# Adaptive Neural Networks Control Using Barrier Lyapunov Functions for DC Motor System with Time-Varying State Constraints

Authors: Lei Ma; Dapeng Li
Journal: Complexity
 (2018)

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## Abstract

This paper proposes an adaptive neural network (NN) control approach for a direct-current (DC) system with full state constraints. To guarantee that state constraints always remain in the asymmetric time-varying constraint regions, the asymmetric time-varying Barrier Lyapunov Function (BLF) is employed to structure an adaptive NN controller. As we all know that the constant constraint is only a special case of the time-varying constraint, hence, the proposed control method is more general for dealing with constraint problem as compared with the existing works on DC systems. As far as we know, this system is the first studied situations with time-varying constraints. Using Lyapunov analysis, all signals in the closed-loop system are proved to be bounded and the constraints are not violated. In this paper, the effectiveness of the control method is demonstrated by simulation results.


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## Body

## 1. Introduction

Due to the requirements of practice and the development of theory, the controller design of uncertain system has become a new research direction and attracted more and more scholars’ attention. The uncertainty of the actual engineering system has been studied in many works [1–4]. The neural networks [5] and fuzzy logic systems [6] have become the two main tools which can effectively deal with the unknown functions in the systems. In [7, 8], these are studies of some actual engineering systems with uncertain parameters. In [9, 10], the NN is used to approximate several random perturbations and unknown functions. In [11–16], several nonlinear system solutions are studied based on neural networks and fuzzy logic systems. In [17], adaptive control schemes based on neural networks were proposed for nonlinear systems with unknown functions. Based on neural networks and fuzzy logic systems, the significant studies proposed the novel adaptive tracking control methods for nonlinear SISO systems in [18–20] and MIMO systems in [21–23]. However, it is worth noting that the constraint problem is worth noting in the above approaches, which lead to the inaccuracy or oscillations of the engineering systems and even cause control systems instability.In fact, there are constraints in most physical systems with various forms, for example, physical stoppages, saturation, performance, and safety specifications, such as restricted robot manipulation system [24], application to chemical process [25], networked surveillance robots systems [26], and nonuniform gantry crane [27]. In recent years, the barrier Lyapunov functions become the main tools to solve the constrained problem which was proposed for the first time in [28]. Based on BLF, some adaptive control methods were presented for nonlinear systems with output constant constraint in [29, 30] and state constant constraint in [31–34]. As we known, the constant constraint is the special case of the time-varying constraint. Subsequently, the authors in [35, 36] proposed some adaptive control approaches to address the stability problem of nonlinear systems with time-varying constraints.Motor is the most important electromechanical energy conversion device, which has been widely used in the industrial and agricultural production, transportation, aerospace, and so on. In particular, the motor system with unknown uncertainties has attracted the attention of many scholars, and the control problem of the motor system becomes more and more important. In [37, 38], the authors proposed two adaptive control methods for systems with unknown functions. The authors in [39, 40] presented an adaptive control with time-varying output constraints for DC motor systems. According to the above descriptions, the urgent problem is how to address the stability problem of the DC motor system with time-varying state constraints.This paper presents an adaptive NN tracking control method for DC motor systems with time-varying state constraints. As far as we know, there is no work dealing with such DC motor systems in the literature at present stage. The contributions of this paper are summarized as follows.(1) The time-varying state constraints are first considered in the DC motor systems; comparing with the existing on DC motor systems, the proposed control method is more general and extensive in the engineering field. (2) To guarantee that the state constraints always remain in the time-varying constrained sets, the asymmetric time-varying BLF is utilized. (3) A novel adaptive tracking controller based on the neural networks and backstepping technique is structured to guarantee that all signals in the closed-loop system are bounded, the tracking errors converge to a small neighborhood of zero and the time-varying state constraints are not transitioned.


## 2. Problem Formulation and Preliminaries

Consider the dynamic system with the DC motor without vibration mode as the following form:(1)α˙1=α2Jα˙2+fα˙1+Tf+d=uy=α1,where α1t is the motor angular position; α2 stands for motor angular velocity; J is a known inertia, f is an unmeasured viscous friction, and Tf is an unmeasured nonlinear friction; dt represents the unknown disturbance but bounded with dt≤dM; y∈R is the system output; and u represents the motor torque. In particular, output yt is required as follows:(2)k_c1t<yt<k-c1t,∀t≥0,where k-c1:R+→R and k_c1:R+→R such that k-c1t>k_c1t, ∀t∈R+.Remark 1.
From (2), the states of DC systems are constrained by the considered time-varying functions. In [35, 36], the constraint problem is omitted, which is the main factor of the oscillations of the engineering systems. The authors in [39] addressed the stability problem of DC motor systems with constant constraint which is the special case of the time-varying constraint. Comparing with the [40], the authors only consider time-varying output constraint; the proposed adaptive control method tries to stabilize the DC motor systems with time-varying state constraints, which cause the difficulty of controller design.In this paper, the control objective is to design an adaptive NN tracking controlleru which adjusts the output of DC motor systems y to track desired trajectory of the reference signal ydt in the range of time-varying constraint functions. Meanwhile, all signals in the closed-loop systems are bounded and the time-varying state constrains are not violated.Assumption 2 (see [35]).
There exist constantsK-ci and K_ci, i=0,1,…,n, such that k-c1t≤K-co, k_c1t≥K_co, and k-c1it≤K-ci, k_c1it≤K_ci, ∀t≥0, i=1,…,n.Assumption 3 (see [32]).
There exist functionsY-0:R+→R+ and Y_0:R+→R+ satisfying Y-0t<k-c1t and Y_0t>k_c1t∀t≥0, and positive constants Yi, i=1,…,n, such that the desired trajectory ydt and its time derivatives satisfy Y_0t≤ydt≤Y-0t and ydit≤Yi, i=1,…,n, ∀t≥0.The following lemma is represented for the establishment of binding compensation and performance limits.Lemma 4 (see [28]).
LetZ≔ξ∈R:ξ<1⊂R and N≔Rl×Z⊂Rl+1 be open sets. Take into account the system(3)α˙=gt,α,where α≔ω,ξT∈N and g:R+×N→Rl+1 in t is piecewise continuous and in α is locally Lipschitz, united in t, on R+×N.Suppose that there are functionsH:Rl×R+→R+ and V1:Z→R+. In their respective domains, they are continuously differentiable and positive definite, such that(4)V1ξ→∞,ξ⟶1λ1ω≤Hω,t≤λ2ω,where λ1 and λ2 are class K∞ functions. Let Vα:=V1ξ+Uω,t, and ε0∈Z. If the inequality is established:(5)V˙=∂V∂αg≤0in ξ∈Z, ξt∈Z∀t∈0,∞.Lemma 5 (see [35]).
For allξ<1 and positive integer p, the inequality log⁡1/1-ξ2p<ξ2p/1-ξ2p.Proof.
Forξ<1, the term ξ2p/1-ξ2p can be rewritten as(6)ξ2p1-ξ2p=log⁡eξ2p/1-ξ2p≥log⁡1+ξ2p1-ξ2p+∑n=2∞ξ2p/1-ξ2pnn!≥log⁡1+ξ2p1-ξ2p=log⁡11-ξ2p.
The proof is completed.


## 3. State Feedback Adaptive Controller Designs

This paper presents an adaptive tracking controller based on a backstepping technique with the asymmetric time-varying BLF for the DC motor systems. The detailed designs process is shown in this section.Denotez1=α1-yd, z2=α2-σ1, where σ1 is the virtual controller which will be given later on. We consider the time-varying asymmetric BLF:(7)V1=qz12plog⁡kb12ptkb12pt-z12p+1-qz12plog⁡ka12ptka12pt-z12p,where p is a positive integer.The time-varying barriers are chosen as(8)ka1t=ydt-k_c1t(9)kb1t=k-c1t-ydtand q· is defined as(10)q·=1,if·>00,if·≤0.Based on Assumptions2 and 3, there are positive constants k_b1, k-b1, k_a1, and k-a1, such that(11)k_b1≤kb1t≤k-b1.k_a1≤ka1t≤k-a1,∀t≥0Through the change of error coordinates,(12)ξai=zikai,ξbi=zikbiξi=qziξbi+1-qziξai,i=1,2.According to (12), (7) can be rewritten as(13)V1=12plog⁡11-ξ12p.Remark 6.
According to (10), we know that when z1>0, we obtain qz1=1, ξ1=ξb1, and V1=1/2plog⁡1/1-ξb12p=1/2plog⁡1/1-ξ12p. When z1<0, we obtain qz1=0, ξ1=ξa1, and V1=1/2plog⁡1/1-ξa12p=1/2plog⁡1/1-ξ12p. From the above, we can get (13) based on (12).Obviously, under the premise ofξ<1, V1 is definite continuously differentiable. The time derivative of V1 is(14)V˙1=qz1ξb12p-1kb11-ξb12pz2+σ1-y˙d-z1k˙b1kb1+1-qz1ξa12p-1ka11-ξa12pz2+σ1-y˙d-z1k˙a1ka1.Choose the virtual controllerσ1 as(15)σ1=-κ1+κ-1tz1+y˙d-2p-12pz1.The time-varying gain is given as(16)κ-it=k˙aikai2+k˙bikbi2+βi,where βi and κi, i=1,2 are any positive constants. Make sure that the time derivative α1 is bounded, when k˙a1 and k˙b1 are both zero. Substituting (15) and (16) into (14) and noting that(17)κ-i+qzik˙bikbi+1-qzik˙aikai≥0we obtain(18)V˙1=qz1ξb12p-1kb11-ξb12p+1-qz1ξa12p-1ka11-ξa12pz2-z1κ-1tqz1ξb12p1-ξb12p+κ-1t1-qz1ξa12p1-ξa12p+k˙b1kb1qz1ξb12p1-ξb12p+k˙a1ka11-qz1ξa12p1-ξa12p-κ1z1qz1ξb12p1-ξb12p+1-qz1ξa12p1-ξa12p-2p-12pμ1z12p,where(19)μ1=qz1kb12p-z12p+1-qz1ka12p-z12p.After finishing it, we get(20)V˙1=qz1kb12p-z12p+1-qz1ka12p-z12pz12p-1z2-ξ12p1-ξ12pk˙b1kb1qz1+k˙a1ka11-qz1+κ-1t-κ1ξ12p1-ξ12p-2p-12pμ1z12p.Based on (12), we obtain (21)V˙1≤-κ1ξ12p1-ξ12p+μ1z12p-1z2-2p-12pμ1z12p.Using Young’s inequality, the following inequality holds:(22)μ1z12p-1z2≤μ12p-12pz12p+12pz22p.Substituting (22) into (21), V˙1 can be further rewritten as(23)V˙1≤-κ1ξ12p1-ξ12p+12pμ1z22p.The Barrier Lyapunov FunctionV2 is given as(24)V2=V1+qz22plog⁡kb22ptkb22pt-z22p+1-qz22plog⁡ka22ptka22pt-z22p.Then, differentiating ofV2 with respect to time is given by(25)V˙2=V˙1+qz2ξb22p-1kb21-ξb22pz˙2-z2k˙b2kb2+1-qz2ξa22p-1ka21-ξa22pz˙2-z2k˙a2ka2.From the definition of the tracking errorz2=α2-σ1, it is easy to obtain z˙2=α˙2-σ˙1, and the derivative of the virtual controller is given as(26)σ˙1=∂σ1∂α1α2+∑j=01∂σ1∂ζ1ζ1j+1,where ζ1=yd,ka1,kb1T.According to (26), (25) can be rewritten as (27)V˙2=α˙2-σ˙11-qz2ka2ξa22p-11-ξa22p+qz2kb2ξb22p-11-ξb22p-z2k˙b2kb2qz2kb2ξb22p-11-ξb22p-z2k˙a2ka21-qz2ka2ξa22p-11-ξa22p+V˙1.Based on (23), we get(28)V˙2≤-κ1ξ12p1-ξ12p+12pμ1z22p+α˙2-σ˙1ξ22pz21-ξ22p-z2k˙b2kb2qz2kb2ξb22p-11-ξb22p-z2k˙a2ka21-qz2ka2ξa22p-11-ξa22p.Substituting (26) into the above formula, we obtain(29)V˙2=-κ1ξ12p1-ξ12p+ξ22pz21-ξ22pJ-1-fα˙1-Tf+u-σ˙1-qz2kb2ξb22p-11-ξb22pz2k˙b2kb2-1-qz2ka2ξa22p-11-ξa22pz2k˙a2ka2-J-1ξ22pz21-ξ22pd+12pμ1z22p.Using Young’s inequality and notingdt≤dM, we obtain(30)-J-1ξ22pz21-ξ22pd≤12γ1J-1ξ22pz21-ξ22p2+12γ1dM2,where γ1 is a positive constant.Based on (30), (29) can be rewritten as(31)V˙2=-κ1ξ12p1-ξ12p+12pμ1z22p+ξ22pz21-ξ22p×J-1-fα˙1-Tf+u-σ˙1+qz2kb2ξb22p-11-ξb22p×-z2k˙b2kb2+1-qz2ka2ξa22p-11-ξa22p-z2k˙a2ka2+12γ1J-1ξ22pz21-ξ22p2+12γ1dM2.For convenience, we define(32)MZ=J-1Tf+fα˙1+σ˙1.In fact, since the parameters off and Tf are not available, M is unknown in practice. In order to solve the uncertainty of this parameter, we designed NN, as shown below to estimate(33)MZ=W∗TSZ+εZ,where Z=α1T,α2TT∈Ωz⊂R3, and similar to [28], we assume that the approximate error εZ satisfies εZ≤ε∗ with the constant ε∗>0.Substituting (33), (31) can be rewritten as(34)V˙2≤-κ1ξ12p1-ξ12p+12pμ1z22p+J-1ξ22pW~TSZz21-ξ22p-J-1ξ22pz21-ξ22pεZ+12γ1J-1ξ22pz21-ξ22p2+12γ1dM2+κ-2ξ22p1-ξ22p+J-1ξ22pz21-ξ22pu.According to Young’s inequality, we can easily obtain(35)-J-1ξ22pεZz21-ξ22p≤12γ2J-1ξ22pz21-ξ22p2+12γ2ε∗2,where γ2 is a positive constant.Based on (35), (34) can be rewritten as(36)V˙2≤-κ1ξ12p1-ξ12p+12pμ1z22p+J-1ξ22pW~TSZz21-ξ22p+12γ2J-1ξ22pz21-ξ22p2+12γ2ε∗2+κ-2ξ22p1-ξ22p+12γ1J-1ξ22pz21-ξ22p2+12γ1dM2+J-1ξ22pz21-ξ22pu.The actual controller is given as(37)u=-12pJμ1z2kb22p-z22p-J-1ξ22p2γ1z21-ξ22p-J-1ξ22p2γ2z21-ξ22p+Jz2W^TSZ-Jz2κ2+κ-2.Substituting (37), we obtain(38)V˙2≤-κ1ξ12p1-ξ12p-κ2ξ22p1-ξ22p+12γ1dM2+ξ22p1-ξ22pW~TSZ+12γ2ε∗2.Design the Lyapunov function candidateV3:(39)V3=V2+12W~TΓ-1W~,where Γ=Γ-1>0 is a constant matrix and W~=W^-W∗.The time derivative ofV3 is(40)V˙3=V˙2+W~TΓ-1W^˙.Based on (38), we obtain(41)V˙3≤-κ1ξ12p1-ξ12p-κ2ξ22p1-ξ22p+12γ1dM2+12γ2ε∗2+W~TΓ-1W^˙+ξ22p1-ξ22pW~TSZ.The adaptive law is given as follows:(42)W^˙=Γ-ξ22p1-ξ22pSZ-ηW^,where η is a positive constant.Substituting (42) into (41), we get(43)V˙3=-κ1ξ12p1-ξ12p-κ2ξ22p1-ξ22p+12γ1dM2+12γ2ε∗2-W~TηW~+W∗.Using Young’s inequality,(44)V˙3≤-κ1ξ12p1-ξ12p-κ2ξ22p1-ξ22p+12γ1dM2+12γ2ε∗2-η2W~2-W∗2.After finishing it, we get(45)V˙3≤-κ1log11-ξ12p-η2W~2+η2W∗2-κ2log11-ξ22p+12γ1dM2+12γ2ε∗2.Then, the above inequality can be rewritten as(46)V˙3≤-ρV3+C,where(47)ρ=min⁡2κ1,2κ2,ηΓ-1.C=12γ1dM2+12γ2ε∗2+η2W∗2Theorem 7.
Consider the unknown DC motor control system (1), based on the assumptions of Assumptions 2 and 3, Lemma 4, actual controller (37), and the adaptive law (42). The following properties guaranteed that the tracking error singles will remain in a compact neighborhood of zero, that is, limt→∞⁡yt-ydt=0, all signals of the closed-loop system are bounded, and all state constraints are never violated.Proof.
With both sides of (46) multiplied by eρt, we obtain(48)ddtV3eρt≤Ceρt.
Integrating (48) over 0,t, we have(49)0≤V3t≤V30e-ρt+Cρ.
Based on (7), (24), and (39), we can obtain(50)V3=12logkb12ptkb12pt-z12p+12logkb22ptkb22pt-z22p+12W~TΓ-1W~.
Then, we have(51)12log⁡kb12ptkb12pt-z12p≤V3At≤V3A0,where (52)V3A0=12log⁡kb12p0kb12p0-z12p+14λmaxΓ-1W^0-W∗+C2ρ.
Therefore, we know that(53)z1kbt2p≤1-e-2V3A0.
Based on the above inequality, the following inequality is obtained:(54)z1t≤D1t,where(55)D1t=kb1t1-e-2V3A02p.
Similar to the derivation ofz1, we can obtain the conclusion that(56)z2t≤D2t,where(57)D2t=kb2t1-e-2V3B02p.
From Assumption2, we can be known that α10<Kc10, and from the definition of kc1t, we have z1t<kb1t. In fact, from α1=z1+yd and α2=z2+σ1, we obtain(58)α1t<kb1t+ydt.
Based on the above inequality, we knowyt≤kc1t, ∀t≥0. Therefore, the output signals are bounded.
Obviously, we can clearly obtain that the virtual controllerσ1 is bounded in (15). Based on z2=α2-σ1 and (56), α2 is bounded. In addition, from (37) and (42), we know the actual controller u and the adaptive law W^ are bounded. Therefore, all the closed-loop system signals are bounded.
The proof is completed.Remark 8.
In the above analysis, it is apparent that the boundedness ofz1 lies on the design parameters γ1, γ2, η, dM, ε∗, W∗, κ1, κ2, and Γ-1. If we fix η>0, it is clear that decreasing γi might result in small C and increasing κi might result in large ρ; thus, it will help to reduce C/ρ. This represents that the tracking errors can be made arbitrarily small by selecting the design parameters appropriately.


## 4. Simulation Results

To illustrate the validity of the proposed adaptive NN control method, a simulation example is provided. Specifically, the following the DC motor system is described by(59)Jα˙2+fα˙1+Tf+d=u,where the inertia is J=0.018 kg·m2, f denotes an unmeasured viscous friction with f=sin⁡t+1.82, Tf is an unmeasured nonlinear friction with Tf=0.987, and d is the external interference with d=0.05sin⁡t. The desired reference signal is given as yd=0.4cos⁡t-0.2. The virtual controller, the actual controller, and the adaptation law are chosen as follows:(60)σ1=-κ1+κ-1tz1+y˙d-2p-12pz1u=-12pJμ1z2kb22p-z22p-Jκ2+κ-2z2-J-1ξ22p2γ1z21-ξ22p-J-1ξ22p2γ2z21-ξ22p+Jz2W^TSZW^˙=Γ-ξ22p1-ξ22pSZ-ηW^.The angular position and the angular velocity of motor systems are bounded bykc1<α1<kc2 and kc3<α2<kc4 with kc1=0.5+0.2cos⁡t, kc2=-1+0.4cos⁡t, kc3=-1.4+0.4cos⁡t, and kc4=0.6+0.2cos⁡t. The NN WTSz contains 20 nodes and the centers μi, i=1,…,20. The design parameters of the proposed control method are chosen as Γ=2.5I, p=2, β1=4, β2=4, κ1=2, κ2=2, c=1, and Tf=0.987 and the initial condition of the system state is chosen as α10=0.35, α20=-0.95, and W^0=0.For the DC motor system, using a method of controlling the program can be obtained by the simulation results shown in Figures1–5. Figures 1 and 2 show the output trajectory. Figure 1 shows the output and the reference signal tracking effect; the figure shows that the two curves almost coincide; that is to say, the tracking error converges to zero. Figure 3 shows the tracking error trajectory of z1t initially from the boundaries kb1t and -ka1t repulsion, but eventually converging to zero. Figure 4 shows a bounded and adaptive law of locus. According to Figure 4, we can see that the track adaptation law is bounded. Thus, we can conclude that a good tracking performance can make all the signals in the closed-loop system bounded. From Figure 5, it can be observed that the control input is bounded by a bounded back and forth reciprocate.Figure 1
The trajectories of outputα1 and the reference signal ydt.Figure 2
Trajectory of stateα2.Figure 3
Phase portrait ofz1 and z2.Figure 4
The trajectory ofW^.Figure 5
The trajectory of inputu.


## 5. Conclusion

In this paper, we propose an adaptive tracking control method for a DC system with full state constraints. The asymmetric time-varying BLF is employed to guarantee that the states always remain in the time-varying constrained sets. In the asymmetric system, neural networks and a backstepping technique are used to construct an adaptive control and adaptation laws to ensure that all signals in the closed-loop system are bounded and the state constraints are not transitioned. The performances of the adaptive control method based asymmetric time-varying BLF are verified by a simulation example.



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