2.1 Fuel supply subsystem model

Fuel supply subsystem consists of frequency converter, frequency conversion motor, quantitative pump, electro-hydraulic proportional throttle valve, pipeline-valve group and gear flowmeter, which provides aviation kerosene for combustor. Its basic working principle is: when the fuel flow rate required by the system is small, the speed of the frequency conversion pump is fixed, and the fuel flow rate is controlled by adjusting the opening of the proportional throttle valve; when the fuel flow rate required by the system is large, the proportional throttle valve is closed, and the fuel flow rate is controlled by controlling the output frequency of the frequency converter. In order to simplify the problem, this paper only considered the case of large flow rate; that is, frequency conversion pump control mode. Under this working mode, the fuel supply subsystem is mainly composed of frequency converter, motor, quantitative pump and pipeline, and the bypass proportional valve does not work. The transfer function between the fuel flow rate ${q_f}$ and the control voltage of the frequency converter can be obtained by referring literature^{(Reference Li, Cai, Lee and Teng33)}.

(1) \begin{equation}{G_{f1}}(s) = \frac{{{q_f}(s)}}{{u(s)}} = \frac{{{b_0}}}{{{a_2}{s^2} + {a_1}s + {a_{\rm{0}}}}}\end{equation}

where ${b_0} = \frac{1}{{60}}{K_1}{K_{\rm{f}}}{K_{\rm int}}{D_{\rm{p}}}$ ; ${a_2} = \frac{\pi }{{30}}{J_{\rm{T}}}{C_p}L$ ; ${a_{1} =\frac{\pi }{30} J_{{\rm T}} +K_{{\rm 3}} L+\frac{\pi }{30} J_{{\rm T}} C_{{\rm p}} R+C_{{\rm p}} L(K_{{\rm 2}} +\frac{\pi }{30} B_{{\rm T}})}$ ; ${a_{0} =(1+C_{p} R)\cdot (K_{2} +\frac{\pi }{30} B_{{\rm T}} {\rm )+}K_{{\rm 3}} R}$ ; ${K_1} = \frac{{{\rm{3}}{m_{\rm{p}}}}}{{{\rm{2}}\pi {R_{{\rm{re}}}}}}{K_{\rm{f}}}$ ; ${K_2} = \frac{{m_{\rm{p}}^{\rm{2}}}}{{{\rm{40}}\pi {R_{{\rm{re}}}}}}$ ; ${K_{\rm{3}}}{\rm{ = }}\frac{{D_{\rm{p}}^{\rm{2}}}}{{120\pi {\eta _{\rm{m}}}}}$ ; $R = {R_{\rm{L}}} + {R_{\rm{N}}}$ ; ${K_{\rm{f}}}$ is the gain between the frequency f and control voltage $u$ ; ${K_{\rm int}}$ is the voltage-frequency ratio; ${m_{\rm{p}}}$ is number of the motor pole pairs; ${R_{{\rm{re}}}}$ is equivalent resistance of the rotor; ${J_{\rm{T}}}$ is the rotational inertia of the motor shaft; ${B_{\rm{T}}}$ is damping coefficient of the motor shaft; ${D_{\rm{p}}}$ is displacement of the pump; ${\eta _{\rm{m}}}$ is the mechanical efficiency of the pump; ${C_{\rm{p}}}$ is the whole leakage coefficient of the fuel circuit; $L = \frac{{4{\rho _f}l}}{{\pi {d^2}}}$ is liquid inductance; ${R_{\rm{L}}} = \frac{{128\mu l}}{{\pi {d^4}}}$ is liquid resistance of the pipeline; ${R_{\rm{N}}} = \frac{{\sqrt {2{\rho _f}p_2^{*}} }}{{{\rm{6}}{C_{\rm{d}}}{A_0}}}$ is liquid resistance of the nozzle; $l$ is the length of the pipeline; d is the diameter of the pipeline; ${\rho _f}$ is the fuel density; $\mu $ is the dynamic viscosity of the fuel; ${C_{\rm{d}}}$ is flow coefficient of the nozzle; ${A_0}$ is the cross-sectional area of the nozzle; $p_2^{*}$ is the operating spout pressure.

Because the gear flowmeter used in the actual system has the time delay of $\tau $ seconds, the actual fuel flow rate ${q_{sf}}$ obtained during the control will lag $\tau $ seconds behind the theoretical fuel flow rate. It can be written as follows.

(2) \begin{equation}{q_{sf}}(s) = {q_f}(s){e^{ - \tau s}}\end{equation}

3.1 Flow rate control strategy

In order to solve the problems of pure time delay, time-varying parameters and disturbance that are not conducive to accurate control in fuel supply subsystem and air supply subsystem, a neural sliding mode predictive control algorithm based on RBF neural network was proposed in this paper, which combines intelligent control, sliding mode control and predictive control. The controller structure is shown in Fig. 1. The core idea of the algorithm is to use Levinson predictor to solve the problem of pure time delay in the system, use sliding mode control algorithm to solve the problem of time-varying parameters and disturbance in the system, and use RBF neural network to improve the bucket vibration phenomenon in sliding mode control.

1. (1) Design of sliding mode controller-based RBF neural network

Figure 1. Controller structure.

Suppose that the state equation of the controlled object is

(13) \begin{equation}\dot x = Ax + Bu\end{equation}

Assuming that the control instruction is $r(t)$ , and the sliding mode switching function is designed as follows:

(14) \begin{equation}s(t) = {c_1}e(t) + {c_2}de(t) + \cdots + {c_{n - 1}}{d^{\:(n - 2)}}e(k) + {d^{\:(n - 1)}}e(t)\end{equation}

where

(15) \begin{equation}e(t) = r(t) - {x_1}\end{equation}

(16) \begin{equation}de(t) = dr(t) - {x_2}\end{equation}

\begin{equation*} \vdots \end{equation*}

(17) \begin{equation}{d^{\:(n - 1)}}e(t) = {d^{\:(n - 1)}}r(t) - {x_n}\end{equation}

Assuming that $X = {[{x_1},{x_1} \cdots {x_n}]^T}$ is the input of RBF neural network, $H = [{h_1},{h_2}\cdots {h_j} \cdots {h_m}]^T$ is the output of hidden layer and ${h_j}$ is the Gauss function, then

(18) \begin{equation}{h_j} = \exp \left( {-} \frac{{{{\left\| {X - {C_j}} \right\|}^2}}}{{2b_j^2}}\right)\ j = 1,2, \cdots\!, m\end{equation}

where ${C_j} = {[{c_{j1}} \cdots {c_{jn}}]^T}$ , ${b_j} = {[{b_{j1}} \cdots {b_{jn}}]^T}$ , $m$ is the number of hidden layer.

Assuming that the weight vector of the network is $W = {[{w_1},{w_2} \cdots {w_j} \cdots {w_m}]^T}$ , then the output of the network is

(19) \begin{equation}u = {w_1}{h_1} + {w_2}{h_2} + \cdots + {w_m}{h_m}\end{equation}

Using sliding mode switching function $s(t)$ as the input of RBF neural network and output of RBF neural network as the output of sliding mode controller, the following result can be obtained.

(20) \begin{equation}u = \sum\limits_{j = 1}^m {{w_j}} \exp \left( {-} \frac{{{{\left\| {s - {c_j}} \right\|}^2}}}{{2b_j^2}}\right)\end{equation}

Since the control objective is to make $s(t)\dot s(t) \to 0$ , the weight adjustment objective function of RBF network can be set as follows:

(21) \begin{equation}E = s(t)\dot s(t)\end{equation}

Therefore, according to the gradient descent method, the weight learning algorithm of the neural network can be obtained:

(22) \begin{equation}d{w_j} = - \frac{{\partial E}}{{\partial {w_j}(t)}} = - \frac{{\partial s(t)\dot s(t)}}{{\partial {w_j}(t)}} = - \frac{{\partial s(t)\dot s(t)}}{{\partial u(t)}}\frac{{\partial u(t)}}{{\partial {w_j}(t)}}\end{equation}

Because

(23) \begin{equation}\frac{{\partial s(t)\dot s(t)}}{{\partial u(t)}} = s(t)\frac{{\partial \dot s(t)}}{{\partial u(t)}} = - s(t)\end{equation}

(24) \begin{equation}\frac{{\partial u(t)}}{{\partial {w_j}(t)}} = \exp \left( {-} \frac{{{{\left\| {s - {c_j}} \right\|}^2}}}{{2b_j^2}}\right)\end{equation}

so

(25) \begin{equation}d{w_j} = s(t)\exp \left( {-} \frac{{{{\left\| {s - {c_j}} \right\|}^2}}}{{2b_j^2}}\right) = s(t){h_j}\end{equation}

(26) \begin{equation}{w_j}(t) = {w_j}(t - 1) + \eta d{w_j} + \alpha [{w_j}(t - 1) - {w_j}(t - 2)]\end{equation}

where $\eta $ is the learning rate, $\alpha $ is the inertia coefficient.

1. (2) Levinson predictor

Levinson predictor is a method of predicting future d-step output variables by using historical data from simple moving average process. Suppose the output of predictor at k-time is

(27) \begin{equation}{y_m}(k) = - \sum\limits_{i = 1}^n {{a_i}} y(k - i)\end{equation}

then the predicted value of d-step ahead can be obtained

(28) \begin{align}{y_m}(k + d|k) = - {a_1}{y_m}(k + d - 1|k) - \cdots - {a_{p - 1}}{y_m}(k + 1|k) - {a_p}y(k) - \cdots - {a_n}y(k + d - n)\nonumber\\[-2pt]\end{align}

The parameters $\left\{ {{a_i}} \right\}$ in Equations (27) and (28) are the best predictive parameters. Usually, after the historical data of system output $\left\{ {y(k - 1),y(k - 2), \cdots y(k - n)} \right\}$ are obtained offline, the parameters can be estimated and determined by Levinson-Durbin algorithm.

4.1 Simulation parameters

According to the mathematic model established in previous section, the simulation model was built in Matlab. The simulation parameters were obtained by looking up tables and calculating, and they are shown in Tables 2 and 3.

Table 2 Model parameters of fuel supply subsystem

Table 3 Model parameters of air supply subsystem

4.2 Single channel flow rate control simulation

Accurate control of single-channel flow rate is the premise of realising coordinated control of double-channel flow rate. Therefore, this paper first used the proposed neural sliding mode predictive control algorithm to simulate the control of single-channel flow rate. In order to illustrate the control effect of the proposed control algorithm directly and clearly, the effectiveness of the control algorithm was only illustrated by the simulation results of fuel flow rate. In the simulation, the control period is 1s, and the target fuel flow rate is 1L/min. The parameters of the neural sliding mode controller are m = 5, C = [−3, −1.5, 0, 1.53], b = 2, and the advanced prediction step of Levinson predictor is five steps, the order of the predictor is six. The optimal parameters are {−1.1159, 0.0826, 0.0237, 0.0068, 0.0019, 0.00086}. In order to show the effect of the proposed neural sliding mode predictive control algorithm on the suppression of pure time delay, the fuel flow rate control simulation was carried out with and without predictor under the same simulation parameters. The simulation results are shown in Fig. 3. It can be seen from the figure that in the case of with predictor, fuel flow rate can be controlled quickly and accurately without overshoot, while in the case of without predictor, fuel flow rate appears overshoot, which shows that the setting of predictor is effective to compensate for the pure time delay of the system.

Figure 3. Control results of fuel flow rate with and without predictor.

From the modeling process of system mentioned above, it can be seen that the parameters of the system have time-varying characteristic. In order to verify the ability of the controller designed in this paper to overcome the time-varying parameters, the control simulation of the fuel flow rate was carried out by using the proposed control algorithm under the condition that the model parameters have changed and the control parameters remain unchanged. In the simulation, the target fuel flow rate is still 1L/min, the simulation results were carried out when the parameter a ₁ of the fuel supply subsystem increased by 20% (a ₁ = 816) and decreased by 20% (a ₁ = 544), and the simulation results are shown in Fig. 4. The simulation results show that although the control effect will be slightly worse when the system parameters have changed, the overall control effect can still be maintained. This shows that the control algorithm designed in this paper has the ability to overcome the time-varying parameters of the system.

Figure 4. Control results of fuel flow rate under model parameters changing.

In order to verify the anti-jamming capability of the proposed control algorithm, a simulation study was carried out under the condition that a voltage disturbance is added to the system at the 50^th second of the system response, and the other parameters of the controller remain unchanged. Figure 5 is the simulation results by using the proposed control algorithm when 0.1V and −0.1V disturbances were added to the control system, respectively. It can be seen from the figure that when disturbance is added to the system, the response of the system will be obviously affected; that is, the control curve deviates from the original state and overshoots occur, but under the action of the controller, the response curve of the system quickly restores to the original state, which shows that the neural sliding mode predictive control algorithm designed in this paper can be very good. It can overcome the disturbance in the system and achieve the predetermined goal of the control system.

Figure 5. Control results of fuel flow rate under disturbance.

4.3 Coordination control simulation

According to the analysis of the previous section, in order to ensure the high quality control of gas temperature of the SHSTS, besides maintaining the optimum proportion of fuel flow rate and air flow rate, the dynamic characteristics of fuel flow rate and air flow rate should be synchronized; that is, the fuel flow rate and air flow rate should be controlled coordinately. In order to achieve the coordinated control of fuel flow rate and air flow rate, the closed-loop response curves of fuel flow rate and air flow rate should be basically the same. Therefore, the response curves of fuel flow rate and air flow rate were basically consistent by adjusting the control parameters; it can be seen in Fig. 6. In order to ensure the optimum combustion ratio of fuel (aviation kerosene) to air (1:14.7), the fuel flow rate was set to 0.068kg/s and the air flow rate was set to 1kg/s in simulation. In order to make the response of the two flow rates consistent and easy to compare, the simulation data of fuel flow rate was multiplied by 14.7, and the simulation results can be found in Fig. 6. It can be seen from the figure that the closed-loop response curves of fuel flow rate and air flow rate are basically the same by adjusting the control parameters, which lay a solid foundation for the coordinated control of fuel flow rate and air flow rate.

Figure 6. The response curves of fuel flow rate and air flow rate tend to be consistent.

Figure 7. Coordinated control result when air supply subsystem with disturbance.

Figure 8. Coordinated control result when fuel supply subsystem with disturbance.

In order to verify that the proposed control algorithm can achieve the coordinated control of fuel flow rate and air flow rate, the coordinated control of fuel flow rate and air flow rate was simulated in two cases. In the first case, when the air supply subsystem was disturbed at the 50^th second of the simulation control and the change of fuel flow rate with air flow rate was observed; in the second case, when the fuel supply subsystem was disturbed at the 50^th second of the simulation control and the change of air flow rate with fuel flow rate was observed. The simulation results are shown in Figs 7 and 8. It can be seen from the figures that when the air flow rate response curve fluctuates due to the system interference, the fuel flow rate will fluctuate accordingly, that is to say, the fuel flow rate can change with the change of the air flow rate; conversely, when the fuel flow rate response curve fluctuates due to the system interference, the air flow rate can also follow. Therefore, the cross-coupling coordinated controller designed in this paper is effective and can realize the coordinated control of fuel flow rate and air flow rate.

5.1 Experimental equipment

In order to realise the control of fuel flow rate and air flow rate of the SHSTS and their coordinated control, based on the existing hardware of fuel supply subsystem and air supply subsystem, the measurement and control system based on the field PLC controller and remote industrial computer was developed. The experimental equipment is shown in Fig. 9.

Figure 9. Experimental equipment.

Figure 10. Control result of fuel flow rate.

Figure 11. Control result of air flow rate.

5.2 Single channel flow rate control experiment

Because the precise control of fuel flow rate and air flow rate is the basis of realising the coordinated control of them, it is necessary to study the separate control of fuel flow rate and air flow rate firstly. In the experiment, the fuel flow rate and air flow rate were adjusted to a fixed value firstly, and then step response experiments were carried out. The experimental results are shown in Figs 10 and 11. It can be seen from the figures that the system runs smoothly with no overshoot and the absolute control precision is ±0.01L/min in the process of the fuel flow rate stepping from 1L/min to 1.5L/min, which meets the predetermined control requirements. In the process of air flow rate stepping from 0.8kg/s to 1.0kg/s, the system achieves a good control effect, that is, the air flow rate is smoothly controlled without overshoot, and the control accuracy is ±2.5g/s.

5.3 Coordinated control experiment

In order to verify the effectiveness of the proposed coordinated control algorithm of fuel flow rate and air flow rate, the coordinated control experiment of fuel flow rate and air flow rate was completed. The control method used in the actual experiment is the same as the control algorithm used in the simulation, that is, the neural sliding mode predictive control algorithm and cross-coupling control were used to complete the coordinated control experiment. The experimental results are shown in Fig. 12. The setting value of fuel flow rate and air flow rate in the experiment are the same as those in the simulation, i.e. the air flow rate is 1.0kg/s and the fuel flow rate is 0.068kg/s. In the experiment, the two-channel flow rates were adjusted to the target flow rate, and then a disturbance was added to the air supply subsystem during the experiment, that is to say, the air flow rate will fluctuate upward and recover. The experimental results show that the change of air flow rate will lead to the change of fuel flow rate when the coordinated control is added, that is to say, the coordinated control of two-channel flow rates is realized. In addition, because the absolute error of fuel flow rate is about ±0.5g/s in the experiment, if multiplied by 14.7, the error will be magnified by 14.7 times, so the fuel flow rate is not multiplied by 14.7 times.

Figure 12. Coordinated control result of fuel flow rate and air flow rate.