Composite control of RBF neural network and PD for nonlinear dynamic plants using U-model

Abstract

In this paper, based on a pole placement PID controller designed, a composite control of RBF and PD is proposed for the tracking control of nonlinear dynamic systems. The proposed scheme combines the stability of PD and the ability of the RBF to approximate any function with any precision, with the control-oriented nature of the U-model to achieve exact tracking of nonlinear plants. The proposed structure has a more general appeal than many other models, such as polynomial NARMAX (Nonlinear Autoregressive Moving Average with Exogenous inputs) model and the Hammerstein model, etc. In addition, the control law is shown to be more simplistic in nature. The effectiveness of the proposed scheme is demonstrated with the help of simulations for the pole placement PID controller and the controller without the Newton-Raphson algorithm.

Keywords

Nonlinear U model system RBF neural networks newton iteration PD

1 Introduction

The area of nonlinear dynamic control has extreme significance in control research, especially with the rapid development of large machinery forward information, intelligence and comprehensiveness. In nonlinear plant research, the first step is how to model a plant. Because the plant has complex non-linear characteristic, it is difficult to describe it by a precisely mathematical model, which brings enormous challenges to model. The quality of the model determines the control effect of the system directly. In a long period of research, scholars have proposed a variety of nonlinear models, such as Hammerstein, Wiener, Bilinear, Nonlinear-FIR model, etc. These models have solved a lot of problems and achieved good results in the production process. However, when they describe the nonlinear plants in the broad-scale application, the versatility of mathematical expressions is not perfect enough. Therefore, putting forward a simple and general nonlinear model is essential to handle with this problem. On this background, the U model, a control oriented expression converted from original linear or nonlinear models, comes into being [1, 2]. It represents a wide class of smooth nonlinear systems. Note that the U model do not handle with the prototype model by any linearization, so that the accuracy of the model can be guaranteed. Furthermore, U-model presents an intuitive appeal and a straightforward algorithm structure to reduce computational burden in controller design with both linear and nonlinear systems. For the biggest advantage of the area of Adaptive Control is that unknown system parameters are tuned online and adjusted adaptively, an adaptive controller, based upon the U-Model, has been suggested by [3]. The authors in [4] have proposed online identification of Air Flow Plant using adaptive U-model to handle with the system parameters tuning adaptively and online. The paper [5] has focused on the design of multivariable underactuated nonlinear adaptive control using U-model methodologies, because U-model enables to include the coupling effect using the inverse Jacobian matrix. U-model based control system design methodology has been expanded into control of nonminimum-phase (NMP) dynamic systems in [6]. The authors in [7] proposes a pole placement PID Controller design procedure, within a general U-model pole placement control framework, for nonlinear systems. MIMO U-model based IMC has been used for the tracking control of multivariable nonlinear systems in [8].

As one of the earliest developed control design, PID possesses lots of advantages, such as simple algorithm, good robustness, high reliability and easy to implement, etc. Therefore, a pole placement PID is proposed based on U model for nonlinear plant. In order to enhance the control precision and response speed, a composite control of RBF and PD is proposed based on U model for nonlinear plant.

A radial basis function (RBF) neural network is trained to perform a mapping from an m-dimensional input space to an n-dimensional output space. RBF network can be used for discrete pattern classification, control, function approximation, signal processing, or any other application which requires a mapping form an input to an output. Because of its strong ability of learning and fault tolerance, RBF network has been attracted much attention in nonlinear control. A novel adaptive control scheme that incorporates fully tuned radial basis function (RBF) neural network has been proposed for the control of MEMS gyroscope with respect to external disturbances and model uncertainties by [9]. The paper [10] has presented an adaptive trajectory tracking neural network control using radial basis function (RBF) for an n-link robot manipulator with robust compensator to achieve the high-precision position tracking. RBF neural network consensus-based distributed control scheme has been proposed for nonholonomic autonomous vehicles in a pre-defined formation along the specified reference trajectory in [11]. Motion velocity and noise level in the sensor are chosen as the selector attributes, and the optimal sensor weightages under different attributes have been approximated using RBF neural network with the reference data from laser interferometer in [12]. The authors in [13] have established a virtual torque controller based on dynamic models of a lower exoskeleton and adopt an approximation of a Radial Basis Function (RBF) neural network to compensate for the dynamic uncertainty error. In the paper [14], an adaptive trajectory tracking controller based on extended normalized radial basis function network (ENRBFN) have been proposed for 3-degree-of-freedom four rotor hover vehicle subjected to external disturbance i.e. wind turbulence. In order to control the pitch angle of blades in wind turbines in the rated power zone, the controller of RBF network has been proposed in [15]. The paper [16] has proposed a model reference adaptive speed controller based on artificial neural network for induction motor drives, where RBF is utilized to adaptively compensate the unknown nonlinearity in the control system.

The rest of the paper are organized as follows. Section 2 presents a basic discussion on the U-model. A pole placement PID is analyzed in Section 3. Section 4 analyzes the working principle of RBF neural network. A composite control of RBF and PD based on the U-model is proposed in Section 5. An upgraded controller based on the controller presented in previous section is proposed in Section 6. Two nonlinear plants are selected to demonstrate the design procedures of three methods and the corresponding simulation results are presented with graphical illustration in Section 7. Section 8 contains some concluding remarks and suggestions for further research.

2 The expression form of U model

In 2002, the concept of U model is formally put forward in document [1]. It establishes a universal mapping, which can transform a wide class of nonlinear plants into plants designed by linear control, and the description forms of plants are as follows: $y (t) = \sum_{j = 0}^{M} a_{j} (t) u^{j} (t - 1) + e (t)$ (1)

In expression (1), y (t) is the output of plant, M is the degree of plant, a_j (t) is a time-varying parameters, including past input and past output, u (t - 1) is the control input of plant, e (t) is the error caused by the uncertain factors of nonlinear systems, such as modeling error, external disturbance and so on [17 –19]. The formula (1) further transforms the nonlinear system model: $U (t) = y (t) = \sum_{j = 0}^{M} a_{j} u^{j} (t - 1) + e (t)$ (2)

The Equation (2) is the U model expression. It is obvious that there is no linearization in the transformation between a wide class of nonlinear plants and the U model, which fully reflects the high precision of the U model. The U model possesses the following advantage:

The U model has a more general appeal as co-mpared to the polynomial NARMAX model, the Hammerstein model etc.

The U model presents all exiting smooth non-linear discrete time models and lots of linear continuous time models.

The U model exhibits a polynomial structure in the current control u (t - 1).

As its polynomial structure, the U model expression, which need to be solved to obtain the output value of the controller, are also polynomials in u (t - 1), unlike other models which lead to complex nonlinear algebraic equations.

3 U model based pole placement PID controller

Due to the universality and versatility of PID, a pole placement PID controller is designed for nonlinear dynamic plant based on U model. Based on the pole placement design of nonlinear U model, it completes this design by further transformed. The pole placement PID controller based on U model is shown in Fig. 1. The system needs to complete the requirements of the performance indicator of the system by the pole assignment based on the U model. According to the requirements of PID controller design, the parameters of pole placement are equivalently transformed into PID parameters. Furthermore, the performance indicator of the nonlinear control system is adjusted according to the adjustment methods of PID parameters. These parameters are also useful to realize online adjustment for system, such as fuzzy-PID, RBF-PID, GA-PID,etc. Meanwhile, the application of the U model reduces the difficulty and computational complexity encountered in the design of nonlinear control systems.

Fig.1

The Design of Pole Placement PID Controller.

The difference between reference and plant output, e (t) = w (t) - y (t), is viewed as the input signal of the pole placement PID controller. Substituting it into express (2), it shows: $U (t) = \frac{T}{R + S - T} e (t) = \frac{T}{A_{c} - T} e (t)$ (3) where w (t) is the reference input, R, T and S are the polynomials of the forward shift operator, which are described as R = qⁿ + r₁q^n-1 + ·· · + r_n, T = t₀q^m + t₁q^m-1 + … + t_m, S = s₀q^l + s₁q^l-1 + … + s_l, where q is the forward operator, n, m and l are the orders of the polynomials R, T and S respectively. Note that l < n, m ≤ n.

Standard digital control PID can be expressed as the ratio of the numerator and denominator polynomial. Make sampling period be unit, it show

$\begin{matrix} U (q) & = & K_{P} [1 + \frac{T_{s} q}{T_{i} (q - 1)} + \frac{T_{d} (q - 1)}{T_{s} q}] E (q) \\ = & K_{P} [\frac{(T_{i} T_{s} + T_{i} T_{d} + T_{s}^{2}) q^{2} - (T_{i} T_{s} + 2 T_{i} T_{d}) q + T_{i} T_{d}}{T_{i} T_{s} (q^{2} - q)}] * E (q) \end{matrix}$ (4) where K_p, T_i, T_d are constants of proportion, integral and differential of PID controller, T_s is sampling period. Make T_s = 1, then expression (4) can be simplified as:

$\begin{matrix} U (q) \\ = K_{P} [\frac{(T_{i} + T_{i} T_{d} + 1) q^{2} - (T_{i} + 2 T_{i} T_{d}) q^{2} + T_{i} T_{d}}{T_{i} T_{s} (q^{2} - q)}] \\ * E (q) \end{matrix}$ (5)

Because the PID controller is the secondary element, a plant with two stable closed-loop dominant poles can be selected to design pole placement controller. According to the requirements of the pole placement, make R = q² + r₁q + r₂, S = s₀q + s₁, T = t₀q² + t₁q + t₂, the expression of pole placement controller shows: $U (q) = \frac{T}{R + S - T} E (q) = \frac{T}{A_{c} - T} E (q)$ (6) where, A_c = R + S = q² + (r₁ + s₀) q + (s₁ + r₂) = q² + a₁q + a₂. By the further transformed, the expression can be showed as: $U (q) = [\frac{t_{0} q^{2} + t_{1} q + t_{2}}{(1 - t_{0}) q^{2} + (a_{1} - t_{1}) q + (a_{2} - t_{2})}] E (q)$ (7)

The parameters of the pole placement PID controller can be designed by the expression (5) and (7), it shows:

$\begin{matrix} K_{P} [\frac{(T_{i} + T_{i} T_{d} + 1) q^{2} - (T_{i} + 2 T_{i} T_{d}) q^{2} + T_{i} T_{d}}{T_{i} T_{s} (q^{2} - q)}] \\ = \frac{t_{0} q^{2} + t_{1} q + t_{2}}{(1 - t_{0}) q^{2} + (a_{1} - t_{1}) q + (a_{2} - t_{2})} \end{matrix}$ (8)

The parameter expression of pole assignment PID controller can be obtained: $\begin{matrix} K_{P} & = & 1 + a_{1} + a_{2} \\ T_{i} & = & - \frac{a_{1} + 2 a_{2}}{2 + a_{1} + a_{2}} \\ T_{d} & = & - \frac{a_{2} (2 + a_{1} + a_{2})}{(1 + a_{1} + a_{2}) (a_{1} + 2 a_{2})} \end{matrix}$

In conclusion, it is clear that the design pole placement PID is only computational equivalent. However, the physical representation of the controller gain K_p, time constants T_i and T_d are definitely different. The design of pole placement PID controller can be demonstrated by simulations.

4 RBF neural network

RBF neural network, including a hidden layer, is a three-layer feedforward neural network, which imitates the human brain features, such as the portion activity of human brain, partially covered with each other, indirectly related with each other and so on. RBF neural network possesses the function approximation, the optimal mapping ability, the rapid convergence features [20, 21]. In this paper, it adopts a SISO RBF neural network, the neural network structure as shown in Fig. 2.

Fig.2

Structure of SISO radial basis function neural network.

RBF neural network is composed of input layer, hidden layer and output layer. The input layer outputs the received value directly, that is, it does not do any processing. The weight between the input layer and the hidden layer is 1, completing the data transmission. The hidden layer deals with the received values by many algorithms, in main, including Gaussian function, reflected sigmoidal and inverse multiquadrics. In this paper, it selects the Gauss function according to the characteristics of the system design. The output layer processes the received values by summing. In Fig. 2, m is the node number of hidden layer, rin (t) is the input of the network, rout (t) is the output of the network, H = [h₁, h₂, ⋯ , h_j, ⋯ h_m] ^T is the radial vector of the network. h_j is the output of Gauss function, W = [w₁ (t) , w (t) ₂, ⋯ , w_j (t) , ⋯ , w_m (t)] ^T is weight matrix of the network. RBF neural network algorithm is as follows: $h_{j} = exp (- \frac{{∥ x - C_{j} ∥}^{2}}{2 b^{2}})$ (9)

C_j = [c_j1, c_j2, ⋯ , c_ji, ⋯ , c_jm] ^T is the vector of the central point of the Gauss function, b is the base-width parameter of Gauss function, The output of the neural network is: $rout (t) = w_{1} (t) h_{1} + w_{2} (t) h_{2} + \dots + w_{m} (t) h_{m}$ (10)

Performance target of network is: $J = \frac{1}{2} {(routm (t) - rout (t))}^{2}$ (11)

routm (t) is target output value.

According to gradient descent method, seeking partial derivative for w_j (t), and getting next formula: $Δ w_{j} (t) = - η \frac{\partial J}{\partial w_{j} (t)} h_{j} = η (routm (t) - rout (t)) h_{j}$ (12)

η, a positive constant number, is the learning rate of the network.

The iterative algorithm of output weights, as follows:

$\begin{matrix} w_{j} (t) & = & w_{j} (t - 1) + Δ w_{j} (t) \\ + α (w_{j} (t - 1) - w_{j} (t - 2)) \end{matrix}$ (13)

α, a positive constant number, is the inertia coefficient.

5 Composite control of RBF neural network and PD

The system adopts parallel control design method of RBF neural network and PD, in which RBF is the feedforward control shown in Fig. 1. The system adopts PD to complete the feedback control, which can ensure the stability of the system and restrain disturbance, and adopts RBF neural network to complete the feedforward control, which can ensure the speed of the system’s control response, decrease the overshoot and enhance the control precision. The control system structure is shown in Fig. 3.

Fig.3

RBF neural network and PID composite control structure diagram.

When the system begins, u₂ (t) = 0, U (t) = u₁ (t), the PD controller plays a major role. Through continuously learning about the output value of the PD, RBF is constantly adjusting the weights to make it become the inverse of the whole system (including the plant, the Newton iteration, and the PD), so that the ideal output can be accurately tracked.

The output of the RBF neural network is: $u_{2} (t) = w_{1} (t) h_{1} + w_{2} (t) h_{2} + \dots + w_{m} (t) h_{m}$ (14)

The PD output: $u_{1} (t) = k_{p} e (t) + k_{d} e (t)$ (15)

The input of system control: $u (t) = u_{1} (t) + u_{2} (t)$ (16)

Performance target of network is: $J = \frac{1}{2 c} {(u (t) - u_{2} (t))}^{2} = \frac{1}{2 c} u_{1}^{2}$ (17)

According to gradient descent method, seeking partial derivative for w_j (t), and getting next formula: $Δ w_{j} (t) = - η \frac{\partial J}{\partial w_{j} (t)} = η \frac{u (t) - u_{2} (t)}{c} = η \frac{u_{1} (t)}{c}$ (18)

η, a positive constant number, is the learning rate of the network.

The iterative algorithm of output weights, as follows:

$\begin{matrix} w_{j} (t) & = & w_{j} (t - 1) + Δ w_{j} (t) \\ + α (w_{j} (t - 1) - w_{j} (t - 2)) \end{matrix}$ (19)

α, a positive constant number, is the inertia coefficient.

6 Composite control of RBF neural network and PD based on model transformation

According to the formula (2), the nonlinear U model expression shows that the coefficient of nonlinear U model is time-varying, and the time-varying speed of different nonlinear system coefficients can not be estimated. RBF neural network algorithm studies continuously to complete system control based on nonlinear plant model, so the tracking speed of nonlinear system based on U model has some limitations, which causes the tracking error. In order to improve the response speed and control accuracy of nonlinear system, the Newton iteration algorithm is adopted to transform the nonlinear U model, which can reduce the time-varying speed of nonlinear model and improve the control accuracy of the system.

The Newton-Raphson iterative algorithm can solve the polynomial, which provides a conversion form the nonlinear plant model based on U expression. In order to adopt the linear control design method to obtain the output nonlinear model of controller, further transforming the formula (1) to the following form: $y (t) = U (t)$ (20)

Here $U (t) = \sum_{j = 0}^{M} α_{j} (t) u^{j} (t - 1) + e (t)$ , U (t) is the output of the controller. In the model transformation section, only to obtain a root of the nonlinear coefficient equation, it should be noted that the transformation of the U model does not lose any characteristics of the original nonlinear model, and improves the design efficiency and precision of the nonlinear control system. The output of Newton iterative formula is u (t - 1). The Newton iterative formula can be described as:

Fig.4

Structure of the improved control system.

$\begin{matrix} u_{k + 1} (t - 1) = u_{k} (t - 1) \\ - {\frac{\sum_{j = 0}^{M} α_{j} (t) u_{k}^{j} (t - 1) - U (t)}{d [\sum_{j = 0}^{M} α_{j} (t) u_{k}^{j} (t - 1)] / du (t - 1)} |}_{u^{j} (t - 1) = u_{k}^{j} (t - 1)} \end{matrix}$ (21)

k is the iteration number. According to the form of nonlinear plant, The Newton iteration formula is the inverse function of the U model. When the structure of the iterative algorithm is same with that of the U model, the system model transformation is right and loses the nonlinear part so that the system output can fully track the desired output. When the iterative algorithm is not completely inversed to the U model or the iteration number of the iterative algorithm is limited, the system model transformation has deviation so that it needs to design a control algorithm to complete the control requirements. But in realistic nonlinear systems, the U model can not describe the nonlinear plant exactly. Therefore, the system has been improved. The improved nonlinear control system is shown in Fig. 3.

It must be noted that the proposed design leads to an extremely simple and general control law. This approach is expected to prove vary useful in the area of nonlinear control.

The output of the RBF neural network is: $u_{2} (t) = w_{1} (t) h_{1} + w_{2} (t) h_{2} + \dots + w_{m} (t) h_{m}$ (22)

The PD output: $u_{1} (t) = k_{p} e (t) + k_{d} e (t)$ (23)

The input of system control: $U (t) = u_{1} (t) + u_{2} (t)$ (24)

The control input is obtained by Newton-Raphson algorithm as:

Performance target of network is: $J = \frac{1}{2 c} {(U (t) - u_{2} (t))}^{2} = \frac{1}{2 c} u_{1}^{2}$ (26)

According to gradient descent method, seeking partial derivative for w_j (t), and getting next formula: $Δ w_{j} (t) = - η \frac{\partial J}{\partial w_{j} (t)} = η \frac{U (t) - u_{2} (t)}{c} = η \frac{u_{1} (t)}{c}$ (27)

η, a positive constant number, is the learning rate of the network.

The iterative algorithm of output weights, as follows:

$\begin{matrix} w_{j} (t) & = & w_{j} (t - 1) + Δ w_{j} (t) \\ + α (w_{j} (t - 1) - w_{j} (t - 2)) \end{matrix}$ (28)

α, a positive constant number, is the inertia coefficient.

7 System simulation

In this section we propose a composite control of RBF and PD based on U model for tracking of two nonlinear system: stirred tank reactor and laboratory level control system. In the following we refer to the block diagrams given in Sections 3 and 4 (Figs. 2 and 3).

Solution to Plant 1

The example is verified by a nonlinear model of a continuous stirred tank reactor. This plant has strong nonlinear characteristics so that it is difficult to control. In order to prove the advantages of U model and the controller proposed in this paper, we take it. The system general model is $\dot{y} = - (1 + 2 a) y + au - uy - {ay}^{2}$ . y, a dimensionless parameter of a component concentration, is the output of the plant. u, a dimensionless parameter of the flow, is the control input. The discretization of the above equation becomes the form of the U model, as follows: $\begin{matrix} U (t) & = & a_{0} (t) + a_{1} (t) u (t - 1) + a_{2} (t) u^{2} (t - 1) \\ + a_{3} (t) u^{3} (t - 1) \end{matrix}$ where, $\begin{matrix} a_{0} (t) & = & 0.8606 y (t - 1) - 0.0401 y^{2} (t - 1) \\ + 0.0017 y^{3} (t - 1) - 0.000125 y^{4} (t - 1) \\ a_{1} (t) & = & 0.0464 - 0.045 y (t - 1) + 0.0034 y^{2} (t - 1) \\ - 0.00025 y^{3} (t - 1) \\ a_{2} (t) & = & - 0.0012 + 0.0013 y (t - 1) \\ - 0.0001458 y^{2} (t - 1) \\ a_{3} (t) & = & 0.00002083 - 0.00002083 y (t - 1) \end{matrix}$

RBF neural network structure adopts 1-7-1, the neural network without Newton iteration: learning rate is η = 0.2, inertial parameter α = 0.01, C initial value is $[\begin{matrix} - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \end{matrix}]$ , the initial value of B is 7, the weighted initial value is 0, the initial values of k_p, k_d are 0.4 and 0.01, the neural network with Newton iteration: learning rate is η = 0.3, inertial parameter α = 0.01, C initial value is $[\begin{matrix} - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \end{matrix}]$ , the initial value of B is 7, the weighted initial value is 0, the initial values of k_p and k_d are 0.1 and 0.02. The input signals are chosen square wave and triangle wave respectively for simulation experiments.

From Figs. 5 and 7, compared with the plant output of pole placement PID, the plant output of composite design of RBF and PD based on the continuous stirred tank reactor using U model is more accurate and faster response speed. From Figs. 6 and 8, compared with the control input of pole placement PID, the control input of composite design of RBF and PD is smoother so that this control design can decrease machine loss and extend its service life.

Fig.5

System response of Plant 1.

Fig.6

Control input of Plant 1.

Fig.7

System response of Plant 1.

Fig.8

Control input of Plant 1.

From Figs. 9 and 11, compared with the output of plant without Newton iteration, the output of plant with Newton iteration based on the continuous stirred tank reactor using U model is more accurate and faster response speed.

Fig.9

System response of Plant 1.

Fig.10

Control input of Plant 1.

Fig.11

System response of Plant 1.

Fig.12

Control input of Plant 1.

Solution to Plant 2

The example is to verify the rationality of the controller design by using the nonlinear model of the laboratory level control system. The device has strong nonlinear characteristics and it is difficult to control. In order to demonstrate the advantages of U model and the controller proposed in this paper, we take it. The U model of the liquid level control system is: $\begin{matrix} a_{0} (t) & = & 0.9722 y (t - 1) - 0.04288 y^{2} (t - 2) \\ + 0.1663 y (t - 2) u (t - 2) \\ + 0.2573 y (t - 2) e (t - 1) \\ - 0.03259 y^{2} (t - 1) y (t - 2) \\ - 0.3513 y^{2} (t - 1) u (t - 2) \\ + 0.3084 y (t - 1) y (t - 2) u (t - 2) \\ + 0.2939 y^{2} (t - 2) e (t - 1) \\ - 0.1295 u (t - 2) \\ + 0.6389 u^{2} (t - 2) e (t - 1) \\ a_{1} (t) & = & 0.3578 - 0.3103 y (t - 1) \\ + 0.1087 y (t - 2) u (t - 2) \\ + 0.4770 y (t - 2) e (t - 1) \end{matrix}$

RBF neural network structure adopts 1-7-1, the neural network without Newton iteration:learning rate is η = 0.3, inertial parameter α = 0.02, C initial value is $[\begin{matrix} - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \end{matrix}]$ , the initial value of B is 7, the weighted initial value is 0, the initial values of k_p, k_d are 0.3 and 0.03, the neural network with Newton iteration: learning rate is η = 0.4, inertial parameter α = 0.01, C initial value is $[\begin{matrix} - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \end{matrix}]$ , the initial value of B is 7, the weighted initial value is 0, the initial values of k_p and k_d are 0.3 and 0.03. The input signals are chosen square wave and triangle wave respectively for simulation experiments.

From Figs. 13 and 15, it clearly shows that compared with the pole placement PID, the composite control of RBF and PD based on the laboratory level system using U model improves the control precision and response speed. From Figs. 14 and 16, it clearly shows that compared with the pole placement PID, the composite design of RBF and PD improves the smoothness of control signal, which can reduce the unnecessary loss of machinery and improve the longevity of plant.

Fig.13

System response of Plant 2.

Fig.14

Control input of Plant 2.

Fig.15

System response of Plant 2.

Fig.16

Control input of Plant 2.

From Figs. 17 and 19, it clearly shows that the controller without Newton iteration can track the reference signal, but there is a large error. However, the controller with Newton iteration can improve the control precision and response speed. It shows that the application of Newton iteration can effectively reduce the nonlinear characteristics of nonlinear U-model, and provide an effective way for the application of linear control design method in nonlinear U-model.

Fig.17

System response of Plant 2.

Fig.18

Control input of Plant 2.

Fig.19

System response of Plant 2.

Fig.20

Control input of Plant 2.

8 Conclusion

In this paper, a model of nonlinear plant is established by adopting the U model of time-varying polynomial. The U-model covers all existing smooth non-linear discrete time model. The parallel control scheme of RBF neural network and PD is proposed based on the nonlinear U model. Considering the time-varying characteristic of nonlinear plant, the Newton iteration algorithm is proposed to transform to weaken the nonlinear characteristic, but it does not handle with the prototype model by any linearization to guarantee the accuracy of the nonlinear plant model. The simulations show that the controller with Newton iteration can improve the control precision and response speed. Meanwhile, the application of newton iterative algorithm transforms the nonlinear model, which reduces the design requirements of nonlinear control system and provides a new way for the application of some mature linear control methods to the design of nonlinear systems. This controller obtains a smooth control input of plants, which can reduce machine wear and improve the use of life cycle of mechanical equipment.

Footnotes

Acknowledgments

This work was supported by Heilongjiang Province Nature Science Foundation under Grant No. LC2015024.

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