In this paper, a new fuzzy dynamic surface control approach based on a state observer is proposed for uncertain nonlinear systems with time-varying output constraints and external disturbances. An adaptive fuzzy state observer is used to estimate the states that cannot be measured in the systems. In our method, a time-varying Barrier Lyapunov Function (BLF) is used to ensure that the output does not violate time-varying constraints. In addition, dynamic surface control (DSC) technology is applied to overcome the problem of “explosion of complexity” in a backstepping control. Finally, the stability and signal boundedness of the system are confirmed by the Lyapunov method. The simulation results show the effectiveness and correctness of the proposed method.
National Natural Science Foundation of China517754631. Introduction
In practical engineering, there are many uncertain nonlinear electromechanical systems, such as robots, for which a mathematical model is difficult to determine. This leads to great difficulty in the design of their control systems [1–3]. Fuzzy logic systems (FLSs) have been widely used in adaptive control of uncertain nonlinear function modeling due to their universal approximation ability [4–8]. FLSs can be combined with backstepping design techniques to overcome the mismatched uncertainties problem. At the same time, backstepping control can provide a symmetric framework for controller design, so fuzzy backstepping control schemes have achieved great success in the control field [4, 9–13]. However, backstepping control needs to do repeated differentiations of the virtual control law. If there are nonlinear functions in the virtual control law, repeated differentiations will lead to the problem of “explosion of complexity” with increasing order of the system. This makes high order systems face great difficulties in controller implementation. Recently, Hedrick et al. proposed a dynamic surface control (DSC) method using first-order low-pass filters to avoid repetitive differential problems. It attracted great interest among researchers [14–18].
There is a lot of literature that focuses on fuzzy backstepping controls, but most of the exiting approaches are based on state feedback, for which all the states of the closed system should be directly measured. In practical engineering, it is impossible to measure all the states directly due to the limitation of sensors, installation positions, or measuring points. Therefore, the control scheme based on state feedback may not be applicable in practical engineering. References [19–24] describe the recent developments in adaptive output feedback control for uncertain nonlinear systems based on a state observer that identifies the unmeasurable states instead of directly measuring them. Reference [19] describes the problem of robust adaptive control for nontriangular stochastic nonlinear systems with unmeasurable states and unmodeled dynamics. Reference [20] describes a study of the problem of output feedback control for a class of SISO stochastic switched nonlinear systems with completely unknown functions, unmodeled dynamics, and arbitrary switching. In [21, 22], under the unified framework of adaptive backstepping control technology, an output feedback tracking control design method based on an adaptive fuzzy observer is proposed for uncertain nonlinear systems. Reference [23] contains a proposal for two adaptive fuzzy output feedback control methods for a class of uncertain stochastic nonlinear strict-feedback systems without state measurement. Reference [24] contains a proposal for a robust H∞ control of an observer-based repetitive-control system. The problems with the control methods in the literature mentioned above are that they are computationally complex and do not take engineering constraints into account.
Output constraints are important engineering constraints for many industrial systems. Without considering the problem of output constraints, equipment may be damaged and accidents can happen. Because a BLF grows to infinity when its related state is close to a certain limit, it has received extensive attention as a way to solve the output constraint problem. Therefore, as long as the BLF is bounded, the related states will not violate the constraints. References [25–27] describe how BLFs have been used to deal with the output constraints. References [25, 27] describe how a BLF can be used to solve the output constraint problem of a robot manipulator system. Reference [26] describes an adaptive neural network control that is designed for the control of a nonlinear affine system subject to external unknown disturbances for the conditions of an input dead zone and output constraints.
References [25–27] all focus on the static output constraints problem, but time-varying output constraints are more in line with practical engineering, leading some researchers to publish literature on this problem. Just as a conventional BLF can handle static output constraints, time-varying output constraints can be tackled by using a time-varying BLF [14, 28, 29]. Reference [14] describes the design of an adaptive state feedback control for uncertain strictly feedback nonlinear systems with asymmetric time-varying output constraints when input saturation occurs. Reference [28] describes how an asymmetric time-varying BLF can be used to prevent the output from exceeding the constraint bounds, and it shows that the output can start anywhere in the initial restricted output space. Reference [29] shows for the first time how time-varying output constraints can be extended to full-state time-varying constraints and describes an adaptive controller based on backstepping technology. However, the control methods in the research mentioned above are all based on state feedback control, for which all the states in closed-loop systems must be measurable.
Because few references consider the output feedback control based on DSC of uncertain nonlinear systems with time-varying constraints, we have tried in this paper to deal with this more difficult and practical problem for the design of an adaptive control based on a state observer for uncertain nonlinear systems with asymmetric time-varying output constraints and unknown external disturbances. Our main contributions lie in two points that contrast with existing works. (1) It is the first time that an adaptive DSC based on a fuzzy state observer has been addressed for uncertain nonlinear systems with time-varying output constraints and external disturbances. The system in this paper is more general and practical, and the control method is simple, which avoids the traditional computational complexity. (2) The control method described in this paper does not require n-order differentiable and bounded conditions for input signals, and it reduces the requirement of hypothetical conditions.
2. System Description and Basic Knowledge
The goal of the study described in this paper was to develop a nonlinear system with a strict-feedback structure that fits the following equations:(1)x˙1=x2+f1x1+d1tx˙2=x3+f2x1,x2+d2t⋮x˙i=xi+1+fix1,x2,…,xi+dit,i=1,2,…,n-1⋮x˙n=ut+fnx1,x2,…,xn+dnty=x1,where x1,x2,…,xn are the state variables and only x1 can be measured. u∈R and y∈R are the input and output of the system, respectively. fi(Xi) (Xi=(x1,x2,…,xi)T, i=1,2,…,n) represents unknown smooth functions. di(t) (i=1,2,…,n) represents the external disturbances with unknown boundaries. The output y(t) requirements meet the boundary constraints:(2)k_c1t≤yt≤k-c1t,∀t>0where k-c1(t):R+→R and k_c1(t):R+→R, such that k-c1(t)>k_c1(t), ∀t≥0.
System (1) can be rewritten as(3)X˙=AX+Ky+∑i=1nBifiXi+di+Buy=CXwhere X=x1,x2,…,xnT, A=-k1⋮I-kn00, K=k1⋮kn, Bi=0…1…0T, B=0⋮1, C=1…0…0, and K is chosen such that A is a Hurwitz matrix. Thus, given a positive definite diagonal matrix Q>0, there exists a positive definite symmetric matrix P>0 satisfying(4)ATP+PA=-2Q.Control Objective. A state observer is designed to estimate the unmeasurable state. An adaptive controller is designed to use this estimate to create the output y(t) tracking the desired trajectory yd(t) and ensure that the output y(t) satisfies time-varying constraints. All signals involved in the closed-loop system are bounded, and the tracking error remains in the sufficiently small range.
Assumption 1 (see [30]).
External disturbance di(t) is bounded by the positive unknown constant diM; that is, di(t)≤diM.
Assumption 2 (see [14]).
There are constants K-ci and K_ci(i=0,1,2,…,n) such that k-c1(t)<K-c0, k_c1(t)<K_c0, and k-c1(i)(t)≤K-ci, k_c1(i)(t)<K_ci(i=1,2,…,n)∀t≥0.
Assumption 3 (see [31]).
There are functions Y-0:R+→R+ and Y_0:R+→R+ that satisfy Y-0<k-c1(t) and Y_0>k_c1(t), ∀t>0, and there is a positive constant Y1 such that the desired trajectory yd(t) and its time derivative satisfy Y_0(t)≤yd(t)≤Y-0(t) and y˙(t)≤Y1, ∀t>0.
3. Fuzzy System and Its Approximation
A Fuzzy system is a universal approximator that is used to approximate unknown nonlinear functions. By defining the fuzzy basis function vector as ξ(x) and the adjustable weight parameter vector as θ∈RN, the general output form of the fuzzy system can be written as follows.(5)f^x∣θ=θTξx
According to the universal approximation theorem of fuzzy systems, if f(x) is a continuous function defined based on the compact set Ω and if a fuzzy system f^(x∣θ) is used to approximate f(x), there exists a parameter vector θ such that supx∈Ω|f(x)-θTξ(x)|≤ε for any given small constant, ε, such that 0<ε<εM [32].
4. Adaptive Control and Observer Design
In this paper, the states x2,…,xn of system (1) are not available for feedback, so a state observer needs to be established to estimate the states. Therefore, we defined the estimate of Xi as X^i, i=1,2,…,n. According to the universal approximation of fuzzy systems, the uncertain nonlinear function fi(Xi) (i=1,2,…,n) can be expressed as(6)fiXi=θiTξiX^i+εi(7)fiXi=θi∗TξiX^i+εi∗where εi is the approximation error, θi∗ is the optimal parameter vector, and εi∗ is the minimal approximation error.
We designed the fuzzy state observer as follows.(8)X^˙=AX^+Ky+∑i=1nBif^iX^i∣θi+Buy^=CX^
By defining the observer error vector as X~=X-X^, from (3) and (8), the observer errors equation becomes(9)X~˙=AX~+∑i=1nBifiXi-f^iX^i∣θi+di=AX~+∑i=1nBiεi+di=AX~+∑i=1nBiδi=AX~+δwhere δ=[δ1,δ2,…,δn]T and δi=εi+di.
Step 1. Define z1=y-ω0 (ω0=yd) as the tracking error and z2=x^2-ω1 as the virtual error for the second step. Define the first virtual control law as α1. Let α1 pass through a first-order filter that has the time constant υ1. We can then obtain ω1:(10)υ1ω˙1+ω1=α1;ω10=α10.
Defining the output error of this filter as e1 leads to e1=ω1-α1 and ω˙1=-e1/υ1, so the time derivative of z1 is(11)z˙1=x2+f1+d1-ω˙0=z2+e1+α1+x~2+θ1∗Tξ1x^1+ε1∗+d1-y˙d.
Let D1=ε1∗+d1. Because ε1∗≤ε1M and d1≤d1M, there exists an unknown constant D1M>0 such that D1≤D1M. Define θ~1=θ1∗-θ1. Now the time-varying asymmetric BLF can be chosen as(12)V0=12X~TPX~+qz12logkb2tkb2t-z12+1-qz12logka2tka2t-z12+12γ1θ~1Tθ~1where γ1>0 is the positive design parameter. The time-varying barriers are defined as(13)kat=ydt-k_c1t(14)kbt=k-c1t-ydt(15)qz1=1z1>00z1≤0.
Define ςa=z1(t)/ka(t), ςb=z1(t)/kb(t), and ς=qςb+(1-q)ςa; then (12) can be rewritten as(16)V0=12X~TPX~+12log11-ς2+12γ1θ~1Tθ~1.
The time derivative of V0 is described as(17)V˙0=12X~˙TPX~+12X~TPX~˙+ςς˙1-ς2-1γ1θ~1Tθ˙1=12X~TPAT+APX~+X~TPδ+ςς˙1-ς2-1γ1θ~1Tθ˙1.
Because(18)ςς˙1-ς2=qςb+1-qςa1-ς2qς˙b+1-qς˙a(19)ς˙b=z˙1kbt-z1k˙btkb2t(20)ς˙a=z˙1kat-z1k˙atka2twe can obtain(21)V˙0=-X~TQX~+X~TPδ+qςb+1-qςa1-ς2qς˙b+1-qς˙a-1γ1θ~1Tθ˙1=-X~TQX~+X~TPδ+qςb1-ς2ς˙b+1-qςa1-ς2ς˙a-1γ1θ~1Tθ˙1=-X~TQX~+X~TPδ+qςbkb1-ςb2z˙1-z1k˙bkb+1-qςaka1-ςa2z˙1-z1k˙aka-1γ1θ~1Tθ˙1.
Assuming that μ=q/(kb2-z12)+(1-q)/(ka2-z12), we get(22)V˙0=-X~TQX~+X~TPδ+μz1z˙1-qz1k˙bkb-1-qz1k˙aka-1γ1θ~1Tθ˙1=-X~TQX~+X~TPδ+μz1x~2+D1+μz1z2+α1+e1+θ1∗Tξ1x^1-y˙d-qz1k˙bkb-1-qz1k˙aka-1γ1θ~1Tθ˙1.
By using the inequality 2ab≤a2+b2, we get(23)X~TPδ+μz1x~2+D1≤12X~2+12Pδ2+12D12+12x~22+μz12≤X~2+12Pδ2+12D12+μz12.
Substituting (23) into (22) results in(24)V˙0≤-λminQ-1X~2+12Pδ2+12D12+μz1z2+μz1+α1+θ1Tξ1x^1-y˙d+qz1k˙bkb+1-qz1k˙aka+1γ1θ~1Tγ1μz1ξ1x^1-θ˙1+μz1e1.
We choose the first virtual control law α1 and the parameter adaptive law θ1 to be(25)α1=-λ1z1-λ1′z1-μz1-θ1Tξ1x^1+y˙d(26)θ˙1=γ1μz1ξ1x^1-2σ1θ1where λ1>0 and σ1>0 are positive design parameters, and λ1′=k˙b/kb2+k˙a/ka2+β with β as a positive design constant.
Substituting (25) and (26) into (24) results in(27)V˙0<-λminQ-1X~2-λ1μz12+μz1z2+μz1e1+12Pδ2+12D12+2σ1γ1θ~1Tθ1.
There are Young’s inequalities in (28) and (29)(28)2σ1γ1θ~1Tθ1≤-σ1γ1θ1Tθ1+σ1γ1θ1∗Tθ1∗(29)z1e1≤z12+14e12.
Substituting (28) and (29) into (27) leads to(30)V˙0<-λminQ-1X~2-λ1-1μz12+μz1z2-σ1γ1θ1Tθ1+σ1γ1θ1∗Tθ1∗+14μe12+12Pδ2+12D12.
By using the following inequality:(31)-12θ~1Tθ~1≥-θ1∗Tθ1∗-θ1Tθ1,we obtain(32)V˙0<-λminQ-1X~2-λ1-1μz12+μz1z2-σ12γ1θ~1Tθ~1+2σ1γ1θ1∗Tθ1∗+14μe12+12Pδ2+12D12.
Now, consider the following Lyapunov function candidate:(33)V1=V0+12e12.
Then we can have(34)V˙1=V˙0+e1-e1υ1-α˙1≤V˙0-e12υ1+e12+14ψ12<-λminQ-1X~2-λ1-1μz12+μz1z2-σ12γ1θ~1Tθ~1+2σ1γ1θ1∗Tθ1∗-1υ1-1-14μe12+12Pδ2+12D12+14ψ12.Here ψ1 is the maximum absolute value of α˙1.
Step 2. Define z3=x^3-ω2. Define the second virtual control law as α2. Let α2 pass through a first-order filter that has the time constant υ2. We can then obtain ω2:(35)υ2ω˙2+ω2=α2;ω20=α20.
By defining the output error of this filter as e2, we get e2=ω2-α2 and ω˙2=-e2/υ2.
So the time derivative of z2 is as follows:(36)z˙2=x^3+k2x~1+f^2-ω˙1=z3+e2+α2+k2x~1+θ2Tξ2X^2-ω˙1.
Define θ~2=θ2∗-θ2, and the Lyapunov Function can be chosen as(37)V2=V1+12z22+12e22+12γ2θ~2Tθ~2where γ2>0 is the positive design parameter.
The time derivative of V2 is described as follows.(38)V˙2=V˙1+z2z3+e2+α2+k2x~1+θ2∗Tξ2X^2+ε2∗-ε2-ω˙1+e2e˙2-1γ2θ~2Tθ˙2=V˙1+z2z3+e2+α2+k2x~1+θ2∗Tξ2X^2+D2-ω˙1+e2e˙2-1γ2θ~2Tθ˙2.Here D2=ε2∗-ε2. D2≤D2M, and D2M is an unknown positive constant.
Then we can obtain(39)V˙2<V˙1+z2z3+e2+α2+k2x~1+D2+θ2Tξ2X^2-ω˙1+e2e˙2+1γ2θ~2Tγ2z2ξ2x^2-θ˙2<V˙1+z2z3+e2+12z2+α2+k2x~1+θ2Tξ2X^2-ω˙1+e2e˙2+1γ2θ~2Tγ2z2ξ2x^2-θ˙2+12D22.
The virtual control law α2 and the parameter adaptive law θ2 can be described as(40)α2=-λ2z2-μz1-12z2-k2x~1-θ2Tξ2X^2-ω1-α1υ1(41)θ˙2=γ2z2ξ2x^2-2σ2θ2where λ2>0and σ2>0 are positive design parameters.
Substituting (40) and (41) into (39) results in(42)V˙2<V˙1-λ2z22-μz1z2+z2z3+z2e2+e2e˙2+2σ2γ2θ~2Tθ2+12D22<V˙1-λ2-1z22-μz1z2+z2z3+14e22+e2-e2υ2-α˙2+2σ2γ2θ~2Tθ2+12D22<-λminQ-1X~2-λ1-1μz12-λ2-1z22+z2z3-∑i=12σi2γiθ~iTθ~i-1υ1-1-14μe12-1υ2-1-14e22+∑i=122σiγiθi∗Tθi∗+12Pδ2+12∑i=12Di2+14∑i=12ψi2where ψ2 is the maximum absolute value of α˙2.
Next, we step i(i=3,4,…,n-1). Define zi=x^i-ωi-1 as the virtual error of the ith step and zi+1=x^i+1-ωi as the virtual error of the (i+1)th step. Define the ith virtual control law as αi. Let αi pass through a first-order filter that has the time constant υi. We can then obtain ωi:(43)υiω˙i+ωi=αi;ωi0=αi0.
Defining the output error of this filter as ei yields ei=ωi-αi and ω˙i=-ei/υi.
Therefore, the time derivative of zi is as follows:(44)z˙i=x^i+1+kix~1+f^i-ω˙i-1=zi+1+ei+αi+kix~1+θiTξiX^i-ω˙i-1.
Define θ~i=θi∗-θi, and choose the Lyapunov Function as(45)Vi=Vi-1+12zi2+12ei2+12γiθ~iTθ~iwhere γi>0 is the positive design parameter.
The time derivative of Vi is described as follows.(46)V˙i=V˙i-1+zizi+1+ei+αi+kix~1+θiTξiX^i-ω˙i-1+eie˙i-1γiθ~iTθ˙i=V˙i-1+zizi+1+ei+αi+kix~1+θi∗TξiX^i+Di-ω˙i-1+eie˙i-1γiθ~iTθ˙i.Here Di=εi∗-εi, Di≤DiM, and DiM is a unknown positive constant.
Then we can obtain(47)V˙i<V˙i-1+zizi+1+ei+αi+kix~1+θiTξiX^i+Di-ω˙i-1+eie˙i+1γiθ~iTγiziξix^i-θ˙i<V˙i-1+zizi+1+ei+αi+kix~1+12zi+θiTξiX^i-ω˙i-1+eie˙i+1γiθ~iTγiziξix^i-θ˙i+12Di2.
The virtual control law αi and the parameter adaptive law θi can be described as(48)αi=-λizi-zi-1-12zi-kix~1-θiTξiX^i-ωi-1-αi-1υi-1(49)θ˙i=γiziξix^i-2σiθiwhere λi>0 and σi>0 are positive design parameters.
Substituting (48) and (49) into (47) results in(50)V˙i<V˙i-1-λizi2-zizi-1+zizi+1+ziei+eie˙i+2σiγiθ~iTθi+12Di2<V˙i-1-λi-1zi2-zizi-1+zizi+1-1υi-1-14ei2-σi2γiθ~iTθ~i+2σiγiθi∗Tθi∗+14ψi2+12Di2<-λminQ-1X~2-λ1-1μz12-∑k=2iλk-1zk2+zkzk+1-∑k=1iσk2γkθ~kTθ~k-1υ1-1-14μe12-∑k=2i1υk-1-14ek2+∑k=1i2σkγkθk∗Tθk∗+12Pδ2+12∑k=1iDk2+14∑k=1iψk2.Here ψi is the maximum absolute value of α˙i.
Step n. Because zn=x^n-ωn-1, the time derivative of zn is(51)z˙n=u+knx~1+f^n-ω˙n-1=u+knx~1+θnTξnX^n-ω˙n-1.
Define θ~n=θn∗-θn, and the Lyapunov function can be chosen as(52)Vn=Vn-1+12zn2+12γnθ~nTθ~nwhere γn>0 is the positive design parameter.
The time derivative of Vn is equal to(53)V˙n=V˙n-1+znu+knx~1+θn∗TξnX^n+εn∗-εn-ω˙n-1-1γnθ~nTθ˙n=V˙n-1+znu+knx~1+θn∗TξnX^n+Dn-ω˙n-1-1γnθ~nTθ˙n.Here Dn=εn∗-εn, Dn≤DnM, and DnM is an unknown positive constant.
Then we can obtain(54)V˙n<V˙n-1+znu+knx~1+θnTξnX^n+12zn-ω˙n-1+1γnθ~nTγnznξnx^n-θ˙n+12Dn.
Choose the control law u and the parameter adaptive law θn as follows:(55)u=-λnzn-zn-1-12zn-knx~1-θnTξnX^n-ωn-1-αn-1υn-1(56)θ˙n=γnznξnx^n-2σnθnwhere λn>0 and σn>0 are positive design parameters.
Substituting (55) and (56) into (54) results in(57)V˙n<V˙n-1-λnzn2-znzn-1+2σnγnθ~nTθn+12Dn<V˙n-1-λnzn2-znzn-1-σn2γnθ~nTθ~n+2σnγnθn∗Tθn∗+12Dn<-λminQ-1X~2-λ1-1μz12-∑i=2n-1λi-1zi2-λnzn2-∑i=1nσi2γiθ~iTθ~i-1υ1-1-14μe12-∑i=2n-11υi-1.25ei2+Mwhere M=∑i=1n(2σi/γi)θi∗Tθi∗+(1/2)PδM2+(1/2)∑i=1nDiM2+(1/4)∑i=1n-1ψi2.
It can be seen from (25), (40), (48), and (55) that the proposed control method not only has overcome the difficulty in backstepping control design due to the “explosion of complexity”, but also has removed the restrictive assumption that is widely used in [33, 34] that the input signal should be n-order differentiable and bounded. Moreover, the proposed control method can easily obtain adaptive control of nonlinear systems with various output constraints and unmeasurable states.
To further illustrate the advantages of our method, we will make some comparisons with previous results that considered adaptive control of nonlinear systems with multiple constraints, but for which all the states of the control system need to be measured [14–16, 29, 33, 34]. This paper describes the design of a fuzzy state observer, such that only the output of the system needs to be measured. Previous papers [19–24, 31] described the development of adaptive control of nonlinear systems based on a fuzzy sate observer, but these adaptive control methods cannot deal with the problem of output constraints. Because of the “explosion of complexity”, these methods have a heavy computation burden. In addition, these control methods all assume that the input signal should be n-order differentiable with bounded derivatives.
5. Stability Analysis
Define V=Vn as the Lyapunov function of the closed-loop system, so the derivation of V is (57).
Select the positive matrix Q and the positive coefficients λi, λn, υ1, and υi as(58)λminQ-1>0(59)λi-1>0,i=1,2,…,n-1(60)λn>0(61)1υ1-1-14μ>0(62)1υi-1.25>0.
Based on lemma 2 in [14], we get(63)V˙<-2λminQ-1λmaxP12X~TPX~-2λ1-112logz12kb2-z12-∑i=2n-12λi-112zi2-2λn12zn2-∑i=1nσi2γiθ~iTθ~i-21υ1-1-14μ12e12-∑i=2n-121υi-1.2512ei2+M.
Define the positive parameter as(64)C=min2λminQ-1λmaxP,2λi-1,2λn,21υ1-1-14μ,21υi-1.25,σi;i=1,2,⋯,n.
Then (63) can be rewritten as(65)V˙≤-CV+M.
The initial condition requirement k_c1(0)≤y(0)≤k-c1(0) implies that -ka(0)<z1(0)<kb(0) and ς(0)<1. Then, based on lemma 1 in [35], we can have ς(t)<1, ∀t>0, where V is bounded in the set of [0,∞). Because -ka(t)<z1(t)<kb(t) and y(t)=z1(t)+yd(t), we can assume that for all t>0, -ka(t)+yd(t)<y(t)<kb(t)+yd(t), and k_c1(t)<y(t)<k-c1(t), ∀t>0, can be deduced.
Multiply both sides of (65) by eCt to obtain(66)eCtV˙≤-CV+MeCt(67)ddtVeCt≤MeCt(68)VeCt-V0≤MCeCt-1(69)0≤Vt≤V0e-Ct+MC1-e-Ct≤V0+MC.
From (69), we can see that if V(0)≤ν and V(t)≤ν+M/C, ∀t>0, the boundedness of ς and V guarantees that all signals of the closed-loop system, such as xi(t), x^i(t), zi(t), ai(t), and u(t), are semiglobally uniformly ultimately bounded (SGUUB) [36, 37]. Based on (69) and the definitions of C and M, it can be seen that z1(t) can be made arbitrarily small by appropriate design parameters.
6. Simulations
Consider a system governed by the following form:(70)x˙1=x2+x1e-0.5x1+0.1sin2tx˙2=ut+x1sinx22+0.01cos10ty=x1where f1(x1)=x1e-0.5x1 and f2(x1,x2)=x1sin(x22) are unknown functions. d1(t)=0.1sin(2t), d2(t)=0.01cos(10t). The input tracking signal is yd=0.5sin(t). k_c1=-0.5+0.4sin(t), k-c1=0.6+0.1cos(t).
By choosing the fuzzy membership function as(71)μF1lx^1=exp-x^1+2.5-l/222,l=1,2,…,9μF2lx^1,x^2=exp-x^1+2.5-l/222×exp-x^2+2.5-l/222,l=1,2,…,9,
defining the fuzzy basis functions as(72)ξ1lx^1=μF1lx^1∑j=19μF1jx^1,l=1,2,…,9ξ2lx^1,x^2=μF1lx^1×μF2lx^2∑j=19μF1jx^1×μF2jx^2,l=1,2,…,9,
and choosing the parameters in the controller and in the adaptive laws as(73)λ1=20,λ2=20,k1=20,k2=10,γ1=γ2=0.1,σ1=σ2=0.1,β=0.01we can obtain(74)K=20,10TandA=-201-100.
Q=diag[5,5] is the given symmetric positive matrix. By solving the Lyapunov equation (4), we can get the symmetric positive matrix P: (75)P=2.75-5-510.275.
The initial conditions of the system and the observer are chosen as(76)x0=0,-0.2Tandx^0=0,0.3T.
Initial values of adaptive parameters are chosen as(77)θ10=0.15,0.15,0.15,0.15,0.15,0.15,0.15,0.15,0.15Tθ20=0.15,0.15,0.15,0.15,0.15,0.15,0.15,0.15,0.15T.
The simulation results are shown in Figures 1–5. Figure 1 shows the output y(t) and the asymmetric constraints, k_c1(t)≤y(t)≤k-c1(t), ∀t≥0. We can see that the output y(t) can track the desired trajectory yd(t) very well. Figure 2 shows the trajectories of tracking error z1(t) and the error boundaries. It shows that z1(t) always satisfies -ka(t)<z1(t)<kb(t), ∀t≥0. Figure 3 shows the trajectories of state x1 and its estimate x^1. Figure 4 shows the trajectories of state x2 and its estimate x^2. Figure 5 shows the control input signal u(t). From Figures 1, 2, and 5, we can see that, when y(t) and z1(t) get close to their constraints, the amplitude of u(t) increases rapidly. This is predictable, and when y and z1 come close to the limit boundaries, the controller will provide a large control effort to keep the output and error away from the constraints. The simulation results show that, in the presence of external disturbances, the proposed output control scheme is capable of guaranteeing the boundedness of all the signals in the closed-loop system, such as, x1, x^1, x2, x^2, and u, without violating the asymmetric time-varying output constraints.
Trajectories of y, yd and the output constraints.
The tracking error z1 and the error bounds.
The trajectories of x1 and x^1.
The trajectories of x2 and x^2.
The control input u(t).
7. Conclusion
This paper has proposed an adaptive DSC scheme based on a fuzzy state observer for uncertain nonlinear systems with asymmetric time-varying output constraints in the presence of external disturbances. As part of this scheme, a fuzzy adaptive state observer has been designed to estimate the unmeasured states, and an asymmetric time-varying BLF is employed to prevent the output from violating the asymmetric time-varying constraints. The problem of “explosion of complexity” is avoided by employing the DSC design. Finally, the stability of the closed-loop system has been confirmed by using Lyapunov method. The semiglobal uniform ultimate boundedness of all the signals can be guaranteed, and the tracking error remains within a sufficiently small boundary. Our future research will include extending of the results described here to nonstrict-feedback MIMO nonlinear systems and stochastic nonlinear systems.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research has been supported by National Natural Science Foundation of China (grant no. 51775463).
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