# Low-Complexity Adaptive Saturated Control of a Class of Nonlinear Systems with Its Application

Authors: Gang Zhang; Deqiang Cheng; Qiqi Kou
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This paper investigates a low-complexity saturated control law for a class of nonlinear systems with consideration of the time-varying output constraint, control constraint, and external disturbance. First, a dead-zone model is employed to transform the control saturation nonlinearity into a linear one with respect to the real input signal. Then, the original system with time-varying output constraint is transformed into a constraint-free one, based on which a novel adaptive saturated control law is devised along the filtered error manifold. By employing minimum learning parameter technique and virtual error concept, only two adaptive parameters are needed to update online, which reduces the computational burdens dramatically. Finally, the applications to Duffing-Holmes chaotic system are organized to validate the effectiveness of the proposed control law.


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## Body

## 1. Introduction

In recent years, considerable attention has been paid to developing systematic methods of adaptive control of nonlinear systems including Duffing-Holmes chaotic system and mechatronic system (e.g., see [1–3] and references therein). As a result, extensive interesting works have been carried out in the existing literature. For example, an adaptive neural network (NN)-based control scheme was proposed for perturbed strict-feedback nonlinear systems in [4]. In [5], barrier Lyapunov function (BLF) technique was applied to solve the output control problem for strict feedback system subject to unknown dynamics.Based on the adaptive control methods in the existing works, one can find that backstepping technique has been widely adopted to develop the relevant control scheme for cascaded nonlinear systems including strict and pure feedback systems. In [6], an adaptive neural control protocol was investigated for a class of unknown pure feedback systems, wherein the unknown nonlinear function is approximated by a radial basis function NN (RBFNN). In terms of nonlinear approximation, fuzzy logic system is also an efficient tool, which is similar with NN. By combing backstepping technique and fuzzy logic system, an effective adaptive fuzzy control scheme was devised for strict feedback system in [7]. However, an inherent drawback, i.e., “explosion of complexity,” is often encountered in the backstepping technique owing to the complex computations for the virtual controller and its derivatives [8, 9]. To conquer this drawback, via introducing a first-order low-pass filter in each step of backstepping controller design, dynamic surface control (DSC) method is proposed to avoid the tedious computations for the derivatives of virtual controller. In this sense, the inherent drawback “explosion of complexity” is conquered to some extent. Thus, DSC method was furthered explored in the nonlinear controller design for strict-feedback systems in [10] and the interconnected nonlinear pure-feedback systems in [11]. Though the computational complexity can be decreased greatly, the computational load in updating the elements of NN or fuzzy logic systems is still very large, especially for high-order nonlinear systems. To conquer this problem, minimal learning parameter (MLP) technique involving less adaptive parameters for strict-feedback systems in [12] provides an alternative way to lower the complexity of adaptive scheme. Thus, how to lower the computational complexity of adaptive control laws deserves further investigations.Apart from the aforementioned discussions, in practical engineering systems such as air conditioner, flexible crane, robot, and hypersonic vehicle [13, 14], state constraints are also often encountered, which makes it pretty difficult to facilitate the corresponding controller. Among the state constraints, output limitation is one typical constraint which potentially degrades the system performance and easily causes serious hazards. To handle this problem, adaptive BLF-based backstepping control schemes attracted extensive attention in the existing works like in [5, 13]. As an alternative, considering the prescribed performance of the output originated from [15], [16] investigated a distinctive handling method to transform the time-varying constrained nonaffine system into a constraint-free strict-feedback one. Then, an efficient adaptive control law was developed by employing the NN-based nonlinear approximation. Compared with BLF-based backstepping control laws, the approach proposed by [16] is more advantageous and easily achievable owing to the fact that the recursive design procedure of backstepping technique is avoided. However, the control input under the aforementioned control laws is always very large to guarantee no violation of the output constraints, which is hard to execute in practical systems. Thus, it deserves further investigations on the constrained control for the unknown nonlinear systems.Motivated by the aforementioned discussions, in this paper, a novel low-complexity adaptive control law is investigated for a class of uncertain nonlinear systems by employing the MLP and RBFNN techniques with consideration of time-varying output constraint, control saturation, and external disturbance. To the authors’ best knowledge, there exist very few works reported on the adaptive saturated control for nonlinear systems with time-varying output constraint using MLP technique and virtual error concept. Compared with the existing works, our main contributions are twofold:(1) a novel low-complexity adaptive neural saturated control law is first proposed for a class of nonlinear systems in the presence of time-varying output constraint, control saturation, and external disturbance, wherein the control saturation nonlinearity is approximated by a dead-zone model. (2) The number of adaptive parameters is reduced to only two, which is not dependent on the system order, hidden nodes of NN, and the rules number of fuzzy logic systems. In this case, the computational load is decreased dramatically and the corresponding control law is more easily achievable in practice.The rest of this paper is organized as follows. Section2 gives the basic description of the controlled system along with some brief introduction for the RBFNN technique. In Section 3, the proposed low-complexity adaptive saturated law is shown. A group of numerical simulations is organized in Section 4. Some conclusions are given in Section 5.


### 1.1. Notations

N , N + represent the set of nonnegative integers and positive integers, respectively. Rn,Rn+ are the set of n-dimensional real numbers, and n-dimensional positive real numbers, respectively. T represents the vector transpose. The Euclidean norm and the absolute value operator are denoted by · and ·, respectively.


## 1.1. Notations

N , N + represent the set of nonnegative integers and positive integers, respectively. Rn,Rn+ are the set of n-dimensional real numbers, and n-dimensional positive real numbers, respectively. T represents the vector transpose. The Euclidean norm and the absolute value operator are denoted by · and ·, respectively.


## 2. Problem Statement and Preliminaries of RBFNN

### 2.1. Model Description

The following system is considered in this paper:(1)x˙i=fx-i+xi+1,i=1,2,…,n-1x˙n=fx-n+bu+dy=x1,where x-n=x1,x2,…,xnT∈Rn,y∈R are the state vector and system output, respectively. x-i=x1,x2,…,xiT∈Ri, f(x-i) is a continuous nonlinear function with at least (n-i)-order derivatives, which is unknown. b∈R is the known control gain, which is not equal to zero with consideration of the system controllability. u,d∈R denote the control input and bounded external disturbance, respectively.In practical engineering applications including robot system and electrical power system, actuator saturation is often encountered. In this work, symmetrical control constraints are considered, i.e.,u≤u0; consequently, the output of the actuator is given by(2)ut=satvt=signvtmin⁡u0,vt where u0 is the upper bound of control input. v(t) is the real input of actuator to be determined in the following sections. For brevity, hereafter, (t) is omitted without inducing any ambiguities. To facilitate the controller design, based on [17], saturation nonlinearity in (2) can be transformed into a linear form with respect to v, which is(3)u=satv=ρ0v-∫0lϕτdzτvdτwhere ρ0=∫0lϕτdτ is a positive constant, ϕτ, a density function, satisfies that ϕτ≥0,∀τ>0,τ>l, and ϕτ=0(l∈R+)dzτv=maxv-τ,min0,v+τ is the dead-zone operator.Assumption 1.
For system (1) subject to input constraint, there exists feasible actual control input v; i.e., v is bounded, such that the expected control objective can be achieved.Remark 2.
According to [17], the actual control input v is bounded due to controllability of a practical system. Moreover, many functions can be utilized as the density function including the Gaussian function. Thus, under Assumption 1, the approximation error -∫0lϕτdzτvdτ is bounded.Accordingly, substituting (3) into (1) yields(4)x˙i=fx-i+xi+1,i=1,2,…,n-1x˙n=fx-n+ρ0bu+d∗y=x1where d∗≔d-∫0lϕτdzτvdτ is the compound disturbance. Accordingly, the control objective of this work is that an expected output reference trajectory yr can be tracked stably with the devised controller. In this case, the output y is evolved within the envelope preplanned by the time-varying asymmetric constraints involving the reference yr, i.e.,(5)y_t<yt<y-twhere y_t,y-t denote the lower and upper bounds and satisfy y_t<yr<y-t. It is assumed that y_,y-,yr are continuous functions with at least (n-1)-order derivatives, which are known and bounded. Note that it is also assumed that the initial output y(0) or x1(0) lies in the envelope described by (5) above.


### 2.2. Radial Basis Function NN Approximation

In the existing works, a continuous function is often approximated with an arbitrary accuracy over a compact set by a radial basis function NN (RBFNN). For an unknown functionG(X), the output of RBFNN with approximation errors is(6)GX=W∗TφX+OXwhere X=[X1,X2,…,XN]T∈RN is the input vector with N∈N+. W∗=[W1∗,W2∗,…,Wm∗]T∈Rm is the optimal weight vector (m is the node number of hidden layer. O(X) is the approximation error. φ(X)=[φ1(X),φ2(X),…,φm(X)]T∈Rm with φi(X)(i=1,2,…,m) being a Gaussian kernel function:(7)φiX=exp-X-γi22εi2where γi=[γi1,γi2,…,γiN]T∈RN and εi are, respectively, the relevant center and width of Gaussian functions.In (6), it is assumed that there is an ideal weight vector W∗ which makes the approximation error O(X) sufficiently small. Moreover, in order to develop the following adaptive saturated controller, some assumptions are imposed.Assumption 3.
There exists an unknown positive constantθ which makes condition W∗≤θ hold.Assumption 4.
The unknown external disturbanced is bounded, i.e., d≤d0 with d0 an unknown positive constant.Remark 5.
Based on Assumption3, the adaptive scheme to be determined can be developed to estimate the unknown constant θ rather than the elements of vector W∗, which decreases the computational load dramatically. This handling way is also referred to as the minimal learning parameter (MLP) scheme [12].Remark 6.
Assumption4 is reasonable owing to the fact that if the external disturbance is unbounded, system (1) will be out of control under bounded control input.According to the foregoing discussion, the following work is about the design of the relevant adaptive saturated controller in the presence of unknown dynamics and output constraints.


## 2.1. Model Description

The following system is considered in this paper:(1)x˙i=fx-i+xi+1,i=1,2,…,n-1x˙n=fx-n+bu+dy=x1,where x-n=x1,x2,…,xnT∈Rn,y∈R are the state vector and system output, respectively. x-i=x1,x2,…,xiT∈Ri, f(x-i) is a continuous nonlinear function with at least (n-i)-order derivatives, which is unknown. b∈R is the known control gain, which is not equal to zero with consideration of the system controllability. u,d∈R denote the control input and bounded external disturbance, respectively.In practical engineering applications including robot system and electrical power system, actuator saturation is often encountered. In this work, symmetrical control constraints are considered, i.e.,u≤u0; consequently, the output of the actuator is given by(2)ut=satvt=signvtmin⁡u0,vt where u0 is the upper bound of control input. v(t) is the real input of actuator to be determined in the following sections. For brevity, hereafter, (t) is omitted without inducing any ambiguities. To facilitate the controller design, based on [17], saturation nonlinearity in (2) can be transformed into a linear form with respect to v, which is(3)u=satv=ρ0v-∫0lϕτdzτvdτwhere ρ0=∫0lϕτdτ is a positive constant, ϕτ, a density function, satisfies that ϕτ≥0,∀τ>0,τ>l, and ϕτ=0(l∈R+)dzτv=maxv-τ,min0,v+τ is the dead-zone operator.Assumption 1.
For system (1) subject to input constraint, there exists feasible actual control input v; i.e., v is bounded, such that the expected control objective can be achieved.Remark 2.
According to [17], the actual control input v is bounded due to controllability of a practical system. Moreover, many functions can be utilized as the density function including the Gaussian function. Thus, under Assumption 1, the approximation error -∫0lϕτdzτvdτ is bounded.Accordingly, substituting (3) into (1) yields(4)x˙i=fx-i+xi+1,i=1,2,…,n-1x˙n=fx-n+ρ0bu+d∗y=x1where d∗≔d-∫0lϕτdzτvdτ is the compound disturbance. Accordingly, the control objective of this work is that an expected output reference trajectory yr can be tracked stably with the devised controller. In this case, the output y is evolved within the envelope preplanned by the time-varying asymmetric constraints involving the reference yr, i.e.,(5)y_t<yt<y-twhere y_t,y-t denote the lower and upper bounds and satisfy y_t<yr<y-t. It is assumed that y_,y-,yr are continuous functions with at least (n-1)-order derivatives, which are known and bounded. Note that it is also assumed that the initial output y(0) or x1(0) lies in the envelope described by (5) above.


## 2.2. Radial Basis Function NN Approximation

In the existing works, a continuous function is often approximated with an arbitrary accuracy over a compact set by a radial basis function NN (RBFNN). For an unknown functionG(X), the output of RBFNN with approximation errors is(6)GX=W∗TφX+OXwhere X=[X1,X2,…,XN]T∈RN is the input vector with N∈N+. W∗=[W1∗,W2∗,…,Wm∗]T∈Rm is the optimal weight vector (m is the node number of hidden layer. O(X) is the approximation error. φ(X)=[φ1(X),φ2(X),…,φm(X)]T∈Rm with φi(X)(i=1,2,…,m) being a Gaussian kernel function:(7)φiX=exp-X-γi22εi2where γi=[γi1,γi2,…,γiN]T∈RN and εi are, respectively, the relevant center and width of Gaussian functions.In (6), it is assumed that there is an ideal weight vector W∗ which makes the approximation error O(X) sufficiently small. Moreover, in order to develop the following adaptive saturated controller, some assumptions are imposed.Assumption 3.
There exists an unknown positive constantθ which makes condition W∗≤θ hold.Assumption 4.
The unknown external disturbanced is bounded, i.e., d≤d0 with d0 an unknown positive constant.Remark 5.
Based on Assumption3, the adaptive scheme to be determined can be developed to estimate the unknown constant θ rather than the elements of vector W∗, which decreases the computational load dramatically. This handling way is also referred to as the minimal learning parameter (MLP) scheme [12].Remark 6.
Assumption4 is reasonable owing to the fact that if the external disturbance is unbounded, system (1) will be out of control under bounded control input.According to the foregoing discussion, the following work is about the design of the relevant adaptive saturated controller in the presence of unknown dynamics and output constraints.


## 3. Main Results

### 3.1. Output Transformation

As (5) shows, the output is constrained, which will increase the difficulty and complexity of designing an efficient controller directly. In order to surmount this problem, the following output transformation is employed, which is(8)y=Pz1,y_,y-or(9)z1t=P-1y,y_,y-where z1 is the transformed output variable. Function P should satisfy the following conditions:(10)∂P·∂z1>0,limz1→-∞Pz1,y_,y-=y_t,limz1→+∞Pz1,y_,y-=y-tInspired by [16], function P is selected as(11)Pz1,y_,y-=y--y_2tanh⁡z1+y-+y_2Then, based on (11), the transformed output z1 equals(12)z1=atanh⁡2y-y--y_y--y_Accordingly, let z2=z˙1,z3=z˙2=z¨1,…,zn=z˙n-1=z1n-1; then one can obtain that(13)z˙i=zi+1,i=1,2,…,n-1z˙n=βfx-+ρ0bv+d∗+gYwith(14)gY≔∑k=1n-1n-1k∂P-1·∂yk·yn-k+∑k=0n-1n-1k∂P-1·∂y_k·y_n-k+∑k=0n-1n-1k∂P-1·∂y-k·y-n-kwhere Y≔[y,y_,y-]T. β≔∂P-1·/∂y=2/(1-α2)>0 with α≔(2y-y--y_)/(y--y_). n-1k is the binomial coefficient.


### 3.2. Adaptive Saturated Controller Design

To proceed, based on transformed system (13), the following filtered state variable is defined as(15)s=c1z1+c2z2+⋯+cn-1zn-1+znwhere there exists a positive constant vector c≔[c1,c2,…,1]T∈Rn such that multinomial λn+cn-1λn-1+⋯+c2λ2+c1=0 is Hurwitz with λ being the Laplace operator [18].Lemma 7 (see [19]).
For the filtered state variable in (15), if s converges to zero as t→∞, then the newly defined tracking error z1 and its derivatives zk(k=2,…,n) converge to zero with the same convergence rate as t→∞.Proof.
Inspired by [19], an equivalent transformation of (15) is expressed by(16)st=χ˙n-1t+ϖn-1χn-1tχn-1t=χ˙n-2t+ϖn-2χn-2t⋮χ2t=χ˙1t+ϖ1χ1t=z˙1t+c1z1tχ1t=z1twhere parameter ϖi is associated with c1,c2,…,cn in (15). Then solving the first equation in (16) yields(17)χn-1t=χn-10exp⁡-ϖn-1t+exp⁡-ϖn-1t∫0tsτexp⁡ϖn-1τdτEquation (17) implies that if term ∫0tsτexpϖn-1τdτ is bounded, then χn-1(t)→0 as t→∞. If term ∫0tsτexpϖn-1τdτ is unbounded, for (17), employing L'Hopital's rule can yield(18)limt→∞χn-1t=0+limt→∞stexpϖn-1tϖn-1expϖn-1t=limt→∞stϖn-1Accordingly, when s(t)→0 as t→∞, χn-1(t)→0 as t→∞ with the same convergence rate as s(t). Likewise, for the second equation in (16), based on (18), employing L'Hopital's rule can yield(19)limt→∞χn-2t=0+limt→∞χn-1texpϖn-2tϖn-2expϖn-2t=limt→∞χn-1tϖn-2=limt→∞stϖn-1ϖn-2As shown in (19), if s(t)→0 as t→∞, χn-2(t)→0 as t→∞ with the same convergence rate as s. For the remaining equations in (16), by employing the same procedure like (17)-(19), one can obtain the same conclusion. Thereby, the proof of Lemma 7 is completed.
Taking the time derivative ofs yields(20)s˙=z˙n+∑i=1n-1cizi+1=f∗x-n,y_,y-+βρ0bv+βd∗+∑i=1n-1cizi+1where f∗x-n,y_,y-≔βfx-n+gY+∑i=1n-1cizi+1. Owing to unknown nonlinearity, fx-n is unknown as well. Thus, according to the preliminaries in Section 2.2, RBFNN is utilized to approximate the lumped unknown function f∗, which is given by(21)f∗Y∗=WTφY∗where Y∗≔[x-nT,y_,y-]T∈Rn+2. According to Assumption 3 above, one can obtain that(22)f∗=f∗=WTφY∗≤WT2+φY∗24≤θ2+φY∗24≤ϑ1ψ1where ϑ1≔max{θ2,1/4}, ψ1≔1+φ(Y∗)2/4. For the compound disturbance d∗, under Remark 2 and Assumption 4, one can obtain(23)βd∗=βd∗≤βd+∫0lϕτdzτvdτ≤ϑ2ψ2where ψ2≔β with β>0 a time-varying positive parameter given in (14). ϑ2 is an unknown positive constant. Based on the foregoing analysis, the adaptive saturated controller is devised as(24)v=-1βρ0bηs+ϑ^1+μ11+exp-∫0tϑ^1τdτψ1signs+ϑ^2+μ21+exp-∫0tϑ^2τdτψ2signswhere η,μ1,μ2 are positive constants. ϑ^1,ϑ^2 are, respectively, the estimated values of unknown constants ϑ1,ϑ2. The corresponding adaptive scheme is(25)ϑ^˙1=-μ1exp-∫0tϑ^1τdτ1+exp-∫0tϑ^1τdτ2ϑ^1+ψ1sϑ^˙2=-μ2exp-∫0tϑ^2τdτ1+exp-∫0tϑ^2τdτ2ϑ^2+ψ2sRemark 8.
The controller (24) has two parts. One is the nominal part: -(1/βρ0b)ηs which is used to stabilize the tracking error system. The other one is the robust part, i.e., -1/βρ0b(ϑ^1+μ1/(1+exp-∫0tϑ^1τdτ))ψ1signs and -1/βρ0b(ϑ^2+μ2/(1+exp-∫0tϑ^2τdτ))ψ2signs which is utilized to further compensate the effects brought by the unknown nonlinearity, external disturbance, and control saturation. Compared with [16], only two parameters are needed to estimate rather than 2m parameters involved in the weight vector of RBFNN approximation. In this sense, the computational load is decreased dramatically.


### 3.3. Stability Analysis

Under the devised controller (24), an important result is given as follows.Theorem 9.
Under the designed controller (24) and adaptive scheme (25), the desired output reference command yr evolved within the envelope preplanned by the time-varying constraints can be tracked with asymptotic stability.Proof.
First, construct the following Lyapunov function as(26)V=V1+V2with(27)V1=12s2,V2=12ϑ~12+ϑ~22where the virtual error ϑ~i is defined as ϑ~i≔ϑi-ϑ^i+μi/(1+exp-∫0tϑ^iτdτ)(i=1,2) which is inspired by [19]. Taking the time derivative of V1 in (27) yields(28)V˙1=ss˙Substituting (20) and (24) into (28) yields(29)V˙1=sf∗x¯n,y_,y-+βρ0bv+βd∗=-ηs2-ϑ^1-μ11+exp-∫0tϑ^1τdτψ1s-ϑ^2+μ21+exp-∫0tϑ^2τdτψ2s+sf∗x¯n,y_,y-+βd∗-ηs2-ϑ^1-μ11+exp-∫0tϑ^1τdτψ1s-ϑ^2+μ21+exp-∫0tϑ^2τdτψ2s+s·f∗x¯n,y_,y-+βd∗≤-ηs2-ϑ^1-μ11+exp-∫0tϑ^1τdτψ1s-ϑ^2+μ21+exp-∫0tϑ^2τdτψ2s+s·f∗x-n,y_,y-+s·βd∗In view of (22) and (23), (29) equals (30)V˙1≤-ηs2-ϑ^1-μ11+exp-∫0tϑ^1τdτψ1s-ϑ^2+μ21+exp-∫0tϑ^2τdτψ2s+ϑ1ψ1s+ϑ2ψ2sTaking the time derivative of V2 in (27) yields(31)V˙2=-ϑ~1ϑ^˙1+μ1exp-∫0tϑ^1τdτ1+exp-∫0tϑ^1τdτ2ϑ^1-ϑ~2ϑ^˙2+μ2exp-∫0tϑ^2τdτ1+exp-∫0tϑ^2τdτ2ϑ^2 Substituting (30) and (31) into the time derivative of V in (26) yields(32)V˙≤-ηs2-ϑ^1-μ11+exp-∫0tϑ^1τdτψ1s-ϑ^2+μ21+exp-∫0tϑ^2τdτψ2s-ϑ~1ϑ^˙1+μ1exp-∫0tϑ^1τdτ1+exp-∫0tϑ^1τdτ2ϑ^1-ϑ~2ϑ^˙2+μ2exp-∫0tϑ^2τdτ1+exp-∫0tϑ^2τdτ2ϑ^2+ϑ1ψ1s+ϑ2ψ2sBased on the definition of virtual error ϑ~i(i=1,2) in (27), then (32) becomes(33)V˙≤-ηs2+ϑ1-ϑ^1-μ11+exp⁡-∫0tϑ^1τdτψ1s+ϑ2-ϑ^2+μ21+exp⁡-∫0tϑ^2τdτψ2s-ϑ~1ϑ^˙1+μ1exp-∫0tϑ^1τdτ1+exp-∫0tϑ^1τdτ2ϑ^1-ϑ~2ϑ^˙2+μ2exp-∫0tϑ^2τdτ1+exp-∫0tϑ^2τdτ2ϑ^2≤-ηs2+ϑ~1ψ1s-ϑ^˙1+μ1exp⁡-∫0tϑ^1τdτ1+exp⁡-∫0tϑ^1τdτ2ϑ^1+ϑ~2ψ2s-ϑ^˙2+μ2exp⁡-∫0tϑ^2τdτ1+exp⁡-∫0tϑ^2τdτ2ϑ^2Substituting (25) into (33) yields(34)V˙≤-ηs2≤0 Only when s=0 can V˙=0. Thus, the tracking error system is asymptotically stable, i.e., s→0 as t→∞. Accordingly, based on (15), by employing Lemma 7, one can obtain that all the newly defined tracking errors z1,z2,…,zn converge to zero as t→∞. This completes the proof of Theorem 9.


## 3.1. Output Transformation

As (5) shows, the output is constrained, which will increase the difficulty and complexity of designing an efficient controller directly. In order to surmount this problem, the following output transformation is employed, which is(8)y=Pz1,y_,y-or(9)z1t=P-1y,y_,y-where z1 is the transformed output variable. Function P should satisfy the following conditions:(10)∂P·∂z1>0,limz1→-∞Pz1,y_,y-=y_t,limz1→+∞Pz1,y_,y-=y-tInspired by [16], function P is selected as(11)Pz1,y_,y-=y--y_2tanh⁡z1+y-+y_2Then, based on (11), the transformed output z1 equals(12)z1=atanh⁡2y-y--y_y--y_Accordingly, let z2=z˙1,z3=z˙2=z¨1,…,zn=z˙n-1=z1n-1; then one can obtain that(13)z˙i=zi+1,i=1,2,…,n-1z˙n=βfx-+ρ0bv+d∗+gYwith(14)gY≔∑k=1n-1n-1k∂P-1·∂yk·yn-k+∑k=0n-1n-1k∂P-1·∂y_k·y_n-k+∑k=0n-1n-1k∂P-1·∂y-k·y-n-kwhere Y≔[y,y_,y-]T. β≔∂P-1·/∂y=2/(1-α2)>0 with α≔(2y-y--y_)/(y--y_). n-1k is the binomial coefficient.


## 3.2. Adaptive Saturated Controller Design

To proceed, based on transformed system (13), the following filtered state variable is defined as(15)s=c1z1+c2z2+⋯+cn-1zn-1+znwhere there exists a positive constant vector c≔[c1,c2,…,1]T∈Rn such that multinomial λn+cn-1λn-1+⋯+c2λ2+c1=0 is Hurwitz with λ being the Laplace operator [18].Lemma 7 (see [19]).
For the filtered state variable in (15), if s converges to zero as t→∞, then the newly defined tracking error z1 and its derivatives zk(k=2,…,n) converge to zero with the same convergence rate as t→∞.Proof.
Inspired by [19], an equivalent transformation of (15) is expressed by(16)st=χ˙n-1t+ϖn-1χn-1tχn-1t=χ˙n-2t+ϖn-2χn-2t⋮χ2t=χ˙1t+ϖ1χ1t=z˙1t+c1z1tχ1t=z1twhere parameter ϖi is associated with c1,c2,…,cn in (15). Then solving the first equation in (16) yields(17)χn-1t=χn-10exp⁡-ϖn-1t+exp⁡-ϖn-1t∫0tsτexp⁡ϖn-1τdτEquation (17) implies that if term ∫0tsτexpϖn-1τdτ is bounded, then χn-1(t)→0 as t→∞. If term ∫0tsτexpϖn-1τdτ is unbounded, for (17), employing L'Hopital's rule can yield(18)limt→∞χn-1t=0+limt→∞stexpϖn-1tϖn-1expϖn-1t=limt→∞stϖn-1Accordingly, when s(t)→0 as t→∞, χn-1(t)→0 as t→∞ with the same convergence rate as s(t). Likewise, for the second equation in (16), based on (18), employing L'Hopital's rule can yield(19)limt→∞χn-2t=0+limt→∞χn-1texpϖn-2tϖn-2expϖn-2t=limt→∞χn-1tϖn-2=limt→∞stϖn-1ϖn-2As shown in (19), if s(t)→0 as t→∞, χn-2(t)→0 as t→∞ with the same convergence rate as s. For the remaining equations in (16), by employing the same procedure like (17)-(19), one can obtain the same conclusion. Thereby, the proof of Lemma 7 is completed.
Taking the time derivative ofs yields(20)s˙=z˙n+∑i=1n-1cizi+1=f∗x-n,y_,y-+βρ0bv+βd∗+∑i=1n-1cizi+1where f∗x-n,y_,y-≔βfx-n+gY+∑i=1n-1cizi+1. Owing to unknown nonlinearity, fx-n is unknown as well. Thus, according to the preliminaries in Section 2.2, RBFNN is utilized to approximate the lumped unknown function f∗, which is given by(21)f∗Y∗=WTφY∗where Y∗≔[x-nT,y_,y-]T∈Rn+2. According to Assumption 3 above, one can obtain that(22)f∗=f∗=WTφY∗≤WT2+φY∗24≤θ2+φY∗24≤ϑ1ψ1where ϑ1≔max{θ2,1/4}, ψ1≔1+φ(Y∗)2/4. For the compound disturbance d∗, under Remark 2 and Assumption 4, one can obtain(23)βd∗=βd∗≤βd+∫0lϕτdzτvdτ≤ϑ2ψ2where ψ2≔β with β>0 a time-varying positive parameter given in (14). ϑ2 is an unknown positive constant. Based on the foregoing analysis, the adaptive saturated controller is devised as(24)v=-1βρ0bηs+ϑ^1+μ11+exp-∫0tϑ^1τdτψ1signs+ϑ^2+μ21+exp-∫0tϑ^2τdτψ2signswhere η,μ1,μ2 are positive constants. ϑ^1,ϑ^2 are, respectively, the estimated values of unknown constants ϑ1,ϑ2. The corresponding adaptive scheme is(25)ϑ^˙1=-μ1exp-∫0tϑ^1τdτ1+exp-∫0tϑ^1τdτ2ϑ^1+ψ1sϑ^˙2=-μ2exp-∫0tϑ^2τdτ1+exp-∫0tϑ^2τdτ2ϑ^2+ψ2sRemark 8.
The controller (24) has two parts. One is the nominal part: -(1/βρ0b)ηs which is used to stabilize the tracking error system. The other one is the robust part, i.e., -1/βρ0b(ϑ^1+μ1/(1+exp-∫0tϑ^1τdτ))ψ1signs and -1/βρ0b(ϑ^2+μ2/(1+exp-∫0tϑ^2τdτ))ψ2signs which is utilized to further compensate the effects brought by the unknown nonlinearity, external disturbance, and control saturation. Compared with [16], only two parameters are needed to estimate rather than 2m parameters involved in the weight vector of RBFNN approximation. In this sense, the computational load is decreased dramatically.


## 3.3. Stability Analysis

Under the devised controller (24), an important result is given as follows.Theorem 9.
Under the designed controller (24) and adaptive scheme (25), the desired output reference command yr evolved within the envelope preplanned by the time-varying constraints can be tracked with asymptotic stability.Proof.
First, construct the following Lyapunov function as(26)V=V1+V2with(27)V1=12s2,V2=12ϑ~12+ϑ~22where the virtual error ϑ~i is defined as ϑ~i≔ϑi-ϑ^i+μi/(1+exp-∫0tϑ^iτdτ)(i=1,2) which is inspired by [19]. Taking the time derivative of V1 in (27) yields(28)V˙1=ss˙Substituting (20) and (24) into (28) yields(29)V˙1=sf∗x¯n,y_,y-+βρ0bv+βd∗=-ηs2-ϑ^1-μ11+exp-∫0tϑ^1τdτψ1s-ϑ^2+μ21+exp-∫0tϑ^2τdτψ2s+sf∗x¯n,y_,y-+βd∗-ηs2-ϑ^1-μ11+exp-∫0tϑ^1τdτψ1s-ϑ^2+μ21+exp-∫0tϑ^2τdτψ2s+s·f∗x¯n,y_,y-+βd∗≤-ηs2-ϑ^1-μ11+exp-∫0tϑ^1τdτψ1s-ϑ^2+μ21+exp-∫0tϑ^2τdτψ2s+s·f∗x-n,y_,y-+s·βd∗In view of (22) and (23), (29) equals (30)V˙1≤-ηs2-ϑ^1-μ11+exp-∫0tϑ^1τdτψ1s-ϑ^2+μ21+exp-∫0tϑ^2τdτψ2s+ϑ1ψ1s+ϑ2ψ2sTaking the time derivative of V2 in (27) yields(31)V˙2=-ϑ~1ϑ^˙1+μ1exp-∫0tϑ^1τdτ1+exp-∫0tϑ^1τdτ2ϑ^1-ϑ~2ϑ^˙2+μ2exp-∫0tϑ^2τdτ1+exp-∫0tϑ^2τdτ2ϑ^2 Substituting (30) and (31) into the time derivative of V in (26) yields(32)V˙≤-ηs2-ϑ^1-μ11+exp-∫0tϑ^1τdτψ1s-ϑ^2+μ21+exp-∫0tϑ^2τdτψ2s-ϑ~1ϑ^˙1+μ1exp-∫0tϑ^1τdτ1+exp-∫0tϑ^1τdτ2ϑ^1-ϑ~2ϑ^˙2+μ2exp-∫0tϑ^2τdτ1+exp-∫0tϑ^2τdτ2ϑ^2+ϑ1ψ1s+ϑ2ψ2sBased on the definition of virtual error ϑ~i(i=1,2) in (27), then (32) becomes(33)V˙≤-ηs2+ϑ1-ϑ^1-μ11+exp⁡-∫0tϑ^1τdτψ1s+ϑ2-ϑ^2+μ21+exp⁡-∫0tϑ^2τdτψ2s-ϑ~1ϑ^˙1+μ1exp-∫0tϑ^1τdτ1+exp-∫0tϑ^1τdτ2ϑ^1-ϑ~2ϑ^˙2+μ2exp-∫0tϑ^2τdτ1+exp-∫0tϑ^2τdτ2ϑ^2≤-ηs2+ϑ~1ψ1s-ϑ^˙1+μ1exp⁡-∫0tϑ^1τdτ1+exp⁡-∫0tϑ^1τdτ2ϑ^1+ϑ~2ψ2s-ϑ^˙2+μ2exp⁡-∫0tϑ^2τdτ1+exp⁡-∫0tϑ^2τdτ2ϑ^2Substituting (25) into (33) yields(34)V˙≤-ηs2≤0 Only when s=0 can V˙=0. Thus, the tracking error system is asymptotically stable, i.e., s→0 as t→∞. Accordingly, based on (15), by employing Lemma 7, one can obtain that all the newly defined tracking errors z1,z2,…,zn converge to zero as t→∞. This completes the proof of Theorem 9.


## 4. Illustrative Example

To validate the effectiveness of the proposed control law, simulations of a second-order Duffing Holmes chaotic system are employed. In this application example, the chaotic system is expressed by [20](35)x˙1=x2x˙2=-ω1x1-ω2x2-ω3x13+qcosξt+satu+dy=x1whereω1=0.3+0.2sin(10t), ω2=0.2+0.2cos(5t), ω3=1, q=5+0.1cos(t), ξ=0.5+0.1sin(t), u≤10, d=0.4sin(0.2πt)+0.3sin(x1x2). The phase image of Duffing Holmes chaotic system is demonstrated in Figure 1. According to Remark 2, Gaussian kernel function with detailed form given in (7) can be utilized as one part of the density function. When the density function is selected as ϕ=10exp(τ2/2), one can obtain ρ0=12.5. The simulation parameters are set: yr=sin(0.5t)+cos(0.25t),y-=yr+exp(-0.3t)+0.45, y_=yr-exp(-0.3t)-0.45 in (5); m=50,εi=0.5+2rand·,γi=[2,-8,6,2]T(i=1,2,…,m) in (6) and (7); c1=2,η=100,μ1=μ2=3, ϑ^10=5+rand·, ϑ^20=3+rand· in (15) and (24). In order to show the effectiveness of the proposed control law, the traditional backstepping control method is employed as the comparative test (the detail introduction can be found in [6], which is omitted for brevity). The number of nodes of RBFNN is 50 with the center evenly spaced on [-1,3]×[-17,1]×[-5,17]×[-2,6]×[-5,7]×[-4,10]. The initial weights Wb=-3+6rand(·) of the RBFNN in the traditional backstepping control technique. The initial simulation conditions are x1=1.7,x2=0. The relevant simulation results are shown in Figures 2–6. Note that ‘ASC’ and ‘TBC’ refer to the proposed control law and the traditional one, respectively.Figure 1
Phase image of system (35) without control input.Figure 2
Output tracking trajectories under two control laws.Figure 3
Output tracking errors under two control laws.Figure 4
Control inputu=sat(v) under two control laws.Figure 5
Adaptive parameters under two control laws.Figure 6
WeightsWb of RBFNN under traditional backstepping control law.Seeing from these simulation results in Figures2–6, one can conclude that (i) after 30 seconds, the output tracking errors under the proposed control law are much smaller than those under the traditional backstepping control law (about 4 times improvement in the tracking accuracy depicted in Figure 3). (ii) From Figures 5 and 6, one can find that only two adaptive parameters are needed to update, while there are 50 elements involved in Wb to be updated in the traditional backstepping control law. In this sense, the computational load is decreased dramatically.


## 5. Conclusion

In this paper, a novel low-complexity adaptive saturated control law has been investigated for a class of nonlinear systems with consideration of time-varying output constraint, control saturation, and external disturbance. Compared with the existing works, the computational load of adaptive schemes is very small and the asymptotic stability of tracking error system can be guaranteed rather than the uniformly ultimately bounded stability. The employed simulations show that about 4 times improvement can be obtained in the tracking accuracy, which demonstrates the effectiveness of the proposed control law.



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*Source: 7135930-2019-06-16.xml*