Two freedom linear parameter varying μ synthesis control for flight environment testbed

Chinese Journal of Aeronautics, (2019), 32(5): 1204–1214

Chinese Society of Aeronautics and Astronautics & Beihang University

Chinese Journal of Aeronautics [email protected] www.sciencedirect.com

Two freedom linear parameter varying l synthesis control for flight environment testbed Meiyin ZHU a,b, Xi WANG a,b,*, Zhihong DAN c, Song ZHANG c, Xitong PEI c a

School of Energy and Power Engineering, Beihang University, Beijing 100083, China Collaborative Innovation Center for Advanced Aero-Engine, Beijing 100083, China c Science and Technology on Altitude Simulation Laboratory, Mianyang 621000, China b

Received 13 April 2018; revised 2 July 2018; accepted 4 September 2018 Available online 15 February 2019

KEYWORDS Altitude ground test facilities; Flight environment testbed; Linear parameter varying; Robust control; Two-degree-of-freedom; l Synthesis

Abstract To solve the problem of robust servo performance of Flight Environment Testbed (FET) of Altitude Ground Test Facilities (AGTF) over the whole operational envelope, a two-degree-offreedom l synthesis method based on Linear Parameter Varying (LPV) schematic is proposed, and meanwhile a new structure frame of l synthesis control on two degrees of freedom with double integral and weighting functions is presented, which constitutes a core support part of the paper. Aimed at the problem of reference command’s rapid change, one freedom feed forward is adopted, while another freedom output feedback is used to meet good servo tracking as well as disturbance and noise rejection; furthermore, to overcome the overshoot problem and acquire dynamic tuning, the integral is introduced in inner loop, and another integral controller is used in outer loop in order to guarantee steady errors; additionally, two performance weighting functions are designed to achieve robust specialty and control energy limit considering the uncertainties in system. As the schedule parameters change over large flight envelope, the stability of closed-loop LPV system is proved using Lyapunov inequalities. The simulation results show that the relative tracking errors of temperature and pressure are less than 0.5% with LPV l synthesis controller. Meanwhile, compared with non-LPV l synthesis controller in large uncertainty range, the proposed approach in this research can ensure robust servo performance of FET over the whole operational envelope. Ó 2019 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).

1. Introduction * Corresponding author at: School of Energy and Power Engineering, Beihang University, Beijing 100083, China. E-mail address: [email protected] (X. WANG). Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

The supersonic and hypersonic flight needs of future advanced aircraft systems are driving the research and development of advanced aircraft engine, especially the research on TurbineBased Combined Cycle (TBCC) propulsion which has greatly been expanding the working envelope of aircraft system.1–3 On the other hand, the research and development of the engine greatly depend on Altitude Ground Test Facilities (AGTF)

https://doi.org/10.1016/j.cja.2019.01.017 1000-9361 Ó 2019 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Two freedom linear parameter varying l synthesis control for flight environment testbed

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Nomenclature cp P Pin1 Pin2 Q R T Tin1 Tin2 V c h

specific heat at constant pressure, J/(kgK) pressure of FET, Pa pressure of the first inlet, Pa pressure of the second inlet, Pa heat, J gas constant, J/(kgK) temperature of FET, K temperature of the first inlet, K temperature of the second inlet, K volume of FET, m3 average flow velocity of gas, m/s enthalpy of air in FET, J/kg

tests (component, overall, assessment and performance tests and so on). The aircraft engine requires AGTF to provide a wider test range to meet its test requirements, which means that AGTF will operate in a larger working envelope. For AGTF, the desire is always to test the engine as close to the conditions (altitude and Mach number) that it will encounter during flight as possible.4 The AGTF needs to simulate the inlet and altitude conditions of the test engine. The inlet condition (total temperature and pressure) is provided by Flight Environment Testbed (FET) where the gas has the same state with the inlet condition. Compared with inlet condition simulation, the altitude condition simulation is much easier. Therefore, the biggest challenge for AGTF is to simulate the inlet condition perfectly over the whole working envelope. However, the conventional controller design method is based on the linear plant of design point, and there must exist large uncertainties when using one linear plant to replace nonlinear system over the whole working envelope. Although many modern control design methods (H1 design, l synthesis, sliding mode control, etc.) can design controller with uncertainty, they cannot ensure that the designed controller has good and robust performance over the whole working envelope.5–9 Considering that the Linear Parameter Varying (LPV) system is good at processing large uncertainties, we propose a new two-degree-of-freedom integral type LPV l synthesis design method to solve the problem of large uncertainties when using a single linear system to design controller and achieve robust performance over the whole working envelope. In Refs.10,11, the LPV system is explained by a family of linear systems whose state-space matrices depend on a set of time-varying parameters. Although the parameters are unknown in advance, it can be measured or estimated upon operations of the systems. Based on the LPV systems, many H1 control schemes have been applied to deal with the attenuation performance for constraining the effect of disturbance on the LPV systems.12–14 Usually, the stability of LPV system is derived by using Lyapunov functions.15,16 Using Lyapunov functions, the gain-scheduled H1 controller design methods have been proposed to guarantee stability and attenuation performance for LPV systems.6,12,16 Ref.11 proposes a gainscheduling of minimax optimal state-feedback controllers for uncertain LPV systems. In Ref.17, the gain-scheduled H1 method is used in a LPV stochastic system. l synthesis theory

hin1 hin2 m_ in1 m_ in2 m_ out Q_ Dr H Ma

enthalpy of air in the first inlet, J/kg enthalpy of air in the second inlet, J/kg mass flow rate of the first inlet, kg/s mass flow rate of the second inlet, kg/s mass flow rate of outlet, kg/s heat transfer rate between FET and surroundings, J/s reference signal set height Mach number

is developed for the first time by Dolye, who introduced the structure singular value to reduce conservative property of robust control design in 1982.18,19 Now, l synthesis theory has become one of the most important techniques in robust control design, which is broadly applied in engineering. A two-degree-of-freedom controller for the distillation column system is proposed in Ref.20 with a reference model and using l synthesis. l synthesis method is used in Ref.21 to achieve robust performance of a flexible-link manipulator in the presence of uncertainties. In Ref.22, a l synthesis controller designed under single flight condition of small commercial aircraft achieved good performance in a large portion of the flight envelope. In Ref.23, l synthesis theory is used to achieve good control of a satellite launch vehicle in presence of parameter uncertainties. l synthesis theory is also applied in aviation field. In Refs.24–26, flight control laws were designed using l synthesis. In Refs.27,28, l synthesis is used to achieve the temperature and pressure synchronous control of FET over a certain working envelope range. However, no literature was found doing research on l synthesis control design based on LPV systems. In order to achieve robust performance over the whole working envelope of FET, this paper will combine the advantage of l synthesis theory and LPV system to do the research on LPV l synthesis design. The paper is organized as follows. First, an augmented LPV system representation of the nonlinear physical system is presented. Then, a brief description of l synthesis theory is given, where the robust stability and performance proof of l synthesis controller is presented. Third, the LPV l synthesis controller design is described, where the stability of closed-loop LPV system is deduced. Finally, the simulation results of LPV l synthesis controller used in FET are presented. To verify the advantage of designed LPV l synthesis controller, we compare simulation results of the l synthesis controller designed on single equilibrium point with simulation results of the LPV l synthesis controller. 2. LPV system description Consider the nonlinear physical system
_ xðtÞ ¼ fðx; u; dÞ yðtÞ ¼ gðx; u; dÞ

ð1Þ

1206

M. ZHU et al.

where x(t) 2 Rn is the system state vector, u(t) 2 Rm is the control input vector, d(t) 2 Rl is the disturbance vector, y(t) 2 Rm is the output vector, f(x, u, d) is an n-dimensional differentiable nonlinear vector function which represents the system dynamics, and g(x, u, d) is an m-dimensional differentiable nonlinear vector function which generates the system outputs. With a desired signal r(t) 2 DrRm given, we intend to devise a feedback control so that the output y(t) tracks r(t) as time goes to infinity and disturbance attenuation with robust ability. It is assumed that, for each r(t) 2 Dr, there is a unique pair (xe, ue, de) that depends continuously on r(t) and satisfies the following equations:


0 ¼ f ðxe ; ue ; de Þ

ð2Þ

rðtÞ ¼ gðxe ; ue ; de Þ

x_ ¼ Ai ðx  xei Þ þ Bi ðu  uei Þ þ Ei ðd  dei Þ y ¼ Ci ðx  xei Þ þ Di ðu  uei Þ þ Fi ðd  dei Þ þ yei

ð3Þ

where Ai 2 Rnn ; Bi 2 Rnm ; Ci 2 Rmn ; Di 2 Rmm ; Ei 2 Rnl ; and Fi 2 Rml ; i 2 I. The matrices are obtained as follows: 8 @f < Ai ¼ @x jðxei ; : Di ¼ @g jðx ; @u ei

uei ; dei Þ ; uei ; dei Þ ;

@f Bi ¼ @u jðxei ;

Ei ¼

uei ; dei Þ ;

@f j ; @d ðxei ; uei ; dei Þ

@g Ci ¼ @x jðx

Fi ¼

ei ;

Corresponding to each linearization at the ith equilibrium point, there exists an ai 2 X, which is a function of equilibrium values of the system, i.e. yei. Then, the nonlinear physical system Eq. (1) can be approximated by the following Linear Parameter Varying (LPV) system:

where

i¼1

Note that aðtÞ is the scheduling parameter for the LPV system and can be measured in real time and ki ðtÞ; i 2 I are the weight values of aðtÞ relative to ai ; i 2 I. Before designing controller with the LPV system, we need to consider the uncertainty of actuators, and consider it as a first-order function with parameter uncertainty as follows: 2 Ka1 3 0  0 Ta1 sþ1 6 7 Ka2 6 0  0 7 Ta2 sþ1 6 7 du ¼ 6 . dv .. 7 .. 6 . 7 4 . . 5 . 0

0



Kam Tam sþ1

Kai 2 ½Kai ; Kai ; Tai 2 ½Tai ; Tai ;

i ¼ 1; 2    ; m

ð7Þ

where dv ¼ v  ve ðaðtÞÞ is the actuator input increment vector, Kai is the gain coefficient of the ith actuator, Kai is the lower bound of Kai ; Kai is the upper bound of Kai ; Tai is the time constant of the ith actuator, Tai is the lower bound of Tai ; and Tai is the upper bound of Tai . Using Laplace transformation in Eq. (7), we obtain 2

 T1a1

6 6 0 6 du_ ¼ 6 . 6 . 4 . 0

0



 T1a2



.. . 0



3

0

2 Ka1

6 7 0 7 6 6 7 du þ 6 7 .. 7 6 . 5 4

 T1am

0



0

Ka2 Ta2



.. .

.. .

0

0

Ta1



0

ð5Þ

3

7 0 7 7 dv .. 7 7 . 5 Kam Tam

ð8Þ Now, we define two metrics as 2

ð4Þ

8 > < dx_ ¼ AðaðtÞÞdx þ BðaðtÞÞdu þ EðaðtÞÞdd aðtÞ 2 X > : dy ¼ CðaðtÞÞdx þ DðaðtÞÞdu þ FðaðtÞÞdd

i¼1

uei ; dei Þ

@g j @d ðxei ; uei ; dei Þ

ð6Þ

i¼1

> > L > X > > > dy ¼ y  ye ðaðtÞÞ ¼ y  ki ðtÞyei > > > > i¼1 > > > > > L L L > X X X > > > AðaðtÞÞ ¼ k ðtÞA ; BðaðtÞÞ ¼ k ðtÞB ; CðaðtÞÞ ¼ ki ðtÞCi > i i i i > > > i¼1 i¼1 i¼1 > > > > > L L L > X X X > > > ki ðtÞDi ; EðaðtÞÞ ¼ ki ðtÞEi ; FðaðtÞÞ ¼ ki ðtÞFi : DðaðtÞÞ ¼ i¼1

where xe is the desired equilibrium point, ue is the steadystate control that is needed to maintain equilibrium at xe, and de is the disturbance at equilibrium point. If we use the linear system of one equilibrium point to represent the nonlinear physical system in the whole working envelope, there must exist large uncertainties when we design controller on the linear system. For reducing the uncertainties, the LPV system will be introduced to approximate the nonlinear physical system. In order to acquire the LPV system, we let W  Rm+n+l be the region of interest for all possible system state, control and disturbance vector (x, u, d) during the system operation and denote xei, uei and dei, i 2 I={1, 2,. . .,L} as a set of steadystate operating points located at some representative and properly separated points inside W. Introduce a set of L regions Wi, i 2 I centered at the chosen operating points (xei, uei, dei), and denote their interiors as Wio, such that Wjo \ Wko = £ for all j – k, and [Li¼1 Wi ¼ W. The linearization of the nonlinear system at each equilibrium point is


8 L L X X > > > aðtÞ ¼ ki ðtÞai ; ki ðtÞ P 0; ki ðtÞ ¼ 1 > > > > i¼1 i¼1 > > > > > > > > L X > > > dx ¼ x  xe ðaðtÞÞ ¼ x  > ki ðtÞxei > > > i¼1 > > > > > > L X > > > ki ðtÞdei > < ddðtÞ ¼ d  de ðaðtÞÞ ¼ d 

6 6 6 Ma ¼ 6 6 4

 T1a1

0



0

 T1a2



.. . 0

.. . 0



0

3

7 0 7 7 ; .. 7 7 . 5

 T1am

2 Ka1 6 6 6 Na ¼ 6 6 4

0



0

Ka2 Ta2



.. .

.. .

0

0

Ta1



0

3

7 0 7 7 .. 7 7 . 5

Kam Tam

ð9Þ

Then, the LPV system Eq. (5) can be augmented as 8 ~ ~ ~ > ~_ ¼ AðaðtÞÞd~ x þ BðaðtÞÞdv þ EðaðtÞÞdd < dx aðtÞ 2 X > : ~ dy ¼ CðaðtÞÞd~ x þ FðaðtÞÞdd where

ð10Þ

Two freedom linear parameter varying l synthesis control for flight environment testbed 

dx









AðaðtÞÞ BðaðtÞÞ 0nm ~ ~ ; AðaðtÞÞ ¼ ; BðaðtÞÞ ¼ 0mn Ma Na   EðaðtÞÞ ~ ~ EðaðtÞÞ ¼ ; CðaðtÞÞ ¼ ½ CðaðtÞÞ DðaðtÞÞ  0ml

d~ x¼



du

Now, we can deduce the design of gain scheduled l controller based on the augmented LPV system Eq. (10). 3. l Synthesis theory description In this section, a brief background l theory, as a framework for robust stability analysis and synthesis, is provided. The l synthesis theory is a robust design method, which is especially good at processing structure uncertainties. The general classification of uncertainty is between parametric uncertainties and unmodeled dynamics. All of these uncertainties can be lumped into one single uncertain block D as shown in Fig. 1, where the standard M-D configuration for l synthesis is illustrated. In Fig. 1, K(s) is the controller, and P(s) is the nominal, open-loop interconnected transfer function matrix, which includes the nominal system model and the weighting functions, as well as the uncertainty weighting functions, which may be portioned as 2 3 P11 ðsÞ P12 ðsÞ P13 ðsÞ 6 7 PðsÞ ¼ 4 P21 ðsÞ P22 ðsÞ P23 ðsÞ 5 ð11Þ P31 ðsÞ P32 ðsÞ P33 ðsÞ M(s) is the lower linear fractional transformation of P(s) and K(s), which can be calculated as     P11 ðsÞ P12 ðsÞ M11 ðsÞ M12 ðsÞ ¼ MðsÞ ¼ M21 ðsÞ M22 ðsÞ P21 ðsÞ P22 ðsÞ   P13 ðsÞ þ KðsÞ½I  P33 ðsÞKðsÞ1 ½ P31 ðsÞ P32 ðsÞ  P23 ðsÞ ð12Þ D is the set of all possible uncertainties. For simplicity, we assume that the uncertainty block D is square. Moreover, w denotes the exogenous input typically including command signals, disturbances, noises, etc.; z denotes error output usually consisting of regulator output, tracking errors, filtered actuator signals, etc.; uD is the output of D; yD is the input of D; uK is the output of controller; yK is the measured output to

1207

controller. To define the set to which the uncertain block D belongs, let nr, nc, nC be three integer numbers and consider the Ns-tuple (Ns: =nr + nc + nC) of positive integers S ¼ fs1 ; s2    ; snr ; snr þ1 ;    ; snr þnc ; snr þnc þ1 ;    ; sNs g

ð13Þ

In the sequel, we refer to S as the uncertainty structure. Based on the structure S, we consider the matrix set n DS :¼ D ¼ diag½dr1 Is1 ; dr2 Is2 ;   ; drnr Isnr ;dc1 Isnr þ1 ; dc2 Isnr þ2 ;    ; dcnc Isnr þnc ; DC1 ; DC2    ; DCnC   ð14Þ : dri 2 R; dci 2 C; DCi 2 Cðsnr þnc þiÞðsnr þnc þiÞ

Definition 1.18 For M 2 Cnn , the structured singular value of M with respect to D is defined as follows: 8 1 > < ðDÞ : detðI  MDÞ ¼ 0g min fr lD ðMÞ :¼ ð15Þ D2DS > : 0 if detðI  MDÞ–0 ; for 8D 2 DS ðDÞ is the maximum singular value of D. When M is an where r interconnected transfer matrix as in Fig. 1, the structure singular value, with respect to D, is defined by lD ½MðsÞ :¼ sup lD ½MðjxÞ

ð16Þ

x2R

Definition 2. The uncertainty set can be defined as   UðDs Þ :¼ DðsÞ 2 RH1 : DðjxÞ 2 DS for all x 2 R; kDk1 < c1 ð17Þ

½DðjxÞ is the 1-norm of D(s), RH1 is the where kDk1 :¼ sup r x2R

set of all the proper and stable functions. Lemma 1.29 For the uncertainty set UðDS Þ, when MðsÞ 2 RH1 , for 8DðsÞ 2 UðDS Þ, we have sup lD ½MðsÞ ¼ sup lD ½MðjxÞ

ReðsÞP0

ð18Þ

x2R

Theorem 1.18,30,31 The controller K(s) in Fig. 1 can stabilize the system for 8DðsÞ 2 UðDS Þ, if and only if sup lD ½M11 ðjxÞ < c

ð19Þ

x2R

Proof. Sufficiently: According to Definition 1 and Lemma 1, we can obtain det½I  M11 ðsÞDðsÞ–0

for ReðsÞ P 0

ð20Þ

In Fig. 1, using upper LFT, we can get h i z ¼ M22 ðsÞ þ M21 ðsÞDðsÞðI  M11 ðsÞDðsÞÞ1 M12 ðsÞ w

Fig. 1

Standard M-D configuration for l synthesis.

where I  M11 ðsÞDðsÞ is the characteristic matrix of the closed-loop transfer function matrix. From Eq. (20), we know that all the poles of the system are located in left hand plane, and therefore K(s) can stabilize the system for 8DðsÞ 2 UðDS Þ. Necessity: If the controller K(s) can stabilize the system for 8DðsÞ 2 UðDS Þ, according to the small-gain therorem29,30,32, we have

1208

M. ZHU et al. Theorem 2.18,30,31 For 8DP ðsÞ 2 UðDSP Þ, the closed-loop system shown in Fig. 2 is well posed and internally stable, and has robust performance if and only if sup lDP ½MðjxÞ < c

ð23Þ

x2R

Therefore, the aim of l synthesis design is to minimize the peak value of the structured singular value lDP ðÞ of the closedloop transfer function matrix M(s) over the set of all stabilizing controllers K(s), which can be illustrated as min

max lDP ½MðjxÞ

Kstabilizing x2R

Fig. 2

Standard M-D configuration with DF for l synthesis.

kM11 ðsÞDðsÞk1 < 1 for 8DðsÞ 2 UðDS Þ sup lD ½M11 ðjxÞ < c x2R

The l synthesis design method not only considers the robust stability, but also takes into account the robust performance of the system. The designed system should perform well (for instance, good tracking) against exogenous disturbances. To ensure that the designed controller achieves robust performance, we introduce a fictitious uncertainty block DF as shown in Fig. 2. DF is unstructured with appropriate dimensions, satisfies kDF k1 6 c1 and is usually called the performance uncertainty block. Therefore, the robust stability and performance test should be performed with the extended uncertain structure
  D 0 nw nz : D 2 DS ; DF 2 C :¼ DP ¼ 0 DF

ð21Þ

Definition 3. The uncertainty set can be defined as  
DðsÞ 0 : DðsÞ 2 RH1 ;DF ðsÞ 2 RH1 ; UðDSp Þ :¼ DP ðsÞ ¼ 0 DF ðsÞ  DðjxÞ 2 DS ; DF ðjxÞ 2 Cnw nz ; 8x 2 R;k D k1 < c1 ; k DF k1 < c1 ð22Þ

Fig. 3

Currently, no analytic method can be used in synthesizing Eq. (24); however, the D-K iteration method that combing the l analysis with l synthesis yields good results. Therefore, in this paper, for synthesizing Eq. (24), we use the D-K iteration method, and the detail can be found in Refs.5,27. 4. Two-degree-freedom integral type LPV l synthesis

So we get

DSP

ð24Þ

In order to adapt consistency control effect in large dynamic parameter changed range of the nonlinear physical system, firstly, a LPV gain schedule dynamic description of the augmented plant of each steady state point is adopted as follows: 8 ~ i Þd~ ~ i Þdd ~ i Þdv þ Eða > ~_ ¼ Aða x þ Bða < dx ð25Þ i2I > : ~ i Þd~ x þ Fðai Þdd dy ¼ Cða Secondly, we present a new structure frame of l synthesis control on two degrees of freedom with double integral and weighting functions, which adopts the combination control law of feed forward and feedback with integral function as in Fig. 3. Aimed at the problem that reference command rapidly changes, one degree of freedom is the feed forward controller Kir ; another degree of freedom, the output feedback controller Kiy ; is used to acquire servo tracking and disturbance and noise rejection. Meanwhile, for overcoming overshoot and acquiring quick response, the integral is introduced in inner loop; for guaranteeing steady state servo tracking property, the integral controller KiI is used in outer loop. Additionally, considering the uncertainties in systems, two performance weighting functions WP and WC are designed to achieve robust specialty and control energy limit, where eP and eC are the performance weighted outputs; de ¼ dr  dy; deI is the integral of de.

Schematic diagram of l synthesis controller design.

Two freedom linear parameter varying l synthesis control for flight environment testbed Utilizing the structure depicted in Fig. 3, all the l synthesis controllers of chosen steady state points are computed by function provided by Robust Control Toolbox of MATLAB called dksyn.30,33 The designed l synthesis controller can be illustrated as 8 i i > < x_ c ¼ Ac xc þ Bc uc ð26Þ i2I > : d_v ¼ Cic xc þ Dic uc where xc is the state vector of l synthesis controller, and uc ¼ ½dr; dy; deI  is the input vector of l synthesis controller. Then, using the scheduling parameter aðtÞ; we obtain the LPV l synthesis controller as 8 > < x_ c ¼ Ac ðaðtÞÞxc þ Bc ðaðtÞÞuc aðtÞ 2 X > : d_v ¼ Cc ðaðtÞÞxc þ Dc ðaðtÞÞuc ð27Þ where 8 L L X X > > > ki ðtÞAic ; Bc ðaðtÞÞ ¼ ki ðtÞBic > Ac ðaðtÞÞ ¼ < i¼1

i¼1

ð28Þ

L L > X X > > > ki ðtÞCic ; Dc ðaðtÞÞ ¼ ki ðtÞDic : Cc ðaðtÞÞ ¼ i¼1




i¼1

ð29Þ

The closed-loop system of the LPV augmented system Eq. (10) with the LPV l synthesis controller Eq. (27) becomes 2

AðaðtÞÞ BðaðtÞÞ 0 dx_ 6 _7 6 0 Ma 0 6 du 7 6 6 7 6 6 x_ c 7 ¼ 6 Bcy ðaðtÞÞCðaðtÞÞ Bcy ðaðtÞÞDðaðtÞÞ Ac ðaðtÞÞ 6 7 6 6 7 6 4 d_v 5 4 Dcy ðaðtÞÞCðaðtÞÞ Dcy ðaðtÞÞDðaðtÞÞ Cc ðaðtÞÞ d_eI |fflfflffl{zfflfflffl} dx_ CL

0

3

0

7 0 7 7 0 BcI ðaðtÞÞ 7 7 7 0 DcI ðaðtÞÞ 5

Na

CðaðtÞÞ DðaðtÞÞ 0 0 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2

dx

3

2

6 7 6 du 7 6 7 7 6 6 xc 7 þ 6 7 4 dv 5 deI

ACL ðaðtÞÞ

EðaðtÞÞ

0

3

7 6 0 0 7  6 7 dr 6 6 Bcr ðaðtÞÞ Bcy ðaðtÞÞFðaðtÞÞ 7 7 dd aðtÞ 2 X 6 7 6 4 Dcr ðaðtÞÞ Dcy ðaðtÞÞFðaðtÞÞ 5 FðaðtÞÞ Imm |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

8t > 0

T

PAiCL þ AiCL P 6 Q

8i 2 I ð33Þ  1  and ACL ðaðtÞÞ 2 Co ACL ; A2CL ;    ; ALCL ; 8aðtÞ 2 X Co means that the space is convex, the system Eq. (30) is stable. Proof. We multiply Eq. (33) by ki 2 ½0; 1 separately and obtain 8   T > > P k1 A1CL þ k1 A1CL P 6 k1 Q > > >  > <  T P k2 A2CL þ k2 A2CL P 6 k2 Q ð34Þ > >    >

 >  > > : P kL ALCL þ kL ALCL T P 6 kL Q Adding all the above inequalities in Eq. (34), we get ! !T ! L L L X X X i i ð35Þ ki ACL þ ki ACL P 6  ki Q P i¼1

i¼1



Since ACL ðaðtÞÞ 2 Co A1CL ; A2CL ;    ; ALCL L X indicates that ACL ðaðtÞÞ ¼ ki AiCL and i¼1

aðtÞ 2 X, then

PACL ðaðtÞÞ þ ACL T ðaðtÞÞP 6 Q

; 8aðtÞ 2 X, which L X ki ¼ 1 for all i¼1

8aðtÞ 2 X

5. Flight environment testbed example We apply the LPV l synthesis controller to a flight environment testbed in Ref.35, whose working envelop is illustrated in Fig. 4. In order to make our controller design cover all the working envelope, we select thirty-six equilibrium points to design LPV l synthesis controller. Furthermore, for the purpose of verifying the robust effectiveness of the designed LPV l synthesis controller, we suppose a test condition to verify the robust performance of the designed LPV l synthesis

ð30Þ

Ci

Di

0

0

0

0 0 Bicr

3 Ei 0 7 7 7 Bicy Fi 7 7 7 Dicy Fi 5

Dicr Imm Fi

ð31Þ

Assumption 1. The matrices ACL ðaðtÞÞ and BCL ðaðtÞÞ are bounded

ð36Þ

Therefore, the closed-loop system Eq. (30) is stable.

BCL ðaðtÞÞ

Note that when aðtÞ ¼ ai , we have 2 2 3 Bi 0 0 0 Ai 6 6 0 Ma 0 Na 0 7 6 6 7 6 6 7 i i i i 7 i 6 B C B D A 0 B ; B ¼ AiCL ¼ 6 i i c cy cy cI CL 6 6 7 6 6 i i i i 7 4 4 Dcy Ci Dcy Di Cc 0 DcI 5

ð32Þ

Theorem 3. 34 Let Assumption1 hold. For a matrix Q ¼ QT > 0, if a single symmetric positive definite matrix P exists and satisfies



Bc ðaðtÞÞ ¼ ½ Bcr ðaðtÞÞ Bcy ðaðtÞÞ BcI ðaðtÞÞ  Dc ðaðtÞÞ ¼ ½ Dcr ðaðtÞÞ Dcy ðaðtÞÞ DcI ðaðtÞÞ 

3

k BCL ðaðtÞÞ k 6 k2

where k1 and k2 are constants.

i¼1

Let

2

k ACL ðaðtÞÞ k 6 k1 ;

1209

Fig. 4

Working envelope of FET.

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controller over the whole working envelope. Finally, we compare the simulation results of a single l synthesis controller designed on one equilibrium point with the simulation results of the LPV l synthesis controller.

Now, we consider the models of the actuators. In FET, the two actuators are the same, and the nominal transfer function of two actuators is taken as Ga ¼

5.1. LPV system of FET The simplified structure diagram of the FET is illustrated in Fig. 5. It has two inlets and one outlet: Inlet 1 is hot flow, whose mass flow rate is controlled by control valve 1; Inlet 2 is cold flow, and its mass flow rate is controlled by control valve 2; the outlet is connected with test engine. The control mechanism of FET is that we achieve temperature and pressure control of FET by regulating the two inlet control valves to regulate the mixing rate of hot and cold flow and the matching relation of import and export flow. In Fig. 5, Tin 1 ; Pin 1 ; m_ in 1 ; c1 are temperature, pressure, mass flow rate and average flow velocity of the first inlet of FET respectively; Tin 2 ; Pin 2 ; m_ in 2 ; c2 are temperature, pressure, mass flow rate and average flow velocity of the second inlet of FET respectively; Tout ; Pout ; m_ out ; c3 are temperature, pressure, mass flow rate and average flow velocity of FET outlet respectively; Q is the heat transferred through the FET wall; Ts is the FET wall temperature; V is the volume of FET. It is assumed that properties (temperature, pressure, density) at the outlet are characterized by mean properties within the FET. The deduction of nonlinear differential equations of FET can be found in Ref.25, and the results on the temperature and pressure differential equations are as follows:

K^a ^ Ta s þ 1

ð39Þ

where K^a ¼ 1; T^a ¼ 0:5. It is assumed that the actual gain coefficient Ka is a constant with relative error of 10% around the nominal value and the time constant Ta with relative error of 20%. We select 36 properly separated equilibrium points for linearizing the differential equations in Eq. (37) at those points. At each point, steady state values and linearization matrix can be obtained. Some of these 36 points are given as follows: Equilibrium point 1 (H = 0 km, Ma = 0)

T

T

xe1 ¼ ½ 288:15 101325  ; ue1 ¼ ½ 98:4641 51:5359  ; de1 ¼ ½ 150 323:15 223:15 200000 200000 T " #  0:2694 0:3448 5:1261106 A1 ¼ ; B1 ¼ 260:1473 121:2596 1:8025103 " 0:1937 0:2258 0:1177 2:3717107 E1 ¼ 233:5484 79:3880 41:3935 8:3397105

0:04011 179:5306



 ; C1 ¼

1:6219108 5:7034106



0 1

ð40Þ

Equilibrium point 16 (H = 10 km, Ma = 1.5)





 h i 8 c2 c2 c2 > ¼ PV RT ðh  RTÞðm_ in 1 þ m_ in 2  m_ out Þ þ hin1 þ 21 m_ in 1 þ hin 2 þ 22 m_ in 2  h þ 23 m_ out þ Q_ < dT dt c R ðp Þ





 h i c23 c21 c22 > dP ¼ R _ : _ _ _ _ _ _ T  h ð m þ m  m Þ þ h þ þ h þ  h þ þ Q c m m m p in 1 in 2 out in 1 in 1 in 2 in 2 out 2 2 2 dt Vðcp RÞ

The state vector, control input vector, disturbance vector, and output vector of FET are defined as follows: 2 3 m_ out 6 7 6 Tin1 7       6 7 T m_ in1 T 7; y ¼ ; d¼6 T x¼ ; u¼ ð38Þ in2 6 7 P m_ in2 P 6 7 4 Pin1 5 Pin2

#

1 0

ð37Þ

T

T

xe16 ¼ ½ 323:568 96966:929  ; ue16 ¼ ½ 113:0803 102:9197  ; de16 ¼ ½ 216 373:15 273:15 200000 200000 T " #     0:3824 0:1134 1 0 0:5869 2:5769105 ; B16 ¼ A16 ¼ ; C16 ¼ 3 300:3662 219:7411 0 1 175:8778 7:722310 " # 0:2651 0:3048 0:2754 5:6026107 2:2653107 E16 ¼ 265:1983 91:3550 82:5402 1:6789104 6:7887105 ð41Þ

Equilibrium point 36 (H = 25 km, Ma = 2.5)

T

T

xe36 ¼ ½ 487:125 42111:234  ; ue36 ¼ ½ 70:8914 14:0086  ; de36 ¼ ½ 84:9 523:15 323:15 100000 100000 T " #     1:6039 0:2378 10 0:7952 1:4573104 ; C36 ¼ ; B36 ¼ A36 ¼ 2 418:3122 259:1023 01 68:7427 1:259810 " # 6 8 1:3637 0:6593 0:1264 7:308210 2:142410 E36 ¼ 397:5449 56:9989 10:9293 6:3178104 1:8520105

Fig. 5

Simplified structure diagram of FET.

ð42Þ

Two freedom linear parameter varying l synthesis control for flight environment testbed

1211

We choose the simulate altitude and Mach number as the scheduling parameter aðtÞ ¼ ½ H Ma T . Two-dimensional linear interpolation has been used to calculate equilibrium values xe ðaðtÞÞ; ue ðaðtÞÞ; de ðaðtÞÞ and linearization matrix AðaðtÞÞ; BðaðtÞÞ; CðaðtÞÞ; EðaðtÞÞ; and then the LPV system Eq. (5) is obtained. After considering the uncertainty actuators, we get the augmented LPV system Eq. (10). 5.2. Two-degree-of-freedom LPV l synthesis controller design We design the performance and controller output weighting functions WP and WC for each equilibrium point. Utilizing the structure depicted in Fig. 3, we obtain 36 l synthesis controllers with 15 orders by using the function provided by Robust Control Toolbox of MATLAB called dksyn.30,33 Two-dimensional linear interpolation has been used to calculate controller matrix Ac ðaðtÞÞ; Bc ðaðtÞÞ; Cc ðaðtÞÞ; Dc ðaðtÞÞ; and then the LPV l synthesis controller Eq. (27) is obtained. The following is the controller design details at Equilibrium point 16. To achieve good tracking of temperature, we not only constrain the error of reference temperature and feedback temperature, but also constrain the integral of the error to ensure good tracking of temperature. The performance weighting functions are designed as follows: 2 0:5sþ3:8 3 0 0 sþ0:00038 6 0:5sþ1 0 7 WP ¼ 4 0 ð43Þ 5 sþ0:0001 0:5sþ1:18 0 0 sþ0:00118 The magnitude response of performance weighting functions is depicted in Fig. 6. To ensure that the controller has a proper output, the control outputs weighting functions are designed by using the principle of low frequency free limit, medium frequency gradually increasing the limit, and high frequency maximum limit, and the weighting function is designed as 2 0:32ðsþ1Þ 3 0 s þ1 100 5 ð44Þ WC ¼ 4 0:18ð0:5sþ1Þ 0 s þ1 120

The magnitude response of control outputs weighting functions is illustrated in Fig. 7. Utilizing the structure depicted in Fig. 3 and the weighting functions in Eqs. (43), (44), the l synthesis controller was computed by MATLAB function dksyn. The iteration result of controller design is shown in Table 1. From Table 1, we can see that the designed l synthesis controller is obtained in the first iteration with 15 order.

Fig. 6

Magnitude response of performance weighting functions.

Fig. 7 Magnitude response of control outputs weighting functions.

Table 1 Iteration result of controller design at Equilibrium point 16. Iteration step

Controller order

Total D-scale order

Gamma achieved

Peak l value

1 2 3 4 5

15 29 55 55 55

0 14 40 40 40

4.581 9.017 8.635 8.885 16.816

3.278 8.994 8.834 8.864 12.731

5.3. Simulation results To verify the effectiveness of the designed LPV l synthesis controller, the MathWorks SIMULINK simulation tool is used to build the simulation platform of FET and its structure is illustrated in Fig. 8. In Fig. 8, xc0 is initial state of l synthesis controller. Then, we assume a test condition depicted in Fig. 4 to verify the robust performance of the designed LPV l synthesis controller over the whole working envelope. Finally, we compare simulation results of the l synthesis controller designed on Equilibrium point 1 with simulation results of the LPV l synthesis controller. The test condition: we assume that simulate condition of engine inlet condition is through a process from Equilibrium point 1 to 8 to 15 to 21 to 28 to 29 and finally to 36, which almost run through the working envelope of FET and can be used to verify the robust performance of the designed LPV l synthesis controller over the whole working envelope. The altitude and Mach number set conditions of the test condition are depicted in Fig. 9. As the altitude and Mach number change, the inlet conditions of test engine change, which means that the temperature and pressure of FET should be controlled to track the change of inlet temperature and pressure of the test engine. The inlet temperatures of the two inlets are illustrated in Fig. 10. The inlet pressures of the two inlets are the same, which is shown in Fig. 11. The simulation result of inlet temperature of test engine in the test condition is illustrated in Fig. 12(a), and the solid line is the actual inlet temperature (Tactual) in real flight environment and the dotted line is the simulate inlet temperature (Tsimulate) provided by FET. In Fig. 12(a), we can see that the temperature of FET can track the inlet temperature change; from the partial enlarged drawing, it can be seen that the relative steady state error is less than 0.1% and the relative tracking error is less than 0.5%, which means that the designed LPV l synthesis controller has good robust performance on temperature control.

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Fig. 8

Schematic structure of LPV l synthesis controller simulation.

Fig. 9

Fig. 10

Test condition of FET.

Temperatures of two inlets.

The simulation result of inlet pressure of test engine in the test condition is illustrated in Fig. 12(b), and the solid line is the actual inlet pressure (Pactual) in real flight environment

Fig. 11

Inlet pressure condition.

and the dotted line is the simulate inlet pressure (Psimulate) provided by FET. In Fig. 12(b), we can see that the pressure of FET can track the inlet pressure change; from the partial

Two freedom linear parameter varying l synthesis control for flight environment testbed

1213

Now, we use the l synthesis controller designed on Equilibrium point 1 to replace the LPV l synthesis controller to do the simulation under the same condition as what we do before. The simulation stops at t = 911 s because the system becomes divergent. The simulation result of inlet temperature with single l synthesis controller is illustrated in Fig. 14(a). Although the controller achieves good tracking performance in early stage of simulation, the simulate temperature has oscillation phenomenon after 620 s and becomes divergent at t = 911 s. The simulation of inlet pressure is similar with inlet temperature and the simulation result is depicted in Fig. 14(b). Fig. 15 illustrates the inlet and outlet mass flow rate with single l synthesis controller. In the early stage of the simulation, the mass flow rate of the two inlets can change with the outlet mass flow rate while we keep the temperature and pressure of FET tracking the inlet condition of test engine; however, after 620 s, the mass flow rate of the two inlets begin to diverge until an unreasonable value which is not permitted in reality.

Fig. 12

Simulation results of test engine.

enlarged drawing, it can be seen that the relative steady state error is less than 0.1% and the relative tracking error is less than 0.3%. At 830 s, when the altitude and Mach number begin to change, the simulate pressure has a slight deviation caused by the controller parameter change with scheduling, but the deviation is very small and can be accepted. It means that the designed LPV l synthesis controller also has good robust performance on pressure control. The inlet and outlet mass flow rate of FET during the test condition is depicted in Fig. 13. In Fig. 13, we can see that the mass flow rate of the two inlets change a lot in the first few seconds, which is caused by the mismatch in the initial state of the controller. In the whole simulation process, the mass flow rate of the two inlets always change with the outlet mass flow rate while we keep the temperature and pressure of FET tracking the inlet condition of test engine. Fig. 14

Fig. 13 Inlet and outlet mass flow rate of FET under test condition.

Simulation results with single l synthesis controller.

Fig. 15 Inlet and outlet mass flow rate with single l synthesis controller.

1214 According to the above analysis, the designed LPV l synthesis controller has robust performance over the whole working envelope of FET and provides better performance than single l synthesis controller. 6. Conclusions This paper presents the method of designing a two-degree-offreedom LPV l synthesis controller and the stability proof of the LPV l synthesis controller over the whole working envelope of FET. A MIMO LPV model of the nonlinear FET system has been developed and the LPV l synthesis controller of FET has been designed. The robust performance of the designed LPV l synthesis controller over the whole working envelope is verified under chosen test condition. The simulation results show that, for temperature control, the relative steady state error is less than 0.1% and the relative tracking error is less than 0.5%; for pressure control, the relative steady state error is less than 0.1% and the relative tracking error is less than 0.3%. Additionally, through comparison with nonLPV, the LPV l synthesis method is good at dealing with large uncertainties. The designed controller achieves robust stability and performance of FET over the whole working envelope. References 1. Walker S, Tang M, Mamplata C. TBCC propulsion for a Mach 6 hypersonic airplane. Reston: AIAA; 2009. Report No.: AIAA2009-7238. 2. Montgomery P, Burdette R, Klepper J, Milhoan A. Evolution of a turbine engine test facility to meet the test needs of future aircraft systems. New York: ASME; 2002. Report No.: GT2002-30605. 3. Saunders JD, Stueber TJ, Thomas SR, Suder KL, Weir LJ, Sanders BW. Testing of the NASA hypersonics project’s combined cycle engine large scale inlet mode transition experiment (CCE LIMX). Washington, D.C.: NASA; 2012. Report No.: NASA/ TM-2012-217217. 4. Davis M, Montgomery P. A flight simulation vision for aeropropulsion altitude ground test facilities. J Eng Gas Turbines Power 2005;127(1):8–17. 5. Doyle JC. Structured uncertainty in control system design. Proceedings of 24th IEEE conference on decision and control; 1985 Dec 11–13; Fort Lauderdale, UAS. Piscataway: IEEE Press; 1985.p. 260–5. 6. Lee SH, Lim JT. Switching control of H1 gain scheduled controllers in uncertain nonlinear systems. Automatica 2000;36 (7):1067–74. 7. Yang S, Wang X. Sliding controller design for aero-engines with the rate limitation of actuators. New York: ASME; 2017. Report No.: GT2017-63571. 8. Yang S, Wang X, Yang B. Adaptive sliding mode control for limit protection of aircraft engines. Chin J Aeronaut 2018;31(7):1480–8. 9. Yang S, Wang X. LPV based sliding mode control for limit protection of aircraft engines. New York: ASME; 2018. Report No.: GT2018-77236. 10. Daafouz J, Bernussou J, Geromel JC. On inexact LPV control design of continuous-time polytopic systems. IEEE Trans Autom Control 2008;53(7):1674–8. 11. Yoon MG, Ugrinovskii VA, Pszczel M. Gain-scheduling of minimax optimal state-feedback controllers for uncertain LPV systems. IEEE Trans Autom Control 2007;52(2):311–7.

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