Wenhao XU, Jinquan HUANG, Jiakun QIN, Muxuan PAN

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

KEYWORDS Command governors;Constrained control;Contractive sets;Gain scheduling;Limit protection;LPV model;

Abstract A new limit protection method based on Scheduling Command Governor(SCG)is proposed for imposing multiple constraints on a turbofan engine during acceleration process. A Gain Scheduling Controller (GSC) is designed for the transient state control and its stability proof is developed using Linear Matrix Inequalities (LMIs). The SCG is an add-on control scheme which manages engine limits effectively based on reference trajectory optimization.Unlike the traditional min–max architecture with switching logic,the SCG method utilizes the Linear Parameter Varying(LPV)closed-loop model to form a prediction of future constraint violation and per instant solves a constraint-admissible reference within an approximate Maximal Output Admissible Set (MOAS).The influence of the variation of engine dynamic characteristics and equilibrium points during transient state control is handled by the design of contractive sets. Simulation results on a turbofan engine component-level model show the applicability and effectiveness of the SCG method. Compared to the traditional min–max method, the SCG method has less conservativeness. In addition,the design of contractive sets makes conservativeness tunable.

1. Introduction

Modern aircraft turbofan engines are complex nonlinear systems that contain various core components. To ensure safe operation of all the components,engine control systems incorporate limiters to prevent critical core component variables from exceeding constraints while providing desired engine thrust, especially for large throttle transients. For a twospool turbofan engine, fan and core shafts speed, High Pressure Compressor (HPC) outlet pressure, low pressure turbine outlet temperature and compressor surge margin are usually considered as critical limits, which means these mechanical,thermodynamic and aerodynamic variables are restricted within a certain range otherwise it will lead to shorter engine life or an appearance of undesirable engine characteristics.1–3In recent years,there has been an increasing interest in improving transient response capability of turbofan engines for handling a variety of emergency maneuvers by aircraft which is needed to operate the engines as close as possible to theirs limits.4

In addition to the idea of increasing the number of actuators and redesigning the thrust controller using advanced methods,5limit protection has a pivotal role in engine control systems for providing ultimate performance close to the limits.6–8One approach to improving engine transient response lies in intelligently relaxing the constraints based on risk functions,4the other is to improve the limit protection method.Therefore, researchers have shown an increasing interest in studying different kinds of limit protection methods recently,such as min–max switching logic,9–11output-based switching logic,12–14barrier Lyapunov function,15Model Predictive Control16–19(MPC)and Command Governor20–22(CG).The min–max switching method is currently the most widely used in turbofan engine limit protection. In this method, the multiple loop control architecture is adopted and the control signals generated by each regulator in the different loop are selected by using min and max selectors to decide whether the thrust control loop is active or any other limitation loops.Several significant issues in the min–max switching method are the stability analysis of the entire controller structure,23–25too conservative limit design,8,26–28risk of constraints violation under non-nominal setpoint,the presence of a disturbance29,30and the failures. Although a great deal of work about the above issues has been carried out by many researchers and the limit protection performance based on multi-loop switching technology has been improved to a certain extent,the cumbersome limiters and switching logic design still makes the design of the turbofan engine limit protection controller a particular challenge.

In order to avoid the complexity of the control system brought by the multi-loop design and obtain high-quality limit protection performance,a series of limit protection approaches based on the idea of prediction and receding horizon optimization have attracted a lot of attention. One of the existing approaches to handle constraints is MPC, which is one of the most widely used advanced control methods in the process control industry.31–33MPC will re-design the controller by online optimization algorithm to achieve not only the limit protection but also the main control tasks,which is not attractive for practitioners who are interested in preserving an existing main controller or are concerned on the stability and computational effort issues.

Another approach known as CG considers augmenting a well-designed controller with constraint handling capability.This kind of approach is attractive by preserving existing controllers which have well-designed control quality. As its name suggests,the CG plays the role of a pre-filter that modifies a reference command when necessary to avoid compromising the system performance.The CG usually requires an explicit model of the system and its constraints to predict future constraints.34Recently investigators have examined the effectiveness of applying the CG to the turbofan engine limit protection based on linear models.21,35–37Many research results show that the CG has a remarkable performance in handling constraints of turbofan engines at certain single operation point. However,turbofan engines are highly nonlinear systems, especially during large transients. Therefore, an effective CG method for nonlinear systems needs to be studied,and the ensuing nonlinear closed-loop systems modeling and nonlinear control stability problems also need to be solved urgently.

In this paper,to deal with the change of dynamic characteristics and equilibrium points of the turbofan engine during large transients,discrete time interpolation-based LPV models of the turbofan engine and the closed-loop system are developed. Both of the LPV models are scheduled by endogenous parameters. Based on the turbofan engine LPV model, a GSC, which is one of the advanced nonlinear controllers, is designed for transient control and the stability analysis for the closed-loop system is presented. A method for finding a single quadratic Lyapunov function for the interpolationbased LPV closed-loop system is essential to proof stability,which is solvable using LMI technique. The design of SCG is based on the closed-loop system LPV model and contractive sets are introduced for guaranteeing limit protection ability by adjusting conservativeness. The main contribution of the new limit protection method includes three points: (A) the SCG can enforce multiple constraints during large transients for nonlinear systems. (B) the SCG achieves excellent speed of response by driving the turbofan engine to work close to the limit value and outperform the traditional min–max method.(C) designers can have more flexibility to achieve the desired performance by adjusting the design parameters of the SCG.

This paper is organized as follows. In Section 2, details of the LPV models,the design of the GSC and its stability verification method are discussed. The design methods and details of the SCG are discussed in Section 3,followed by the simulation results and discussion in Section 4. Conclusions are given in Section 5.

2. Turbofan engine model and nominal controller

A high degree of confidence component level model of a twinspool turbofan engine has been achieved.The turbofan engine mainly consists of the following components: inlet, fan, HPC,combustor, High Pressure Turbine (HPT), Low Pressure Turbine (LPT), bypass, nozzle.38Each component is modeled by aerothermodynamics calculations and solving a set of balance equations.The engine design operation data and characteristic maps of rotating components are used to construct the turbofan engine nonlinear model. This turbofan engine nonlinear model has been also used in other extensive researches.39–43The detailed nonlinear model can calculate steady state data and response data at an arbitrary operating point.The nonlinear model of the turbofan engine is given by

where x(t )∊Rnis the engine state vector,u(t)∊R is the input to the engine, ym(t)∊R is the engine main output to be tracked, yc(t)∊Rmis the engine constrained output vector,f( ∙), g(∙) and h(∙) are differentiable nonlinear functions. A GSC is designed such that ym(t) tracks a desired reference r(t)as time goes to infinity.For each trackable reference,there is an equilibrium point which satisfies the equations

where Ai, Bi, Cmi, Dmi, Cci, Dciare system matrices of the ith equilibrium point. The linear model Eq. (5) is preprocessed by discretization using a zero-order hold of sampling period T = 0:02 s to bring the engine model into discrete-time form,convenient for follow-up design and digital control implementation:

2.1. Gain-scheduling control design

Fig.1 is the schematic of the state and output dependent GSC system without the SCG.

which is an LQ-Proportional Integral(PI)controller where K1(a(k)) is the proportional control gain matrix and K2(a(k))is the integral control gain parameter. Combining the LPV augmented system in Eq. (11) with the GSC in Eq. (13) yields

Fig. 1 Structure diagram of GSC without SCG.

Note that fclis an m-dimensional differentiable nonlinear vector function which represents the closed-loop system dynamics, gcland hclare differentiable nonlinear functions,which generate the engine main tracking output and the engine constrained output vector.

The equilibrium family of the closed-loop system in Eq.(17) match the engine equilibrium family in Eq. (3) so that the closed-loop system maintains suitable scheduling parameters the same as the LPV model in Eq. (6). In general, the Eq. (15) can well describe the states and outputs of the closed-loop system. However, for the turbofan engine, due to its dynamic nonlinear characteristics, the Eq. (15) fails to achieve the expected effect, so a method of establishing a closed-loop system LPV model based on the nonlinear closed-loop system is used in this article. To obtain an LPV closed-loop system model with sufficient accuracy, each linear closed-loop system model of the corresponding equilibrium point is established by the hybrid fitting method and the matrices Acl0, Bcl0, Cclm0and Cclc0are used as incipient matrices.Hence, the LPV closed-loop system model can be written as

The LPV model Eq. (18) can be used to safely predict the behavior of the closed-loop system Eq.(17),which will be used to predict the constrained outputs and the input rate in the SCG.

2.2. Stability analysis of closed-loop system

Relying on the design of the LQ tracking controllers,the local stability around the equilibrium points of the system is guaranteed. However, However, the stability of the GSC needs to prove the global stability of the system. In this section, a stability verification method for the closed-loop system is described which is a necessary condition for designing SCG.Then the global stability is verified by proposing a numerical optimization process.

Lemma 147. Eq. (18) is asymptotically stable if, and only if,there exits symmetric positive definite matrices P(a) and Q(a)such that

According to Lemma 1, the closed-loop system in Eq. (18)is asymptotically stable.

To prove necessity, assume the closed-loop system in Eq.(18) is asymptotically stable. Then, the Eq. (26) is feasible,which is equivalent, by Schur complement, to Eq. (25). Set one of βi= 1 in turn and others equal to 0,the Eq.(22)is get.

By computing a single Lyapunov matrix P in Eq.(22)for a symmetric positive definite matrix Q using convex optimization tools, the global stability is verified.

3. Scheduling command governor design

In this section, the detailed design method of the SCG is presented,the structure diagram of the SCG can be seen in Fig.2.Since the turbofan engine limitation is not considered in the GSC design, the SCG is considered to be used as an add-on part to handle constraints of the turbofan engine. The command governor is a dynamical system that adjusts the reference trajectory to follow the desired reference system capturing a desired closed-loop dynamical system behavior in transient time. Most treatments of command governors assume Eq.(17)is linear and make use of constraint admissible positively invariant sets. The advantage of linearity is that the Maximal Output Admissible Set (MOAS) can be determined constructively. Considering the nonlinearity of the Eq. (17),it is difficult to make the construction of such sets. A possible approach to avoid this difficulty is to embed the Eq. (17) into an LPV model.

Fig. 2 Structure diagram of SCG.

Fig. 3 Ad(a), Bd(a), Cdc(a) and Ddc(a) components as functions of scheduling parameter a.

where j=0;1;∙∙∙. Each Njcontains safety pairs(xcl(0);v(0))that do not exceed the limit within j sample time.Every time before increasing the index j, the following condition will be checked

If the Eq.(34)is confirmed,O∞=Nj=Nj+1,hence,the set O∞is said to be finitely determined and j is called the finite index. With the finite index j, the MOAS will be computable.The construction of the MOAS makes the more safe pairs,which makes the SCG has larger constrained domains of attraction.

Fig. 4 Steady state values of equilibrium points as functions of scheduling parameter a.

To reduce the complexity of the algorithm for constructing the MOAS for the given LPV system, the varying of the equilibrium point is ignored in the prediction process. However,the update of the equilibrium point will be performed on the moving horizon.

Fig. 5 K1(a) and K2(a) components as functions of scheduling parameter a.

The algorithm for constructing the MOAS for the given LPV system can be formulated:

Algorithm 1. Given an LPV system Eqs. (27)-(28) subject to constraints Eq. (30), perform the following operations:

is not strictly larger than the number of rows in AN:

Fig. 6 Acl(a), Bcl(a) and Cclc(a) components as functions of scheduling parameter a.

In the existing approaches,a functional characterization of the set N(or of its subset)is computed offline or,alternatively,an online response prediction may be used.The computational overhead of calculating the N offline or predicting system constraints output online based on the set N is mainly affected by the number of the equilibrium points we select. Reducing the number of equilibrium points selected in Algorithm 1 can effectively reduce the computational cost, the scope of application and the conservativeness of the set N. However,some strong properties such as positive-invariance properties and feasibility may be lost because of the lack of a global perspective when designing the set N.

In this paper,scheduling local constraint set is designed offline, which is equivalent to considering a single initial stage of the equilibrium point in Algorithm 1.Then,an SCG method is used online solving a similar optimization problem in Eq.(36).The SCG schedules local constraint set and contractive index according to the current system state.

Fig. 7 Comparisons of tracking simulations with LPVCLM and NCLM.

Set a constant reference r to the Eqs. (17) and (37) starting from the same equilibrium points,two system state and output sequence from initial instant to instant k are obtained until the reference is tracked:

where P is a positive definite matrix and ρ(Acli) is the spectral radius of the matrix Acli.

Fig. 8 Related design parameters of SCG.

Remark 3. Relying on Algorithm 2, the calculation of the contractive index λinot only contract the local constraint sets,but also improves the accuracy of the closed-loop system LPV model in describing the acceleration process.

With the use of a scheduling local constraint set,which does not satisfy appropriate positive-invariance properties, the feasibility of the SCG optimization problem may occasionally be lost. In this case, the Eq. (43) is modified to

When the online optimization problem does not admit a solution,the used strategy is to continue applying the previous reference, which could result in the stop of updating reference and the tracking rate will slow down. Nevertheless, the scheduling method reduces the duration of the unsolved situation to a certain extent.

4. Simulation results and discussion

To illustrate the effectiveness of the proposed SCG scheme for limit protection,numerical simulations on the turbofan engine component level model described above have been carried out.The engine model has two states(fan speed NLand compressor rotor speed NH), a single input (fuel flow Wf), a single main output NL, two constrained outputs (compressor outlet pressure P3and low pressure turbine outlet temperature T6)and an introduced constraint input rate (rate of fuel flow Wfr). All variables are normalized by their design values. The fan speed NLis chosen as the scheduling parameter of the LPV model,controller and constraint set in this paper.

Consider taking an equilibrium point every 2% of the fan speed in the range from 0.65 to 1.05 at sea level condition.The linearization matrices for these twenty-one equilibrium points and steady state values are shown in Figs. 3 and 4 respectively. Piecewise linear interpolation has been used to compute matrices components and steady state values of each pair of adjacent equilibrium points.

The linear controller is designed for each linearization model and selected equilibrium point in Figs. 3 and 4. Then,the GSC is interpolated with respect to the scheduling parameter. The controller parameters K1(a) and K2(a) are shown in Fig. 5.

To verify the stability of the closed-loop system and design the SCG, the LPV closed-loop system model is established. A part of related matrices components are shown in Fig. 6 and the others are set to 0.

By solving LMIs in Eq. (22), we have

In order to verify the stability and accuracy of the LPV Closed-Loop system Model (LPVCLM) with simulation,experiments are conducted respectively based on LPVCLM and Nonlinear Component-Level Model (NCLM) to increase the fan speed from 0.65 to 1, Fig. 7 shows the results.

Fig. 9 Comparison of SCG-GSC and GSC.

Since the engine dynamic state is far away from the equilibrium manifold in the simulation without considering constraints, the prediction accuracy of LPVCLM is reduced.However,the model will have higher prediction accuracy when constraints are enforced, which leads the engine closer to the equilibrium manifold.

To enforce constraints to the engine,the SCG has been well designed.Some details such as the spectral radius of the matrix Acl(a), the finite index j(a) and the contractive index λ(a) are shown in Fig. 8.

In the following simulations,the SCG strategy is applied to control the high fidelity turbofan engine NCLM. This case study simulates the engine acceleration from 65% to 100%of the fan speed for the standard day sea level condition.The constraints below are considered in the simulations.

Then, the following three scenarios have been used for different simulation purposes. In Scenario 1, to verify the effectiveness of the developed SCG scheme, comparisons put in two phenomenon that whether the SCG is added on the GSC. The simulation results are shown in Fig. 9.

From Fig. 9, one can find that the acceleration of the fan speed obtained by the SCG-GSC method is slower because of the strict enforcement of constraints that make a lower command before the virtual reference reaches 100%.In Scenario 2,to verify the superiority of the developed SCG scheme, we make a comparison between the proposed SCG method and the traditional Min-Max approach. The simulation results are shown in Fig. 10.

Fig. 10 tells that the acceleration of the fan speed obtained by the SCG is slightly faster than the traditional Min-Max approach since the former uses a more aggressive fuel regulation method in the early stage of acceleration. Meanwhile, the constrained outputs T6and P3obtained by the SCG are much more closer to the limit values during acceleration than the traditional Min-Max approach, which shows that the former method becomes less conservative due to the introduction of the optimization idea. In order to verify the design effect of the contraction set, we make a comparison for the different contractive index of the SCG method, the simulation can be seen in Fig. 11.

Fig. 10 Comparison of SCG and traditional Min-Max approach.

Fig. 11 Comparison of SCG using different contractive index.

It can be observed from Fig.11 that the acceleration of the fan speed becomes faster when the contractive index is increased. Setting λ=1, which means using local constraint set during acceleration, leads to the temperature T6slightly exceeds the limit value. Another situation that setting λ=0.975 causes a too slow tracking speed. Unlike the first two,the interpolation within contractive indexes in Algorithm 2 successfully avoids constraint violations while providing a satisfactory dynamic effect. Generally speaking, the value of the contractive index represents the degree of contraction of the local constraint set corresponding to each equilibrium point,which can characterize the robustness of the contractive set to the dynamic and equilibrium point change of the engine.

5. Conclusions

In this paper, the SCG design for limit protection of turbofan engines during acceleration is discussed. The turbofan engines are embedded into a discrete time LPV system.The stability of the GSC designed based on the LPV system has been proved theoretically. The algorithm of calculation scheduling local constraint set based on the general algorithm of calculation MOAS for the LPV system reduces the design complexity and calculation burden. The design of the contractive set supports is conducted for the regulation of restricting conservativeness. The limit protection method based on the SCG proposed in this paper can handle the constraints during acceleration while providing faster and less conservativeness response than the traditional Min-Max approach.The simulations on the turbofan engine NCLM demonstrated the effectiveness and superiority of the proposed SCG method for turbofan engines. In the future, the acceleration schedule will be considered for a better acceleration and a robust CG will be studied for handling the complex uncertainty of the engine.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This research was supported by National Science and Technology Major Project of China (No. 2017-V-0004-0054).