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Rapid-convergent nonlinear differentiator
Mechanical Systems and Signal Processing 28 (2012) 414–431
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Rapid-convergent nonlinear differentiator Xinhua Wang n, Bijan Shirinzadeh Robotics and Mechatronics Research Laboratory, Department of Mechanical and Aerospace Engineering, Monash University, Clayton, VIC, 3800, Australia
a r t i c l e in f o
abstract
Article history: Received 28 February 2011 Received in revised form 27 September 2011 Accepted 28 September 2011 Available online 29 October 2011
A nonlinear differentiator being fit for rapid convergence is presented, which is based on singular perturbation technique. The differentiator design can not only sufficiently reduce the chattering phenomenon of derivative estimation by introducing a continuous power function, but the dynamical performances are also improved by adding linear correction terms to the nonlinear ones. Moreover, strong robustness ability is obtained by integrating nonlinear items and the linear filter. The merits of the rapidconvergent differentiator include the excellent dynamical performances, restraining noises sufficiently, avoiding the chattering phenomenon and being not based on system model. The theoretical results are confirmed by computer simulations and an experiment. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Differentiator Singular perturbation Rapid convergence Chattering phenomenon
1. Introduction Differentiation of signals is an old and well-known problem [1–3] and has attracted more attention [4–6] in recent years. Rapidly and accurately obtaining the velocities and accelerations of tracked targets is crucial for several kinds of systems with correct and timely performances, such as the missile-interception systems in defense systems [22] and underwater vehicle systems [23]. Therefore, the convergence rates, robustness and computation time of differentiators are very significant. Difference method is used to estimate approximately the derivatives of signals. Because disturbances exist in almost all the signals, the satisfying estimation precision cannot be obtained by difference method. Kalman filter can reduce the disturbance and obtain the derivatives of signals. However, the model of plant must be required. This constrains the types of signals. Therefore, it is very important to construct an estimator with filtering ability and without knowing the model of plant. Usually the uncertainties of nonlinear systems are approximated by some intelligent algorithms, such as fuzzy systems or radial-based-function (RBF) networks. Over the last decade, control researchers tried to apply intelligent methodologies to the design of estimation and control for uncertain nonlinear systems [7–17]. In paper [7], a conclusion was given that fuzzy systems are universal approximation. The Stone–Weierstrass Theorem is used to prove that fuzzy systems with product inference, centroid defuzzification, and Gaussian membership function are capable of approximating any real continuous function on a compact set to arbitrary accuracy. Paper [8] reported on a related study of radial-basis-function (RBF) networks, and it was proved that RBF networks having one hidden layer are capable of universal approximation. The main difficulties with estimating uncertainties by these algorithms are: (1) the parameters or the neural network weights are difficult to be regulated; (2) membership function and Gaussian function are selected by experiences; (3) all of the system states must be required for estimation and feedback; (4) noises cannot be restrained by the above approximation
n
Corresponding author. E-mail address: [email protected] (X. Wang).
0888-3270/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2011.09.026
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algorithms. In [18–20], the differential transformation method (DTM) and the finite difference method (FDM) are used to analyze the nonlinear dynamic behavior of a rigid rotor supported by a micro gas bearing system (MGBS), and in [21], a nonlinear rule based fuzzy logic controller is proposed to control the nonlinear dynamic behavior of the probe tip of an atomic force microscope system (AFMs). However, the abilities of restraining noise were not considered. Differentiators are usually used to obtain the information of tracked targets or the information of the plants, which are not modeled easily. For instance, the position of a tracked target is measured by radar, and its velocity can be carried out by a differentiator. Differentiators are the estimators, which are not based on the model of plant. The popular high-gain differentiators [4] provide for an exact derivative when their gains tend to infinity. Unfortunately, their sensitivity to small high-frequency noise also infinitely grows. With any finite gain values such a differentiator has also a finite bandwidth. Thus, being not exact, it is, at the same time, insensitive with respect to high-frequency noise. Such insensitivity may be considered both as advantage or disadvantage depending on the circumstances. In [5,6], a differentiator via second-order (or high-order) sliding modes algorithm has been proposed. The information one needs to know on the signal is an upper bound for Lipschitz constant of the derivative of the signal. Although second-order sliding mode is introduced and there exists no chattering phenomenon in signal tracking. However, for the robust exact differentiator, in the second dynamical equation, a switching function exists. The output of derivative estimation is continuous but not smooth. Therefore, chattering phenomenon still exists in derivative estimation. In [34], we presented a finite-time-convergent differentiator that is based on singular perturbation technique [24–29]. The merits of this differentiator exist in three aspects: rapidly finite-time convergence compared with other typical differentiators; no chattering phenomenon; and besides the derivatives of the derivable signals, the generalized derivatives of some classes of signals can be obtained. However, the differentiators in [34] are complicated and difficult to be used in practice. Moreover, the computation time is still large and its robustness is not considered. In this paper, we expand our work in [34]. The proposed rapid-convergent differentiator is an integration of a nonlinear term (comprising of continuous power function) and a linear correction term. The structure of the designed differentiator is simple and the computation time is short. In the nonlinear part, a continuous power function with a perturbation parameter is used. Therefore, chattering phenomenon can be avoided in the output of derivative estimation. Strong robustness ability is obtained by integrating sliding mode items and the linear filter. The linear part can restrain highfrequency noises by giving a suitable nature frequency, and small bounded noises can be restrained by the nonlinear items. Moreover, the dynamical performances are also improved. This paper is organized in the following format. In Section 2, problem statement is given, and preliminaries are given in Section 3. In Section 4, our main results are presented. In Section 5, the main results are proved in detail. In Section 6, robustness analysis of finite-time-convergent differentiator is given. In Section 7, reduction of peaking phenomenon is given. In Section 8, the simulations and experiments are given, and our conclusions are made in Section 9. 2. Problem statement When we design a controller for a system, state feedback is used to place the poles. In order to use the system states to design feedback controller, sensors must be adopted to measure the states. However, not all states can be measured directly. Therefore, state observer is presented to estimate the system states. These observers are designed under the condition that the system model must be known. We present the following three questions. Question 1. For arbitrary signals, which cannot be expressed by mathematical expressions, such as the signal shown in Fig. 1, how can we estimate the derivate of the signal? Question 2. How can we obtain the derivative of a signal with noise? For instance, the following signal in Fig. 2. Question 3. For system x_ 1 ¼ x2 x_ 2 ¼ u y ¼ x1
ð1Þ
we require that the output y tracks the desired trajectory yd(t) shown in Fig. 3. Let e1 ¼ x1 yd ðtÞ,
e2 ¼ x2 y_ d ðtÞ
ð2Þ
Fig. 1. Stochastic signal.
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Fig. 2. Signal with noises.
v(t)
y Fig. 3. Trajectory tracking.
we get the tracking error system as follows: e_ 1 ¼ e_ 2 e_ 2 ¼ uy€ d
ð3Þ
Before designing contoller u, how can we obtain the information of y_ d and y€ d ? Design of usual observer is based on the model of system. However, most signals are difficult to be described by a given model. Therefore, using usual observer to estimate derivative of signal is restricted. We expect to construct the following form: x_ 1 ¼ x2 x_ 2 ¼ f ðx1 vðtÞ,x2 Þ
ð4Þ
Under the condition that the solution exists for the above differential equation, a differentiator can be constructed if the _ following relation is satisfied: x1 converges to v(t) and x2 converges to vðtÞ. We are interested in designing a rapid-convergent differentiator to estimate the derivative of signal, and highfrequency noise can be restrained sufficiently. 3. Preliminaries First of all, the related concepts are given. In order to demonstrate the finite-time convergence, we introduce the following definitions, assumptions and theorems from the corresponding references. Definition 1. [35] ‘‘Generalized derivative’’ denotes that the left and right derivatives of a point in a function trajectory both exist, and they may be not equal to each other. Definition 2. [33] Consider a time-invariant system in the form of x_ ¼ f ðxÞ,
f ð0Þ ¼ 0,
x 2 Rn
ð5Þ
where f:D-Rn is continuous on open neighborhood DCRn of the origin. The origin is said to be a finite-time-stable equilibrium of the above system if there exists an open neighborhood N DD of the origin and a function Tf:N\{0}-(0,N), called the settling-time function, such that the following statements hold: (1) Finite-time-convergence: For every xAN\{0}, cx is the flow starting from x and defined on [0,Tf(x)), cx(t)AN\{0} for all x tA[0,Tf(x)), and limt-T f ðxÞ c ðtÞ ¼ 0. (2) Lyapunov stability: For every open neighborhood Ue of zero, there exists an open subset Ud of N containing zero such that, for every xAUd\{0}, cx(t)AUe for all tA[0,Tf(x)). The origin is said to be a globally finite-time-stable equilibrium if it is a finite-time-stable equilibrium with D ¼ N¼ Rn. Then the system is said to be finite-time-convergent with respect to the origin.
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Assumption 1. Suppose the origin is a finite-time-stable equilibrium of [33, theorem 4.3] of z_~ 1 ¼ z~ 2 ... z~_ ¼ z~ n n1
z_~ n ¼ f ðz~ 1 , z~ 2 ,. . ., z~ n Þ
ð6Þ
And the settling-time function Tf is continuous at zero, where f(U) is continuous and f(0). Let N be as in Definition 2 and let yA(0,1). Then there exists a continuous scalar function V such that (1) V is positive definite and (2) V_ is real valued and continuous on N and there exists c 40 such that V_ þ cV y r0
for any y 2 ð0,1Þ
ð7Þ
Assumption 2. There exists a Lipschitz Lyapunov function V satisfying (4) with Lipschitz constant M. Assumption 3. For (6), there exists riA(0,1], i¼0,1,y,n 1, and a nonnegative constant a such that 9f ðz~ 1 , z~ 2 ,. . ., z~ n Þf ðz1 ,z2 ,. . .,zn Þ9 ra
n X
ri 1
9z~ i zi 9
ð8Þ
i¼1
where z~ i ,zi 2 R, i ¼ 1,. . .,n. Assumption 4. v(t) is a continuous and piecewise n-order derivable signal with the following properties. The derivatives of v(t) up to order n 2 exist on the whole time domain and v(t) is not (n 1)-order differentiable at some instants tj,j ¼1,y,k. However, the (n 1)-order left derivative vðn1Þ ðt j Þ and right derivative vðn1Þ ðt j Þ exist [35], respectively, and þ ðn1Þ ðt j Þavðn1Þ ðt j Þ, j ¼ 1,. . .,k, may be satisfied. v þ
Remark 1. There are a number of nonlinear functions actually satisfying this assumption. For example, one such function r
r
is xi since 9xri xri 9 r 21ri 9xx9 i , ri 2 ð0,1. Moreover, there are smooth functions also satisfying this property. In fact, it is easy to verify that 9sin xsin x9 r 29xx9
ri
for any riA(0,1].
Theorem 3.1. in [30] Let the differential equation z_ ðtÞ ¼ f ðz1 ðtÞ,. . .,zn ðtÞÞ
ð9Þ
be locally homogeneous of degree qo0 with respect to dilation (r1,y,rn) and let the equilibrium point z ¼0 of (9) be globally uniformly asymptotically stable. Then the differential Eq. (9) is globally uniformly finite time stable. Theorem 1. in [34] If z_~ 1 ¼ z~ 2 ... ¼ z~ n z_~ n1
z_~ n ¼ f ðz~ 1 , z~ 2 ,. . ., z~ n Þ
ð10Þ
is satisfied with Assumptions 1–3, signal v(t) is satisfied with Assumption 4. Then for system dx1 ¼ x2 dt ... dxn1 ¼ xn dt dx en n ¼ f ðx1 vðtÞ, ex2 ,. . ., en1 xn Þ dt
ð11Þ
there exist g 40 (where rg 4n) and G 40 such that xi vði1Þ ðtÞ ¼ Oðergi þ 1 Þ
ð12Þ
for tj 4t Ztj 1 þ eG(X(e)e þ (tj 1)), j¼1,y,kþ1, (let t0 ¼0 and tk þ 1 ¼ N) i¼1,y,n with xn ðt j Þvðn1Þ ðtÞ ¼ Oðergn þ 1 Þ,
j ¼ 1,. . .,k
ð13Þ rg i þ 1
rg i þ 1
where e 40 is the perturbation parameter and O(e ) denotes the approximation of e order [28] between xi and T v(i 1)(t); and g ¼(1 y)/y, y 2 ð0,minfr=ðr þ nÞ,1=2gÞ, nZ2. ei ¼xi v(i 1),i¼1,y,n. e ¼ e1 en , e þ ðt j1 Þ ¼ h iT e1 ðt j1 Þ . . . en1 ðt j1 Þ enþ ðt j1 Þ , enþ ðt j1 Þ ¼ xn vðþn1Þ ðt j1 Þ, and X(e) ¼diag{1,e,y,en 1}.
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The practical form of differentiator in [34] is x_ 1 ðtÞ ¼ x2 ðtÞ
e2 x_ 2 ðtÞ ¼ u a=ð2aÞ
u ¼ sateb fsignðfa ðx1 vðtÞ, ex2 Þ9fa ðx1 vðtÞ, ex2 Þ9
a
Þgsatev fsignðx2 Þ9ex2 9 g
where (
2a
fa ðx1 vðtÞ, ex2 Þ ¼ x1 vðtÞ þ
signðx2 Þ9ex2 9 2a
,
sateb ðxÞ ¼
x,
x,
eb signðxÞ, 9x9 Z eb
Its structure is too complicated. In the following, we will design more simple differentiators based on Theorem 1 in [34], and the designed rapid-convergent differentiator can keep more rapidly convergent at all times. Denotations: a sigðyÞa ¼ 9y9 sgnðyÞ, a 40. It is obvious that sigðyÞa ¼ ya only if a ¼q/p. where p,q are positive odd numbers. Moreover, d aþ1 9y9 ¼ ða þ 1ÞsigðyÞa , dy
d a sigðyÞa þ 1 ¼ ða þ 1Þ9y9 dy
ð14Þ
In the following, we will give our main results of rapid-convergent differentiator. 4. Rapid-convergent differentiator Here, we will design a nonlinear differentiator being fit for rapid convergence. The proposed differentiator consists of linear and nonlinear parts. It is shown that for arbitrary signal, there exists a differentiator achieving finite-time error decay. Moreover, high-frequency noise can be restrained sufficiently. We design a rapid-convergent differentiator as follow: Theorem 1. The rapid-convergent differentiator is designed as: x_ 1 ¼ x2
e2 x_ 2 ¼ a10 ðx1 vðtÞÞa11 sigðx1 vðtÞÞa=ð2aÞ a20 ex2 a21 sigðex2 Þa y ¼ x2
ð15Þ
For a continuous and piecewise two-order derivative signal v(t), there exists g 4 0 (where rg 42 and r ¼ minfa, a=ð2aÞg ¼
a=ð2aÞ ), such that xi vði1Þ ðtÞ ¼ Oðergi þ 1 Þ
ð16Þ
for t Z eG(X(e)e(0)),i¼1, and tj 4t 4 Ztj 1 þ eG(X(e)e þ (tj 1)), j ¼1,y,kþ 1, i¼2, respectively, with x2 ðt j Þv0 ðt j Þ ¼ Oðerg1 Þ, j ¼ 1,. . .,k , where aA(0,1), e 40 is the perturbation parameter and O(erg i þ 1) denotes the approximation of erg i þ 1 h iT T þ order [28] between xi and v(i 1)(t); ei ¼xi v(i 1),i ¼1,2. e ¼ e1 e2 , e þ ðt j1 Þ ¼ e1 ðt j1 Þ e2 ðt j1 Þ , e2þ ðt j1 Þ ¼ x2 v0þ ðt j1 Þ , and X(e) ¼diag{1,e}. Eq. (16) denotes that variable xi track (i 1)-order derivative of signal v(t). The rapid-convergent differentiator (15) consists of linear and nonlinear differentiator given, respectively, as follows: x_ 1 ¼ x2
e2 x_ 2 ¼ a10 ðx1 vðtÞÞa20 ex2 y ¼ x2
ð17Þ
and x_ 1 ¼ x2
e2 x_ 2 ¼ a11 sigðx1 vðtÞÞa=ð2aÞ a21 sigðex2 Þa y ¼ x2
ð18Þ
For linear differentiator (17), we have the following conclusion: Theorem 2. For linear differentiator (17) and a signal v(t) (where v(t) is a continuous and piecewise second-order derivable signal, we have that _ ¼ OðeÞ x2 ðtÞvðtÞ
for t-1
_ where a10, a20 40, e 40 is perturbation parameter, O(e) denotes the approximation of e order [28] between x2 and vðtÞ.
ð19Þ
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For nonlinear differentiator (18), we have the following conclusion: Theorem 3. For nonlinear differentiator (18) and a continuous and piecewise two-order derivative signal v(t), there exists g 4 0 (where rg 4 2 and r ¼ minfa, a=ð2aÞg ¼ a=ð2aÞ,), such that xi vði1Þ ðtÞ ¼ Oðergi þ 1 Þ
ð20Þ
for t Z eG(X(e)e(0)),i ¼1, and tj 4t 4 Ztj 1 þ eG(X(e)e þ (tj 1)), j¼ 1,y,kþ 1, i¼2, respectively, with j ¼ 1,. . .,k , where aA(0,1).
x2 ðt j Þv0 ðt j Þ ¼ Oð rg1 Þ,
e
Remark 2. From the analysis of linear systems and non-Lipschitz nonlinear systems, linear differentiator (17) and nonlinear differentiator (18), two notable differences can be observed. First, the two algorithms have quite different converging properties: the linear system converges exponentially, whereas the trajectories of the non-Lipschitz nonlinear algorithm converge in finite time. This is due to the lack of local Lipschitzness of the nonlinear algorithm at the origin, that is, its behavior around the zero state is very strong compared to the linear case. On the other side, the linear correction terms are stronger than the ones of the nonlinear algorithm far from the origin. These differences cause another striking difference between both algorithms: the kind of perturbations that each one is able to tolerate. The main difference is that the linear system can deal with perturbations that are stronger very far from the origin and weaker near the origin than the ones that are endured by the nonlinear algorithm. So, for example, the nonlinear algorithm is not able to endure (globally) a bounded perturbation with linear growth in time, but the linear algorithm can deal with it easily. However, the linear algorithm is not able to support a strong perturbation near the origin, what is one of the main advantages of non-Lipschitz nonlinear algorithm. In order to integrate the merits of linear differentiator (17) and nonlinear differentiator (18), and to restrain their shortcomings, respectively, rapid-convergent differentiator (15) is integrated. The proposed rapid-convergent differentiator is an integration of a nonlinear term (comprising of continuous power function) and a linear correction term. The overall design is an integration of nonlinear non-Lipschitz differentiators (18) and linear differentiator (17) in that a perturbation parameter a is introduced, which takes values (0,1) and an additional linear correction term is appended, so that it inherits the best properties of both. In the nonlinear part, a continuous power function with a perturbation parameter is used. Therefore, chattering phenomenon can be avoided in the output of derivative estimation. Strong robustness ability is obtained by integrating sliding mode items and the linear filter. The linear part can restrain highfrequency noises by giving a suitable nature frequency, and small bounded noises can be restrained by the nonlinear items, at the same time, the nonlinear items can compensate the delay brought by the linear filter. 5. Proof of Theorems 1–3 5.1. Rapid convergence In order to analyze differentiators (15), (17) and (18), firstly, we give the following system convergence Lemmas 1–3: Lemma 1 (Convergence of linear system). System 8 < dðz1 ðtÞÞ ¼ z2 ðtÞ dt
ð21Þ
: dðz2 ðtÞÞ ¼ a10 z1 ðtÞa20 z2 ðtÞ dt is exponentially convergent with respect to the origin (0, 0). Where a10 40 and a20 40 are constants. 2
2
Proof. If the Lyapunov function candidate is selected as V(z1(t), z2(t)) ¼(a10z1(t) þz2(t))/2, then the result of the convergence can be obtained. & The process of convergent velocity for the system (21) is shown in Fig. 4.
Convergent velocity
States
Fig. 4. Convergent velocity in linear form.
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From Fig. 4, the convergent velocities of the states are rapid when they are far away from the equilibrium point. However, the convergent velocities are slow when they approach the equilibrium point. It is required to improve the process of convergent velocities when the states approach the equilibrium point. Lemma 2 (Convergence of nonlinear system). System dðz1 ðtÞÞ ¼ z2 ðtÞ dt dðz2 ðtÞÞ ¼ a11 sigðz1 ðtÞÞa=ð2aÞ a21 sigðz2 ðtÞÞa dt
ð22Þ
is finite time convergent with respect to the origin (0, 0), i.e., z1 ðtÞ 0 and z2 ðtÞ 0 for t Z ts
ð23Þ
where 0 o a o1, a11 and a21 are positive constants, ts is time constant. Proof. Let the Lyapunov function candidate be: a11 ð2aÞ 1 2=ð2aÞ 9z1 ðtÞ9 þ z22 ðtÞ Vðz1 ðtÞ,z2 ðtÞÞ ¼ 2 2
ð24Þ
Therefore, we have dðVðz1 ðtÞ,z2 ðtÞÞÞ 1þa ¼ a11 sigðz1 ðtÞÞa=ð2aÞ z2 ðtÞ þ z2 ðtÞða11 sigðz1 ðtÞÞa=ð2aÞ a21 sigðz2 ðtÞÞa Þ ¼ a21 9z2 ðtÞ9 o0 dt
ð25Þ
Note that the uniqueness of the solution of the system can be obtained from [31]. Then based on LaSalle invariant theory, the system (22) is asymptotically stable. Therefore, based on the negative homogeneity [30, Theorem 3.1], we have that er2 z2 ðtÞ ¼ er1 þ k z2 ðtÞa11 sigðer1 z1 ðtÞÞa=ð2aÞ a21 sig er2 z2 ðtÞ a ¼ er2 þ k ða11 sigðz1 ðtÞÞa=ð2aÞ a21 sigðz2 ðtÞÞa Þ ð26Þ Therefore we have that r 2 ¼ r 1 þk,
r1
a 2a
¼ r2 a ¼ r2 þ k
ð27Þ
i.e., r1 ¼ k
2a 1 , r2 ¼ k a1 a1
ð28Þ
Therefore, system (22) is finite time convergent with respect to the origin (0, 0), i.e., z1 ðtÞ 0 and z2 ðtÞ 0 for t Z ts
ð29Þ
Remark 3. From [32], system dðz1 ðtÞÞ ¼ z2 ðtÞ dt dðz2 ðtÞÞ ¼ sigðz1 ðtÞÞa1 sigðz2 ðtÞÞa2 dt
ð30Þ
is finite time convergent with respect to the origin (0, 0), i.e., z1 ðtÞ 0 and z2 ðtÞ 0 for t Z ts
ð31Þ
where 0 o a2 o1, and 14 a1 4 a2/(2 a2). The process of convergent velocity for the system (22) or (30) is shown by solid line in Fig. 2. From Fig. 5, although the convergent velocities of the states are rapid when they approach the equilibrium point, the convergent velocities of the states are slow when they are far away from the equilibrium point. Therefore, it is required to improve the convergent velocities whether the states are near to the equilibrium point or not, i.e., rapid convergence should be guaranteed at all times. Lemma 3 (Convergence of rapid-convergent system). System dðz1 ðtÞÞ ¼ z2 ðtÞ dt dðz2 ðtÞÞ ¼ a10 z1 ðtÞa11 sigðz1 ðtÞÞa=2a a20 z2 ðtÞa21 sigðz2 ðtÞÞa dt
ð32Þ
is finite time convergent with respect to the origin (0, 0), i.e., z1 ðtÞ 0 and z2 ðtÞ 0
for t Z ts
where 0 o a o1, a10, a20, a11 and a21 are positive constants, ts is time constant.
ð33Þ
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Convergent velocity
States
Fig. 5. Convergent velocity in nonlinear form.
Convergent velocity
States
Fig. 6. Convergent velocity in rapid-convergent form.
Proof. We select the Lyapunov function candidate as: Vðz1 ðtÞ,z2 ðtÞÞ ¼
a11 ð2aÞ 1 2=ð2aÞ 9z1 ðtÞ9 þ ða10 z21 þ z22 ðtÞÞ 2 2
ð34Þ
Therefore, we have dðVðz1 ðtÞ,z2 ðtÞÞÞ ¼ a11 sigðz1 ðtÞÞa=ð2aÞ z2 ðtÞ þ z2 ðtÞða10 z1 ðtÞa11 sigðz1 ðtÞÞa=ð2aÞ dt a20 z2 ðtÞa21 sigðz2 ðtÞÞa Þ þ a10 z1 ðtÞz2 ðtÞ ¼ a21 9z2 ðtÞ9
1þa
a20 z22 ðtÞ o 0
ð35Þ
Note that the unique of the solution of the system can be obtained from [16]. Then based on LaSalle invariant theory, the system (32) is asymptotically stable. From [30], a10z1(t) a20z2(t) is locally uniformly bounded, whereas the right-hand side of the nominal model (22) is continuous and globally homogeneous of degree q¼ko0 with respect to dilation r ¼(r1, r2). Hence, the condition kþr2 r0, required by Theorem 3.2 in [30], is satisfied, and Theorem 3.2 in [30] is applicable to the globally equiuniformly asymptotically stable (32). By applying Theorem 3.2 in [30], (32) is thus globally equiuniformly finite time stable. The proof of Lemma is completed. & Remark 4. From [32] and [30], system dðz1 ðtÞÞ ¼ z2 ðtÞ dt dðz2 ðtÞÞ ¼ a10 z1 ðtÞa11 sigðz1 ðtÞÞa1 a20 z2 ðtÞa21 sigðz2 ðtÞÞa2 dt
ð36Þ
is finite time convergent with respect to the origin (0, 0), i.e., z1 ðtÞ 0 and z2 ðtÞ 0
for t Z ts
ð37Þ
where 0o a2 o1, and 1 4 a1 4 a2/(2 a2), a10, a20, a11 and a21 are positive constants, ts is time constant. The process of convergent velocity for the system (32) or (36) is shown by solid line in Fig. 6. From Fig. 6, we can find that the convergent velocities keep rapid whether the states approach to the equilibrium point or not, that is to say, the rapid-convergent velocities can be guaranteed at all times.
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5.2. Proof of Theorem 2 Taking Laplace transformation of linear differentiator (17), we have sX 1 ðsÞ ¼ X 2 ðsÞ
e2 sX 2 ðsÞ ¼ a10 ðX 1 ðsÞVðsÞÞa20 eX 2 ðsÞ Therefore,
e2 sX 2 ðsÞ ¼ a10
X 2 ðsÞ VðsÞ a20 eX 2 ðsÞ s
ð38Þ
ð39Þ
Then, we have sVðsÞ e2 s2 þa20 es ¼ 1þ X 2 ðsÞ a10
ð40Þ
X 2 ðsÞ s ¼ VðsÞ 1 þ ððe2 s2 þ a20 esÞ=a10 Þ
ð41Þ
i.e.,
It is clear that lim e-0
X 2 ðsÞ s ¼ lim ¼s VðsÞ e-0 1þ ððe2 s2 þa20 esÞ=a10 Þ
ð42Þ
_ It means that x2(t) approximates the derivative vðtÞ. This ends the proof of Theorem 2.U Remark 4. In fact, the differentiator (17) is equivalent to the one presented in [4]. Select x1 ¼ w1 ea2 w2 =a1 ,
x2 ¼ w2
ð43Þ
we have _ 1 ¼ w2 a2 ðw1 vðtÞÞ=e w _ 2 ¼ a1 ðw1 vðtÞÞ=e2 w y ¼ w2
ð44Þ
From (41), we find that the differentiation is achieved over a limited frequency range 1/e. State x2(t) of differentiator (17) is the output of a concatenated ideal 1-order differentiator and a low-pass filter of order 2. By increasing the differentiation order 2, noise is more attenuated, but bandwidth becomes smaller than the usual one. The controllable canonical form (17) seems to be so interesting when the differentiator is used in closed-loop configurations. In addition, we see that v(t) just appears in differentiator (17), so a great amount of eventual additive noise shall be eliminated because of the presence of the successive 2 integrators. Linear differentiator (17) is the boundary layer of system (21), and the convergent velocities of the state variables are slow in the nonlinear region of the system dynamics, the lagging phenomenon is inevitable. This is confirmed by Fig. 4. For linear differentiator (17), we also find that when the tracking error is large, i.e., far from the origin, its behavior is strong. On the other hand, when tracking error is small, i.e., around the origin, its behavior is weak. In order to remove the slow convergence in the nonlinear region of the system dynamics, we present an algorithm of nonlinear differentiator (18), and we given the convergence proof of differentiator (18) in the following. 5.3. Proof of Theorem 3. From Lemma 2, we know that system (22) is finite-time-convergent with respect to the origin with a finite time Tf. And from Theorem 1 in [34], we have the result (20) for nonlinear differentiator (18). Remark 5. Based on finite-time convergence of (22) (in Remark 3), we have another simple differentiator as follows: x_ 1 ¼ x2
e2 x_ 2 ¼ sigðx1 vðtÞÞa1 a21 sigðex2 Þa2 y ¼ x2
ð45Þ
where 0 o a2 o1, and 14 a1 4 a2/(2 a2). Differentiators (36) and (38) are the boundary layers of system (18) and (30), respectively. However, rapid convergence is local from Fig. 2. We can find that when the tracking error of (18) is large, i.e., far from the origin, the gains in (18) are small. On the other hand, when the tracking error is small, i.e., around the origin, the gains are large. Therefore, the behavior of nonlinear differentiator (18) is weak far from the origin, and contrarily, the behavior around the origin is strong (see Fig. 5).
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The differentiator (15) can not only reduce sufficiently chattering phenomenon of derivative estimation, but also its dynamical performances are improved by adding linear correction terms to the nonlinear ones. In the following, we will give the convergence of differentiator (15). 5.4. Proof of Theorem 1 From Lemma 3, we know that system (32) is finite-time-convergent with respect to the origin with a finite time Tf. And from Theorem 1 in [34], we have the result (16) for rapid-convergent differentiator (15). Remark 6. Based on finite-time convergence of (36) (in Remark 4), we have another simple differentiator as follows: x_ 1 ¼ x2 e2 x_ 2 ¼ a10 ðx1 vðtÞÞa11 sigðx1 vðtÞÞa1 a20 ex2 a21 sigðex2 Þa2 y ¼ x2
ð46Þ
where 0o a2 o1, and 1 4 a1 4 a2/(2 a2), a10, a20, a11 and a21 are positive constants. Differentiators (15) and (39) are the boundary layers of system (32) and (36), respectively. We can find that differentiator (15) has the rapid convergence at all times from Fig. 6, and due to the simple construct, the computation time is very short. Moreover, because of the continuity of the system, no chattering phenomena happen. This type of differentiator is adapted to the systems requiring rapid convergence. 6. Robustness analysis of finite-time-convergent differentiator Here, we will give the robustness of finite-time-convergent differentiator (11). Theorem 4. For finite-time-convergent differentiator (11), if there exists a noise in signal v(t), that is v(t)¼v0(t) þd(t), where v0(t) is the desired second-order derivative signal, and d(t) is a bounded noise and is satisfied with 9d(t)9rhd, we have xi vði1Þ ðtÞ ¼ Oðergi þ 1 Þ under the condition that e o(c/2Md)1/r and hd o ðc2ep M d=2MaÞ1=pd . Proof. Differentiator (11) can be written as dx1 ¼ x2 dt dxn1 ¼ xn dt dx en n ¼ f ðx1 vðtÞdðtÞ, ex2 ,. . ., en1 xn Þ dt
ð47Þ
Let e1 ¼ x1 v0 ðtÞ,e2 ¼ x2 v00 ðtÞ,. . .,en ¼ xn vðnÞ 0 ðtÞ The error system is de1 ¼ e2 dt den1 ¼ en dt ! n1 n dv v d v n den n1 n1 d ,. . ., e e2 þ e ¼ f e1 dðtÞ, ee2 þ e e en n n1 dt dt dt dt
ð48Þ
System (41) can be written as de1 ¼ ee2 dt=e den2 en1 ¼ en1 en dt=e
! n1 n den1 en dv d v d v ,. . ., en1 en þ en1 n1 en n ¼ f e1 dðtÞ, ee2 þ e dt dt dt=e dt
ð49Þ
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Let t
t¼ , e z1 ðtÞ ¼ e1 ðtÞ,z2 ðtÞ ¼ ee2 ðtÞ,. . .,zn ðtÞ ¼ en1 en ðtÞ, z ¼ z1
...
zn
T
,dðtÞ ¼ dðtÞ
we have z¼ X(e)e, where X(e) ¼diag{1,e,y,e
ð50Þ n1
}. From (43), system (42) can be transferred as
dz1 ¼ z2 dt dzn1 ¼ zn dt
! n1 n dzn dv d v d v ¼ f z1 dðtÞ,z2 þ ,. . .,zn þ n1 n dt dt dt dt We find that system (44) is the perturbation system of (10). Select Lyapunov be ðV3zÞðtÞ and let z0 ¼ z2 Assumption 2, we know that ðV3zÞðtÞ is Lipschitz, and its’ Lipschitz constant is M. Therefore, we have
n T n1 @V 0 f z1 ,z2 þ dv ,. . .,zn þ ddtn1v ddtvn ðzÞ z D þ ðV3zÞðtÞ ¼ dt @z (
n T iT @V n1 @V h 0 d v z0 f z1 dðtÞ,z2 þ dv ,. . .,zn þ ddtn1v f ðz1 ,z2 ,. . .,zn Þ þ ðzÞ z ðzÞ ¼ dt @z @z dtn
!
n n1 i ri h iT X
d v r
d v r V_ þ Ma 9dðtÞ9 d þ z0 f ðz1 ,z2 ,. . .,zn Þ þ M
n
i
dt dt i¼1
ri " # " #
i n
n 1 n 1 X X r
rd rd iri d v n d v n i iri _ _ ¼ V þ M a9dðtÞ9 þa e i þ e n r V þ M ahd þ a hi e þ hn e
dt dt i¼1 i¼1
ð51Þ ...
zn . From
ð52Þ
where hi 40 is the up boundness of 9dvi0 =dt i 9. Set r ¼ minfiri g. From Assumption 3, we know rdA(0,1] and P ri d ¼ a n1 i ¼ 1 hi þ hn . Therefore, we can get " # n1 X r r D þ ðV3zÞðtÞ r V_ þ er M a hi i þ hn en ¼ V_ þ er M d þMahd d ð53Þ i¼1
It is required that the perturbation in system (44) be continuous, and this requirement hold in each [tj 1,tj), j¼1,y,kþ1. Therefore, there exists a constant G 40(G 4Tf), for tj/e 4 t Ztj 1/e þ G(z(tj 1/e), j¼ 1,y,kþ 1, satisfied with [30, Theorem 5.2 and Proposition 2.4] r ð1yÞ=y r ð2e M d þ2Mahd d Þ=c ðVðzðtÞÞÞ1y r ð54Þ :zðtÞ: r rcð1yÞ rcð1yÞ
y can be chosen to be sufficiently small and (1 y)/y can be sufficiently large, therefore, we have r
2er M d þ2Mahd d o1 c Therefore, z(t) is sufficiently small. From (55), we have 1=rd c2er M d hd o 2Ma
ð55Þ
ð56Þ
Therefore, when e o(c/2Md)1/r and hd oðc2er M d=2MaÞ1=rd , noise can be restrained sufficiently. 7. Reduction of peaking phenomenon In rapid-convergent differentiator, peaking phenomena happen due to the infinity e. In order to reduce peaking phenomena sufficiently, we choose e as ( mt if 0 rt r t max 1=e ¼ ð57Þ mtmax otherwise where m and tmax are chosen according to the desired maximum error that depends on the value of gmax ¼ mtmax.
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8. Simulations and experiments 8.1. Convergences of three types of systems Because (17), (18) and (15) are the boundary layer of systems (21), (22) and (32), respectively, we give the convergences of (21), (22) and (32) firstly, and their convergences are shown in Fig. 7. ( ( ( z_ 1 ¼ z2 z_ 1 ¼ z2 z_ 1 ¼ z2 , , z_ 2 ¼ sigðz1 Þ0:5 sigðz2 Þ0:5 z_ 2 ¼ z1 sigðz1 Þ0:5 z2 sigðz2 Þ0:5 z_ 2 ¼ z1 z2 8.2. Three types of differentiator In the following, we select the functions of sin t and triangular wave, respectively, as the input signal v(t). (1) Linear differentiator (differentiator (17)). Parameters: a10 ¼ 5, a11 ¼2, e ¼1/300. The derivative estimations of linear differentiator are shown in Fig. 8. x_ 1 ¼ x2 1 2 x2 x_ 2 ¼ 5ðx1 vðtÞÞ 300 3002 y ¼ x2 (2) Nonlinear differentiator (differentiator (18)). Parameters: e ¼1/300, a11 ¼5, a21 ¼2, a ¼ 0.5. The derivative estimations of nonlinear differentiator are shown in Fig. 9. x_ 1 ¼ x2 1 2
x_ 2 ¼ 5sigðx1 vðtÞÞ0:5 2 sig
300 y ¼ x2
x 0:5 2 300
Fig. 7. Convergences of three systems: (a) linear convergence, (b) nonlinear convergence and (c) linear-nonlinear convergence.
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Fig. 8. Linear differentiator: (a) tracking ‘‘sin t’’ wave, (b) the magnified figure of (a) and (c) tracking ‘‘triangular’’ wave.
Fig. 9. Nonlinear differentiator: (a) tracking ‘‘sin t’’ wave, (b) the magnified figure of (a) and (c) tracking ‘‘triangular’’ wave.
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(3) Rapid-convergent differentiator (differentiator (15)). Parameters: a10 ¼5, a11 ¼2, a20 ¼ 0.5, a21 ¼0.5, e ¼1/300, a1 ¼ a2 ¼0.5. x_ 1 ¼ x2 x 0:5 1 x2 2 0:5sig x_ ¼ 5ðx1 vðtÞÞ0:5sigðx1 vðtÞÞ0:5 2 2 2 300 300 300 y ¼ x2
The derivative estimations of rapid-convergent differentiator are shown in Fig. 10. From the simulations above, for the representative input signal v(t)¼ sin t, we can see that the convergent time ts ¼0.025 s in linear differentiator based on singular perturbation technique, computation time: 0.8 s, however, the vibrating phenomenon is obvious; the convergent time ts ¼0.18 s in nonlinear differentiator based on singular perturbation technique, computation time: 1 s, the lagging phenomenon happen; the convergent time ts ¼0.01 s in rapidconvergent differentiator based on singular perturbation technique, computation time: 1.2 s, the effect of tracking has high precision and no chattering phenomenon happen. 8.3. Estimation of uncertainty with noise If there exists noise in an uncertain system, we can use the differentiator to estimate the uncertainty, and noise can be restrained sufficiently. We consider the following uncertain system: x_ ¼ x þu þ d yun ¼ x þ x
Fig. 10. Rapid-convergent differentiator: (a) tracking ‘‘sin t’’ wave, (b) the magnified figure of (a) and (c) tracking ‘‘triangular’’ wave.
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X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 28 (2012) 414–431
where xAR is the state, u is the control input, yun is the measurement output, d is the uncertainty, x is the noise. We use the differentiator to estimate the uncertainty d, i.e.,
d x_ þxu Suppose u¼0.1 sin t, and d ¼ cos t. We use the following rapid-convergent differentiator: x_ 1 ¼ x2 x_ 2 ¼ 0:05 452 ðx1 yun Þ0:015 452 sigðx1 yun Þ0:6 0:3 45x2 0:015 452 sig
x 0:6 2
45
y ¼ x2 The measurement output with noise is shown in Fig. 11, and the estimation of uncertainty by rapid-convergent differentiator is shown in Fig. 12. We can find that the uncertainty can be estimated with satisfying precision in spite of noise existing in the measurement output.
Fig. 11. Measurement output with noise.
Fig. 12. Estimation of uncertainty.
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8.4. Experiment for rapid-convergent differentiator In the following, we use rapid-differentiator to estimate, respectively, the rotor acceleration from rotor speed and velocity of convertor voltage from convertor voltage for a servo system with direct current double loop speed driving. The simulink of control system is given in Fig. 13, and the equipments of servo system are shown in Fig. 14. The rotor speed and convertor voltage are set sinusoidal and ‘‘triangular’’ waveforms, respectively, and the rapidconvergent differentiator is compiled in Digital Signal Processor 6713 (DSP6713), the differential equation is carried out by the method of 4-order RungeKutta. For the apparatus in Fig. 14, through an oscillograph, we can obtain the derivative signals shown in Fig. 15. From Fig. 15, we can find that though the input signals have much noise due to the effect of mechanical vibration and other reasons, the derivative estimations obtained from rapid-convergent differentiator still have satisfying quality, and the noises are restrained sufficiently by the differentiator. Not only chattering phenomenon and noises can be reduced sufficiently, but also dynamical performances are improved effectively.
9. Conclusions In this paper, we present an algorithm of rapid-convergent differentiator based on singular perturbation technique and compare it with other approaches of differentiation. Because of its continuous structure, and consists of linear and
Fig. 13. Simulink of servo system.
DSP6713
Fig. 14. Equipment of servo system.
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Fig. 15. An experiment for rapid-convergent differentiator: (a) tracking ‘‘sin t’’ wave and (b) tracking ‘‘triangular’’ wave.
nonlinear parts, not only chattering phenomenon and noises can be reduced sufficiently, but also dynamical performances are improved effectively. However, the parameters in rapid-convergent differentiator are not optimal, and the differentiator is not suitable to delayed signal. Our future work is to design a robust differentiator for delayed signal and use adaptive methods to optimize differentiators. References [1] S.C. Pei, J.J. Shyu, Design of FIR Hilbert transformers and differentiators by eigenfilter, IEEE Trans. Acoust. Speech Signal Process. 37 (1989) 505–511. [2] B. Kumar, S.C.D. Roy, Design of digital differentiators for low frequencies, Proc. IEEE 76 (1988) 287–289. [3] L.R. Rabiner, K. Steiglitz, The design of wide-band recursive and nonrecursive digital differentiators, IEEE Trans. Audio Electroacoust. 18 (1970) 204–209. [4] H. Kahlil, Robust servomechanism output feedback controller for feedback linearizable systems, Automatica 30 (1994) 1587–1599. [5] A. Levant, Robust exact differentiation via sliding mode technique, Automatica 34 (1998) 379–384. [6] A. Levant, High-order sliding modes, differentiation and output-feedback control, Int. J. Control 76 (2003) 924–941. [7] Li-Xin Wang, Fuzzy systems are universal approximation, in: Proceedings of the IEEE International Conference on Fuzzy Systems, San Diego, 1992, pp. 1163–1170. [8] J. Park, I.W. Sandberg, Universal approximation using radial-basis-function networks, Neural Comput. (1991) 246–257. [9] H.Kim Young, Frank L. Lewis, Chaouki T. Abdallahs, A dynamic recurrent neural network-based adaptive observer for a class of nonlinear systems, Automatica 33 (No. 8) (1997) 1539–1543. [10] Shuzhi S. Ge, C.C. Hang, L.C. Woon, Adaptive neural network control of robot manipulators in task space, IEEE Trans. Ind. Electron. 44 (6) (1997) 746–752. [11] Gang Feng, A compensating scheme for robot tracking based on neural networks, Robotics Auton. Syst. 15 (1995) 199–206. [12] Shaocheng Tong, Jiantao Tang, Tao Wang, Fuzzy adaptive control of multivariable nonlinear systems, Fuzzy Sets Syst. 111 (2000) 153–167. [13] Yuzheng Guo, Peng-Yung Woo, An adaptive fuzzy sliding mode controller for robotic manipulators, IEEE Trans. Syst. Man Cybern.—Part A: Syst. Hum. 33 (2) (2003) 149–159. [14] F.L. Lewis, Woo Kam Tim, Li-Zin Wang, Z.X. Li, Deadzone compensation in motion control systems using adaptive fuzzy logic control, IEEE Trans. Control Syst. Technol. 7 (6) (1999) 731–742. [15] Byung Kook Yoo, Woon Chul Ham, Adaptive control of robot manipulator using fuzzy compensator, IEEE Trans. Fuzzy Syst. 8 (NO. 2, April) (2000) 186–199. [16] Rastko R. Selmic, Frank L. Lewis, Deadzone compensation in motion control systems using neural networks, IEEE Trans. Autom. Control 45 (No. 4, April) (2000) 602–613. [17] S. Purwar, I.N. Kar, A.N. Jha, Adaptive control of robot manipulators using fuzzy logic systems under actuator constraints, Fuzzy Sets Syst. 152 (2005) 651–664. [18] Cheng-Chi Wang, Her-Terng Yau, Application of a hybrid numerical method to the bifurcation analysis of a rigid rotor supported by a spherical gas journal bearing system, Nonlinear Dyn. 51 (2008) 515–528. [19] Cheng-Chi Wang, Her-Terng Yau, Chaotic analysis and control of microcandilevers with PD feedback using differential transformation method, Int. J. Nonlinear Sci. Numer. Simulation 10 (2009) 425–444. [20] Cheng-Chi Wang, Application of a hybrid numerical method to the nonlinear dynamic analysis of a micro gas bearing system, Nonlinear Dyn. 59 (2010) 695–710. [21] Her-Terng Yau, Cheng-Chi Wang, Dynamics analysis and fuzzy logic controller design of atomic force microscope system with uncertainties, J. Optoelectron. Adv. Mater. 11 (2009) 1178–1184. [22] J. Moffat, The system dynamics of future warfare, Eur. J. Oper. Res. 90 (1996) 609–618. [23] A. Pisano, E. Usai, Output-feedback control of an underwater vehicle prototype by higher-order sliding modes, Automatica 40 (2004) 1525–1531. [24] P.V. Kokotovic, H.K. Khalil, J.O. Reilly, Singular Perturbation Methods in Control: Analysis and Design, Academic, New York, 1998. [25] Isidori, S.S. Sastry, P.V. Kokotovic, C.I. Byrnes, Singular perturbed zero dynamics of nonlinear systems, IEEE Trans. Autom. Control 37 (1992) 1625–1631. [26] A. Saberi, H. Khalil, Quadratic-type Lyapunov functions for singularly perturbed systems, IEEE Trans. Autom. Control 29 (1984) 542–550. [27] J.I. Lee, I.J. Ha, A novel approach to control of nonminimum-phase nonlinear systems, IEEE Trans. Autom. Control 47 (2002) 1480–1486. [28] H.K. Khalil, Nonlinear Systems, Second ed., Prentice-Hall, New York, 1996. [29] S.J. Wilson, P.A. Webley, Perturbation techniques for accelerated convergence of cyclic steady state (CSS) in oxygen VSA simulations, Chem. Eng. Sci. 57 (2002) 4145–4159.
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Contents lists available at SciVerse ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Rapid-convergent nonlinear differentiator Xinhua Wang n, Bijan Shirinzadeh Robotics and Mechatronics Research Laboratory, Department of Mechanical and Aerospace Engineering, Monash University, Clayton, VIC, 3800, Australia
a r t i c l e in f o
abstract
Article history: Received 28 February 2011 Received in revised form 27 September 2011 Accepted 28 September 2011 Available online 29 October 2011
A nonlinear differentiator being fit for rapid convergence is presented, which is based on singular perturbation technique. The differentiator design can not only sufficiently reduce the chattering phenomenon of derivative estimation by introducing a continuous power function, but the dynamical performances are also improved by adding linear correction terms to the nonlinear ones. Moreover, strong robustness ability is obtained by integrating nonlinear items and the linear filter. The merits of the rapidconvergent differentiator include the excellent dynamical performances, restraining noises sufficiently, avoiding the chattering phenomenon and being not based on system model. The theoretical results are confirmed by computer simulations and an experiment. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Differentiator Singular perturbation Rapid convergence Chattering phenomenon
1. Introduction Differentiation of signals is an old and well-known problem [1–3] and has attracted more attention [4–6] in recent years. Rapidly and accurately obtaining the velocities and accelerations of tracked targets is crucial for several kinds of systems with correct and timely performances, such as the missile-interception systems in defense systems [22] and underwater vehicle systems [23]. Therefore, the convergence rates, robustness and computation time of differentiators are very significant. Difference method is used to estimate approximately the derivatives of signals. Because disturbances exist in almost all the signals, the satisfying estimation precision cannot be obtained by difference method. Kalman filter can reduce the disturbance and obtain the derivatives of signals. However, the model of plant must be required. This constrains the types of signals. Therefore, it is very important to construct an estimator with filtering ability and without knowing the model of plant. Usually the uncertainties of nonlinear systems are approximated by some intelligent algorithms, such as fuzzy systems or radial-based-function (RBF) networks. Over the last decade, control researchers tried to apply intelligent methodologies to the design of estimation and control for uncertain nonlinear systems [7–17]. In paper [7], a conclusion was given that fuzzy systems are universal approximation. The Stone–Weierstrass Theorem is used to prove that fuzzy systems with product inference, centroid defuzzification, and Gaussian membership function are capable of approximating any real continuous function on a compact set to arbitrary accuracy. Paper [8] reported on a related study of radial-basis-function (RBF) networks, and it was proved that RBF networks having one hidden layer are capable of universal approximation. The main difficulties with estimating uncertainties by these algorithms are: (1) the parameters or the neural network weights are difficult to be regulated; (2) membership function and Gaussian function are selected by experiences; (3) all of the system states must be required for estimation and feedback; (4) noises cannot be restrained by the above approximation
n
Corresponding author. E-mail address: [email protected] (X. Wang).
0888-3270/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2011.09.026
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415
algorithms. In [18–20], the differential transformation method (DTM) and the finite difference method (FDM) are used to analyze the nonlinear dynamic behavior of a rigid rotor supported by a micro gas bearing system (MGBS), and in [21], a nonlinear rule based fuzzy logic controller is proposed to control the nonlinear dynamic behavior of the probe tip of an atomic force microscope system (AFMs). However, the abilities of restraining noise were not considered. Differentiators are usually used to obtain the information of tracked targets or the information of the plants, which are not modeled easily. For instance, the position of a tracked target is measured by radar, and its velocity can be carried out by a differentiator. Differentiators are the estimators, which are not based on the model of plant. The popular high-gain differentiators [4] provide for an exact derivative when their gains tend to infinity. Unfortunately, their sensitivity to small high-frequency noise also infinitely grows. With any finite gain values such a differentiator has also a finite bandwidth. Thus, being not exact, it is, at the same time, insensitive with respect to high-frequency noise. Such insensitivity may be considered both as advantage or disadvantage depending on the circumstances. In [5,6], a differentiator via second-order (or high-order) sliding modes algorithm has been proposed. The information one needs to know on the signal is an upper bound for Lipschitz constant of the derivative of the signal. Although second-order sliding mode is introduced and there exists no chattering phenomenon in signal tracking. However, for the robust exact differentiator, in the second dynamical equation, a switching function exists. The output of derivative estimation is continuous but not smooth. Therefore, chattering phenomenon still exists in derivative estimation. In [34], we presented a finite-time-convergent differentiator that is based on singular perturbation technique [24–29]. The merits of this differentiator exist in three aspects: rapidly finite-time convergence compared with other typical differentiators; no chattering phenomenon; and besides the derivatives of the derivable signals, the generalized derivatives of some classes of signals can be obtained. However, the differentiators in [34] are complicated and difficult to be used in practice. Moreover, the computation time is still large and its robustness is not considered. In this paper, we expand our work in [34]. The proposed rapid-convergent differentiator is an integration of a nonlinear term (comprising of continuous power function) and a linear correction term. The structure of the designed differentiator is simple and the computation time is short. In the nonlinear part, a continuous power function with a perturbation parameter is used. Therefore, chattering phenomenon can be avoided in the output of derivative estimation. Strong robustness ability is obtained by integrating sliding mode items and the linear filter. The linear part can restrain highfrequency noises by giving a suitable nature frequency, and small bounded noises can be restrained by the nonlinear items. Moreover, the dynamical performances are also improved. This paper is organized in the following format. In Section 2, problem statement is given, and preliminaries are given in Section 3. In Section 4, our main results are presented. In Section 5, the main results are proved in detail. In Section 6, robustness analysis of finite-time-convergent differentiator is given. In Section 7, reduction of peaking phenomenon is given. In Section 8, the simulations and experiments are given, and our conclusions are made in Section 9. 2. Problem statement When we design a controller for a system, state feedback is used to place the poles. In order to use the system states to design feedback controller, sensors must be adopted to measure the states. However, not all states can be measured directly. Therefore, state observer is presented to estimate the system states. These observers are designed under the condition that the system model must be known. We present the following three questions. Question 1. For arbitrary signals, which cannot be expressed by mathematical expressions, such as the signal shown in Fig. 1, how can we estimate the derivate of the signal? Question 2. How can we obtain the derivative of a signal with noise? For instance, the following signal in Fig. 2. Question 3. For system x_ 1 ¼ x2 x_ 2 ¼ u y ¼ x1
ð1Þ
we require that the output y tracks the desired trajectory yd(t) shown in Fig. 3. Let e1 ¼ x1 yd ðtÞ,
e2 ¼ x2 y_ d ðtÞ
ð2Þ
Fig. 1. Stochastic signal.
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Fig. 2. Signal with noises.
v(t)
y Fig. 3. Trajectory tracking.
we get the tracking error system as follows: e_ 1 ¼ e_ 2 e_ 2 ¼ uy€ d
ð3Þ
Before designing contoller u, how can we obtain the information of y_ d and y€ d ? Design of usual observer is based on the model of system. However, most signals are difficult to be described by a given model. Therefore, using usual observer to estimate derivative of signal is restricted. We expect to construct the following form: x_ 1 ¼ x2 x_ 2 ¼ f ðx1 vðtÞ,x2 Þ
ð4Þ
Under the condition that the solution exists for the above differential equation, a differentiator can be constructed if the _ following relation is satisfied: x1 converges to v(t) and x2 converges to vðtÞ. We are interested in designing a rapid-convergent differentiator to estimate the derivative of signal, and highfrequency noise can be restrained sufficiently. 3. Preliminaries First of all, the related concepts are given. In order to demonstrate the finite-time convergence, we introduce the following definitions, assumptions and theorems from the corresponding references. Definition 1. [35] ‘‘Generalized derivative’’ denotes that the left and right derivatives of a point in a function trajectory both exist, and they may be not equal to each other. Definition 2. [33] Consider a time-invariant system in the form of x_ ¼ f ðxÞ,
f ð0Þ ¼ 0,
x 2 Rn
ð5Þ
where f:D-Rn is continuous on open neighborhood DCRn of the origin. The origin is said to be a finite-time-stable equilibrium of the above system if there exists an open neighborhood N DD of the origin and a function Tf:N\{0}-(0,N), called the settling-time function, such that the following statements hold: (1) Finite-time-convergence: For every xAN\{0}, cx is the flow starting from x and defined on [0,Tf(x)), cx(t)AN\{0} for all x tA[0,Tf(x)), and limt-T f ðxÞ c ðtÞ ¼ 0. (2) Lyapunov stability: For every open neighborhood Ue of zero, there exists an open subset Ud of N containing zero such that, for every xAUd\{0}, cx(t)AUe for all tA[0,Tf(x)). The origin is said to be a globally finite-time-stable equilibrium if it is a finite-time-stable equilibrium with D ¼ N¼ Rn. Then the system is said to be finite-time-convergent with respect to the origin.
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Assumption 1. Suppose the origin is a finite-time-stable equilibrium of [33, theorem 4.3] of z_~ 1 ¼ z~ 2 ... z~_ ¼ z~ n n1
z_~ n ¼ f ðz~ 1 , z~ 2 ,. . ., z~ n Þ
ð6Þ
And the settling-time function Tf is continuous at zero, where f(U) is continuous and f(0). Let N be as in Definition 2 and let yA(0,1). Then there exists a continuous scalar function V such that (1) V is positive definite and (2) V_ is real valued and continuous on N and there exists c 40 such that V_ þ cV y r0
for any y 2 ð0,1Þ
ð7Þ
Assumption 2. There exists a Lipschitz Lyapunov function V satisfying (4) with Lipschitz constant M. Assumption 3. For (6), there exists riA(0,1], i¼0,1,y,n 1, and a nonnegative constant a such that 9f ðz~ 1 , z~ 2 ,. . ., z~ n Þf ðz1 ,z2 ,. . .,zn Þ9 ra
n X
ri 1
9z~ i zi 9
ð8Þ
i¼1
where z~ i ,zi 2 R, i ¼ 1,. . .,n. Assumption 4. v(t) is a continuous and piecewise n-order derivable signal with the following properties. The derivatives of v(t) up to order n 2 exist on the whole time domain and v(t) is not (n 1)-order differentiable at some instants tj,j ¼1,y,k. However, the (n 1)-order left derivative vðn1Þ ðt j Þ and right derivative vðn1Þ ðt j Þ exist [35], respectively, and þ ðn1Þ ðt j Þavðn1Þ ðt j Þ, j ¼ 1,. . .,k, may be satisfied. v þ
Remark 1. There are a number of nonlinear functions actually satisfying this assumption. For example, one such function r
r
is xi since 9xri xri 9 r 21ri 9xx9 i , ri 2 ð0,1. Moreover, there are smooth functions also satisfying this property. In fact, it is easy to verify that 9sin xsin x9 r 29xx9
ri
for any riA(0,1].
Theorem 3.1. in [30] Let the differential equation z_ ðtÞ ¼ f ðz1 ðtÞ,. . .,zn ðtÞÞ
ð9Þ
be locally homogeneous of degree qo0 with respect to dilation (r1,y,rn) and let the equilibrium point z ¼0 of (9) be globally uniformly asymptotically stable. Then the differential Eq. (9) is globally uniformly finite time stable. Theorem 1. in [34] If z_~ 1 ¼ z~ 2 ... ¼ z~ n z_~ n1
z_~ n ¼ f ðz~ 1 , z~ 2 ,. . ., z~ n Þ
ð10Þ
is satisfied with Assumptions 1–3, signal v(t) is satisfied with Assumption 4. Then for system dx1 ¼ x2 dt ... dxn1 ¼ xn dt dx en n ¼ f ðx1 vðtÞ, ex2 ,. . ., en1 xn Þ dt
ð11Þ
there exist g 40 (where rg 4n) and G 40 such that xi vði1Þ ðtÞ ¼ Oðergi þ 1 Þ
ð12Þ
for tj 4t Ztj 1 þ eG(X(e)e þ (tj 1)), j¼1,y,kþ1, (let t0 ¼0 and tk þ 1 ¼ N) i¼1,y,n with xn ðt j Þvðn1Þ ðtÞ ¼ Oðergn þ 1 Þ,
j ¼ 1,. . .,k
ð13Þ rg i þ 1
rg i þ 1
where e 40 is the perturbation parameter and O(e ) denotes the approximation of e order [28] between xi and T v(i 1)(t); and g ¼(1 y)/y, y 2 ð0,minfr=ðr þ nÞ,1=2gÞ, nZ2. ei ¼xi v(i 1),i¼1,y,n. e ¼ e1 en , e þ ðt j1 Þ ¼ h iT e1 ðt j1 Þ . . . en1 ðt j1 Þ enþ ðt j1 Þ , enþ ðt j1 Þ ¼ xn vðþn1Þ ðt j1 Þ, and X(e) ¼diag{1,e,y,en 1}.
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The practical form of differentiator in [34] is x_ 1 ðtÞ ¼ x2 ðtÞ
e2 x_ 2 ðtÞ ¼ u a=ð2aÞ
u ¼ sateb fsignðfa ðx1 vðtÞ, ex2 Þ9fa ðx1 vðtÞ, ex2 Þ9
a
Þgsatev fsignðx2 Þ9ex2 9 g
where (
2a
fa ðx1 vðtÞ, ex2 Þ ¼ x1 vðtÞ þ
signðx2 Þ9ex2 9 2a
,
sateb ðxÞ ¼
x,
x,
eb signðxÞ, 9x9 Z eb
Its structure is too complicated. In the following, we will design more simple differentiators based on Theorem 1 in [34], and the designed rapid-convergent differentiator can keep more rapidly convergent at all times. Denotations: a sigðyÞa ¼ 9y9 sgnðyÞ, a 40. It is obvious that sigðyÞa ¼ ya only if a ¼q/p. where p,q are positive odd numbers. Moreover, d aþ1 9y9 ¼ ða þ 1ÞsigðyÞa , dy
d a sigðyÞa þ 1 ¼ ða þ 1Þ9y9 dy
ð14Þ
In the following, we will give our main results of rapid-convergent differentiator. 4. Rapid-convergent differentiator Here, we will design a nonlinear differentiator being fit for rapid convergence. The proposed differentiator consists of linear and nonlinear parts. It is shown that for arbitrary signal, there exists a differentiator achieving finite-time error decay. Moreover, high-frequency noise can be restrained sufficiently. We design a rapid-convergent differentiator as follow: Theorem 1. The rapid-convergent differentiator is designed as: x_ 1 ¼ x2
e2 x_ 2 ¼ a10 ðx1 vðtÞÞa11 sigðx1 vðtÞÞa=ð2aÞ a20 ex2 a21 sigðex2 Þa y ¼ x2
ð15Þ
For a continuous and piecewise two-order derivative signal v(t), there exists g 4 0 (where rg 42 and r ¼ minfa, a=ð2aÞg ¼
a=ð2aÞ ), such that xi vði1Þ ðtÞ ¼ Oðergi þ 1 Þ
ð16Þ
for t Z eG(X(e)e(0)),i¼1, and tj 4t 4 Ztj 1 þ eG(X(e)e þ (tj 1)), j ¼1,y,kþ 1, i¼2, respectively, with x2 ðt j Þv0 ðt j Þ ¼ Oðerg1 Þ, j ¼ 1,. . .,k , where aA(0,1), e 40 is the perturbation parameter and O(erg i þ 1) denotes the approximation of erg i þ 1 h iT T þ order [28] between xi and v(i 1)(t); ei ¼xi v(i 1),i ¼1,2. e ¼ e1 e2 , e þ ðt j1 Þ ¼ e1 ðt j1 Þ e2 ðt j1 Þ , e2þ ðt j1 Þ ¼ x2 v0þ ðt j1 Þ , and X(e) ¼diag{1,e}. Eq. (16) denotes that variable xi track (i 1)-order derivative of signal v(t). The rapid-convergent differentiator (15) consists of linear and nonlinear differentiator given, respectively, as follows: x_ 1 ¼ x2
e2 x_ 2 ¼ a10 ðx1 vðtÞÞa20 ex2 y ¼ x2
ð17Þ
and x_ 1 ¼ x2
e2 x_ 2 ¼ a11 sigðx1 vðtÞÞa=ð2aÞ a21 sigðex2 Þa y ¼ x2
ð18Þ
For linear differentiator (17), we have the following conclusion: Theorem 2. For linear differentiator (17) and a signal v(t) (where v(t) is a continuous and piecewise second-order derivable signal, we have that _ ¼ OðeÞ x2 ðtÞvðtÞ
for t-1
_ where a10, a20 40, e 40 is perturbation parameter, O(e) denotes the approximation of e order [28] between x2 and vðtÞ.
ð19Þ
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For nonlinear differentiator (18), we have the following conclusion: Theorem 3. For nonlinear differentiator (18) and a continuous and piecewise two-order derivative signal v(t), there exists g 4 0 (where rg 4 2 and r ¼ minfa, a=ð2aÞg ¼ a=ð2aÞ,), such that xi vði1Þ ðtÞ ¼ Oðergi þ 1 Þ
ð20Þ
for t Z eG(X(e)e(0)),i ¼1, and tj 4t 4 Ztj 1 þ eG(X(e)e þ (tj 1)), j¼ 1,y,kþ 1, i¼2, respectively, with j ¼ 1,. . .,k , where aA(0,1).
x2 ðt j Þv0 ðt j Þ ¼ Oð rg1 Þ,
e
Remark 2. From the analysis of linear systems and non-Lipschitz nonlinear systems, linear differentiator (17) and nonlinear differentiator (18), two notable differences can be observed. First, the two algorithms have quite different converging properties: the linear system converges exponentially, whereas the trajectories of the non-Lipschitz nonlinear algorithm converge in finite time. This is due to the lack of local Lipschitzness of the nonlinear algorithm at the origin, that is, its behavior around the zero state is very strong compared to the linear case. On the other side, the linear correction terms are stronger than the ones of the nonlinear algorithm far from the origin. These differences cause another striking difference between both algorithms: the kind of perturbations that each one is able to tolerate. The main difference is that the linear system can deal with perturbations that are stronger very far from the origin and weaker near the origin than the ones that are endured by the nonlinear algorithm. So, for example, the nonlinear algorithm is not able to endure (globally) a bounded perturbation with linear growth in time, but the linear algorithm can deal with it easily. However, the linear algorithm is not able to support a strong perturbation near the origin, what is one of the main advantages of non-Lipschitz nonlinear algorithm. In order to integrate the merits of linear differentiator (17) and nonlinear differentiator (18), and to restrain their shortcomings, respectively, rapid-convergent differentiator (15) is integrated. The proposed rapid-convergent differentiator is an integration of a nonlinear term (comprising of continuous power function) and a linear correction term. The overall design is an integration of nonlinear non-Lipschitz differentiators (18) and linear differentiator (17) in that a perturbation parameter a is introduced, which takes values (0,1) and an additional linear correction term is appended, so that it inherits the best properties of both. In the nonlinear part, a continuous power function with a perturbation parameter is used. Therefore, chattering phenomenon can be avoided in the output of derivative estimation. Strong robustness ability is obtained by integrating sliding mode items and the linear filter. The linear part can restrain highfrequency noises by giving a suitable nature frequency, and small bounded noises can be restrained by the nonlinear items, at the same time, the nonlinear items can compensate the delay brought by the linear filter. 5. Proof of Theorems 1–3 5.1. Rapid convergence In order to analyze differentiators (15), (17) and (18), firstly, we give the following system convergence Lemmas 1–3: Lemma 1 (Convergence of linear system). System 8 < dðz1 ðtÞÞ ¼ z2 ðtÞ dt
ð21Þ
: dðz2 ðtÞÞ ¼ a10 z1 ðtÞa20 z2 ðtÞ dt is exponentially convergent with respect to the origin (0, 0). Where a10 40 and a20 40 are constants. 2
2
Proof. If the Lyapunov function candidate is selected as V(z1(t), z2(t)) ¼(a10z1(t) þz2(t))/2, then the result of the convergence can be obtained. & The process of convergent velocity for the system (21) is shown in Fig. 4.
Convergent velocity
States
Fig. 4. Convergent velocity in linear form.
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From Fig. 4, the convergent velocities of the states are rapid when they are far away from the equilibrium point. However, the convergent velocities are slow when they approach the equilibrium point. It is required to improve the process of convergent velocities when the states approach the equilibrium point. Lemma 2 (Convergence of nonlinear system). System dðz1 ðtÞÞ ¼ z2 ðtÞ dt dðz2 ðtÞÞ ¼ a11 sigðz1 ðtÞÞa=ð2aÞ a21 sigðz2 ðtÞÞa dt
ð22Þ
is finite time convergent with respect to the origin (0, 0), i.e., z1 ðtÞ 0 and z2 ðtÞ 0 for t Z ts
ð23Þ
where 0 o a o1, a11 and a21 are positive constants, ts is time constant. Proof. Let the Lyapunov function candidate be: a11 ð2aÞ 1 2=ð2aÞ 9z1 ðtÞ9 þ z22 ðtÞ Vðz1 ðtÞ,z2 ðtÞÞ ¼ 2 2
ð24Þ
Therefore, we have dðVðz1 ðtÞ,z2 ðtÞÞÞ 1þa ¼ a11 sigðz1 ðtÞÞa=ð2aÞ z2 ðtÞ þ z2 ðtÞða11 sigðz1 ðtÞÞa=ð2aÞ a21 sigðz2 ðtÞÞa Þ ¼ a21 9z2 ðtÞ9 o0 dt
ð25Þ
Note that the uniqueness of the solution of the system can be obtained from [31]. Then based on LaSalle invariant theory, the system (22) is asymptotically stable. Therefore, based on the negative homogeneity [30, Theorem 3.1], we have that er2 z2 ðtÞ ¼ er1 þ k z2 ðtÞa11 sigðer1 z1 ðtÞÞa=ð2aÞ a21 sig er2 z2 ðtÞ a ¼ er2 þ k ða11 sigðz1 ðtÞÞa=ð2aÞ a21 sigðz2 ðtÞÞa Þ ð26Þ Therefore we have that r 2 ¼ r 1 þk,
r1
a 2a
¼ r2 a ¼ r2 þ k
ð27Þ
i.e., r1 ¼ k
2a 1 , r2 ¼ k a1 a1
ð28Þ
Therefore, system (22) is finite time convergent with respect to the origin (0, 0), i.e., z1 ðtÞ 0 and z2 ðtÞ 0 for t Z ts
ð29Þ
Remark 3. From [32], system dðz1 ðtÞÞ ¼ z2 ðtÞ dt dðz2 ðtÞÞ ¼ sigðz1 ðtÞÞa1 sigðz2 ðtÞÞa2 dt
ð30Þ
is finite time convergent with respect to the origin (0, 0), i.e., z1 ðtÞ 0 and z2 ðtÞ 0 for t Z ts
ð31Þ
where 0 o a2 o1, and 14 a1 4 a2/(2 a2). The process of convergent velocity for the system (22) or (30) is shown by solid line in Fig. 2. From Fig. 5, although the convergent velocities of the states are rapid when they approach the equilibrium point, the convergent velocities of the states are slow when they are far away from the equilibrium point. Therefore, it is required to improve the convergent velocities whether the states are near to the equilibrium point or not, i.e., rapid convergence should be guaranteed at all times. Lemma 3 (Convergence of rapid-convergent system). System dðz1 ðtÞÞ ¼ z2 ðtÞ dt dðz2 ðtÞÞ ¼ a10 z1 ðtÞa11 sigðz1 ðtÞÞa=2a a20 z2 ðtÞa21 sigðz2 ðtÞÞa dt
ð32Þ
is finite time convergent with respect to the origin (0, 0), i.e., z1 ðtÞ 0 and z2 ðtÞ 0
for t Z ts
where 0 o a o1, a10, a20, a11 and a21 are positive constants, ts is time constant.
ð33Þ
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Convergent velocity
States
Fig. 5. Convergent velocity in nonlinear form.
Convergent velocity
States
Fig. 6. Convergent velocity in rapid-convergent form.
Proof. We select the Lyapunov function candidate as: Vðz1 ðtÞ,z2 ðtÞÞ ¼
a11 ð2aÞ 1 2=ð2aÞ 9z1 ðtÞ9 þ ða10 z21 þ z22 ðtÞÞ 2 2
ð34Þ
Therefore, we have dðVðz1 ðtÞ,z2 ðtÞÞÞ ¼ a11 sigðz1 ðtÞÞa=ð2aÞ z2 ðtÞ þ z2 ðtÞða10 z1 ðtÞa11 sigðz1 ðtÞÞa=ð2aÞ dt a20 z2 ðtÞa21 sigðz2 ðtÞÞa Þ þ a10 z1 ðtÞz2 ðtÞ ¼ a21 9z2 ðtÞ9
1þa
a20 z22 ðtÞ o 0
ð35Þ
Note that the unique of the solution of the system can be obtained from [16]. Then based on LaSalle invariant theory, the system (32) is asymptotically stable. From [30], a10z1(t) a20z2(t) is locally uniformly bounded, whereas the right-hand side of the nominal model (22) is continuous and globally homogeneous of degree q¼ko0 with respect to dilation r ¼(r1, r2). Hence, the condition kþr2 r0, required by Theorem 3.2 in [30], is satisfied, and Theorem 3.2 in [30] is applicable to the globally equiuniformly asymptotically stable (32). By applying Theorem 3.2 in [30], (32) is thus globally equiuniformly finite time stable. The proof of Lemma is completed. & Remark 4. From [32] and [30], system dðz1 ðtÞÞ ¼ z2 ðtÞ dt dðz2 ðtÞÞ ¼ a10 z1 ðtÞa11 sigðz1 ðtÞÞa1 a20 z2 ðtÞa21 sigðz2 ðtÞÞa2 dt
ð36Þ
is finite time convergent with respect to the origin (0, 0), i.e., z1 ðtÞ 0 and z2 ðtÞ 0
for t Z ts
ð37Þ
where 0o a2 o1, and 1 4 a1 4 a2/(2 a2), a10, a20, a11 and a21 are positive constants, ts is time constant. The process of convergent velocity for the system (32) or (36) is shown by solid line in Fig. 6. From Fig. 6, we can find that the convergent velocities keep rapid whether the states approach to the equilibrium point or not, that is to say, the rapid-convergent velocities can be guaranteed at all times.
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5.2. Proof of Theorem 2 Taking Laplace transformation of linear differentiator (17), we have sX 1 ðsÞ ¼ X 2 ðsÞ
e2 sX 2 ðsÞ ¼ a10 ðX 1 ðsÞVðsÞÞa20 eX 2 ðsÞ Therefore,
e2 sX 2 ðsÞ ¼ a10
X 2 ðsÞ VðsÞ a20 eX 2 ðsÞ s
ð38Þ
ð39Þ
Then, we have sVðsÞ e2 s2 þa20 es ¼ 1þ X 2 ðsÞ a10
ð40Þ
X 2 ðsÞ s ¼ VðsÞ 1 þ ððe2 s2 þ a20 esÞ=a10 Þ
ð41Þ
i.e.,
It is clear that lim e-0
X 2 ðsÞ s ¼ lim ¼s VðsÞ e-0 1þ ððe2 s2 þa20 esÞ=a10 Þ
ð42Þ
_ It means that x2(t) approximates the derivative vðtÞ. This ends the proof of Theorem 2.U Remark 4. In fact, the differentiator (17) is equivalent to the one presented in [4]. Select x1 ¼ w1 ea2 w2 =a1 ,
x2 ¼ w2
ð43Þ
we have _ 1 ¼ w2 a2 ðw1 vðtÞÞ=e w _ 2 ¼ a1 ðw1 vðtÞÞ=e2 w y ¼ w2
ð44Þ
From (41), we find that the differentiation is achieved over a limited frequency range 1/e. State x2(t) of differentiator (17) is the output of a concatenated ideal 1-order differentiator and a low-pass filter of order 2. By increasing the differentiation order 2, noise is more attenuated, but bandwidth becomes smaller than the usual one. The controllable canonical form (17) seems to be so interesting when the differentiator is used in closed-loop configurations. In addition, we see that v(t) just appears in differentiator (17), so a great amount of eventual additive noise shall be eliminated because of the presence of the successive 2 integrators. Linear differentiator (17) is the boundary layer of system (21), and the convergent velocities of the state variables are slow in the nonlinear region of the system dynamics, the lagging phenomenon is inevitable. This is confirmed by Fig. 4. For linear differentiator (17), we also find that when the tracking error is large, i.e., far from the origin, its behavior is strong. On the other hand, when tracking error is small, i.e., around the origin, its behavior is weak. In order to remove the slow convergence in the nonlinear region of the system dynamics, we present an algorithm of nonlinear differentiator (18), and we given the convergence proof of differentiator (18) in the following. 5.3. Proof of Theorem 3. From Lemma 2, we know that system (22) is finite-time-convergent with respect to the origin with a finite time Tf. And from Theorem 1 in [34], we have the result (20) for nonlinear differentiator (18). Remark 5. Based on finite-time convergence of (22) (in Remark 3), we have another simple differentiator as follows: x_ 1 ¼ x2
e2 x_ 2 ¼ sigðx1 vðtÞÞa1 a21 sigðex2 Þa2 y ¼ x2
ð45Þ
where 0 o a2 o1, and 14 a1 4 a2/(2 a2). Differentiators (36) and (38) are the boundary layers of system (18) and (30), respectively. However, rapid convergence is local from Fig. 2. We can find that when the tracking error of (18) is large, i.e., far from the origin, the gains in (18) are small. On the other hand, when the tracking error is small, i.e., around the origin, the gains are large. Therefore, the behavior of nonlinear differentiator (18) is weak far from the origin, and contrarily, the behavior around the origin is strong (see Fig. 5).
X. Wang, B. Shirinzadeh / Mechanical Systems and Signal Processing 28 (2012) 414–431
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The differentiator (15) can not only reduce sufficiently chattering phenomenon of derivative estimation, but also its dynamical performances are improved by adding linear correction terms to the nonlinear ones. In the following, we will give the convergence of differentiator (15). 5.4. Proof of Theorem 1 From Lemma 3, we know that system (32) is finite-time-convergent with respect to the origin with a finite time Tf. And from Theorem 1 in [34], we have the result (16) for rapid-convergent differentiator (15). Remark 6. Based on finite-time convergence of (36) (in Remark 4), we have another simple differentiator as follows: x_ 1 ¼ x2 e2 x_ 2 ¼ a10 ðx1 vðtÞÞa11 sigðx1 vðtÞÞa1 a20 ex2 a21 sigðex2 Þa2 y ¼ x2
ð46Þ
where 0o a2 o1, and 1 4 a1 4 a2/(2 a2), a10, a20, a11 and a21 are positive constants. Differentiators (15) and (39) are the boundary layers of system (32) and (36), respectively. We can find that differentiator (15) has the rapid convergence at all times from Fig. 6, and due to the simple construct, the computation time is very short. Moreover, because of the continuity of the system, no chattering phenomena happen. This type of differentiator is adapted to the systems requiring rapid convergence. 6. Robustness analysis of finite-time-convergent differentiator Here, we will give the robustness of finite-time-convergent differentiator (11). Theorem 4. For finite-time-convergent differentiator (11), if there exists a noise in signal v(t), that is v(t)¼v0(t) þd(t), where v0(t) is the desired second-order derivative signal, and d(t) is a bounded noise and is satisfied with 9d(t)9rhd, we have xi vði1Þ ðtÞ ¼ Oðergi þ 1 Þ under the condition that e o(c/2Md)1/r and hd o ðc2ep M d=2MaÞ1=pd . Proof. Differentiator (11) can be written as dx1 ¼ x2 dt dxn1 ¼ xn dt dx en n ¼ f ðx1 vðtÞdðtÞ, ex2 ,. . ., en1 xn Þ dt
ð47Þ
Let e1 ¼ x1 v0 ðtÞ,e2 ¼ x2 v00 ðtÞ,. . .,en ¼ xn vðnÞ 0 ðtÞ The error system is de1 ¼ e2 dt den1 ¼ en dt ! n1 n dv v d v n den n1 n1 d ,. . ., e e2 þ e ¼ f e1 dðtÞ, ee2 þ e e en n n1 dt dt dt dt
ð48Þ
System (41) can be written as de1 ¼ ee2 dt=e den2 en1 ¼ en1 en dt=e
! n1 n den1 en dv d v d v ,. . ., en1 en þ en1 n1 en n ¼ f e1 dðtÞ, ee2 þ e dt dt dt=e dt
ð49Þ
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Let t
t¼ , e z1 ðtÞ ¼ e1 ðtÞ,z2 ðtÞ ¼ ee2 ðtÞ,. . .,zn ðtÞ ¼ en1 en ðtÞ, z ¼ z1
...
zn
T
,dðtÞ ¼ dðtÞ
we have z¼ X(e)e, where X(e) ¼diag{1,e,y,e
ð50Þ n1
}. From (43), system (42) can be transferred as
dz1 ¼ z2 dt dzn1 ¼ zn dt
! n1 n dzn dv d v d v ¼ f z1 dðtÞ,z2 þ ,. . .,zn þ n1 n dt dt dt dt We find that system (44) is the perturbation system of (10). Select Lyapunov be ðV3zÞðtÞ and let z0 ¼ z2 Assumption 2, we know that ðV3zÞðtÞ is Lipschitz, and its’ Lipschitz constant is M. Therefore, we have
n T n1 @V 0 f z1 ,z2 þ dv ,. . .,zn þ ddtn1v ddtvn ðzÞ z D þ ðV3zÞðtÞ ¼ dt @z (
n T iT @V n1 @V h 0 d v z0 f z1 dðtÞ,z2 þ dv ,. . .,zn þ ddtn1v f ðz1 ,z2 ,. . .,zn Þ þ ðzÞ z ðzÞ ¼ dt @z @z dtn
!
n n1 i ri h iT X
d v r
d v r V_ þ Ma 9dðtÞ9 d þ z0 f ðz1 ,z2 ,. . .,zn Þ þ M
n
i
dt dt i¼1
ri " # " #
i n
n 1 n 1 X X r
rd rd iri d v n d v n i iri _ _ ¼ V þ M a9dðtÞ9 þa e i þ e n r V þ M ahd þ a hi e þ hn e
dt dt i¼1 i¼1
ð51Þ ...
zn . From
ð52Þ
where hi 40 is the up boundness of 9dvi0 =dt i 9. Set r ¼ minfiri g. From Assumption 3, we know rdA(0,1] and P ri d ¼ a n1 i ¼ 1 hi þ hn . Therefore, we can get " # n1 X r r D þ ðV3zÞðtÞ r V_ þ er M a hi i þ hn en ¼ V_ þ er M d þMahd d ð53Þ i¼1
It is required that the perturbation in system (44) be continuous, and this requirement hold in each [tj 1,tj), j¼1,y,kþ1. Therefore, there exists a constant G 40(G 4Tf), for tj/e 4 t Ztj 1/e þ G(z(tj 1/e), j¼ 1,y,kþ 1, satisfied with [30, Theorem 5.2 and Proposition 2.4] r ð1yÞ=y r ð2e M d þ2Mahd d Þ=c ðVðzðtÞÞÞ1y r ð54Þ :zðtÞ: r rcð1yÞ rcð1yÞ
y can be chosen to be sufficiently small and (1 y)/y can be sufficiently large, therefore, we have r
2er M d þ2Mahd d o1 c Therefore, z(t) is sufficiently small. From (55), we have 1=rd c2er M d hd o 2Ma
ð55Þ
ð56Þ
Therefore, when e o(c/2Md)1/r and hd oðc2er M d=2MaÞ1=rd , noise can be restrained sufficiently. 7. Reduction of peaking phenomenon In rapid-convergent differentiator, peaking phenomena happen due to the infinity e. In order to reduce peaking phenomena sufficiently, we choose e as ( mt if 0 rt r t max 1=e ¼ ð57Þ mtmax otherwise where m and tmax are chosen according to the desired maximum error that depends on the value of gmax ¼ mtmax.
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8. Simulations and experiments 8.1. Convergences of three types of systems Because (17), (18) and (15) are the boundary layer of systems (21), (22) and (32), respectively, we give the convergences of (21), (22) and (32) firstly, and their convergences are shown in Fig. 7. ( ( ( z_ 1 ¼ z2 z_ 1 ¼ z2 z_ 1 ¼ z2 , , z_ 2 ¼ sigðz1 Þ0:5 sigðz2 Þ0:5 z_ 2 ¼ z1 sigðz1 Þ0:5 z2 sigðz2 Þ0:5 z_ 2 ¼ z1 z2 8.2. Three types of differentiator In the following, we select the functions of sin t and triangular wave, respectively, as the input signal v(t). (1) Linear differentiator (differentiator (17)). Parameters: a10 ¼ 5, a11 ¼2, e ¼1/300. The derivative estimations of linear differentiator are shown in Fig. 8. x_ 1 ¼ x2 1 2 x2 x_ 2 ¼ 5ðx1 vðtÞÞ 300 3002 y ¼ x2 (2) Nonlinear differentiator (differentiator (18)). Parameters: e ¼1/300, a11 ¼5, a21 ¼2, a ¼ 0.5. The derivative estimations of nonlinear differentiator are shown in Fig. 9. x_ 1 ¼ x2 1 2
x_ 2 ¼ 5sigðx1 vðtÞÞ0:5 2 sig
300 y ¼ x2
x 0:5 2 300
Fig. 7. Convergences of three systems: (a) linear convergence, (b) nonlinear convergence and (c) linear-nonlinear convergence.
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Fig. 8. Linear differentiator: (a) tracking ‘‘sin t’’ wave, (b) the magnified figure of (a) and (c) tracking ‘‘triangular’’ wave.
Fig. 9. Nonlinear differentiator: (a) tracking ‘‘sin t’’ wave, (b) the magnified figure of (a) and (c) tracking ‘‘triangular’’ wave.
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(3) Rapid-convergent differentiator (differentiator (15)). Parameters: a10 ¼5, a11 ¼2, a20 ¼ 0.5, a21 ¼0.5, e ¼1/300, a1 ¼ a2 ¼0.5. x_ 1 ¼ x2 x 0:5 1 x2 2 0:5sig x_ ¼ 5ðx1 vðtÞÞ0:5sigðx1 vðtÞÞ0:5 2 2 2 300 300 300 y ¼ x2
The derivative estimations of rapid-convergent differentiator are shown in Fig. 10. From the simulations above, for the representative input signal v(t)¼ sin t, we can see that the convergent time ts ¼0.025 s in linear differentiator based on singular perturbation technique, computation time: 0.8 s, however, the vibrating phenomenon is obvious; the convergent time ts ¼0.18 s in nonlinear differentiator based on singular perturbation technique, computation time: 1 s, the lagging phenomenon happen; the convergent time ts ¼0.01 s in rapidconvergent differentiator based on singular perturbation technique, computation time: 1.2 s, the effect of tracking has high precision and no chattering phenomenon happen. 8.3. Estimation of uncertainty with noise If there exists noise in an uncertain system, we can use the differentiator to estimate the uncertainty, and noise can be restrained sufficiently. We consider the following uncertain system: x_ ¼ x þu þ d yun ¼ x þ x
Fig. 10. Rapid-convergent differentiator: (a) tracking ‘‘sin t’’ wave, (b) the magnified figure of (a) and (c) tracking ‘‘triangular’’ wave.
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where xAR is the state, u is the control input, yun is the measurement output, d is the uncertainty, x is the noise. We use the differentiator to estimate the uncertainty d, i.e.,
d x_ þxu Suppose u¼0.1 sin t, and d ¼ cos t. We use the following rapid-convergent differentiator: x_ 1 ¼ x2 x_ 2 ¼ 0:05 452 ðx1 yun Þ0:015 452 sigðx1 yun Þ0:6 0:3 45x2 0:015 452 sig
x 0:6 2
45
y ¼ x2 The measurement output with noise is shown in Fig. 11, and the estimation of uncertainty by rapid-convergent differentiator is shown in Fig. 12. We can find that the uncertainty can be estimated with satisfying precision in spite of noise existing in the measurement output.
Fig. 11. Measurement output with noise.
Fig. 12. Estimation of uncertainty.
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8.4. Experiment for rapid-convergent differentiator In the following, we use rapid-differentiator to estimate, respectively, the rotor acceleration from rotor speed and velocity of convertor voltage from convertor voltage for a servo system with direct current double loop speed driving. The simulink of control system is given in Fig. 13, and the equipments of servo system are shown in Fig. 14. The rotor speed and convertor voltage are set sinusoidal and ‘‘triangular’’ waveforms, respectively, and the rapidconvergent differentiator is compiled in Digital Signal Processor 6713 (DSP6713), the differential equation is carried out by the method of 4-order RungeKutta. For the apparatus in Fig. 14, through an oscillograph, we can obtain the derivative signals shown in Fig. 15. From Fig. 15, we can find that though the input signals have much noise due to the effect of mechanical vibration and other reasons, the derivative estimations obtained from rapid-convergent differentiator still have satisfying quality, and the noises are restrained sufficiently by the differentiator. Not only chattering phenomenon and noises can be reduced sufficiently, but also dynamical performances are improved effectively.
9. Conclusions In this paper, we present an algorithm of rapid-convergent differentiator based on singular perturbation technique and compare it with other approaches of differentiation. Because of its continuous structure, and consists of linear and
Fig. 13. Simulink of servo system.
DSP6713
Fig. 14. Equipment of servo system.
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Fig. 15. An experiment for rapid-convergent differentiator: (a) tracking ‘‘sin t’’ wave and (b) tracking ‘‘triangular’’ wave.
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