Passivity-based control for rocket launcher position servo system based on improved active disturbance rejection technology

Abstract

In order to achieve high motion accuracy and better robustness of the rocket launcher position servo system driven by a permanent magnet synchronous motor, a passivity-based controller based on improved active disturbance rejection control is proposed in this article. The convenient method of interconnection and damping assignment and passivity-based control is adopted to establish the port-controlled Hamiltonian system with dissipation model of permanent magnet synchronous motor. To further enhance the robustness and adaptability of the traditional active disturbance rejection controller, an improved active disturbance rejection control strategy–based radical basis function neural network is introduced to on-line update the proportional and derivative gains of improved active disturbance rejection controller. The results of numerical simulation and bench test indicate that the proposed improved active disturbance rejection control passivity–based control algorithm has advantages of smaller overshoot, fast response, small steady-state error, and strong robustness. It proves that the proposed control scheme is effective and suitable.

Keywords

Passivity-based control active disturbance rejection control rocket launcher position servo system permanent magnet synchronous motor radical basis function neural network

Introduction

Permanent magnet synchronous motors (PMSMs) have been widely applied as servo motors in high-dynamic and high-precision applications, owing to their advantages over other motors, such as simple mechanical structure, high power density, high torque-to-inertia ratio, high efficiency, fast response, and reliable operation.^1–4 Meanwhile, alternating current (AC) position servo system of rocket launcher is an extremely complicated system with strong nonlinearities and many uncertainties, such as widely changing moment of inertia and load moment, strong impact moment, and variable parameters of control system induced by its varying working conditions. The motion accuracy and the robustness of the rocket launcher position servo system are regarded as two important challenges to achieve better performance.^5,6 However, due to the inherent nonlinearities and the unavoidable uncertainties of the rocket launcher position servo system, it is very difficult to achieve higher control precision and higher quality effect by means of a traditional vector control method.^7,8

Along with the development of nonlinear theory, various effective control theories have already been investigated for the position servo system of PMSM, such as sliding mode control, fuzzy-sliding mode control, nonlinear sliding mode control, passivity-based control (PBC), adaptive control, predictive functional control, explicit model predictive control, iterative learning control, robust control, and recurrent Elman neural network.^9–19 However, the above-mentioned advanced control methods have the merits and drawbacks, respectively, which can significantly improve the robustness of control system, but at the expense of its motion accuracy. For instance, the robustness of sliding mode control comes at the cost of the well-known chattering and the phase delay.

In order to obtain high-dynamic tracking accuracy, a novel effective control strategy is needed to resolve the conflicts between the motion accuracy and the robustness of the servo control system. Fortunately, active disturbance rejection control (ADRC) is a novel robust control method, which has been successfully applied in motor drive system for about 10 years.^8,20–22 ADRC does not depend on the accurate plant model, and it has some advantages, such as fast response speed, better robust, and high control precision. However, although ADRC has strong robustness and good adaptability, the control parameters of ADRC controller are needed to be continuously adjusted to obtain better performances during the control process, which lead to poor operation. Then, the ADRC-based control schemes for PMSM have been reported in the control research literatures, such as fuzzy-adaptive disturbance rejection control and sliding mode–adaptive disturbance rejection control.^7,13

Ortega et al.²³ first developed a new PBC theory, which is called interconnection and damping assignment and passivity-based control (IDA-PBC). Petrovic et al.²⁴ first applied the IDA-PBC controller for PMSMs. It can be easy to design and analyze stability of the system controller with IDA-PBC, for its “almost” global stability. Nowadays, a passive control method receives more and more recognition and application.²⁵

The remaining part of this article is organized as follows: The port-controlled Hamiltonian system with dissipation (PCHD) model of PMSM is established in section “The PCHD model of PMSM.” The primary concept of ADRC is introduced briefly in section “A preliminary to ADRC.” A passivity-based controller is designed for PMSM in section “The PBC system of PMSM.” A passivity-based controller based on improved active disturbance rejection control (IADRC) is applied for rocket launcher position servo system. Aiming to enhance the control performance, an on-line radial basis function neural network supervision (RBFNNS) is adopted to optimize the control parameters of IADRC in section “The IADRC-PBC controller design.” The proposed approach is verified using MATLAB/Simulink in section “Simulation result and analysis.” The effectiveness of the proposed method is verified by a series of tracking experiments, which is conducted on the semi-physical simulation platform in section “Semi-physical simulation test.” Finally, some conclusions are presented in section “Conclusion.”

The PCHD model of PMSM

Principle of the PCHD system

The structure of diagram of an AC servo system is given in Figure 1, where the bad effects of working conditions, disturbances, and environmental changes are not described. The mathematical model properties of multivariate, nonlinear, and strong coupling are directly determined by the physical structure of AC motor. However, the complex function of electromagnetic torque is composed of the stator current, the flux linkage of the stator and the rotor, and the electromagnetic parameters of the stator and the rotor, which vary with motor temperature. However, it is very difficult to reach better control performance for PMSM with traditional vector control methods.

Figure 1.

The structure diagram of AC servo system.

PCHD system is a kind of passive system, which has attracted more and more attentions by the researchers in control field for its “almost” global stabilization objectives.²⁶ Port-controlled Hamiltonian (PCH) system is defined as

{\begin{matrix} \overset{\cdot}{x} = J (x) \frac{\partial H (x)}{\partial x} + g (x) u \\ y = g^{T} (x) \frac{\partial H (x)}{\partial x} \end{matrix}

(1)

Energy dissipation is imported in the framework of PCH system by terminating some ports with resistive elements.

Then, PCHD can be presented by

{\begin{matrix} \overset{\cdot}{x} = [J (x) - R (x)] \frac{\partial H (x)}{\partial x} + g (x) u \\ y = g^{T} (x) \frac{\partial H (x)}{\partial x} \end{matrix}

(2)

where $x \in R^{n}$ is the state vector; $u, y \in R^{m}$ is the input and output vectors, respectively; $J (x)$ is the matrix of internal interconnection, $J (x) = - J^{T} (x)$ ; $R (x)$ is the dissipation matrix; and $H (x)$ is the stored energy function.

PCHD model of PMSM

Regardless of hysteresis losses, damping torque coefficients, and friction coefficient, the model of PMSM can be described in a synchronously rotating $d - q$ reference frame as

{\begin{matrix} L_{d} \frac{d i_{d}}{dt} = - R_{s} i_{d} + n_{p} ω_{r} L_{q} i_{q} + u_{d} \\ L_{q} \frac{d i_{q}}{dt} = - R_{s} i_{q} + n_{p} ω_{r} L_{d} i_{d} - n_{p} ω_{r} ψ_{r} + u_{q} \\ J \frac{d ω_{r}}{dt} = T_{e} - T_{L} = n_{p} [ω_{r} i_{q} + (L_{d} - L_{q}) i_{d} i_{d}] - T_{L} \end{matrix}

(3)

where $u_{d}$ , $u_{q}$ , $i_{d}$ , and $i_{q}$ are the stator voltages and the stator currents of the d- and q-axes components, respectively; $ψ_{r}$ is the rotor flux of the permanent component; $L_{d}$ and $L_{q}$ are the rotor inductance of the d- and q-axes components, respectively; $R_{S}$ is the stator winding resistance; $n_{p}$ is the polar pair; J is the rotor inertia; $T_{e}$ is the electromagnetic torque; and $T_{L}$ is the load torque.

The state vector and the input and output vector of the PMSM system can be defined as

\begin{matrix} x = [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] = [\begin{matrix} L_{d} i_{d} \\ L_{q} i_{q} \\ J ω_{r} \end{matrix}] = D [\begin{matrix} i_{d} \\ i_{q} \\ ω_{r} \end{matrix}] \\ u = [\begin{matrix} u_{d} \\ u_{q} \\ T_{L} \end{matrix}] \\ y = [\begin{matrix} i_{d} \\ i_{q} \\ ω_{r} / n_{p} \end{matrix}] \\ D = diag (L_{d}, L_{q}, J) \end{matrix}

The energy of the PMSM system is given by

H (x) = \frac{1}{2} (\frac{1}{L_{d}} x_{1}^{2} + \frac{1}{L_{q}} x_{2}^{2} + \frac{1}{J} x_{3}^{2})

(4)

The PCHD model of PMSM can be obtained by formula (2), which is shown as

{\begin{matrix} \overset{\cdot}{x} = [J (x) - R_{s} (x)] \frac{\partial H (x)}{\partial x} + g (x) u \\ y = g^{T} (x) \frac{\partial H (x)}{\partial x} \end{matrix}

(5)

where $J (x) = [\begin{matrix} 0 & 0 & n_{p} x_{2} \\ 0 & 0 & - n_{p} (ψ_{r} + x_{1}) \\ - n_{p} x_{2} & n_{p} (ψ_{r} + x_{1}) & 0 \end{matrix}]$ , $R_{s} (x) = [\begin{matrix} - R_{s} & 0 & 0 \\ 0 & - R_{s} & 0 \\ 0 & 0 & 0 \end{matrix}]$ , and $g (x) = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & - 1 \end{matrix}]$ .

A preliminary to ADRC

ADRC systematically proposed by Han is a novel robust control method developed from nonlinear proportional–integral–derivative (PID).^27,28 As illustrated in Figure 2, the conventional ADRC is composed of a tracking differentiator (TD), an extended state observer (ESO), and a nonlinear state error feedback (NLSEF) control law. The key technology of ADRC is its strong robustness, which can automatically compensate for internal and external disturbance estimations.

Figure 2.

Schematic diagram of the ADRC.

Nonlinear TD

There are many advantages of TD employed in the ADRC, such as smooth regulation of tracking reference signal, rapidity and stability for tracking without overshoots, and elimination of outliers for avoiding the rapid fluctuations of the control signal. Compared to the traditional TD, the structure of nonlinear tracking differentiator (NTD) is nonlinear, and the tracking efficiency of NTD is far better than that of TD.^20,28 The definition of the NTD process can be described as

{\begin{matrix} x_{1} (k + 1) = x_{1} (k) + h x_{2} (k) \\ x_{2} (k + 1) = x_{2} (k) + hfst (x_{1} (k) - y_{d} (k), x_{2} (k), r, h_{0}) \end{matrix}

(6)

where $y_{d} (k)$ is the desired trajectory of the control system, h is the sampling step, $h_{0}$ is the filtering factor, r is the speed factor, and $x_{1} (k)$ and $x_{2} (k)$ track $y_{d} (k)$ and ${\overset{\cdot}{y}}_{d} (k)$ , respectively.

In general, large $h_{0}$ helps to reduce noise and big r helps to raise tracking efficiency. The function of $fst (\cdot)$ is a nonlinear operator, which can be expressed as

{\begin{matrix} fst (x_{1} (k), x_{2} (k), r, h) = {\begin{matrix} - r (a / d), | a | \leq d \\ - r sgn (a), | a | > d \end{matrix} \\ a = {\begin{matrix} x_{2} (k) + \frac{a_{0} - d}{2} sgn (y (k)), | y (k) | > d_{0} \\ x_{2} (k) + \frac{y (k)}{h}, | y (k) | \leq d_{0} \end{matrix} \\ d = rh, d_{0} = dh, a_{0} = \sqrt{d^{2} + 8 r | y (k) |} \\ y (k) = x_{1} (k) + h x_{2} (k) \end{matrix}

(7)

Extended state observer

Under the action of ESO, the ADRC can actively estimate and compensate for the effects of the unknown dynamics and disturbances in real time, so as to make an otherwise unknown plant to run like a nominal one.²⁹ A commonly employed third-order ESO can be built as

{\begin{matrix} e (k) = z_{1} (k) - y (k) \\ z_{1} (k + 1) = z_{1} (k) + h [z_{2} (k) - β_{01} e (k)], \\ z_{2} (k + 1) = z_{2} (k) + h [z_{3} (k) - β_{02} fal (e (k), α_{1}, δ_{0})] + b_{0} u (k) \\ z_{3} (k + 1) = z_{3} (k) - h β_{03} fal (e (k), α_{2}, δ_{1}) \end{matrix}

(8)

where $z_{1} (k)$ and $z_{2} (k)$ represent the tracking signal of $y (k)$ and the differentiation of $z_{1} (k)$ , respectively; $z_{3} (k)$ is the extended state, which represents the estimated total disturbance; and $α_{1}$ , $α_{2}$ , $δ_{0}$ , $δ_{1}$ , and $b_{0}$ are the parameters decided by the control system.

The nonlinear function $fal (e (k), α, δ)$ can be described as

fal (e (k), α, δ) = {\begin{matrix} {| e (k) |}^{α} sgn (e (k)) | e (k) | > δ \\ \frac{e (k)}{δ^{1 - α}} | e (k) | \leq δ \end{matrix}

(9)

Under three orders, $β_{01}$ , $β_{02}$ , and $β_{03}$ are the observer gains, which can be determined according to equation (10)²⁰

β_{01} = 1, β_{02} = 1 / 2 \sqrt{h}, β_{03} = 2 / 5^{2} h^{1.2}

(10)

NLSEF controller

Comparing the difference between the outputs of NTD and the outputs of ESO, the NLSEF is applied to drive the state error to zero point. The NLSEF controller is a nonlinear proportional derivative (PD) controller and can be given as

{\begin{matrix} e_{1} (k + 1) = x_{1} (k + 1) - z_{1} (k + 1) \\ e_{2} (k + 1) = x_{2} (k + 1) - z_{2} (k + 1) \\ u_{0} (k + 1) = β_{1} fal (e_{1} (k + 1), α_{3}, δ_{2}) \\ + β_{2} fal (e_{2} (k + 1), α_{4}, δ_{3}) \end{matrix}

(11)

where $α_{3}$ , $α_{4}$ , $δ_{2}$ , and $δ_{3}$ are the parameters decided by the control system and $β_{1}$ and $β_{2}$ represent the proportional and derivative gains, respectively.

The PBC system of PMSM

PBC controller design

According to the principle of maximum torque per ampere (MTPA), the desired equilibrium of the PMSM system equation (3) can be acquired by

x^{*} = {(\begin{matrix} x_{1}^{*}, & x_{2}^{*}, & x_{3}^{*} \end{matrix})}^{T} = {(\begin{matrix} L_{d} i_{d}^{*}, & L_{q} i_{q}^{*}, & J ω^{*} \end{matrix})}^{T}

(12)

To the implicit PMSM, it is well known that $L_{d} = L_{q}$ . According to equation (3), the PMSM runs steady at the desired speed, when the control strategy of $i_{d} = 0$ is adopted. ${\overset{\cdot}{x}}_{3}^{*} = 0$ can be obtained that

{\overset{\cdot}{x}}_{3}^{*} = n_{p} [ω_{r} i_{q}^{*} + (L_{d} - L_{q}) i_{d}^{*} i_{d}^{*}] - T_{L} = 0

(13)

According to equation (13), when the control strategy of $i_{d}^{*} = 0$ is adopted, we can obtain

i_{q}^{*} = \frac{T_{L}}{n_{p} ω_{r}}

by the above analysis, and the desired equilibrium of the PMSM system equation (3) can be expressed as

x^{*} = {(\begin{matrix} x_{1}^{*}, & x_{2}^{*}, & x_{3}^{*} \end{matrix})}^{T} = {(\begin{matrix} 0, & L_{q} \frac{T_{L}}{n_{p} ω_{r}}, & J ω^{*} \end{matrix})}^{T}

(14)

According to the desired equilibrium equation (12), the desired Hamiltonian function of the closed-loop system is built as

\begin{matrix} H_{d} (x) = H (x) + H_{a} (x) \\ = \frac{1}{2} {(x - x^{*})}^{T} D^{- 1} (x - x^{*}) \end{matrix}

(15)

In order to satisfy the bound equation (16) at x, where x is at any location in the domain of $x^{*}$ and $x \neq x^{*}$ , the feedback control law is chosen as in the form of equation (17). Then, the state-space function of the closed-loop system can be described as in the form of equation (18)

H_{d} (x) > H_{d} (x^{*})

(16)

u = - Ky

(17)

\overset{\cdot}{x} = [J_{d} (x) - R_{d} (x)] \frac{\partial H_{d} (x)}{\partial x}

(18)

where $J_{d} (x) = J_{a} (x) + J (x)$ and $R_{d} (x) = R_{a} (x) + R (x)$ .

Proposition 1

Defining v as a new input to the system equation (2) and considering the feedback, the control law is described as

u = - Ky + v

(19)

For the implicit PMSM, equation (20) is satisfied as

[J (x) - R (x)] \frac{\partial H_{a} (x)}{\partial x} = g (x) ν

(20)

$H_{a} (x)$ meets

\begin{matrix} \frac{\partial^{T} H_{a}}{\partial x} (x) J_{d} (x) \frac{\partial H_{a} (x)}{\partial x} = 0 \\ \frac{\partial^{T} H_{a}}{\partial x} (x^{*}) = - \frac{\partial H (x^{*})}{\partial x}, that is, \frac{\partial^{T} H_{d}}{\partial x} (x^{*}) = 0 \\ \frac{\partial^{T} H_{a}}{\partial x} (x^{*}) > - \frac{\partial H (x^{*})}{\partial x}, that is, \frac{\partial^{T} H_{d}}{\partial x} (x^{*}) > 0 \end{matrix}

As mentioned above, under the control of the control rule (equation (19)), the state-space function of the closed-loop system can be described as in the form of equation (21) and $x^{*}$ is the desired equilibrium of the closed-loop system

\overset{\cdot}{x} = [J (x) - R (x)] \frac{\partial H_{d} (x)}{\partial x}

(21)

According to Proposition 1, we can obtain the following:

ν = - g^{- 1} (x) [J (x) - R (x)] D^{- 1} x^{*}

(22)

where $R (x) = R_{s} (x) + g^{T} (x) Kg (x)$ .

In particular, we define

K = [\begin{matrix} γ_{1} & 0 & 0 \\ 0 & γ_{2} & 0 \\ 0 & 0 & γ_{3} \end{matrix}]

(23)

Through solving equation (19), the controller can be defined as

\begin{matrix} u = {[\begin{matrix} u_{d}, & u_{q}, & T_{L} \end{matrix}]}^{T} \\ {\begin{matrix} u_{d} = - γ_{1} i_{d} + (R_{s} + γ_{1}) i_{d}^{*} - n_{p} L_{q} i_{q} ω^{*} \\ u_{q} = - γ_{2} i_{q} + (R_{s} + γ_{2}) i_{q}^{*} - n_{p} (L_{d} i_{d} + ψ_{a}) ω^{*} \\ T_{L} = γ_{3} ω + n_{p} (L_{d} i_{d} + ψ_{a}) i_{q}^{*} - γ_{3} ω^{*} \end{matrix} \end{matrix}

(24)

Due to the load disturbance and the unknown $T_{L}$ , $T_{L}$ can be estimated by the torque observer, which is described as

{\bar{T}}_{L} = n_{p} [(L_{d} - L_{q}) i_{d} i_{q} + ψ_{a} i_{q}] - J \frac{d ω}{dt}

(25)

According to formulas (24) and (25), the design of IDA-PBC controller is shown in the form of equation (26)

{\begin{matrix} u_{d} = - γ_{1} i_{d} + (R_{s} + γ_{1}) i_{d}^{*} - n_{p} L_{q} i_{q} ω^{*} \\ u_{q} = - γ_{2} i_{q} + (R_{s} + γ_{2}) i_{q}^{*} - n_{p} (L_{d} i_{d} + ψ_{a}) ω^{*} \\ {\bar{T}}_{L} = n_{p} [(L_{d} - L_{q}) i_{d} i_{q} + ψ_{a} i_{q}] - J \frac{d ω}{dt} \\ i_{q}^{*} = \frac{[T_{L} + γ_{3} (ω^{*} - ω)]}{n_{p} (L_{d} i_{d} + ψ_{a})} \end{matrix}

(26)

where $γ_{1}$ , $γ_{2}$ , and $γ_{3}$ are the interconnection and damping parameters, respectively.

Stability analyses of the PBC system

According to the feedback control law (equation (17)), we can obtain

\begin{matrix} u = - K g^{T} (x) \frac{\partial H_{d} (x)}{\partial x} \\ \frac{d H_{d} (x)}{dt} = \frac{\partial H_{d}}{\partial x} (x) \frac{dx}{dt} \\ = \frac{\partial H_{d}}{\partial x} (x) [J_{d} (x) - R_{d} (x)] \frac{\partial H_{d} (x)}{\partial x} + \frac{\partial H_{d}}{\partial x} (x) g (x) u \end{matrix}

The damping matrix is defined as $R_{d} (x) = g (x)^{T} Kg (x)$ and then

\begin{matrix} \frac{d H_{d} (x)}{dt} = \frac{\partial H_{d}}{\partial x} (x) J_{d} (x) \frac{\partial H_{d} (x)}{\partial x} \\ \frac{\partial H_{d} (x)}{\partial x} = D^{- 1} [x - x^{*}] \\ \frac{\partial^{2} H_{d} (x)}{\partial x^{2}} = D^{- 1} \end{matrix}

By the above study, we can obtain

\begin{matrix} \frac{\partial H_{d} (x)}{\partial x} = 0 \\ \frac{\partial^{2} H_{d} (x)}{\partial x^{2}} > 0 \end{matrix}

and the positive definition of the Hessian matrix $H_{d} (x)$ , when $x = x^{*}$ .

Therefore, the closed-loop system is asymptotically stable at equilibrium point.

The IADRC-PBC controller design

RBFNNS-IADRC control principle

The NLSEF of the traditional ADRC is optimized by the single neuron adaptive controller (SNAC). The nonlinear functions $fal [e_{1} (k), α_{3}, δ_{2}]$ and $fal [e_{2} (k), α_{4}, δ_{3}]$ are the inputs of SNAC. The diagram of the IADRC-PBC controller is shown in Figure 3, and its control object is the passivity-based control system (PBCS). Compared with the other control parameters of IADRC, the proportional and derivative gains $β_{1}$ and $β_{2}$ are the most key parameters, which have significant effects on the performance of the control system. For the sake of improving the robustness of the control system, the RBFNNS adopted in this article is able to adjust the weights $β_{1}$ and $β_{2}$ of SNAC on-line automatically.

Figure 3.

Diagram of the IADRC-PBC.

The output of the SNAC can be calculated as

u_{0} (k) = u_{0} (k - 1) + μ \sum_{i = 1}^{2} β_{i} x_{i} (k)

(27)

where $μ$ represents the ratio of single neurons and $μ > 0$ .

Adaptive updating of the proportional and derivative gains

The parameters’ adjustment goal of the RBFNNS-IADRC controller is to seek the control signal which can minimize the difference between the actual output and the desired output. The quadratic performance index of RBFNNS applied in this article for the parameter updating is defined as

E (k) = \frac{{[θ_{m} (k) - θ (k)]}^{2}}{2} = \frac{e^{2} (k)}{2}

(28)

With the gradient descent approach, the proportional and derivative gains $β_{1}$ and $β_{2}$ are calculated as

{\begin{matrix} Δ β_{i} (k) = - η_{1} \frac{\partial E}{\partial β_{i}} = - η_{1} \frac{\partial E}{\partial θ} \frac{\partial θ}{\partial u} \frac{\partial u}{\partial β_{i}} \\ = η_{1} e (k) \frac{\partial θ}{\partial u} x_{i} (k) \\ β_{i} = β_{i} (k - 1) + Δ β_{i} (k) \end{matrix}

(29)

where $i = 1, 2$ ; $η_{1}$ is the learning rate; and $\partial θ / \partial u$ denotes the Jacobian matrix of the control plant identified by the RBFNNS.

Its input vector is $R = [r_{1}, r_{2}]$ . We can define $r_{1} (k) = θ (k)$ and $r_{2} (k) = u (k)$ . The performance index of RBFNNS for its weight adjustments is defined as

\hat{E} (k) = \frac{{[θ (k) - \hat{θ} (k)]}^{2}}{2} = \frac{{\hat{e}}^{2} (k)}{2}

(30)

where $θ (k)$ and $\hat{θ} (k)$ represent the output and the identification value of RBFNNS, respectively.

With the gradient descent approach, the iterative functions for the weights of the RBFNNS are calculated as

\begin{matrix} ω_{j} (k) = & ω_{j} (k - 1) + Δ ω_{j} (k) \\ + α [ω_{j} (k - 1) - ω_{j} (k - 2)] \\ Δ ω_{j} (k) = & - η_{2} \frac{\partial E (k)}{\partial ω_{j} (k)} = η_{2} \hat{e} (k) h_{j} (k) \end{matrix}

(31)

where $ω_{j} (k)$ is the connection weight between the $j th$ $(j = 1, 2, 3, 4, 5)$ hidden node and the output node; the Gaussian basis function $h_{j} (k)$ is the $j th$ hidden node; $η_{2}$ is a learning factor of the RBFNNS; and $α$ is the momentum factor.

The iterative functions for the node center of the RBF hidden layer is defined as

\begin{matrix} b_{j} (k) = & b_{j} (k - 1) + Δ b_{j} (k) \\ + α [b_{j} (k - 1) - b_{j} (k - 2)] \\ Δ b_{j} (k) = & η_{2} \hat{e} (k) ω_{j} h_{j} \frac{{‖ X - C_{j} ‖}^{2}}{b_{j}^{3}} \end{matrix}

(32)

where $b_{j}$ is the width of the RBF node, $C_{j} = {[C_{j 1}, C_{j 2}]}^{T}$ is the center vector of the jth hidden node; and $X = {[r_{1}, r_{2}]}^{T}$ is the input vector.

The iterative functions for the node width of the RBF hidden layer is defined as

\begin{matrix} c_{ji} (k) = & c_{ji} (k - 1) + Δ c_{ji} (k) \\ + α [c_{ji} (k - 1) - c_{ji} (k - 2)] \\ Δ c_{ji} (k) = & η_{2} \hat{e} (k) ω_{j} \frac{r_{i} - c_{ji}}{b_{j}^{2}} \end{matrix}

(33)

where $r_{i}$ is the input of the RBFNNS and $r_{1} = y (k)$ and $r_{2} = u (k)$ .

According to the analysis above, the Jacobian matrix $\partial θ / \partial u$ of the controlled plant can be calculated as

\frac{\partial θ (k)}{\partial u (k)} \approx \frac{\partial \hat{θ} (k)}{\partial u (k)} = \sum_{j = 1}^{5} ω_{j} h_{j} \frac{c_{ji} - r_{1}}{b_{j}^{2}}

(34)

Simulation result and analysis

The main selected parameters in the AC system are as follows: load converted to the motor output shaft moment of inertia: $J = 5.556 \times 10^{- 3} kg m^{2}$ , reduction ratio $i = 1039$ , friction coefficient of the system is $B = 1.43 \times 10^{- 4} N \cdot m / (rad / s)$ , friction moment of the load: $T_{f} = 850 kg \cdot m^{2}$ , system load disturbing moment: $T_{L} = 9.32 \times 10^{3} kg \cdot m^{2}$ , motor torque coefficient: $K_{t} = 0.195 N \cdot m / A$ , the PMSM polar pair: $n_{p} = 3$ , the stator inductance of the implicit PMSM: $L_{d} = L_{q} = 21.24 mH$ , and motor flux magnetic flux: $ψ_{a} = 0.65 Wb$ .

In order to examine the control performance of IADRC-PBC for the rocket launcher system, a series of simulation was conducted using MATLAB/Simulink. The parameters in each part of the designed controller are estimated and shown as follows: the TD parameters: $h = 0.01$ , $h_{0} = 0.1$ , and $r = 1000$ ; the ESO parameters: $α_{1} = 0.5$ , $α_{2} = 0.25$ , $δ_{0} = 0.01$ , $δ_{1} = 0.01$ , $β_{01} = 1$ , $β_{02} = 15.8$ , $β_{03} = 318.5$ , and $b_{0} = 526.5$ ; the preset parameters of SNAC: $β_{1} = 10$ , $β_{2} = 5$ , $α_{3} = 0.6$ , $α_{4} = 0.3$ , $δ_{2} = 0.01$ , and $δ_{3} = 0.01$ ; the initial interconnection and damping parameters of the PBC controller: $γ_{1} = 10$ , $γ_{2} = 8$ , and $γ_{3} = 1$ ; the RBFNNS parameters: $η_{1} = 0.3$ , $μ = 0.25$ , $η_{2} = 0.4$ , and $α = 0.05$ ; and the optimized weights and thresholds of RBFNNS were taken as initial values for the control system, which is obtained by off-line identification of pseudorandom binary signal. The simulation results are shown in Figures 4 –6.

Figure 4.

Step response of load disturbance.

Figure 5.

System dynamic response curves.

Figure 6.

Tracking error curves of the system.

Figure 4 compares the step response of ADRC, ADRC-PBC, and IADRC-PBC controller through the position response curve added with a 300 N·m step disturbance at 3.0 s. The results of simulation data are shown in Table 1.

Table 1.

Step input simulation data.

Control strategy	Adjustment time (s)	Maximum disturbance value (°)	Recovery time (s)
ADRC	1.18	0.68	0.37
ADRC-PBC	1.03	0.36	0.32
IADRC-PBC	0.94	0.17	0.25

ADRC: active disturbance rejection control; PBC: passivity-based control; IADRC: improved active disturbance rejection control.

As it can be seen from Figure 4 and Table 1, the ADRC step response needs a long time to adjust; when there is a load disturbance, using the ADRC algorithm in response to a large location occurs, it needs 0.37 s to recover the reference position. At the stability and rapidity, the ADRC-PBC controller relative to the ADRC controller has better improvements, but poor accuracy. However, the IADRC-PBC controller has the shortest adjustment time and the fastest response speed; when disturbed, it can reach a steady state only in need of 0.25 s. The results of experiment indicate that IADRC-PBC controller has good interference ability and better robust.

A dynamic system is shown in Figure 5 to track the location of random interference, which added to the response curve. At the stability and accuracy, when the system parameters are uncertain and external disturbances appear, the ADRC-PBC controller relative to the ADRC controller has only minor improvements. The maximum tracking error of ADRC and ADRC-PBC controller is 0.079° and 0.056°, respectively. However, at the same working condition, the IADRC-PBC controller can enable the tracking servo to reach the reference position and suppress the impact of various uncertainties rapidly by an on-line RBF neural network supervision, which is designed to on-line update the proportional and derivative gains of IADRC. The maximum tracking errors of IADRC-PBC controller is just 0.026°, with an improved tracking accuracy by 115.4% than that of the ADRC-PBC controller. The results of experiment indicate that the IADRC-PBC controller has strong anti-interference ability and good dynamic performance.

Under the control of the ADRC, ADRC-PBC, and IADRC-PBC, the tracking errors in the constant speed tracking of the system are illustrated in Figure 6. A sinusoidal disturbance is added to the constant speed reference signal, which has a frequency of 0.5 Hz and amplitude of 0.4°. After entering the steady state, the maximum tracking error of the ADRC and ADRC-PBC controller is about 0.109° and 0.085°, respectively, while that of the IADRC-PBC controller is only about 0.019°, with an improved tracking accuracy by 3.47 times than that of the ADRC-PBC controller.

By comparing step responses and constant speed tracking of the ADRC, ADRC-PBC, and IADRC-PBC controllers, it demonstrates that the IADRC-PBC controller has advantages of higher accuracy, faster response, and stronger robustness than that of ADRC and ADRC-PBC controllers.

Semi-physical simulation test

To investigate the efficiency of the proposed IADRC-PBC algorithm as a strategy in establishing a rocket launcher control system, a semi-physical simulation platform is constructed to simulate the working conditions of the position servo control system. The structure diagram of semi-physical simulation platform is shown in Figure 7. The photograph of the semi-physical simulation platform is illustrated in Figure 8. Based on the components illustrated in Figure 8, the platform is composed of seven parts, including the test bed, the control computer, the power amplifier (PA), the loading fixture (LF), the precision reduction gearbox (PRG), the actuating motor (AM), and the sensor system for measurement. The LF consists of the rotational inertia plate (RIP) and the magnetic powder brake (MPB). The function of LF is to simulate the load torque, the rotational inertia, and the frictional resistance moment of the control system. Through changing the RIP, the rotational inertia variations in the loads can be simulated well. Similarly, the frictional resistance moment and the variations in the load torque can be simulated well by controlling the output torque of the MPB.

Figure 7.

Schematic of the semi-physical simulation platform.

Figure 8.

Photograph of the semi-physical simulation platform.

In order to verify the tracking accuracy of the position servo system with the IADRC-PBC controller, the sinusoidal command tracking with a frequency of 0.2656 Hz and amplitude of 45° is experimented on the semi-physical simulation platform. The corresponding tracking curves of ADRC, ADRC-PBC, and IADRC-PBC controllers are shown in Figure 9. From Figure 9, the maximum sinusoidal tracking error of the ADRC and ADRC-PBC controller is about 0.161° and 0.098°, respectively, while that of the IADRC-PBC control system is only about 0.054°.

Figure 9.

Sinusoidal tracking error curves.

On the basis of the calculation and analysis of the sinusoidal tracking errors of the ADRC, ADRC-PBC, and IADRC-PBC controller, the mean square error of the ADRC and ADRC-PBC controller is about 0.127° and 0.067°, respectively; while that of the IADRC-PBC controller is only about 0.038°, which can better meet that of the dynamic error indexes for system design requirements 0.216° or 3.6 mil ( $360^{\circ} = 6000 mil$ ).

By comparison, the IADRC-PBC controller can be able to increase the accuracy and robustness of control system effectively, which has a better steady-state and dynamic performance than that of the ADRC and ADRC-PBC controllers.

Conclusion

In this article, the IADRC-PBC control method has been proposed due to the inherent nonlinearities and the unavoidable uncertainties of the rocket launcher position servo system.

The numerical simulations show that compared with traditional ADRC controller, the RBFNNS-IADRC controller can better enhance the robustness and accuracy performance of the IADRC-PBC controller. When the step response is affected by a 300 N·m step disturbance, the maximum deviation of the IADRC-PBC controller is only about 47.2% of the ADRC-PBC controller. In terms of tracking the location of random interference, under the same working conditions, the tracking accuracy of the IADRC-PBC controller is improved by 115.4% than that of the ADRC-PBC controller. When the constant speed tracking is affected by a sinusoidal signal disturbance, the maximum tracking error of the IADRC-PBC controller is only about 19.4% of the ADRC-PBC controller.

The semi-physical simulation test results show that compared with ADRC and ADRC-PBC, the IADRC-PBC controller has a better performance of excellent dynamic and steady state. When the tracking reference is a sinusoidal signal with the frequency of 0.2656 Hz and the amplitude of 45°, the maximum sinusoidal tracking error of the IADRC-PBC controller is only about 33.5% and 55.1% of the ADRC and ADRC-PBC controllers, respectively.

From the simulation and prototype test results, it can be concluded that the IADRC-PBC controller has stronger robustness and better steady-state performance. So, it is proved that the proposed control scheme in this article is effective and suitable.

Footnotes

Handling Editor: Fakher Chaari

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work described in this article was supported by the National Natural Science Foundation of China (51305205 and 51275245).

ORCID iD

Ronglin Wang

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