In this paper, we dealt with the tracking control problem of a class of fractional-order uncertain systems with time delays. In order to handle the effects brought by the uncertainties, external disturbances, time-delay terms, and to overcome the obstacles caused by inputs saturation, the tracking controller, which consisted of linear control law, nonlinear law, and robust control law proposed in this paper, was designed by combining the composite nonlinear feedback control method and the properties of fractional order operators. Furthermore, the validation of this tracking controller was proved.
Citation: Guijun Xing, Huatao Chen, Zahra S. Aghayan, Jingfei Jiang, Juan L. G. Guirao. Tracking control for a class of fractional order uncertain systems with time-delay based on composite nonlinear feedback control[J]. AIMS Mathematics, 2024, 9(5): 13058-13076. doi: 10.3934/math.2024637
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In this paper, we dealt with the tracking control problem of a class of fractional-order uncertain systems with time delays. In order to handle the effects brought by the uncertainties, external disturbances, time-delay terms, and to overcome the obstacles caused by inputs saturation, the tracking controller, which consisted of linear control law, nonlinear law, and robust control law proposed in this paper, was designed by combining the composite nonlinear feedback control method and the properties of fractional order operators. Furthermore, the validation of this tracking controller was proved.
There are many studies on fractional calculus and related topics [1,2,3,4,5,6,7,8], such as Podlubny [2] who talked about several classical definitions of fractional order operators; Miller [5] introduced the general theory of fractional differential equations; a new fractional derivatives with nonlocal and non-singular kernel were created by Atangana and Baleanu [8], to name but a few. In recent years, relying on the fact that many complex phenomenon can be simplified and accurately described by fractional-order operators, fractional-order systems have attracted great attention in applied sciences [9,10,11]. The control problem is one of the important issues in theory and applications of fractional order systems. Recently, varieties of fractional-order control methods have been designed, such as sliding mode control [12,13], adaptive control[14], feedback control [15], and so on. It is mentioned that the sliding mode control method can effectively ensure the stability and robustness of a nonlinear fractional order system; alternatively, it can switch the motion to the sliding mode surface through the switching control law, so as to ensure rapid response and robustness. In addition, the combinations of several controllers are effective ways to achieve better control effects by taking the advantages of different control methods [16,17,18,19,20,21]. However, to our best knowledge, there are few results on the tracking control of fractional-order systems based on the composite nonlinear feedback (CNF) control method, particularly for systems with time delays and actuator saturation constraints. On the other hand, due to the presence of uncertainties and external disturbances in the system, it is necessary to identify unknown nonlinear terms, which should be compensated in the process of designing the controller. Furthermore, the time delays bring some obstacles in designing the controller and proving the stability.
The systems with time delays are basic mathematical models to describe the practical problems, for example, chemical reaction, mechanical vibration, power system, and so on (for more details, one can refer to Ref. [22]). When the control problems for systems with time delays are considered, the time delays lead to the complex of designing control and the proof for the system controlled (for more details, see [23,24,25,26]). In addition, the phenomenon of actuator saturation usually happens in controlled systems. Usually, the input saturations restrict the system's performance, which result in the inaccuracies and instabilities of the system considered. To deal with control problems for the time-delay system with actuator saturation, many control methods have been developed [27,28,29]. In Ref. [30], a class of linear systems with input saturation constraints and time delay is studied, and Lyapunov-Razumihkin and Lyapunov-Krasovskii functional approaches are used to analyze the domain of attraction problem and stability problem of the system. In [31], a state feedback controller design method was proposed for a class of uncertain discrete time-delay systems with control input saturation and bounded external disturbances, which guarantee the trajectories of systems to converge to the desired state.
In the above control methods, most of the control inputs depend on the sign function, which results in that the control law is not smooth. In order to improve the transient performance of the tracking ability of the closed-loop system, the composite nonlinear feedback control method was established in [32], and developed by Mobayen and Tchier [33], Chen et al. [34], Lin et al. [35], He et al. [36], and so on. The CNF control method is often used to deal with tracking control problems of systems with input saturation and it can improve the transient performance of the closed-loop system while maintaining a small overshoot or no overshoot. Jafari et al. [37] designed a CNF controller based on a disturbance observer, which can effectively guarantee the tracking performance of the system. Based on the CNF control method, a discrete integral sliding mode controller, which can produce the superior transient performance, was proposed by Mondal S. et al. [38]. In Ref. [39], employing the CNF control method, Jafari et al. considered the control problem for the system with a singular time delay. In terms of the CNF control method, a novel controller for nonlinear time-delay systems with saturation constraints was given by Ghaffari et al. [40]. For more details, one can refer to [41,42,43] and the references therein. It must be mentioned that most investigations that considering control problems for differential systems by the CNF control method were focused on the integer order differential systems with time delay. Thus, it is necessary to develop composite nonlinear feedback control to deal with the control problem for fractional-order systems.
Relying on CNF control methods, we consider the control problems for fractional-order uncertain systems with time delay and external disturbances. The rest of the paper is organized as follows. In Section 2, we describe the fractional-order system investigated in this paper. Section 3 is devoted to give the major results and the associated proofs.
The following are the definitions of Caputo-fractional order derivative adopted in this paper.
Definition 2.1. [2] For a continuous function x(t):[0,∞)→R, the Caputo-type fractional order derivative with the order α of the function x(t) is defined as
C0Dαtx(t)=1Γ(1−α)∫t0(t−s)−αx′(s)ds,0<α<1. |
Definition 2.2. [2] The Caputo-type fractional integral with the order α of function x(t) is defined as
0Iαtx(t)=1Γ(α)∫t0(t−s)α−1x(s)ds,0<α<1. |
Some properties of fractional calculus operators are introduced as follows.
Proposition 2.1. [16] Let x∈Ck[a,b] for some a<b and some k∈N. Moreover, let n,ε>0 such that there exists some ℓ∈N with ℓ≤k and n,n+ε∈[ℓ−1,ℓ]. Then,
C0Dεt(C0Dntx(t))=C0Dε+ntx(t). |
Proposition 2.2. [2] If the Caputo fractional differential C0Dαtx(t) is integrable, then
0Iαt(C0Dαtx(t))=x(t)−x(0), |
if the function x(t)∈C1[0,t], and 0<α<1.
Consider the following multi-input and multi-output fractional-order uncertain system with actuator saturation
{C0Dαtx(t)=(A+ΔA(ν(t)))x(t)+ˉA(ς(t))x(t−τ(t))+(B +ΔB(σ(t)))sat(u(t))+D(θ(t)),y(t)=Cx(t),0<t<+∞, | (2.1) |
where x(t)∈Rn,y(t)∈Rm,m<n and u(t)∈Rn are the system state vector, the system output vector and the control input vector respectively. The matrix A denotes the system matrix, B is the input matrix and C represents the output matrix, they are both the constant matrices with the appropriate dimensions. τ(t)∈R+ is the time delay. The terms ΔA(⋅) and ΔB(⋅) represent the uncertainties of the system, and D(⋅) denotes the perturbation, the uncertain terms ν(⋅):R+→D,σ(⋅):R+→D and θ(⋅):R+→D are Lebesgue measurable functions, where D is a compact bounded set.
The control input vector is constrained by a saturation function sat:Rn→Rn with the following form
sat(u(t))=[sat(u1(t))sat(u2(t))⋮sat(un(t))], | (2.2) |
where the operator
sat(ui(t))=sign(ui(t))min(|ui|,ˉui),i=1,2,⋯,n, | (2.3) |
and ˉui represents the saturation level of the i-th control channel.
The objective in this paper is to derive the composite controller u(t), which leads to the output vector y(t) of the system (2.1) can track the output vector yr(t) of the reference system rapidly and smoothly. The reference system is defined as following
{C0Dαtxr(t)=Arxr(t),yr(t)=Crxr(t), | (2.4) |
where Ar∈Rn×n and Cr∈Rn×n are both constant matrices. xr(t)∈Rn denotes the reference state vector and yr(t)∈Rm is the reference output vector. For the purposes of the tracking control, it is required that there exists a constant d>0 such that ||xr(t)||⩽d for all t⩾0.
It is turned to list some hypothesises about the system (2.1) and system (2.4).
Hypothesis 2.1. There exist two constant matrices G and H which satisfy
[ABC0][GH]=[GArCr]. | (2.5) |
Moreover, for any positive-definite matrix Q∈Rn×n, there exists an unique positive-definite matrix P∈Rn×n satisfying the following Riccati algebraic equation [44]
ATP+PA−ηPBBTP=−Q. | (2.6) |
Hypothesis 2.2. The fractional derivative of the unknown time delay τ(t) is bounded, which means there is a positive constant ϑ such that |C0Dαtτ|⩽ϑ. Furthermore, suppose ϑ<1.
Hypothesis 2.3. The matrices ΔA(ν(t)), ΔB(σ(t)) and D(θ(t)) are matched, and there exist continuous and bounded functions N1(⋅), N2(⋅) and N3(⋅) with the boundary
ρ1=maxν∈D‖N1(ν)‖,ρ2=maxσ∈D‖N2(σ)‖,ρ3=maxθ∈D‖N3(θ)‖, | (2.7) |
such that
ΔA(ν(t))=BN1(ν),ΔB(σ(t))=BN2(σ),D(θ(t))=BN3(θ). | (2.8) |
Moreover, assume the time-delay matrix ˉA is matched and
ˉA(ς)=BˉN. | (2.9) |
Hypothesis 2.4. The pair {A,B} from the system (2.1) is completely controllable.
The next lemma is very important in deriving the main results of this paper.
Lemma 2.1. [45] (Schur Complement) The following LMI condition
[F11(t)F12(t)F21(t)F22(t)]<0 | (2.10) |
holds if and only if
{F22(t)<0,F11(t)−F12(t)F−122(t)FT21(t)<0, |
or is equivalent to
{F11(t)<0,F22(t)−F21(t)F−111(t)FT12(t)<0, |
where F11(t)=FT11(t), F12(t)=FT21(t) and F22(t)=FT22(t).
This section is devoted to obtain the main results and the proof associated. Initially, we transform the system (2.1) to the error system.
Consider the following tracking error vector e(t) and the auxiliary state vector defined by
e(t)=y(t)−yr(t), | (3.1) |
and
˜x(t)=x(t)−Gxr(t), | (3.2) |
where the matrix G satisfies the Hypothesis 2.1. Thus, combining the system (2.1) with the reference system (2.4) gives
e(t)=C(x(t)−Gxr(t))=C˜x(t), | (3.3) |
then
‖e(t)‖=‖C˜x(t)‖⩽‖C‖‖˜x(t)‖, | (3.4) |
which implies that
limt→+∞‖e(t)‖⩽limt→+∞‖˜x(t)‖. |
Thus, we obtain limt→+∞‖e(t)‖=0 when limt→+∞‖˜x(t)‖=0, which means that ‖˜x(t)‖→0 with t→∞ can guarantee the output y(t) can be forced to track the reference output yr(t) asymptotically.
The following Lemmas and Definitions are very important to obtain the main results in this paper.
Lemma 3.1. [46] Suppose x(t) is continuously differentiable function, then, for any time variable t⩾0, the following inequality holds
12C0Dαtx2(t)⩽x(t)(C0Dαtx(t)),0<α<1. |
Lemma 3.2. [47] Let x(t) be a vector and xT(t)Px(t) is continuously differentiable function for any symmetric matrix P, then, for each time t⩾0, the following can be obtained.
12C0Dαt(xT(t)Px(t))⩽xT(t)P(C0Dαtx(t)),∀α∈(0,1],∀t⩾0. |
Definition 3.1. [48] If the continuous function α(⋅):[0,t)→[0,∞) is strictly increasing and α(0)=0, then, it belongs to K−class function.
Lemma 3.3. [49] (Fractional order Mittag-Leffer asymptotical stability) Let x=0 be an equilibrium point of the fractional system (2.1). Assume that there exists a Lyapunov function V(x(t)) and K−class functions αi(⋅)(i=1,2,3) satisfying
α1(‖x(t)‖)⩽V(x(t))⩽α2(‖x(t)‖), |
C0DαtV(x(t))⩽−α3(‖x(t)‖), |
where 0<α⩽1. Then, the equilibrium point of system (2.1) is asymptotically stable.
Lemma 3.4. [50] (Integer-order Barbalat's Lemma) If η:R→R is a uniformly continuous function for t⩾0 and limt→∞∫t0η(ω)dω, 0<q<1 exists and is finite, then limt→∞η(t)=0.
The objective in this part is to design a tracking control law based on the CNF control approach without large overshoot and unfavorable actuator saturation effect.
The process of the controller design can be divided into the following four steps.
(1) The design of a linear state feedback controller.
(2) The design of a nonlinear feedback controller.
(3) The design of a robust tracking controller.
(4) The design for the CNF controller needed.
The exact process is as following.
Step 1: The linear feedback controller is designed as
uL(t)=Fx(t)+(H−FG)xr(t)=F˜x(t)+Hxr(t), | (3.5) |
where F represents a gain matrix which is determined later. The linear part can ensure the closed-loop system possesses the properties of fast response and enough small damping ratio.
Step 2: The nonlinear feedback controller is expressed as
uN(t)=μ(t)BTP˜x(t), | (3.6) |
where P is a positive definite matrix, and
μ(t)=−κ2(t)κ(t)‖BTP˜x(t)‖+ϱ(t), | (3.7) |
where κ(t)>0 is a function which is needed to be designed and the bounded function ϱ(t) is an any non-negative and uniform continuous function. Moreover, ϱ(⋅) satisfies
supt∈[0,+∞)∫t0[ϱ(˜x,s)]ds⩽ˉϱ, | (3.8) |
where ˉϱ>0, then one can have
limt→+∞∫t0[ϱ(˜x,s)]ds⩽ˉϱ<+∞. | (3.9) |
Obviously, μ(t) formulated by (3.7) is non-positive and satisfies the local Lipschitz condition.
Remark 3.1. The value of ϱ(t) which is depended on the error signal e(t) would increase with the output signal y(t) far from the reference signal yr(t). Moreover, the value of |μ(t)| would decrease, which can leads to that the effect of the nonlinear part can be eliminated, and vice versa.
Step 3: Consider a fractional-order sliding mode surface as following
s(t)=k1˜x(t)+k2(C0Dαt˜x(t))+⋯+kn(C0D(n−1)αt˜x(t))=n∑i=1ki(C0D(i−1)αt˜x(t)), | (3.10) |
where ki(i=1,2,⋯,n) is a constant row vector. Taking the fractional-order derivative with respect to t in both sides of (3.10) implies
C0Dαts(t)=k1(C0Dαt˜x(t))+k2(C0D2αt˜x(t))+⋯+kn(C0Dnαt˜x(t))=n∑i=1ki(C0Diαt˜x(t)). | (3.11) |
On the other hand, when the states of the system arrive the sliding mode surface s(t), then s(t)=0, thus, the robust control law can be constructed as
us(t)=−(k1B)−1[n∑i=2ki(C0Diαt˜x(t))+k1(A+BF+μ(t)BBTP)˜x(t)+ls(t)+ksgn(s)], | (3.12) |
where k1B is non-vanishing, and l and k are two positive constants. This robust controller can guarantee the process of tracking for the output signal to the reference signal can not be affected by external disturbances and uncertainties, and the tracking ability of the system can be further improved.
Step 4: The CNF controller is comprised of the linear, nonlinear and robust control laws, which are derived in Step 1, Step 2 and Step 3 respectively, with the following form
u(t)=F˜x(t)+Hxr(t)+μ(t)BTP˜x(t)+us(t), | (3.13) |
where
μ(t)=−(ρ1(‖˜x(t)‖+‖Gxr(t)‖)+ρ3+ρ2ˉu+2ˉu+˜ρ(ˉu))2(ρ1(‖˜x(t)‖+‖Gxr(t)‖)+ρ3+ρ2ˉu+2ˉu+˜ρ(ˉu))‖BTP˜x(t)‖+ϱ(˜x(t)), | (3.14) |
here ˜ρ(ˉu) is a positive constant and satisfies ‖u(t)‖≤˜ρ(ˉu).
Remark 3.2. Because ˜x(t), xr(t) and s(t) are all bounded, the input of controller formulated by (3.13) is also bounded.
Set
ω(t)=sat(u(t))−F˜x(t)−Hxr(t), | (3.15) |
which together with (3.13) implies
ω(t)=sat(F˜x(t)+Hxr(t)+μ(t)BTP˜x(t)+us(t))−F˜x(t)−Hxr(t). | (3.16) |
Taking the fractional-order derivative with respect to t in both sides of (3.2) along the trajectories of (2.1) and (2.4), we can get
C0Dαt˜x(t)=C0Dαtx(t)−G(C0Dαtxr(t))=(A+ΔA)x(t)+ˉAx(t−τ)+(B+ΔB)sat(u)+D−GArxr(t)=(A+ΔA)˜x(t)+(A+ΔA)Gxr(t)+ˉA˜x(t−τ)+ˉAGxr(t−τ)+(B+ΔB)sat(u)+D−GArxr(t). | (3.17) |
Substituting ω(t) into (3.17) yields that
C0Dαt˜x(t)=(A+ΔA+BF)˜x(t)+BHxr(t)+Bω(t)+(A+ΔA)Gxr(t) +ˉA˜x(t−τ)+ˉAGxr(t−τ)+D−GArxr(t)+ΔBsat(u)=(A+ΔA+BF)˜x(t)+Bω(t)+ˉA˜x(t−τ)+ˉAGxr(t−τ) +D+ΔAGxr(t)+ΔBsat(u). | (3.18) |
Remark 3.3. The matrix A is a negative definite matrix if and only if the even order principal sub-formula Di>0, and the order principal sub-formula of odd order Di<0. Then, the quadratic f(x1,x2,⋯,xn)=XTAX is a negative quadratic.
The main results of this paper are represented by the coming Theorem 3.1.
Theorem 3.1. Consider the fractional-order uncertain system (2.1) and the reference system (2.4). Suppose the Hypothesises 2.1, 2.2 and 2.3 hold, and for any δi∈(0,1)(i=1,2), let cδ is the largest positive scalar such that ˜x∈Xδ with Xδ={˜x:˜xTP˜x⩽cδ}, the following inequalities hold,
‖F˜x(t)‖⩽(1−δ1−δ2)ˉu, | (3.19) |
‖Hxr(t)‖⩽δ1ˉu, | (3.20) |
‖us(t)‖⩽δ2ˉu. | (3.21) |
If there exist a matric Z>0 with adequate dimensions, and satisfy the following condition:
Λ=[Λ11PˉA∗−(1−ϑ)Z]<0, | (3.22) |
where Λ11=(A+BF)TP+P(A+BF)+(1−ϑ)−1P2+Z+Q+FTWF, and Q+FTWF is a positive definite matrix. Then, under the controller formulated by (3.13), the error e(t) defined by (3.1) converges to zero asymptotically with t→+∞.
Proof. The whole proof is divided into four situations.
S1: The input signal is unsaturated which means the values of inputs are less than the supremum of saturation function and more than the infimum of saturation function
S2: The values of all input channels of control are more than the supremum of saturation function.
S3: The values of input channels of control are less than he infimum of saturation function.
S4: Some of the inputs channels are unsaturated, and the others are saturated
Proof for S1. In this case, we have
|ui(t)|⩽ˉui,i=1,2,⋯,n, | (3.23) |
then sat(u)=u(t), therefore, it can be obtained that
ω(t)=sat(F˜x(t)+Hxr(t)+μ(t)BTP˜x(t)+us(t))−F˜x(t)−Hxr(t)=μ(t)BTP˜x(t)+us(t). | (3.24) |
Given the following Lyapunov function
V1(˜x(t))=12s2(t). | (3.25) |
Taking the fractional-order derivative with respect to t in both sides of (3.25) along the trajectories of the sliding mode surface (3.10), which together with Lemma 3.1 yields
C0DαtV1(t)⩽s(t)(C0Dαts(t))=s(t)[k1(C0Dαt˜x(t))+n∑i=2ki(C0Diαt˜x(t))]. | (3.26) |
Substituting (3.18) into (3.26) gives
C0DαtV1(t)⩽s(t)[k1(A+ΔA+BF)˜x(t)+k1Bω(t)+k1ˉA˜x(t−τ)+k1D +k1ˉAGxr(t−τ)+k1ΔAGxr(t)+k1ΔBsat(u)+n∑i=2ki(C0Dαt˜x(t))]=s(t)[k1(A+ΔA+BF+ΔBF)˜x(t)+k1ˉA˜x(t−τ)+k1Bω(t) +k1μ(t)ΔBBTP˜x(t)+k1χ(t)+n∑i=2ki(C0Dαt˜x(t))], |
where
χ(t)=ˉAGxr(t−τ)+D+ΔAGxr(t)+ΔBHxr(t)+ΔBus(t)=Bξ(t), | (3.27) |
along with Hypothesis 2.3, we have
χ(t)=Bξ(t), | (3.28) |
here
ξ(t)=ˉNGxr(t−τ)+N3+N1Gxr(t)+N2Hxr(t)+N2us(t). | (3.29) |
With robust control law (3.12) and Hypothesis 2.3, from (3.24), we can get
C0DαtV1(t)⩽s(t)[k1(ΔA+ΔBF)˜x(t)+k1ˉA˜x(t−τ)+k1μ(t)ΔBBTP˜x(t) +k1χ(t)]−ls2(t)−k|s(t)|=s(t)[k1B(N1+N2F)˜x(t)+k1BˉN˜x(t−τ)+k1N2μ(t)BBTP˜x(t) +k1Bξ(t)]−ls2(t)−k|s(t)|, |
then
C0DαtV1(t)⩽|s(t)|‖k1B‖[(ρ1+ρ2‖F‖)‖˜x(t)‖+‖ˉN‖‖˜x(t−τ)‖ +ρ2|μ(t)|‖BTP‖‖˜x(t)‖+ρξ]−ls2(t)−k|s(t)|, |
where ρξ=max‖ξ(t)‖.
Thus, when the system parameters satisfy the following switching condition
k⩾‖k1B‖[(ρ1+ρ2‖F‖)‖˜x(t)‖+‖ˉN‖‖˜x(t−τ)‖+ρ2|μ(t)|‖BTP‖‖˜x(t)‖+ρξ], |
it can be asserted that
C0DαtV1(t)⩽−ls2(t). |
Therefore, using Lemma 3.3, we can derive the equilibrium point of the system (2.1) is asymptotically stable and the trajectories converge to the sliding surface.
Conducting the following discussion requires an alternative approach, thus, we need another Lyapunov functional candidate as follows
V2(˜x(t),xr(t))=0I1−αt[˜xT(t)P˜x(t)]+∫tt−τ˜xT(β)Z˜x(β)dβ+0I1−αt[xTr(t)Prxr(t)]+∫tt−τxTr(β)GTˉATˉAGxr(β)dβ, | (3.30) |
where the matrix Z and Pr are positive definite which can be determined later.
Taking derivative in both sides of (3.30), along with Hypothesis 2.2, we can find
˙V2(t)⩽[C0Dαt˜x(t)]TP˜x(t)+˜xT(t)P(C0Dαt˜x(t))+˜xT(t)Z˜x(t)+[C0Dαtxr(t)]TPrxr(t) −(1−ϑ)˜xT(t−τ)Z˜x(t−τ)+xTr(t)Pr(C0Dαtxr(t))+xTr(t)GTˉATˉAGxr(t) −(1−ϑ)xTr(t−τ)GTˉATˉAGxr(t−τ). |
According to (2.4) and (3.18), we have
˙V2(t)⩽˜xT(t)[(A+ΔA+BF)TP+P(A+ΔA+BF)+Z]˜x(t)+˜xT(t−τ)ˉATP˜x(t)+˜xT(t)PˉA˜x(t−τ)+xTr(t−τ)GTˉATP˜x(t)+˜xT(t)PˉAGxr(t−τ)+xTr(t)GTΔATP˜x(t)+˜xT(t)PΔAGxr(t)+ωT(t)BTP˜x(t)+˜xT(t)PBω(t)+DTP˜x(t)+[sat(u)]TΔBTP˜x(t)+˜xT(t)PΔBsat(u)+˜xT(t)PD−(1−ϑ)˜xT(t−τ)Z˜x(t−τ)+xTr(t)PrArxr(t)−(1−ϑ)xTr(t−τ)GTˉATˉAGxr(t−τ)+[Arxr(t)]TPrxr(t)+xTr(t)GTˉATˉAGxr(t), | (3.31) |
together with the Hypothesis 2.3, we get
h(t)=D+ΔAGxr(t)+ΔBsat(u)=Bγ(t), | (3.32) |
where
γ(t)=N1Gxr(t)+N2sat(u)+N3. |
Since, for any given ε>0, the following holds
MTN+NTM⩽εMTM+ε−1NTN, |
where M and N are any matrices with the appropriate dimensions, then we have
xTr(t−τ)GTˉATP˜x(t)+˜xT(t)PˉAGxr(t−τ)⩽ε˜xT(t)P2˜x(t)+ε−1xTr(t−τ)GTˉATˉAGxr(t−τ). | (3.33) |
Employing the inequality (3.33), the inequality (3.31) can be written as
˙V2(t)⩽˜xT(t)[(A+BF)TP+P(A+BF)+εP2+Z]˜x(t)+˜xT(t)PBω(t) +˜xT(t)PˉA˜x(t−τ)+˜xT(t−τ)ˉATP˜x(t)−(1−ϑ)˜xT(t−τ)Z˜x(t −τ)+ε−1xTr(t−τ)GTˉATˉAGxr(t−τ)+xTr(t)(ATrPr+PrAr +GTˉATˉAG)xr(t)−(1−ϑ)xTr(t−τ)GTˉATˉAGxr(t−τ) +˜xT(t)[ΔATP+PΔA]˜x(t)+˜xT(t)Ph(t)+hT(t)P˜x(t)+ωT(t)BTP˜x(t). |
Let ε=(1−ϑ)−1, we get
˙V2(t)+˜xT(t)(Q+FTWF)˜x(t)≤slant˜xT(t)[(A+BF)TP+P(A+BF)+(1−ϑ)−1P2+Z+Q+FTWF]˜x(t)+˜xT(t)PˉA˜x(t−τ)+˜xT(t−τ)ˉATP˜x(t)−(1−ϑ)˜xT(t−τ)Z˜x(t−τ)+xTr(t)(PrAr+ATrPr+GTˉATˉAG)xr(t)+˜xT(t)[ΔATP+PΔA]˜x(t)+˜xT(t)Ph(t)+hT(t)P˜x(t)+ωT(t)BTP˜x(t)+˜xT(t)PBω(t), | (3.34) |
where Q and W are positive definite matrixes, and the matrix Pr satisfies the following Riccati algebraic equation
GTˉATˉAG+PrAr+ATrPr⩽0. |
By using the matrix inequality (3.22), the inequality (3.34) can be simplified as
˙V2(t)+˜xT(t)(Q+FTWF)˜x(t)⩽ΨTΛΨ+˜xT(t)[ΔATP+PΔA]˜x(t)+˜xT(t)Ph(t)+hT(t)P˜x(t)+ωT(t)BTP˜x(t)+˜xT(t)PBω(t), | (3.35) |
here Ψ=[˜x(t)˜x(t−τ)]T, and
Λ=[Λ11PˉA∗−(1−ϑ)Z], |
where Λ11=(A+BF)TP+P(A+BF)+(1−ϑ)−1P2+Z+Q+FTWF.
Proof for S2. When the values of control input ui(t) of all input channels overbear their upper boundaries, which means ui(t)≥ˉui, then we have sat(ui)=ˉui and
˜ρi(ˉui)⩾ui(t)=Fi˜x(t)+Hixr(t)+μ(t)BiTP˜x(t)+uis(t)⩾ˉui, |
where ˜ρi(ˉui) is the maximum value of ui(t). By (2.3) and (3.16), we find
ωi(t)=ˉui−Fi˜x(t)−Hixr(t). | (3.36) |
Using (3.19), (3.20) and (3.21), we get
Fi˜x(t)+Hixr(t)+uis(t)⩽|Fi˜x(t)+Hixr(t)+uis(t)|⩽|Fi˜x(t)|+|Hixr(t)|+|uis(t)|⩽(1−δ1−δ2)ˉui+δ1ˉui+δ2ˉui⩽ˉui. | (3.37) |
From (2.3), (3.16) and (3.37), we have
ωi(t)=ˉui−Fi˜x(t)−Hixr(t)⩾0. | (3.38) |
According to Eq (3.13), we can obtain
Fi˜x(t)+Hixr(t)=ui(t)−μ(t)BTiP˜x(t)−uis(t). | (3.39) |
Therefore, applying (3.38) and (3.39), we get
ωi(t)=ˉui−ui(t)+μ(t)BTiP˜x(t)+uis(t). | (3.40) |
Since the μ(t)⩽0 and μ(t)BTiP˜x(t)⩾0, it can be asserted that
BiTP˜x(t)=˜xT(t)PBi⩽0. |
Proof for S3. When the control input ui(t) of all input channels are less than the lower bounds, alternatively,
−˜ρi(ˉui)⩽ui(t)=Fi˜x(t)+Hixr(t)+μ(t)BiTP˜x(t)+uis(t)⩽−ˉui, |
which implies sat(ui)=−ˉui. From (2.3) and (3.16), we have
ωi(t)=−ˉui−Fi˜x(t)−Hixr(t)⩽0. | (3.41) |
Following the similar manner of obtaining (3.40), we find
ωi(t)=−ˉui−ui(t)+μ(t)BTiP˜x(t)+uis(t). |
Since μ(t)⩽0 and μ(t)BTiP˜x(t)⩽0, we get
BiTP˜x(t)=˜xT(t)PBi⩾0. |
Proof for S4. When the values of some control input ui(t) are unsaturated, but the others are saturated. As for the unsaturated inputs, we can obtain ˜xT(t)PBiωi(t)⩽0, and
ωi(t)=μ(t)BTiP˜x(t)+uis(t). |
With respect to saturated inputs the values of which are more than the supremum of saturation function, the results in S2 imply ωi(t)⩾0 and ˜xT(t)PBi⩽0, then we have ˜xT(t)PBiωi(t)⩽0, thus
ωi(t)=ˉui−ui(t)+μ(t)BTiP˜x(t)+uis(t). |
As for the saturated inputs the values of which are less than the infimum of saturation function, the assertions of S3 indicate ωi(t)⩽0 and ˜xT(t)PBi⩾0, then we can get ˜xT(t)PBiωi(t)⩽0, and
ωi(t)=−ˉui−ui(t)+μ(t)BTiP˜x(t)+uis(t). |
As indicated above, together with the inequality (3.35), we can assert
˙V2(t)+˜xT(t)(Q+FTWF)˜x(t)⩽ΨTΛΨ+˜xT(t)Ph(t)+hT(t)P˜x(t)+˜xT(t)(ΔATP+PΔA)˜x(t)+2˜xT(t)PB(ˉu−u(t)+μ(t)BTP˜x(t)+us(t)), | (3.42) |
combining with hypothesis 2.3, we can obtain
˙V2(t)+˜xT(t)(Q+FTWF)˜x(t)⩽ΨTΛΨ+2‖BTP˜x(t)‖[ρ1(‖˜x(t)‖+‖Gxr(t)‖)+ρ3+ρ2ˉu+2ˉu+˜ρ(ˉu)]+2‖BTP˜x(t)‖2μ(t). | (3.43) |
By (3.14) and (3.43), we can get
˙V2(t)+˜xT(t)(Q+FTWF)˜x(t)⩽ΨTΛΨ+2(ρ1(‖˜x(t)‖+‖Gxr(t)‖)+ρ3+ρ2ˉu+2ˉu+˜ρ(ˉu))‖BTP˜x(t)‖ϱ(˜x(t))(ρ1(‖˜x(t)‖+‖Gxr(t)‖)+ρ3+ρ2ˉu+2ˉu+˜ρ(ˉu))‖BTP˜x(t)‖+ϱ(˜x(t)). | (3.44) |
Obviously, the following inequality holds
0⩽ϱ(˜x(t))ϕϱ(˜x(t))+ϕ⩽ϱ(˜x(t)),∀ϱ(˜x(t))>0,ϕ>0. | (3.45) |
Then, it can be obtained that
(ρ1(‖˜x(t)‖+‖Gxr(t)‖)+ρ3+ρ2ˉu+2ˉu+˜ρ(ˉu))‖BTP˜x(t)‖ϱ(˜x(t))(ρ1(‖˜x(t)‖+‖Gxr(t)‖)+ρ3+ρ2ˉu+2ˉu+˜ρ(ˉu))‖BTP˜x(t)‖+ϱ(˜x(t))⩽ϱ(˜x(t)). | (3.46) |
Combined (3.44) and (3.46), it's obtained that
˙V2(t)+˜xT(t)(Q+FTWF)˜x(t)⩽ΨTΛΨ+2ϱ(˜x(t)). |
If there exist some matrices X>0 and Z>0 such that
Λ=[Λ11PˉA∗−(1−ϑ)Z]<0, |
then, λ(Λ)<0. Thus
˙V2(t)+˜xT(t)(Q+FTWF)˜x(t)⩽λmin(Λ)‖Ψ(t)‖2+2ϱ(˜x(t)). |
Here, we choose
ϱ(˜x(t))⩽12˜xT(t)(Q+FTWF)˜x(t)⩽12λmax(Q+FTWF)‖˜x(t)‖2. |
Moreover, according to the representation of the Lyapunov function V2(t), there exist two K−class functions α1(⋅), α2(⋅) such that
α1(‖˜x(t)‖)⩽V2(˜x(t))⩽α2(‖˜x(t)‖), | (3.47) |
which implies
α1(‖˜x(t)‖)=∫t0˙V2(˜x(s))ds+V2(˜x(0))⩽α2(‖˜x(0)‖)+∫t0λmin(Λ)‖Ψ(s)‖2ds+2∫t0ϱ(˜x(s))ds, | (3.48) |
which together with (3.8) gives
α1(‖˜x(t)‖)⩽α2(‖˜x(0)‖)+2∫t0ϱ(˜x(s))ds⩽α2(‖˜x(0)‖)+2¯ϱ. | (3.49) |
Then, we can conclude that for any t>0,
−∫t0λmin(Λ)‖Ψ(s)‖2ds⩽α2(‖˜x(0)‖)+2∫t0ϱ(˜x(s))ds⩽α2(‖˜x(0)‖)+2ˉϱ, | (3.50) |
which implies that
−limt→+∞[∫t0λmin(Λ)‖Ψ(s)‖2ds]⩽α2(‖˜x(0)‖)+2ˉϱ<+∞. | (3.51) |
Hence, it follows from Barbalat's Lemma that
limt→+∞[∫t0λmin(Λ)‖Ψ(t)‖2ds]=0, |
furthermore
limt→+∞‖Ψ(t)‖=0. |
As indicated above, the auxiliary state ˜x(t) converges to zero asymptotically. Thus, based on the relationship of ˜x(t) and e(t), it can be asserted that the system output y(t) can be forced to track the reference state yr(t) asymptotically.
Compared with the results in other studies [33,35,42,51,52,53], the system considered in this paper is a fractional-order uncertain system with time delays and saturation function, which is very complex. The tracking controller is designed by the CNF control approach. Furthermore, based on the fractional-order Mittag-Leffer asymptotical stability theorem, the asymptotical tracking and stability of the controller proposed is proven by designing a fractional-order Lyapunov function and the fractional Barbalat's Lemma.
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.
This work is supported by National Natural Science Foundation of China (No.12002194; No.12072178) and Natural Science Foundation of Shandong Province (No. ZR2020QA037; No. ZR2020MA054).
The authors declare no conflicts of interest.
[1] |
R. L. Bagley, R. A. Calico, Fractional order state equations for the control of viscoelasticallydamped structures, J. Guid. Control Dynam., 14 (1991), 304–311. https://doi.org/10.2514/3.20641 doi: 10.2514/3.20641
![]() |
[2] | I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, San Diego: Academic Press, 1999. https://doi.org/10.1016/s0076-5392(99)x8001-5 |
[3] | R. Hilfer, Applications of fractional calculus in physics, Singapore: World Scientific, 2000. https://doi.org/10.1142/3779 |
[4] |
K. Diethelm, N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (2002), 229–248. https://doi.org/10.1006/jmaa.2000.7194 doi: 10.1006/jmaa.2000.7194
![]() |
[5] | A. Loverro, Fractional calculus: History, definitions and applications for the engineer, 2004. |
[6] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, New York: Elsevier, 2006. https://doi.org/10.1016/S0304-0208(06)80001-0 |
[7] | J. S. Bardi, The calculus wars: Newton, Leibniz, and the greatest mathematical clash of all time, New York: Thunder's Mouth Press, 2006. |
[8] |
A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. https://doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
![]() |
[9] |
Y. Di, J. X. Zhang, X. Zhang, Robust stabilization of descriptor fractional-order interval systems with uncertain derivative matrices, Appl. Math. Comput., 453 (2023), 128076. https://doi.org/10.1016/j.amc.2023.128076 doi: 10.1016/j.amc.2023.128076
![]() |
[10] |
Z. Ma, H. Ma, Adaptive fuzzy backstepping dynamic surface control of strict-feedback fractional-order uncertain nonlinear systems, IEEE. T. Fuzzy. Syst., 28 (2019), 122–133. https://doi.org/10.1109/TFUZZ.2019.2900602 doi: 10.1109/TFUZZ.2019.2900602
![]() |
[11] |
H. L. Li, J. Cao, C. Hu, H. Jiang, F. E. Alsaadi, Synchronization analysis of discrete-time fractional-order quaternion-valued uncertain neural networks, IEEE. T. Neur. Net. Lear., 2023. https://doi.org/10.1109/TNNLS.2023.3274959 doi: 10.1109/TNNLS.2023.3274959
![]() |
[12] |
H. Delavari, R. Ghaderi, A. Ranjbar, S. Momani, Fuzzy fractional order sliding mode controller for nonlinear systems, Commun. Nolinear Sci., 15 (2010), 963–978. https://doi.org/10.1016/j.cnsns.2009.05.025 doi: 10.1016/j.cnsns.2009.05.025
![]() |
[13] |
J. Jiang, H. Chen, D. Cao, J. L. Guirao, The global sliding mode tracking control for a class of variable order fractional differential systems, Chaos Solition Fract., 154 (2022), 111674. https://doi.org/10.1016/j.chaos.2021.111674 doi: 10.1016/j.chaos.2021.111674
![]() |
[14] |
S. Ladaci, J. J. Loiseau, A. Charef, Fractional order adaptive high-gain controllers for a class of linear systems, Commun. Nolinear Sci., 13 (2008), 707–714. https://doi.org/10.1016/j.cnsns.2006.06.009 doi: 10.1016/j.cnsns.2006.06.009
![]() |
[15] | I. Petráš, Fractional-order feedback control of a dc motor, J. Electr. Eng., 60 (2009), 117–128. |
[16] |
H. Delavari, H. Heydarinejad, Fractional-order backstepping sliding-mode control based on fractional-order nonlinear disturbance observer, J. Comput. Nonlinear Dynam., 13 (2018), 111009. https://doi.org/10.1115/1.4041322 doi: 10.1115/1.4041322
![]() |
[17] |
N. Bigdeli, H. A. Ziazi, Finite-time fractional-order adaptive intelligent backstepping sliding mode control of uncertain fractional-order chaotic systems, J. Fanklin Inst., 354 (2017), 160–183. https://doi.org/10.1016/j.jfranklin.2016.10.004 doi: 10.1016/j.jfranklin.2016.10.004
![]() |
[18] |
X. Li, C. Wen, Y. Zou, Adaptive backstepping control for fractional-order nonlinear systems with external disturbance and uncertain parameters using smooth control, IEEE Trans. Syst. Man Cybern., 51 (2020), 7860–7869. https://doi.org/10.1109/TSMC.2020.2987335 doi: 10.1109/TSMC.2020.2987335
![]() |
[19] |
J. Jiang, H. Li, K. Zhao, D. Cao, J. L. Guirao, Finite time stability and sliding mode control for uncertain variable fractional order nonlinear systems, Adv. Differ. Equ., 2021 (2021), 127. https://doi.org/10.1186/s13662-021-03286-z doi: 10.1186/s13662-021-03286-z
![]() |
[20] |
J. Jiang, X. Xu, K. Zhao, J. L. Guirao, T. Saeed, H. Chen, The tracking control of the variable-order fractional differential systems by time-varying sliding-mode control approach, Fractal Fract., 6 (2022), 231. https://doi.org/10.3390/fractalfract6050231 doi: 10.3390/fractalfract6050231
![]() |
[21] |
M. Karami, A. Kazemi, R. Vatankhah, A. Khosravifard, Adaptive fractional-order backstepping sliding mode controller design for an electrostatically actuated size-dependent microplate, J. Vib. Control, 27 (2021), 1353–1369. https://doi.org/10.1177/1077546320940916 doi: 10.1177/1077546320940916
![]() |
[22] | E. Fridman, Introduction to time-delay systems: Analysis and control, Springer, 2014. https://doi.org/10.1007/978-3-319-09393-2 |
[23] |
S. E. Hamamci, An algorithm for stabilization of fractional-order time delay systems using fractional-order pid controllers, IEEE Trans. Autom. Control, 52 (2007), 1964–1969. https://doi.org/10.1109/TAC.2007.906243 doi: 10.1109/TAC.2007.906243
![]() |
[24] |
M. P. Lazarević, A. M. Spasić, Finite-time stability analysis of fractional order time-delay systems: Gronwall's approach, Math. Comput. Model., 49 (2009), 475–481. https://doi.org/10.1016/j.mcm.2008.09.011 doi: 10.1016/j.mcm.2008.09.011
![]() |
[25] |
I. Birs, C. Muresan, I. Nascu, C. Ionescu, A survey of recent advances in fractional order control for time delay systems, IEEE Access, 7 (2019), 951–965. https://doi.org/10.1109/ACCESS.2019.2902567 doi: 10.1109/ACCESS.2019.2902567
![]() |
[26] |
Z. Y. Nie, Y. M. Zheng, Q. G. Wang, R. J. Liu, L. J. Xiang, Fractional-order pid controller design for time-delay systems based on modified bode's ideal transfer function, IEEE Access, 8 (2020), 103500–103510. https://doi.org/10.1109/ACCESS.2020.2996265 doi: 10.1109/ACCESS.2020.2996265
![]() |
[27] |
H. Min, S. Xu, Q. Ma, B. Zhang, Z. Zhang, Composite-observer-based output-feedback control for nonlinear time-delay systems with input saturation and its application, IEEE Trans. Ind. Electron., 65 (2017), 5856–5863. https://doi.org/10.1109/TIE.2017.2784347 doi: 10.1109/TIE.2017.2784347
![]() |
[28] |
Y. Wu, X. J. Xie, Adaptive fuzzy control for high-order nonlinear time-delay systems with full-state constraints and input saturation, IEEE Trans. Fuzzy Syst., 28 (2019), 1652–1663. https://doi.org/10.1109/TFUZZ.2019.2920808 doi: 10.1109/TFUZZ.2019.2920808
![]() |
[29] |
H. Min, S. Xu, B. Zhang, Q. Ma, Output-feedback control for stochastic nonlinear systems subject to input saturation and time-varying delay, IEEE Trans. Autom. Control, 64 (2018), 359–364. https://doi.org/10.1109/TAC.2018.2828084 doi: 10.1109/TAC.2018.2828084
![]() |
[30] |
Y. Y. Cao, Z. Lin, T. Hu, Stability analysis of linear time-delay systems subject to input saturation, IEEE Trans. Circuits Syst. I, 49 (2002), 233–240. https://doi.org/10.1109/81.983870 doi: 10.1109/81.983870
![]() |
[31] |
S. Xu, G. Feng, Y. Zou, J. Huang, Robust controller design of uncertain discrete time-delay systems with input saturation and disturbances, IEEE Trans. Autom. Control, 57 (2012), 2604–2609. https://doi.org/10.1109/TAC.2012.2190181 doi: 10.1109/TAC.2012.2190181
![]() |
[32] |
Z. Lin, M. Pachter, S. Banda, Toward improvement of tracking performance nonlinear feedback for linear systems, Int. J. Control, 70 (1998), 1–11. https://doi.org/10.1080/002071798222433 doi: 10.1080/002071798222433
![]() |
[33] |
S. Mobayen, F. Tchier, Composite nonlinear feedback control technique for master/slave synchronization of nonlinear systems, Nonlinear Dynam., 87 (2017), 1731–1747. https://doi.org/10.1007/s11071-016-3148-8 doi: 10.1007/s11071-016-3148-8
![]() |
[34] |
B. M. Chen, T. H. Lee, K. Peng, V. Venkataramanan, Composite nonlinear feedback control for linear systems with input saturation: Theory and an application, IEEE Trans. Autom. Control, 48 (2003), 427–439. https://doi.org/10.1109/TAC.2003.809148 doi: 10.1109/TAC.2003.809148
![]() |
[35] |
D. Lin, W. Lan, M. Li, Composite nonlinear feedback control for linear singular systems with input saturation, Syst. Control Lett., 60 (2011), 825–831. https://doi.org/10.1016/j.sysconle.2011.06.006 doi: 10.1016/j.sysconle.2011.06.006
![]() |
[36] |
Y. He, B. M. Chen, C. Wu, Composite nonlinear control with state and measurement feedback for general multivariable systems with input saturation, Syst. Control Lett., 54 (2005), 455–469. https://doi.org/10.1016/j.sysconle.2004.09.010 doi: 10.1016/j.sysconle.2004.09.010
![]() |
[37] |
E. Jafari, T. Binazadeh, Observer-based improved composite nonlinear feedback control for output tracking of time-varying references in descriptor systems with actuator saturation, ISA Trans., 91 (2019), 1–10. https://doi.org/10.1016/j.isatra.2019.01.035 doi: 10.1016/j.isatra.2019.01.035
![]() |
[38] |
S. Mondal, C. Mahanta, Composite nonlinear feedback based discrete integral sliding mode controller for uncertain systems, Commun. Nonlinear Sci., 17 (2012), 1320–1331. https://doi.org/10.1016/j.cnsns.2011.08.010 doi: 10.1016/j.cnsns.2011.08.010
![]() |
[39] |
E. Jafari, T. Binazadeh, Low-conservative robust composite nonlinear feedback control for singular time-delay systems, J. Vib. Control, 27 (2021), 2109–2122. https://doi.org/10.1177/1077546320953736 doi: 10.1177/1077546320953736
![]() |
[40] |
V. Ghaffari, An improved control technique for designing of composite non-linear feedback control in constrained time-delay systems, IET Control Theory Appl., 15 (2021), 149–165. https://doi.org/10.1049/cth2.12018 doi: 10.1049/cth2.12018
![]() |
[41] |
Z. Sheng, Q. Ma, Composite-observer-based sampled-data control for uncertain upper-triangular nonlinear time-delay systems and its application, Int. J. Robust Nonlinear Control, 31 (2021), 6699–6720. https://doi.org/10.1002/rnc.5637 doi: 10.1002/rnc.5637
![]() |
[42] |
S. Mobayen, Robust tracking controller for multivariable delayed systems with input saturation via composite nonlinear feedback, Nonlinear Dyn., 76 (2014), 827–838. https://doi.org/10.1007/s11071-013-1172-5 doi: 10.1007/s11071-013-1172-5
![]() |
[43] |
S. Rasoolinasab, S. Mobayen, A. Fekih, P. Narayan, Y. Yao, A composite feedback approach to stabilize nonholonomic systems with time varying time delays and nonlinear disturbances, ISA Trans., 101 (2020), 177–188. https://doi.org/10.1016/j.isatra.2020.02.009 doi: 10.1016/j.isatra.2020.02.009
![]() |
[44] |
H. Wu, Robust tracking and model following control with zero tracking error for uncertain dynamical systems, J. Optim. Theory Appl., 107 (2000), 169–182. https://doi.org/10.1023/A:1004665018593 doi: 10.1023/A:1004665018593
![]() |
[45] | F. Zhang, The Schur complement and its applications, New York: Springer, 2005. https://doi.org/10.1007/b105056 |
[46] |
N. Aguila-Camacho, M. A. Duarte-Mermoud, J. A. Gallegos, Lyapunov functions for fractional order systems, Commun. Nonlinear Sci., 19 (2014), 2951–2957. https://doi.org/10.1016/j.cnsns.2014.01.022 doi: 10.1016/j.cnsns.2014.01.022
![]() |
[47] |
M. A. Duarte-Mermoud, N. Aguila-Camacho, J. A. Gallegos, R. Castro-Linares, Using general quadratic lyapunov functions to prove lyapunov uniform stability for fractional order systems, Commun. Nonlinear Sci., 22 (2015), 650–659. https://doi.org/10.1016/j.cnsns.2014.10.008 doi: 10.1016/j.cnsns.2014.10.008
![]() |
[48] |
J. Jiang, D. Cao, H. Chen, Sliding mode control for a class of variable-order fractional chaotic systems, J. Fanklin Inst., 357 (2020), 10127–10158. https://doi.org/10.1016/j.jfranklin.2019.11.036 doi: 10.1016/j.jfranklin.2019.11.036
![]() |
[49] |
Y. Li, Y. Chen, I. Podlubny, Mittag-leffler stability of fractional order nonlinear dynamic systems, Automatica, 45 (2009), 1965–1969. https://doi.org/10.1016/j.automatica.2009.04.003 doi: 10.1016/j.automatica.2009.04.003
![]() |
[50] |
Z. Wu, Y. Xia, X. Xie, Stochastic barbalat's lemma and its applications, IEEE Trans. Autom. Control, 57 (2011), 1537–1543. https://doi.org/10.1109/TAC.2011.2175071 doi: 10.1109/TAC.2011.2175071
![]() |
[51] |
D. Lin, W. Lan, Output feedback composite nonlinear feedback control for singular systems with input saturation, J. Fanklin Inst., 352 (2015), 384–398. https://doi.org/10.1016/j.jfranklin.2014.10.018 doi: 10.1016/j.jfranklin.2014.10.018
![]() |
[52] |
S. Mobayen, Chaos synchronization of uncertain chaotic systems using composite nonlinear feedback based integral sliding mode control, ISA Trans., 77 (2018), 100–111. https://doi.org/10.1016/j.isatra.2018.03.026 doi: 10.1016/j.isatra.2018.03.026
![]() |
[53] |
S. Mobayen, F. Tchier, Composite nonlinear feedback integral sliding mode tracker design for uncertain switched systems with input saturation, Commun. Nonlinear Sci., 65 (2018), 173–184. https://doi.org/10.1016/j.cnsns.2018.05.019 doi: 10.1016/j.cnsns.2018.05.019
![]() |
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