# A Nonlinear Exact Disturbance Observer Inspired by Sliding Mode Techniques

Authors: Xinkai Chen
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

Inspired by sliding mode techniques, a nonlinear exact disturbance observer is proposed. The disturbance and its derivatives up to the second order are assumed to be bounded. However, the bounds of the disturbance and its derivatives are unknown, and they are adaptively estimated online during the observation of the disturbances. The exact convergence of the disturbance observer to the genuine disturbance is assured theoretically. The convergence speed of the disturbance estimation error is controlled by design parameters. The proposed method is robust to the type of disturbance and is easy to be implemented. Computer simulation results show the superiority and effectiveness of the proposed formulation.


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## Body

## 1. Introduction

In all practical control systems, the existence of unknown disturbances, model uncertainties, and uncertain nonlinearities is inevitable. In the past decades, disturbance observation has been one of the major topics in control engineering. The motivation is suggested by the fact that if the disturbances can be estimated, then control of the uncertain dynamic systems with disturbances may become easier. For example, the controller with disturbance cancellation functions can be easily constructed by using the estimated disturbances [1–6]. The disturbance observer approaches in the literature have been developed either in the transfer function domain or in the state space domain.Due to the intuitive and simple logic of the disturbance observers based on the inversion of transfer function, it has been applied in the tracking controllers for practical motion control systems [7–9]. In this kind of design, an “inner loop” around the controlled system is closed in order to reject the disturbances and to urge the relation between the input and the output to approximate the so-called “nominal model” of the controlled system at low frequency domain. The inner loop is tuned by adjusting a low pass filter. One typical shortcoming of this formulation is that the control performance is not satisfactory when the type of the disturbance is unknown and there exist unmodeled dynamics [10].Another approach which is based on the state space domain is also extensively studied in the literature due to its allowance of transient behavior analysis [11–15]. Among the many presented methods, a typical formulation of estimating the disturbance which is based on the variable structure system (VSS) techniques is proposed in [14]. By estimating the unknown states and their derivatives by using the input-output information, eventually, the disturbances in the state space expression are estimated. One shortcoming of this kind of formulation is that a high order observer structure needs to be constructed.From the robustness point of view, the advantages of variable structure control system are well addressed in [16–18]. Thus, by nature, the VSS theory is to be applied to the robust disturbance observer formulation. In the literature, it is well known that, for minimum phase dynamical systems with relative degree one, the disturbance can be estimated by using the VSS “equivalent control theory” [17, 18]. However, this approach cannot cope with systems with higher relative degrees. Furthermore, the VSS equivalent control approach is not theoretically strict since, on the sliding surfaceS(t)=0, the derivative ofS(t)cannot be proved to be zero. To overcome these difficulties, approximate nonlinear disturbance observers are proposed in [19, 20] for systems with arbitrary relative degrees, where strict analysis has been achieved for the disturbance estimating precision.In this paper, inspired by the sliding mode control theory, a nonlinear exact disturbance observer is proposed based on state space approach. The “disturbance” may be the model uncertainties, the external disturbance, the combination of them, and so forth. The disturbance and its derivatives up to the second order are assumed to be bounded. Since these bounds are usually unknown in practice,a priori information of the value of the bounds is not required. In the disturbance estimating process, the bounds relating to the disturbances are simultaneously estimated online by using adaptive method. The contribution of the paper is that an exact asymptotic estimation of the disturbance can be achieved theoretically, whereas only approximate estimate could be obtained in the literature [1, 7–11, 13, 19, 20]. Convergence of the proposed observer is strictly analyzed and proved for arbitrary initial conditions of the disturbance observer. The convergence speed of the disturbance estimation error is controlled by design parameters. The proposed disturbance observer is robust to the types of the disturbances [7, 8, 10] and is easy to be implemented. Comparison with a nonlinear disturbance observer in the literature [19] is conducted, where the transient time and the robustness to measurement noises are considered. Computer simulation results illustrate the effectiveness and superiority of the proposed formulation.


## 2. Problem Statement

Consider an uncertain system of the form(1)x˙t=fxt,ut,t+dt,where x(t)=x1t,…,xn(t)T∈Rnis the state;u(t)∈Ris the input;f(x(t),u(t),t)∈Rnis a known function of t, xt, andu(t);d(t)=d1(t),…,dn(t)T∈Rnis an unknown vector which is composed of external disturbances, model uncertainties, and so forth. For simplicity,d(t)is called “disturbance” in the following part of this paper.This paper tries to propose an exact nonlinear method to estimate the disturbanced(t). For this purpose, the following two assumptions are made.(A1)
The statex(t)is available.(A2)
The signald(t)and its derivatives up to the second order are bounded, where the upper bounds can be unknown (at the undifferentiable points, the right- and left-hand derivatives are meant).


## 3. The Nonlinear Robust Disturbance Observer

The proposed nonlinear disturbance observer in this paper is inspired by the sliding mode method.Based on (1), the following dynamical systems are constructed:(2)x^˙t=fxt,ut,t+wt,x^0=x^0,wherex^(t)=x^1(t),…,x^n(t)T∈Rnis the reconstructed state which can be obtained by solving (2);x^0is the initial value ofx^(t);w(t)=w1(t),…,wn(t)T∈Rnis a vector which will be determined later.Inspired by the robustness of sliding mode control technique, we try to find a formulation ofw(t)such thatx(t)-x^(t)→0andx˙(t)-x^˙(t)→0ast→∞. The componentswi(t)(fori=1,…,n) are given by the following equations: (3)w˙it=-αi+βiwit+χ^it·signxit-x^it+αiβixit-x^it,whereχ^i(t)are defined as(4)χ^it=γixit-x^it+αiqitwith(5)q˙it=xit-x^it;αi,βi, and γiare positive constants;wi(0)can be any values;qi(0)should be positive constants.Theorem 1.
Fori=1,…,n, the generated signalswi(t)andχ^i(t)are uniformly bounded;xi(t)-x^i(t)∈L10,∞ anddi(t)-wi(t)∈L20,∞. Furthermore, it holds that(6)limt→∞xit-x^it=0,(7)limt→∞dit-wit=0.Proof.
Define(8)eit=xit-x^it,(9)σit=e˙it+αieit.Then, differentiatingei(t)andσi(t)yields(10)e˙it=dit-wit,(11)σ˙it=d˙it-w˙it+αidit-wit=ηit-βiσit-χ^it·signeitwith(12)ηit=d˙it+αi+βidit.By Assumption (A2), there exist positive constantsχisuch that(13)ηit+1βiη˙it≤χi.Consider the Lyapunov candidate (14)Vit=12σi2t+12γiχ^it-χi2.DifferentiatingVi(t)yields(15)V˙it=σitηit-βiσit-χ^it·signeit+χ^it-χiddteit+αieit=-βiσi2t+σitηit-e˙it+αieitχ^it·signeit+χ^it-χiddteit+αieit=-βiσi2t+σitηit-χi·ddteit-αiχieit.Integrating both sides of (15) from 0 totyields (16)Vit=Vi0-βi∫0tσi2τdτ+∫0te˙iτ+αieiτηiτdτ-χi∫0tddteiτdτ-α1χ1∫0teiτdτ=Vi0-βi∫0tσi2τdτ+eitηit-ei0ηi0+∫0teiτ-η˙iτ-αiηiτdτ-χieit-αiχi∫0teiτdτ≤Vi0-βi∫0tσi2τdτ+eitηit-χi+∫0teiτη˙iτ+αiηiτ-αiχidτ+χiei0-ei0ηi0≤Vi0-βi∫0tσi2τdτ+χiei0-ei0ηi0,where (13) is employed in the last step. Thus, relation (16) implies thatVi(t)and∫0tσi2(τ)dτare uniformly bounded (i.e.,σi(t)∈L20,∞). Since the uniform boundedness ofVi(t)implies the uniform boundedness ofσi(t)andχ^i(t), the uniform boundedness ofσ˙i(t)can be concluded by observing the expression ofσ˙i(t)in (11) and Assumption (A2). Therefore, by applying Corollary  1.2.2 of Barbalat’s Lemma in [21], it yieldslimt→∞⁡σi(t)=0by observing the uniform boundedness ofσ˙i(t)and∫0tσi2(τ)dτ.
On the other hand, by (4) and (5), it is obvious that the uniform boundedness ofχ^i(t)meansei(t)∈L∞0,∞andei(t)∈L10,∞. Thus, from (9), it can be seen thate˙i(t)=σi(t)-αiei(t)is uniformly bounded. Now, by applying Barbalat’s lemma again, it giveslimt→∞⁡ei(t)=0. Then,limt→∞⁡e˙i(t)=limt→∞⁡σi(t)-αiei(t)=0. Therefore, from (10), relation (7) is proved.
By the fact thatei(t)∈L∞0,∞∩L10,∞, it givesei(t)∈L20,∞(see [21]). By noticinge˙i(t)=σi(t)-αiei(t), it yieldse˙i(t)∈L20,∞. Therefore,di(t)-wi(t)∈L20,∞. The theorem is proved.Remark 2.
The upper boundsχiin (13) need not be known. They are updated by the adaptive algorithms defined in (4)-(5). The parametersαiandγishould be chosen as large constants so that a fast adjustment ofχ^i(t)can be obtained.Remark 3.
σ
i
(
t
)
=
0can be regarded as the sliding surface inei(t),e˙i(t)plane. By the definition ofσi(t)in (9), it can be seen that the parametersαishould be chosen to be large enough in order to get fast convergences ofei(t). By observing (16), it can be seen that the parametersβishould also be chosen to be large enough in order to get a fast convergence ofσi(t).Remark 4.
The initial valuex^0ofx^(t)can be chosen to be any value. Certainly, if the initial valuex(0)of the statex(t)is known, it is convenient to choosex^0=x(0).


## 4. Simulation Results

Consider the system described by(17)x˙t=-0.2xt+dt,where x(t) is the state;d(t)is the disturbance.In simulation study, supposex(0)=2and the disturbanced(t)is (18)dt=cosπt1+0.8x2txt+0.5.Since system (17) is one dimension, we will omit the subscripts in the following. The parameters are chosen asα=β=γ=500. The initial values are chosen asx^(0)=w(0)=0.5, q(0)=0.The computer simulation is conducted by MATLAB, where the sampling period is set to 0.001 seconds. Figure1 shows the disturbanced(t). The estimation errord(t)-w(t)is shown in Figure 2. It can be seen that the convergence is very fast and the estimation error in the steady state is very small. A much smaller estimation error can be achieved by choosing much larger parametersα,β, andγ.Figure 1
The disturbance signald(t).Figure 2
The estimation errord(t)-w(t).
(a)
The estimation error ast is small
(b)
The estimation error at steady stateIn order to test the robustness to measurement noise, suppose that a measurement noisev(t)is accompanied with the state. In the simulation study, suppose the noisev(t), which is shown in Figure 3, is a band-limited random noise whose sampling period is 0.001. By using the same design parameters, the disturbance estimation error is given in Figure 4. Due to the existence of the measurement noises, the estimation error becomes larger.Figure 3
The measurement noisev(t).Figure 4
The estimation errord(t)-w(t)when measurement noise exists.
(a)
The estimation error ast is small
(b)
The estimation error at steady stateNow, let us compare the proposed method with the formulation in [19]. The disturbance observer is given by(19)x^^˙t+λx^t=λ-0.2xt+mt,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiix^^0=0.5,wherem(t)is determined by(20)mt=xt-x^^tC^txt-x^^t+δ;C^(t)is updated by(21)C^˙t=κxt-x^^tifxt-x^^t>2δC^tλ0otherwise,,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiC^0=0.5.In order to get a similar estimation error in the steady state as shown in Figure 2, the parameters are chosen asλ=2,κ=5, and δ=0.006. The estimation errord(t)-m(t)is shown in Figure 5. It can be seen that transient time of the new observer is much shorter than that of the observer in [19] in order to get the same estimation error.Figure 5
The estimation errord(t)-m(t)by using the observer in [19].
(a)
The estimation error for0≤t≤8
(b)
The estimation error at steady stateNow, let us investigate the robustness to measurement noises of the observer in [19]. Suppose the same measurement noise which is shown in Figure 3 exists in the state. By using the same design parameters, the disturbance estimation error is shown in Figure 6. It can be seen that the estimation error becomes much larger than that in Figure 4. By comparing Figures 4 and 6, it can be seen that the new disturbance observer is much more robust to the measurement noises.Figure 6
The estimation errord(t)-m(t)by using the observer in [19] when measurement noise exists.
(a)
The estimation error for0≤t≤8
(b)
The estimation error at steady state


## 5. Conclusions

In this paper, inspired by the sliding mode control theory, a nonlinear exact disturbance observer is proposed based on state space approach. The disturbance and its derivatives up to the second order are assumed to be bounded. In the disturbance estimation, the bounds relating to the disturbances are simultaneously updated online by an adaptive algorithm. The exact estimation of the disturbance can be achieved theoretically. The convergence speed of the disturbance estimation error is controlled by design parameters. The proposed method is robust to the type of disturbance and is easy to be implemented. Computer simulation results show that the proposed method is effective and is superior to the method in [19] in the sense of transient time and robustness to measurement noises. The proposed observer is expected to be developed for the systems in transfer function domain.



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*Source: 651601-2015-05-21.xml*