input (K -K (t)) X8 and output ey, has the following transfer function: (2.36) As stated in the previous section, the adaptive controller must make the augmented system approach the ideal target system asymptotically. In other worqs, K(t) 15
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has to be chosen so that ex approaches zero as t approaches infinity. Xe ... K(l)-K + Fig. 2.3 Equivalent error system. Such an adaptive control law, which is developed based on the principles presented in references [9], [10] and [11], would have the following form Kp( t) e XT T Y e (2.37) (2.38) (2.39) (2.40) where T and T are selected (constant or time-variant) to be positive definite symmetric adaptation coefficient matrices. It will be shown that positive definiteness and symmetricity of these matrices is required for global stability of the adaptive system. The memory of the adaptive mechanism is provided by the integral term K1(t), which is similar to the adaptive gain proposed in 16
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reference [8]. Also, the proportional term Kp(t) and the constant gain matrices T and T make the new MRAC system much faster than the MRIC system. Furthermore, as will be shown in the next section, the proportional term Kp(t) (refer to equation (2.39)) facilitates direct control of the output error ey. Therefore, as indicated in reference [12], the output error can be ultimately reduced to zero under the assumption of a disturbance free environment. A final observation is that the new adaptive algorithm, given by equations (2.37) -(2.40) is extremely simple as compared with many other adaptive control designs. Having developed the generalized error equation and the adaptive algorithm the stability properties of the new model-reference adaptive control system are developed next. 2.3.2 Stability Analysis of the Hew MRAC System The first step in the analysis using a Lyapunov approach is to form a quadratic function, which is positive definite in the state variables of the adaptation algorithm, namely ex and K1(t). A good choice of a positive definite function is V( t) eJ Pex + tx [ ( Kz K) T-1 ( Kz K) T] (2.41) 17
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where P is a n x n positive definite symmetric matrix, T-1 is assumed to be positive definite symmetric matrix (which is the first sufficient condition for stability), and K is a 1 x n unspecified matrix. Note that K is a dummy gain matrix because it appears only in the function V(t) and not in the adaptive control algorithm. Before the time derivative of V is taken, its partition of the following form, is defined (2.42) where (2.43) and (2.44) Note that V (t) is given by (2.45) The derivative of V1(t) is (2.46) Substituting ex from (2.35) into (2.46) gives (2.47) el (A T p + PA.) ex -2 el PB. (K( t) K) xe 18
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The derivative of V2(t) is (2.48) T T But K1 -ey Xe T (equation ( 2. 40)), and T = T since T is symmetric. With this in mind, equation (2.48) becomes (2.49) Since ey is a scalar, equation (2.49) can be rewritten as follows (2.50) Combining equation (2.38) and (2.39) gives (2.51) Substitution of (2.51) into (2.50) gives (2.52) Since tr[M] = M if M is a scalar, equation (2.52) can be written as (2.53) 19
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Noting that ey is scalar the following equation: ey-Cex-(Cex> TefcT (2.54) is obtained from (2.30). Using equation (2.54), equation (2.53) can be rewritten as (2.55) The substitution of (2.47) and (2.55) into (2.45) gives or V(t) -eT(A*rP+PA*)e -2eT(PB*-CT) X X X '(2.56) The next step in the analysis is to state the remaining two sufficient conditions that guarantee the stability of the new MRAC system. Recall that the first condition is imposed on the gain matrix T (T must be positive definite symmetric because V(t) has to be a positive definite quadratic function). The other two conditions ensure that V(t) is negative definite in the error ex. The second condition is that T should be a positive semidefinite symmetric matrix. This requirement constrains the third term of equation (2.56) to be 20
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negative semidefinite. The third condition is that there must exist a gain matrix K (refer to equation (2.28)), which is not needed for the controller implementation, such that the following two equations: A 2' P + P A -Q (2.57) and PB* CT (2.58) are satisfied. In the above equations, P and Q are positive definite symmetric matrices. According to the Kalman-Yakubovitch-Popov's Lemma [13], which is presented in Chapter 4, equations (2.57) and (2.58) are satisfied if and only if the transfer function Z(S) = C ( SI -A*)_, a*, of the error system shown in Fig. 2. 3, is strictly positive real (SPR). Assuming that the conditions stated in the paragraph above hold, and substituting equations (2.57) and (2.58) into (2.56) gives (2.59) From the last equation it is seen that V(t) is negative definite in the error ex(t) and it is negative semidefinite in the augmented state [ex, K1(t)]. Since V(t) is positive definite, the new MRAC system is asymptotically stable. The asymptotic stability of the 21
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adaptive system can be seen using the following lemma of Barbalat [14]. Lemma 2.1. If g is a real function of the real variable t defined and uniformly continuous for t > 0 and if the limit of the integral fat g(t) d't (2.60) as t tends to infinity exists and is a finite number, then limg ( t) -o (2.61) The function V(t) (refer to equation (2.41)) is bounded from below because it is positive definite. V(t) is also a nonincreasing function since V(t) is negative semidefinite. Therefore, V(t) converges to a finite value v. as t approaches og. Then, there exists lim foe V ( t) dt -lim V( t) 1: -v .. V0 (2.62) which is a finite number. To use Lemma 2.1 it still remains to be shown that V(t) is uniformly continuous. In other words, it has to be shown that V(t) is bounded. It can be seen that V (t) is bounded if ex, X8 and X8 are bounded. Xe is the solution vector of a stable linear differential equation (refer to equation (2.14)). 22
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Therefore, Xe and X8 are bounded. In addition, the fact that ex is bounded follows from equation (2.35) because by assumption A* is a stable matrix. Hence, by Lemma 2.1, (2.63) or lim ex -0 and lim ey -o t- t- (2.64) Also, by (2.39), (2.40) and (2.64), since Xe is bounded, it follows that lim Kp(t) -o and lim KI( t) -o (2.65) t- t- Therefore, K(t) given by (2.38) is a bounded gain matrix. Summarizing, it is shown that the new MRAC system is asymptotically stable and that the output error ey tends to zero as t approaches infinity. In other words, the output of the augmented system approaches the output of the model asymptotically. However, as shown above, asymptotic stability is guaranteed if Z(S) = C(SI is strictly positive real. This condition can be relaxed and it is still possible to have global stability, which for all practical purposes is sufficient. The adaptive system is globally stable under the following conditions: 23
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(a) T is a positive definite symmetric matrix (b) T is a positive definite symmetric matrix (c) There exists a gain matrix K (refer to equation (2.28)) such that A* is a stable matrix that satisfies equation (2.57), namely (2.57) where Q and P are positive definite symmetric matrices. It should be noted that the second condition, (refer to equation (2.58)) for a SPR error transfer function, has been relaxed. This condition could be satisfied by the proper choice of the output matrix c. The first condition, for the gain matrix T, assure that V(t) is a positive definite quadratic The last two conditions, stated above, give the following expression for V(t): (2.66) The first and third terms in (2.66) are negative definite. Therefore, if either ex or ey becomes large, one or two of the three terms in (2.66) become negative and dominant, such that V(t) becomes negative. Also V(t) is assumed to be positive definite. Therefore, as it was shown using the Lemma of Barbalat, all states, gains, and 24
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errors are bounded. These values are ultimately bounded by the set (2.67) The third term in (2.66), (2.68) is, as can be seen, a direct result of the proportional gain term Kp(t). Also, the third term is negative definite quadratic with respect to the output error ey, and proportional to the constant adaptation matrix T. Therefore, by selecting relatively large T this term can be dominant even when the output error ey becomes really small. In other words, by choosing relatively large gain matrices T, the error ey can be controlled and reduced arbitrarily. In addition, as stated in reference [10], the effect of the third term is so strong that in some instances the other positivity conditions may not be necessary. Finally, note that the plant used for the simulation is such that the resulting error system cannot satisfy the SPR conditions. However, as will be seen by the simulation results, ey is rapidly reduced to zero and the plant together with the new MRAC system form a globally stable system. 25
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2 4 SWIIIIUlry In this chapter a new MRAC system was developed. The complete new adaptive system is shoWn in Fig. 2.4. REFERENCE Ym ...... / }IODEL +\ e y -/i\ r + u Yp ...... ..... PI.OO / ,.. / +' 'v ADAPTIVE / Xe STATE CAJNS :E:i'TWATOR II\ Fig. 2.4 New model reference adaptive control The equations that describe the operation of the system shown in the above figure are given in section 2.3 and section 2.3.1. Conditions for asymptotic stability and for global stability are given in section 2.3.2. The conditions for global stability are easily met, in theory, by numerous augmented systems. This makes the new MRAC system really versatile. In addition, from sLmulation results, which are presented in Chapter 5, it is shown that the new adaptive controller results in a system, which is robust under extreme plant parameter 26
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Another advantage is the simplicity of the new adaptive algorithm. Finally, in spite of its simplicity, as it will be shown by comparing the new MRAC system to the AMFC system, the new adaptive algorithm is as effective as other more complex adaptive controllers. 27
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CHAPTER 3 ADAPTIVE MODEL-FOLLOWING CONTROL SYSTEM 3.1 Introduction As stated in the previous chapter, the effectiveness of the new MRAC system will be determined by comparing it to an adaptive control system whose design is based on the hyperstability theory. This chapter contains a brief description of an adaptive model-following control (AMFC) system, which was developed by Landau [5]. The AMFC system has two parts: a) a linear part, which is a solution to the perfect model-following control problem (the linear part of the AMFC system is designed for nominal values of the plant parameters), and b) an adaptive nonlinear feedback loop (the design of which is based on the hyperstability theory) that is used to overcome the problem of uncertainty in or variations of the process parameters. 3.2 Linear Part of the AMFC System The linear part of this adaptive control system is developed based on the ideas of linear model-following control (LMFC) systems. The LMFC systems use a reference model (that specifies the required plant dynamical characteristics) for generating part of the control law
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as shown in Fig. 3.1. In this figure Kp and Ku are the feedback and feedforward gains respectively. The signal e is the generalized state error and r is the reference input. Note, that Fig. 3.1 shows a simplified LMFC system. This simplification is based on the assumption that all the states of the plant are available. REFERENCE Xm MODEL + \I/ e II\ r +, u PLANT Xp Ku -/i\ Kp Fig. 3.1 Linear control system. The objective of the LMFC system is to minimize the error, e, between the states of the model and those of the controlled plant. In order to proceed with the design of the controller the principle of perfect modelfollowing (PMF) can be used. The PMF problem is stated 29
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as follows: given the system shown in Fig. 3.1 does there exist gain matrices KP and Ku such that they will force the generalized state error and its derivative to be zero for any input r. To complete the design the formal statement of the PMF problem has to be expressed in mathematical terms (refer to Fig. 3.1). The plant is described by (3.1) and the reference model is given by ( 3. 2) The control input in the equation (3.1) is equal to (3.3) In the equations, XP and Xm are n-dimensional vectors, r is the reference input (m-dimensional), U is the plant input
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(3.4) Second, substituting (3.3) into (3.1) and subtracting the resulting expression from (3.2) gives 0 e -XmXp (3.5) For PMF e = 0 and e ... 0 for all r with Xp eRn. Hence, it is required that ( 3 0 6) Equation (3.6) is satisfied if (3.7) (3.8) Equations (3.7) and (3.8) can be solved for KP if and only if (3.9) or, in other words, if and only if the column vectors of the difference matrix and of the matrix Bm are linearly dependent on the column vectors of the matrix A class of solutions can be obtained by using the left Penrose pseudo-inverse of Bp defined as 31
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(3.10) Combining equations (3.10) with (3.7) and (3.8) gives (3.11) and (3.12) By introducing the expressions for Kp and Ku into equations (3.7) and (3.8) the conditions for PMF are obtained. These conditions, known as the Erzberger conditions are: (I -BpB;) (Am -Ap) 0 (I -BpB;) Bm 0 (3.13) (3.14) Note, that Bp, usually, is a singular square or a rec_tangular matrix. In this case, the conditions ( 3.13) and ( 3 14) mean that (I Bp a;) is orthogonal to Ap) and Bm. Finally, before closing this section some observations should be made. The matrices Ap and Bp used for the design of the LMFC system are constant, nominal matrices. The Erzberger conditions are satisfied if the reference model and plant are structurally related. This concludes the discussion of the linear part of the AMFC system. The next section presents the nonlinear adaptive part of the controller. 32
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3.3 Adaptive Part of the AMFC System As mentioned previously the LMFC systems do not overcome the problem of uncertainty in or variations of the plant parameters. Therefore, the linear controller has to be supplemented by an adaptive mechanism that will make the generalized state error vector approach zero asymptotically when the plant parameters differ from their nominal values. The combination of the linear part of the controller design with the adaptive mechanism results in an AMFC system. For the rest of the analysis the AMFC system that uses the signal synthesis adaptation principle will be considered. This control system is shown in Fig. 3.2. The purpose of the adaptive feedback loop is to produce the gains that result in perfect model-following. But the AMFC system can achieve this task only if such gains exist. In other words, the plant and model system matrices have to satisfy the PMF conditions. Hence, in the following mathematical analysis the assumption of the existence of a solution for perfect is central. Consider the parallel AMFC system with signal synthesis shown in Fig. 3.2. The differential equations that describe the operation of this system are given next. The linear time invariant plant is described by (3.15) 33
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REFERENCE Xm MODEL -,1, + Ul + u Xp r -"" .... PLANT "' Ku ... ,; -'I' +I "u2 Kp ;' -.____ ADAPTATION fE-MECHANISM ..... Fig. 3.2 Parallel AMFC system with signal synthesis. the reference model is given by (3.16) the generalized state error is defined as e-X -X m P (3.17) and the input U to the plant consists of two parts (3.18) u1 is the linear part, which is equal to (3.19) 34
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where KP and Ku are constant matrices designed for some nominal plant parameter values. u2 is the nonlinear adaptive part, which is expressed as U2 !iKp (e, t)Xp + !iKu (e, t) I (3.20) where !l.Kp and !iKu are used to assure that the generalized state error e goes to zero under certain conditions. In the above equations Xp and Xm are n-dimensional vectors, r is an m-dimensional reference input, u1 and u2 are m, dimensional vectors, and Bm, Ap, Bp, Kp, Ku, !i Kp and ll Ku are matrices of appropriate dimensions. The general form of the adaptation mechanism, which is developed by Landau [5], may be expressed by the following equations: v-De !J.Kp (e, t) !J.Kp (V, t) e (V, t, 't) d't + (V, t) and !l.Ku (e, t) !l.Ku (V, t) L t 1Jr1 (V, t, 't) d't + 1Jr2 (V, t) (3.21) (3.22) (3.23) where the constant matrix D and the vector functions 1Jr1 1Jr2 are picked so that the hyperstability conditions, presented later, are satisfied. Note that the integral terms of equations (3.22) and (3.23) constitute the memory of the adaptation mechanism, while and w2 are transient terms used to speed up the 35
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adaptation process. These transient ter.ms vanish when e is equal to zero. Before the development of the AMFC system design is further considered the hypotheses that make the design possible are stated. These two hypotheses are: 1. Am, Bm, Ap and BP belong to the class of matrices, which satisfy the perfect modelfollowing conditions. 2. Ap and Bp are assumed to be constant during the adaptation process. The next step in the analysis is the development of the equivalent block of the AMFC system. To formulate this block the differential equation of the generalized state error has to be developed. By differentiating equation (3.17) and by substituting equations (3.15) and (3.16) into the resulting equation from the differentiation, it follows that 0 0 0 e Xm Xp AmXm + Bmi ApXp BpU (3.24) Substituting equations (3.18), (3.19) and (3.20) into (3.24) results in 0 e-Ame +(AmAp)XpBp(U1 + U2 ) + Bmr (3.25) 36
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As stated in the hypotheses presented in the previous paragraph it is assumed that a solution exists for the PMF problem. Therefore, (3.26) (3.27) where Kp and Ku are the unknown values of Kp and Ku that assure PMF. Now, the combination of (3.25) with (3.26) and (3.27) gives e -A.me + Bp [Kp Kp 11Kp (e, t)] Xp (3.28) + Bp [Ku Ku ll Ku ( e, t)] I Let Aa K!'-Kp (3.29) A1 Ku Ku (3.30) With the above two expression equation (3.28) becomes Equation (1) defines an equivalent feedback system described by the following equations: e -A.m e + B P W1 ( 3 3 2 ) v-De (3.33) W--W1 [llKp(e, t) .. -A0]Xp + [11Ku(e, t) (3.34) where the vector functions ll Kp ( e, t) and 11 Ku ( e, t) are given by equations (3.22) and (3.23). It is worth noting 37
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that the generalized state vector is the forcing signal of the equivalent feedback system. Now that the equivalent feedback system has been developed, the hyperstability theory can be used to find an appropriate constant output matrix D and vector functions A Kp { e, t ) and A Ku { e, t ) which will guarantee that the generalized state error e goes to zero asymptotically. Before proceeding with the hyperstability analysis of the equivalent system described by equations (3.32) -(3.34) some basic concepts concerning hyperstability will be presented. 3.3.1 Basic Principles of Byperstahility Theory Consider the closed-loop system (shown in Fig. 3.3) having a linear time-invariant feedforward block and a \Yl LINEAR v / "w BLOCK NONLINEAR / BLOCK ..... Fig. 3.3 Standard multivariable nonlinear feedback system. 38
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nonlinear feedback block. The feedforward block is described by X-AX+ BW1 (3.35) v-ex (3.36) and the feedback block is given by w --W1 -f < v, -; t > (3.37) where X is a n-vector, w1 and V are the input and the output, respectively, of the feedforward block (both mdi.mensional) ; A, B and C are matrices of appropriate dimension; the pair (A, B) is completely controllable; the pair (C, A) is completely observable; and f(V, -;, t) is a vector functional. Furthermore, the feedback block is such that it satisfies the Popov integral inequality fo:r all t1 > o ( 3. 38) where y0 is a finite positive constant. The following theorem gives the condition that the linear feedforward block (shown in Fig. 3.3) must satisfy for the closedloop system to be asymptotically stable. Theorem 3.1 [15]. The necessary and sufficient condition for the feedback system described by equations (3.35), (3.36), (3.37) and (3.38) to be hyperstable is as follows: The transfer matrix 39
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H ( S) C ( S I -A) -l B (3.39) must be strictly positive real. A transfer matrix is strictly positive real if it satisfies the conditions stated in the following Kalman -Yakubovitch-Popov's Lemma [13]. Lemma 3.1. The transfer matrix H(S) is strictly positive real if and only if c-BTP (3.40) where P, a symmetric positive definite matrix, is a solution of the Lyapunov equation PA +A Tp -Q where Q is a symmetric positivedefinite matrix. 3.3.2 Conditions for the Byperstability of the AMFC System (3.41) Now that the basic hyperstability tools have been presented the stability of the equivalent feedback system of the AMFC system can be proven. First, vector functions A Kp ( e, t) and A Ku ( e, t) (refer to equations (3.22) and (3.23)) have to be found, which will result in a feedback block that satisfies the Popov integral inequality. Landau [5], developed vector functions that satisfy equation (3.38). These vector functions are 40
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ll.Ku (e, t) ll.Ku (V, t) t MV(Nr) Tdt + MV(Nr) T (3.43) where F, M, G and N are symmetric positive definite matrices and F, M are positive semidefinite matrices with appropriate dimensions. This type of vector function results in an adaptation scheme, which is called proportional plus integral adaptation. The complete equivalent feedback system when ll.Kp and ll.Ku are given by equations (3.42) and (3.43) is shown in Fig. 3.4. The expression for ll. Kp and ll. Ku can be used to show that the resulting feedback satisfies the Popov integral inequality. Substituting equation (3.34) into (3.38) gives (3.44) Jtl + 0 v T ( ll. Ku ( e I t ) -Al ) r d t -y2o t > 0 where Vis given by equation (3.33). Combining equation (3.44) with equations (3.42) and (3.43) results in 11 (0, t1 ) vT[fo e (FV(GXp) Tdt A0 ]xpdt 41 (3.45)
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The last two integrals in (3.45) are greater than or equal to zero because F, M are positive semidefinite matrices and G, N are positive definite symmetric matrices. Therefore, for Popov inequality to hold it is + Bp e D T M r F Xp G Fig. 3.4 Equivalent feedback representation of AMFC system. 42
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sufficient that each of the first two integrals in (3.45) be greater than a negative finite constant. Using the properties of the positive definite symmetric matrices, the first integral of (3.45) can be written as: where X G X E ( F1T) -1 Ao ( G1) -1 p... 1 pi But the integral I1 can be described also by: (3.47) The second integral in (3.45) also verifies such an inequality and therefore, the feedback block satisfies the Popov integral inequality. The other requirement, for the closed loop to be hyperstable, is that the transfer matrix of the feedforward block must be strictly positive real (SPR). The feedforward transfer matrix, see Fig. 3.4, is 43
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(3.48) This transfer matrix can be made SPR if an appropriate output matrix D is chosen. Assuming that Bp has the same structure as Bm the matrix 0 DBpP (3.49) where is the nominal value of Bp, makes H(S) strictly positive real [5]. The Pin (3.49) is the solution of the Lyapunov equation (3.50) where P and Q are positive definite symmetric matrices. In conclusion, the equivalent feedback system that is shown in Fig. 3.4 and is described by equations (3.32) -( 3. 34) is hyperstable if ll Kp and ll Ku have the form expressed by equations (3.42) and (3.43) and if D P. Finally, the fact that the closed-loop system is hyperstable means that the generalized state error approaches zero asymptotically. 44
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CHAPTER 4 DYNAMICAL EQUATIONS OF THE PLANT 4.1 Introduction As mentioned earlier, the dynamic performance of the new model reference adaptive control system will be compared to the performance of the adaptive modelfollowing control system. Both the new MRAC system and the AMFC system will be used to control a plant with uncertain parameters. The plant consists of a single link robotic manipulator driven by a d.c. motor. The uncertainty of the plant parameters arises from the fact that the manipulator dynamics change when the payload changes. In this design study an adaptive controller is used to compensate for the resulting plant parameter uncertainties. This chapter, which presents the derivation of the plant equations, is organized as follows. Section 4.2 deals with the dynamic equation that describes the motion of the single link manipulator. Section 4.3 contains a brief discussion of the dynamics of a d.c. motor. Finally, section 4.4 presents the state space equation of the plant.
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4.2 Dynamic Equations of the Single Link Manipulator The plant to be controlled is shown in Fig. 4.1. It consists of a rigid link coupled through a gear train to a d. c. motor. As seen from the figure e l and em denote Fig. 4.1 Single link robot. the angular position of the link and the angular position of the motor shaft respectively. Also, the gear train has a gear ratio of n : 1. Therefore, the relation between eland em can be expressed as l 81-e n (4.1) Having these basic definitions the equation of motion of the single link manipulator can be obtained by using the method of Lagrangian mechanics. Before developing the 46
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dynamic equation of the plant, a brief discussion on Lagrangian mechanics is presented. The fundamental principle of the Lagrange equation is the representation of the system by a set of generalized coordinates 8 1 ( i -1, 2 . r) one for each independent degree of freedom of the system. By the right selection of the generalized coordinates the Lagrange equation can be to incorporate completely the constraints that exist due to the different, interconnected parts of the mechanical system. After having defined the generalized coordinates, the kinetic energy K is expressed in terms of these coordinates and their derivatives. Also the potential energy V may be expressed in terms of the generalized coordinates. Note that the potential energy V is not a function of the derivatives of these coordinates. Next, given K and V the Lagrangian function may be written as follows . LK(01 Or, 01 Or)-V(01 82 ) (4.2) Finally, the desired equations of motion are derived using Euler-Lagrange equation i-1,2, ... ,r (4.3) where 0; denotes generalized forces (forces and torques) that are external to the system. Note that these 47
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generalized forces, O;, are not related to the scalar potential function. Now, the analysis of the system shown in Fig. 4.1 can proceed. Note, that the mechanical system selected has one degree of freedom; therefore, one generalized coordinate is needed, and this coordinate is chosen to Referring to Fig. 4.1 it is seen that the kinetic energy of the system may be expressed as (4.4) where Jm and J1 are the rotational inertias of the motor and link, respectively. Combining equation (4.1) with (4.4) one obtains (4.5) The potential energy of the system is given by v -M g 1 < 1 -cos e 1 > (4.6) where M is the total mass of the link (link plus payload), 1 is the distance from the joint axis to the link center-of-mass (refer to Fig. 4.1) and g is the acceleration due to gravity. Now, by substituting 48
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equations (4.5) and (4.6) into equation (4.2) the Lagrangian: is obtained. Keeping in mind that the generalized coordinate of the mechanical system is 61 equation (4.3) can be expressed as (4.7) (4.8) where is the sum of the torques acting on the system. Then the partial derivative of equation (4.7) with respect to 61 is (4.9) Taking the time derivative of equation (4.9) results in (4.10) Also, the partial derivative of the Lagrangian L with respect to 61 is equal to aL 1 e ae--Mg Sl.n 1 1 (4.11) Substituting (4.10) and (4.11) into (4.8) gives the equation of motion of the mechanical system, which is expressed as 49
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(n2 Jm + J1 ) a 1 + Mg 1 sin a 1 1: (4.12) The generalized force 1: consists of the motor torque 1:m . and the damping torques Bmam and B1a1 Reflecting the terms 'T:m and Bmam to the link shaft yields . 1:-n1:mnBmamB1a1 (4.13) or, by using equation (4.1), (4.14) Combining equation (4.14) with (4.12) the following equation of motion is obtained (4.15) Equation (4.15) is nonlinear in the variable a1 However, it can be linearized and simplified by making the following three reasonable assumptions: a) the nonconservative damping torques are small compared to 1:m (the motor torque); therefore, they can be neglected b) 61 is assumed to be in the range for which the approximation sina 1 .. a 1 is reasonable c) the mass of the link is concentrated at the center of mass at a distance 1 from the joint; 50
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therefore, the rotational inertia of the link is given by (4.16) Based on the above three assumptions (4.15) can be expressed as (4.17) As it was stated at the beginning of this section it is assumed that the single-link manipulator is driven by a d.c. motor. Therefore, in order to complete the dynamic equation of the plant the dynamics of the d.c. motor have to be included. The next section deals with the analysis of the dynamics of a d.c. motor. 4.3 Dynamic Equation of a D.C. Motor For the following analysis a separately excited d.c. motor is considered. The simplified, ideal model of such a motor is shown in the Fig. 4.2. It is usually assumed +0------Vv I Ra + Vs If ___,. I Fig. 4.2 Schematic diagram of a separately excited d.c. motor. 51 R!
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that Lag and Lff (the self-inductance of the armature circuit and the field circuit respectively) can be neglected. Assuming that the field current If is constant, the torque developed in the motor and the back e.m.f. are given by (4.18) (4.19) where ia and are the armature current and angular velocity of the motor and Km = Kfif is a constant. Note that If is the field current of the motor. The voltage equation for the armature circuit is where R8 is the armature resistance. Substituting the expression of ea from equation (4.19) into equation (4.20) results in or i a Substituting the expression of i8 given by equation (4.22) into equation (4.18) gives 't m 52 (4.21) (4.22) (4.23)
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. Using the equality Calmem (where em is the angular position of the armature) equation (4.23) becomes (4.24) The last equation represents the dynamics of the d.c. motor. Having this equation the complete dynamics of the plant can be obtained. 4.4 Dynamic Equations of the Plant in the State Space The complete dynamic equation of the process can be obtained by combining equation (4.17) with (4.24). As a result the following equation is obtained: Using (4.1) equation (4.25) can be written in the following form or Let F-G-Mgl 53 (4.25) (4.26) (4.27)
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Then (4.27) can be written as E6 1 + F6 1 + G6 1 -HVs (4.28) or (4.29) Defining a state space vector X = [X1 X2]T with X1 = em and x2 -em, equation ( 4. 29). can be expressed in the following state space form: 0 G E 0 (4.30) Note that E and G are functions of M, which is the weight of the link plus the payload. Therefore; when the load that the manipulator has to lift changes, the values of E and G will change. In addition, by referring to equation (4.29) it is seen that all the coefficients of this differential equation will be affected by the load changes. In other words, the dynamical behavior or the plant is directly related to the payload. Therefore, there is a need of an adaptive controller that will make the plant behave in a predetermined way, independently of the weight of the link. 54
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CBAP'rER 5 SIMULA'riON RESUL'rS 5.1 Introduction In this chapter several simulation results are given to illustrate the effectiveness of the new MRAC system with that of an AMFC system. Both adaptive systems will be used to control a plant consisting of a single link manipulator driven by a d.c. motor through a gear train. The purpose of the adaptive controller is to constrain the plant to respond in a prescribed manner despite the uncertainty in the plant parameters. 5.2 Numerical Equations of the Plant and the Reference Model As mentioned in Chapter 4 the dynamical behavior of the single link manipulator changes when the weight of the payload varies. The plant equations given below show how the variations of the weight of the payload affect the plant dynamics. It is assumed that the d.c. motor, which drives the single link manipulator, has the following constants: Km 0 071 N m I A
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Ra-1. 6 Q Also, it is assumed that the ratio of the gear train is 20 : 1. The numerical equations of the plant are derived for three different payloads. The distance of the center of mass of the link to the joint is assumed to be 1 = 0.15 m for all three cases. In the first case it is assumed that the total mass of the link is M = 1 Kg. Then equation (4.30) becomes [ 0 x-37.69 (5.1) Since the output of the plant is the angular position of the link, the output equation is Y-[1 0] X (5.2) Using equations (5.1) and (5.2) the transfer function of the plant is e 1 ~~ V8 (S) 22.82 82 + 32. 38 + 37.69 ( 5. 3) The step response of this transfer function is shown in Fig. 5 .1. In the second case it is assumed that M = 3 Kg. Then equation (4.30) becomes 56 ~~
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[ 0 x--52.5 (5.4) and, for this case, the transfer function of the system is 10.6 (5.5) S2 + 15S + 52 5 Also, the step response of this transfer function is shown in Fig. 5.2. Finally, in the third case considered here, the total mass of the link is assumed to be M = 7 Kg. Then, equation (4.30) gives [ 0 x--59.14 (5.6) Equation (5.6) together with the output equation (5.2) can be expressed in transfer function form as e 1 (s) V9 (S) 5.11 S2 + 7 24S + 59 .14 (5.7) The step response of this transfer function is shown in Fig. 5.3. As it is seen from Figs. 5.1 through 5.3 the dynamical behavior of the single link manipulator changes drastically whenever the total mass of the link changes. However, in practical use it is usually required that the manipulator link respond in a specified manner regardless of the changes in payload. In other words, the 57
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manipulator has to have a predetermined step response, which is independent of the mass of the payload. The ideal step response of the manipulator can be expressed as the step response of a well-designed reference model. In developing the equations of the reference mode,! the designer should keep in mind that the manipulator must lift the payload in a relatively short time with no overshoot. Therefore, the transfer function of the reference model is chosen to be e 1 ~~ V8 (S) (1)2 ll 25 S2 + lOS+ 25 (5.8) where the damping ratio is equal to one ({1) and the natural frequency is equal to 5 rad/s = 5 rad/sec). Equation (5.8) can be written in state space form as follows: [ 0 x--25 with an output equation given by (5.2). The step (5.9) response of the reference model is given in Fig. 5.4. In conclusion, it should be stated that the purpose of the adaptive mechanism is to force the output of the plant to track the output of the model. In other words, the adaptive controller tries to minimize the error, which is defined as the output of the reference model minus the output of the plant. 58 ~~
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co 0 IJ") v n N ..- . . 0 0 0 0 0 (pcJ) JCJn5uy 59 I c c
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; I n 0 CD &t'l n 0 CD &t'l n c N N ...-...-...-...-0 0 0 c . . 0 ci ci 0 0 0 0 0 0 c (pcJ) UO!t!SOd JOjn5uy 60
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Ltn L.....: r-,. Ftn I 0'1 ... ro ::.::: c ""' i II r (.) Etn! ::=: rj Q) 112 c 0 c. 112 Q) Sol I c. Q) .., a:r c ra I"""! 1""'1 t.n Eo 0'1 ..... 1":1:.4 0 N 0 DO tO N 0 oor-oor-0 q 0 0 0 . . . 0 0 0 0 0 0 0 (poJ) .Jo1nouv 61
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Lr.n c f-o i c IIJ") Ln G.l co io s: Fn.-0 c. i tJ co c C1) G.l en ,.. c. NC1) G.l F oE .j,J &a i-.-f G.l eN '0 0 e G.l F..; tJ c Eo G.l ,.. G.l r...; G.l c:: r c tt.: Ill ,o r I 0'1 r o..j ro s:r.. I o 0 N 0 co tO v N 0 0 0 0 0 0 0 0 ..... ..... 0 0 0 0 0 (pcJ) JoJn5uv 62
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5.3 Simulation Results for the Hew MRAC System Before presenting the simulation results, a brief discussion is given on how the feedback gains of the state estimator should be chosen. The choice of the feedback gain matrix L (refer to equation (2.14)) must guarantee the existence of a constant gain matrix K, which will make A* a stable matrix that satisfies equation (2.57). In the case of a full order estimator A* can be written as (5.10) where Cis the identity matrix C = diag [1, 1, 1], L = diag [ 11 12 . In order to clarify the procedure for choosing the gain matrix L a numerical example is given, which uses the equations of the plant and the reference model presented in this chapter. -When the total mass of the link is 1 Kg the numerical expression for A* can be written by using equations (5.1), (5.9) and (5.10). This expression is 0 1 0 0 -37.69 -32.3 22. e2.1C1 A -(5.11) 11 0 -11 1 0 12 -25 -10-1 2 63
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First, the values of 11 and 12 must be such that the submatrix A;2 (note A;2 -Au. -LC) is a stable matrix. If L is chosen to be equal to diag [10, 2] (or 11 = 10, 12 = 2) the matrix which is equal to (5.12) is a stable matrix. The eigenvalues of are -11.00 j 4.89. Next, the following question has to be addressed. Given the values for 11 and 12 does there exist values for K1 that will make A* a stable matrix? For example, with 11 = 10 and 12 = 2 there are many possible gain matrices R:-[K1 K2 ] that will result in a stable matrix A*. With K =[2, 2] A* is equal to the following matrix: 0 -37.69 10 0 1 -32.3 0 2 0 0 45.64 45.64 -10 1 (5.13) -25 -12 The A* matrix given above is stable, because the eigenvalues of A* are located in the left half part of the complex plane. That is the eigenvalues of A* are -14.475, -33.849, -2.988 j 3.786. Next, the designer must verify that A* satisfies equation (2.57). This may be accomplished by selecting a symmetric positive definite matrix Q. Then, equation (2.57) can be solved 64
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for P, which is a positive definite symmetric matrix. By choosing Q equal to diag [2, 2, 2, 2] and substituting for A* the numerical expression given by (5.13) the following equation (which is equation (2.57)) (5.14) can be solved for P. The matrix P, which satisfies equation (5.14), is the following positive definite symmetric matrix: 5.8673 0.1573 P-0.4927 0.3036 0.1573 0.0396 0.0099 0.0616 0.4927 0.0099 0 I 5715 -0.1704 0.3036 0.0616 -0.1704 0.3035 (5.15) Finally, the designer must make sure that the chosen gain matrix L guarantees the existence of the required K for all possible plant matrices and Bp (refer to equation (5.10)). Therefore, it must be determined if the gain matrix L is adequate when the total mass of the link has the values 3 Kg and 7 Kg. For the 3 Kg case (equation (5.4)) and with the same L and K gain matrices the A* is found to be equal to: 0 1 0 0 A --52.5 -15 21.2 21.2 (5.16) 10 0 -10 1 0 2 -25 -12 The matrix given above is stable because its eigenvalues are located in the left hand plane. The eigenvalues of 65
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A* are: -2.635 j 5.473 and -15.865 j 5.412. Also, by substituting A* and Q = diag [2, 2, 2, 2] in equation (5.14) it is found that there exists a symmetric positive definite matrix P that satisfies equation (5.14). This matrix P is equal to: 7.7615 0.2006 0.9533 -0.0594 0.2006 0 0 0917 0.0179 0.0872 (5.17) P -0.1342 0.9533 0.0179 0.4735 -0.0594 0 0 0872 -0.1342 0.2262 Finally, it must be checked if the matrix A* satisfies the two conditions when the total mass of the link is 7 Kg. The A* matrix given below, 0 1 0 0 -59.14 -7.24 10.22 10.22 A* 10 0 -10 1 (5.18) 0 2 -25 -12 was obtained by substituting the appropriate matrices from equations (5.6) and (5.9) into equation (5.10). Also the gain matrices K and L used were the same as those used for the 3 Kg and 1 Kg cases. The matrix A* is a stable matrix because the eigenvalues of A* are -1.884 j 6.842 and -12.736 j 5.868. In addition, there exists a positive definite symmetric matrix P that satisfies equation (5.14) where matrix A* is given by the expression (5.18) and Q is equal to diag [2, 2, 2, 2]. The resultant matrix P is equal to: 66
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P 13.975 0.2358 1.2946 -0.3i82 0.2358 0.2007 0.0238 0.1068 1.2946 0.0238 0.3988 -0.1098 -0.3182 0.1086 -0.1098 0.1667 Summarizing, it can be said that this analysis (5.19) indicates that with L = diag [10, 2] the controlled system will be globally stable if the constant gain matrices T and T of the adaptation mechanism are symmetric positive definite. Having shown that the controlled system is stable, the system can be simulated on a computer. The adaptive gain matrices used for the following simulation results were T = diag [200, 600] and T = [600, 600]. The estimator feedback gain matrix was L = diag [10, 2]. Furthermore, the command input to the controlled plant was a step of amplitude 1 radian, or in other words, it was required that the manipulator link would rotate from the initial angular position e 1 0 rads to the final angular position e 1 -1 :rad. The simulation results for the plant when the total mass of the link is 1 Kg (equation (5.1)) are shown in Fig. 5. 5 and Fig. 5. 6.. Fig 5. 5 shows the position response of the plant and the reference model. From this figure it is seen that the output of the plant tracks perfectly the output of the model. Fig. 5.6 shows the output error (output of the plant minus the output of the model). From this it is observed that the maximum 67
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position error is 3. 8 x 10-3 radians and that the adaptation time is approximately 1.5 seconds. The simulation results for the plant when the total mass of the link is 3 Kg (equation (5.4)) are shown in Fig. 5.7 and Fig. 5.8. From Fig. 5.7 it is seen that the output of the plant tracks really closely the output of the model. The maximum position error is 9.1 x 10-3 radians and the adaptation time is around 2.2 seconds (refer to Fig. 5.8). The third and last simulation was performed for a plant with a link of total mass of 7 Kg (equation (5.6)). The simulation results for this case are shown in Fig. 5.9 and Fig. 5.10. As seen from Fig. 5.9 the output of the plant is almost the same as the output of the model. Also, it can be observed from Fig. 5.10, that the maximum position error is 23 x 10-3 radians and that the adaptation time is 2.6 seconds. The simulation results indicate that the new MRAC system performs really well. Also, it should be noted that the maximum position error and the adaptation time can be further reduced by choosing adaptive gain matrices T and T with larger coefficients. On the other hand, if the coefficients of the matrix T are smaller the time response of the controlled plant is smoother. 68
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0'1 \0 ,-.... -o 0 1.20 1.00 00.80 c 0 u; 0.60 0 a. ... 0 "5 0.40 Ol c c( 0.20 Model Output Plant Output 0.00 -rrrrr)-111-,--JI-r-rt-,-rr-1 I I I I rry-1-1-r-rr-rrrrT-rrrq-y-rrrrrrJ-q 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 lime (sec) Fig. 5.5 Angular position response (M = 1 Kg).
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0 0 0 0 0 0 CD tO N -0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (poJ) JOJJa 0 0 0 0 0 N tO 0 0 0 0 0 0 0 0 0 0 0 0 0 I I I uoH!sod JoJn6uv 70 0 CD 0 0 0 I r I" r f [ I I f f f c c ... c c I
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-...J ...... ""'"' "U 0 1.20 1.00 ...=.o.ao -c 0 --tJ c;; 0.60 0 a. '-0 3 0.40 Ol c <( 0.20 Model Output Plant Output 0.00 -r I I I I I I I I II I I I ...,-r-1 I I I I I I ljl-11,-rrl I I I I Ill I I I I I 1,-1-rTl 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 lime (sec) Fig. 5.7 Angular position response (M = 3 Kg).
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0 0 0 0 0 0 c N (() 0 (() C'o ..... 0 0 0 0 0 .... 0 0 0 0 0 0 c . . . 0 0 0 0 0 0 c I I I (poJ) JOJJe UO!t!SOd JOfn5U'v' 72
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-.J w """"' "U 0 1.20 1.00 00.80 -c 0 0.60 a. L 0 :J 0.40 {J1 c -<( 0.20 Model Output Plant Output 0. 00 -r 1 n-..,-. 1 1 1 1 1 1 -rrrrrrrr-rl_,-,-,-,-,-r-re 1 1 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Time (sec) Fig. 5.9 Angular position response (M = 7 Kg).
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0.0480 0.0320 ,........ "U 0 L. ..._ 0.0160 L. 0 L. \.. Cl) c 0.0000 0 4::i --...I IJ) 0 0. -0.0160 L. 0 :J Ol !i -0.0320 i -0.0480 1-rrrrr-rr.,..,.-rrr I I I 1....-rr I I I I I 11-1-1-1-rrrrrr-rrrTTTill I I I q 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Time (sec) Fig. 5.10 Angular position tracking error (M = 7 Kg).
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5.4 Simulation Results for the AMFC System The simulation results will be presented after a brief discussion on how the gains Kp, Ku and D were calculated. For these calculations the nominal plant parameter values were the values of the plant with link of total mass of 2 Kg. This plant is given by the following state space equation: [ 0 x--47.88 (5.20) The reference model equation was developed in section 5.2 and it is given by equation (5.9). Using equation (3.10) the pseudo-inverse B; of Bp can be calculated as follows: -[ 0 0. 0685] 14 59 ] [ 0 ll-l [ 0 14 59 ] 14.59 (5.21) Then, using equation (3.12) the gain Ku is calculated (5.22) Also, the gain matrix KP can be found as follows: 75
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... -[0 0.0685]([ 0 -25 -[-1.567 -0.721] (5.23) Finally, in order to find the gain matrix D, equation (3.50) must be solved for P which is a positive definite symmetric matrix. With [ 0 A -m -25 and Q-[: it is found from (3.50) that P is equal to Now the matrix D is [0.375 P-0.02 0 0 02] 0.007 [0.375 D BJ P -[ 0 14. 59 ] 0.02 -[ 0 0 292 0 0 102] 0 0 02] 0.007 (5.24) (5.25) In simulating the AMFC system, the input to the controlled plant was a step of amplitude unity. In other words, it was required that the link would move from the initial angular position a 1 0 radians to the final angular position 81 -1 radian. Also, for all three simulation case studies the gain N was picked to be equal to one (N = 1), the matrix G was chosen to be equal to diag [1, 0.001] (G = diag [1, 0.001]), and the rest four adaptive gain matrices were: F = M = 500, F-M-28. 76
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(F, F, M, M, G and N are the gain matrices of the adaptation mechanism described by equations (3.42) and (3.43f). The simulation results for a plant with a link of total mass of 1 Kg are shown in Fig. 5.11 and Fig. 5.12. From Fig. 5.11 it is seen that the output of the plant tracks the output of the model closely. From Fig. 5.12 it can be seen that the position error (which is defined as the output of the model minus the output of the plant) has a maximum of -7.9 x 10-3 radians. The minus sign indicates that, in this case, the manipulator will lift the payload with a small overshoot. Also, from Fig. 5.12 it is seen that the adaptation time is around 2.1 seconds. The results of the simulation for a 3 Kg link (total mass) are shown in Fig. 5.13 and Fig. 5.14. As seen from Fig. 5.13 the output of the plant is almost the same as the output of the model. Also, it can be observed, from Fig. 5.14, that the maximum position error is 8 x 10-3 radians and that the adaptation time is 2.15 seconds. Finally, the simulation results for a plant with a link of total mass of 7 Kg are shown in Fig. 5.15 and Fig. 5.16. Fig. 5.15 indicates that the output of the plant tracks the output of the model fairly well. From Fig. 5.16 it can be seen that the maximum position error is 40 x 10-3 radians and that the adaptation time is 2.45 seconds. 77
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..., (X) _... o 0 1.20 1.00 c 0 :p .iii 0.60 0 a. L 0 :; 0.40 (]) c -<( 0.20 -----Model Output Plant Output 0.00 -lrrlll I I I rrl.,.--.--rJ"'"rrrrr-rrr-1 I I I I 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Time (sec) Fig. 5.11 Angular position response (M = 1 Kg).
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0 0 0 0 0 0 0 0 0 0 c 0 co tO -.:t' N 0 N -.:t' tO co c -0 0 0 0 0 0 0 0 0 .... 0 0 0 0 0. 0 0 0 0 0 c . . . . 0 0 0 0 0 0 0 0 0 0 c I I I I I (poJ) JOJJe UO!i!SOd JOJnOU'v' 79
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Q) 0 ,-.. "0 0 1.20 1.00 00.80 c 0 0.60 a. L. 0 :; 0.40 o 0.20 -Model Output Plant Output 0.00 -r I I I 1,--rt_l_'l I I I I I I I I I rr1 I I I I I I I I I I 1.,.,-,-.-,-,--r.-.-rrTTl 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Time (sec) Fig. 5.13 Angular position response (M = 3 Kg).
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0 0 0 0 0 0 c N cc ...r 0 ...r a:J C' ...... 0 0 0 0 0 ... 0 0 0 0 0 0 c . . 0 0 0 0 0 c I I I (pcJ) UOfHSOd JCJnOU\( 81
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co 1\J -1.20 1.00 "U 0 .t:-0.80 c 0 :iJ 0.60 a. L 0 :; 0.40 Ot c < 0.20 J II Model Output Plant Output 0.00 -p-rrrrrrrrr,....--n-[1-rrrl 1 1 1 1 1 n-1.-,-,-rn-rn--rrn-r-,-fll--. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4-.0 4.5 5.0 Time (sec) Fig. 5.15 Angular position response (M = 7 Kg).
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--0.0480 0.0320 ,........ "'0 u L '--' 0.0160 L 0 L L. GJ c 0.0000 0 IX) 1/) 0 w a. -0.0160 L 0 :J {J1 .!i -0.0320 -0.0480 -J-1--rl"l"! I I I I I I I I I I I I I I I I I I I I I rr1 I I I I rrrrrTflllll I I I I I 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Time (sec) Fig. 5.16 Angular position tracking error (M = 7 Kg).
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5.5 Comparison of the Two Adaptation Methods Based on the Simulation Results Before, the simulation results of the new MRAC system are compared to those of the AMFC system, it should be noted that the new adaptive algorithm is simpler than the algorithm of the AMFC system. Specifically, for implementing the adaptive algorithm of the AMFC system a set of seven gains, and three integrators are needed. However, the new adaptive algorithm uses only a set of four gains and two integrators. Keeping this in mind the simulation results can be compared. From the simulation results it can be concluded that the effectiveness of the new MRAC system is similar to that of the AMFC system. When the plant is controlled by the new MRAC system the maximum position error, on the average, is smaller than that obtained when the plant is controlled by the AMFC system. Also, the adaptation times are almost the same with those of the AMFC system. On the other hand, the position response of the plant, when it is controlled by the AMFC system, is smoother than that obtained by the new MRAC system. However, since the amplitude of the oscillations that occur at the beginning of the adaptation (for the new MRAC system) are small they will not affect the performance of the controlled plant. In conclusion, it can be said that the 84
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new MRAC system (which employs a simpler adaptation mechanism than that of the AMFC system) performs at least as good as the AMFC system. 85
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6 CONCLUSIONS In this thesis the development and testing of a new model reference adaptive control system for singleinput/single-output systems have been presented. For the application of this controller it is not required that the perfect model-following conditions be satisfied. Therefore, the plant does not have to be structurally similar to the reference model. Furthermore, the new adaptive algorithm is simpler as compared with already existing adaptive schemes. The design of the new control algorithm is based on Lyapunov techniques. Asymptotic stability is assured, provided that the transfer function of the equivalent error system is strictly positive real. Also, the new control mechanism guarantees stability, in terms of a bounded error, for a much less restrictive condition of positivity. Also, it must be noted that the positivity conditions, in both stability cases, are imposed on the equivalent error system and not on the controlled plant. Hence, the new adaptive controller is quite versatile because it can be applied to a great number of plants. Despite its simplicity the new MRAC system is robust and it performs very well. Actually, as shown from the
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simulation results presented in Chapter 5, the new MRAC system is as effective as the AMFC system that employs a more complex adaptive algorithm. In addition, the simulation results indicate that the new adaptive controller is highly robust to the variations of the plant dynamics. The simplicity and robustness of the algorithm presented in this thesis, indicates that the new MRAC system shows promise as a superior controller. However, in order to fully take advantage of this new adaptive mechanism further research should be considered. Therefore, the following recommendations for future research are proposed. First, research should be done to investigate how a modified MRAC system, which uses error feedback, will affect the performance of the controlled plant. It is expected that this modification will reduce the adaptation time. Second, the effect of the state estimator dynamics on the overall system performance has to be studied. In other words, general guidelines should be developed, which will help the designer place the poles of the estimator in order to maximize the performance of the controlled plant. Third, the maximum allowable order-difference (defined as the order of the plant minus the order of the state estimator) should be 87
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determined so that the resulting augmented system will have an A* matrix, which satisfies the required positivity conditions. Finally, research should be extended to include application of the new MRAC system to multi-input/multi-output systems. 88
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BIBLIOGRAPHY 1. P.V. Osborn, H.P . Whitaker, and A. Keezer, "New Developments in the Design of Adaptive Control Systems," Inst. Aeronaut. Sci., Paper 61-39 (1961). 2. P.C. Parks, "Lyapunov Redesign of Model Reference Adaptive Control Systems," IEEE Trans. Autom. Control AC-11, pp. 362-367, July 1966. 3. L.P. Grayson, "The Status of Synthesis Using Lyapunov's Method," Automatica, Vol. 3, pp. 91-121, December 1965. 4. C.A. Winsor and R.J. Roy, "Design of Model Reference Adaptive Control Systems by Lyapunov's Second Method," IEEE Trans. Autom. Control AC-13, No. 2, p. 204, April 1968. 5. I.D. Landau, Adaptive Control: The Model Reference Approach, New York: Marcel Dekker, 1979. 6. J.R. Broussard and M.J. O'Brien, "Feed-Forward Control to Track the Output of a Forced Model," Proc. 17th IEEE Conf. Decision Control, pp. 1149-1155, January 1979. 7. L. Mabius and H. Kaufman, "An Implicit Adaptation Algorithm for a Linear Model Reference Control System," Proc. 1975 IEEE Conf. Decision Control, Houston, pp. 864-865, December 1975. 8. J.T. Bialasiewicz and J.C. Proano, "Model-Reference Intelligent Control System," Kybernetika, Vol. 25, No. 2, pp. 95-103, 1989. 9. K.M. Sobel and H. Kaufman, "Direct Model Reference Adaptive Control for a Class of MIMO Systems," Control and Dynamic Systems, Academic Press, Vol. 24, Part 3, pp. 246-311, 1986. 10. I. Bar-Kana, "Adaptive Control: A Simplified Approach," Control and Dvnamic Systems, Academic Press, Vol. 25, Part 3, pp. 187-232, 1987.
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11. P. Ioannou and P.V. Kokotovic, "Singular Petrubations and Robust Redesign of Adaptive Control,".Proc. 21st IEEE Conf. Decision Control, Orlando, pp. 24-29 (1982). 12. I Bar-Kana and H. Kaufman, "Global Stability and Performance of a Control," Int. J. Control 42, pp. 1491-1501 (1985). 13. B.D.O. Anderson and s. Vongpanitlerd, Network Analysis and Synthesis, Englewood Cliffs, New Jersey: Prentice-Hall, 1973. 14. K.S. Narendra and L.S. Valavani, "A Comparison of Lyapunov and Hyperstability Approaches to Adaptive Control of Continuous Systems," IEEE Trans. Autom. Control AC-25, No. 2, April 1980. 15. V.M. Popov, Hyperstability of Automatic Control Systems, New York: Springer, 1973. ." :" . . :. 90
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