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Abstract

In this article, a nonlinear model predictive control algorithm for a micro-turboshaft engine is designed. The control effect is verified by a bench test. First, a micro-turboshaft engine test bench is built, and the open-loop control experiment was carried out on it. Based on experiment data, a linear parameter varying prediction model is established. Then, by online rolling optimization based on multistep output prediction, together with feedback correction, a nonlinear model predictive control algorithm is obtained. The influence of algorithm parameters on the control effect is studied, and reasonable prediction period M, control period N, and control coefficient R are designed. Finally, the application of nonlinear model predictive control in micro-turboshaft engine is verified by bench test. The results show that with the changing of pitch angle, nonlinear model predictive control algorithm has a faster adjustment speed and smaller overshoot, compared with the conventional cascade proportional–integral control with feedforward. It is proofed that nonlinear model predictive control can be applied to a real turboshaft engine and has a better control effect.

Keywords

Micro-turboshaft engine nonlinear model predictive control linear parameter varying model online optimization bench test

Introduction

Model predictive control (MPC) is a model-based closed-loop optimization control strategy, which has been used for more than 15 years in the industry as an effective means to deal with multivariable constrained control problems.¹ Its basic principle is to use prediction model to predict the output of controlled object at multiple moments in the future according to historical information of control system and future control quantity and then calculate the optimal control quantity in the prediction period by rolling optimization method. MPC has achieved much progress on online optimization, stability, and performance issues for nonlinear systems and shows good robustness in the actual complex control process.² In view of unavoidable existence of many physical constraints and uncertainties in aero-engine control systems, and the increasing demands for accuracy and performance of aero-engine control systems, more and more researchers pay attention to MPC in aero-engine control field.^3,4

At present, MPC algorithm research is mainly for turbojet and turboprop engines, and conventional control of turboshaft engines still uses cascade proportional–integral (PI) control.^5,6 Cascade PI control cannot consider the influence of system time lag, so it often results in slow response and large fluctuation of power turbine speed. Turboshaft engine is a strong nonlinear system. Due to the huge amount of calculation, the MPC of turboshaft engine is always in numerical simulation stage.^7–12 There is almost no engine bench verification. The actual control effect needs to be further demonstrated.

This article takes the turboshaft engine as the research object. Based on an SPT-10 micro-engine made by JetCat Company in Germany, a micro-turboshaft engine test bench is built. Based on open-loop experimental data, a numerical fitting model of the engine is established. It describes the operating state above the engine idling speed. Based on this model, the research on nonlinear model predictive control (NMPC) of the turboshaft engine is carried out. A linear parameter varying (LPV) model near the engine power turbine command speed is established as the prediction model. An online optimization module is designed to continuously optimize the control quantity. A feedback correction module is designed to correct the control commands. A full digital simulation experiment of NMPC algorithm is carried out, and the influence of algorithm parameters on control effect is studied. Based on this, a bench test is carried out. The results show that the application of NMPC on turboshaft engine is feasible. Its control effect is better than that of cascade PI control with pitch angle feedforward.

Research platform establishment

Test bench establishment

The micro-turboshaft test bench built in this article is shown in Figure 1. The main sensors of the experimental system include speed, temperature, torque, and pressure sensors. The actuators include micro-engine’s electron magnetic valves of main and auxiliary fuel supply circuits, starting motor, igniter, electric fuel pump, and step motor and screw mechanism of variable pitch system. The electronic controller consists of a core controller and electric fuel pump controller. It is responsible for sensor signal acquisition, control law calculation, and actuator control. The test bench is equipped with a host computer for monitoring data and giving control instructions.

Figure 1.

Control system scheme of micro-turboshaft engine.

In the experimental platform established, operators can obtain all the data of the sensors, have full control over the actuators, and realize free design of control algorithm and control law.

Simulation model establishment

In order to design a control algorithm for a real engine, it is necessary to perform simulation verification first. It is difficult to obtain the specifications of micro-turboshaft engine and establish a component-level model. Based on the data obtained from open-loop test, this article establishes a fitting model using numerical steady-state model in series with Auto Regressive with External Input (ARX) dynamic model.^13,14 The model structure is shown in Figure 2. $u (k)$ is the model input, $v (k)$ is the output steady-state value corresponding to the input, and $y (k)$ is the output dynamic value.

Figure 2.

Numerical steady-state ARX series model.

The numerical fitting model of core engine speed $n_{g}$ and power turbine speed $n_{p}$ of micro-turboshaft engine are established using the above method, and multiple sets of verification data are prepared to verify the accuracy and generalization ability of the model.

The initial value of $n_{g}$ , the initial value of $n_{p}$ , and the dynamic data of $n_{pump}$ (electric fuel pump speed) in actual test are input into the model, and the model simulation results are compared with the actual data. The specified model simulation error is calculated by equation (1)

error = \frac{n_{s} - n_{t}}{n_{t}} \times 100 %

(1)

where $n_{s}$ is the simulation speed and $n_{t}$ is the actual speed.

Figure 3 shows a verification of the model. Figure 3(a) and (b) shows the fitting effects of $n_{g}$ and $n_{p}$ , respectively. This article uses the input of additional experimental data to obtain the output of the model, and comparing with the experimental data to verify the generalization ability of the model. It can be seen that the model simulation results can accurately reflect $n_{g}$ change. The maximum error between simulation and actual data is less than 3%, and root mean square error (RMSE) is 252.0 r/min, indicating that the model has a high fitting accuracy. The pitch angle $θ$ is $32.8125 °$ at 50–180 s, when the simulated $n_{p}$ is highly consistent with the actual data. $θ$ is $39.375 °$ at 180–300 s, when the simulated $n_{p}$ can accurately reflect trends of test data. The maximum error does not exceed 5%, and RMSE is 7.85 r/min. The model built has a high fitting accuracy.

Figure 3.

Comparison of experimental and fitting data: (a) $n_{g}$ fitting effect and (b) $n_{p}$ fitting effect.

It is worth mentioning that $n_{g}$ is only linked to $n_{pump}$ . The steady-state calculation can be realized only by one-dimensional interpolation, and the fitting dynamic effect can be more accurate. The $n_{p}$ speed is linked to both $n_{pump}$ and $θ$ . The steady-state point should be applied by two-dimensional interpolation. However, due to the limited computing ability of the embedded system, it is obtained by three one-dimensional interpolation. The accuracy is slightly reduced.

This article validates the model with multiple sets of data. The simulation results show that the proposed modeling method is feasible and can meet the simulation requirements of micro-turboshaft engine control system.

Contrastive control algorithms establishment

To verify the control effect of NMPC, the traditional cascade PI control algorithm is first established as a contrast. Cascade PI control algorithm is the most commonly used control algorithm for turboshaft engines. The algorithm is divided into two internal and external loops. The external loop adopts $n_{p}$ PI control, and the internal loop adopts $n_{g}$ PI control. The control plan is to control $n_{pump}$ so that the real $n_{g}$ follows command $n_{g}$ . Cascade PI control has an inner $n_{g}$ negative feedback loop compared to single PI control, so that the internal interference can be suppressed in time, and the influence of external loop is reduced. The entire system can get better control performance.^14–16 The structure of the cascade PI control algorithm designed in this article is shown in Figure 4.

Figure 4.

Cascade PI control algorithm structure.

In Figure 4, the feedforward compensation of pitch angle is added to the output instruction of external loop. The feedforward value of pitch angle $Δ n_{gr}$ is determined by $θ$ interpolation. The feedforward interpolation table is determined by test data, which has following relationship

Δ n_{gr} = f (θ)

(2)

The feedforward compensation of speed–fuel is added to the inner instruction of internal loop. The feedforward value of fuel $Δ W_{f}$ is determined by core engine command speed $n_{gr}$ interpolation. The feedforward interpolation table is determined by test data, which has following relationship

Δ W_{f} = g (n_{gr})

(3)

When pitch angle $θ$ changes, feedforward $Δ n_{gr}$ changes instantly. $Δ n_{gr}$ will create a new fuel feedforward compensation $Δ W_{f}$ . The relationship between $θ$ and $Δ W_{f}$ is as follows

Δ W_{f} = g (f (θ))

(4)

After the two feedforward are connected in series, the fuel quantity $W_{f}$ is compensated directly according to the pitch angle $θ$ , which can improve dynamic response ability of control algorithm. Separating one $θ - W_{f}$ feedforward into two feedforward compensations ensures that the internal loop structure when setting inner PI parameters is consistent with that of the actual control. The internal loop structure is shown by the dotted line in Figure 4. This ensures the reliability of PI parameters.

Closed-loop control of real micro-turboshaft engine must avoid engine speed oscillation. Otherwise, the oil-rich combustion in combustion chamber will cause nozzle fire and even engine fire. Therefore, the designed PI controller has the function of integral separation and anti-integral saturation. The control algorithm also needs to limit the speed and range of electric fuel pump.

Based on the open-loop test data, two feedforward interpolation tables for $θ - Δ n_{gr}$ and $Δ W_{f} - n_{gr}$ are established, and then the PI parameters of internal and external loop are adjusted through multiple simulation tests. The cascade PI control algorithm is verified by numerical simulation and hardware-in-the-loop simulation and finally verified by bench test. The final internal PI loop parameters $k_{p} = 0.4$ , $k_{i} = 0.15$ , and external PI loop parameters $k_{p} = 60$ , $k_{i} = 30$ . Figure 5 shows the comparison of cascade PI control and traditional single PI control. It can be seen in Figure 5(b) that when $θ$ changes rapidly, the speed response of both cascade PI control and single PI control changes with maximum acceleration, but the range of $n_{pump}$ under cascade PI control is larger. In Figure 5(d), the response time of cascade PI control is shorter and the recovery speed of target $n_{p}$ is faster. Cascade PI control is better than single PI control because it has an internal regulation loop, which can speed up the adjustment speed and further reduce the influence of internal interference on the system.

Figure 5.

Comparison between cascade PI control and single PI control when $θ$ increase: (a) changing curve of pitch angle $θ$ ,(b) response of electric fuel pump speed $n_{pump}$ , (c) response of core engine speed $n_{g}$ , and (d) response of power turbine speed $n_{p}$ .

NMPC algorithm design

Algorithmic structure

The control objective of micro-turboshaft engine is to keep $n_{p}$ unchanged by adjusting $W_{f}$ when $θ$ changes, and to minimize the overshoot of $n_{p}$ .¹⁷

According to the principle of NMPC, the structure of NMPC algorithm is shown in Figure 6. It mainly includes online rolling optimization module, prediction model, feedback correction module, and control object. At time $k$ , based on the current turboshaft engine state $n_{g} (k - 1)$ , $n_{p} (k - 1)$ and the control quantities of next $N$ moments $W_{f} (k), W_{f} (k + 1) \dots W_{f} (k + N)$ output by the online rolling optimization module, the prediction model can predict the state quantities for next $N$ moments

n_{gm} (k), n_{gm} (k + 1) \dots n_{gm} (k + N)

(5)

n_{pm} (k), n_{pm} (k + 1) \dots n_{pm} (k + N)

(6)

Figure 6.

NMPC algorithmic structure.

The online rolling optimization module iteratively optimizes its output control quantity according to predicted value of the prediction model until optimal control quantity under the constraints is calculated. There is a certain error between any model and real engine. The feedback correction module corrects the command value at time $k + 1$ according to the difference between actual engine output $n_{p}$ and model predicted output $n_{pm} (k)$ . The correction is $δ (k)$ .

Prediction model

Nonlinear predictive model is the basis of NMPC. The effectiveness of NMPC is mostly determined by the accuracy and real-time performance of prediction model. A numerical fitting model has been established in section “Simulation model establishment,” but the NMPC controller which uses it as a predictive model has the problem of excessive computation and does not meet the real-time requirements. Therefore, piecewise linearization method is used to establish a micro-turboshaft engine LPV model as the prediction model.^18–22 LPV model is simpler and can transform the optimization problem into a standard quadratic programming (QP) problem, which can be solved by QP algorithm without using a computationally intensive sequence quadratic programming (SQP) algorithm, can greatly improve real-time performance of NMPC.^23,24

NMPC control target of micro-turboshaft engine is to ensure that the power turbine speed $n_{p}$ is constant, so the established LPV prediction model should be a linear model of power turbine near the command speed. The power turbine command speed set in this article is $600 r / \min$ . First, establish a state space model of $n_{p}$ at $600 r / \min$ under different pitch angles. Then use $θ$ as interpolation object to obtain the state space model parameters and steady-state point parameters corresponding to other $θ$ values. Finally, combining these parameters to establish an LPV model over the entire range of $θ$ variation. Under a certain stable value of $θ$ , the state space model of the micro-turboshaft engine at steady-state point $(x_{0}, y_{0}, u_{0})$ is

{\begin{matrix} Δ x (k + 1) = A Δ x (k) + B Δ u (k) \\ Δ y (k) = C^{T} Δ x (k) \end{matrix}

(7)

In the formula, $x_{0}$ , $y_{0}$ , and $u_{0}$ are the state quantity, output quantity, and input quantity of steady-state point, respectively; A and B are parameter matrices to be determined. The state quantity $x = [n_{g}, n_{p}]^{T}$ , the output quantity $y = n_{p}$ , the input quantity $u = n_{pump}$ , so obviously $C = [0, 1]^{T}$ . Since the order of magnitude of each parameter of the model differs greatly, in order to improve the accuracy of parameter fitting, the model data are normalized by equation (8)

Δ x = \frac{x - x_{0}}{x_{d}}, Δ y = \frac{y - y_{0}}{y_{d}}, Δ u = \frac{u - u_{0}}{u_{d}}

(8)

In the formula, $x_{d}$ , $y_{d}$ , and $u_{d}$ are the state quantity, output quantity, and input quantity of the design point of the micro-turboshaft engine, respectively. Because power turbine speed of the design point is also $600 r / \min$ , so $y_{0} = y_{d}$ .

The state space model is established using the fitting model data of section “Simulation model establishment.” First, a small step instruction is given to control quantity $n_{pump}$ in the fitting model, and dynamic response data of the state quantity and output quantity of the model are obtained. Then, the model parameters A and B are determined by the least square fitting method. Using the above method, the steady-state point corresponding to $n_{p} = 600 r / \min$ at four different $θ$ makes $n_{pump}$ step 2%. The four $θ$ values are 32.8125°, 35°, 37.1875°, and 39.375°, respectively. The coefficient matrices of state space model established under each $θ$ are as follows

\begin{matrix} θ = 32.8125 °, A = [\begin{matrix} 0.9896 & 6.6971 e - 7 \\ 0.02181 & 0.9879 \end{matrix}], \\ B = [\begin{matrix} 0.01587 \\ 8.2668 e - 4 \end{matrix}] \\ θ = 35 °, A = [\begin{matrix} 0.9896 & 8.6357 e - 7 \\ 0.02010 & 0.9880 \end{matrix}], \\ B = [\begin{matrix} 0.01587 \\ 7.5238 e - 4 \end{matrix}] \\ θ = 37.1875 °, A = [\begin{matrix} 0.9896 & 5.3535 e - 6 \\ 0.01891 & 0.9879 \end{matrix}], \\ B = [\begin{matrix} 0.01291 \\ 4.7523 e - 4 \end{matrix}] \\ θ = 39.375 °, A = [\begin{matrix} 0.9895 & 2.9137 e - 6 \\ 0.01663 & 0.9878 \end{matrix}], \\ B = [\begin{matrix} 0.01417 \\ 2.6328 e - 4 \end{matrix}] \end{matrix}

Establishing a complete LPV model also requires four steady-state point data corresponding to $θ$ , because the state quantity, input quantity, and output quantity of the established state space model are normalized by equation (8). Use these four steady-state point data to establish an interpolation table, normalize and denormalize the model parameters. LPV model is composed of linear interpolation module and state space module. Its structure is shown in Figure 7.

Figure 7.

LPV model structure.

Do the same $θ$ step for numerical steady-state ARX series model and LPV model to verify the fitting effect. The comparison between output quantity of LPV model and numerical steady-state ARX series model is shown in Figure 8. LPV model has a good fitting effect in $n_{p} = 600 r / \min$ , which meets the accuracy requirement of prediction model.

Figure 8.

Fitting effect of LPV model: (a) fitting effect of core engine speed and (b) fitting effect of power turbine speed.

The LPV model established in this article can obtain steady-state point $(x_{0}, y_{0}, u_{0})$ and state space model parameters $A$ and $B$ of micro-turboshaft engine according to $θ$ interpolation at each time. If the input of future multiple times is known, the output of micro-turboshaft engine in future multiple times can be calculated. Regardless of denormalization of LPV model, the state quantity in future $M$ $(M \geq N)$ times under the action of $u (k), u (k + 1) \dots u (k + N - 1)$ inputs is

{\begin{matrix} Δ x (k + 1) = A Δ x (k) + B Δ u (k) \\ Δ x (k + 2) = A^{2} Δ x (k) + AB Δ u (k) + B Δ u (k + 1) \\ ⋮ \\ Δ x (k + N) = A^{N} Δ x (k) + A^{N - 1} Bu (k) + \dots + \\ B Δ u (k + N - 1) \\ Δ x (k + N + 1) = A^{N + 1} Δ x (k) + A^{N} Bu (k) + \dots + \\ (AB + B) Δ u (k + N - 1) \\ ⋮ \\ Δ x (k + M) = A^{M} Δ x (k) + A^{M - 1} Bu (k) + \dots + \\ (A^{M - N} B + \dots + B) Δ u (k + N - 1) \end{matrix}

(9)

According to the relationship between output and state quantity in equation (7), the following formula can be derived

Δ Y = F Δ x + G Δ U

(10)

In the formula, $Δ Y$ is the output quantity of prediction model in future $M$ times. $Δ U$ is the input quantity in future $N$ times. The formula is expressed as follows

\begin{matrix} Δ Y = {[\begin{matrix} Δ y (k + 1) \\ ⋮ \\ Δ y (k + M) \end{matrix}]}_{(M \times 1)}, Δ U = {[\begin{matrix} Δ u (k) \\ ⋮ \\ Δ u (k + N - 1) \end{matrix}]}_{(N \times 1)}, \\ F = {[\begin{matrix} C^{T} A \\ ⋮ \\ C^{T} A^{M} \end{matrix}]}_{(M \times 2)}, G = {[\begin{matrix} C^{T} B & \dots & 0 \\ ⋮ & ⋱ & 0 \\ C^{T} A^{N - 1} B & \dots & C^{T} B \\ ⋮ & ⋱ & ⋮ \\ C^{T} A^{M - 1} B & \dots & C^{T} \sum_{I = 0}^{M - N} A^{i} B \end{matrix}]}_{(M \times N)} \end{matrix}

(11)

Online optimization module

The NMPC algorithm usually uses the minimum of the quadratic objective function as the optimization target. The main control objective of micro-turboshaft engine is to keep power turbine speed constant. When the speed is disturbed, it needs to be adjusted and stabilized as soon as possible. The quadratic objective function proposed in this article is as follows

min J (k) = \sum_{i = 1}^{M} {[Δ y (k + i)]}^{2} + R \sum_{i = 0}^{N - 1} {[Δ u (k + i)]}^{2}

(12)

In the formula, the first term $\sum_{i = 1}^{M} {[Δ y (k + i)]}^{2}$ represents the deviation between power turbine speed and target speed of micro-turboshaft engine. Term $R \sum_{i = 0}^{N - 1} {[Δ u (k + i)]}^{2}$ represents the degree of change in control quantity through control process. $R$ is a factor, which is far less than 1. The control quality of NMPC can be changed by adjusting the value of $R$ . Combining equations (9)–(11), equation (12) can be transformed into following form

\begin{matrix} min J = Δ U^{T} (G^{T} G + RI) Δ U + 2 Δ x^{T} F^{T} G Δ U \\ + Δ x^{T} F^{T} F Δ x \end{matrix}

(13)

In the formula, $Δ x^{T} F^{T} F Δ x$ is a constant related to current state quantity, which can be ignored in optimization problem. Therefore, the QP objective function can be further transformed into

min J = \frac{1}{2} Δ U^{T} H Δ U + c^{T} Δ U

(14)

In the formula, $H = 2 (G^{T} G + RI)$ , $c^{T} = 2 Δ x^{T} F^{T} G$ . This converts the output optimization problem into standard QP problem. In micro-turboshaft engine control, the limitation of fuel variation and maximum value must be taken into consideration, so the following constraints need to be added when solving the QP problem

s . t . {\begin{matrix} u_{0} - u_{\min} \leq Δ u (k) \leq u_{\max} - u_{0} \\ {\overset{\cdot}{u}}_{\min} \leq Δ u (k + 1) - Δ u (k) \leq {\overset{\cdot}{u}}_{\max} \end{matrix}

(15)

Transform equation (15) to get matrix expression

\begin{matrix} I_{N \times N} \times {[\begin{matrix} Δ u (k) \\ ⋮ \\ Δ u (k + N - 1) \end{matrix}]}_{N \times 1} \leq {[\begin{matrix} u_{\max} - u_{0} \\ ⋮ \\ u_{\max} - u_{0} \end{matrix}]}_{N \times 1} \\ - I_{N \times N} \times {[\begin{matrix} Δ u (k) \\ ⋮ \\ Δ u (k + N - 1) \end{matrix}]}_{N \times 1} \leq {[\begin{matrix} u_{\min} - u_{0} \\ ⋮ \\ u_{\min} - u_{0} \end{matrix}]}_{N \times 1} \end{matrix}

(16)

\begin{matrix} {[\begin{matrix} - 1 & 0 & \dots & \dots & 0 \\ 1 & - 1 & 0 & \dots & 0 \\ 0 & 1 & ⋱ & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & ⋱ & 0 \\ 0 & \dots & 0 & 1 & - 1 \end{matrix}]}_{N \times N} \times {[\begin{matrix} Δ u (k) \\ ⋮ \\ Δ u (k + N - 1) \end{matrix}]}_{N \times 1} \\ \leq [\begin{matrix} {\overset{\cdot}{u}}_{\max} - Δ u (k - 1) \\ {\overset{\cdot}{u}}_{\max} \\ ⋮ \\ {\overset{\cdot}{u}}_{\max} \end{matrix}] \\ {[\begin{matrix} 1 & 0 & \dots & \dots & 0 \\ - 1 & 1 & 0 & \dots & 0 \\ 0 & - 1 & ⋱ & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & ⋱ & 0 \\ 0 & \dots & 0 & - 1 & 1 \end{matrix}]}_{N \times N} \times {[\begin{matrix} Δ u (k) \\ ⋮ \\ Δ u (k + N - 1) \end{matrix}]}_{N \times 1} \\ \leq [\begin{matrix} {\overset{\cdot}{u}}_{\max} - Δ u (k - 1) \\ {\overset{\cdot}{u}}_{\max} \\ ⋮ \\ {\overset{\cdot}{u}}_{\max} \end{matrix}] \end{matrix}

(17)

Combine these matrices to obtain inequality constraints for QP problems. Use QP algorithm to solve the constrained QP problem. $Δ \bar{U}$ is a vector consisting of the input of next $N$ moments. Find input $Δ \bar{U}$ that minimizes the performance index J, and output its first component $Δ \bar{u} (k)$ as speed instruction of electric fuel pump.

Feedback correction module

NMPC calculates control quantity based on prediction model. The accuracy of model determines the control precision. However, no matter what modeling method is used, it is difficult to avoid the difference between model and real micro-turboshaft engine. In order to enhance robustness and reduce control error, it is necessary to add a feedback correction module in controller.^22,25–28

Feedback correction module generally has two ways of correction: one is to modify the prediction model and the other is to modify the control instructions. Since it is difficult to change nonlinear model online, almost all the existing feedback correction modules modify the control instructions, as shown in Figure 6. The LPV prediction model established in this article is a linear model near command speed. If the control instructions are modified, the deviation of the model will be increased. The correction method adopted in this article is shown in Figure 9. The prediction model consists of linear interpolation module and state space module. The feedback correction module can modify the prediction model by modifying linear interpolation output of the LPV model. It first calculates the error of command speed $n_{pr}$ and micro-turboshaft engine $n_{p}$ and then uses PI term of the error to correct linear interpolation output value $u_{0}$ of the LPV model, eliminating the steady-state error existing in NMPC control system.

Figure 9.

NMPC algorithm structure based on LPV model.

Algorithm analysis and verification

Simulation analysis

All modules of NMPC algorithm for micro-turboshaft engine have been designed. The simulation experiment is carried out on MATLAB with the numerical steady-state ARX series model established in section “Simulation model establishment” as the control object.

The online optimization module performs the optimal control quantity calculation of NMPC by solving the target optimization function. Different solving algorithms lead to different time consuming, which will have a great impact on the real-time performance of the control system. Therefore, this article first explores the computational speed of different optimization algorithms applied to NMPC. Quadprog (Quadratic Programming) functions of QP algorithm and fmincon (Find Minimum of Constrained Nonlinear Multivariable Function) functions of SQP algorithm in MATLAB are used to calculate the optimal control quantity. A total of 9000 control periods are calculated on a 2.2 GHz computer. While QP algorithm takes 130.356 s, each control period is 14.484 ms, while SQP algorithm takes 423.977 s, each control period is 47.109 ms. It can be seen that QP algorithm is more real-time and meets the requirements of 20 ms control period. SQP algorithm takes a long time and has exceeded 20 ms control period. The control has obvious delay. Therefore, this article chooses QP algorithm as optimization algorithm.

NMPC optimization objective function is shown in equation (12). The equation contains three control parameters: prediction period $M$ , control period $N$ , and control coefficient $R$ . In this article, the optimal control parameters of NMPC algorithm are explored through simulation experiments to obtain the best control effect.

The prediction period $M$ represents the NMPC algorithm to predict system output of following $M$ steps. The longer prediction period $M$ is, the better control quality will be, but it will also greatly increase the calculation and affect real-time performance of NMPC algorithm. This article uses different $M$ values for control simulation. When $M$ is increased, the proportion of first term of NMPC target optimization function will increase, so the corresponding $R$ value should be reduced to ensure that the proportion remains unchanged. Two sets of simulation parameters selected in this article are $M = 10, R = 0.01$ and $M = 5, R = 0.005$ . The comparison of control effects is shown in Figure 10. It can be seen that the longer prediction period is, the smaller overshoot of NMPC algorithm is, and the faster adjustment speed is. This is because NMPC only looks for the optimal control quantity in the prediction period. This optimal value is actually a local optimal solution. A larger prediction period can make local optimal solution close to global optimal solution, so that control effect of NMPC can be better. In order to balance real-time performance and control effect of the algorithm, this article sets $M = 10$ , which is to predict output of the system in next 10 periods.

Figure 10.

Comparison of NMPC control effects using different $M$ : (a) response of power turbine speed and (b) response of electric fuel pump speed.

The control period $N$ also affects control effect of NMPC. The closer $N$ approaches $M$ , the better the control effect. Two sets of simulation parameters selected in this article are $M = 10, N = 1$ and $M = 10, N = 10$ . The comparison of control effects is shown in Figure 11. When $N$ becomes larger, the variable range of control quantity can be increased. The larger $N$ is, the smaller overshoot is, the faster adjustment speed is. However, improvement of control effect caused by the increase of $N$ is very limited. In order to reduce calculation, we can take a smaller $N$ . The $N$ selected in this article is 5.

Figure 11.

Comparison of NMPC control effects using different $N$ : (a) response of power turbine speed and (b) response of electric fuel pump speed.

The second term of optimization objective function represents the degree of change in the control quantity during the prediction period. The larger the coefficient $R$ is, the smaller the change degree of the output control quantity is. This article uses $R = 0.005$ and $R = 0.02$ for control simulation. The comparison of control effects is shown in Figure 12.

Figure 12.

Comparison of NMPC control effects using different $R$ : (a) response of power turbine speed and (b) response of electric fuel pump speed.

In Figure 12, the speed change of the electric fuel pump at $R = 0.005$ is obviously greater than that at $R = 0.02$ . The maximum overshoot of the NMPC controller is smaller and the response time is slightly better. When R becomes smaller, the degree of change of control quantity increases, and NMPC can output control quantity closer to the optimal solution. However, $R$ should not be too small, because NMPC can only predict the output of controlled system at a limited time in future. If the change of control quantity is not limited, the system can easily enter into oscillation.

When $R = 0.0001$ , the control effect is shown in Figure 13. When $θ$ changes greatly, the speed of the electric fuel pump varies sharply, and it is difficult to restore stability under the constraint. The micro-turboshaft engine enters the oscillation state. Therefore, the $R$ value set in this article is 0.005.

Figure 13.

NMPC control effects when $R = 0.0001$ : (a) response of power turbine speed and (b) response of electric fuel pump speed.

The parameters of the NMPC algorithm are determined by simulation experiments above. Figure 14 shows a comparison of the control effects of NMPC and cascade PI with pitch angle feedforward. The model pitch angle changes as shown in Figure 14(a). Figure 14(b) shows the response of $n_{pump}$ with two control algorithms to keep $n_{p}$ unchanged. It can be seen from Figure 14(c) that the final stable state of $n_{p}$ controlled by two controllers is the same. When $θ$ suddenly increases, $n_{pump}$ of both controllers accelerates with maximum acceleration. The cascade PI controller does not show lag. This article considers that the pitch angle feedforward makes cascade PI controller have a certain predictive ability. Under the constraints of $n_{pump}$ maximum acceleration and maximum speed, the response of two controllers to sudden change of $θ$ is consistent, so the maximum overshoot is basically the same. As can be seen in the whole adjustment process and local enlargement in Figure 14(d), NMPC controller has smaller overshoot and faster adjustment speed. Its response is obviously better than that of the cascade PI controller.

Figure 14.

Simulation comparison between NMPC and cascade PI control: (a) changing curve of pitch angle $θ$ , (b) response of electric fuel pump speed $n_{pump}$ , (c) response of power turbine speed $n_{p}$ , and (d) local enlargement of (c).

Bench test

In order to verify the effectiveness of NMPC algorithm for real micro-turboshaft engines, a bench test verification was carried out on the test bench described in section “Test bench establishment.”

The calculation of NMPC algorithm is large. The numerical simulation in section “Simulation analysis” is performed in a high-performance computer to meet real-time requirements. It is difficult to achieve such high real-time computing tasks in embedded systems with limited resources. Therefore, this article realizes NMPC using host computer to assist calculation. In test experiment, 10,000 periods of control instructions were computed on a 2.2 GHz host computer with a total time of 70 s. The average calculation time is 7 ms, which meets the real-time requirements. In addition, using this solution needs to ensure real-time communication, otherwise, it will cause control delay. The embedded system used in this article uses lightweight IP (lwIP) protocol stack, which is lightweight and efficient. After testing, the communication delay between core controller and host computer is not more than 20 ms, which meets the real-time requirement of communication.

Before the bench experiment, hardware-in-the-loop simulation is required for the reliability and safety of experiment. It is used to verify controller function and control algorithm effects. After verification, the controller has complete monitoring, control, and protection functions. The results of NMPC hardware-in-the-loop simulation are similar to those of the bench tests below, so it is unnecessary to elaborate on them.

The results of the bench test are shown in Figures 15 and 16. The two diagrams are respectively the response comparison of NMPC and cascade PI control with pitch angle feedforward when pitch angle is taking ascend and descent step. Similar to the simulation results, it can be seen that in Figures 15(b) and 16(b), when the pitch angle suddenly changes, NMPC and cascade PI’s electric fuel pump have the same speed response, and both change at the maximum acceleration limit. The overshoot of two control algorithms is basically the same at the beginning, as shown in Figures 15(c) and 16(c). However, the overshoot of NMPC is less than 2% and the cascade PI is about 5% in the follow adjustment process. The smaller overshoot of NMPC also reduces the range of nozzle temperature variation, avoiding over temperature problem during engine control, as shown in Figures 15(d) and 16(d). NMPC has a faster adjustment speed. Its adjustment time is about 3–4 s less than the cascade PI, and its control effect is obviously better than the cascade PI control.

Figure 15.

Comparison between NMPC and cascade PI control when $θ$ decreases: (a) changing curve of pitch angle $θ$ , (b) response of electric fuel pump speed $n_{pump}$ , (c) response of power turbine speed $n_{p}$ , and (d) variation of nozzle temperature.

Figure 16.

Comparison between NMPC and cascade PI control when $θ$ increases: (a) changing curve of pitch angle $θ$ , (b) response of electric fuel pump speed $n_{pump}$ , (c) response of power turbine speed $n_{p}$ , and (d) variation of nozzle temperature.

Summary

This article studied the NMPC algorithm of turboshaft engine based on the test bench of micro-turboshaft engine and its numerical fitting model. The structure of NMPC algorithm is designed, and the LPV model near the command speed of engine power turbine is established as the prediction model. The online optimization module is designed to optimize the objective function. The feedback correction module is designed to correct the output of control algorithm. The simulation experiment of NMPC algorithm is carried out on MATLAB, and the control effect is compared with the cascade PI control with pitch angle feedforward. The simulation results show that although NMPC cannot further reduce the maximum overshoot of power turbine speed, it can speed up the adjustment and reduce the speed fluctuation in the adjustment. Its control effect is better than the traditional cascade PI controller.

This article successfully applied the NMPC algorithm to a real micro-turboshaft engine. The results show that compared to cascade PI control with pitch angle feedforward, the NMPC can predict the future response of the engine and achieve stronger control effect. It can overcome the disadvantage of cascade PI which cannot consider the influence of time-delay system, resulting in slow response and large fluctuation of power turbine speed. NMPC has a similar overshoot with PI control when pitch angle suddenly changes. Through the whole control process, NMPC has smaller overall overshoot and faster adjustment speed. It can effectively reduce the temperature change rate before the engine turbine, avoid over temperature and extend engine life.

In this article, NMPC algorithm is calculated on monitoring computer. Some scholars have developed QP algorithm library quadprog++ based on C++ under Linux, which can realize most functions of quadprog function in MATLAB and calculate faster. In the future, NMPC test of complex turboshaft engines and airborne test can be realized through a high-performance embedded system.

Footnotes

Appendix 1

Handling Editor: Jianjun Feng

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: This work was supported by the National Natural Science Foundation of China (No.51976089, No.51576097).

ORCID iD

Mengwei Zhang

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Design and verification of model predictive control for micro-turboshaft engine

Abstract

Keywords

Introduction

Research platform establishment

Test bench establishment

Simulation model establishment

Contrastive control algorithms establishment

NMPC algorithm design

Algorithmic structure

Prediction model

Online optimization module

Feedback correction module

Algorithm analysis and verification

Simulation analysis

Bench test

Summary

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

ORCID iD

References