Weak Convergence of Nonlinear High-Gain Tracking Differentiator

Transcription

1 1074 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 4, APRIL 2013 Weak Convergence of Nonlinear High-Gain Tracking Differentiator Bao-Zhu Guo and Zhi-Liang Zhao In applications, the signal may be only locally integrable like the piecewise continuous signal or bounded measurable, and hence its classical (pointwise) derivative may not exist but its th generalized derivative, still denoted by, always exist in the sense of distribution which is defined as a functional of for any as Abstract In this technical note, the weak convergence of a nonlinear high-gain tracking differentiator based on finite-time stable system is presented under some easy checkable conditions. An example is constructed by using homogeneity. Numerical simulation shows that this tracking differentiator takes advantages over the existing ones. This result relaxes the strict conditions required in existing literature that the Lyapunov function satisfies the global Lipschitz condition and the setting-time function is continuous at zero, both of them seem very restrictive in applications. Index Terms Finite-time stability, homogeneity, tracking differentiator. (1.3),. The above is the standard definition of the generalized derivative ([1, Eq.(15), page 21]). From this definition, we see that any order of the generalized derivative always exists provided that is bounded measurable. Suppose that (1.2) holds true. Then considering as a function of ([1, Eq.(13), p.20]) I. INTRODUCTION The differential tracking for a given signal is a well known yet challenging problem in control theory and practice. The numerous researches have been contributed to differential trackers like high-gain observer based differentiator [6], the super-twisting second-order sliding-mode algorithm [8], linear time-derivative tracker [16], robust exact differentiation [17], name just a few. For a nice comparison with different differential trackers, we refer to [21]. In this technical note, we study the following tracking differentiator first proposed in [12]: (1.4) (1.1) Comparing the right-hand sides of (1.3) and (1.4), we see that is an identity matrix. It is claimed without proof in [12] that if the free system (i.e., and in (1.1)) is globally attractive, that is, every solution of the free system tends to zero as time goes to infinity, then for any given measurable signal and any given time, the solution of (1.1) satisfies (1.2) Manuscript received April 28, 2012; revised August 28, 2012; accepted September 05, Date of publication September 11, 2012; date of current version March 20, This work was supported by the National Natural Science Foundation of China, the National Basic Research Program of China (2011CB808002), and the National Research Foundation of South Africa. Recommended by Associate Editor A. Ferrara. B.-Z. Guo is with the Academy of Mathematics and Systems Science, Academia Sinica, Beijing , China and also with the School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg Wits 2050, South Africa ( bzguo@iss.ac.cn). Z.-L. Zhao is with the Department of Mathematics, University of Science and Technology Hefei, Anhui , China and also with the School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg Wits 2050, South Africa ( gsdxzzl@mail.ustc.eu.cn). Color versions of one or more of the figures in this technical note are available online at Digital Object Identifier /TAC in the sense of distribution ([1, pages 20 21]. So can be regarded as an approximation of the th generalized derivative of in. For more details, we refer to [14]. Definition 1: For any given initial value, the tracking differentiator (1.1) is said to be weak convergent if for almost all (and hence (1.2) is valid for any ). Although the first weak convergence is given in [13] and re-appeared later in [19], it is indicated in [11] that the proofs of [13], [19] are incorrect. A rigorous proof for weak convergence of the linear tracking differentiator is given in [6], [10], and recently the weak convergence for nonlinear one in [11]. A convergence result with the more accurate error estimation than [13], [11] is given in [20], which is based on a strict condition that there is a Lyapunov function satisfying I. in for some,. II. is globally Lipschitz continuous or the gradient is bounded in. The condition I above is to guarantee that the free system of (1.1) (i.e.,, ) is globally finite-time stable. The differentiator based on such a system has its independent significance due to its fast convergence. The second condition is quite strong and is actually not necessary as we shall see later in present technical note. A differentiator based on finite-time stable system given in [20] is as follows: (1.5) /$ IEEE

2 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 4, APRIL with, for, for. (1.5) is a special tracking differentiator (1.1) with,. The conclusion of [20] is as follows: For the given signal, there exists a with and such that for all for some positive number. There is no direct verification of the conditions I, II (finite-time stability and global Lipschitz continuous of Lyapunov function) in [20] for system (1.5) but instead, is refereed to [5]. Although the finite-time stability of the free system of (1.5) (i.e., ) is studied in [5], the Lyapunov function is not available in [5] too. So it is not clear for us how to verify the global Lipschitz condition for Laypunov function required in [20] for the system (1.5), which, to our knowledge, is far from simple. For instance, a very simple Lyapunov function like does not satisfy the global Lipschitz condition. Moreover, the choice of the parameter is also a big deal. According to the proof in [20],, is the power exponent in its assumption I. However since the Lyapunov function is not given explicitly in both [20] and [5], we are not clear why the required parameter condition is satisfied. Notice that the finite-time stability of the free system of (1.5) is concluded from the following system: (1.6) which is equal to (1.5) in some neighborhood of zero in.forsystem (1.6), the Lyapunov function satisfying assumption I is given in [5] with, and.byasimple computation, we get and. So it seems impossible to choose satisfying even for system (1.6). In this technical note, we first generalize, in Section II, the stability result for the perturbed finite-time stable systems studied in [3], and then apply the result to the proof of weak convergence of the finite-time stable based tracking differentiator without assuming the global Lipschitz continuous for Lyapunov function. In Section III, a first order tracking differentiator based on finite-time stable system is constructed using homogeneity, which is simpler than (1.5). All required conditions are verified. In Section IV, numerical experiments are performed to illustrate the advantage of the proposed differentiator over the existing ones. Finally, in Section V, we apply the tracking differentiator presented in Section III to stabilize a one-dimensional wave equation under the boundary control for which the derivative of the output signal is no more than locally square integrable. Numerical simulations are illustrated the effectiveness of the differentiator. II. WEAK CONVERGENCE OF TRACKING DIFFERENTIATOR The main purpose of this section is to prove the weak convergence of (1.1) by removing the global Lipschitz continuity on Lyapunov function required in [20]. Before going on, we give the definition and some preliminary results on finite-time stability. For notational simplicity, we drop the subscript in all state variables by abuse of notation. Definition 2: The following system: (2.7) is said to be globally finite-time stable, if a) it is Lyapunov stable; b) for any, there exists a such that the solution of (2.7) satisfies,and for all. The function in Definition 2 is called the settling-time function. Remark 1: In [20], another additional restrictive assumption is imposed on setting-time function that is continuous at zero, which is also almost impossible to check. This assumption is removed in present technical note. The following Lemma 2.1 follows from (iii) of proposition 2.4 and theorem 4.2 of [3]. Lemma 2.1: Suppose that there exists a continuous, positive definite function, constants, such that (2.8) denotes the th component of. Then (2.7) is globally finitetime stable. Furthermore, there exists a such that for any,ithas (2.9) We remark that in theorem 4.2 of [3], it is shown that for all in a neighborhood of origin, and in proposition 2.4 of [3], for all in another neighborhood of origin. These two facts together lead to (2.9). The following Lemma 2.2 is a generalization of theorem 5.2 of [3] for the case of attract basin being the whole space by removing the global Lipschitz continuity for Lyapunov function. Lemma 2.2: Consider the following perturbed system of (2.7): (2.10) is a constant. If there exist a continuous, positive definite and radially unbounded (i.e., ) function with all continuous partial derivatives in its variables, and constants, such that (2.8) holds, then for any, there exists a -dependent constant such that for any continuous function with the solution of (2.10) is bounded and (2.11) (2.12) the constants, depend on the initial value. Proof: We split the proof into two steps. Step 1. Thereexistsa such that for any, is defined in (2.11), the solution of (2.10) is bounded. Let be a -dependent constant, and with. For the radially unbounded positive definite function, by lemma 4.3 of [15] on page 145, there are strictly increase functions such that,, and.since,,, it is known that the set is bounded. This together with the continuity of concludes that.hence is a positive number (even for ). We assume that the claim in Step 1 is not true and obtain a contradiction. From the definition of,. This together with the continuity of guarantees that for, there exist such that the solution of (2.10) satisfies (2.13)

3 1076 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 4, APRIL 2013 Finding the derivative of in gives (2.14) which contradicts with (2.13). So the claim of Step 1 is true. Step 2. There exists a -dependent constant, such that for any, is defined in (2.11), the solution of (2.10) satisfies for all, is the same as that in Lemma 2.1, is a positive constant depending on the initial value. Let,,, and are the same as that in Step 1. Then the derivative of along the solution of (2.10) is found to satisfy, when,that with,,and is the vector field:. Then for any initial value of (1.1) and constant, there exists a depending on the initial value of (1.1), input and its derivatives, such that (2.21) is a positive constant depending on the initial value of (1.1), input and its derivatives,,, are the solutions of (1.1). Proof: Let (2.22) (2.15) Then satisfies the following system of differential equations Consider the following scalar differential equation: Its solution can be found as (2.16) (2.23). (2.17) If for all,, the complementary set of,thenby comparison principle, with,, which is actually a contradiction. This is because on the one hand, implies that, and on the other hand, since, it follows from (2.17) that is identical to zero for. Therefore, there exists an initial value dependent constant, such that.since for,itmusthave for all.however,forany,ithas.thisistheclaiminstep2. By Lemma 2.1, and Steps 1 2, we get that (2.18) for all. Setting completes the proof. Remark 2: In the proof of theorem 5.2 of [3], the inequality (2.15) is obtained by the Lipschitz continuity of the Lyapunov function on the attraction basin. Such a Lyapunov function is hard to construct in applications when attraction basin is the whole space.herewederive it from the boundedness of the solution claimed in Step 1 in the proof of Lemma 2.2. Theorem 1: Suppose that (i), ; (ii) the nonlinear function in (1.1) satisfies and (2.24) By conditions (i) (ii), there exists a constant, depending on input and its derivatives, but independent of,suchthat For any,let By Lemma 2.2, there exist constants, and depending on initial value, input and its derivatives such that, By (2.22), it follows that for all,, (2.25) (2.26) (2.27) (2.28) This completes the proof. Theorem 1 can be extended to the piecewise continuous signal, which can be considered as a consequence of Theorem 1. Corollary 2.1: Suppose that there are such that is -times differentiable in,andis left and right differentiable at. Assume that (2.19) (iii) there exists a continuous, positive definite function, with all continuous partial derivatives in its variables, satisfying (2.20) and the function in (1.1) satisfies conditions (ii) (iii) of Theorem 1. Then for any initial value of (1.1) and

4 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 4, APRIL , there exists a, depending on initial value, and, such that for all and or,ithas (2.29) denotes the left derivative and the right one, is some positiveconstant dependingon theinitialvalue of (1.1), input and its derivatives,,, are the solutions of (1.1). III. A FIRST ORDER TRACKING DIFFERENTIATOR In this section, we construct a first order differentiator based on finite-time stable system by the help of homogeneity. Definition 3: A function is called homogeneous of degree with weights,,if, The finite-time stability of (3.3) is also studied in [2] on page 191. But here we are more interested in inequality (3.9) since it means that our condition (2.20) in Theorem 1 is valid. Now, we show that satisfies condition (2.19) in Theorem 1. Let be defined by for some,. is decreasing since for all.itfollowsthat (3.10) Furthermore, since for all, is convex on. By Jessen s inequality, we have Combining inequalities (3.10), (3.11), we get (3.11) Avectorfield is called homogeneous of degree with weights,ifforany,,, is the th component of. Consider the second order system: (3.1) (3.2) (3.3) (3.4) (3.12) That is, the condition (2.19) of Theorem 1 is satisfied for. Based on finite-time stable system (3.3), we can construct the following tracking differentiator: By Theorem 1, (3.13) is weak convergent. For differentiator (3.13), using the notation (2.22) for error equation becomes (3.13),the (3.14) with It is seen that for any,and (3.5) for some positive constant.since is homogeneous of degree with weights, it follows from lemma 4.2 of [4] that. Let the Lyapunov function be defined in (3.9). Finding its derivative along the solution of (3.14) gives (3.6) Therefore, the vector field is homogeneous of degree with weights. Let be given by (3.7) It is easy to verify that if, then along the solution of (3.14). Hence there exist an initial value dependent constant such that for all. Since the functions are homogeneity of degree and with weights and respectively, using lemma 4.2 of [4] again, we get that A direct computation shows that (3.8) By LaSalle s invariance principle, system (3.3) is globally asymptotically stable. From theorem 6.2 of [4], there is a continuous, positive definite Lyapunov function such that is continuous on. Moreover, is homogeneous of degree,and is homogeneous of degree, both with the same weights. We also know from theorem 2 of [18] that is radially unbounded. By virtue of lemma 4.2 of [4], there exists a such that (3.9) We have thus proved the following Theorem 2. Theorem 2: If the signal satisfies for,2,thenthefirst order high-gain finite-time stable based differentiator (3.13) is convergent in the sense that for any initial value of (3.13) and,thereexistsa depending on initial value and such that, are constants depending on initial value and,and by (3.5). (3.15)

5 1078 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 4, APRIL 2013 Fig. 1. Derivative tracking for by DI (for interpretation of the references to color of the figures legend in this section, we refer to the PDF version of this article). (a) red, green, and blue; (b) magnification of Fig. 1(a). Fig. 3. Derivative tracking for by DIII.(a) red, green, blue, ;(b) red, green, blue, ;(c)magnification of Fig. 3(a); (d) red, green, blue,, with time delay. Fig. 2. Derivative tracking for by DII.(a) red, green, blue, ;(b) red, green, blue,. IV. NUMERICAL SIMULATIONS In this section, we give some numerical simulations to compare the following three differentiators. DI. Robust exact differentiator using sliding mode technique from [17]: (4.1) DII. Linear tracking differentiator in [10], which is equivalent to the high-gain observer presented in [6] under the coordinate transformation: (4.2) DIII. High-gain finite-time stable system based tracking differentiator (3.13) (taking,,,, in (3.13)): (4.3) The Matlab program of Euler method is adopted in investigation. We choose the same zero initial value and in all simulations. The results by differentiator DI are plotted in Fig. 1 in Fig. 1(a), and integral step. Fig. 1(b) is the magnification of Fig. 1(a). The results by differentiator DII are plotted in Fig. 2 in Fig. 2(a), and integral step, while in Fig. 2(b) and integral step.theresultsbydifferentiator DIII are plotted in Fig. 3 in Fig. 3(a), and integral step, while in Fig. 3(b) and integral step. Fig. 3(c) is the magnification of Fig. 3(a). Fig. 3(d) plots the results of differentiator DIII with delayed signal of delay 0.05, Fig. 4. Derivative tracking of DIII for disturbed by noise. (a) red, green, blue, ;(b) red, green, blue,. and integral step. Fig. 4 plots the numerical results of differentiator DIII with signal disturbed by its 1% uniform white noise. In Fig. 4(a), we take,integralstep, while in Fig. 4(b),,. From Figs. 1, 2 and 3, we see that our tracking differentiator based on finite-time stable system DIII is smoother than differentiator DI in which the discontinuous of function produces problems like chattering. And differentiator DIII tracks faster than linear differentiator DII. Moreover, it seems that the our finite-time stable system based tracking differentiator DIII is more accurate than linear DII with the same high-gain parameters. Finally, the finite-time stable system based tracking differentiator is tolerant to small time delay and noise. From Fig. 4 we see that the tuning parameter in differentiator DIII plays a significant role in convergence and noise tolerance: the larger the is, the more accurate the tracking effect would be, but more sensitive to the noise. This suggests that the choice of parameter in DIII is a tradeoff between tracking accuracy and noise tolerance in practice. V. APPLICATION TO STABILIZATION OF A STRING EQUATION Consider the following one-dimensional wave equation (5.1) which describes the vibration of string, is the amplitude, is the velocity, is the vertical force,,

6 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 4, APRIL Fig. 5. Numerical simulations for and from (4.3) with and (5.2). Fig. 6. Numerical simulations for in (4.3) with. is the initial value, and is the boundary control input. It is well-known that with the boundary feedback,, the system (5.1) is exponentially stable in the sense that for some positive constants,, is the energy of the vibrating string. The feedback controller requires the velocity which is hard to measure ([7]). Instead, the amplitude is easily measurable. However, in order to stabilize the system (5.1) by the output feedback with the measurement, we need to track from. But the regularity of is no more than that,thatis, may not exist every in the classical sense. In this case, the weak differentiator (4.3) (with ) plays the role in the stabilization of (5.1). Actually, consider as an approximation of in the sense of distribution. Under the output feedback,the closed-loop system of (5.1) becomes (5.2) We use numerical method to study the stability of the system (5.2). The finite difference method is adopted in simulation. Let be steps along the and axis, respectively. We set in (4.3) with, (5.2), and,. The numerical result for amplitude from (4.3), (5.2) is plotted in Fig. 5(a). The velocity is obtained from by difference interpolation with time and the result is plotted in Fig. 5(b). It is seen that both of them are convergent satisfactorily. The corresponding from (4.3) with the same parameters as that used for Fig. 5 are plotted in Fig. 6. It is seen that is convergent also but seems complicated, which shows the significance of the weak convergence. The study of the stabilization of system (5.2) remains an interesting theoretical problem. VI. CONCLUSION In this technical note, we have proved the weak convergence for the high-gain nonlinear tracking differentiator based on finite-time stable systems. An unnecessary restrictive condition of the global Lipschitz continuity for Lyapunov function in literature is removed, by which the conditions become simple and practically checkable. All the conditions are implemented in the construction of a first order finite-time stable system based tracking differentiator. By this example, numerical experiment is carried out to show the fast tracking, accuracy, smooth, anti-chattering, small time delay and small noise tolerance of this differentiator compared with the existing ones. We also apply this nonlinear differentiator to the boundary stabilization of a one-dimensional wave equation. The numerical simulation shows that it is very effective and noise tolerance even for infinite-dimensional systems. However, the other issues with a differentiator design like evaluating its accuracy in the presence of a measurement noise (robustness) and the settling-time function estimation are needed for further investigations theoretically, although it is believed that our differentiator is robust to noise as proved in linear differentiator of the same kind in [6], [10] and confirmed by numerical simulations in this technical note. A recent effort on this aspect can be found in [9]. ACKNOWLEDGMENT The authors would like to thank the anonymous referees for their very careful reading and constructive suggestions that improve substantially the manuscript. REFERENCES [1] A.AdamsandJ.J.F.Fournier, Sobolev Spaces, 2nd ed. New York: Academic, [2] A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory. Berlin, Germany: Springer-Verlag, [3] S. P. Bhat and D. S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM J. Control Optim., vol. 38, no. 3, pp , [4] S. 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7 1080 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 4, APRIL 2013 [17] A. Levant, Robust exact differentiation via sliding mode technique, Automatica, vol. 34, no. 3, pp , [18] L. Rosier, Homogeneous Lyapunov function for homogeneous continuous vector field, Syst. Control Lett., vol. 19, no. 6, pp , [19] Y. X. Su, C. H. Zheng, and D. Sun, A simple nonlinear velocity stimatiom for high-performance motion control, IEEE Trans. Ind. Electron., vol. 52, no. 4, pp , Aug [20] X. Wang, Z. Chen, and G. Yang, Finite-time-convergent differentiator based on singular perturbation technique, IEEE Trans. Autom. Control, vol. 52, no. 9, pp , Sep [21] W. Xue, Y. Huang, and X. Yang, What kinds of system can be used as tracking-differentiator, in Proc. Chinese Control Conf., 2010, pp Low-Frequency Learning and Fast Adaptation in Model Reference Adaptive Control Tansel Yucelen and Wassim M. Haddad Abstract While adaptive control has been used in numerous applications to achieve system performance without excessive reliance on dynamical system models, the necessity of high-gain learning rates to achieve fast adaptation can be a serious limitation of adaptive controllers. This is due to the fact that fast adaptation using high-gain learning rates can cause high-frequency oscillations in the control response resulting in system instability. In this note, we present a new adaptive control architecture for nonlinear uncertain dynamical systems to address the problem of achieving fast adaptation using high-gain learning rates. The proposed framework involves a new and novel controller architecture involving a modification term in the update law. Specifically, this modification term filters out the high-frequency content contained in the update law while preserving asymptotic stability of the system error dynamics. This key feature of our framework allows for robust, fast adaptation in the face of high-gain learning rates. Furthermore, we show that transient and steady-state system performance is guaranteed with the proposed architecture. Two illustrative numerical examples are provided to demonstrate the efficacy of the proposed approach. Index Terms Adaptive control, command following, fast adaptation, high-gain learning rate, low-frequency learning, nonlinear uncertain dynamical systems, stabilization, transient and steady state performance. I. INTRODUCTION While adaptive control has been used in numerous applications to achieve system performance without excessive reliance on system models, the necessity of high-gain learning rates for achieving fast adaptation can be a serious limitation of adaptive controllers [1]. Specifically, in certain applications fast adaptation is required to achieve stringent tracking performance specifications in the face of large system uncertainties and abrupt changes in system dynamics. Manuscript received April 28, 2012; revised August 28, 2012 and August 21, 2012; accepted August 30, Date of publication September 11, 2012; date of current version March 20, This research was supported in part by the Air Force Office of Scientific Research under Grant FA Recommended by Associate Editor A. Astolfi. The authors are with the School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, USA ( tansel. yucelen@aerospace.gatech.edu; wm.haddad@aerospace.gatech.edu). Color versions of one or more of the figures in this technical note are available online at Digital Object Identifier /TAC This, for example, is the case for high performance aircraft systems that are subjected to system faults or structural damage which can result in major changes in aerodynamic system parameters. In such situations, adaptive control with high-gain learning rates is necessary in order to rapidly reduce and maintain system tracking errors. However, fast adaptation using high-gain learning rates can cause high-frequency oscillations in the control response resulting in system instability [2] [4]. Hence, there exists a critical trade-off between system stability and adaptation learning rate (i.e., adaptation gain). In this note, we present a new adaptive control architecture for nonlinear uncertain dynamical systems to address the problem of achieving fast adaptation using high-gain learning rates. The proposed framework involves a new and novel controller architecture involving a modification term in the update law. Specifically, this modification term filters out the high-frequency content contained in the update law while preserving asymptotic stability of the system error dynamics. This key feature of our framework allows for robust, fast adaptation in the face of high-gain learning rates. We further show that transient and steady-state system performance is guaranteed with the proposed architecture. Two illustrative numerical examples are provided to demonstrate the efficacy of the proposed approach. The notation used in this technical note is fairly standard. Specifically, denotes the set of real numbers, denotes the set of real column vectors, denotes the set of real matrices, denotes transpose, denotes inverse, denotes the Euclidian norm, and denotes the Frobenius matrix norm. Furthermore, we write (resp., ) for the minimum (resp., maximum) eigenvalue of the Hermitian matrix and for the trace operator. II. MODEL REFERENCE ADAPTIVE CONTROL We begin by presenting a brief review of the model reference adaptive control problem. Specifically, consider the nonlinear uncertain dynamical system given by,,isthestatevector,,,isthe control input, and are known matrices such that the pair is controllable, and is a matched system uncertainty. We assume that the full state is available for feedback and the control input is restricted to the class of admissible controls consisting of measurable functions such that,.in addition, we consider the reference system given by,, is the reference state vector,,, is a bounded piecewise continuous reference input, is Hurwitz, and. Assumption 2.1: The matched uncertainty in (1) is linearly parameterized as is an unknown constant weighting matrix is a basis function of the form. Here, the aim is to construct a feedback control law,, and such that the state of the nonlinear uncertain dynamical system given by (1) asymptotically tracks the state of the reference model given by (2) in the presence of matched uncertainty satisfying (3). (1) (2) (3) /$ IEEE