Non-affine parameter dependent LPV model and LMI based adaptive control for turbofan engines

Bei YANG,Xi WANG,Penghui SUN

aSchool of Energy and Power Engineering,Beihang University,Beijing 100083,China

bSchool of Aircraft Engineering,Nanchang Hangkong University,Nanchang 330063,China

KEYWORDS Adaptive control;Linear matrix inequalities;Linear parameter varying;Neural networks;Turbofan engines

Abstract The precise control of turbofan engines thrust is an important guarantee for an aircraft to obtain good flight performance and a challenge due to complex nonlinear dynamics of engines and time-varying parameters.The main difficulties lie in the following two aspects.Firstly,it is hard to obtain an accurate kinetic model for the turbofan engine.Secondly,some model parameters often change in different flight conditions and states and even fluctuate sharply in some cases.These variable parameters bring huge challenge for the turbofan engine control.To solve the turbofan engine control problem,this paper presents a non-affine parameter-dependent Linear Parameter Varying(LPV)model-based adaptive control approach.In this approach,polynomial-based LPV modeling method is firstly employed to obtain the basis matrices,and then the Radial Basis Function Neural Networks(RBFNN)is introduced for the online estimation of the non-affine model parameters to improve the simulation performance.LPV model-based Linear Matrix Inequality(LMI)control method is applied to derive the control law.A robust control term is introduced to fix the estimation error of the nonlinear time-varying model parameters for better control performance.Finally,the Lyapunov stability analysis is performed to ensure the asymptotical convergence of the closed loop system.The simulation results show that the states of the engine can change smoothly and the thrust of the engine can accurately follow the desired trajectory,indicating that the proposed control approach is effective.The contribution of this work lies in the combination of linear system control and nonlinear system control methods to design an effective controller for the turbofan engine and to provide a new way for turbofan engine control research.

1.Introduction

The turbofan engine is highly complex nonlinear system whose performance characteristics change with working states and flight conditions.1-3It is difficult to obtain a dynamic model that can completely describe the dynamic behavior under any cases.The linearization at equilibrium points is needed to obtain a linear model for control design.However,the nonlinear properties are quite different under different working conditions.It is hard to ensure the control performance in the whole flight envelope with the controller designed based on linear model.To improve the control performance of the gas turbine engine,the common method is to divide the whole flight envelope into several flight subenvelopes and to design the controllers for each flight subenvelope.Next,a gain scheduling strategy is used to switch the controllers to generate the control signals.In fact,the gain scheduling strategy combines these controllers of the same structure and different gains into a super controller.4-7Several control methods such as PI control,H∞robust control,and model predictive control8-11are often used to derive the controllers for the distinct flight sub-envelopes.However,the main drawback of gain scheduling methods is that it needs to select many equilibrium points in the working space.Each equilibrium point is to ensure the stability of the flight process.In addition,it requires the switch between the controllers to be unperturbed.12,13However,if the real working condition is quite different from the reference working condition,the designed controller cannot ensure the desired control performance.In addition,the more flight sub-envelopes there are,the more complex the gain scheduling strategy is.The reliability of the closed-loop system will become worse so as to impact the flight performance.Moreover,the response speed of the switch between the controllers may be slower than the speed change of the system dynamics.This will directly influence the stability of the system.

To overcome the drawbacks of the gain scheduling method,several variable-gain control approaches such as LPV robust variable-gain control and multi-objective robust variable-gain control have been proposed recently.14-18These variable-gain control approaches can adapt the gain of the controller with time-varying parameters,while the gain scheduling control performs the switch between the controllers using interpolation methods.When the working point of the system deviates from the equilibrium point,the variable-gain controller can still ensure the asymptotical convergence of the system.This is because the LPV model can adequately describe the nonlinear dynamics of the system.We can construct a LPV polytopic system in the time-varying parameter space,and use linear control theory and LMI method to derive a suitable controller for the system.17However,the disadvantage is that the robust variable-gain controller may be conservative in most cases if the parameters change significantly.8,16,19

Although the conservative design strategy of the robust variable-gain control can ensure the system stability,it may worsen the control performance.Although some approaches such as robust variable-gain control based on affine parameter-dependent LPV model can reduce the conservation of the robust controller,17,20it cannot accurately describe the nonlinear dynamics of the gas turbine engine due to the rapid change of the working environment.There are two key problems to be addressed in turbofan engine control based on the LPV model.One is to construct the LPV model that accurately describes the system for a turbofan engine.The other one is to develop a suitable adaptive controller to deal with the model error and the time-varying parameters.

To estimate the nonlinear dynamics of the turbofan engine,this paper proposes a non-affine parameter-dependent LPV model based on the affine parameter-dependent LPV model.In the proposed model,the equivalent affine parameters are considered as an unknown nonlinear function with respect to the time-varying parameters and the state variables.Additionally,the Radial Basis Function(RBF)network is adopted to estimate these unknown nonlinear functions.Using the proposed LPV model,an adaptive control approach for a turbofan engine is presented based on the Lyapunov stability analysis to improve the control performance.

2.Non-affine parameter-dependent model for turbofan engines

Generally,a turbofan engine can be described as

where x∈Rnis the state vector,u∈Rmis the control vector,and y∈Rpdenotes the output vector which usually includes the thrust and fan rotation speed of the engine,f（·）and g（·）are the n dimensional nonlinear functions which represent the engine dynamics and the p dimensional measurement output function,respectively.However,due to the complex aerodynamic characteristics of the engine,it is difficult to obtain both the functions.Therefore,the Eq.(1)is transformed into a linear model using the Taylor series expansion method at the equilibrium point（xe，ue）which satisfies the following equations

If ignoring the high order terms,the linearization approximation model of a turbofan engine can be expressed as

where A，B，C，D are the parameter matrices of the system.For convenience,let Δx=x-xe， Δu=u-ue,and Δy=y-ye.Since there are many equilibrium points in a flight process,Δx，Δu can be further rewritten as x，y for simplification,and the Eq.(4)can be described as

Generally,the parameter matrices A，B，C，D depend on the measurable time-varying parameters and the system state variables,and then the system can be rewritten as

where α=（N2-N2，min）/（N2，max-N2，min）is the time-varying parameter,in which N2represents the high pressure shaft speed,N2，maxis the high pressure shaft speed under the full power condition,and N2，minis the idle speed.Assuming there is a nonlinear map θ:=ψ（α）to transform the Eq.(6)into affine LPV form with respect to the equivalent parameter θ,the system can be expressed as

where θ0is fixed as 1,and Ai，Bi，Ci，Diare the base matrices,which can be obtained by solving the following optimization problem

where Qa，Qb，Qc，Qdare the weighting coefficient,and Np∈R+is the number of linearized models.Solving this optimization problem requires that the equivalent time-varying parameter θiis exactly given.Since the nonlinear transformation function ψ（α）is unknown,it is difficult to obtain the base matrices Ai，Bi，Ci，Di.Many studies indicate that a polynomial may approximate any nonlinear function with arbitrary accuracy if the degree is large enough.21-23Therefore,one way to obtain the base matrices Ai，Bi，Ci，Diis to use polynomials to estimate the unknown function ψ（α）with the sampling data of the system and then the equivalent time-varying parameter θiis calculated by

where kiis the multinomial coefficient.Substituting Eq.(10)into Eq.(9),the base matrices can be determined.It should be noted that the estimation of the function ψ（α）may be unavailable because the explosive behavior of polynomial may be unavailable outside the estimated region.Therefore,in some cases,the estimation of θicalculated by Eq.(10)may be far from its real value,which greatly impacts the stability of the control process.To avoid this problem,these unknown functions are estimated online in the present work.Based on the obtained basis matrices Ai，Bi，Ci，Di,RBF network is employed to online estimate the nonlinear transformation ψi（α）,yielding

where Wi∈Rd×1is the weight matrix corresponding to the parameter θi,δ（α） ∈Rd×1is the activation function vector,εidenotes the estimation error,θiand ψi（α）are the ith element of θ， ψ（α）,respectively.Then the estimation of θican be expressed by

Substituting Eqs.(8)and(12)into Eq.(7),the system can be rewritten as

where εy，i（x，u）denotes the estimation error of system output,εT，i（x，u） denotes the estimation error which satisfies‖ εT，i（x，u） ‖ ＜ ε*T，i＜ ρ*（1+‖x‖）,and ε*T，i， ρ*∈R are the unknown constants.

3.LPV model based adaptive control algorithm

3.1.Adaptive control algorithm design

To deal with the time-varying parameters of LPV systems,the possible worst cases of the parameter change should be considered when designing a robust controller for the system.However,if these worst cases only appear occasionally,the robust controller would be excessive conservative in most cases.To address this issue,this paper proposes an adaptive control approach for the turbofan engine system.

Firstly,we consider the parameter-dependent autonomous system with the following form

where A（θ）=θ0A0+θ1A1+···+θLAL.The quadratic stability of the Eq.(16)means that there exist L+1 positive definite symmetric matrices P0，P1，···，PLto satisfy the following inequalities

The following theorem gives the necessary and sufficient conditions to guarantee the stability of the Eq.(6).

Theorem 124.The parameter-dependent Eq.(15)is asymptotically stable if there exist the positive definite symmetric matrices P0，P1，···，PLand all n（n+1）/2 matrices Hij，0 ≤ i＜ j≤ L to satisfy the following conditions:

where His symmetric negative definitive matrix,

Proof.For affine parameter-dependent LPV system,the system matrix A（θ）and the positive matrix P（θ）can be written as

Using the inequalities Eq.(18)and Hi=ATiPi+PiAi,Eq.(22)can be written as

where η = ［θ0，θ1，···，θL］T.Since θ0=1,it implies η≠0,then the following inequality holds

The inequality Eq.(25)means the asymptotical stability of the Eq.(15).

Theorem 1 implies that there exists a state feedback matrix K（θ）=to construct the control law u=K（θ）x,then we can obtain the closed-loop system

Theorem 224.Consider the system characterized by A（θ），B（θ），C（θ），D（θ）,its close-loop system Eq.(26)is stable if there exist positive definite symmetric matricesreal symmetric matrixand n（n+1）/2 matricesto satisfy the following inequalities

The detailed proof of Theorem 2 can be found in Ref.21.

3.2.Neural network design

The state feedback matrix Eq.(31)can be constructed by the scheduled parameter θ if the equivalent time-varying parameter θ1，θ2，···，θLcan be directly measured.However,these parameters are not measurable.Consequently,in this paper,RBF neural network is used for online estimation of these equivalent parameters,and the update law of the RBF weight is derived to ensure the asymptotical convergence of the closed-loop system.

Since θ cannot be directly measured,it is difficult to estimate these parameters by minimizing the residual identifier error which is based on gradient25or modified Robust Integral of Sign of the Error(RISE)algorithm,26and it is also hard to guarantee the identifier weight convergence.27To ensure the convergence of the system control process with parameter estimation,Na presented a novel adaptive law to‘direct'estimate the unknown NN weights28in Eq.(14).

In the ‘direct” estimation method,two auxiliary filters are introduced to estimate the NN weight,which are de fined respectively as follows where xf，δfare the filtered variables with respect to x，δ,δiand δf，iare the ith element of δ and δf,respectively;0＜k∈R is filter constant.From Eqs.(14)and(32),we can obtain

Additionally, two auxiliary matrices Φi∈Rd×dand Qi∈Rd×1are de fined,which satisfy the following equations

where ηfis a positive constant selected by designers.Then we have

Furthermore,matrix Mi∈Rd×1can be de fined as

It should be noted that if the activation function vector δiis Persistently Excited(PE),we can conclude that the matrix Φiis positive definite and satisfies σ ＜ λmin（Φi）， σ ∈R+.Next,we can de fine the following update law for ˙^Wi.

where Γi＞0，i=1，2，···，L are constant learning gain matrixes.To deal with the estimation error εT，i,a smooth robust term uris introduced to guarantee the asymptotical convergence of the closed-loop system,and the adaptive laws are de fined as

where η1，η2，η3are positive scalars given by designers,ρ is the estimation of ρ*,and ρ～=ρ*-ρ denotes the estimation error.

Using Eq.(31)and the robust term ur,we can obtain the control law of

3.3.Lyapunov stability analysis

Based on Eqs.(13)and(14),the system can be rewritten as

Theorem 2 indicates that there exist L+1 positive definite symmetric matricesand matricessatisfy the following inequalities

and the closed-loop system can be rewritten as

Considering the following Lyapunov function candidate

whose time derivative is given by

Assume that the states and control are bounded,then the following inequalities hold

where βi∈R+，i=1，2，···.L.Then we can obtain

Combining Eqs.(34),(35)and(36),we obtain

Substitute Eq.(59)into Eq.(57),we find that

Similarly,with substitution of Eqs.(41)-(43)into Eq.(60),it is found that

4.Simulation and results

As stated in Section 2,in the non-affine parameter-dependent LPV model,weneed to determine the basis matrices Ai，Bi，Ci，Di，i=0，1，···，L,and to estimate the unknown nonlinear parameters of the basis matrices by RBFNN.To obtain these basis matrices, firstly we need to measure the dynamics of the turbofan engine to obtain sufficient sample data,then use polynomial to approximate the unknown nonlinear parameters instead of the RBFNN.

In fact,several accurate nonlinear models for turbofan engine have been developed in the past years.They use Newton-Raphson's method to calculate the turbofan engine dynamics.However,these nonlinear models usually cannot be interpreted by differential equations.As a result,they cannot be directly used for the nonlinear control design of the turbofan engine.This is the reason why we need to reconstruct the LPV model for the turbofan engine.We use the data to validate the proposed control method.Considering the common situation of turbofan engine control practice,this uses engine fan speed N1and engine core speed N2as the state variables,and the fuel flow and the thrust are regarded as the control variable and the system output,respectively.However,since it is usually hard to directly measure the thrust,the thrust can be usually controlled indirectly by controlling the low pressure rotor speed according to the turbofan engine control practice.8Additionally,other parameters such as the turbine inlet temperature and the compressor discharge pressure are used as performance indicators.

According to the Taylor expansion,nonlinear system can be approximated by a linear model in the neighborhood of the equilibrium points.If we select some working points in the flight envelope,and linearize the system at the working points,then a set of linear models can be obtained,which can represent the real system.Obviously,the system matrix and the control matrix of the linear models change with time and system state,and we can use the components of the matrices in the obtained linear models to evaluate the basis matrices of the proposed LPV model.If the components of the LPV model agree well with the corresponding components of the linear models,we can conclude that the proposed LPV model is effective.

In this simulation,the number of the basis matrices L is 4,and the design parameters of NN are listed in Table 1,where k and ηfare the filter constant and the positive constant,respectively,which are selected by the designers,η1,η2,and η3are the design parameters for the adaptive law,and Γ1,Γ2,Γ3,and Γ4are the design parameters for the Lyapunov function.The thirteen working points in the flight envelope are selected to linearize the system.These equilibrium points are selected to describe the large-scale engine operation from idle to full power for the condition H=0 and Ma=0(H and Ma denote flight altitude and flight mach number,respectively).The fuel flow and shaft speed of these equilibrium points are listed in Table 2.

Table 1 Values of design parameters.

Table 2 Working conditions for the selected 13 equilibrium points.

To evaluate the basis matrices Ai，Biof the proposed LVP model,we first linearize the real turbofan engine system at the selected 13 working points,and obtain 13 matrices A and B,as shown in Fig.1.According to Eqs.(8)and(9),the polynomial is used to approximate the 13 obtained matrices A and B.Three indices including Root-Mean-Square Error(RMSE),Average Error(AE)and Maximum Error(ME)are introduced to evaluate the simulation performance.The RMSE,AE and ME of each element of the basic matrix are de fined respectively as follows:

where Xejdenotes the jth actual value of the set Xe,and Xjdenotes the estimated value of Xej.Then,the basis matrices Ai，Biof the non-affine parameter dependent LPV model based on the polynomial estimation can be found in Table 3.The basis matrices of Kiare shown in Table 3.

The estimation results of the components of A and B in the LPV model are shown in Figs.1 and 2,where the scale of the horizontal axis is the serial number of the selected working points,and that of the vertical axis is the normalized value of the elements in A and B.

Fig.1 Elements of system matrix A in LPV model estimated by polynomial and obtained by linearizing real system at 13 working points,respectively.

Fig.1 shows that the components of the matrix A in the LPV model estimated by polynomial agree well with the real system,indicating that the estimated basis matrices Ai，Biare available.In some cases,however,there are large tracking errors such as A11due to the difficulty of determining the order of the polynomial.If the order is insufficient,it is hard for the polynomial to approximate the nonaffine parameters,and it may even not be able to ensure the calculation convergence.Table 4 shows that both the RMSE and AE of the components of the matrix A are small,and the total RMSE and AE of the matrix A are 0.2703 and 0.1602 respectively,implying the effectiveness of the obtained basis matrix Ai.Likewise,Fig.2 illustrates that the matrix B in the LPV model agrees well with the control matrix obtained by linearizing the real system at the 13 working points.The three evaluation indices are small,as shown in Table 4.The RMSE and AE are 0.0117 and 0.013,respectively.Therefore,it can be concluded that the proposed LPV model is effective.

Nevertheless,the LPV model error still impacts the control performance of the turbofan engine in a certain degree,and further improvement of simulation performance of the LPV model is required.This is the reason why this work adopts the neural network to online estimate the time-varying parameters instead of using the polynomial based on the obtained basis matrices Ai，Bi.Consequently,the proposed adaptivecontrol method allows us to regulate the low pressure rotor speed so as to indirectly control the thrust.

Table 3 Basis matrices of A,B and K.

Fig.2 Elements of system matrix B in LPV model estimated by polynomial and obtained by linearizing real system at 13 working points,respectively.

Table 4 Evaluation indices of the matrices A and B.

Fig.3 Thrust of the turbofan engine during operation.

To illustrate the effectiveness of the adaptive control method,a comparative experiment with the PI based gain schedule method is conducted for the turbofan engine.The simulation results of the engine thrust are shown in Fig.3,which shows that both control methods can achieve similar thrust with the step input,but the obtained thrust with the proposed method changes more smoothly than that with the PI based gain schedule method.The controlled thrust with PI based gain schedule method is easy to overshoot and vibrate.In addition,the turbine inlet temperature T4and the exhaust gas temperature T5also have large oscillations in some cases,as shown in Fig.4.This is because the proportion integral parameters of the PI controller for the two adjacent working points are different.Therefore,the non-smooth switch of the PI controller parameters occurs,and the transition process between the two adjacent working points shows certain fluctuation.In contrast,the two controlled parameters of the turbofan engine change more smoothly with the proposed control method.The reason is that the time-varying parameters of the system can be accurately estimated by the NN,and the adaptive law can also compensate the estimation error on the system dynamics.Likewise,similar control results of the output temperature T3,pressure of the compressor P3,N1and N2can be obtained,as shown in Figs.5 and 6.

Fig.4 Turbine inlet temperature T4and the discharge temperature T5.

Fig.5 Compressor outlet temperature T3and pressure P3.

Since the proposed LPV model for the turbofan engine is a deviation model about the equilibrium point,i.e.,the values of the system states and control inputs in the LPV model are the relative values at the equilibrium points.5The real control signal consists of three parts,i.e.,u=Kx+ur+u0,where u0is the normal control input at the equilibrium point.The real fuel flow is shown in Fig.7.It can be seen that as compared to the PI based gain schedule method,the real fuel flow with the proposed adaptive control method changes more smoothly,indicating that the fuel flow swing of the engine can be restrainted.In addition,sometimes it exhibits a steep decline in term of the surge margin of the booster in the PI based gain schedule control process,which usually leads to the surge and thus impacts the stability of the system,as shown in Fig.8,where the surge margin changes smoothly in the adaptive control case.In summary,the adaptive controller can ensure the convergence of the closed loop system,and the desired control performance can be achieved.These suggest that the proposed control method is feasible and effective.

Fig.6 Fan speed N1and core speed N2.

Fig.7 Fuel flow during operation.

Fig.8 Surge margin.

Fig.9 Simulation results of LPV models and coefficients of basis matrices online estimated by RBFNN.

A set of given control signals are used to drive the real system and the LPV model which is estimated by the RBFNN and polynomial is used to evaluate the simulation of the LPV models.The results are illustrated in Fig.9.From Fig.9(a),it can be seen that the controlled output N1of the RBFNN LPV model agrees with the real output of the system well,while the simulation performance of the polynomial LPV model is worse than that of the RBFNN LPV model.The estimated coefficients of the basis matrices are stable,as shown in Fig.9(b).These results indicate that the estimated LPV model is effective.

5.Conclusions

In essence,the LPV model of the turbofan engine is a kind of nonlinear model.If the LPV model is accurate enough,it can predict the system dynamics well.If the turbofan engine system flatly changes,the controller designed with an accurate LPV model is satisfactory.However,due to the quick change of the system dynamics and the local optimum of the NN,the estimated time-varying parameters with the LPV method do not always converge to their real value in finite time interval,which results in additional error.

To minimize the effect of the model error,this paper proposes an adaptive control method for the turbofan engine based on the state feedback control theory for linear systems.The Lyapunov stability analysis is performed to ensure the asymptotical convergence of the closed system.The simulation results indicate that the proposed control approach can ensure a smooth change of the controlled parameters,such as the thrust and the turbofan inlet temperature.

The novelty of this work is that the method combining the LMI control method for linear systems and the adaptive control method for nonlinear systems is proposed to develop a suitable controller for the turbofan engine.The LMI control laws used to ensure the major tracking behavior of the system and the adaptive law is used to offset the bad effect of the LPV model error on the control performance.In other words,the proposed control approach combines the merits of the LMI control and adaptive control methods,and effectively ensures the control accuracy of the controlled outputs and stability of the control process.The proposed approach also provides a new way for turbofan engine control design.

Acknowledgements

This study was supported by the National Natural Science Foundation of China(No.51766011)and the Aeronautical Science Foundation of China(No.2014ZB56002).