# Nonlinear Backstepping Control Design for Coupled Nonlinear Systems under External Disturbances

Authors: Wonhee Kim; Chang Mook Kang; Young Seop Son; Chung Choo Chung
Journal: Complexity
 (2019)

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## Abstract

A nonlinear backstepping control is proposed for the coupled normal form of nonlinear systems. The proposed method is designed by combining the sliding-mode control and backstepping control with a disturbance observer (DOB). The key idea behind the proposed method is that the linear terms of state variables of the second subsystem are lumped into the virtual input in the first subsystem. A DOB is developed to estimate the external disturbances. Auxiliary state variables are used to avoid amplification of the measurement noise in the DOB. For output tracking and unmatched disturbance cancellation in the first subsystem, the desired virtual input is derived via the backstepping procedure. The actual input in the second subsystem is developed to guarantee the convergence of the virtual input to the desired virtual input by using a sliding-mode control. The stability of the closed-loop is verified by using the input-to-state stable (ISS) property. The performance of the proposed method is validated via numerical simulations and an application to a vehicle system based on CarSim platform.


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## Body

## 1. Introduction

Control for nonlinear systems has attracted considerable attention, and, therefore, various control methods have been investigated for nonlinear systems. Control methods using the Lyapunov function were proposed for nonlinear systems [1–4]. Input-output linearization and feedback linearization were proposed to transform nonlinear systems to the normal form and to control the nonlinear systems [5, 6]. Sliding-mode control techniques were developed for nonlinear systems owing to the decoupling and invariance properties [7, 8]. Control algorithms based on the singular perturbation theory were developed for nonlinear systems involving fast and slow dynamics [9, 10]. Backstepping method is one of the breakthroughs for the control of nonlinear systems. This method is a recursive procedure using a Lyapunov function and a systematic design approach for special forms of the nonlinear systems (the strict feedback form or the normal form or both) [11]. It can guarantee global stability and improvement of tracking and transient performances. In the past decades, various backstepping methods were widely used to solve the control problems of nonlinear systems. A backstepping control method was developed to improve the force control performance for an electro-hydraulic actuator [12]. An adaptive control technique was implemented to the backstepping controller for unknown disturbance or parameters [13]. An adaptive backstepping sliding-mode controller was designed to improve the tracking performance in the sliding and presliding phases [14]. In [15], an output feedback nonlinear control was proposed for a hydraulic system with mismatched modeling uncertainties; in this control, an extended state observer (ESO) and a nonlinear robust controller are synthesized via the backstepping method. A recurrent fuzzy neural network backstepping control was proposed for the prescribed output tracking performance of nonlinear dynamic systems [16]. An adaptive robust control using ESO was developed for the DC motor control [17]. An ESO-based backstepping was proposed to improve the output-tracking performance with external disturbance using only output feedback [18]. Active disturbance rejection adaptive control schemes were proposed for both parametric uncertainties and uncertain nonlinearities of the nonlinear systems [19, 20]. An observer-based backstepping control method using reduced lateral dynamics was developed for autonomous lane-keeping system [21].Let us consider a class of nonlinear systems coupled with two normal form subsystems as follows:(1)x˙1=x2⋮x˙r-1=xrx˙r=frxa+∑i=r+1naixi+d1x˙r+1=xr+2⋮x˙n-1=xnx˙n=fnx+gxu+d2y=x1where x=[x1,x2,…,xn]T=[xa,xb]T∈Rn is the state variable vector, xa=[x1,x2,…,xr]T∈Rr, xb=[xr+1,xr+2,…,xn]T∈Rn-r, y=x1 denotes the output, an is nonzero constant, the input function g(x) is always positive or negative for all x(t), and d1(t) and d2(t) are disturbances. In this paper, this class of systems (1) is called the coupled normal form in nonlinear systems. This nonlinear system (1) is not in the general normal form because xr+1 cannot be regarded as the virtual input in xr dynamics. Furthermore, the disturbance d1 is in xr dynamics. Thus, previous backstepping control methods cannot be used directly for tracking control of these systems. Several methods were presented to solve the control problem of nonlinear systems [22, 23]. In [22], two nonlinear control techniques using backstepping and sliding mode techniques were applied to an autonomous microhelicopter. Recently, a robust nonlinear control was developed for the synchronization control of cross-strict feedback hyperchaotic systems [23]. Unfortunately, because systems in [22, 23] have multi-inputs, these techniques cannot solve the control problems for the coupled normal form in a nonlinear system (1).In this paper, we propose a nonlinear backstepping control for the coupled normal form of nonlinear systems. The proposed method combines a sliding mode technique and backstepping control with the disturbance observer (DOB). The key idea of the proposed method is that the terms∑i=r+1n‍aixi are lumped into the virtual input in the first subsystem. A DOB is developed to estimate the external disturbances. Auxiliary state variables are used to avoid amplification of the measurement noise in the DOB. For output tracking and unmatched disturbance cancellation in the first subsystem, the desired virtual input is derived via the backstepping procedure. The actual input in the second subsystem is developed to guarantee the convergence of the virtual input to the desired virtual input by using a super twisting algorithm (STA). The stability of the closed-loop is proven by using the input-to-state stable (ISS) property. The performance of the proposed method is validated via numerical simulations and an application to a vehicle system based on CarSim platform.


## 2. Disturbance Observer Design

In this paper, we describe the situation that satisfies the following Assumptions1 and 2.Assumption 1.
d 1, d2 and their derivatives are also bounded.In general, prior information about the derivative of the disturbances is unknown but bounded, at least locally [24]. The unknown constants d˙1max and d˙2max exist such that (2)d˙1≤d˙1maxd˙2≤d˙2max.Assumption 2.
The polynomialansn-r-1+an-1sn-r-2+…+ar+1 is Hurwitz.Most real systems that are in the form of (1) satisfy Assumption 2. For example, the lateral dynamics of a vehicle with differential braking force input satisfy Assumption 2 [25]. Thus, this assumption is not strict for actual physical systems.From (1), the disturbances, d1 and d2, can be rewritten as (3)d1=x˙r-frxa-∑i=r+1naixid2=x˙n-fnx-gxu.We define the estimations of the disturbances, d^1 and d^2. The dynamics of d^1 and d^2 are designed as (4)d^˙1=1ε1x˙r-frxa-∑i=r+1naixi-d^1d^˙2=1ε2x˙n-fnx-gxu-d^2where 1/ε2 and 1/ε3 are the observer gains. The estimation errors are defined as (5)d~1=d1-d^1d~2=d2-d^2.From, (3), (4), and (5), the estimation error dynamics can be derived as (6)d~˙1=d˙1-d^˙1=d˙1-1ε1x˙r-frxa-∑i=r+1naixi-d^1=d˙1-1ε1d1-d^1=-1ε1d~1+d˙1d~˙2=d˙2-d^˙2=d˙2-1ε2x˙n-fnx-gxu-d^2=d˙2-1ε2d2-d^2=-1ε2d~2+d˙2.To suppress the bounded derivatives of the disturbances, the high gains, i.e., the low values of ε1 and ε2, are required. In practice, measurement noises do appear in sensors. The dynamics of d^1 and d^2 (4) employ the derivative of the state. If high observer gains are used, the noise is amplified by the high gains. Thus, the observer is not practical for implementation. To avoid the use of the derivative of the state, we use the auxiliary state variables, ξ1, ξ2.Theorem 3.
With Assumption1, given the auxiliary state variables, ξ1, ξ2 such as (7)ξ1=d^1-xrε1ξ2=d^2-xnε2the dynamics of the auxiliary state variables are (8)ξ˙1=-1ε1ξ1+xrε1+1ε1-frxa-∑i=r+1naixiξ˙2=-1ε2ξ2+xnε2+1ε2-fnx-gxu.Then, |d~i(t)|≤exp-1/2εit|d~i(0)|+2εid˙imax for i∈[1,2].Proof.
Differentiating the auxiliary state variables with respect to time gives(9)ξ˙i=d^˙i-x˙jεifor i∈[1,2] and j∈{r,n}. From (1), (5), (7), and (8), the disturbance estimation error dynamics are obtained as (10)d~˙i=-1εid~i+d˙i.In (10), the dynamics of d~i2 are (11)ddtd~i22=-1εid~i2+d~d˙i≤-12εid~i2-12εid~id~i-2εid˙iThe following result is thus derived from (11), using Lemma 6.20 and Theorem C.2 in [11]: (12)d~it≤exp⁡-12εitd~i0+2εisup0≤τ≤td˙iτ≤exp⁡-12εitd~i0+2εid˙imax.The upper bound of |d~i(∞)| thus decreases as εi gets smaller.Remark 4.
The proposed DOB (8) with the auxiliary state variable (7) does not require the derivatives of states, i.e., x˙r and x˙n, to obtain d˙1 and d˙2. Thus, if (7) and (8) are used to estimate the disturbances instead of (4), amplification of the measurement noise by the high gain can be reduced, such that it is negligible in practice.


## 3. Sliding Mode Backstepping Controller Design

In this paper, the control goal is to determineu that makes the output y=x1 track the desired reference trajectory yd=x1d, which is assumed to have continuous derivatives up to the nth order. The tracking error eii∈[1,r] is defined as (13)e1=x1-yde2=x˙1-y˙de3=x¨1-y¨d⋮er=x1r-1-ydr-1.To eliminate the steady-state error, the integral error e0 is defined as (14)e0=∫0te1τdτ.The error dynamics from e1 to er can be written as (15)e˙0=e1e˙1=e2⋮e˙r-1=ere˙r=frxa+∑i=r+1naixi+d1-ydr.The linear combination of the tracking errors, s1, is designed in terms of the error (16)s1=er+∑i=0r-1σieiwhere the coefficients σ0,…,σr-1 are chosen such that the polynomial sr+σr-1sr-1+⋯+σ0 is Hurwitz. From (15) and (16), we obtain s˙1 as (17)s˙1=frxa+∑i=r+1naixi+d1-ydr+∑i=0r-1σiei+1.We define the terms, ∑i=r+1n‍aixi, as the virtual input of the first subsystem in (1): (18)s2=∑i=r+1naixi.Equation (17) then becomes (19)s˙1=frxa+d1-ydr+∑i=0r-1σiei+1+s2d+z2where s2d is the desired virtual input and the sliding surface z2=s2-s2d. The desired virtual input, s2d, is designed as (20)s2d=-frxa-d1+ydr-∑i=0r-1σiei+1-ks1s1where ks1 is positive and constant. The derivative of z2 with respect to time is (21)z˙2=s˙2-s˙2d=∑i=r+1n-1aixi+1+anx˙n-s˙2d=∑i=r+1n-1aixi+1+anfnx+gxu+d2-s˙2d.The input is designed using STA as (22)u=-1angx∑i=r+1n-1aixi+1-s˙2d+ϕ1z2-ϕ2z2-1gxfnx+d2where ϕ1(z2)=kz1|z2|1/2sgn(z2), ϕ˙2(z2)=-kz2sgn(z2), and kz1 and kz2 are positive constants. In order to avoid the chattering problem, STA [26, 27] is applied to the controller (22).Theorem 5.
With Assumption2, suppose that the control law, (20) and (22), is applied to system (1). Then the output tracking error e1 converges to zero and xb is ultimately bounded.Proof.
Step  1. From (19) and (20), we have (23)s˙1=-ks1s1+z2.By defining the positive-definite Lyapunov function V1 as (24)V1=12s12we obtain (25)V˙1=-ks1s12+z2s1.Using z2 as the input and s1 as the output in (23) gives (26)z2︸inputs1︸output=V˙1+ks1s12︸≥0.Then (26) shows that the relationship between s1 and z2 is strictly output passive [9] and s˙1=-ks1s1 is zero-state observable. Therefore, s1 system is ISS. With control law (22), the dynamics of z2 and ϕ2 become (27)z˙2=-kz1z21/2sgnz2+ϕ2ϕ˙2=-kz2sgnz2.We define the vector ζ=ζ1ζ2T=[|z2|1/2sgn(z2),ϕ2]T. The derivative of ζ with respect to time is (28)ζ˙=1ζ1Aζζwhere (29)Aζ=-12kz112-kz20and |ζ1|=|z2|1/2. Because kz1>0 and kz2>0, Aζ is Hurwitz. We define the Lyapunov candidate function Vζ as (30)Vζ=ζTPζζwhere Pζ is positive definite. The derivative of ζ with respect to time is given by (31)V˙ζ=-1ζ1ζTQζζwhere Qζ is positive definite such that AζTPζ+PζAζ=-Qζ. From [27], the origin ζ=0 is finite-time stable. Consequently z2 is equal to zero, identically, after a finite time interval.
Step  2. With z2=0, (32)s˙1=-ks1s1.Then, (16) can be rewritten as (33)e˙0=e1⋮e˙r-1=erer=-∑i=0r-1σiei+s1.Equation (33) is simplified as (34)e˙a=Aaea+Bas1where ea=[e0,e1,…,er-1]T∈Rr, (35)Aa=010⋯00001⋯00000⋯00⋮⋮⋮⋱⋮⋮000⋯01-σ0-σ1-σ2⋯-σr-2-σr-1,Ba=0,0,…,0,1T.Because Aa is Hurwitz, ea is bounded-input bounded output (BIBO) stable. With the convergence of s1 to zero, e0, e1, …, er-1 converge to zeros. Consequently, er=-σ0e0-σ1e1-⋯-σr-1er-1 also converges to zero.
Step  3. With z2=0, (36)s2=s2d.Because e0, e1, …, er-1 converge to zeros and yd has continuous derivatives up to the nth order, ξ=-fr(xa)+yd(r)-d1-∑i=1r-1‍σiei+1 is bounded. Thus, a positive constant ξmax exists such that ξmax=suptξ(t). From (1), (18), and (36), we obtain (37)x˙r+1=xr+2⋮x˙n-1=xnanxn=-∑i=r+1n-1aixi+ξ.Equation (37) is simplified as (38)x˙bz=Abxbz+Bbξwhere xbz=[xr+1,xr+2,…,xn-1]T∈Rn-r-1, (39)Ab=010⋯00001⋯00000⋯00⋮⋮⋮⋱⋮⋮000⋯01-ar+1an-ar+2an-ar+3an⋯-an-2an-an-1an,Bb=0,0,…,0,1anT.We define the positive-definite Lyapunov function Vb as (40)Vb=xbzTPbxbz.Because Ab is Hurwitz, a positive definite matrix Pb exists such that (41)AbTPb+PbAb=-I.The derivative of Vb is (42)V˙b≤-xbzTxbz+2xbzTPbBbξmax≤-xbz22+2λmaxPbξmaxanxbz2≤-1-θ1xbz22-θ1xbz22+2λmaxPbξmaxanxbz2where 0<θ1<1, and λmin(A) and λmax(A) are the minimum and maximum eigenvalues of the matrix A, respectively. Then, (43)V˙b≤-1-θ1xbz22for all xbz2≤μ1 where μ1=2λmax(Pb)ξmax/θ1an. There exists T1 such that (44)xbz2≤λmaxPbλminPbμ1for all t≥T1. Consequently, xb is also ultimately bounded with the ultimate bound of xbz.Remark 6.
Owing toξ in (38), only the boundedness of xb is guaranteed. Furthermore, the convergence rate of xb is fixed by the system parameters. Only the convergence of z2 to z2d is sufficient to make e1 converge to zero, regardless of the convergence rate of xb.Remark 7.
Whenea converges to zero and yd(∞) and d are zero, ξmax becomes zero. Consequently, xb also converges to zero.Remark 8.
In [22, 23], the coupled systems with multi-inputs were dealt; these techniques cannot solve the control problems for the coupled nonlinear system with single input (1). On the other hand, the proposed method (20) and (22) can solve the tracking control problems for the coupled nonlinear system with one input (1).


## 4. Closed-Loop Stability Analysis

Actually, controller (20) and (22) uses the estimated disturbances d^1 and d^2 instead of the disturbances d1 and d2. The controller becomes (45)s2d=-frxa+ydr-d^1-∑i=0r-1σiei+1-ks1s1u=-1angx∑i=r+1n-1aixi+1-s˙2d+ϕ1z2-ϕ2z2-1gxfnx+d^2where ϕ1(z2)=kz1|z2|1/2sgn(z2), ϕ˙2(z2)=-kz2sgn(z2), and ks1, kz1, and kz2 are positive constants. The closed-loop system including controller (45) and observer (4) is given as follows: (46)s˙1=-ks1s1+z2+d~1z˙2=-kz1z21/2sgnz2+ϕ2+d~2ϕ˙2=-kz2sgnz2d~˙1=-1ε1d~1+d˙1d~˙2=-1ε2d~2+d˙2Theorem 9.
With Assumptions1 and 2, suppose that controller (45) and disturbance observer (7) and (8) are used in (1). Further, μz2 and Tz2 exist such that (47)z2≤μz2for all t≥Tz2. The overall tracking error system (46) is the serial interconnected system of the ISS system. As t→∞, (48)s1t≤s1maxd~1t≤2ε1d˙1maxd~2t≤2ε2d˙2maxwhere s1max=2/ks1μz2+2ε1d˙1max. Consequently, ea and xb are ultimately bounded.Proof.
Step  1. In (46), the dynamics of z2 and ϕ2 are (49)z˙2=-kz1z21/2sgnz2+ϕ2+d~2ϕ˙2=-kz2sgnz2.We define the vector ζ=ζ1ζ2T=[|z2|1/2sgn(z2),ϕ2]T. The derivative of ζ with respect to time is (50)ζ˙=1ζ1Aζζwhere (51)Aζ=-12kz112-kz20and |ζ1|=|z2|1/2. Because kz1>0 and kz2>0, Aζ is Hurwitz. We define the Lyapunov candidate function Vζ as (52)Vζ=ζTPζζwhere Pζ is positive definite. The derivative of ζ with respect to time is given by (53)V˙ζ=-1ζ1ζTQζζ+1ζ11/2d~20Pζζwhere Qζ is positive definite such that AζTPζ+PζAζ=-Qζ. From (10) and the definition of Vd~2=d~22, V˙d~2 is given by (54)V˙d~2=-1ε2d~22+d˙2d~2=-1ε2d~22+d˙2maxd~2=-1ε2-θd~2d~22-θd~2d~22+d˙2maxd~2where 0<θd~2<1. Then, (55)V˙d~2≤-1ε2-θd~2d~22for all |d~2|≥μd~2 where μd~2=d˙2max/θd~2. There exists Td~2 such that (56)d~2≤μd~2for all t≥Td~2 When t≥Td~2, (53) becomes (57)V˙ζ≤-1ζ1ζTQζζ+μd~2λPζζ2≤1ζ1-λminQζ-θζζ2-θζζ2+μd~2λPζζ2where 0<θζ<1. Then, (58)V˙ζ≤1ζ1-λminQζ-θζζ2for all ζ2≥μζ where μζ=μd~2λ(Pζ)/θζ. There exists Tζ such that (59)ζ2≤λmaxPζλminPζμζfor all t≥Tζ. Consequently, μz2 and Tz2 exist such that (60)z2≤μz2for all t≥Tz2. In (46), s1 dynamics can be obtained as (61)s˙1=-ks1s1+z2+d~1.Then we obtain the dynamics of s12 as (62)ddts122=-ks1s12+s1z2+d~1≤-ks12s12-ks12s1s1-2ks1z2+d~1.From (62), the following result is derived using Lemma 6.20 and Theorem C.2 in [11]: (63)s1t≤exp-ks12ts10+2ks1sup0≤τ≤tz2τ+d~1τ.Equation (63) shows that the relationship between z2 and d~1, and ei+1 has ISS property. The overall tracking error system (46) is the serial interconnected system of the ISS system. As t→∞, (64)s1t≤s1maxd~1t≤2ε1d˙1maxd~2t≤2ε2d˙2maxwhere s1max=2/ks1μz2+2ε1d˙1max.
Step  2. Equation (16) can be rewritten as (65)e˙0=e1⋮e˙r-1=erer=-∑i=0r-1σiei+s1.Equation (65) is simplified as (66)e˙a=Aaea+Bas1where ea=[e0,e1,…,er-1]T∈Rr, (67)Aa=010⋯00001⋯00000⋯00⋮⋮⋮⋱⋮⋮000⋯01-σ0-σ1-σ2⋯-σr-2-σr-1,Ba=0,0,…,0,1T.Because Aa is Hurwitz, ea is BIBO stable. In (63) it was shown that s1 is ultimately bounded and that |s1(t)|≤s1max as t→∞. Thus, as t→∞, (66) can be rewritten as follows: (68)e˙a≤Aaea+Bas1maxWe define the positive-definite Lyapunov function Va as (69)Va=eaTPaea.Because Aa is Hurwitz, a positive definite matrix Pa exists such that (70)AaTPa+PaAa=-I.The derivative of Va is (71)V˙a≤-eaTea+2eaTPaBas1max≤-ea22+2λmaxPas1maxea2≤-1-θeea22-θeea22+2λmaxPas1maxea2.Then, (72)V˙a≤-1-θeea22for all ea2≤μ1 where μe=2λmax(Pa)s1max/θe. Consequently, ea converges to the bounded ball, Bea as (73)Bea=ea∣ea2≤λmaxPaλminPaμeas t→∞.
Step  3. With |z2|≤μz2, we obtain (74)x˙r+1=xr+2⋮x˙n-1=xnanxn≤-∑i=r+1n-1aixi+ξ+μz2.Because e0, e1, …, er-1 converge to the bounded ball Ba and because yd has continuous derivatives up to the nth order, ξ=-fr(xa)+yd(r)-d1-∑i=1r-1‍σiei+1 is bounded. Equation (74) is simplified as (75)x˙bz≤Abxbz+Bbξmax+μz2.We define the positive-definite Lyapunov function Vb as (76)Vb=xbzTPbxbz.Because Ab is Hurwitz, a positive definite matrix Pb exists such that (77)AbTPb+PbAb=-I.The derivative of Vb is (78)V˙b≤-xbzTxbz+2xbTPbBbξmax+μz2≤-xbz22+2anλmaxPbξmax+μz2xbz2≤-1-θ2xbz22-θ2xbz22+2λmaxPbanξmax+μz2xbz2.where 0<θ2<1. Then, (79)V˙b≤-1-θ1xbz22for all xbz2≤μ2 where μ2=2λmax(Pb)/θ2anξmax+μz2. From Theorem 4.18 of [9], there exists T2 such that (80)xbz2≤λmaxPbλminPbμ2for all t≥T2. Consequently, xb is also ultimately bounded with the ultimate bound of xbz.Remark 10.
As the controller gainsk1 and k2, and the observer gains 1/ε1 and 1/ε2 increase, ea2 becomes smaller. If the disturbances d1 and d2 are constant, we see that the disturbance estimation errors d~1 and d~2 converge to zeros from (12). Then, ea converges to zero. Consequently, the output tracking error e1 converges to zero.


## 5. Performance Analysis

### 5.1. Numerical Simulation Study

Simulations were performed to analyze the performance of the proposed method. In these simulations, we used the system(81)x˙1=x2x˙2=3x1x2+sin⁡x2+20x3+4x4+d1x˙3=x4x˙4=x12+2x2+3sin⁡2x3+x4+1+0.5cos⁡0.1x1u+d2where d1=sin(πt) and d2=2+cos(5x1). The desired reference trajectory x1d=(1-e-2t)sin(0.2t) was used. The controller was designed as (82)ξ1=d^1-x2ε1ξ2=d^2-x4ε2ξ^˙1=-1ε1ξ1+xrε1+1ε1-3x1x2-sin⁡x2-2x3-4x4ξ^˙2=-1ε2ξ2+xnε2-1ε2-x12-2x2-3sin⁡2x3-x4+1ε21+0.5cos⁡0.1x1us1=σ0e0+σ1e1+e2s2=2x3+4x4s2d=-3x1x2+sin⁡x2+y¨d-d^1-σ0e1+σ1e2-ks1s1u=-141+0.5cos⁡0.1x1a3x4-s˙2d+ϕ1z2-ϕ2z2-11+0.5cos⁡0.1x1x12+2x2+3sin⁡2x3+x4+d^2where ϕ1(z2)=kz1|z2|1/2sgn(z2) and ϕ˙2(z2)=-kz2sgn(z2). In controller (82), the following parameters were used: ks1=1000, kz1=20, kz2=10, σ0=100, σ1=20, ε1=0.1, and ε2=0.05.The estimation performance of the DOB is shown in Figure1. The disturbances were well estimated by the DOB. The tracking performances of s1 and s2 are shown in Figure 2. s1 and s2 converged to the neighborhood of zero and neighborhood of s2d, respectively, because the controller and observer gains were sufficiently high to suppress the effect of the estimation error. The tracking performance and state variables are shown in Figures 3 and 4. We see that both tracking errors e1 and e2 converged to almost zero because of the proposed controller (82). Because of the disturbance and nonzero trajectory, state variables x3 and x4 did not converge to zeros, but were bounded. Figure 5 shows the control input. Because the control method was designed using STA, there was no chattering problem.Figure 1
Estimated disturbances.
(a)
d 1 and d^1 (b)
d 2 and d^2Figure 2
Tracking performances ofs1 and s2.
(a)
s 1 (b)
s 2 d and s2Figure 3
x 1 tracking performance and errors.
(a)
x 1 d and x1 (b)
e 1 (c)
e 2Figure 4
x 3 and x4.
(a)
x 3 (b)
x 4Figure 5
Control input.


### 5.2. Application to Differential Braking Control in Vehicle Lateral Dynamics

To evaluate the performance of the proposed method in a practical system, the proposed method was applied to the differential braking control system in a vehicle. In the vehicular control system, the lateral position is controlled for avoiding collisions using differential brake forces when the driver changes the lane under collision risk or with a vehicle in the blind spot [25]. The lateral control system with the differential braking is (83)e˙1=e2e˙2=a22e2+a23e3+a24e4+bδ2δ+bw2ψ˙d+d1e˙3=e4e˙4=a42e2+a43e3+a44e4+bδ4δ+bw4ψ˙d+bF4Fbs+d2.where e1 is the lateral offset, e2 is the derivative of the lateral offset, e3 is the yaw error, e4 is the yaw error rate, Fbs is the differential braking force control input, δ is the steering wheel angle, ψ˙d is the desired yaw rate, d1 is the disturbance including the driver torque and modeling error, and d2 is the disturbance including self-aligned torque, modeling error, etc. The detailed definitions of the parameters can be found in [28]. The aim of controller design is to determine the brake steer force Fbs that makes (84)limt→∞e1t=0.when the driver attempts the lane change under a collision risk or with a vehicle in the blind spot. This system (83) satisfies Assumption 2. The controller is designed as (85)ξ1=d^1-xrε1ξ2=d^2-xnε2ξ^˙1=-1ε1ξ1+xrε1+1ε1-a22e2-a23e3-a24e4-ξ^˙2=-1ε2ξ2+xnε2+1ε2-a42e2-a43e3-a44e4-bF4Fbss1=σ0e0+σ1e1+e2s2=a23e3+a24e4s2d=-σ0e1-σ1e2-a22e2-bδ2δ-bw2ψ˙d-d1-ks1s1Fbs=-1a24bF4a23e4+a24a42e2+a43e3+a44e4+d2-1a24bF4a24bδ4δ+bw4ψ˙d-s˙2d+ϕ1z2-ϕ2z2where ϕ1(z2)=kz1|z2|1/2sgn(z2) and ϕ˙2(z2)=-kz2sgn(z2), σ0=1000000, σ1=200, σ3=50, σ4=0, ks1=5, kz1=1, kz2=0.1, ε1=0.5, and ε2=0.5. The velocity of the vehicle is 80 km/h on a straight road. The test scenario is as follows: (1) at 5 sec., the driver attempts lane change under collision risk with the object vehicle in the target lane; (2) as soon as the driver attempts lane change, the differential braking control (DBC) system is activated with a warning against the collision risk; (3) the driver attempts to keep the original lane with the help of the DBC; (4) the DBC system operates to move the vehicle to the center of the original lane.Simulations were performed using the vehicle dynamic software CarSim and Matlab/Simulink as shown in Figure6. The S-function coded in C language was used for implementing the proposed sliding mode backstepping control method. The output of CarSim consists of vehicle motion data such as steer angle, lateral velocity, and brake force. We also modeled the lane camera sensor to obtain the lane coefficients [29]; c0 denotes the lateral lane center offset at c.g., c1 denotes the in-lane heading slop, the heading angle error at c.g., c2 denotes curvature/2 at s=0, and c3 denotes the curvature-rate/6. The steering wheel angle used in the simulations is shown in Figure 7.Figure 6
Vehicle and camera models used in the simulations.
(a)
Overall simulation structure that consists of CarSim vehicle model (b)
Vehicle part. The output of lane camera is lane coefficients;c0 denotes the lateral lane center offset at c.g., c1 denotes the in-lane heading slop, the heading angle error at c.g., c2 denotes curvature/2 at s=0, and c3 denotes the curvature-rate/6Figure 7
Steering wheel angle.Two cases were tested:(1) driving with proposed controller (85); (2) driving without proposed controller (85). The simulation results are shown in Figure 8. The lateral offset errors of the two cases are depicted in Figure 8(a). Figure 8(b) shows the control input. Because the control method was designed using STA, there was no chattering problem. In case 2 (without DBC), for the given steering wheel angle, the lateral offset error e1 becomes 1.1 m owing to the steering wheel angle. On the other hand, in case 1 (with DBC), the lateral offset error e1 was maintained to nearly zero because the brake steer force compensated for the steering wheel angle using the proposed method (85). The yaw rate error e4 was also kept to nearly zero. Figure 9 shows the estimated disturbance. External disturbances appeared owing to the bank angle, road reaction force, the assumptions for this modeling, etc. The external disturbances were compensated by using utilizing the proposed method. Consequently, the lateral offset e1 converged to zero despite the steer angle and the disturbances.Figure 8
Control performance.
(a)
Lateral offset errore1 (b)
Brake steer force inputFbsFigure 9
Estimated disturbances.
(a)
d 1 (b)
d 2


## 5.1. Numerical Simulation Study

Simulations were performed to analyze the performance of the proposed method. In these simulations, we used the system(81)x˙1=x2x˙2=3x1x2+sin⁡x2+20x3+4x4+d1x˙3=x4x˙4=x12+2x2+3sin⁡2x3+x4+1+0.5cos⁡0.1x1u+d2where d1=sin(πt) and d2=2+cos(5x1). The desired reference trajectory x1d=(1-e-2t)sin(0.2t) was used. The controller was designed as (82)ξ1=d^1-x2ε1ξ2=d^2-x4ε2ξ^˙1=-1ε1ξ1+xrε1+1ε1-3x1x2-sin⁡x2-2x3-4x4ξ^˙2=-1ε2ξ2+xnε2-1ε2-x12-2x2-3sin⁡2x3-x4+1ε21+0.5cos⁡0.1x1us1=σ0e0+σ1e1+e2s2=2x3+4x4s2d=-3x1x2+sin⁡x2+y¨d-d^1-σ0e1+σ1e2-ks1s1u=-141+0.5cos⁡0.1x1a3x4-s˙2d+ϕ1z2-ϕ2z2-11+0.5cos⁡0.1x1x12+2x2+3sin⁡2x3+x4+d^2where ϕ1(z2)=kz1|z2|1/2sgn(z2) and ϕ˙2(z2)=-kz2sgn(z2). In controller (82), the following parameters were used: ks1=1000, kz1=20, kz2=10, σ0=100, σ1=20, ε1=0.1, and ε2=0.05.The estimation performance of the DOB is shown in Figure1. The disturbances were well estimated by the DOB. The tracking performances of s1 and s2 are shown in Figure 2. s1 and s2 converged to the neighborhood of zero and neighborhood of s2d, respectively, because the controller and observer gains were sufficiently high to suppress the effect of the estimation error. The tracking performance and state variables are shown in Figures 3 and 4. We see that both tracking errors e1 and e2 converged to almost zero because of the proposed controller (82). Because of the disturbance and nonzero trajectory, state variables x3 and x4 did not converge to zeros, but were bounded. Figure 5 shows the control input. Because the control method was designed using STA, there was no chattering problem.Figure 1
Estimated disturbances.
(a)
d 1 and d^1 (b)
d 2 and d^2Figure 2
Tracking performances ofs1 and s2.
(a)
s 1 (b)
s 2 d and s2Figure 3
x 1 tracking performance and errors.
(a)
x 1 d and x1 (b)
e 1 (c)
e 2Figure 4
x 3 and x4.
(a)
x 3 (b)
x 4Figure 5
Control input.


## 5.2. Application to Differential Braking Control in Vehicle Lateral Dynamics

To evaluate the performance of the proposed method in a practical system, the proposed method was applied to the differential braking control system in a vehicle. In the vehicular control system, the lateral position is controlled for avoiding collisions using differential brake forces when the driver changes the lane under collision risk or with a vehicle in the blind spot [25]. The lateral control system with the differential braking is (83)e˙1=e2e˙2=a22e2+a23e3+a24e4+bδ2δ+bw2ψ˙d+d1e˙3=e4e˙4=a42e2+a43e3+a44e4+bδ4δ+bw4ψ˙d+bF4Fbs+d2.where e1 is the lateral offset, e2 is the derivative of the lateral offset, e3 is the yaw error, e4 is the yaw error rate, Fbs is the differential braking force control input, δ is the steering wheel angle, ψ˙d is the desired yaw rate, d1 is the disturbance including the driver torque and modeling error, and d2 is the disturbance including self-aligned torque, modeling error, etc. The detailed definitions of the parameters can be found in [28]. The aim of controller design is to determine the brake steer force Fbs that makes (84)limt→∞e1t=0.when the driver attempts the lane change under a collision risk or with a vehicle in the blind spot. This system (83) satisfies Assumption 2. The controller is designed as (85)ξ1=d^1-xrε1ξ2=d^2-xnε2ξ^˙1=-1ε1ξ1+xrε1+1ε1-a22e2-a23e3-a24e4-ξ^˙2=-1ε2ξ2+xnε2+1ε2-a42e2-a43e3-a44e4-bF4Fbss1=σ0e0+σ1e1+e2s2=a23e3+a24e4s2d=-σ0e1-σ1e2-a22e2-bδ2δ-bw2ψ˙d-d1-ks1s1Fbs=-1a24bF4a23e4+a24a42e2+a43e3+a44e4+d2-1a24bF4a24bδ4δ+bw4ψ˙d-s˙2d+ϕ1z2-ϕ2z2where ϕ1(z2)=kz1|z2|1/2sgn(z2) and ϕ˙2(z2)=-kz2sgn(z2), σ0=1000000, σ1=200, σ3=50, σ4=0, ks1=5, kz1=1, kz2=0.1, ε1=0.5, and ε2=0.5. The velocity of the vehicle is 80 km/h on a straight road. The test scenario is as follows: (1) at 5 sec., the driver attempts lane change under collision risk with the object vehicle in the target lane; (2) as soon as the driver attempts lane change, the differential braking control (DBC) system is activated with a warning against the collision risk; (3) the driver attempts to keep the original lane with the help of the DBC; (4) the DBC system operates to move the vehicle to the center of the original lane.Simulations were performed using the vehicle dynamic software CarSim and Matlab/Simulink as shown in Figure6. The S-function coded in C language was used for implementing the proposed sliding mode backstepping control method. The output of CarSim consists of vehicle motion data such as steer angle, lateral velocity, and brake force. We also modeled the lane camera sensor to obtain the lane coefficients [29]; c0 denotes the lateral lane center offset at c.g., c1 denotes the in-lane heading slop, the heading angle error at c.g., c2 denotes curvature/2 at s=0, and c3 denotes the curvature-rate/6. The steering wheel angle used in the simulations is shown in Figure 7.Figure 6
Vehicle and camera models used in the simulations.
(a)
Overall simulation structure that consists of CarSim vehicle model (b)
Vehicle part. The output of lane camera is lane coefficients;c0 denotes the lateral lane center offset at c.g., c1 denotes the in-lane heading slop, the heading angle error at c.g., c2 denotes curvature/2 at s=0, and c3 denotes the curvature-rate/6Figure 7
Steering wheel angle.Two cases were tested:(1) driving with proposed controller (85); (2) driving without proposed controller (85). The simulation results are shown in Figure 8. The lateral offset errors of the two cases are depicted in Figure 8(a). Figure 8(b) shows the control input. Because the control method was designed using STA, there was no chattering problem. In case 2 (without DBC), for the given steering wheel angle, the lateral offset error e1 becomes 1.1 m owing to the steering wheel angle. On the other hand, in case 1 (with DBC), the lateral offset error e1 was maintained to nearly zero because the brake steer force compensated for the steering wheel angle using the proposed method (85). The yaw rate error e4 was also kept to nearly zero. Figure 9 shows the estimated disturbance. External disturbances appeared owing to the bank angle, road reaction force, the assumptions for this modeling, etc. The external disturbances were compensated by using utilizing the proposed method. Consequently, the lateral offset e1 converged to zero despite the steer angle and the disturbances.Figure 8
Control performance.
(a)
Lateral offset errore1 (b)
Brake steer force inputFbsFigure 9
Estimated disturbances.
(a)
d 1 (b)
d 2


## 6. Conclusions

In this paper, we proposed a sliding-mode backstepping control for the coupled normal form of nonlinear systems. The proposed method was developed by combining backstepping and sliding-mode control. The key idea of the proposed method is that the linear terms of the state variables of the second subsystem are lumped into the virtual input in the first subsystem. To compensate for the disturbances, a DOB was developed. The stability of the closed-loop is validated by using the ISS property. Through numerical simulations and application to a vehicle system, the proposed method was observed to lead to convergence of the output to the desired output trajectory under the described disturbances.The main drawback is the use of the derivative of the measured signal in the controller. It may result in the amplification of the measurement noise. Generally, the filter technique is widely used to obtain the derivatives of the measured signals without the amplification of the measurement noise [30, 31]. However, the use of the filter may cause the phase lag in the feedback loop. In future works, we will develop the control method with the consideration of the amplification of the measurement noise in the derivatives of the measured signals.



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*Source: 7941302-2019-02-07.xml*