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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.

In most input-output control systems, an alteration of one input signal may trigger the change of multiple output signals [

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 [

Recently, great advances for adaptive output feedback control and robust control of nonlinear systems have been achieved [

Polynomial control method is an intuitive controller design approach. Its design parameters are of clear physical sense [

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 [

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:

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.

Even with modeling errors, the MTNC proposed in this paper can still be able to satisfy certain performance indexes with excellent dynamic performance.

Compared with the existing control approaches, the proposed control method is simple and accurate.

Consider the following MIMO nonlinear system:

After differential homeomorphic transformation, (

Equation (

From the concept of the dominant input, the

Select a dominant input in the

Set

Assuming only output can be measured, and

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 (

Define the filtering error as

Assuming system (

Taking the Lyapunov function candidate as

Substituting

According to the Lyapunov theorem, this result means

That completes the proof.

However,

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

For the convenience of writing, denote the number of elements of

The basic structure of MTN is shown in Figure

The basic structure of MTN.

The output of MTN

Setting

The optimal parameter

Then the MTN output feedback controller can be written as

The control structure is shown in Figure

The control structure.

To ensure that the output

Specific adjustments go as follows:

The system structure is unknown. However, the influence of the real system by output

This alternative is feasible, as the numerical size of

Step

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.

The initial parameters of the system are

For the proposed MTNC, when the dimension

Output comparison.

The unit step response curve of the neural network self-adaption reconstitution controller is presented in Figure

Absolute error curve.

For demonstration of the antijamming capability of the controller, when

Output comparison.

From Figure

The absolute error curve is displayed in Figure

Absolute error curve.

When

Output comparison.

As indicated by Figure

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

Output comparison.

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

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.

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

The authors declare that they have no conflicts of interest.

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

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.