Adaptive control of non‐affine MIMO systems with input non‐linearity and unmodelled dynamics

In this study, an adaptive neural network control scheme is proposed for a class of multi-input multi-output (MIMO) non-affine systems with unmodelled dynamics and dead-zone non-linear input. This scheme solves the complexity of computation problem, broadens the variables of unmodelled dynamics and cancels the assumption of the neural network approximation error to be bounded. Using the mean value theorem and Young's inequality, only one adaptive parameter is adjusted for the whole MIMO system. By theoretical analysis, all the signals in the closed-loop systems are proved to be semi-globally uniformly ultimately boundedness. The numerical simulation illustrates the effectiveness of the proposed scheme.


Introduction
Dead-zone non-linearity in actuators is the most common actuator non-linearities in many industrial processes, and many researchers are studying on it. In [1, 2], based on the dynamic surface control (DSC) technique and using radial basis function neural networks (RBFNNs), two adaptive schemes for a class of pure-feedback non-linear systems were proposed, respectively. While the uncertain multi-input multi-output (MIMO) non-linear timevarying delay systems with dead-zone were studied in [3]. In [4], a decentralised variable structure controller was presented for a class of system with dead-zone and time-delay. However, the dead-zone parameters u i − and u i + were needed to be known. The problem of over-parametrisation still existed in [5]. The DSC was first proposed to deal with the 'explosion of complexity' problem in [6], and [7,8] expanded the DSC technology to backstepping design. It is well known that unmodelled dynamics exists in some non-linear systems, such as modelling errors, external disturbances, i.e. they may severely limit system performance. In [9], combining backstepping with DSC, a novel controller was used in the system with unmodelled dynamics. While in [10], the idea was used for the stochastic system with unmodelled dynamics. In [11], the main contributions lied in that a control strategy was provided for a class of strict-feedback non-linear systems with unmodelled dynamics and input saturation, and the scheme does not require any information for the parameter of input saturation non-linearity.
In practice, input non-linearity and unmodelled dynamics often appear simultaneously. Inspired by the previous articles, a class of MIMO non-affine systems with unmodelled dynamics and deadzone non-linear input are discussed in this paper. The main contributions lie in: (i) It is the first time to deal with such MIMO non-affine systems with unmodelled dynamics, non-linear input and dynamic disturbances. (ii) Using Young's inequality, only one adaptive parameter needs to be tuned online for the whole MIMO systems, and the complexity of control system design is greatly reduced. (iii) The assumptions of the unmodelled dynamics are broadened, and the functions in hypothesis 4 and hypothesis 6 are unnecessary to be known.

Problem description
A class of MIMO non-linear systems with unmodelled dynamics and dead-zone input are considered Dead-zone model is as follows: According to Assumption 1, the dead-zone (2) can be rewritten as follows: where Based on mean value theorem, there exist parameters 0 < λ j, i j < 1, 0 < λ j, ρ j < 1 such that The control objective is that control law v j (t) for the system (1) is designed to make the output y j tracking the expected trajectory y jd .
Assumption 2: g m and g M are positive constants which satisfy where Ω jd = y¯j: y jd 2 + y˙j d 2 + y¨j d 2 ≤ Q j0 is known compact set, Q j0 > 0 is known constant.

Radial basis function neural network
In this paper, we will use RBFNNs to approximate unknown smooth function. Suppose Ω Z i j i ⊂ R i j + 2 be a compact set, and ω ji j T ξ ji j (Z ji j ) be the approximation of the RBFNNs on the compact set Ω Z¯j i j to L ji j (Z ji j ), where the unknown continuous function L ji j (Z ji j ) will be given later. Then, we have where being chosen as the commonly used Gaussian function, T is the centre of the receptive field with q i j = i + 2 and φ i j is the width of the Gaussian function.
To facilitate design, we agree on the following symbols:

Adaptive DCS design
In the every step of n − 1 steps, we design virtual control, and we design the control law at the nth step.
Step 1: Consider the first dynamic surface of the jth subsystem, we have According to Assumption 4, and using Young's inequality, we obtain where δ > 0 is a constant. Substituting (6) into (5), we get Using the RBFNNs to approximate the unknown L j, 1 (Z j, 1 ), and Young's inequality, we obtain J. Eng where α j1 is a positive design constant.
D j1 = (α j1 2 /2g m ) + (1/2)p j *2 + (δ 2 /4c 3 2 )p j *4 . The virtual control is designed as follows: where k j, 1 > 0 is a design constant. Substituting (8) into (7), we get Step i j (2 ≤ i ≤ ρ j − 1): Consider the i j th dynamic surface of the jth subsystem, we have Similar to step 1, we have with design constant α ji j > 0 will be given later, The virtual control is designed as follows: where k j, i j > 0 is a constant. Substituting (13) into (12), we obtain Step ρ j : The control law will be designed in this step. Consider the ρ j th dynamic surface, we have S˙j , ρ j = F j, ρ j (x¯j , ρ j , w j − 1 (t), 0) + g j, ρ j (x¯j , ρ j , w j − 1 (t), λ j, ρ j w j (t)) Using the similar method, we can get Using similar method, we obtain The control law is designed as follows: where k j, ρ j > 0 is a design constant.

Stability analysis
Define the first-order filter as follows: where τ j, i j + 1 is a design constant given later. From (18), we have Noting Assumption 4, we have and where Φ j, i j + 1 ( ⋅ ) is a continuous function.
Using (20) and (21) And let Define a compact set as follows: where χ > 0 is a constant. Let Φ j, i j + 1 , ε¯j , i j and ψ 0 4 ( x ) have maximum M j, i j + 1 , N j, i j and ψ max on the compact set Ω.
Theorem 1: Consider the closed-loop system shown as (1) under Assumptions 1-6, the controller (15) and adaptation law (17). For any bounded initial condition, k j, i j , τ j, i j + 1 and α 0 are the existed positive constants, and satisfying V(0) ≤ χ, such that the overall closed-loop control system is semi-globally stable, namely, all the signals in the closed-loop system are bounded, and the tracking error can be more smaller, and k j, i j and τ j, i j + 1 satisfy (24) Proof: Choose the Lyapunov function as follows: Differentiating V(t) to the time, we obtain Substituting (9), (13), (16), (22) and (23) into (26), and utilising (17), we get

Conclusion
An adaptive neural network control has been presented for a class of MIMO non-affine systems. Using the mean value theorem and Young's inequality, only one adaptive parameter is needed to be adjusted online for the whole MIMO system. The restrictions of unmodelled dynamics are relaxed by utilising RBFNNs approximating the unknown constructed function. The numerical simulation has verified the scheme effectiveness. In the future research work, we will extend the proposed results to a class of stochastic non-linear systems.

Acknowledgment
This study was partially supported by the National Natural Science Foundation of China (61573307).