# Observer-Based Finite-Time Inverse Optimal Output Regulation for Uncertain Nonlinear Systems

Authors: Guizhi Meng; Yan Lv
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

This paper deals with the finite-time inverse optimal output regulation problem for a class of uncertain nonlinear systems that are subjected to an exosystem. The nonlinear systems not only contain external disturbances in the exosystem but also take the unknown nonlinear functions and unmeasured states into account. The output regulation problem is first converted into a stabilization problem through the internal model. Then, fuzzy logic systems (FLSs) are employed to approximate the unknown nonlinear functions. An auxiliary system is constructed and, based on the auxiliary system, a fuzzy state observer is designed to estimate the unmeasured states. Furthermore, a novel adaptive fuzzy finite-time inverse optimal output feedback controller and adaptive law are designed by combining backstepping technology, adaptive control technology, finite-time stability theory, and the inverse optimal control method. The control algorithm ensures that all the signals of the closed-loop system are semiglobally practical finite-time stable (SGPFS), so the newly well-defined cost function that is connected with the auxiliary system can be minimized. Finally, the validity of the approach is confirmed by virtue of the simulation results.


---

## Body

## 1. Introduction

During the past several years, the output regulation problem has been widely studied. The target of the output regulation problem is to design a controller when the system is disturbed by external disturbances. The controller can not only guarantee the stability of the system but also ensure the output of the system tracks reference signals and rejects disturbances. Reference signals and disturbances are both generated by an exosystem. Over the past few decades, with the help of the internal model [1], some results have been obtained regarding the nonlinear output regulation problems [2, 3]. However, as the process of control becomes increasingly complex, some uncertainties such as measuring error and unmodeled dynamics can be produced in some practical systems, for instance, rigid spacecraft systems, mobile robot systems, and quadrotor systems [4–7] are uncertain. So, it is significant to study the output regulation problem of uncertain nonlinear systems. More recently, in [8, 9], a controller was presented by using an adaptive internal model and iterative procedure without knowing the frequencies of the exosystem. Chen and Huang proposed that it was possible to find a controller when the exogenous signals were produced by nonlinear exosystems [10, 11]. Moreover, the adaptive backstepping control technique and the Nussbaum gain method were employed to design an output feedback controller [12]. But the above references cannot solve the unknown nonlinear functions in the controlled system. FLSs were considered to be an effective way to deal with this kind of problem [13–15]. By utilizing FLSs with adaptive backstepping control, the controllers were designed in state feedback and output feedback form, respectively [16, 17]. To solve the output regulation problem of nonlinear systems with unknown nonlinear functions, Meng et al. [18, 19] design an adaptive fuzzy controller. However, some communication systems and multiagent systems require finite-time control to achieve transient performance and rapid response [20], but the above-designed controllers are unsuitable for finite-time control problems.Finite-time control aims to achieve the performance of the system within a finite time. This method has good antidisturbance performance and fast convergence speed because of the fractional power term. Recently, for a class of nonlinear systems, some achievements have been made in finite-time control by using the Lyapunov theory and dynamic surface technique [21–23]. But the above research results have strict restrictions on nonlinear functions, so adaptive fuzzy finite-time control methods were proposed for nonlinear systems with unknown nonlinear functions [24–26]. Liu et al. [27] extended finite-time control to the output regulation problem. But the references mentioned above do not consider optimality in the control schemes, which is an important problem in modern control fields.Recently, many fruitful results have been obtained concerning optimal control. To solve the linear optimal control problem, the off-policy integral reinforcement learning (IRL) approach, which is an artificial intelligence algorithm, is employed [28]. But for nonlinear systems, seeking the solutions of the Hamilton–Jacobi–Bellman (HJB) equation is an inevitable and difficult process when the optimal control problem is considered. In order to overcome this difficulty, neural networks were applied to approximate the solutions of the HJB equation, then the optimal control problem of nonlinear systems was addressed [29, 30]. However, the approximate errors can be generated in this way, and the optimal control target will be affected if the errors are not small enough. To this end, the inverse optimal control method was proposed, which is an approach that does not need to solve the HJB equation [31, 32]. The controller and the solutions of the HJB equation are both obtained with the help of the control Lyapunov function (CLF), and the controller can also minimize the cost functional. On the basis of [32], for nonlinear systems with unknown parameters, the inverse optimal control problem was investigated in [33]. When the nonlinear functions were unknown, an adaptive fuzzy inverse optimal controller was designed in [34, 35] by using an adaptive backstepping design technique and an inverse optimal control method. The applications of the inverse optimal control method on Quanser 2 DOF helicopters and unmanned aerial helicopters can be found in references [36, 37]. It can be found that the methods above mainly address the stabilization problem of the nonlinear systems. At present, there are some references to the optimal output regulation problem. The reinforcement learning (RL) method was extended to the linear optimal output regulation problem [38, 39]. For nonlinear systems with minimum phases and exosystems, the optimal output regulation problem was investigated first by the inverse optimal control method [40]. But there are two main limitations in [40]. One is that the controlled system does not consider unknown dynamics and all the states are measured directly; the other is that the performance of the system will not be achieved in finite time.Motivated by the above investigations, the finite-time inverse optimal output regulation problem for a class of uncertain nonlinear systems with unknown nonlinear functions and unmeasured states is investigated; that is, the proposed controller not only ensures that all the signals of the closed-loop system are semiglobally practical finite-time stable but also minimizes the cost functional. It should be noted that the nonlinear systems in our paper contain unmeasured information. Because this is also an optimal problem, the internal model and the state observer in [12, 13, 19] are not applicable. The main challenge of this paper is how to give an appropriate cost functional. Because of the exosystem, we must employ the internal model such that the exosystem can be immersed in it. Thus, the internal model must be considered in the cost functional, and the existing cost functionals [30–34] are invalid. Note that the reference [40] gives the cost functional for a nonlinear optimal output regulation problem, but because unknown nonlinear functions and unmeasured states are not taken into account in [40], the proposed cost functional in [40] is also unsuitable.To deal with the above problems, compared with the existing results, we provide our main contributions as follows:(i)
A novel adaptive fuzzy finite-time inverse optimal output feedback controller is designed. It is proved that all the signals of the closed-loop system are SGPFS, and the controller can minimize the cost functional. Compared with the reference [18], by using inverse optimal control and finite-time control, the newly proposed controller can make the output of the controlled system track the reference signals faster and the tracking error smaller.(ii)
An auxiliary system is constructed. A new internal model and fuzzy state observer are given related to the auxiliary system, and the fuzzy state observer can be designed to estimate the unmeasured states.(iii)
The newly well-defined cost functional, which is connected with the auxiliary system, is given. Although the form of the functional cost is similar to [31–35], the functions lx and γ are designed differently.The rest of this paper is organized as follows. Section2 provides a brief problem formulation and preliminary knowledge on the output regulation problem. Section 3 demonstrates the design of the adaptive fuzzy finite-time inverse optimal controller. Section 4 proves the finite-time stability of the closed-loop system and the minimization of cost functional. Section 5 includes the numerical simulation results, and the conclusion is drawn in Section 6.


## 2. Problem Formulation and Preliminary Knowledge

### 2.1. Problem Description

Consider the following strict feedback nonlinear systems(1)ξ˙i=ξi+1+fiξ1,ξ2,…,ξi+diw,i=1,2,…,n−1,ξ˙n=u+fnξ1,ξ2,…,ξn+dnw,y=ξ1,e=y−qw,where ξ=ξ1,ξ2,⋯,ξnT∈Rn,u, and y are the state vector, the control input, and the output, respectively. fii=1,2,⋯,n is an unknown smooth nonlinear function, diwi=1,2,⋯,n and qw represent the undesired disturbance and reference input, where diw is an unknown smoooth function and qw is a known smooth function. e is the tracking error. w is the state of the exosystem generated by the following system:(2)w˙=Sw.In this paper, we assume that onlyy=ξ1 is measured directly.It is worth mentioning that systems (1) and (2) can be obtained for certain practical problems. For example, a dynamic model is established to solve the attitude tracking and disturbance rejection problem of rigid spacecraft, and the model can be disturbed by external disturbance torque, this problem is a special example of an output regulation problem [4].The main objective of the output regulation problem with the strict feedback nonlinear system (1) and linear exosystem (2) is to find an adaptive fuzzy finite-time inverse optimal controller, which depends on the error output and the state estimation vector so that for any x0∈Ω⊆Rn,w0∈W⊆Rm, where Ω and W are any known tight sets containing the origin in Rn and Rm, all the signals of the closed-loop system are SGPFS, and this controller can achieve inverse optimization in relation to the following cost functional.To study the output regulation problem, the following assumptions are introduced.Assumption 1.
(see [3]): The matrix S has distinct eigenvalues on the imaginary axis.
Assumption1 means that w is bounded and persistent, and the exosystem is neutrally stable. This is a common assumption for the output regulation problem.Assumption 2.
(see [3]): For nonlinear systems like(3)ξ˙=fξ,w+gξ,wu,e=hξ,w,w=S˜w,there exists a continuously differentiable mapping ξ=πw, with π0=0,∀w∈W, and a continuous mapping αw that solve the equations(4)∂πw∂wS˜w=fπ,w+gπ,wαw,0=hπ,w.
Based on Assumption2, by setting π1w=qw, we have(5)π1w=qw,πi+1w=π˙iw−fiπ1,π2,⋯,πi−diw,i=2,⋯,n−1,αw=π˙nw−fnπ1,π2,⋯,πn−dnw.
Define a state transformation asxi=ξi−πiwi=1,2,⋯,n. Then, we get the following systems:(6)x˙i=xi+1+FiXi,i=1,2,⋯,n−1,x˙n=u−αw+FnXn,e=x1,where Xi=x1,x2,⋯,xi,FiXi=fiξ1,ξ2,⋯,ξi−fiπ1,π2,⋯,πi,i=1,2,⋯,n. Thus, the output regulation problem of (1) and (2) is converted into a stabilization problem of (6).
The following definition needs to be introduced to solve the stabilization of the system (6).Definition 1 (see [21]).
Letv=0 be the equilibrium point of v˙=fv. It is said to be SGPFS if for any initial condition vt0=v0, there exists a constant δ>0 and a settling time Tδ,v0<+∞ such that(7)vt<δ,∀t>t0+T.
In order to prove SGPFS for the system (6), the following lemmas are used in this paper.Lemma 1 (see [21]).
Consider the nonlinear systemx˙=fx, if there exists a smooth positive define function Vx and some scalars a>0,β∈0,1 and b>0 such that(8)V˙x≤−aVβx+b,t≥0,then, the system x˙=fx is SGPFS.Lemma 2 (see [26]).
Letq∈0,1 be a real number and xi∈R,i=1,2,⋯,n. Then, the following inequality holds(9)∑i=1nxiq≤∑i=1nxiq≤n1−q∑i=1nxiq.Lemma 3 (see [26]).
Letb1,b2, and b3 be positive constants. Then, for any real variables x and y, one has(10)xb1yb2≤b1b3b1+b2xb1+b2+b2b1+b2b3−b1/b2yb1+b2.


### 2.2. Fuzzy Logical Systems and Auxiliary System

In this section, FLSs are introduced to approximate the unknown nonlinear functions in the controlled system (6) and an auxiliary system is constructed.The main part of FLSs is a collection of fuzzy If-then rules, which can be expressed as: ifx1 is F1l and x2 is F2l and ⋯ and xn is Fnl, then, y is Gl,l=1,2,⋯,L, where x=x1,x2,⋯,xnT∈Rn and y∈R are input and output of FLSs, Fil and Gli=1,2,⋯,n are fuzzy sets whose membership functions are μFilxi and μGlyi=1,2,⋯,n, respectively. L is the rule member.Based on the knowledge of fuzzy mathematics, the output of FLSs is presented as follows:(11)y=∑l=1Lyl¯∏i=1nμFilxi∑l=1L∏i=1nμFilxi,where yl¯=maxy∈RμGly.The fuzzy basis functions are defined as follows:(12)φlx=∏i=1nμFilxi∑l=1L∏i=1nμFilxi,we can see obviously that 0≤φT/lxφlx≤1.LetθT=y¯1,y¯2,⋯,y¯L and φx=φ1x,⋯,φLxT, then, the output of FLSs can be rewritten as yx=θTφx.Lemma 4 (see [18]).
Letfx be a continuous function defined on a compact set Ω∈Rn, then for any constant ε>0, there exist an FLS such that(13)supx∈Ωfx−θTφx≤ε.
Furthermore, a lot of knowledge about auxiliary system is given. For the nonlinear system(14)x˙=fx+gxu+dx,where x∈Rn and u∈R are the state vector and the control input, fx and gx are smooth functions, dx is an unknown bounded disturbance vector. It is shown in [32] that the auxiliary system of (14) can be built as(15)x˙=fx+gxu+ℓγ2LVLVTLV2,where ℓγ is defined according to Lemma A1 in [32], Vx is the CLF, LV is Lie derivative.
According to Lemma4, the function FiXi in (6) can be approximated by FLS, so we have(16)FiXi=θi∗TφiXi+δiXi,i=1,2,⋯,n,where θi∗=argminθ^i∈UisupXi∈ΩiFiXi|θ^i−FiXi,δiXii=1,2,⋯,n, δiXi is a bounded error, Ui,Ωi are compact regions for θi∗ and Xi. θ^i is the estimation of θi.
Therefore, system (6) can be converted into(17)x˙i=xi+1+θi∗TφiXi+δiXi,i=1,2,⋯,n−1,x˙n=u−αw+θi∗TφnXn+δnXn,e=x1.
Defineγv=v2/μ,μ≠0. Based on (15), an auxiliary system of (17) can be built as follows:(18)x˙i=xi+1+θi∗TφiXi+μLVi¯T,i=1,2⋯n−1,x˙n=u−αw+θi∗TφnXn+μLVn¯T,e=x1,where Vi¯=Vi−Vi−1,LVi¯=∂Vi−Vi−1/∂zi,i=1,2,⋯,n,Vi and zi will be given later.Definition 2.
(see [32]). The inverse optimal gain assignment problem for system (14) is solvable if there exists a class K∞ function γ whose derivative γ′ is also a K∞ function, and a matrix value function Tx such that Tx=TTx>0 for all x, positive define radially unbounded functions lx and Dx, and a feedback law u=u∗ continuous away from the origin with u∗0=0, which minimizes the cost functional(19)Ju=supd∈Ulimt⟶∞Dx+∫0tlx+uTTxu−γddϖ.Where U is the set of locally bounded functions of x.


### 2.3. Internal Model

In this section, an internal model is constructed such that the exosystem can be immersed in it.Assumption 3.
(see [3]): There exists a set of real numbers a1,…,an−1 such that αw satisfies the equation(20)LSpαw=a0αw+a1LSαw+ap−1LSp−1αw.Where LSαw=∂αw/∂wSw and the characteristic polynomial xp−ap−1xp−1−⋯−a0 has distinct roots on the imaginary axis.
According to Assumption3, we have(21)∂χw∂wSw=ψχw,αw=Γχw,where ψ=01⋯0⋮⋮⋱⋮00⋯1a0a1⋯ap−1,χw=αwLSαw⋮LSp−2αwLSp−1αw,Γ=10⋯0.
Based on the above analysis, (2) can be immersed in the normalized form of an internal model as follows:(22)η˙=M+Nψη,αw=ψη,where η∈Rn×1,N∈Rn×1,M∈Rn×n is a Hurwitz matrix, and M,ψ is controllable.
For the normalized form (22), according to the principle of deterministic equivalence, the error form of the internal model is given as follows:(23)η^˙=M+Nψη^+σ⋅,where σ⋅ is a function which will be designed in the following part.


## 2.1. Problem Description

Consider the following strict feedback nonlinear systems(1)ξ˙i=ξi+1+fiξ1,ξ2,…,ξi+diw,i=1,2,…,n−1,ξ˙n=u+fnξ1,ξ2,…,ξn+dnw,y=ξ1,e=y−qw,where ξ=ξ1,ξ2,⋯,ξnT∈Rn,u, and y are the state vector, the control input, and the output, respectively. fii=1,2,⋯,n is an unknown smooth nonlinear function, diwi=1,2,⋯,n and qw represent the undesired disturbance and reference input, where diw is an unknown smoooth function and qw is a known smooth function. e is the tracking error. w is the state of the exosystem generated by the following system:(2)w˙=Sw.In this paper, we assume that onlyy=ξ1 is measured directly.It is worth mentioning that systems (1) and (2) can be obtained for certain practical problems. For example, a dynamic model is established to solve the attitude tracking and disturbance rejection problem of rigid spacecraft, and the model can be disturbed by external disturbance torque, this problem is a special example of an output regulation problem [4].The main objective of the output regulation problem with the strict feedback nonlinear system (1) and linear exosystem (2) is to find an adaptive fuzzy finite-time inverse optimal controller, which depends on the error output and the state estimation vector so that for any x0∈Ω⊆Rn,w0∈W⊆Rm, where Ω and W are any known tight sets containing the origin in Rn and Rm, all the signals of the closed-loop system are SGPFS, and this controller can achieve inverse optimization in relation to the following cost functional.To study the output regulation problem, the following assumptions are introduced.Assumption 1.
(see [3]): The matrix S has distinct eigenvalues on the imaginary axis.
Assumption1 means that w is bounded and persistent, and the exosystem is neutrally stable. This is a common assumption for the output regulation problem.Assumption 2.
(see [3]): For nonlinear systems like(3)ξ˙=fξ,w+gξ,wu,e=hξ,w,w=S˜w,there exists a continuously differentiable mapping ξ=πw, with π0=0,∀w∈W, and a continuous mapping αw that solve the equations(4)∂πw∂wS˜w=fπ,w+gπ,wαw,0=hπ,w.
Based on Assumption2, by setting π1w=qw, we have(5)π1w=qw,πi+1w=π˙iw−fiπ1,π2,⋯,πi−diw,i=2,⋯,n−1,αw=π˙nw−fnπ1,π2,⋯,πn−dnw.
Define a state transformation asxi=ξi−πiwi=1,2,⋯,n. Then, we get the following systems:(6)x˙i=xi+1+FiXi,i=1,2,⋯,n−1,x˙n=u−αw+FnXn,e=x1,where Xi=x1,x2,⋯,xi,FiXi=fiξ1,ξ2,⋯,ξi−fiπ1,π2,⋯,πi,i=1,2,⋯,n. Thus, the output regulation problem of (1) and (2) is converted into a stabilization problem of (6).
The following definition needs to be introduced to solve the stabilization of the system (6).Definition 1 (see [21]).
Letv=0 be the equilibrium point of v˙=fv. It is said to be SGPFS if for any initial condition vt0=v0, there exists a constant δ>0 and a settling time Tδ,v0<+∞ such that(7)vt<δ,∀t>t0+T.
In order to prove SGPFS for the system (6), the following lemmas are used in this paper.Lemma 1 (see [21]).
Consider the nonlinear systemx˙=fx, if there exists a smooth positive define function Vx and some scalars a>0,β∈0,1 and b>0 such that(8)V˙x≤−aVβx+b,t≥0,then, the system x˙=fx is SGPFS.Lemma 2 (see [26]).
Letq∈0,1 be a real number and xi∈R,i=1,2,⋯,n. Then, the following inequality holds(9)∑i=1nxiq≤∑i=1nxiq≤n1−q∑i=1nxiq.Lemma 3 (see [26]).
Letb1,b2, and b3 be positive constants. Then, for any real variables x and y, one has(10)xb1yb2≤b1b3b1+b2xb1+b2+b2b1+b2b3−b1/b2yb1+b2.


## 2.2. Fuzzy Logical Systems and Auxiliary System

In this section, FLSs are introduced to approximate the unknown nonlinear functions in the controlled system (6) and an auxiliary system is constructed.The main part of FLSs is a collection of fuzzy If-then rules, which can be expressed as: ifx1 is F1l and x2 is F2l and ⋯ and xn is Fnl, then, y is Gl,l=1,2,⋯,L, where x=x1,x2,⋯,xnT∈Rn and y∈R are input and output of FLSs, Fil and Gli=1,2,⋯,n are fuzzy sets whose membership functions are μFilxi and μGlyi=1,2,⋯,n, respectively. L is the rule member.Based on the knowledge of fuzzy mathematics, the output of FLSs is presented as follows:(11)y=∑l=1Lyl¯∏i=1nμFilxi∑l=1L∏i=1nμFilxi,where yl¯=maxy∈RμGly.The fuzzy basis functions are defined as follows:(12)φlx=∏i=1nμFilxi∑l=1L∏i=1nμFilxi,we can see obviously that 0≤φT/lxφlx≤1.LetθT=y¯1,y¯2,⋯,y¯L and φx=φ1x,⋯,φLxT, then, the output of FLSs can be rewritten as yx=θTφx.Lemma 4 (see [18]).
Letfx be a continuous function defined on a compact set Ω∈Rn, then for any constant ε>0, there exist an FLS such that(13)supx∈Ωfx−θTφx≤ε.
Furthermore, a lot of knowledge about auxiliary system is given. For the nonlinear system(14)x˙=fx+gxu+dx,where x∈Rn and u∈R are the state vector and the control input, fx and gx are smooth functions, dx is an unknown bounded disturbance vector. It is shown in [32] that the auxiliary system of (14) can be built as(15)x˙=fx+gxu+ℓγ2LVLVTLV2,where ℓγ is defined according to Lemma A1 in [32], Vx is the CLF, LV is Lie derivative.
According to Lemma4, the function FiXi in (6) can be approximated by FLS, so we have(16)FiXi=θi∗TφiXi+δiXi,i=1,2,⋯,n,where θi∗=argminθ^i∈UisupXi∈ΩiFiXi|θ^i−FiXi,δiXii=1,2,⋯,n, δiXi is a bounded error, Ui,Ωi are compact regions for θi∗ and Xi. θ^i is the estimation of θi.
Therefore, system (6) can be converted into(17)x˙i=xi+1+θi∗TφiXi+δiXi,i=1,2,⋯,n−1,x˙n=u−αw+θi∗TφnXn+δnXn,e=x1.
Defineγv=v2/μ,μ≠0. Based on (15), an auxiliary system of (17) can be built as follows:(18)x˙i=xi+1+θi∗TφiXi+μLVi¯T,i=1,2⋯n−1,x˙n=u−αw+θi∗TφnXn+μLVn¯T,e=x1,where Vi¯=Vi−Vi−1,LVi¯=∂Vi−Vi−1/∂zi,i=1,2,⋯,n,Vi and zi will be given later.Definition 2.
(see [32]). The inverse optimal gain assignment problem for system (14) is solvable if there exists a class K∞ function γ whose derivative γ′ is also a K∞ function, and a matrix value function Tx such that Tx=TTx>0 for all x, positive define radially unbounded functions lx and Dx, and a feedback law u=u∗ continuous away from the origin with u∗0=0, which minimizes the cost functional(19)Ju=supd∈Ulimt⟶∞Dx+∫0tlx+uTTxu−γddϖ.Where U is the set of locally bounded functions of x.


## 2.3. Internal Model

In this section, an internal model is constructed such that the exosystem can be immersed in it.Assumption 3.
(see [3]): There exists a set of real numbers a1,…,an−1 such that αw satisfies the equation(20)LSpαw=a0αw+a1LSαw+ap−1LSp−1αw.Where LSαw=∂αw/∂wSw and the characteristic polynomial xp−ap−1xp−1−⋯−a0 has distinct roots on the imaginary axis.
According to Assumption3, we have(21)∂χw∂wSw=ψχw,αw=Γχw,where ψ=01⋯0⋮⋮⋱⋮00⋯1a0a1⋯ap−1,χw=αwLSαw⋮LSp−2αwLSp−1αw,Γ=10⋯0.
Based on the above analysis, (2) can be immersed in the normalized form of an internal model as follows:(22)η˙=M+Nψη,αw=ψη,where η∈Rn×1,N∈Rn×1,M∈Rn×n is a Hurwitz matrix, and M,ψ is controllable.
For the normalized form (22), according to the principle of deterministic equivalence, the error form of the internal model is given as follows:(23)η^˙=M+Nψη^+σ⋅,where σ⋅ is a function which will be designed in the following part.


## 3. Design of Adaptive Fuzzy Finite-time Inverse Optimal Controller

In this section, a new controller is designed by combining adaptive control technology, finite-time stability theory, and an inverse optimal control method such that all the signals in the closed-loop system are SGPFS. A state observer is first designed to estimate the unmeasured states based on the auxiliary system (18).The fuzzy state observer can be designed as follows:(24)x^˙i=x^i+1+θ^iTφiX^i+μLV¯iT+kie−x^1,i=1,2,⋯,n−1,x^˙n=u−αw+θ^nTφnX^n+μLV¯nT+kne−x^1,where X^i=x^1,x^2,⋯,x^i.Rewriting (18) and (24) as follows:(25)x˙=Ax+Ke+∑i=1nBiθi∗TφiXi+μLVi¯T+Bu−αw,x^˙=Ax^+Ke+∑i=1nBiθ^iTφiX^i+μLVi¯T+Bu−αw,where A=−k11⋯0⋮⋮⋱⋮−kn−10⋯1−kn0⋯0,x=x1,⋯,xnT,x^=x^1,x^2⋯,x^nT,Bi=0,⋯,0,1,0,⋯,0T, K=k1,k2,⋯,knT,B=0,⋯,0,1T, the parameter ki is chosen such that the matrix A is Hurwitz, so there will exist positive definite matrices P=PT>0,Q=QT>0, such that(26)ATP+PA=−2Q.Define the observer error ase˜=x−x^. According to (25), we get(27)e˜˙=Ae˜+θ˜Tφ⋅+∑i=1nBiθi∗TφiXi−φiX^i,where θ˜Tφ⋅=θ˜1φ1X^1,θ˜2φ2X^2,⋯,θ˜nφnX^nT,θ˜i=θi∗−θ^i,i=1,2,⋯,n.In the following section, an adaptive fuzzy finite-time inverse optimal output feedback controller is designed.Define the coordinate transformations as follows:(28)z1=x1,zi=x^i−αi−1x^1,x^2,⋯,x^i−1,θ^1,⋯,θ^i−1,e,i=2,⋯,n,where αi is a virtual controller. The controller u will be designed in the last step, and the detailed process is as follows.Step 1: DesignV1=V0+Vη+1/2z12+1/2κ1θ˜1Tθ˜1, where V0=1/2e˜TPe˜,Vη=1/2η˜TRη˜,η˜=η^−η−Ne,κ1 is a positive constant to be designed, R=RT>0 is a positive definite matrix satisfying(29)MTR+RM=−2I.From (26) and (27), we obtain(30)V˙0=−e˜TQe˜+e˜TPθ˜Tφ⋅+e˜TP∑i=1nBiθi∗TφiXi−φiX^i.Here, by employing Young’s inequality, the last two terms of (30) are given as follows:(31)e˜TPθ˜Tφ⋅≤12e˜Te˜+12P2∑k=1nθ˜kTθ˜k,e˜TP∑i=1nBiθi∗TφiXi−φiX^i≤ne˜Te˜+P2∑i=1nθi∗2.Due to (30) and (31), the derivative of V0 is(32)V˙0≤−Π0e˜Te˜+12P2∑k=1nθ˜kTθ˜k+D0,whereΠ0=λminQ−1/2−n,D0=P2∑i=1nθi∗2.Based on the definition ofη˜, we get(33)η˜˙=M+Nψη^−M+Nψη+σ⋅−Ne˜2+x^2+θ1∗Tφ1x1+μz1.Designσ⋅=−M+NψNe+Nx^2+μz1. From (23) and (33), we obtain(34)V˙η=−η˜Tη˜+12η˜TNψTRη˜+12η˜TRNψη˜−η˜TRNe˜2+θ1∗Tφ1X1.By using Young’s inequality, the following inequalities are given:(35)−η˜TRNe˜2≤12η˜Tη˜+12RN2e˜2Te˜2,−η˜TRNθ1∗Tφ1x1≤12η˜Tη˜+12RN2θ1∗2.Consequently, the derivative ofVη can be further expressed as follows:(36)V˙η≤−Λη˜Tη˜+12RN2e˜Te˜+12RN2θ1∗2,whereΛ=−λmaxRNψ.According to (18) and (28), we obtain(37)z˙1=e˜2+z2+α1+θ1∗Tφ1X1−φ1X^1+θ^1Tφ1X^1+μz1+θ˜1Tφ1X^1,then, the derivative ofV1 is(38)V˙1=V˙0+V˙η+z1e˜2+z2+α1+θ1∗Tφ1X1−φ1X^1+θ^1Tφ1X^1+μz1+1κ1θ˜1Tκ1z1φ1X^1−θ^˙1.The following inequalities are obtained by using Young’s inequality(39)z1e˜2≤12z12+12e˜2Te˜2,z1θ1∗Tφ1X1−φ1X^1≤z12+θ1∗2.Design the virtual controlα1 and adaptive law θ^1 as(40)α1=−c1z12β−1−32+μz1−θ^1Tφ1X^1,θ^˙1=κ1z1φ1X^1−r1θ^1,wherec1 and r1 are positive parameters to be designed.Substituting (32), (36), (39), and (40) into (38) yields(41)V˙1≤−c1z12β+z1z2+r1κ1θ˜1Tθ^1−Π1e˜Te˜−Λη˜Tη˜+12P2∑k=1nθ˜kTθ˜k+D1,whereD1=D0+1+1/2RN2θ1∗2,Π1=Π0−1/2−1/2RN2.Stepii=2,3,⋯,n−1: Similar to Step 1, define Vi as(42)Vi=Vi−1+12zi2+12κiθ˜iTθ˜i,whereκi is a positive constant to be designed.From (24) and (28), we get(43)z˙i=zi+1+αi+Hi+kie˜1−∂αi−1∂ee˜2−∂αi−1∂eθ1∗Tφ1X1,whereHi=μzi+θ^iTφiX^i−∑k=1i−1∂αi−1/∂x^kx^˙k−∑k=1i−1∂αi−1/∂θ^kθ^˙k−∂αi−1/∂ex^2+μz1, then, V˙i is(44)V˙i=V˙i−1+zizi+1+kie˜1+αi+Hi−∂αi−1∂ee˜2−∂αi−1∂eθ1∗Tφ1X1+1κiθ˜iTκiziφiX^i−θ^˙i−θ˜iTziφiX^i.We have the following inequalities by using Young’s inequality:(45)−θ˜iTziφiX^i≤12zi2+12θ˜iTθ˜i,zikie˜1≤12zi2+12ki2e˜1Te˜1,−zi∂αi−1∂eθ1∗Tφ1X1≤12∂αi−1∂e2zi2+12θ1∗2,−zi∂αi−1∂ee˜2≤12∂αi−1∂e2zi2+12e˜2Te˜2.Substituting (45) into (44), we obtain(46)V˙i≤−∑k=1i−1ckzk2β+zizi+1+Hi+zi−1+αi+zi+∂αi−1∂e2zi−Πie˜Te˜−Λη˜Tη˜+12∑k=2iθ˜kTθ˜k+12P2∑k=1nθ˜kTθ˜k+∑k=1i−1rkκkθ˜kTθ^k+Di+1κiθi˜TκiziφiX^i−θ^˙i,whereΠi=Π1−i−1/2+∑k=2i1/2kk2,Di=D1+i−1/2θ1∗2.Design the virtual controlαi and adaptive law θ^i as(47)αi=−cizi2β−1−zi−1−Hi−zi−∂αi−1∂e2zi,θ^˙i=κiziφiX^i−riθ^i,whereci and ri are positive constants to be designed.Substituting (47) into (46), we have(48)Vi.≤−∑k=1ickzk2β+zizi+1−Πie˜Te˜−Λη˜Tη˜+12∑k=2iθ˜kTθ˜k+12P2∑k=1nθ˜kTθ˜k+∑k=1irkκkθ˜kTθ^k+Di.Step n: Choose CLFVn as(49)Vn=V,=Vn−1+12zn2+12κnθ˜nTθ˜n,where κn is a positive constant to be designed.From (25) and (28), we have(50)z˙n=u−αw+θ^nTφnXn+μzn+kne˜1−∑k=1n−1∂αn−1∂x^kx^˙k−∑k=1n−1∂αn−1∂θ^kθ^˙k−∂αn−1∂ee˜2−∂αn−1∂ex^2+μz1−∂αn−1∂eθ1∗Tφ1X1,the derivative of Vn is(51)V˙n=V˙n−1+znu−αw+θ^nTφnXn+μzn+kne˜1−∑k=1n−1∂αn−1∂x^kx^˙k−∑k=1n−1∂αn−1∂θ^kθ^˙k−∂αn−1∂ee˜2−∂αn−1∂ex^2+μz1−∂αn−1∂eθ1∗Tφ1X1+1κnθ˜nTκnznφnX^n−θ^˙n−θ˜nTznφnX^n.From Young’s inequality, we get(52)−θ˜nTznφnX^n≤12zn2+12θ˜nTθ˜n,znθ^nTφnXn≤zn2+12θn∗2+12θ˜nTθ˜n,znkne˜1≤12zn2+12kn2e˜1Te˜1,−zn∂αn−1∂eθ1∗Tφ1X1≤12∂αn−1∂e2zn2+12θ1∗2,−zn∂αn−1∂ee˜2≤12∂αn−1∂e2zn2+12e˜2Te˜2.Based on (28), we can conclude that each αi will vanish when z=0,x=0. On the basis of the mean value theorem, there exists a smooth function ςk such that(53)∑k=1n−1∂αn−1∂x^kx^˙k−∑k=1n−1∂αn−1∂θ^kθ^˙k−∂αn−1∂ex^2+μz1=∑k=1nςkzkβ.Taking adaptive lawθ^n as(54)θ^˙n=κnznφnX^n−rnθ^n,where rn is a positive constant to be designed.From (52), (53), (54), and (51) can be written as(55)Vn.≤−∑k=1n−1ckzk2β+znu−αw+μzn+2zn+∑k=1nϕkzkβ+∂αn−1∂e2zn−Πne˜Te˜−Λη˜Tη˜+∑k=1nmkθ˜kTθ˜k+∑k=1nrkκkθ˜kTθ^k+Dn,where(56)Dn=D1+n−12θ1∗2+12θn∗2,Πn=Π1−n−12−∑k=2n12kk2>0,m1=12P2,mi=12P2+12,i=2,⋯,n−1,mn=12P2+1,ϕk=ςk,k=1,2,⋯,n−2,ϕn−1=ςn−1+zn−11−β,ϕn=ςn.Design the controller as(57)u=ψη^−ψNe−cnzn2β−2+μ+52+∂αn−1∂e2+∑k=1nϕk22ckzn=ψη^−ψNe−Ψx^zn,where cn is designed positive constant, Ψx^=cnzn2β−2+μ+5/2+∂αn−1/∂e2+∑k=1nϕk2/2ck.Substituting (57) into (55) yields(58)Vn.≤−12∑k=1nckzk2β+znψη˜−12zn2−Πne˜Te˜−12∑k=1nckzkβ−ϕkckzn2−Λη˜Tη˜+∑k=1nmkθ˜kTθ˜k+∑k=1nrkκkθ˜kTθ^k+Dn.By employing Young’s inequality, the following inequalities are obtained:(59)znψη˜≤12zn2+12ψ2η˜Tη˜,rkκkθ˜kTθ^k≤rk2κkθk∗2−rk2κkθ˜kTθ˜k,then, we have(60)Vn.≤−12∑k=1nckzk2β−Πne˜Te˜−Υη˜Tη˜−∑k=1nσkθ˜kTθ˜k+D,where D=Dn+∑k=1nrk/2κkθk∗2,Υ=Λ−1/2ψ2>0,σk=rk/2κk−mk>0.


## 4. Finite-Time Stability and Performance Analysis

In this part, the feasibility of the controller is proved, including the finite-time stability of the closed-loop system and the minimization of the cost functional.Theorem 1.
For the auxiliary system (18), if Assumptions 1–3 hold, there exists an internal model (23), a fuzzy state observer (24), a virtual control, an adaptive law (47), and an adaptive fuzzy finite-time inverse optimal control input (57) such that all the signals in the closed-loop system (6) are SGPFS. Moreover, the control input u∗=−ϱu can minimize the following cost functional:(61)Ju=supδkXk∈Ulimt⟶∞2ϱVx+∫0tlx+uTTx^u−∑k=1n4γϱδkXk4dϖ,where ϱ≥2,Tx^=ϱ+12ΨTx^Ψx^, and(62)lx=−2ϱ−∑k=1n∂V∂θ˜kθ^˙k+∂V∂e˜e˜˙+∂V∂η˜η˜˙+LFV+∑k=1nℓγ2LV¯k+LBVu−αw+∂V∂wSw+2ϱ−2∑k=1nℓγ2LV¯k.Proof of Theorem 1.
Based on the analysis ofVn, for inequality (60), let(63)a=min2−1+βck,2κkσk,2Υ|λmaxR,2Πn|λmaxP,k=1,2,⋯,n,we have(64)V˙≤−a∑k=1n12zk2β−a12e˜TPe˜−a∑k=1n12κkθ˜kTθ˜k−a12η˜TRη˜−a∑k=1n12κkθ˜kTθ˜kβ+a∑k=1n12κkθ˜kTθ˜kβ−a12η˜TRη˜β+a12η˜TRη˜β−a12e˜TPe˜β+a12e˜TPe˜β+D.
According to Lemma3, we obtain(65)a∑k=1n12κkθ˜kTθ˜kβ≤a∑k=1n12κkθ˜kTθ˜k+a1−βββ/1−β,a12η˜TRη˜β≤a12η˜TRη˜+a1−βββ/1−β,a12e˜TPe˜β≤a12e˜TPe˜+a1−βββ/1−β.
Substituting (65) into (64), and employing Lemma 2, the following inequality holds:(66)V˙≤−a∑k=1n12zk2β−a∑k=1n12κkθ˜kTθ˜kβ−a12η˜TRη˜β−a12e˜TPe˜β+b≤−aVβ+b,where b=3a1−βββ/1−β+D.
From (8), it can be seen that V˙≤−aαVβ when Vβ≤b/1−αa, where 0<α<1. Since V is continuous, we integrate V˙≤−aαVβ and obtain(67)T=11−βaαV1−βx0−b1−αa1−ββ.
Up to now, the proof of SGPFS is completely based on Lemma1 and (66), and the time boundary is given as (67).
Next, it is proved that the controller can also minimize the cost functional. Rewriting the system (18) as the following form:(68)x˙=Fx,θ+δx+Bu−αw,where Fx,θ=x2+θ1∗Tφ1X1,⋯,θn∗TφnXnT,δx=δ1X1,⋯,δnXnT,B=⋅0,0,⋯,1T.
The auxiliary system of (68) is designed as follows:(69)x˙=Fx,θ+ℓγ2LV¯LV¯TLV¯2+Bu−αw,where LV¯=LV¯1,LV¯2,⋯,LV¯nT.
Since system (6) is stable, based on the Lasalle–Yoshizawa invariant set theorem, we know that there exists a continuous positive function Wx^, such that(70)LFV+∑k=1nℓγ2LV¯k+LBVu−αw−∑k=1n∂V∂θ˜kθ^˙k+∂V∂e˜e˜˙+∂V∂η˜η˜˙+∂V∂wSw≤−Wx^.
Based on (70), lx satisfies(71)lx≥2ϱWx^+2ϱ−2∑k=1nℓγ2LV¯k,since ϱ≥2,Wx^>0,ℓγ is K∞ function, then lx>0. Therefore, the proposed cost functional Ju is meaningful.
Taking (62) into (61), Ju can be converted into(72)Ju=supδkXk∈Ulimt⟶∞2ϱVx−2ϱ∫0tdV+∫0tu−u∗TTx^u−u∗dϖ+∫0tϱ∑k=1nΔδkXk,δk∗Xkdϖ,where ΔδkXk,δk∗Xk=−4/ϱγϱδkXk/4+γ′ϱδk∗Xk/4δk∗XkT/δk∗XkδkXk−4/ϱℓγ(73)2LV¯k,δk∗Xk=4ϱγ′−12LVk¯LVk¯TLVk¯.
According to Lemma A1 and Lemma A2 in [32], we obtain(74)ΔδkXk,δk∗Xk≤−4ϱγϱδkXk4−4ϱℓγγ′ϱδk∗Xk4+4ϱγϱδkXk4+4ϱℓγγ′ϱδk∗Xk4=0,where ΔδkXk,δk∗Xk=0 if and only if ϱδkXk/4=γ′−1γ′ϱδk∗Xk/4δk∗XkT/δk∗Xk, that is δkXk=δk∗Xk. Then, we have(75)supδkXk∈Ulimt⟶∞∫0tϱ∑k=1nΔδkXk,δk∗Xkdϖ=0.
Thus, we can conclude that the inverse optimal controlleru=u∗ can minimize the cost functional Ju, and the minimum value of Ju is(76)Jminu=2ϱV0.
According to the above process, the proof of the theorem is completed.
In order to minimize the cost functional, and based onϱ≥2, the parameter ϱ can be chosen as ϱ=2. Then, the cost functional reaches the minimum value Ju=4V0.


## 5. Simulation

In this section, a simulation example is shown to verify the effectiveness of the controller.Example 1.
Consider the following strict feedback nonlinear systems(77)ξ1.=ξ2+2w2,ξ2.=u−w2−ξ2+ξ1−2ξ1ξ2,y=ξ1,e=y−w1,where ξ=ξ1,ξ2T represents the state vector, u and y are the control input and the output, respectively, w=w1,w2T is generated by the following exosystem:(78)w˙1=w2,w˙2=−w1.
Based on (3), the solutions of regulator equation are given as π10=w1,π2w=−w2, let xi=ξi−πi,i=1,2, a closed-loop system is generated(79)x˙1=x2,x˙2=u−αw+fx1,x2,e=x1,where αw=−2w1w2,fx1,x2=−2x1x2−x2+x1+2x1w2−2x2w1.
Choosing the fuzzy membership as follows:(80)μF1X^2=11+exp5x^1+0.711+exp5x^2+0.7,μF2X^2=exp−x^1+0.42exp−x^2+0.42,μF3X^2=exp−x^1+0.32exp−x^2+0.32,μF4X^2=exp−x^12exp−x^22,μF5X^2=exp−x^1−0.32exp−x^2−0.32,μF6X^2=exp−x^1−0.42exp−x^2−0.42,μF7X^2=11+exp−5x^1+0.711+exp−5x^2+0.7,where X^2=x^1,x^2. From (12), the fuzzy basis function is defined as follows:(81)φ2,lX^2=μFlx^1,x^2∑l=17μFlx^1,x^2,l=1,2,…,7.
The state observer of (79) is(82)x^˙1=x^2+k1e˜1+μz1,x^˙2=u+2w1w2+θ^2Tφ2X^2+k2e˜1+μz2,where φ2X^2=φ2,1X^2,⋯,φ2,7X^2T,θ^2=θ^21,θ^22,⋯,θ^27T.
According to (21) and (22), we choose ψ=13,M=01−2−3,N=01. Then, the internal model can be given as follows:(83)η^˙1=η^2−e,η^˙2=−η^1+x^2+μe.
The optimal controller and adaptive law for system (77) and (78) respond to(84)u=ψη^−ψNe−c2z22β−2+μ+32+2β−1c1z12β−2−522+ϕ122c1+ϕ222c2z2,θ^˙2=κ2z2φ2X^2−r2θ^2,where ϕ1=2β−1c1z1β−2x^2+2β−3/2c1z1β−1+19/4z11−β,ϕ2=−5/2z21−β.
The parameters of fuzzy state observer, internal model, virtual controls, adaptive laws, and control input are chosen ask1=3,k2=24,β=97/101,c1=2,c2=1,κ1=2,κ2=2,r1=1,r2=1,μ=1,ϱ=2, the initial values are chosen as ξ10=0,ξ20=0.3,w10=0.9,(85)w20=0.5,η^10=0,η^20=0,θ^20=0,0,0,0,0,0,0T,x^10=0,x^20=0.
According to Figures1–4, we conclude that the designed controller in this paper can make all the signals of the closed-loop system be SGPFS. The output of (77) can track the reference signal, as shown in Figure 3. The state observer (82) can estimate the state of the system (77) in Figure 4. Compared with [18], the output in this paper can track the reference signal faster, and the tracking error is smaller, as they are shown in Figures 1 and 2.Figure 1
The tracking error of this paper.Figure 2
The tracking error with the method of [18].Figure 3
The output and reference signal of this paper.Figure 4
The statee and it’s estimation x^1.


## 6. Conclusion

In this paper, the problem of finite-time inverse optimal output regulation is addressed for a class of nonlinear systems that are driven by a linear exosystem. The considered nonlinear systems contain unknown functions and unmeasured states. By state transformation, a closed-loop system is obtained, then an auxiliary system is designed. On the basis of the auxiliary system, a state observer and a novel adaptive fuzzy finite-time inverse optimal output feedback controller are designed by employing adaptive control technology, finite-time stability theory, and an inverse optimal control method. We also prove that the new controller makes all the signals of the closed-loop system SGPFS, and the newly well-defined cost functional can be minimized. From the simulation results, the feasibility of the newly raised controller and state observer can be seen clearly. Compared with the results in [18], the output can track the reference signal faster and the tracking error is smaller. Inspired by [5–7], we will consider combining the proposed method with mobile robots and quadrotors control in future work.



---

*Source: 5828376-2022-10-10.xml*