Multi-Dimensional Taylor Network-Based Adaptive Output-Feedback Tracking Control for a Class of Nonlinear Systems

In this paper, the output feedback adaptive multi-dimensional Taylor network (MTN) tracking control for a class of nonlinear systems with unmeasurable states is investigated. Firstly, a nonlinear state observer is designed to estimate the unmeasurable states, and then an adaptive MTN-based output-feedback control approach is developed via backstepping technique. Secondly, in view of the simple structure of MTN, the controller based on MTN has the advantages of simple structure and fast calculation speed. Thirdly, in order to avoid the “differential explosion” problem inherited from the backstepping design, dynamic surface control (DSC) technique is introduced in the process of controller design. The results demonstrate that this scheme guarantees the stability and tracking performance of the closed-loop system. Finally, simulation examples are given to reveal the viability of the proposed method.


I. INTRODUCTION
In recent years, more and more scholars have begun to pay attention to the stability analysis and controller design of nonlinear systems, and many interesting results have been reported [1], [2]. Due to the output-feedback control is more suitable for practical engineering systems [3], significant progress has been made in the design of output-feedback controllers for nonlinear systems, such as uncertain nonlinear systems [4], input-delayed systems with time-varying uncertainties [5], Markovian jump systems [6], and large-scale stochastic nonlinear systems [7]. However, compared with full state feedback control, for example, strict-feedback [8], pure-feedback [9] and non-strict feedback [10], the design of output feedback control is more difficult and challenging, the results of controller design for nonlinear systems are relatively few. Consequently, it remains a significant and The associate editor coordinating the review of this manuscript and approving it for publication was Bohui Wang . interesting task to put forward a state observer with good estimation performance and design an output feedback controller with good control performance for nonlinear systems.
In view of the excellent performance of neural networks (NNs) and fuzzy logic systems (FLSs), especially the traits of nonlinear, capacity of study and self adapting, the approximation-based adaptive neural or fuzzy control schemes have become a useful approach to deal with uncertain nonlinear systems [11]- [26]. Meanwhile, NNs-based or FLSs-based control approaches have been applied to uncertain discrete-time nonlinear systems [11], dynamic parameters adjustment nonlinear systems [12], dynamic uncertainties nonlinear systems [13], strict-feedback nonlinear systems [14]- [16], pure-feedback nonlinear systems [17], [18], switched nonlinear systems [19]- [22], MIMO nonlinear systems [23], [24] and stochastic nonlinear systems [25], [26]. Although the adaptive neural or fuzzy backstepping control has achieved great progress, three aspects can not be ignored: (i) the training time of most NNs or FLSs are usually too long and there also exists local minimum. (ii) Most of the NNs can not be applied to actual dynamic systems because their neurons have limited functions. (iii) The accuracy of fuzzy control is not high enough and oscillation may occur. This encourages us to investigate new approximation-based adaptive control approaches for the control of nonlinear systems to solve the above problems. In this context, the idea of multi-dimensional Taylor network (MTN) emerged.
MTN is a three-layer feedback network, and includes the input layer, middle layer and output layer. MTN-based approach was first proposed to solve the problem of prediction control. Later, it was successfully extended to the control of nonlinear systems, and significant results have been achieved, for instance, based on account of discrete MTN, Yan and Kang [27] studied the asymptotic tracking and dynamic regulation of SISO nonlinear systems. Kang and Yan [28] proposed a MTN controller to stabilize the nonlinear time-varying delay systems with an inaccurate model. Han and Yan [29] studied the problem of adaptive tracking control for SISO uncertain stochastic nonlinear systems based on MTN. Yan and Han [30] investigated the problem of adaptive MTN decentralized tracking control for a class of large-scale stochastic nonlinear systems. Yan et al. [31] proposed an optimal output-feedback tracking control approach for SISO stochastic nonlinear systems. However, to the best of the authors' knowledge, fewer efforts have been devoted to the MTN-approximation-based adaptive output-feedback tacking control for nonlinear systems [32], [33]. Therefore, the construction of adaptive MTN tracking control algorithm for nonlinear systems is still an interesting and challenging subject, which has some inspiration for our research.
For the above-mentioned observations, this paper tries to study the adaptive output-feedback tracking control design problem for a class of nonlinear systems with unmeasurable states, and proposes an output-feedback control scheme based on adaptive MTN. Firstly, using the method by references [34], [35], a nonlinear state observer is designed to estimate the unmeasurable states. Secondly, the backstepping technique and MTN are combined to construct an adaptive output-feedback control scheme. Meanwhile, in order to avoid the ''differential explosion'' problem inherited from the backstepping design, DSC technique is introduced in the process of controller design. Thirdly, the stability of the closed-loop control system, the boundedness of the tracking error and control signals are ensured by Lyapunov stability theory. Finally, simulation results are presented to demonstrate the effectiveness of the design approach. The contributions of this paper are highlighted as follows: (i) A novel adaptive output feedback control method based on MTN is proposed for a class of nonlinear systems with unmeasurable states. The proposed method can obtain accurate tracking results with low computational cost, and has good real-time performance and convergence.
(ii) The computational complexity of the designed MTN-based controller is greatly minimizes through the following two aspects: a) Because of the simple structure of MTN, the controller based on MTN has the advantages of simple structure and fast calculation speed. b) At every step of backstepping, combining MTN method with DSC technique, the calculating amount is reduced as well as the problem of the nonlinear is effectively handled.
Throughout this paper, the following notations are used. R indicates the set of all real numbers, R n denotes the real n dimensional space. In formula θ T P m n (s), n denotes the input number of MTN, m represents the highest power of the polynomials in the middle layer of MTN, θ T is the weight vector of MTN.

II. SYSTEM DESCRIPTIONS AND PRELIMINARY A. PROBLEM DESCRIPTION
Consider the following nonlinear system with external disturbances: The objective of this paper is to design an adaptive controller ensuring that y tracks y d , where y d is a given continuous reference signal. Rewriting the nonlinear system (1) into the following form . . .
The study of this paper is based on following assumptions. Assumption 1: The given reference signal y d and its time derivatives up to the n-th order are continuous and bounded.
Assumption 3: [34], [35] There exist a matrix H and a function h(x), such that F(x) = Hh(x), and h(x) satisfies: where h (x) and F (x) are vector-valued function with F (0) = 0. Remark 1: It should be noted that there are some physical systems satisfy Assumption 3, such as single link flexible VOLUME 8, 2020 joint robot systems [36] and omnidirectional intelligent navigation systems [37].
Assumption 4: [34], [35] Matrices A, C and H defined in (2) and (3) satisfy the following linear matrix inequality (LMIs): B. MULTI-DIMENSIONAL TAYLOR NETWORK Figure 1 shows the structure of MTN with n inputs and the highest power of the polynomials in the middle layer is m, where s 1 , · · · , s n are the input vector of the MTN, θ = [θ 1 , · · · , θ n ] T is the weight vector of the MTN. In this paper, the unknown nonlinear functions in the system will be approximated by the MTN. In particular, suppose f (s) is defined on a compact set S ∈ R n , then we have Lemma 1: [29] Assume that φ (s) is a continuous function defined on a compact set s . Then, for any given desired level of accuracy ε > 0, there exists a MTN, such that where θ * is the ideal weight vector and defined as and δ (s) denotes the approximation error and satisfies |δ (s)| ≤ ε.

III. MTN-BASED ADAPTIVE OUTPUT-FEEDBACK CONTROLLER DESIGN A. NONLINEAR OBSERVER DESIGN
First of all, the following observer [34], [35] is used to estimate the unmeasured stateṡ wherex = x 1 , · · · ,x n T is the observer state vector and matrices K and L satisfy Assumption 4. Define the observer error asx = x −x, from (2), we havė Consider the following Lyapunov function , and by taking into consideration of Assumption 3, Lemma 1 and formula F(x) = Hh(x). Similar to the literature [35], we havė From Assumption 2, there exist constant matrixD = [d 1 , · · · ,d n ] T such that Then, by the Young's inequality, we havẽ By (7), (8) and (9), we havė where λ = λ min (Q 1 )λ min (Q 2 ).

B. MTN-BASED CONTROLLER DESIGN
According to (1) and (5), we have following entire system First of all, a change of coordinates is introduced as follows 77300 VOLUME 8, 2020 where α i,f is the output of the first-order filter with α i−1 as the input.
The time derivative of V 1 iṡ 1θ 1 (15) By the Young's inequality, we have substituting (16) and (17) into (15) giveṡ According to Lemma 1, for any ε 1 > 0, there exists a MTN θ T 1 S(z 1 ), such that Based on (18) and (19), taking the virtual control signal α 1 as where k 1 > 0 is a design parameter. Form (19) and (20), and by the Young's inequality, we have Substituting (21) into (18) giveṡ In order to avoid the repetitive differential of α 1 , a new variable α 2,f is introduced and let α 1 pass through a first-order filter whose time constant is τ 2 , and α 2,f is where τ 2 > 0 is time constant. Define the output error of the filter as Due to z 2 =x 2 − α 2,f , and by (23) and (24), we have substituting (25) into (22) giveṡ Step 2: A new state variable α 3,f is introduced, and α 2 is input into a first-order low-pass filter with a time constant of τ 3 to obtain a new variable α 3,f as where τ 3 > 0 is time constant. Define the output error of the filter as The time-derivative of χ 3 iṡ Consider the following Lyapunov function whereθ 2 = θ 2 −θ 2 is the parameter error, and 2 = T 2 > 0 is any constant matrix.
The time-derivative of V 2 iṡ Step 1, a new MTN θ T 2 S 2 (z 2 ) is employed to approximate the unknown functionf 2 , for any given ε 2 > 0, we havef where z 2 = [z 1 , z 2 ] T , and σ 2 (z 2 ) is approximation error. Taking the virtual control signal α 2 as By (32) and (33), we have Due to z 3 =x 3 − α 3,f , and by (34),we havė Step i(3 ≤ i ≤ n − 1). A new state variable α i+1,f is introduced, and α i,f is input into a first-order low-pass filter with a time constant of τ i+1 to obtain a new variable α i+1,f as where τ i+1 > 0 is time constant. Due to z i+1 =x i+1 − α i+1,f , define the output error of the first-order low-pass filter as The time-derivative of χ i+1 iṡ

Consider the following Lyapunov function
whereθ i = θ i −θ i is the parameter error, and i = T i > 0 is any constant matrix.
The time-derivative of V i iṡ wheref i = φ i − l ix1 −α i,f and l i > 0. Similar to Step 2, for any given ε i > 0, we havẽ where z i = [z 1 , · · · , z i ] T , and σ i (z i ) is approximation error. Take the virtual control signal α i as By (41) and (42), we have Due to z i+1 =x i+1 − α i+1,f and (43), we havė Step n: Consider the following Lyapunov function whereθ n = θ n −θ n is the parameter error. According to (45) with i = n, we havė wheref n = φ n − l nx1 −α n,f and l n > 0. Similarly, by the Lemma 2.1, for any given ε n > 0, we havẽ where z n = [z 1 , · · · , z n ] T , and σ n (z n ) is approximation error. Take controller u as u = −k n z n −θ T n S n (z n ), (k n > 0) By the Young's inequality, we have z n (u +f n ) ≤ z nθ T n S n − k n z 2 n + Substituting (49) into (46) giveṡ (50) By the Young's inequality, we have where ξ i , λ i (i = 1, · · · , n − 1) is any constant greater than zero. Substituting (51), (52) and (53) into (50), we havė In summary, the design procedure of the MTN-based controller is shown in Figure 2.

C. STABILITY ANALYSIS
Theorem 1: Considering the nonlinear system (1), if design the observer in the form of (5), design the control law u in the form of (48), the intermediate virtual control signals α i (i = 1, · · · , n − 1) described as (42), and the adaptive lawṡ θ i (i = 1, · · · , n − 1) defined aṡ where constants k i > 0 and η i > 0 are designed parameters, and constants matrices i = T i > 0. Then, under bounded initial conditions, all the signals in the closed-loop system are bounded, and the tracking error converges to a small neighborhood of the origin.
Proof: For the stability analysis of the closed-loop system, we choose the following Lyapunov equation: By (54) and (56), we havė By the Lemma 1, we have Let then inequality (59) can be rewritten in the following forṁ Using the similar arguments in [32], it is easy concluded that the conclusions of Theorem 1 is valid.
Remark 2: The inequality (60) implies that According to (61), we know that V (t),x i , z i , θ i are bounded. Thus, to guarantee that the tracking error converges to a small residual set around the origin in the sense of mean quartic value, we can properly adjust the parameters a 0 and b 0 .
Remark 3: Recalling (56) and (61), we have Thus, for given > 2b 0 a 0 , there exists a time T , for all t ≥ T , such that which means that θ i converge to zero by properly adjusting the parameters, such as k i , τ i , ξ i , η i . Remark 4: Theoretically speaking, based on Theorem 1, choosing appropriately the design parameters, such as k i , η i and i , can make the tracking error arbitrarily small. In practical application, however, these parameters should be selected appropriately to meet specific requirements.

IV. SIMULATION RESEARCH
In this section, we will demonstrate the effectiveness of the proposed adaptive MTN control method through two simulation examples.
According to (2) and (62), we have Let h (x) = Design the following state observeṙ According to Theorem 1, the virtual control laws, the actual control law and the adaptive control laws are designed as In the simulation, the parameters are chosen as follows: k 1 = 15, k 2 = 10, η 1 = 0.5, η 2 = 1.5, 1 = 20I 4 , 2 = 5I 9 , τ 2 = 0.005. The reference signal y d = sin t. The simulation results are shown in Figures 3-8.  The simulation results indicate that a good tracking control performance has been achieved. Figure 5 indicates the tracking error converges to a small neighbourhood around the origin. Figures 6-7 show that all signals of the closed-loop system, such as state x 1 , x 2 and their estimationx 1 ,x 2 are bounded. Figure 8 depicts that the adaptive parameters θ 1 77304 VOLUME 8, 2020   Example 2: On a similar method to [39], [40], a type of closed, continuously stirred tank, chemical reactor with one mode of feed stream and disturbances can be described as follows:     ẋ 1 = x 2 + 0.5x 1 + 0.1 sin ṫ x 2 = u + 0.1 cos t y = x 1 (65) Using the same process of Example 1, design the following state observeṙ    The simulation results are shown in Figures 9-14. The simulation results further verify the effectiveness of the control method proposed in this paper. VOLUME 8, 2020

V. CONCLUSION
In this paper, the problem of adaptive output-feedback tracking control has been investigated for a class of nonlinear systems with unmeasurable states based on multi-dimensional Taylor network approach. A nonlinear observer is designed to estimate the unmeasurable states of the system. By combining backstepping approach and dynamic surface control technique, a novel MTN-based adaptive output-feedback control scheme has been proposed. The designed MTN-based controller in this paper has the advantages of simple structure and fast computation speed, and the proposed approach can overcome the problem of ''explosion of complexity''. The simulation results show that the proposed control scheme can keep all signals of the closed-loop system bounded and the tracking error converges to any small neighborhood around the origin.
Our future work will be directed at further extending the proposed methodology to switching nonlinear systems and MIMO nonlinear systems.