Yong WANG, Qian’gang ZHENG, Ziyan DU, Haibo ZHANG

College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

KEYWORDS Double turboshaft engines;Fast response control;Helicopter;Nonlinear model predictive control;Torques matching method

Abstract In order to reach a compromise between fast response control and torques matching control in double turboshaft engines, research on nonlinear model predictive control for turboshaft engines based on double engines torques matching is conducted. Meanwhile, a Nonlinear Model Predictive Control (NMPC) method is proposed, which combines the control index of the power turbine speed with torques matching of double engines creatively.In addition to the control index,the difference of output torques between each engine is also incorporated in the objective function as a penalty term to ensure constant speed control and short torques matching time. Simulation results demonstrate that relative to unilateral torques matching,the settling time of the bidirectional matching method can be reduced by nearly 30.8%. Nevertheless, compared with the bidirectional torques matching method under the cascade PID controller, the NMPC method can decrease the overshoot of the power turbine speed by 65% and reduce the matching time by 15.5% synchronously. Besides fast response control of turboshaft engines, fast torques matching control of double engines is accomplished as well.

1. Introduction

Since the 1990s,the electronic systems of helicoptershave been developed towards a highly integrated direction. Nevertheless,the combat situation of modern armed helicopters is becoming more and more severe. Therefore, how to improve firepower and maneuverability of helicopters has been an objective to improve their survivability. The turboshaft engine has proven to be the best choice for modern helicopters as a propulsion device owing to its high power margin and wide flight envelope.1,2Because of the load requirement of the main rotor,helicopters are often equipped with two or more turboshaft engines. However, even for the same kind of engines, the performance of each engine is hardly identical due to the manufacture and performance degradation in practice. Therefore,in the case of double or even multiple engines driving a helicopter, the conventional cascade PID control structure can barely guarantee the identical output power of each engine in the meantime.3-5Under this circumstance, it is necessary for the engine control system to accomplish an even load distribution through an appropriate matching control strategy.

The matching strategy of multiple engines needs to take into account the service lives of transmissions and engines,the performance degradation of a single engine,and other factors. Therefore, how to obtain a more effective matching control method of double or multiple engines has been a core problem in the helicopter field,6which has attracted the attention of some talented scholars. Gaulmin et al.7invented a method and an associated fuel metering system for balancing the power delivered by two aircraft turboshaft engines. The method determines and transforms first and second limiting margins of the engines. Thereafter, the values of the power margins are compared, and the engine having the greater power margin is accelerated by minimizing the primary difference as much as possible. Shi et al.8established an integrated dynamic model of a helicopter rotor with three turboshaft engines and proposed unilateral and bidirectional matching methods.Simulation results show that the settling time of bidirectional matching is shorter than that of unilateral matching.In order to deal with the unbalanced output power in a helicopter with multiple turboshaft engines caused by the performance degradation of a single engine, Yang et al.9replaced the outer loop of the conventional cascade PID controller with direct power control. Combined with the onboard model of rotor-demanded power, a power matching control system for multiple turboshaft engines was constructed, which enables two engines with different performance degradations to output coincident power.

It is difficult to compromise between power-matching control and constant-speed control of multiple turboshaft engines.For conventional cascade PID control of a turboshaft engine,it is hardly beneficial to obtain a high-quality control effect because of the lack of predictive and decoupling abilities.Therefore, in order to realize power-matching control of multiple turboshaft engines,it is inevitable to sacrifice the dynamic performance of the power turbine speed. On the contrary,Model Predictive Control (MPC), as a model-based control method,10-13can be adopted to cope with the coupled control problem of time-varying nonlinear systems,14,15such as robust MPC and Nonlinear MPC(NMPC).These methods can effectively deal with the control problem of coupled nonlinear systems with complex constraints and disturbances.16,17Wang et al.18conducted a study on the NMPC technology for turboshaft engines with constrained optimization based on an integrated helicopter/turboshaft engine simulation platform.Meanwhile, a model predictive controller for a single turboshaft is available through Recursive Reduced-Least Squares Support Vector Regression (RR-LSSVR). However, this research only involved a single engine,so it is scarcely applicable to speed closed-loop control for turboshaft engines based on double engines power matching.

In order to compensate for the lack of research on fast response control methods of turboshaft engines under double engines matching, an NMPC method for speed closed-loop control of turboshaft engines based on double engines torques matching is proposed. Firstly, based on the conventional cascade PID controller of turboshaft engines, a bidirectional torques-matching strategy is proposed and verified. Secondly,according to the bidirectional matching strategy, the Minbatch Gradient Descent-Neural Network (MGD-NN) is adopted to obtain an onboard model of double engines,which can simulate the coupling dynamic characteristics quite precisely. Then,in addition to the speed control index, the difference of output torques between each engine is also incorporated in the objective function, and the Sequential Quadratic Programming (SQP) optimization algorithm is utilized to cope with the constrained problem.Finally,under typical flight missions, simulation verification is carried out, and results are compared with those of bidirectional torques matching applied in double turboshaft engines based on conventional cascade PID control.

2. Bidirectional torques matching of double turboshaft engines based on cascade PID control

In order to operate a helicopter conveniently and reliably,it is necessary to keep the rotor speed constant. The reason is that the gear ratio from the power turbine to the main rotor is unchangeable currently,and the specific fuel consumption of a modern turboshaft engine reaches the optimum in a relatively narrow speed range.Fig.1 summarizes the mechanical connection between a helicopter and double turboshaft engines.

Fig. 1 Schematic diagram of a simplified helicopter rotor drive train system.

In terms of double turboshaft engines, the existing matching method is unilateral torques matching, and the control structure is shown in Fig. 2, where turboshaft engines A and B have identical design parameters. pnpr and pnp denote the reference signal and relative speed of the power turbine,respectively. Then, the relative speed of the power turbine is defined as the ratio of the rotational speed to its design point value. TQe,Aand TQe,Brepresent the output torques of turboshaft engines A and B,respectively.As shown,the unilateral torques matching method can only adjust the fuel flow of a single engine according to the torques difference,thus eliminating the difference between the output torques of double engines and realizing torques matching control. In addition, Fig. 3 summarizes the bidirectional torques matching Scheme.8Different from the unilateral matching method, bidirectional matching can simultaneously change the fuel flows of double turboshaft engines to achieve torques matching as soon as possible.

In order to validate the bidirectional torques matching scheme,simulation verification is conducted at a flight altitude of H=600 m and a forward speed of νc=118 m/s. At t=10 s, the compressor mass flow of turboshaft Engine A increases by 5%to adjust the output torques of double engines in different flight states. During the adjustment process, the helicopter rotor speed always keeps closed-loop control.Simulation results are shown in Fig. 4.

As shown in Fig. 4(b), the output torque of turboshaft Engine A increases sharply at 10 s due to an instantaneous increase in the compressor mass flow. The helicopter rotor speed adopts constant-speed control.Therefore,in the absence of any torques matching control strategy,the output torque of turboshaft Engine B should decrease to meet the power requirements of the helicopter rotor. Because there is a response delay between the turboshaft engine output and the main rotor, an instantaneous overshoot and sag of the power turbine speed occurs inevitably, and the overshoot reaches about 2% unfortunately. When the unilateral matching method is applied, the output torques of double engines keep coincident 9.3 s later,which realize torques matching of double turboshaft engines.Nevertheless,the output torques of double engines match each other at t=16.44 s through the bidirectional torques matching method, which reduces the settling time by nearly 30.8%. Fig. 4(d) shows that the change of the gas turbine speed is opposite to those of output torques.Fig. 4(a) summarizes that no matter which matching method is adopted based on the cascade PID controller,the overshoot and sag of the power turbine speed can hardly decrease effectively, and the dynamic response performance of the power turbine speed in the matching process can barely be polished up.Therefore,in order to shorten the settling time and reduce the overshoot and sag of the power turbine speed remarkably in the matching process,it is necessary to develop an advanced fast response control method for turboshaft engines based on the bidirectional torques matching strategy.

Fig. 2 Scheme of unilateral matching method.

Fig. 3 Scheme of bidirectional matching method.

Fig. 4 Comparison between bidirectional matching and unilateral matching.

3. Design of a nonlinear model predictive controller based on bidirectional torques matching

3.1. Onboard model of double turboshaft engines

Fig. 5 Back-propagation principle diagram of a neural network.

A back-propagation Neural Network(NN)can learn independently according to sample data, so that it can express the mapping relationship between input and output samples more accurately12,19.The principle is shown in Fig.5.The min-batch gradient descent method divides a training set into groups randomly, which reduces the computational cost remarkably,making the NN applicable to large sample data.

Eq.(1)shows the output of each neuron in Fig.5.Here,σ(·)denotes the activation function,andrepresent the gradient of the NN weight and the bias in the lth layer,respectively, and η is the learning rate.

The Min-Batch Gradient Descent (MB-GD) method divides the training set data into M groups randomly,and each group has Nbtraining sets. J(W,b;x,y) denotes the loss function of each group, which is the training objective of the network. x, y are training and testing samples. W and b represent the weight matrix and the bias matrix of NN. Eq.(2)summarizes the derivative (δl)of the nodes in the lth layer,where ?denotes the Hadamard or Schur product.

where ∑bi=N, N denotes the number of training samples;hW，b（xbi，j）represents the mapping model of NN.The loss function is a subset of the training sets.

In order to reflect the coupling dynamic characteristics of the original double turboshaft engines, the Nonlinear Auto-Regressive Moving-Average (NARMA) model is applied.For the onboard model of double turboshaft engines, in addition to engine output torques(TQe,A,TQe,B),the flight altitude(H), the forward speed (νc), fuel flows (Wfb,A, Wfb,B), the relative speeds of gas turbines (pncA, pncB), the relative speed of the power turbine (pnp), and turbine inlet temperatures(T41,A,T41,B)are also requisite.To represent the rotor dynamic characteristics of double engines in the matching process accurately, the main rotor demanded torque TQrshould be taken into account as well. The order of input variables will affect the accuracy of the onboard model. Since an aero-engine can be simplified as a second-order object, the order is set to 2.The onboard model of double turboshaft engines is available as follows.

As shown in Eq. (3), the output variables are current pnc,T41, TQeof each engine, TQr, and pnp. The input variables are historical H, νc, Wfb, pnc, T41, TQe, TQr, pnp and current H, νc, Wfb. Therefore, the onboard model consists of 28 input variables,30 hidden layer nodes,and 8 output variables.In the same condition as in Fig. 4, double engines are adequately motivated. Then, the output data is normalized and utilized as the training data. Fig. 6 shows the relative errors of the onboard model. All relative errors are less than 1%, which can be adopted to predict the core performance parameters of double turboshaft engines.

3.2. State estimator based on an unscented Kalman filter

The input variables of the onboard model shown in Eq. (3)contain turbine inlet temperature T41. Nevertheless, temperature sensors can hardly measure turbine inlet temperature due to the restriction of material. Therefore, it is necessary to estimate the high temperature accurately according to some measurable state variables. The Kalman filter has remarkable advantages over other state estimators because of its mature theory and ideal robustness. An Unscented Kalman Filter(UKF) can not only avoid calculation of a Jacobian matrix in an Expanded Kalman Filter (EKF), but also improve the estimation accuracy and convergence speed significantly without increasing the amount of calculation20.

A UKF is a nonlinear Kalman filter that replaces the estimation of the linear propagation mode of the statistical properties in a Kalman filter or an extended Kalman filter with unscented transformation. In order to estimate the turbine inlet temperature of double turboshaft engines accurately,two appropriate measurable state parameters are essential.Here, the relative speed of the power turbine (pnp) and the main rotor demanded torque (TQr) are selected.

According to Eq. (3), the UKF algorithm is as follows,wherein xk=[pncA(k), pncB(k), T41,A(k), T41,B(k), TQe,A(k),TQe,B(k), TQr(k), pnp(k)]T.

Step 1. Filter initialization

Step 2. χk-1calculation, here, n denotes the dimensions of the output in Eq. (3), and λ is the proportional factor.

Step 4. Measurement update

Modification is available according to the measured output yk,estimated,and the estimation error covariance Pxk，k-1.

Based on the above algorithm,the unmeasurable state variables T41of double turboshaft engines can be estimated online according to the errors between predicted state parameters and corresponding measured values.

3.3. Nonlinear model predictive control

Traditional optimal control algorithms often take the minimum quadratic objective function as the performance index,so does model predictive control.As known,the control objective of turboshaft engines is to keep the relative speed of the power turbine constant as much as possible. Meanwhile, the difference between the main rotor demanded torque transferred through transmissions and engines output torques should be as small as possible. In order to keep the relative speed of the power turbine constant and decrease the settling time of double engines torques matching in the meantime,the difference of output torques between each turboshaft engine is incorporated as a penalty term in the objective function. In addition, it is also necessary to ensure satisfying some indispensable constraints,such as no overheating and no overturning.Therefore,the complete objective function is shown in Eq. (8), in which P is set to 5.

where NTdenotes the gear ratio of the power turbine to the main rotor, TQe,A,dsrepresents the design-point value of Engine A output torque, μ is the penalty factor, and ε is the threshold. ω1and ω2represent weight coefficients. The sat function is defined as follows:

In Eq.(8),the first term of the objective function keeps pnp at about 100%. The second term is beneficial to decrease the difference between the main rotor demanded torque transferred through transmissions and engines output torques,which can be utilized to reduce the overshoot and sag of the power turbine speed in the matching process. The third one is adopted to realize double engines torques matching. It will take effect if and only if the difference between the output torques of double turboshaft engines can hardly meet the threshold, that is, the output torques of double engines are difficult to match each other. Many methods can cope with this kind of optimization problem. Among them, SQP is accessible.

3.4. Simulation verification

Fig. 7 summarizes the structure of NMPC for helicopter/engines based on double engines torques matching. As shown, the onboard model of double turboshaft engines can predict the future output according to historic and current inputs. The UKF is able to estimate the unmeasurable turbine inlet temperature online according to the power turbine speed,the rotor demand torque from the onboard model, and their measured values,which will be utilized in the nonlinear model predictive controller. SQP optimization algorithm and penalty function are available to cope with the objective function and output the fuel flow online, which realize speed closed-loop control of turboshaft engines and double engines torques matching control synchronously.

Fig. 7 Structure diagram of NMPC for helicopter/engines under double engines torques matching.

In the same flight conditions and engines states as in Fig.4,simulation verification of the NMPC method for turboshaft engines based on double engines torques matching is carried out, and compared with that of the bidirectional torques matching method based on a cascade PID control loop.Therein, the operating environment of the two cases is Windows 7 Ultimate with Service Pack 1 (x64). The CPU is Intel(R) Core (TM) i5-4590 with a main frequency of 3.30 GHz.The memory is 8 G,and the software is MATLAB 2016b.Simulation results are as follows.

As shown in Fig. 8(a), the compressor mass flow of turboshaft Engine A increases instantaneously at t=10 s, and an overshoot of the power turbine speed occurs immediately.Nevertheless, NMPC can reduce the overshoot of pnp by about 65% with the steady-state error no more than 0.2% in the double engines matching process, which improves the response speeds of turboshaft engines significantly. Fig. 8(b)summarizes that NMPC can improve the matching speed remarkably and decrease the settling time of double engines torques matching by about 15.5% compared with those of the bidirectional torques matching control method based on the cascade PID controller. The nonlinear model predictive control method based on double engines torques matching can significantly shorten the matching time and realize fast response control of turboshaft engines synchronously. Figs. 8(c)-(e)show the fuel flow,the relative speed of the gas turbines,and the turbine inlet temperature of double engines, respectively. As shown, the fuel flow curves are smoother because NMPC needs to satisfy the constraints shown in Eq. (8) in the optimization process.in Fig. 8(f) denotes the normalized value of the turbine inlet temperature. It shows that the estimated turbine inlet temperature of turboshaft Engine A has a steady-state error, but the error does not exceed 0.2%.Nevertheless, the estimated value of turboshaft Engine B is consistent with the measured value, which proves the satisfactory parameter estimation accuracy of the UKF. Fig. 8(g)summarizes that except for several sampling points, the time cost of the optimization program adopted in NMPC remains within 1.5 ms at each step, which has a good real-time performance.

In order to validate the robustness performance of NMPC,another simulation experiment is conducted. Different from the former, at t=10 s, the compressor mass flow of turboshaft Engine A increases by 4% and the compressor efficiency of turboshaft Engine B decreases by 2% to simulate different degradation scenarios of double turboshaft engines.Simulation results are shown in Fig. 9.

As shown in Fig. 9(a), although the degradation scenarios of turboshaft engines become inconsistent, NMPC can still reduce the overshoot of the power turbine speed remarkably with a steady-state error less than 0.25%in the double engines matching process. Meanwhile, Fig. 9(b) summarizes that NMPC can also decrease the settling time of double engines torques matching by nearly 20.4%,which proves a satisfactory robustness performance of the nonlinear model predictive control method. Unlikein Fig. 8(f), Fig. 9(f) shows that the estimated turbine inlet temperatures of turboshaft engines A and B both have a steady-state error, but the errors hardly exceed 0.5%. Fig. 9(g) shows that the time cost of the optimization program adopted in NMPC stays no more than 1.0 ms at most sampling points.

4. Conclusions

Research on nonlinear model predictive control based on double engines torques matching is conducted. A novel nonlinear model predictive control method is proposed, which combines the control index of the power turbine speed with torques matching of double engines. The following conclusions can be drawn:

(1) Based on the cascade PID controller, the bidirectional torques matching control method can shorten the settling time effectively, but can hardly reduce the overshoot and sag of the power turbine speed significantly in the matching process.

Fig. 8 Comparison results of NMPC with the cascade PID controller based on bidirectional matching.

Fig. 9 Comparison results of NMPC with the cascade PID controller based on bidirectional matching under different degradations.

(2) Compared with the bidirectional torques matching control method under the cascade PID controller, the nonlinear model predictive control method based on double engines torques matching can decrease the overshoot of pnp by 65% and reduce the settling time of torques matching by more than 15.5%. It realizes fast-response control of turboshaft engines and fast double engines torques matching control in the meantime.

(3) The unscented Kalman filter can be accessible to estimate the unmeasurable parameters of the nonlinear model with high precision.

(4) The design method can be extended to fast response control based on triple turboshaft engines torques matching.

Acknowledgements

The work has been co-supported by the National Natural Science Foundation of China (No. 51576096) and Qing Lan and 333 Project and Research Funds for Central Universities(No. NF2018003).