Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1548095, 9 pages

https://doi.org/10.1155/2017/1548095

## Multidimensional Taylor Network Optimal Control of MIMO Nonlinear Systems without Models for Tracking by Output Feedback

^{1}School of Automation, Southeast University, Nanjing 210096, China^{2}Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Nanjing 210096, China

Correspondence should be addressed to Hong-Sen Yan; nc.ude.ues@naysh

Received 15 June 2017; Accepted 14 September 2017; Published 30 October 2017

Academic Editor: Quanxin Zhu

Copyright © 2017 Qi-Ming Sun and Hong-Sen Yan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The actual controlled objects are generally multi-input and multioutput (MIMO) nonlinear systems with imprecise models or even without models, so it is one of the hot topics in the control theory. Due to the complex internal structure, the general control methods without models tend to be based on neural networks. However, the neuron of neural networks includes the exponential function, which contributes to the complexity of calculation, making the neural network control unable to meet the real-time requirements. The newly developed multidimensional Taylor network (MTN) requires only addition and multiplication, so it is easy to realize real-time control. In the present study, the MTN approach is extended to MIMO nonlinear systems without models to realize adaptive output feedback control. The MTN controller is proposed to guarantee the stability of the closed-loop system. Our experimental results show that the output signals of the system are bounded and the tracking error goes nearly to zero. The MTN optimal controller is proven to promise far better real-time dynamic performance and robustness than the BP neural network self-adaption reconstitution controller.

#### 1. Introduction

In most input-output control systems, an alteration of one input signal may trigger the change of multiple output signals [1, 2]. However, in the actual control process, the system’s stability is highly demanded [3–5], along with its desirable dynamic performance and multiple tracking with less or even zero errors [6]. Because of the complex internal structure of the multi-input and multioutput (MIMO) system, satisfactory results are hardly gained as a result [7].

Support vector machine stemming from statistical theory is good at handling classification and regression problems. Its great progress has been observed in the fields of nonlinear control [8], fault diagnosis [9], and process modeling [10]. However, at present, it is only applicable to the single variable system due to its complexity.

Recently, great advances for adaptive output feedback control and robust control of nonlinear systems have been achieved [11–13]. Good approximation property of the neural networks provides a new access to adaptive output feedback control [14–16]. For general MIMO nonlinear systems, especially the systems without models, the control methods are always based on neural networks [17–19]. The neuron of neural networks includes the exponential function, which contributes to the complexity of calculation, making the neural network control unable to meet the real-time requirements. Though the larger the number of neural network nodes, the smaller the approximation error, neural network with a large hidden node number tends to complicate the control and the computation of the control system. So, it is difficult to realize the system’s real-time control.

Polynomial control method is an intuitive controller design approach. Its design parameters are of clear physical sense [20]. However, it requires an accurate model of the controlled object for solving MIMO nonlinear control problems.

The multidimensional Taylor network (MTN, whose idea was proposed by Hong-Sen Yan in 2010 and realization was completed by Bo Zhou who is Yan’s Ph.D. student) is commonly applied to the analysis of time series prediction [21–28]. MTN, a simple function of the state and input, is good at analyzing and solving the problems in point due to its polynomials. Besides, MTN only involves multiplication and addition, so its simple computation makes desirable real-time control possible. In fact, its computation complexity is nearly equal to that of Taylor expansion of a single neuron of neural network. The idea of MTN optimal control was proposed by Yan [29]. The optimal adjustment controller based on MTN was then developed [30]. However, the parameters of the controller are fixed. The graduate students supervised by Yan have used the simple MTN to solve some practical control engineering problems successfully, such as the plane flight [31], the tank firing control in high speed motion [32], the axisymmetric cruise missile flight for attacking static target [33, 34], and ship roll stabilization [35], all involving MIMO nonlinear systems with strong disturbance. Simulation results show that the simple MTN promises faster response speed, stronger anti-interference capability, and better external stability than PID [31–35], PID neural network [31, 33], neural network [34], sliding mode control [31, 34, 35], and active disturbance rejection controller [33].

As has already been pointed out, the existing control methods are known to have some shortcomings, such as needing an accurate model of the controlled object, mainly aiming at the SISO nonlinear system. Some MTN control strategies have also been developed. However, they are just based on the simple MTN (i.e., PID plus the sum of their second-order monomials plus PID, each item of which is multiplied by its corresponding parameter), and the PID parameters are chosen as the initial parameters of MTN controller via the minimum principle, mainly targeting the SISO nonlinear systems. Proposed in the present study is the MIMO feedback control, which, unlike other approaches, does not require the specific format of MTN. Each input of the controlled object corresponds to a subset of internal weights of Multidimensional Taylor Network Controller (MTNC). Even without knowing the internal characteristic parameters of the controlled object, and only by adjusting the internal weights of MTNC, the outputs of the closed-loop system can be made to track the desired signals effectively. Superior to the BP neural network self-adaption reconstitution controller, the MTNC tracks the expectation output curves more satisfactorily as well as suppressing the disturbance more efficiently, promising a better real-time dynamic performance.

The main contributions of this paper are listed as follows:(1)For the first time, the MTN approach is used to solve the tracking control problem of MIMO nonlinear systems. The computation burden is released due to the simple structure of MTN.(2)Even with modeling errors, the MTNC proposed in this paper can still be able to satisfy certain performance indexes with excellent dynamic performance.(3)Compared with the existing control approaches, the proposed control method is simple and accurate.

#### 2. Problem Statement

Consider the following MIMO nonlinear system:

After differential homeomorphic transformation, (1) can be converted into (2) [36]. Without losing generality, consider the following MIMO nonlinear system:where is the relative order of the system, and ; is the output vector of the system; is the state vector of the system; is the input vector of the system; , is the unknown bounded nonlinear mapping; , is the known bounded nonlinear mapping.

Equation (1) can be rewritten aswhere is the control gain matrix; is the unknown and bounded disturbance.

From the concept of the dominant input, the MIMO system can be seen as multi-input single output system, and the th subsystem is

Select a dominant input in the input, and set . The rest is treated as interference; then we have

Setand (4) is equivalent to

Assuming only output can be measured, andthe expected output of the system isthe desired state vector of the th subsystem isand the tracking error is

To prove that the tracking error converges to 0, set the tracking error vector

Control target: for the affine nonlinear MIMO systems, without state observation, design the adaptive MTN optimal output feedback controller to make the nonlinear system (2) stable, enabling its output to track the desired signal and its parameters and tracking error to be uniformly bounded.

Define the filtering error aswhere , is the design constant.

Theorem 1. *Assuming system (5) with and is known and , the tracking error then converges to 0 if the optimal control input is set aswhere , is the design constant, and .*

*Proof. *Taking the Lyapunov function candidate aswe haveSubstituting and into (7) yieldsAccording to the Lyapunov theorem, this result means .

That completes the proof.

However, . So other methods are needed to suppress the interference. As is unknown, is unobtainable. Thus, the MTN is used to approximate , and is added to interference suppression.

#### 3. Multidimensional Taylor Network

MTN can approximate any nonlinear functions with the finite point of discontinuity. Compared with the existing methods based on neural networks, MTN has the following merits: (1) being neatly structured; (2) being good at representing or approximating to nonlinear dynamical systems; (3) guaranteeing real-time control by only addition or multiplication operations allowable. In addition, as with the neural networks, only the internal weights of MTNC are required to be adjusted to make the outputs of the closed-loop system track the desired signals effectively.

By the function approximation of MTN, we obtainwhere approximation error satisfies the following conditions:where is the given normal number, and

For the convenience of writing, denote the number of elements of as , and we have

The basic structure of MTN is shown in Figure 1.