Adaptive Neural Control of Uncertain MIMO Nonlinear Pure-Feedback Systems via Quantized State Feedback

We present an adaptive quantized state feedback tracking methodology for a class of uncertain multiple-input multiple-output (MIMO) nonlinear block-triangular pure-feedback systems with state quantizers. Uniform quantizers are considered to quantize all measurable state variables for feedback. Compared with the existing tracking approaches of MIMO lower-triangular nonlinear systems, the main contributions of the proposed strategy are developing (1) a quantized-state-feedback-based adaptive tracker in the presence of nonaffine interaction of states and control variables of MIMO systems and (2) an analysis strategy for quantized feedback stability using adaptive compensation terms to derive bounded quantization errors. In addition, the stability of the closed-loop system with quantized state feedback is analyzed based on the Lyapunov stability theorem. Finally, simulation examples, including interconnected inverted pendulums, are presented to validate the effectiveness of the proposed control strategy.


I. INTRODUCTION
In the field of nonlinear control, adaptive recursive control techniques based on backstepping [1], dynamic surface design [2], and command-filtered backstepping design [3], [4] have been regarded as powerful tools for dealing with unmatched and uncertain nonlinearities. Adaptive control and filter designs using neural-network function approximators have been actively studied for systems with unknown nonlinearities (see [5]- [8] and references therein). To deal with more complex nonlinear systems, these techniques have been extended to multiple-input multiple-output (MIMO) nonlinear systems that include nonlinearities with couplings among system states and inputs. In [9], an adaptive fuzzy output feedback controller was designed for MIMO strict-feedback systems with unknown dead-zone inputs. In [10], an adaptive fuzzy tracking approach for MIMO nonlinear switched strict-feedback systems was studied. Adaptive neural control problems of MIMO pure-feedback nonlinear systems The associate editor coordinating the review of this manuscript and approving it for publication was Jinquan Xu . were addressed in [11] and [12]. In [13] and [14], fuzzy adaptive control approaches were developed for uncertain nonlinear multivariable systems. Moreover, the problems of output constraints [15] and event-triggered control [16] have been addressed for adaptive neural control designs of MIMO nonlinear systems. Contrary to these control designs using continuous feedback, industrial control systems based on digital networks require the transmission of quantized signals with finite values owing to band-limited communication channels [17]. Linear systems [18]- [20] and nonlinear control systems [21], [22] have been considered for quantized control. Several adaptive recursive control strategies have been investigated for uncertain lower-triangular single-input single-output nonlinear systems with quantized input signals [23]- [28]. Furthermore, these quantized control approaches have been adopted to address several control problems of input-quantized MIMO nonlinear systems. In [29], a tracking control problem using an observer was investigated for MIMO time-delay nonlinear systems in purefeedback form. In [30], a prescribed performance control was proposed for interconnected MIMO nonlinear systems. VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ An adaptive neural control problem of switched MIMO nonlinear systems with quantized dead-zone input was addressed in [31]. In [32], an adaptive input-quantized control approach was proposed for MIMO nonlinear systems with underactuated faults and time-varying output constraints. However, these successful quantized control approaches [23]- [32] are only feasible in the presence of input quantization. In network-based control systems, the quantization of measured state variables must be considered for feedback control design because the controller and MIMO systems are connected by band-limited network channels. To the best of our knowledge, quantized state feedback control design of uncertain MIMO pure-feedback nonlinear systems has not been investigated yet.
On the other hand, quantized-state-feedback-based recursive control design problems in the presence of state quantizers have been recently addressed for nonlinear systems with parametric uncertainties [33], [34]. Neural-networkbased quantized feedback control designs were developed for strict-feedback nonlinear systems with time delays [35]. Furthermore, a distributed control strategy was presented for multi-agent nonlinear systems in strict-feedback form [36]. However, these approaches fail to provide a quantized state feedback control solution for MIMO nonlinear systems with coupling of state and control variables in pure-feedback form.
A primary difficulty in addressing this problem is to deal with unknown control coefficient matrices resulting from unknown nonaffine nonlinearities via quantized feedback information of states of all subsystems coupled in nonaffine nonlinear form. Owing to unknown control coefficient matrices induced from unknown nonaffine nonlinearities, the quantized-state-based neural network adaptive control strategies reported in [35], [36] cannot be applied to ensure boundedness of close-loop quantization errors for the stability analysis of the closed-loop system. Therefore, it is important to establish quantized-state-based adaptive compensation strategies for tuning neural network approximators to guarantee that close-loop quantization errors are bounded in the presence of unknown control coefficient matrices.
To address this difficulty, we propose an adaptive quantized state feedback control methodology for uncertain MIMO nonlinear pure-feedback systems with state quantizers and external disturbances. All measurable state variables for feedback are quantized via state quantizers. An adaptive tracking scheme using quantized states is constructed in the presence of nonaffine nonlinear vectors and unknown bounds of gain matrices derived using the mean value theorem. In the proposed scheme, adaptive neural compensation terms using quantized state feedback are introduced to analyze quantization errors between unquantized and quantized signals in a closed-loop system. In addition, we analyze the boundedness of estimated parameters and quantization errors by establishing technical lemmas. Finally, the stability of the proposed control system is proved using the Lyapunov stability theorem. The main contributions of this study are as follows: (i) Contrary to previous results [23]- [32] in which the input quantization problems were only considered in quantized control for MIMO lower-triangular nonlinear systems, this study addresses the state quantization problem for uncertain MIMO nonlinear pure-feedback systems. A neural-networkbased quantized state feedback control strategy is developed to ensure the boundedness of quantization errors in the presence of unknown pure-feedback nonlinearities. In addition, adaptive function approximation terms and adaptive tuning laws using quantized states are constructed to compensate for unknown nonaffine nonlinearities and quantization errors.
(ii) In contrast to previous recursive tracker designs using quantized state feedback [33]- [36], this paper firstly deals with nonaffine nonlinearities interacting state variables and inputs in uncertain MIMO nonlinear systems. Moreover, a design difficulty caused by unknown gain matrices derived from the mean value theorem is resolved by introducing new adaptive approximation terms using quantized state feedback, contrary to [33]- [36]. The closed-loop stability based on quantized state feedback is investigated by analyzing the quantization errors.
The rest of the paper is organized as follows. The neuralnetwork-based quantized state feedback tracking problem of MIMO nonlinear pure-feedback systems with state quantizers is described in Section II. The proposed adaptive quantized control design and stability analysis are discussed in Section III. Section IV presents simulation results including a practical example. Finally, we draw our conclusions in Section V.

II. PROBLEM FORMULATION
Consider the following uncertain MIMO block-triangular pure-feedback nonlinear system with external disturbances: . . , f n,m (x n , u m )] ∈ R m are the unknown C 1 nonaffine nonlinear function vectors, and d i (t) ∈ R m , i = 1, . . . , n, is the external time-varying disturbance vector.
In this study, a network-based control problem using quantized state feedback is considered for system (1). In the network-based control problem, the system (1) and the controller are assumed to be connected through a network with a limited bandwidth. Thus, the measured state feedback information is transmitted to the controller after state quantization. For state quantization, the uniform quantizer is selected as follows: where i = 1, . . . , n and j = 1, . . . , m, µ ∈ Z + , ρ is the quantization level, Z 1 = ρ, and Z µ+1 = Z µ + ρ. We use the definition x q i,j Q(x i,j ) for notation simplicity. Then, using the property of the uniform quantizer, the state quantization Remark 1: Owing to the uniformity of the quantization levels and simple structure, uniform quantizers facilitate the analysis of the quantization effect. Therefore, they are frequently utilized in analog-to-digital signal conversion [18]. For this reason, we use uniform quantizers (7) to quantize all measurable states for feedback. However, hysteresis-uniform or logarithmic-uniform quantizers can be also applied to the proposed approach.
Assumption 2 [33]: The quantized state vector . . , n, is available for feedback, rather than the unquantized state vector x i . Assumption 3 [2]: The desired signal vector x d ∈ R m and its time derivativesẋ d andẍ d are bounded.

Assumption 4:
The time-varying external disturbance vector d i , i = 1, . . . , n, satisfies d i ≤d i with an unknown constantd i > 0.
Lemma 1 [39]: For a Hurwitz matrix A ∈ R m×m and a symmetric positive definite matrix S ∈ R m×m , the inequality e At ≤ a 1 e −b 1 t is ensured where a 1 = √ λ max (S)/λ min (S), b 1 = 1/λ max (S), and λ max (S) and λ min (S) are the maximum and minimum eigenvalues of S, respectively.
Problem 1: The control objective is to design a neuralnetwork-based adaptive quantized state feedback control law u for uncertain MIMO pure-feedback nonlinear systems (1) with state quantizers (7) so that the output vector y(t) tracks the desired trajectory x d (t), while all the closed-loop signals are bounded.
Remark 2: The validity of Assumptions 1-4 is explained as follows.
(i) The matrix G i (·) plays the role of the coefficient matrix for the virtual and actual control laws in the control design steps. G i (·) = 0 should be assumed for controllability, which leads to 0 < g i m ≤ |λ(G i )| ≤ g i M in Assumption 1. This implies that G i (·) is strictly either positive or negative definite. Without loss of generality, it is assumed that G i (·) > 0. Thus, Assumption 1 is reasonable.
(ii) Assumption 2 means that the state quantization problem is considered in this study. Thus, quantized state variables are available for feedback control design. This assumption is given for the problem formulation.
(iii) Assumption 3 implies that the desired signal and its first two derivatives are bounded. This assumption is reasonable for the recursive tracking control design objective.
(iv) Assumption 4 indicates that external disturbances may not grow arbitrarily large. This is common in existing control results.
Remark 3: For Problem 1, it is necessary to design a control law and adaptive tuning laws using quantized states in the presence of unknown nonaffine nonlinearities in MIMO form while ensuring the boundedness of quantization errors between the original and quantized signals. Furthermore, although quantized signals are used as inputs for neural network approximators, the boundedness of quantization errors and the closed-loop stability of the proposed neural-networkbased control system should be proved. However, the existing adaptive control designs [23]- [32] dealing with input quantization of MIMO nonlinear systems use continuous state feedback information without state quantization. Thus, owing to the presence of quantized state feedback signals, Problem 1 cannot be addressed by the solutions proposed in [23]- [32].

III. NEURAL-NETWORK-BASED ADAPTIVE QUANTIZED STATE FEEDBACK CONTROL A. RADIAL BASIS FUNCTION NEURAL NETWORKS
Unknown continuous nonlinear function vectors N i (ν i ) ∈ R m , i = 1, . . . , n, can be approximated via radial basis function neural networks (RBFNNs) [40] in the compact set VOLUME 10, 2022 i ⊂ R p i as follows: where ν i ∈ i is the input vector of the RBFNN, . . , m, denotes the Gaussian function vector. Using the inherent property of Gaussian basis functions, it is ensured that [42]. Assumption 5: [41] θ i and φ i are bounded as θ i ≤θ i and φ i ≤φ i , respectively, whereθ i > 0 andφ i > 0 are constants.

B. ADAPTIVE CONTROL DESIGN USING QUANTIZED STATE FEEDBACK
The quantized state variables cannot be utilized directly in a Lyapunov-based systematic design because they are discontinuous. Thus, our design strategy is based on (i) designing the intermediate control signals using the unquantized state variables in Step 1, (ii) expressing the actual adaptive control input vector u using quantized state feedback in Step 2, and (iii) analyzing the quantization errors between unquantized and quantized signals to ensure the stability of the closed-loop system in the stability analysis part (see Section III-C).
Step 1: For the systematic control design of intermediate control signals, the command-filtered backstepping technique using second-order low-pass filters is employed. The error surfaces are expressed as follows: ∈ R m are their corresponding filtered signals provided by the following second-order lowpass filters:α with  (6) is given bẏ Let us consider the Lyapunov function V 1 as follows: Differentiating V 1 with respect to time yieldṡ where From Assumptions 1 and 5, it holds that The intermediate signal α 1 is designed as follows where ζ 1 is a design parameter, tanh(z 1 / 1 ) = [tanh(z 1,1 / 1,1 ), . . . , tanh(z 1,m / 1,m )] ∈ R m ; 1,j > 0, j = 1, . . . , m, are design parameters,ε is a design parameter satisfyingε ≥ 1/4, andŴ 1 is the estimate of W 1 , andδ 1 is the estimate of an unknown constant δ 1 > 0, which is defined subsequently.
Substituting (33) into (32) and using the following inequality − z n G n z n g n M (z n z n +ε) B n 2 W n + g n m g n M B n 2 W n ≤ g n mε B n 2 W n g n M (z n z n +ε) . (34) VOLUME 10, 2022 whereδ n =δ n − δ n andW n =Ŵ n − W n are the estimation errors.
Step 2: In this step, we present a quantized-state-based control law u based on the structures of the error surfaces and the intermediate signals designed in Step 1. Owing to the recursive design property of a quantized-state-based control law u, all error surfaces and intermediate signals designed in Step 1 are redefined using quantized states.
The error surfaces using state quantization are defined as where i = 1, . . . , n − 1, and α * i,f are the command-filtered signals of the virtual control laws α * i using quantized states expressed aṡ with α * i,f (0) = α * i (0) and β * i (0) = 0. Then, we introduce the quantized-state-feedback-based virtual control laws α * i and the adaptive actual control law u as follows: where i = 1, . . . , n − 1, j = 1, . . . , n, . . , n, W ,j > 0 and δ,j > 0 are tuning gains, and σ W ,j and σ δ,j are positive constants for σ -modification. A block diagram of the proposed state feedback adaptive tracking system is shown in Fig. 1.
Remark 4: Compared with previous recursive tracking results using quantized state feedback [33]- [36], we introduce the neural-network-based adaptive compensation (38) and (39). Thus, we enable the analysis of the boundedness of the quantization errors in the presence of unknown control coefficient matrices G i , which will be presented in the next section. The stability of the quantized state feedback adaptive tracking system is analyzed based on these terms. Furthermore, the adaptive laws (40) and (41) using quantized states are derived to ensure that the estimatesŴ j andδ j are bounded, as proved in the next section.
Remark 5: Using the second-order command filter (37), α * i,f , i = 1, . . . , n − 1, are replaced by the continuous signals β * i in the control laws (38) and (39) where β * i are included in the inputs of neural networks. Thus, from the recursive design procedure, we attenuate the unexpected chattering effects caused byα * i,f in the control law u.
(iii) Similar to (52), it is satisfied that where z,n = ρ √ m + χ,n−1 . Then, from (33) and (39), we obtain Then, using z n /(z n z n +ε) ≤ 1, µ u is bounded as where u (ζ n + 1) z,n + 2B nWn + 2δ n √ m with positive unknown constantsW n andδ n satisfying |Ŵ n | ≤W n and |δ n | ≤δ n , respectively. This completes the proof. Choose the overall Lyapunov function candidate V as where S j > 0 is a symmetric matrix. Remark 6: Contrary to the existing neural-network-based adaptive control approaches considering state quantization [34]- [36], this paper deals with the problem of unknown control coefficient function matrices G i in the control design procedure. Specifically, the minimal parameter technique is employed to tune the unknown parameters W i , i = 1, . . . , n including the norm of the weights of RBFNNs. The minimal parameter technique is commonly used in the form in the existing studies for uncertain nonlinear pure-feedback systems [12], [15], [16], [28], [29]. However, this form cannot be directly adopted in our quantized state feedback controller because the boundedness of the quantization errors i cannot be analyzed based on the boundedness of µ z,i and B * i in Lemma 3. Thus, we introduce adaptive neural compensation terms z i B i 2Ŵ i /(z i z i +ε) in the proposed tracker (i.e., (38) and (39)) and the inequality iW i is employed in the proof of Lemma 3.
Theorem 1: Consider an uncertain MIMO pure-feedback nonlinear system (1) with state quantizers (7) where state variables and control inputs are fully interconnected in the block-triangular pure-feedback form. Then, for any initial conditions satisfying V (0) ≤ ψ with a constant ψ > 0, the neural-network-based quantized adaptive tracking scheme consisting of (38)-(41) ensures uniform ultimate boundedness of all closed-loop signals using quantized feedback and the convergence of the tracking error z 1 to an arbitrarily small neighborhood of the origin.
Remark 7: The control performance can be improved by reducing the compact set ϒ in the proof of Theorem 1. In this direction, the guidelines for the design parameters of the proposed adaptive quantized state feedback controller are as follows.
(i) Increasing the control gains ζ i , i = 1, . . . , n, helps in increasing ζ and consequently ϒ can be reduced.
(ii) C in ϒ can be reduced by reducing the quantization level ρ as long as the network resources allow, and by decreasing the design parameters i,j , i = 1, . . . , n, j = 1, . . . , m. Then, the compact set ϒ can be reduced.
(iii) Selecting the filter constants i and ξ i appropriately helps in increasing the minimum eigenvalue q i m of Q i , i = 1, . . . , n − 1, which subsequently increases ζ . Consequently, the compact set ϒ is then reduced.
(iv) Increasing the tuning gains W ,i and δ,i , i = 1, . . . , n and fixing σ W ,i and σ δ,j as small values help to increase the learning speed of the estimated vectorsŴ i and parametersδ j .
Remark 8: In [43], a fuzzy adaptive output feedback control approach was presented for MIMO nonlinear systems with full-states prescribed performance in finite-time. In [44], a finite-time adaptive quantized control problem was investigated for stochastic systems in presence of input quantization. The finite-time controllers designed in [43], [44] are based on non-quantized state variables (i.e., continuous state feedback information). To apply the proposed state-quantized design approach to the finite-time control problem, a finitetime control design ensuring bounded quantization errors, as reported in Lemma 3, should be newly developed. Therefore, it will be interesting to address the finite-time control problem based on quantized state feedback of MIMO nonlinear pure-feedback systems in future studies.

IV. SIMULATION RESULTS
To demonstrate the effectiveness of our theoretical tracking strategy in the presence of state quantization, we present two examples including interconnected inverted pendulums. In addition, we compare the tracking performance of the   proposed state-quantized controller and the existing tracking controller [11] designed without state quantization.
The quantized state feedback tracking results and errors of the proposed state-quantized adaptive tracker and the previous adaptive tracker [11] are compared in Figs. 2 and 3. The time responses of the system outputs and desired signals are compared in Fig. 2 and those of the tracking errors are displayed in Fig. 3. The tracking errors rapidly converge to nearly zero. Although the proposed tracker is designed under state quantization, its tracking performance is similar to that of the unquantized state feedback tracker [11] for MIMO pure-feedback nonlinear systems. Fig. 4 shows the control inputs u 1 and u 2 of the proposed approach. The estimates of W i and δ i , i = 1, 2 of the proposed approach are depicted in Fig. 5. These results demonstrate that satisfactory tracking performance is achieved by the proposed neural-networkbased quantized state feedback tracker while ensuring that all the state variables and tracking errors are bounded.
Example 2: In this example, two inverted pendulums interconnected by a spring are taken as an example to verify the effectiveness of the proposed control strategy. The model is expressed as follows [45]: where θ 1 and θ 2 are pendulum angular positions, u 1 and u 2 are control torques, m 1 = 1.5 kg and m 2 = 1.2 kg are the masses, L p = 0.4 m is the pendulum length, J 1 = m 1 L 2 p and J 2 = m 2 L 2 p are the moments of inertia, D = 0.3 m denotes the ground distance between two pendulums, S c = 45 N/m and S = 0.5 m represent the spring constant and spring natural length, respectively, and L d is the distance between the connection points, which is expressed as follows (1 − cos(θ 2 − θ 1 )).  The relative angular position θ 0 can be defined as θ 0 = tan −1 0.5L p (cos θ 2 − cos θ 1 ) D + 0.5L p (sin θ 1 − sin θ 2 ) .
The tracking results and errors for Example 2 are compared in Figs. 6 and 7. The tracking results of the position angles y 1 and y 2 are compared in Fig. 6. Fig. 7 shows a comparison of the tracking errors. Despite the quantization of state feedback signals, the proposed approach has a similar tracking performance to that of the previous adaptive tracker [11]. Fig. 8 illustrates the control inputs u 1 and u 2 of the proposed approach. The estimates of W i and δ i , i = 1, 2, of the proposed approach are displayed in Fig. 9. From these results, we conclude that the proposed adaptive neural tracker using quantized state feedback is effective in coping with unknown nonaffine nonlinearities of MIMO pure-feedback systems and state quantization.

V. CONCLUSION
We have presented a neural-network-based adaptive quantized state feedback control design and stability analysis strategies for uncertain MIMO block-triangular purefeedback nonlinear systems with state quantization. The key aspect of the proposed strategy is that adaptive neural compensation terms using quantized states are derived to ensure the boundedness of the quantization errors of the closed-loop signals and to deal with unknown control coefficient functions derived from recursive designs. The adaptive neural tracker and its adaptive laws have been designed via quantized states, and the quantization errors have been compensated by the adaptive laws. By constructing technical lemmas for quantization errors and adaptive parameters, the uniform ultimate boundedness of all signals of the closed-loop system using quantized feedback is proved based on the Lyapunov stability analysis. Finally, the simulation results demonstrate that the proposed theoretical strategy provides effective control with good tracking performance. The neural-networkbased quantized feedback tracking problem in the presence of measurement noise or faults will be investigated in future studies.