Adaptive asymptotic stabilization of uncertain nonstrict feedback nonlinear HOFA systems with time delays

We consider the adaptive control for the nonstrict-feedback uncertain nonlinear high-order fully actuated (HOFA) systems with either uncertain constant delay or time-varying delay. Different from the previous works, the most difficult point is that the uncertain nonlinear function in the HOFA systems contains nonstrict feedback form and time delay in system states. In order to overcome this difficulty, the arbitrary approximation property of the radial basis function neural network (RBF NN) is used to estimate the unknown nonlinear functions. What’s more, the nonstrict-feedback form is transformed into strict-feedback strcture by using the property of Gaussian function. In addition, the appropriate Lyapunov-Krasovskii functions and separation techniques are used to compensate the effect of the unknown constant time delay or time-varying delay. Based on the HOFA system approaches, two adaptive NN controllers are proposed to ensure the system with constant time-delay and time-varying delay to be stabilized asymptotically, respectively. Finally, the proposed control strategies are applied to the thermoacoustic instability systems to demonstrate the feasibility and effectiveness of the obtained results.


Introduction
Most physical systems can be described by secondorder or high-order differential equations. However, in the long-term development of control system theory, most of the second-order or high-order systems are transformed into first-order systems and processed in the framework of first-order systems. After more than half a century of research, the first-order state space methods have achieved more attention and a lot of fruitful results. The convenience required by the control design is not present in the state space representation, despite the fact that it is useful for generating the state response solution and observation. In contrast, the fully actuated system can easily give the controller to ensure system stability through its structural characteristics. The primary obstacle is that fully actuated systems are only a subset of the total number of physical systems that make up control systems. In this regard, Duan introduced the idea of HOFA systems as well as a recursive method to transform the generalised second-order and high-order systems with strict-feedback form into the HOFA systems under certain conditions [1]. Based on the HOFA system model, it is simple to design a controller that will eliminate any measurable nonlinear term and convert the system into a linear system with the required characteristic structure [2]. In [3], the attitude control model of a single input spherical liquid-filled spacecraft is transformed into a HOFA system. Then, the control law can be designed immediately to ensure that each Euler angle is on a given target. In [4], combined with the backstepping scheme, the controller is designed under the framework of the HOFA system, which can ensure the closed-loop system to achieve stability and a certain degree of output disturbance attenuation. In fact, most practical systems will be affected by uncertain nonlinearities. Therefore, robust and adaptive controllers are proposed for uncertain nonlinear HOFA systems [5,6].
In addition, time delay is one of the important factors that affects control performance and even leads to system instability in practical engineering, such as chemical processes [7], fluid dynamics [8] and communication networks [9], etc.. Therefore, time-delay systems have been widely studied and achieved remarkable results [10][11][12][13][14]. In [10], for the nonlinear time-delay system with lower triangle structure, the influence of time-delay can be effectively dealt with by changing the supply rate of the iterative algorithm. Recently, a dynamic feedback control scheme using dynamic gain to counteract the effect of time-delay nonlinearity is proposed in [14]. It cannot be ignored that the system with time delay can also be transformed from the state space description to the HOFA system. At this point, the controller of HOFA timedelay system is given, so that the linear system with ideal characteristic structure is obtained [15]. Subsequently, for the strict feedback nonlinear time-delay system, an adaptive controller based on the HOFA system method is proposed to ensure the system asymptotically stable [16].
Unlike strict feedback systems, nonstrict feedback systems contain the entire state variable, so the current state cannot be used as an approximator to directly compensate. In order to overcome this difficulty caused by the nonstrict feedback form, many related results are given in [17][18][19][20][21]. Among them, the unknown function with all state variables is converted to the sum of smooth functions containing only each error dynamics by using the variable separation method in [17]. The structure of RBF NN [19] is proposed to deal with nonstrict form. However, according to the survey, how to solve the algebraic ring problem caused by the nonstrict structure by using the HOFA system method has not been investigated. This will be challenging for design a robust controller for nonstrict feedback nonlinear time-delay systems based on the HOFA system method.
In this paper, for uncertain nonlinear HOFA systems with unknown constant and time-varying delays, two adaptive controllers are proposed respectively. Different from the previous works, the unknown nonlinear function system has nonstrict feedback structure and with time delays. Therefore, RBF NN is adopted to approximate the unknown nonlinear function, and Gaussion function is used to handle nonstrict feedback structure. In addition, combined with the variable separation technology, a appropriate Lyapunov-Krasovskii function is selected to counteract the effect of delay. Then, based on the HOFA system method and Barbalat lemma, two adaptive controllers are given to ensure the asymptotic stability of the systems with constant time-delay or time-varying delay, respectively. As we all known, the unknown nonlinear with time-varying delay are more complex with respect to the case of the constant delay, and the design of the controller is also more complex and difficult.
The problem description and preparation are given in Sect. 2. In Sect. 3, two adaptive controllers and stability analysis are proposed for nonstrict nonlinear HOFA systems with constant time delays and timevarying delays, respectively. In Sect. 4, the thermoacoustic instability system is used as an effective simulation example to verify the feasibility and effectiveness of the above control strategies. The Sect. V gives the conclusion of this paper.

System statements and preliminaries
Consider a nonstrict feedback uncertain nonlinear HOFA time-delay system, where t À s ð Þ : The above assumption about the unknown function can be summarized as the variable separation technique, which is adopted to handle unknown nonlinear functions with time delay in (2). On this basis, we will choose the appropriate Lyapunov-Krasovskii function to compensate the effect of time delays. In addition, in what follows RBF NN [24] will be utilized to approximate the unknown nonlinear function related to g il n n À 1. There exists similar assumption proposed in [22]. Next, we provide lemmas necessary to prove the main results.
and S n 0 $ nÀ1 ð Þ 1 $ n be the basic function vector of the RBF NN. For 8i, if i n, then 3 Main results

Controller design: unknown constant time delay
For the nonlinear HOFA system (1) with the unknown constant time delay, an adaptive NN controller is proposed to ensure the stability of the system. Denote P in as the abbreviation of P in A 0 $ nÀ1 i À Á , and U i denotes For the system (1), we first consider the unknown constant time-delay case, where the upper bound of s is s max .
Different from the state-space approach, the most important of the HOFA approach in [5] and [15] is to provide the nonlinear controller for the system (1) directly, i.e., _ h¼ and l i ; k i are parameters, h i is an exponential decay function, Proof Choose the Lyapunov-Krasovskii vector functional candidate For simplicity, we take V i ði ¼ 1; 2; Á Á Á ; nÞ as an example to give a detailed proof. Choose the Lyapunov functional candidate, parameter and estimation errors, respectively. Then, the derivative of (8) is Then, we have Based on (2), the unknown nonlinear function with Substituting (10)- (11) and (6) into (9) leads to an unknown nonlinear function.
Then, for 8e i [ 0, the unknown continuous func- Substituting (13) into (12), we have Based on Lemma 2, we have where where (17) and (7) into (14) yields Then, one haŝ Substituting (19) and (20) into (18), one has where Based on the Barbalat Lemma, we know that then the asymptotically convergence feature is achieved. h Remark 2 Based on the HOFA method, an adaptive control law (7) is designed to eliminate the unknown nonlinearity. In addition, the controller is related to the maximum value of time delay. It is less conservative than the case of time-delay independent conditions.

Remark 3
Since the radial basis function S i n 0 $ nÀ1 ð Þ 1 $ n is the form of nonstrict feedback, it contains the whole system state and cannot be directly compensated by the current state approximators. Therefore, using Lemma 2 can deal with the nonstrict feedback form.

Controller design: time-varying delay
Next, consider the time-varying delay for the nonstrict feedback uncertain nonlinear HOFA system, where s t ð Þ satisfies 0\s t ð Þ s; ð26Þ where s and h are known constants. For any 1 i n, we assume an unknown positive function The following theorem provides the controller design method to stabilize the nonstrict feedback HOFA system with unknown nonlinear vector function and time-varying delay.
can guarantee the all states of the system (25) converge to zero asymptotically, where ; and l i ; k i are parameters, h i is an exponential decay function, P i1 ; P i2 ; Á Á Á ; P in 2 R nÂ1 ; i ¼ 1; . . .; n.
Proof Similar to Theorem 1, choose the Lyapunov functional vector candidate Àb i are parameter and estimation errors, respectively. Then, the derivative of (30) is Then, one has Based on (28), the unknown nonlinear function with By substituting (32)-(33) and (6) into (31), one leads to where G i n is an unknown nonlinear function.
Similarly, we can obtain where From the Barbalat Lemma, we can deduce which indicates the system (25) is asymptotically converge to zero under the adaptive NN control law (29). h

Remark 4
The controller (29) has the same structural characteristics as the controller (7) in Theorem 1. The difference is that the controller is related to the maximum value of delay and the maximum of delay derivative. When the derivative of time-varying delay h ! 0, the controller (29) tends to the controller (7). Although the controller gain increases with the increasing of h, the controller has less conservatism and can ensure the system to achieve asymptotic stability.
Remark 5 Different from [16], this paper considers the nonstrict feedback systems with constant and timevarying delay. Moreover, the assumption of uncertain nonlinear functions is different. In [16], the uncertain nonlinear functions are assumed to be the product of known functions and unknown parameters. In contrast, the assumption in this paper can be an unknown function, which is less conservative.
Based on the HOFA method, for the uncertain nonlinear time-delay system, an adaptive controller is given to counteract the known nonlinearity in the system, adjust the influence of uncertain parameters online, and allocate the linear dominant part of the system. As for the parametric solution of the coefficient matrix A 0 $ nÀ1 i in the controllers (7) and (29), the following results are obtained.
Proposition 1 [4] For8F i 2 R nÂn , the matrix A 0 $ nÀ1 i and the nonsingular matrix V i 2 R nÂn satisfying where the parameter matrix Z i 2 R 1Ân satisfies Then,the matrix Àa i Àb i b i Àa i ! is selected as a diagonal block of F i , which has complex eigenvalues Àa i AE b i j, where a i and b i are two positive scalars.
For the Hurwitz matrix U i 2 R nÂn and the positive definite matrix 8Q i 2 R nÂn , the unique solution of Lya punov equation U T i P i þ P i U i ¼ ÀQ i can be given by

Simulation results
The thermoacoustic instability system is used as a classic practical example to verify the effectiveness of the proposed control strategy. The thermoacoustic instability is a kind of thermoacoustic oscillation phenomenon caused by the interaction between acoustic and the heat release when air flows through the tube. According to the reference [17], a nonstrict feedback nonlinear coupling model with N modes is presented. Since the physical model satisfies the physical meaning of the second-order fully actuated system, it can be transformed into a HOFA system. Without loss of generality, we consider the influence on the thermoacoustic unstable system for N=2 with the unknown nonlinear heat release rate. As shown in Fig. 1, the two ends of the horizontal Rijke tube are open.
From [17], the thermoacoustics instability system is given ð39Þ where x f and x ak k ¼ 1; 2; . . .; K ð Þrepresent the location of the heat source and the k-th actuator, respectively. a ak ¼ S ak =S, S and S ak are the cross-sectional area of the tube and k-th actuator, respectively. j denotes the damping coefficient, c represents the specific heat ratio of the medium, _ Q s is the heat release rate with time-delay s. R k ðtÞ and S k ðtÞ are control gains.
Based on [25,26], the thermoacoustics instability system can be rewritten as: where nðtÞ ¼ gðtÞ, n According to [25,26], the acoustic velocity and the pressure are respectively expressed as vðn; tÞ ¼ X 2 j¼1 cosðjpxÞn j ðtÞ; pðn; tÞ ¼ À X 2 j¼1 sinðjpxÞ jp _ n j ðtÞ: Among them, the time delay s caused by heat transfer and flow velocity is usually difficult to measure the actuated values. Therefore, we discuss it from the following two aspects:
The initial values are selected asĥ 1 0 ð Þĥ 2 0 ð Þ Â Ã T ¼ 3:5 3 ½ T , Fig. 1 Schematic diagram of a horizontal Rijke tube with two actuators Fig. 2 The trajectory of pressure p and velocity t with two modes: without the control law   Figure 7 shows the fluctuation of pressure and velocity. Obviously, it can be seen from Fig. 7 that the system is unstable without a controller. Unlike the constant amplitude oscillation in Fig. 2, the amplitude fluctuation in Fig. 7 changes with the time t due to the influence of time-varying delay. From Fig. 8, It is obvious that the velocity and pressure trajectory of the Rijke tube can quickly converge to zero under controlled by the controller. Figure 9 and Fig. 10 show the curves of the adaptive laws and the controllers, respectively.

Conclusion
The main topic of this study is to propose a reasonable controller to make the system asymptotically stable for a class of nonstrict uncertain HOFA systems with time Fig. 7 The trajectory of pressure p and velocity t with two modes: without the control law  The trajectory of states p and t with two modes: with the control law in [28] delay. Different from the previous works, the HOFA system is nonstrict feedback from with unknown nonlinearity and time delay, which is the most challenging in the design process. The RBF NN of Gaussian functions as basis functions is used to approximate the unknown nonlinear functions with nonstrict feedback structures. Then, an appropriate Lyapunov Krasovskii function and separation technique are proposed to compensate for the influence of unknown constant time delay or time-varying time delay. Based on the HOFA systems method, two adaptive NN controllers are presented to ensure the system with constant time-delay and time-varying delay asymptotically stable, respectively. Finally, the thermoacoustic instability systems is taken into consideration to prove the feasibility and effectiveness of this research method. Because of the application requirements, it is critical that the constraints are properly addressed in the control design. Therefore, this work can be extended to the HOFA system with output or state constraints. It is worth discussing whether the system with output or state constraints can also be transformed from state space representation to HOFA system representation. And a reasonable control strategy is proposed to ensure that the system is asymptotically stable. These issues are currently being studied.