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Design of Fast Variable Structure Adaptive Fuzzy Control for Nonlinear State-Delay Systems with Uncertainty
M. Montazeri, M. R. Yousefi, K. Shojaei & G. Shahgholian
To cite this article: M. Montazeri, M. R. Yousefi, K. Shojaei & G. Shahgholian (2022) Design of Fast Variable Structure Adaptive Fuzzy Control for Nonlinear State-Delay Systems with Uncertainty, IETE Journal of Research, 68:6, 4577-4589, DOI: 10.1080/03772063.2020.1800522 To link to this article: https://doi.org/10.1080/03772063.2020.1800522
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https://doi.org/10.1080/03772063.2020.1800522
Design of Fast Variable Structure Adaptive Fuzzy Control for Nonlinear State-Delay Systems with Uncertainty
M. Montazeri1, M. R. Yousefi 1,2, K. Shojaei 1,2,3and G. Shahgholian 1,2
1Department of Electrical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran;2Smart Microgrid Research Center, Najafabad Branch, Islamic Azad University, Najafabad, Iran;3Digital Processing and Machine Vision Research Center, Najafabad Branch, Islamic Azad University, Najafabad, Iran
ABSTRACT
In this study, a new fast variable structure adaptive fuzzy controller is presented for nonlinear state- delay systems which are subjected to external disturbances and uncertainties. The undesirable chattering and singularity of the variable structure scheme are eliminated by using a novel fast robust high-precision continuous nonsingular control law which is able to accelerate the finite-time conver- gence both in reaching and sliding phases of the motion. A fuzzy logic system with a neural network adaptive law is used to approximate the dynamics of the nonlinear system containing the current state and the delayed state. The superiority of the proposed fuzzy neural network in online adjusting the weights of the network is the fast convergence rate of the approximation error to the optimum value in a very short time. The stability of the closed-loop system is proved by using an extended finite-time Lyapunov criterion such that the convergence of the position tracking error, velocity tracking error, and the estimation error to the bounded region is guaranteed in a very short time.
Two second-order uncertain nonlinear simulation examples with external disturbances are given to evaluate the efficacy of the proposed control technique. The simulation results show that faster and high-precision tracking performance is obtained compared with the existing recent works focused on robust control of nonlinear state-delay systems with uncertainties.
KEYWORDS Adaptive control;
Continuous; Finite-time;
Fuzzy neural network;
Nonsingular; State-delay;
Variable structure scheme
1. INTRODUCTION
The time delay exists in many practical uncertain nonlin- ear systems with external disturbances such as biological systems, chemical processes, hydraulic systems,etc. The delay in the state factors can reduce the performance of the tracking as well as the dynamics and the stabilization problem of the closed-loop system become much more complicated. Therefore, studying the controller design procedures for such systems has attracted a great deal of attentions. In the recent decades, many researchers have investigated the control problem of the state-delay uncer- tain nonlinear systems using robust control techniques based on the sliding-mode technique [1–4]. In [1], an appropriate transformation was utilized at first to explore the system in a regular form to facilitate the design of the sliding surface. Then, a robust sliding mode controller was employed to stabilize the system structure asymp- totically. In [2], a novel dynamic sliding mode controller (SMC) has been introduced to control a class of stochas- tic time-delay nonlinear systems. It was shown that a set of linear matrix inequalities (LMIs) should be solved to design the dynamics of SMC. Similar condition has been provided in [3,4] to guarantee the SMC based on the LMI technique. The main limitation of these works is that the
control law only drives the system trajectories toward the sliding surface in finite time and maintains the sliding motion in the neighbourhood of the origin. Chenet al.
have addressed the problem of the robust stabilization for uncertain systems with the distributed delays under satu- rated state feedback [5]. Recently, Zhanget al.have used SMC for the robust control of a class of uncertain non- linear systems with the time delay [6]. By transforming the control law to a virtual feedback, the LMI technique was applied to guarantee the reaching condition of the sliding-mode in the finite time and to keep the system dynamics on it. The necessary requirement of all the mentioned works to design the control law is that they need known dynamics of the plant.
In recent decades, some researchers have focused on the design of robust control methods in combination with the adaptive control laws based on the fuzzy logic systems (FLSs) and neural networks (NNs). In highly nonlin- ear dynamic systems with unknown dynamics, FLSs and artificial NNs are two main universal approximators to estimate the unknown nonlinear continuous functions of the plant subjected to external disturbances and uncer- tainties. Based on the ability of these approximators,
© 2022 IETE
the main advantage of FLS- and NN-based estimators is that they can estimate any continuous function with the sufficient fuzzy rules [7–9] or hidden neurons [10].
Therefore, the unknown dynamics of the plant being controlled can be identified online to any prescribed accuracy. In [11–19], indirect adaptive fuzzy controllers have been utilized to control a class of nonlinear systems with time delays subjected to system uncertainties such as parameter variations and external disturbances. Using the adaptive fuzzy laws, the unknown nonlinear system dynamics were estimated using FLSs. In some works, the researchers have proposed adaptive NNs controllers (Dynamic surface control [20–22], backstepping pro- cedures [23] and leader-following consensus approach [24]) to control a class of time-delay nonlinear systems with dynamic uncertainties. Radial basis function NNs were utilized to learn the unknown nonlinear terms in the control law. Geet al. presented a practical robust adap- tive controller based on the backstepping design for a class of parametric-strict-feedback nonlinear unknown state-delay systems by employing two adaptive laws for estimating the functions in the control law [25]. A similar work has been presented based on continuous RISE- based adaptive control in [26]. The key feature of all of these works is that the controller doesn’t need any prior information about the nonlinear time-varying dynamics of the system being controlled. However, all the afore- mentioned controllers suffer from a major drawback in which only the asymptotic convergence of the closed- loop system can be assured. Moreover, all the adapta- tion laws in these works have been derived from the integer-order Lyapunov or Lyapunov–Krasovskii func- tions which can be very slow.
Motivated to deal with these limitations and to improve the convergence rate of the global asymptotic stabil- ity, a finite-time variable structure controller (VSC) has been proposed to achieve the convergence of the sys- tem dynamics to the origin in a finite time [27–30]. The advantage of this method is that the finite-time VSC accomplishes faster convergence and higher-precision control in the finite time by using nonlinear sliding sur- faces over linear conventional VSC scheme with linear variable structure surfaces [31–33]. However, the main difficulties of the conventional discontinuous finite-time VSC are the singularity and chattering problems which are undesirable in practice. The singularity occurs in some areas of the state space in which the derivative of the sliding surface results in terms with negative (frac- tional) powers so that the finite-time VSC may require to be infinitely large in order to maintain the ideal finite- time variable structure motion in both reaching and slid- ing phases [31]. The discontinuous control action across
the variable structure surface causes chattering which involves extremely high control activity, and further- more, may excite high-frequency dynamics neglected in the course of modelling.
To resolve all the above limitations in this paper, we present a robust continuous nonsingular VSC adaptive fuzzy controller for tracking problem of the second- order nonlinear uncertain state-delay system. The non- linear time-varying dynamics of the system (current and delayed states) are completely assumed to be unknown and a FLS approximator with NN learning algorithm is employed to estimate these dynamics. The main contri- butions of the paper lie in the following items:
(1) From the continuity property of the control law and designing the nonsingular fast finite-time variable structure surface, the chattering phenomena and singularity have been avoided, respectively.
(2) In the reaching phase of the VSC in the litera- ture [34,35], in order to accelerate the speed of the motion of the state trajectory toward the sliding surface in the finite time, a fast finite-time variable structure reaching law has been developed. In con- trast, in the current study, a new faster finite-time reachability condition is designed for the first time, as far as the authors know. The two nonlinear terms with fractional power in the new continuous variable structure-type reaching law amplify the speed of the motion both in far away from the surface and in the neighbourhood of the manifold.
(3) In the sliding phase of the VSC, the finite-time con- vergence of the position and velocity tracking errors to the neighbourhood of zero is ensured by design- ing a fast continuous nonsingular finite-time vari- able structure surface.
(4) Initial values of the adaptive laws based on the FLSs or NNs, in the current study, a fuzzy inference sys- tem trained by a terminal-based neural network learning algorithm which is called fuzzy neural net- work (FNN) is used for on-line approximation of the unknown continuous nonlinear functions con- taining current state and the delayed state. The syn- ergy of the two universal approximators combines the advantages of both FLSs and NNs. Moreover, in [11], the bounds of the uncertainties are necessary to be known in the process of the controller design whereas in the current study, the uncertainties are completely assumed unknown and approximated with FNN.
(5) The adaptive robust fuzzy control schemes in [11,19]
have used the conventional slow rate gradient descent (GD) algorithm as the adaptation rule to
Table 1:Convergence time of the tracking error and its first derivative for the Example 2
Control method te1(s) te˙1(s) te2(s) te˙2(s)
Proposed method 0.23 0.27 0.30 0.38
Variable structure adaptive fuzzy controller [19] 0.30 0.40 0.70 0.65
tackle system uncertainties. But, the superiority of the proposed FNN in online adjusting the weights of the network is the faster convergence rate of the approximation error to the optimum value in a very short time in comparison with the GD learning algorithm. By increasing the fuzzy rules or hidden neurons of the network, the estimation error can be arbitrarily small so that the adaptation parameters of the approximators converge to the optimal values in the finite time.
Computer simulation examples are presented to eval- uate the effectiveness of the proposed control method through the control of a second-order nonlinear sys- tem and a two-degree-of-freedom mass-spring-damper system. The results of a second-order nonlinear sys- tem showed fast disturbance rejection and high-accurate tracking of the controllers under time-varying system parameters. For the benchmark example of the nonlin- ear mechanical system, the results of Figure5and Table1 in comparison with the results of Figures 3–6 in [19]
demonstrated that the faster convergence speed can be acquired by using the proposed variable structure adap- tive fuzzy controller than other schemes found in the literature.
2. FAST FINITE-TIME VARIABLE STRUCTURE ADAPTIVE FUZZY CONTROL
In this paper, the following uncertain nonlinear second- order state-delay system is considered:
˙
x1(t)=x2(t),
˙
x2(t)=f(x,t)+g(xτ(t),t)+u(t), (1) where x=x1,x2]T, and xτ(t) =x1(t−τ(t)),x2(t− τ(t))]T are vectors of current and delayed states, respectively,u(t) is the control input, τ(t) is a known bounded time delay and f(x,t) and g(xτ(t),t) are unknown lumped functions containing the uncertainties of current and delayed states, respectively.
Assumption 1: The desired trajectory xd(t)is a known bounded function of time which its first and second deriva- tives exist and are bounded by known positive constant.
Control objective:The controller objective is to generate an adaptive robust control law for system (1) to force the trajectory of the system states to track the referencexd(t) in the presence of model uncertainties. In this work, we design the following continuous nonsingular finite-time variable structure switching surface function [36]:
s(t)=e(t)+σ1sig(e(t))μ1+σ2sig(˙e(t))μ2, (2) where
sig(e(t))μ1= |e(t)|μ1sgn(e(t)), (3) sig(˙e(t))μ2= |˙e(t)|μ2sgn(˙e(t)). (4) sgnis the sign function,σ1,σ2 are design positive con- stants,μ1,μ2 are positive integers satisfying 1< μ2<
2,μ1> μ2ande(t)=x1(t)−xd(t)is the tracking error.
It can be easily proven that the sliding surface (2) has a globally finite-time stable attractor ate=0 [36].
2.1 Fuzzy Neural Network Approximator
Since in the real applications, the nonlinear functions f(x,t)andg(xτ(t),t) are unknown, in order to design the adaptive control input for system (1), these func- tions cannot be applied to the system directly and should be estimated online. We take a FNN to approximate the functions f(x,t) and g(xτ(t),t) in (1). The fuzzy sys- tem uses the fuzzy IF–THEN rules to perform a mapping from an input vectorx=[x1,. . .,xn]T ∈Rnto an output functiony∈R. Then,ith fuzzy rule is represented as Ri:ifx1isAij(x1)and . . .andxnisAin(xn), thenyisBi whereAijandBiare fuzzy sets with membership functions μAi
j(xj)andμBi(y), respectively. By using the product- inference rule, singleton fuzzifier and centre-average defuzzifier, the output of FLS can be described as y=
N
i= 1υin
j= 1μAij(xj) N
i = 1
n
j= 1μAij(xj) =υTφ(x), (5) where N is the number of total fuzzy rules, υi is the fuzzy singleton for the output in the ith rule, μAi
j(xj) is the membership function of the fuzzy vari- able xj characterized by the Gaussian function, υ= [υ1,υ2,. . .,υN]T is an adjustable parameter vector and φ=[φ1,φ2,. . .,φN]T is a fuzzy basis vector, whereφi is defined as
φi=
n
j= 1μAij(xj) N
i= 1
n
j= 1μAij(xj). (6)
By introducing the fuzzy systems in (5), the approxima- tion of functionsf(x,t)andg(xτ(t),t)can be expressed
as follows:
fˆ(x,υ1)=υT1φ(x), ˆ
g(xτ(t),υ2)=υT2φ(xτ(t)), (7) whereυ1Tandυ2Tare the corresponding tunable param- eter vectors for each FNN (see details in [37]).
2.2 Adaptive Law
To derive a neural network adaptive law, the parameters υ∗1 andυ∗2 must be adjusted with the following optimal parameter vectors:
υ1∗=arg min
υ1 {sup|ˆf(x,υ1)−f(x,t)|}, υ2∗=arg min
υ2 {sup|ˆg(xτ(t),υ2)−g(xτ(t),t)|}. (8) The fuzzy approximation error can be defined as ε(t)=υTφ(x,xτ(t))−(f(x,t)+g(xτ(t),t)), (9) where
υTφ(x, xτ(t))=ˆf(x,υ1)+ ˆg(xτ(t),υ2), φ(x, xτ(t))=[φ(x)Tφ(xτ(t))T]T
=[φ1(x),. . .,φN(x),φN+ 1(xτ(t)), . . .,φ2N(xτ(t))]T (10) and
υT =
υT1 υT2
. (11)
According to (7), the optimal approximation error is defined as
ε∗(x, xτ(t))=υ∗Tφ(x, xτ(t))−(f(x,t)+g(xτ(t),t)).
(12)
Combining (8) and (11) gives:
ε(t)= ˜υTφ(x,xτ(t))+ε∗(x,xτ(t)), (13) where υ˜ =υ−υ∗ is the optimal approximation error vector. The mechanism of the on-line adaptation is given by
˙
υ= −λ1|ε(t)|sgn(ε(t))φ(x, xτ(t))
−λ2|ε(t)|αα12sgn(ε(t))φ(x, xτ(t)), (14) whereλ1,λ2>0 and the positive parametersα1andα2
(α2>α1) should be odd integers. Then, the control input is designed as follows in this paper:
u(t)= −ˆf(x,υ1)− ˆg(xτ(t),υ2)+ ¨xd(t)
− 1 μ2σ2
(1 +μ1σ1|e|μ1−1)(sig(˙e(t))2−μ2
−k1sig(s)ω1−k2sig(s)ω2, (15) where 1< ω1<3 and 0< ω2<1 are two design parameters. Figure 1shows a proposed block diagram of the closed-loop finite-time variable structure adap- tive fuzzy control for uncertain nonlinear second-order state-delay system.
Theorem 1: For the nonlinear uncertain system (1), under Assumption 1, the continuous control input (15) with the finite-time switching surface (2), the nonlinear approximated functionsfˆ(x,υ1)andg(ˆ xτ(t),υ2)in (7), and the adaptation law (14) can guarantee that all sig- nals in the closed-loop system are bounded, the parameters υ1 andυ2converge to theυ∗1 andυ∗2 respectively with a fast convergence rate, and the tracking error and its first derivative converge to a bounded region in the finite time.
Figure 1:Block diagram for proposed finite-time variable structure adaptive fuzzy control
Lemma 1: Consider an extended Lyapunov function as V(˙ x)+β1Vω1(x)+β2Vω2(x)≤0, (16) whereβ1,β2 >0, ω1≥1and0< ω2<1.The settling time tsis given by
ts≤ V(x0)1−ω1 β1(ω1−1)
·F
1, ω1−1 ω1−λ2
; ω1−1
ω1−ω2 +1;−β2
β1V(x0)ω2−ω1 , (17) where F(a,b;c;z) denotes Gauss’ Hypergeometric func- tion [38]. With the conditions ofβ1,β2,ω1,ω2, absolute convergence of F(.) will be kept.
Proof of Lemma 1: Equation (16) can be represented as
V˙ ≤ −β1Vω1−β2Vω2. (18)
It can be seen that (18) has an equilibrium point atV=0.
From (18), and the fact that 0 < ω2< 1, we can write:
∂V˙
∂V ≤ −β1ω1Vω1−1−β2ω2Vω2−1
= −
β1ω1Vω1−1+ β2ω2
V1−ω2 . (19)
Since ∂V/∂V˙ → −∞ when V →0, the equilibrium point is a finite-time attractor of (16) with the initial conditionV=V(x0)[39]. By dividing two sides of the inequality (18) to −β1Vω1(x)−β2Vω2(x), and inte- grating both sides of it from 0 tots, we have:
ts≤ V(x0)1−ω1 β1(ω1−1)
·F
1, ω1−1 ω1−ω2
; ω1−1 ω1−ω2
+ 1;− β2
β1V(x0)ω2−ω1 , (20) which completes the proof of Lemma 1.
Lemma 2: [35]: Suppose a1,a2,. . .,an and0<p<2, are all positive numbers, then, the following inequality holds true:
(a12+. . .+an2)p< (a1p+. . .+anp)2. (21)
Proof of Theorem 1: Using e(t)¨ = ¨x1(t)− ¨xd(t) and (1), the first derivative of Equation (2) can be written as
˙
s= −μ2σ2|˙e|μ2−1(k1sig(s)ω1
−(f(x,t)+g(xτ(t),t)− ˆf(x,υ1)− ˆg(xτ(t),υ2))
+k2sig(s)ω2). (22)
By choosing the Lyapunov function candidate as V=V1+V2
=0.5˜υTυ˜+0.5s2, (23) and by differentiatingVwith respect to time, it yields:
V˙ = ˜υTυ˙ +s˙s= ˙V1+ ˙V2. (24) By substituting (12) and (14) into the first term of (24), we have:
V˙1= −(ε(t)−ε∗(x,xτ(t)))
λ1ε+λ2|ε|αα12sgn(ε)
= −(|ε(t)| − |ε∗(x,xτ(t))|)
λ1|ε(t)| +λ2|ε|α1α2 . (25) Since the estimation error meets|ε(t)|>|ε∗(x, xτ1(t))|, (25) indicates thatV˙1 < 0, and therefore, the adapted weights of the FNN estimator converge to the optimal one until
|ε(t)|<|ε∗(x, xτ1(t))|.
Remark 1: Using(12),υ˙˜ican be written as
˙˜
υi= ˙υi− ˙υi∗ = ˙υi, i=1,. . ., 2N
= −λ1ε(t)φi(x,xτ(t))
−λ2sig(ε(t))(α1/α2)φi(x,xτ(t)). (26) From(26), we can consider the Jacobian around the mini- mumυ˜i=0, i.e.
J= ∂υ˙˜i
∂υ˜i = −λ1φi(x, xτ(t))2−λ2α1φi(x, xτ(t))2 α2|ε(t)|(α2−α1)/α2. (27) The introduction of the nonlinearity item|ε(t)|α1/α2sgn(ε) in(14)amplifies the convergence rate of the approximation error in the neighbourhood ofε=0. This leads to the fact that the closer to the minimum, the faster the convergence speed toυ˜i=0.
Substituting (22) into the second term of (24) gives V˙2= −μ2σ2|˙e|μ2−1
×s((k1−(f(x,t)+g(xτ(t),t)− ˆf(x,υ1)
− ˆg(xτ(t),υ2))sig(s)−ω1)sig(s)ω1+k2sig(s)ω2) (28)
or
V˙2= −μ2σ2|˙e|μ2−1
×s((k2−(f(x,t)+g(xτ(t),t)− ˆf(x,υ1)
− ˆg(xτ(t),υ2))sig(s)−ω2)sig(s)ω2+k1sig(s)ω1).
(29) If k1 is chosen so that k1−(f(x,t)+g(xτ(t),t)− fˆ(x,υ1)− ˆg(xτ(t),υ2))sig(s)−ω1 is positive, then, there exists a bounded region ofs, beyond whichV˙2<0 holds true. Then, (28) can be written as
V˙2= −s(t)k¯1sig(s(t))ω1−s(t)k¯2sig(s(t))ω2 (30) wherek¯1=μ2σ2|˙e(t)|μ2−1(k1−(f(x,t)+g(xτ(t),t)− fˆ(x,υ1)− ˆg(xτ(t),υ2))sig(s)−ω1) and k¯2=μ2σ2
|˙e(t)|μ2−1k2. From Lemma 1, we have:
V˙2≤ −2(1+ω1/2)k¯1 minV2(1+ω1/2)
−2(1+ω2/2)k¯2 minV2(1+ω2/2), (31) where k¯1 min and k¯2 min represent the minimum val- ues ofk¯1andk¯2, respectively, and 1< (1+ω1)/2<2, 0.5< (1 +ω2)/2 < 1. According to the finite-time sta- bility criterion defined in Lemma 1, the finite-time reach- ability to the boundary layer will be achieved in the finite time:
tr ≤ 2(1−ω1/2)V2(x0)(1−ω1/2) k¯1 min(ω1−1)
·F
1, ω1−1
ω1−ω2; ω1−1 ω1−ω2 +1;
−2(ω2−ω1/2)k¯2 min
k¯1 min
V2(x0)(ω2−ω1/2)
. (32)
If the following condition holds true:
k1− |(f(x,t)+g(xτ(t),t)− ˆf(x,υ1)
− ˆg(xτ(t),υ2))|/|s|ω1>0, the following region
|s| ≤
⎡
⎢⎢
⎢⎣
|f(x,t)+g(xτ(t),t)
−ˆf(x,υ1)− ˆg(xτ(t),υ2)|
k1
⎤
⎥⎥
⎥⎦
(1/ω1)
=δ1 (33)
is reached in the finite time from Lemma 2. For (29), similar to the analysis of (28), the region:
|s| ≤
⎡
⎢⎢
⎢⎣
|f(x,t)+g(xτ(t),t)
−ˆf(x,υ1)− ˆg(xτ(t),υ2)|
k2
⎤
⎥⎥
⎥⎦
(1/ω2)
=δ2 (34)
is reached in the finite time. By virtue of (33) and (34), the region|s| ≤δ= min (δ1,δ2)can be reached in the finite time. Because|s| ≤δ, the finite-time variable structure surface (2) can be rewritten as
e(t)+
σ1− ψ
|e|μ1sgn(e)
|e(t)|μ1sgn(e)+σ2|˙e(t)|μ2sgn(˙e)=0,|ψ| ≤δ (35) Then, whenσ1−ψ/(|e|μ1sgn(e)) >0, (35) is still kept in the form of the Equation (2) which means that the tracking error will converge to the following region:
|e(t)| ≤ δ
σ1 (1/μ1)
(36) in the finite time. Furthermore, by considering the finite- time variable structure surface (35) and Equation (36), the first derivative of the tracking error will converge to the following region:
|˙e(t)| ≤((2δ+(δ/σ1)(1/μ1))/σ2)(1/μ2) (37) in the finite time. According to (25), (33) and (34), the parametersυ1andυ2converge to theυ1∗andυ2∗respec- tively, and all signals in the closed-loop control system are bounded. This completes the proof.
Remark 2: By using the sig function instead of sgn func- tion in the Equation(15), the control input is continuous (see details in [35], Appendix A) and therefore, the chatter- ing is terminated. Moreover, due to the positive fractional power μ21σ2(1+μ1σ1|e|μ1−1)sig(˙e(t))2−μ2, the control input(15)doesn’t have any singularity.
Remark 3: According to(33)and(34), the larger chosen parameters k1and k2results in smaller the boundary layer δ. However, increasing the value of these parameters can amplify the amplitude of the control input which may cause some problems in the implementation.
Remark 4: The control law in (15) can be applied for the tracking problem of the special class of the second- order nonlinear systems in the canonical form(1)such as a planar model of n-link rigid robotic manipulators. Future work will focus on the exploitation of the current controller design for multi-input multi-output nth-order nonlinear systems.
3. SIMULATION RESULTS
To assess the effectiveness of the control method, two nonlinear state-delay systems are considered below. The
controller parameters are selected heuristically for each system to achieve the best controller performance. The algorithm is simulated in MATLAB Simulink (The Math- works, R2014a) with the sampling period of 1ms. The root mean square(RMS) error is used as the performance index to measure the tracking accuracy as follows:
RMS =
1/TT
i= 1(x(i)−xd(i))2, (38) whereTis the total running time of the simulation.
Example 1: Consider the following second-order non- linear system:
¨
x(t)= f(x,t)+ d(t)+ g(˙xτ(t),t)+ u(t), (39) where the functions f(x,t), g(˙xτ(t),t) and d(t) are defined as follows:
f(x,t)=
0.5sin(x1)˙x2
−0.3sin(x2)˙x1
,
g=
g11 g12
g21 g22
˙
x1(t−τ(t))
˙
x2(t−τ(t))
g11=0.4˙x1(t−τ(t))sin(x1(t−τ(t))) + 3,
g12=0.1sin(x1(t−τ(t)))(˙x1(t−τ(t))+ ˙x2(t−τ(t))), g21= −0.4 sin(x2(t−τ(t)))x1(t−τ(t)),
g22=5 cos(x2(t−τ(t)))−2,
d(t)=
0.5 sin(2t)e−0.5t 0.5 cos(2t)e−0.3t
= d1
d2
. (40)
The state-delay is selected to be time-varying(0.7cos(t)+ 2). The control objective is to force the system outputsx1 andx2to track the following desired trajectories:
xd1(t)=(0.3 sin(5t)+0.6)(2+0.5e−0.5t2) (rad) xd2(t)= −(0.4 sin(3t)+0.8)(2−0.5e−0.5t2)(rad). (41) The nonlinear functions f(x,t) and g(xτ(t),t) are assumed completely unknown. The states x1, x˙1 (sys- tem 1) and x2, x˙2 (system 2) are controlled with separated controllers designed in the previous section and the interaction between both systems are con- sidered as external disturbances. Two fuzzy systems are used to estimate the unknown systems dynam- ics for the implementation of the variable struc- ture adaptive fuzzy controllers. The fuzzy systems are used to describe the function f(x,t), have x1,x˙1,x2
and x˙2 as inputs, and the ones used to approximate g(˙xτ(t),t)havex1,τ(t),x˙1,τ(t),x2,τ(t) andx˙2,τ(t) as input.
For each state variable x=[x1,x˙1,x2,x˙2]T and xτ(t)= [x1,τ(t),x˙1,τ(t),x2,τ(t),x˙2,τ(t)]T, we define three Gaussian membership functions as
μA1
j =exp
−1 2
(xjorxj,τ(t))+1.5 1
2 ,
μA2j =exp
−1 2
(xjorxj,τ(t)) 1
2 ,
μA3
j =exp
−1 2
(xjorxj,τ(t))−1.5 1
2
, j=1,. . ., 4 (42) The centres and the width of the membership func- tions of the fuzzy estimators are chosen such that the full possible range of the states with different initial conditions can be covered. The initial conditions are x(0) = [0.2 0.5 0.6 0.4]Tand the initial values of the esti- mated parameter υ for each controller are set equal to random values uniformly distributed between [0,1].
The designed parameters of the controllers used in this simulation are selected as follows: μ1= 1.2, μ2= 1.1, k1= 170, k2= 60, σ1=0.5, σ2=0.01, λ1=λ2=0.05, ω1= 1.1, andω2=0.3, for the controller 1, andμ1= 1.2, μ2= 1.1,k1= 250,k2= 40,σ1= 1,σ2=0.15,λ1=λ2= 0.05, ω1= 1.1, and ω2=0.4, for the controller 2. The gains of the controllerk1andk2are selected large enough to accelerate the motion toward the sliding surface and therefore, the small boundary layerδis achieved. But the maximum values of these parameters are limited accord- ing to Remark 3. The parametersω1,ω2,σ1,σ2,μ1and μ2are chosen heuristically in the defined ranges to obtain the chattering elimination and finite-time reachability to the boundary layer as well as decreasing the tracking time. The simulation results are shown by Figure2. The desired and actual trajectories almost overlap with each other after about 0.4 s and the control signals are fortu- nately chattering-free. The tracking errors for the outputs x1andx2are 0.1152 rad and 0.1591 rad, respectively. The results of Figure2demonstrate that the performance of the tracking is relatively less sensitive to the frequency and the magnitude of the delay signal and the controller could efficiently track the desired trajectories even with a large state-delay in system.
To verify the performance of the proposed controller under time-varying system parameters, the values of these parameters were varied randomly±50% from their
Figure 2:Simulation results of a second-order nonlinear system using proposed variable structure adaptive fuzzy controller
Figure 3:Simulation results of a second-order nonlinear system using proposed variable structure adaptive fuzzy controller under time-varying system parameters
nominal values during 10 s of simulation time. The ran- dom variations (uniform distribution) were generated by filtering random sequences through a low-pass Butter- worth filter (1 Hz cutoff frequency). The results of the simulation are shown by Figure3. The RMS errors for the outputsx1andx2are 0.1168 rad and 0.1621 rad, respec- tively. It is clearly observed that a very accurate track- ing occurs with the proposed variable structure adaptive fuzzy control under time varying uncertainties. Figure4 shows the response of the controllers under time-varying system parameters and a sudden step-like external dis- turbance, simultaneously whose RMS is 0.1170 rad forx1
Figure 4:Simulation results of sudden step-like external distur- bance rejection under the time-varying system parameters for proposed variable structure adaptive fuzzy controller
and 0.1627 rad forx2. The results clearly demonstrate the fast disturbance rejection and high-accurate tracking of the controllers under time-varying system parameters.
Example 2: Consider a multi-input multi-output non- linear system comprised of two inverted pendulums mounted on two carts, interconnected by a moving spring [19] as follows:
θ(t)¨ = f(θ,θ,˙ t)+ g(θτ1(t),t)+ u(t)+ d(t), (43) whereθ,θ,˙ θ¨ ∈R2are the vectors of the angular displace- ments, velocities and accelerations of the pendulums respectively, measured from the vertical on two carts and
f(θ,θ˙,t)=
f1(θ1,θ˙1,θ2) f2(θ1,θ2,θ˙2)
=
⎡
⎢⎢
⎢⎢
⎢⎢
⎢⎣ g
cLθ1− m
Mθ˙12sin(θ1)+k(s−cL) cmL2 (s(t)(θ2−θ1)+z2−z1)
g
cLθ2− m
Mθ˙22sin(θ2) +k(s−cL)
cmL2 (s(t)(θ1−θ2)+z1−z2)
⎤
⎥⎥
⎥⎥
⎥⎥
⎥⎦
g(θτ1(t),t)=
g1(θ1(t−τ1)) g2(θ2(t−τ1))
=
sin(t)θ1(t−τ1) cos(t)θ2(t−τ1)
, (44) where s(t)=sin(5t),z1(t)=sin(2t),z2(t)=sin(3t) +2, and c=m/(m+M). The external disturbances
are selected as 0 for t∈ [0,3] and are random noise uniformly distributed in the interval [0,0.5] for t∈ (3,10]. In the simulation, the parameter values are selected as g= 9.8 m/s2,τ1=0.3 s and 0.2 s for joint 1 and joint 2, respectively, m=10kg,M=10Kg, and L=1m. The boundary conditions for the system are restricted to –π/3≤θ1,θ2≤π/3(rad),−2π/3≤ ˙θ1≤ 2π/3(rad/s), and−2π/3≤ ˙θ2≤2.7π/2(rad/s). The initial conditions are chosen as [θ1(0),θ˙1(0),θ2(0), θ˙2(0)]T =[0.5, 0.5,−1, 2]Tand the desired trajectory for two links isθd1(t)=θd2(t)=0.1 sin(t).
Within this simulation, the nonlinear functions f(x,t) and g(xτ,t) for each system are assumed completely unknown. Two independent controllers are designed, one for each system. For each controller, two fuzzy sys- tems in the form of (7) are used to approximate the functionsf(x,t)andg(xτ,t). The fuzzy systems used to approximatef(x,t)haveθ1(t),θ˙1(t),θ2(t)andθ˙2(t)for the first and second carts as inputs and the ones used to describe g(xτ,t)have θ1(t−τ),θ˙1(t−τ),θ2(t−τ) and θ˙2(t−τ)for the first and second carts as inputs.
For each state variable x=[θ1,θ˙1,θ2,θ˙2]T, and xτ = [θ1,τ,θ˙1,τ,θ2,τ,θ˙2,τ]T, three Gaussian-type membership functions are described as follows:
μAij=exp
⎛
⎝−1 2
xj−cij δj
2⎞
⎠,
i=1,. . ., 3,j=1,. . ., 4 (45) where c1j = −1,j=1,. . ., 4, c2j =0,j=1,. . ., 4 andc3j
=1,j=1,. . ., 4 are the centres and δ1=δ2=δ3= δ4=1 are the widths of the membership functions. The initial values of the estimated parametersυfor each con- troller are set to random values uniformly distributed between [0,1], and the design parameters of the con- trollers are chosen as μ1=1.4, μ2=1.3, k1=230, k2=185, σ1=0.06, σ2=0.01, λ1=λ2=0.01, ω1= 1.1, and ω2=0.4, for the cart 1, and μ1=1.4, μ2= 1.3,k1=25,k2=86.5,σ1=0.09,σ2=0.04,λ1=λ2= 0.05,ω1=1.1, and ω2=0.63, for the cart 2. In order to demonstrate the faster convergence of the proposed control method in comparison with the robust tracking design based on variable structure adaptive fuzzy control for nonlinear state-delay system [19], in the simulation, the parameters of the system, the boundary conditions for the states of the system, the initial condition and the desired trajectories have been chosen based on the data in [19]. The simulation results are shown by Figure 5.
It is observed that the tracking errors of the system can be decreased to the boundary layer in the neighbour- hood of the zero in the finite time in the presence of the
Figure 5:Simulation results of a two inverted pendulums mounted on two carts, interconnected by a moving spring system for proposed variable structure adaptive fuzzy controller.
(a), (b) Actual and desired trajectory, (c) switching surface, (d) position tracking error, (e) velocity tracking error, and (f ) control input
uncertainties and external disturbances. The boundary layer||s|| ≤5×10−3reaches in the finite timet=0.30 s and the tracking errors and their first derivative converge to the layers|e1| ≤5×10−3and|˙e1| ≤5×10−3in the finite timet= 0.23s andt =0.27s, respectively, for the outputθ1and|e2| ≤5×10−3and|˙e2| ≤5×10−3in the finite timet= 0.30s andt=0.38s respectively, for the outputθ2. Due to the limitation of the space, only the con- vergence time of the tracking error and its first derivative has been summarized in Table1in comparison with the work [19]. The results of Figure5and Table1in com- parison with the results of Figures 3–6 in [19] showed that faster convergence speed can be acquired by using the proposed variable structure adaptive fuzzy controller.
4. CONCLUSION AND FUTURE WORKS
In this paper, a new fast finite-time variable struc- ture adaptive fuzzy control technique was proposed to overcome the effects of the state-delay in the nonlin- ear second-order systems with model uncertainties and
external disturbances. In the implementation process of the controller design outside the sliding surface, the robust reachability condition embedded in the control law is needed to force the state trajectory to reach the sliding surface in the finite time. In the literature, two general sliding reaching laws, i.e. strong sliding [40] and fast terminal sliding mode-type [35] reach- ability conditions have been presented. In this paper, a new continuous variable structure-type reaching law
−k1sig(s)ω1−k2sig(s)ω2 was designed. The superiority of the new continuous variable structure-type reaching law in comparison with the two conventional reachabil- ity laws [35,40], is the faster convergence far away from and in the neighbourhood of the surface due to using nonlinear terms with fractional powers. The main dis- advantage of the presented robust control methods in the literature [1–3,11,19,41] in the sliding phase of the motion is the asymptotic stability of the closed-loop sys- tem. In the current study, the finite-time convergence of the tracking error and its derivative to the bound- ary layer in the neighbourhood of the zero was ensured by using finite-time variable structure switching sur- face. Moreover, the optimal approximation error in the Equation (12) can be arbitrarily small by increasing the fuzzy rules which results in the convergence of the track- ing error and its first derivative to the zero in the finite time.
In the current study, we have investigated the control problem of the state-delay second-order nonlinear uncer- tain system with the matched disturbance. In the absence of mismatched disturbance, the nonlinear finite-time switching surface for such system is usually defined as (2) and the control law (15) with appropriately cho- sen parameters can force the tracking error and its first derivative to converge to the neighbourhood of the zero in a very short time. In the presence of the matched and mismatched disturbances simultaneously, it is not easy to determine the control law to guarantee the finite-time stability of the closed-loop system. One solution is to address the disturbance debilitation problem of the adap- tive variable structure system controller with a finite- time disturbance observer [32]. Thus, an extension of the proposed method to control the mismatched time- varying uncertain systems can be considered as the future research study. Moreover, in this study, the states vector of the system including the current and delayed states is assumed to be available for the measurement. In many practical applications, the state variables are unknown and only the output of the system is available for the feedback. The observer-based output feedback variable structure system control strategy can be considered as a direction for the future study.
The main objective of the current study for the devel- opment of the control method, is the learning ability of the control system to overcome the rapidly time-varying nonlinear properties of the plant to be controlled such as neuromusculoskeletal system in Functional electri- cal stimulation (FES) applications. FES is a promising technique for the rehabilitation of the movement after spinal cord injury. In this method, the movement in hind limbs can be restored by controlling the electri- cal current pulses by stimulating the intraspinal motor neurons or muscular fibres under the spinal lesion. By changing the pulse width or pulse amplitude of the cur- rent pulses (control input), the level of the contraction for generating the functional movement of the hindlimb joints can be altered. The major challenges for the devel- opment of an appropriate control method for stimulat- ing paralyzed limbs include high-order nonlinear and time-varying properties of the neuromusculoskeletal sys- tem with unknown dynamics. Currently, we are working on the control of the neuromusculoskeletal model dur- ing weight-bearing walking through FES using robust adaptive control techniques include fast finite-time VSC adaptive fuzzy control algorithms.
ACKNOWLEDGEMENTS
The authors would like to thank the anonymous reviewers and Associate Editor for their valuable comments and suggestions to improve the original manuscript.
ORCID
M. R. Yousefi http://orcid.org/0000-0002-5126-8459 K. Shojaei http://orcid.org/0000-0003-1799-2141 G. Shahgholian http://orcid.org/0000-0003-2774-4694
REFERENCES
1. X. G. Yan, S. K. Spurgeon, and C. Edwards, “Memoryless static output feedback sliding mode control for nonlinear systems with delayed disturbances,”IEEE Trans. Autom.
Control, Vol. 59, no. 7, pp. 1906–12,2014.
2. Q. Gao, G. Feng, Z. Xi, Y. Wang, and J. Qiu, “A new design of robustH∞sliding mode control for uncertain stochas- tic T–S fuzzy time-delay systems,”IEEE Trans. Cybern., Vol. 44, no. 9, pp. 1556–66,2014.
3. Y. Han, Y. Kao, and C. Gao, “Robust sliding mode control for uncertain discrete singular systems with time-varying delays and external disturbances,” Automatica, Vol. 75, pp. 210–16,2017.
4. H. Cherni, I. Boulaabi, A. Sellami, and F. Ben Hmida,
“Robust actuator and sensor faults reconstruction based on new sliding mode observer for a class of output time-delay systems,”IETE J. Res., Vol. 62, no. 6, pp. 812–21,2016.
