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

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.


Introduction
During the past several years, the output regulation problem has been widely studied. e target of the output regulation problem is to design a controller when the system is disturbed by external disturbances. e 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][5][6][7] are uncertain. So, it is signi cant 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 nd 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 e ective way to deal with this kind of problem [13][14][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 nite-time control to achieve transient performance and rapid response [20], but the abovedesigned controllers are unsuitable for nite-time control problems.
Finite-time control aims to achieve the performance of the system within a finite time.
is 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][22][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][25][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]. e 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. e 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. e 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. e 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. us, the internal model must be considered in the cost functional, and the existing cost functionals [30][31][32][33][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) e 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][32][33][34][35], the functions l(x) and c are designed differently. e rest of this paper is organized as follows. Section 2 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 finitetime 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.

Problem Formulation and Preliminary Knowledge
where ξ � [ξ 1 , ξ 2 , · · · , ξ n ] T ∈ R n , u, and y are the state vector, the control input, and the output, respectively. f i (i � 1, 2, · · · , n) is an unknown smooth nonlinear function, d i (w)(i � 1, 2, · · · , n) and q(w) represent the undesired disturbance and reference input, where d i (w) is an unknown smoooth function and q(w) is a known smooth function. e is the tracking error. w is the state of the exosystem generated by the following system: In this paper, we assume that only y � ξ 1 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]. e 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 x(0) ∈ Ω⊆R n , w(0) ∈ W⊆R m , where Ω and W are any known tight sets containing the origin in R n and R m , 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]): e matrix S has distinct eigenvalues on the imaginary axis.
Assumption 1 means that w is bounded and persistent, and the exosystem is neutrally stable.
is is a common assumption for the output regulation problem.
Assumption 2 (see [3]): For nonlinear systems like there exists a continuously differentiable mapping ξ � π(w), with π(0) � 0,∀w ∈ W, and a continuous mapping α(w) that solve the equations Based on Assumption 2, by setting π 1 (w) � q(w), we have n). en, we get the following systems: where us, the output regulation problem of (1) and (2) is converted into a stabilization problem of (6). e following definition needs to be introduced to solve the stabilization of the system (6).
In order to prove SGPFS for the system (6), the following lemmas are used in this paper.

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. e main part of FLSs is a collection of fuzzy If-then rules, which can be expressed as: if x 1 is F l 1 and x 2 is F l 2 and · · · and x n is F l n , then, y is G l , l � 1, 2, · · · , L, where x � [x 1 , x 2 , · · · , x n ] T ∈ R n and y ∈ R are input and output of FLSs, F l i and G l (i � 1, 2, · · · , n) are fuzzy sets whose membership functions are μ F l i (x i ) and μ G l (y)(i � 1, 2, · · · , n), respectively. L is the rule member.

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Based on the knowledge of fuzzy mathematics, the output of FLSs is presented as follows: where y l � max y∈R μ G l (y). e fuzzy basis functions are defined as follows: we can see obviously that 0 ≤ φT/l(x)φ l (x) ≤ 1.
Let θ T � [y 1 , y 2 , · · · , y L ] and φ(x) � [φ 1 (x), · · · , φ L (x)] T , then, the output of FLSs can be rewritten as Lemma 4 (see [18]). Let f(x) be a continuous function defined on a compact set Ω ∈ R n , then for any constant ε > 0, there exist an FLS such that Furthermore, a lot of knowledge about auxiliary system is given. For the nonlinear system where x ∈ R n and u ∈ R are the state vector and the control input, f(x) and g(x) are smooth functions, d(x) is an unknown bounded disturbance vector. It is shown in [32] that the auxiliary system of (14) can be built as where ℓc is defined according to Lemma A1 in [32], V(x) is the CLF, LV is Lie derivative. According to Lemma 4, the function F i (X i ) in (6) can be approximated by FLS, so we have where θ * i � arg min i and X i . θ i is the estimation of θ i . erefore, system (6) can be converted into Define c(v) � v 2 /μ, μ ≠ 0. Based on (15), an auxiliary system of (17) can be built as follows: , · · · , n, V i and z i will be given later.
Definition 2 (see [32]). e inverse optimal gain assignment problem for system (14) is solvable if there exists a class K ∞ function c whose derivative c ′ is also a K ∞ function, and a matrix value function T(x) such that T(x) � T T (x) > 0 for all x, positive define radially unbounded functions l(x) and D(x), and a feedback law u � u * continuous away from the origin with u * (0) � 0, which minimizes the cost functional Where U is the set of locally bounded functions of x.

Internal Model. In this section, an internal model is constructed such that the exosystem can be immersed in it.
Assumption 3 (see [3]): ere exists a set of real numbers a 1 , . . . , a n− 1 such that α(w) satisfies the equation Where L S α(w) � (zα(w)/zw)Sw and the characteristic polynomial x p − a p− 1 x p− 1 − · · · − a 0 has distinct roots on the imaginary axis. According to Assumption 3, we have where ψ � Based on the above analysis, (2) can be immersed in the normalized form of an internal model as follows: where η ∈ R n×1 , N ∈ R n×1 , M ∈ R n×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: where σ(·) is a function which will be designed in the following part.

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). e fuzzy state observer can be designed as follows: where (X i ) � (x 1 , x 2 , · · · , x i ).
Rewriting (18) and (24) as follows: where · · · , 0, 1] T , the parameter k i is chosen such that the matrix A is Hurwitz, so there will exist positive definite matrices P � P T > 0, Q � Q T > 0, such that Define the observer error as e � x − x. According to (25), we get In the following section, an adaptive fuzzy finite-time inverse optimal output feedback controller is designed.
Define the coordinate transformations as follows: where α i is a virtual controller. e controller u will be designed in the last step, and the detailed process is as follows.
Step 1: is a positive constant to be designed, R � R T > 0 is a positive definite matrix satisfying From (26) and (27), we obtain Here, by employing Young's inequality, the last two terms of (30) are given as follows: Due to (30) and (31), the derivative of V 0 is Design σ(·) � − (M + Nψ)Ne + N(x 2 + μz 1 ). From (23) and (33), we obtain By using Young's inequality, the following inequalities are given: Consequently, the derivative of V η can be further expressed as follows: where Λ � − λ max (RNψ). According to (18) and (28), we obtain then, the derivative of V 1 is Mathematical Problems in Engineering e following inequalities are obtained by using Young's inequality Design the virtual control α 1 and adaptive law θ 1 as where c 1 and r 1 are positive parameters to be designed. Substituting (32), (36), (39), and (40) into (38) yields where Step i(i � 2, 3, · · · , n − 1): Similar to Step 1, define V i as where κ i is a positive constant to be designed. From (24) and (28), we get where We have the following inequalities by using Young's inequality: Substituting (45) into (44), we obtain where Design the virtual control α i and adaptive law θ i as where c i and r i are positive constants to be designed. Substituting (47) into (46), we have Step n: Choose CLF V n as V n � V,

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where κ n is a positive constant to be designed.

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In order to minimize the cost functional, and based on ϱ ≥ 2, the parameter ϱ can be chosen as ϱ � 2. en, the cost functional reaches the minimum value J(u) � 4V(0). □

(85)
According to Figures 1-4, we conclude that the designed controller in this paper can make all the signals of the closedloop system be SGPFS. e output of (77) can track the reference signal, as shown in Figure 3. e 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.

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. e 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][6][7], we will consider combining the proposed method with mobile robots and quadrotors control in future work.

Data Availability
All data used to support the findings of the study are included within the article.

Conflicts of Interest
e authors declare that there is no conflict of interest.