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

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:

x (k + 1) = f (x (k), u (k)) y (k) = h (x (k), u (k)). (1)

After differential homeomorphic transformation, (1) can be converted into (2) [36]. Without losing generality, consider the following MIMO nonlinear system:

[mathematical expression not reproducible], (2)

where [[r.sub.1], [r.sub.2], [r.sub.m]] is the relative order of the system, and [r.sub.1] + [r.sub.2] +... + [r.sub.m] = n; y = [[y.sub.1], [y.sub.2]..., [y.sub.m]].sup.T] [member of] [R.sup.m] is the output vector of the system; [mathematical expression not reproducible] is the state vector of the system; u = [[u.sub.1], [u.sub.2], ..., [u.sub.m]].sup.T] [member of] [R.sup.m] is the input vector of the system; [f.sub.i] (x(k)), i = 1, 2, ..., m is the unknown bounded nonlinear mapping; [g.sub.ij] (x(k)), i, j = 1, 2, ..., m is the known bounded nonlinear mapping.

Equation (1) can be rewritten as

[mathematical expression not reproducible], (3)

where [mathematical expression not reproducible] is the control gain matrix; d = [[[d.sub.1], [d.sub.2], ..., [d.sub.m]].sup.T] is the unknown and bounded disturbance.

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

[mathematical expression not reproducible]. (4)

Select a dominant input in the m input, and set [u.sub.i](k). The rest is treated as interference; then we have

[mathematical expression not reproducible], (5)

[mathematical expression not reproducible], (6)

and (4) is equivalent to

[mathematical expression not reproducible]. (7)

Assuming only output can be measured, and

[g.sub.ii] (x (k)) [greater than or equal to] [g.sub.0] > 0,

[absolute value of dsi] [less than or equal to] D, where D is an unknown constant,

the expected output of the system is

[mathematical expression not reproducible], (9)

the desired state vector of the ith subsystem is

[mathematical expression not reproducible], (10)

and the tracking error is

[e.sub.i] (k) = [y.sub.di], (k) - [y.sub.i] (k). (11)

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

[e.sub.i] (k) = [x.sub.di] (k) - [x.sub.i] (k). (12)

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 as

[s.sub.i] (k) = [LAMBDA] * [e.sub.i] (k) = [[tau], [tau], ...,[tau]] * [e.sub.i] (k),

i = 1, 2, ... m, (13)

where [LAMBDA] = [[tau], [tau], ..., [tau]], [tau] > 0 is the design constant.

Theorem 1. Assuming system (5) with [f.sub.i] (x(k)) and [g.sub.ii](x(k)) is known and [d.sub.si] = 0, the tracking error then converges to 0 if the optimal control input is set as

[u.sup.*.sub.i] (k) = Ts (k) * [e.sub.i] (k) + [u.sup.*.sub.in] (k), (14)

where [u.sup.*.sub.in] (k) = ([y.sub.di](k + [r.sub.i] - [f.sub.i](x(k)) - [e.sub.i](k))/[g.sub.ii](x(k)), Ts(k) is the design constant, and Ts(k) * [e.sub.i] (k) * [g.sub.ii](x(k)) > 0.

Proof. Taking the Lyapunov function candidate as

[s.sub.i] (k) = [LAMBDA] * [e.sub.i] (k) = [[tau], [tau], ..., [tau]]* [e.sub.i] (k), (15)

we have

[mathematical expression not reproducible]. (16)

Substituting [u.sup.*.sub.i] (k) and [d.sub.si] = 0 into (7) yields

[DELTA][s.sub.i] (k) = -Ts (k) * [e.sub.i] (k) * [g.sub.ii] (x (k)) < 0. (17)

According to the Lyapunov theorem, this result means [lim.sub.k[right arrow][infinity]] [s.sub.i](k) = 0.

That completes the proof.

However, [d.sub.si] [not equal to] 0. So other methods are needed to suppress the interference. As [f.sub.i] (x(k)) is unknown, [u.sup.*.sub.in] (k) is unobtainable. Thus, the MTN is used to approximate [u.sup.*.sub.in] (k), and [u.sub.si](k) 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 obtain

[mathematical expression not reproducible], (18)

where approximation error [epsilon] satisfies the following conditions:

[absolute value of [epsilon]] [less than or equal to] [[epsilon].sub.0], (19)

where [[epsilon].sub.0] is the given normal number, and

[eta] (k)

= [[[e.sub.1] (k), [e.sub.i] (k-1), ..., [e.sub.i] (k), ..., [e.sub.m] (k),...].sup.T]. (20)

For the convenience of writing, denote the number of elements of [eta](k) as [n.sub.z], and we have

[mathematical expression not reproducible]. (21)

The basic structure of MTN is shown in Figure 1.

The output of MTN [u.sub.in](k) can be expressed as

[mathematical expression not reproducible], (22)

where N([n.sub.z], t) is the total number of the expansions, [w.sub.ij] (k) is the weight of the jth product term, [[lambda].sub.(s,j)] is the power of [z.sub.s] (k) in the jth product term, and [[summation].sup.n.sub.s=1] [[lambda].sub.s,j] [less than or equal to] t.

Setting [mathematical expression not reproducible] gives

[u.sup.*.sub.in](k) = [W.sup.*.sub.i] (k) * [THETA] ([eta]/(k)) + [epsilon]. (23)

The optimal parameter [W.sup.*.sub.i] (k) is defined as

[mathematical expression not reproducible], (24)

where [mathematical expression not reproducible] is the design constant.

Then the MTN output feedback controller can be written as

[mathematical expression not reproducible], (25)

where [mathematical expression not reproducible] is the output of the MTN, [[??].sub.i] (k) is the estimated value of the optimal weight [W.sup.*.sub.i] (k), and [u.sub.si] is the robust control term:

[u.sub.si] (k) = D + [g.sub.0] x [[epsilon].sub.0]/[g.sup.0] x sgn ([s.sub.i](k)). (26)

The control structure is shown in Figure 2.

4. Parameter Adjustment

To ensure that the output [y.sub.i](k) of the controlled object follows the desired input [r.sub.i](k), the parameters [W.sub.i](k) of MTN controller are adjusted by back-propagation learning algorithm.

Specific adjustments go as follows:

[mathematical expression not reproducible]. (27)

And we get

[w.sub.ij] (k+1) = [w.sub.ij] (k) - [[alpha].sub.j] (k) [partial derivative]E/[w.sub.ij](k) (28)

where

[mathematical expression not reproducible]. (29)

The system structure is unknown. However, the influence of the real system by output [u.sub.i](k) of controller can be obtained, so [partial derivative][y.sub.i](k)/[partial derivative][u.sub.i](k) can be replaced by the sign function

sgn [[y.sub.i] (k+1) - [y.sub.i] (k)/[u.sub.i]](k) - [u.sub.i](k-1)]. (30)

This alternative is feasible, as the numerical size of [partial derivative][y.sub.i] (k)/ [partial derivative][y.sub.i] (k), as an intermediate product term, can be adjusted by step [[alpha].sub.i](k). But it does not change the direction of the adjustment. Simultaneously, it can prevent the numerical size from going too large by the two-step output of the MTN convergence.

Step [[alpha].sub.i](k) uses the following adaptive adjustment strategy:

[[alpha].sub.j] (0) = [[alpha].sub.j0], (31)

where [[alpha].sub.j0] is a given constant as the initial value,

[mathematical expression not reproducible], (32)

where [beta] > 1, 0 < [gamma] < 1, and [parallel][partial derivative]E/[w.sub.ij](k)[parallel] is the norm of [partial derivative]E/[w.sub.ij](k).

5. Simulation Results

For demonstration of the effectiveness of the proposed method, consider the following nonlinear system. No specific parameters of the controlled object are needed in the control process.

[mathematical expression not reproducible], (33)

where Ts = 0.1.

The initial parameters of the system are

[mathematical expression not reproducible]. (34)

5.1. Step Response. For the proposed MTNC, when the dimension n of the controller is equal to 2, the unit step response curve is shown in Figure 3.

The unit step response curve of the neural network self-adaption reconstitution controller is presented in Figure 3, which reveals that the BP neural network self-adaption reconstitution controller responds slowly, while the MTNC has a much faster response speed. Figure 4 illustrates the absolute tracking error curve.

5.2. Antijamming Capability. For demonstration of the antijamming capability of the controller, when k is greater than 60, a sinusoidal signal with an amplitude of 0.12 is superimposed on [x.sub.2] and [x.sub.4], respectively, and the response curve is shown in Figure 5.

From Figure 5, it can be seen that the MTNC is of higher antijamming capability.

The absolute error curve is displayed in Figure 6.

5.3. Tracking Ability. When k > 10, with the expected output curves [y.sub.d1] and [y.sub.d2] overlaying the cosine signal with an amplitude of 0.1 to verify the follow-up response performance of the controller, the response curves are shown in Figure 7.

As indicated by Figure 7, the proposed design approach tracks the desired signal more quickly.

5.4. Robustness. The amplitude of disturbance is expanded 5 times to verify the robustness of the controller, and the response curve is given in Figure 8.

Experimental results demonstrate that the MTNC based on the multidimensional Taylor network is of better robustness for disturbance.

5.5. Summary. In short, by realizing the optimal closed-loop tracking control of the MIMO nonlinear systems through output feedback with no model involved, the MTNC guarantees its faster response speed, robustness, and antidisturbance capability.

Besides, the MTNC, with good real-time capability and low computational complexity, does not contain the exponential function, so it is easy to achieve real-time control. Taking ten nodes as an example and supposing that the highest order of the exponential function expansion of the neural network is three, the controller of the neural network needs 98 additions and 160 multiplications, while the MTNC needs only 18 additions and 30 multiplications. Therefore, the reduction of calculation complexity greatly improves the real-time performance of the proposed controller.

6. Conclusions

The MIMO nonlinear system control method based on the multidimensional Taylor network has been proposed here. Compared with the traditional methods, MTN, as a kind of dynamic model, can be seen as a system model. In the control process, the closed-loop tracking control of nonlinear system was realized by output feedback without state observation. Due to the Lyapunov function, the tracking error turned out to be nearly zero. The MTNC parameter adjustment algorithm without the specific parameters of the controlled object was developed as well.

During the experiments, the MTNC tracked the expected output curves satisfactorily, capable of suppressing the disturbance efficiently when a sinusoidal disturbance was superimposed on [x.sub.2] and [x.sub.4]; the real-time dynamic performance of the controller has been significantly improved due to the reduction of calculation complexity.

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors' Contributions

Qi-Ming Sun and Hong-Sen Yan contributed equally to this work.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grants 61673112 and 60934008, the Fundamental Research Funds for the Key Universities of China under Grants 2242017K10003 and 2242014K10031, and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). The authors thank Professor Li Lu for valuable comments and suggestions.

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Qi-Ming Sun (1,2) and Hong-Sen Yan (1,2)

(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; hsyan@seu.edu.cn

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

Academic Editor: Quanxin Zhu

Caption: Figure 1: The basic structure of MTN.

Caption: Figure 2: The control structure.

Caption: Figure 3: Output comparison.

Caption: Figure 4: Absolute error curve.

Caption: Figure 5: Output comparison.

Caption: Figure 6: Absolute error curve.

Caption: Figure 7: Output comparison.

Caption: Figure 8: Output comparison.
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Title Annotation:Research Article
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Publication:Mathematical Problems in Engineering
Date:Jan 1, 2017
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