# Networked Control System Design for Turbofan Aeroengines with Aging and Deterioration

Authors: Ruichao Li; Sing Kiong Nguang; Yingqing Guo; Yifeng Chen
Journal: Complexity
 (2018)

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## Abstract

This paper is concerned with designing a networked controller for a mixed flow two-spool turbofan aeroengine with aging and deterioration. Firstly, the state-space representation of the aeroengine considering aging and deterioration is identified, by which the engine system with aging and deterioration is modeled as an uncertain linear system. Then based on this uncertain linear system, theoretical results from the networked control systems and the regional pole assignment are introduced to formulate the networked engine control design in the form of linear matrix inequalities (LMIs). By solving these LMIs simultaneously, a networked engine controller is obtained which guarantees both the robustness against delay/dropout and the satisfactory dynamic performance. Finally, the proposed method is applied to an aerothermodynamic component-level engine simulator to demonstrate its validity and applicability. The corresponding delay/dropout margin is also calculated, which provides reference for the future development of the distributed engine control system.


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## Body

## 1. Introduction

The next revolution in turbine engine control systems would be the physical distribution of control functionality, known as distributed engine control (DEC) [1]. The concept of DEC is borrowed from the distributed control system in automated industry, which is featured by the application of advanced data buses, smart sensors, and smart actuators [2, 3]. The realization of DEC is beneficial to the implementation of the advanced control mode and control algorithm, such as active control, fuzzy control, and model-based control, which bring advantages of weight reduction, modularity, high reliability, high intelligence, etc. [4–10].However, due to immatureness of the high-temperature electronics [11, 12], transition from the centralized control system to the fully distributed control system will be a gradual process. An intermediate distributed system approach is called the networked engine control (NEC) system, in which the data bus is introduced to replace the analog signal transmission. The A/D conversion functions (sampling, shaping, and quantization) of the centralized controller are moved into smart sensors while the control logic still remains as the central form (single controller).Due to the introduction of the data bus, transmission delay and packet dropout are inevitable in a NEC system. Depending on how network-induced delays and packet dropouts are handled, a number of methods are developed on stability analysis and control design for networked control systems (NCSs). Among them, time-delay system approach and switched system approach are the two mainstream methods. The time-delay system approach yue 2004 state, Jiang 2008A, e.g., [13, 14], models the packet dropout as input-delay, and therefore, the maximum allowable number of consecutive packet dropouts is calculated. The switched system approach models the NCS as a switched system with arbitrary switching Zhang 2008 modelling, Zhang 2009 robust, Donkers 2011 stability, Kruszewski 2012 switched, e.g., [15–18], or known probability switching [19, 20], which is usually applied in the discrete-time domain.As a safety-critical power device, it is crucial to figure out to what degree the control system could tolerate the delay and the packet dropout. To settle this problem, different kinds of NEC system models are established by scholars in the aeroengine control field. Constant delay and deterministic dropout are considered by [21, 22] who calculate the worst case bound on the number of consecutive dropouts. The case of stochastic dropout has also been considered. Reference [23] models the packet dropout as a Bernoulli process and calculates the maximum allowable packet dropping probability for the decentralized NEC system. Random delays in the NEC system are considered by [24, 25].Although a lot of work has been done on the NEC systems, the requirement of the dynamic performance has not been fully considered. Since previous results are developed based on the Lyapunov stability theory, the overall system is Lyapunov stable under the network-induced factors but it may still have a very small decay rate if no constraints are imposed. Besides, the effect of the uncertainty has also not been considered in the design stage; thus, previous results are still far away from practical application. Based on the discussion above, the goal of this paper is to design a NEC system satisfying the following control requirements:(1)
Robustness against the transmission delay and packet dropout
(2)
Satisfactory dynamical performance under different degrees of aging and deterioration
(3)
Zero tracking errorTo satisfy these control requirements, theoretical results from the networked control systems and the regional pole placement [26, 27] which have been studied previously are considered as solutions. In this paper, these theories are further extended to an uncertain case and formulated in the form of LMIs which are numerically tractable. Besides, efforts are also made to establish the uncertain description for aeroengines to make the proposed method applicable.This paper is organized as follows. Section2 introduces the modeling process of the aeroengine. Section 3 presents the stability theory and controller design of the NEC system. In Section 4, the robust networked controller and its corresponding delay/dropout margin are calculated. Then simulations are performed to demonstrate the effectiveness of the proposed method. Discussions are also extended on further simulation results. Finally, Section 5 presents conclusions. The symbols used in this paper are listed in Table 1.Table 1
Nomenclature.
Symbol
Type
Meaning
Function
Euclidean vector norm
∗
Function
Ellipsis for the symmetric terms
diag⋯
Function
Yielding a block diagonal matrix
symA
Function
DenotingA+AT
I
Matrix
Identity matrix of appropriate dimensions
u, x, y, w
Vector
Input, state, output, perturbation
xss, yss, uss, wss
Vector
Steady-state values ofu, x, y, w
δu, δx, δy
Vector
Deviation betweenu, x, y and xss, yss, uss
ui, xi, yi, wi
Scalar
Theith element in u, x, y, w
z
Vector
State of the integrator
x¯
Vector
Augmented state
A, B, C, D
Matrix
Identified system matrix
A¯, B¯
Matrix
Augmented identified system matrix
Al, Bl, Cl, Dl
Matrix
Identified system matrix under thelth health condition
A¯l, B¯l
Matrix
Augmented identified system matrix
A, B, C, D
Matrix
Nominal system matrix
ΔA, ΔB, ΔC, ΔD
Matrix
Norm-bounded uncertainty
Kx,Kz,K
Matrix
Feedback gain matrix
L, M
Set
Set whose members contain a group ofAl, Bl, Cl, Dl
L¯, M¯
Set
Set whose members contain a group ofA¯l, B¯l
N
Matrix
Interval matrix
Gi, Eii∈a,b,c,d
Matrix
Constant matrix
G¯, E¯a, E¯b
Matrix
Augmented constant matrix
Fii∈a,b,c,d
Matrix
Uncertain matrix satisfyingFiTFi≤I
F¯
Matrix
Augmented uncertain matrix satisfyingF¯TF¯≤I
h
Scalar
Sampling period
η
Scalar
Delay/dropout margin
τmax
Scalar
Maximum allowable transmission delay
nmax
Scalar
Maximum allowable number of consecutive packet dropouts
C−q,r
Set
A disk region of which the center is−q and the radii is r


## 2. Modeling of the Aeroengine

In this section, the working principle of the two-spool turbofan aeroengine and its commercial simulator is introduced first. Then by treating the complex simulator as a black box, a group of linear models is identified. These linear models are further redescribed as a nominal model with norm-bounded uncertainty to facilitate the controller design.


### 2.1. Mixed Flow Two-Spool Turbofan Aeroengine

The cross-section of a mixed flow two-spool turbofan aeroengine is given in Figure1. In a two-spool aeroengine, the compressors and turbines are linked by concentric shafts which rotate independently, namely, the low-pressure compressor (LPC) is driven by the low-pressure turbine (LPT) and the high-pressure compressor (HPC) is powered by the high-pressure turbine (HPT).Figure 1
Cross-section of a mixed flow two-spool turbofan aeroengine.In [28], using GasTurb modeling software, an aerothermodynamic component-level aeroengine simulator is established for the mixed flow two-spool turbofan aeroengine with realistic structural parameter and features of compressors/turbines. The turbine engine modeling software, GasTurb, is a commercial software [29, 30] which is widely used for modeling and simulating gas turbine aeroengines. For the convenience in controller design, the aeroengine simulator developed in [28] is complied under the MATLAB/Simulink environment further. However, due to the limited space and the modeling of simulators being not the main focus of the paper, the modeling process of the simulator is not included in this paper. Readers who are interested could obtain a general knowledge about the modeling process from the gas turbine textbook (see chap. 2.5 of [28, 31]).The nonlinear state and output equations of the aeroengine simulator are expressed as(1)ẋ=fx,u,w,y=gx,u,w,where f and g are nonlinear functions of the state vector x, the input vector u, and the health parameter vector w as given in Table 2. The degradation is due to usage and aging results in the higher fuel consumption and exhaust gas temperature, which further yields a shorter component life and a higher cost. Generally, the degradation is reflected as changes in flow characteristics and efficiencies of the rotational components. Therefore, a group of multipliers, which are referred as “health parameters,” are introduced. For instance, wi=1i=1,…,8 denotes a new component while a value being lower than 1 denotes a degraded one. For the engine which needs an overhaul, the health parameters may degrade to a maximum of 0.99.Table 2
Structure and physical meaning of the identified model.
Signal type
Notation
Designed value
Physical meaning
Inputu
u1
5813 kg/h
Main combustor fuel flow
u2
0.26m2
Nozzle exit area
Statex
x1
8880 rpm
Rotation speed of the LP shaft
x2
16419 rpm
Rotation speed of the HP shaft
Outputy
y1
8880 rpm
Being the same withx1
y2
10.65
Turbine pressure ratio (ratio of static pressure at the HPC exit and total pressure at the LPT exit)
Health parameterw
w1,…,w8
1.00,…,1.00
Flow modifiers and efficiency modifiers of the LPC, HPC, HPT, and LPT


### 2.2. State-Space Models Obtained by the System Identification

Considering the complexity of the simulator, the system identification technique is utilized to obtain a state-space representation. At the steady-state operating pointxss,yss,uss,wss, the simulator is identified as
(2)ẋt=A⋅δxt+ℬ⋅δut,δyt=C⋅δxt+D⋅δut,where symbol δ denotes the deviation from the steady-state point.Take the “full health” statusw=1,…,1T as an example. To apply the system identification technology, outputs of the aeroengine simulators y1 and y2 are first driven to their designed value given in Table 2; then two excitation signals δultl=1,2 are added separately to the input channels to generate δxlt and δyltl=1,2. These two excitation signals are taken, respectively, as
(3)δu1t=chirpt0T+ωt,δu2t=0chirptT+ωt,where chirpt is a swept-frequency cosine signal with amplitude being 1% udesigned. As stated in chap. 3.1 of [31], frequencies of the dominant dynamics are generally not higher than 2 Hz for aeroengines. Therefore, the frequency of chirpt is set to increase linearly from 0 to 2 Hz within 20 s, which is adequate to extract the spectral characteristics of rotors. Moreover, zero-mean white Gaussian noises ωt are also added to the excitation signals such that the signal-to-noise ratio (SNR) satisfies
(4)SNR=σsignal2σnoise2=100,in which σ denotes the standard deviation.The excitation signalsδul=u1lt,u2ltTl=1,2 and the deviations δx1l, δx2l, and δy2ll=1,2 generated from the aeroengine simulator are depicted in Figure 2, which have already been normalized by their designed value. With these two samples, the system matrices A,B,C,D under the “full-health” status are identified by the method proposed in [32] as
(5)A=−5.7924.8180.612−4.390,ℬ=0.6280.3630.7340.579,C=10−0.5781.900,D=000.0330.707,Figure 2
Data generated for the system identification.For simplicity, it is assumed that all health parameterswii=1,…,8 degrade simultaneously. Then by varying wi from 1 to 0.99 evenly and still maintaining y at its designed value ydesigned given in Table 3, a group of matrices Al, Bl, Cl, and Dll=1,2,…,11 is obtained via the system identification method above, which constitutes the set L below
(6)ℒ≜ΛΛ=AlℬlClDl,l=1,2,…,11.Table 3
Delay/dropout margin of each disk regionr=0.6q.
−q
−6
−7
−8
−9
−10
−11
−12
−13
ηmax
0.143
0.121
0.104
0.090
0.079
0.070
0.062
0.056
−q
−14
−15
−16
−17
−18
−19
−20
ηmax
0.050
0.046
0.041
0.038
0.034
0.031
0.028Due to the limited space, the values of these system matrices are not listed here.


### 2.3. Derivation of the Norm-Bounded Uncertainty

In this paper, degradations of the health parametersw (representing aging and deterioration) are viewed as the source of uncertainties for the identified state-space models. The degradation has two adverse effects on the identified linear models: On the one hand, the degradation is reflected in the drift of the steady-state value, and on the other hand, it causes the perturbation of the system matrices.Since the identified system matricesA, B, C, and D vary with w, they are supposed to be decomposed as
(7)A=A+ΔA,ℬ=B+ΔB,C=C+ΔC,D=D+ΔD,in which ΔA, ΔB, ΔC and ΔD are norm-bounded uncertainties satisfying
(8)ΔA=GaFaEa,ΔB=GbFbEb,ΔC=GcFcEc,ΔD=GdFdEd,where Gi and Eii∈a,b,c,d are constant matrices of appropriate dimensions. Fi is the uncertain matrix with Lebesgue measurable elements that satisfy FiTFi≤I.When obtaining the uncertain state-space representation (7), the difficulty lies in how to decompose a group of matrices Al further into a uniform norm-bounded description A+GaFaEa, namely, transforming L to the set M defined below.
(9)ℳ≜Λ∣Λ=ABCD+GaFaEaGbFbEbGcFcEcGdFdEd,FiTFi≤I,i∈a,b,c,d.To the best of our knowledge, there exists no transformation method such thatL equals M.Therefore, efforts are made to find a setM which could encompass L, namely, L⊆M. This is handled via the concept of interval matrix introduced below.Definition 1 [33].
With the given matricesP≜pijn×n, Q≜qijn×n, and A≜aijn×n, notation NP,Q is used to denote the set of matrix A which satisfies pij≤aij≤qij, namely,
(10)NP,Q≜A∈ℝn×n∣pij≤aij≤qij,i,j=1,2,…,n.
ThenNP,Q is called interval matrix, in which P and Q are the lower- and upper-bound matrices, respectively.
Based on the lemma given below, the interval matrixNP,Qcan be converted equivalently to a matrix with norm-bounded uncertainty. Therefore, the uncertain representation is obtained by seeking a lower-bound matrix P and an upper-bound matrix Q such that Al∈NP,Ql=1,2,…,11.Lemma 1 [34].
The interval matrixNP,Q is equivalent to the set Ma defined below.
(11)ℳa≜A∈ℝn×nA=A+GaFaEa,FaTFa≤I,in which matrices Ga∈ℝn×n2, Fa∈ℝn2×n2, and Ea∈ℝn2×n are taken as
(12)A=P+Q2,H≜hijn×n=Q−P2,Ga=h11e1⋯h1ne1⋯hn1en⋯hnnen,Ea=h11e1⋯h1nen⋯hn1e1⋯hnnenT,ei=0,…,0,1,⏟i0,…,0T∈ℝn,i=1,2,…,n,Fa=diagε11,…,ε1n,…,εn1,…,εnn,εij≤1.Remark 1.
The equivalence ofNP,Q and Ma has been proved in [34] and thus is omitted here. Readers could verify equalities P=A−GaIn2Ea and Q=A+GaIn2Ea for ease of understanding.
With the given matricesAll=1,2,…,11, by selecting “tight” bound matrices P and Q such that Al∈NP,Q, matrices A, Ga, and Ea are calculated according to Lemma 1 as
(13)A=−5.2484.0801.012−4.933,Ga=0.7380.85900000.6320.737,Ea=0.73800.632000.85900.737T.
Similarly, matricesB, C, D, Gi, and Eii∈b,c,d in (7) are calculated as
(14)B=0.6310.3890.7430.576,Gb=0.0610.16000000.0910.060,Eb=0.06100.091000.16000.060T,C=10−0.7592.107,Gc=0000000.4250.455,Ec=000.42500000.455T,D=000.0340.700,Gd=0000000.0420.080,Ed=000.04200000.080T.
Therefore, the setM encompassing the set L is obtained to redescribe a group of linear models (6) as a nominal matrix (9) with norm-bounded uncertainties.


## 2.1. Mixed Flow Two-Spool Turbofan Aeroengine

The cross-section of a mixed flow two-spool turbofan aeroengine is given in Figure1. In a two-spool aeroengine, the compressors and turbines are linked by concentric shafts which rotate independently, namely, the low-pressure compressor (LPC) is driven by the low-pressure turbine (LPT) and the high-pressure compressor (HPC) is powered by the high-pressure turbine (HPT).Figure 1
Cross-section of a mixed flow two-spool turbofan aeroengine.In [28], using GasTurb modeling software, an aerothermodynamic component-level aeroengine simulator is established for the mixed flow two-spool turbofan aeroengine with realistic structural parameter and features of compressors/turbines. The turbine engine modeling software, GasTurb, is a commercial software [29, 30] which is widely used for modeling and simulating gas turbine aeroengines. For the convenience in controller design, the aeroengine simulator developed in [28] is complied under the MATLAB/Simulink environment further. However, due to the limited space and the modeling of simulators being not the main focus of the paper, the modeling process of the simulator is not included in this paper. Readers who are interested could obtain a general knowledge about the modeling process from the gas turbine textbook (see chap. 2.5 of [28, 31]).The nonlinear state and output equations of the aeroengine simulator are expressed as(1)ẋ=fx,u,w,y=gx,u,w,where f and g are nonlinear functions of the state vector x, the input vector u, and the health parameter vector w as given in Table 2. The degradation is due to usage and aging results in the higher fuel consumption and exhaust gas temperature, which further yields a shorter component life and a higher cost. Generally, the degradation is reflected as changes in flow characteristics and efficiencies of the rotational components. Therefore, a group of multipliers, which are referred as “health parameters,” are introduced. For instance, wi=1i=1,…,8 denotes a new component while a value being lower than 1 denotes a degraded one. For the engine which needs an overhaul, the health parameters may degrade to a maximum of 0.99.Table 2
Structure and physical meaning of the identified model.
Signal type
Notation
Designed value
Physical meaning
Inputu
u1
5813 kg/h
Main combustor fuel flow
u2
0.26m2
Nozzle exit area
Statex
x1
8880 rpm
Rotation speed of the LP shaft
x2
16419 rpm
Rotation speed of the HP shaft
Outputy
y1
8880 rpm
Being the same withx1
y2
10.65
Turbine pressure ratio (ratio of static pressure at the HPC exit and total pressure at the LPT exit)
Health parameterw
w1,…,w8
1.00,…,1.00
Flow modifiers and efficiency modifiers of the LPC, HPC, HPT, and LPT


## 2.2. State-Space Models Obtained by the System Identification

Considering the complexity of the simulator, the system identification technique is utilized to obtain a state-space representation. At the steady-state operating pointxss,yss,uss,wss, the simulator is identified as
(2)ẋt=A⋅δxt+ℬ⋅δut,δyt=C⋅δxt+D⋅δut,where symbol δ denotes the deviation from the steady-state point.Take the “full health” statusw=1,…,1T as an example. To apply the system identification technology, outputs of the aeroengine simulators y1 and y2 are first driven to their designed value given in Table 2; then two excitation signals δultl=1,2 are added separately to the input channels to generate δxlt and δyltl=1,2. These two excitation signals are taken, respectively, as
(3)δu1t=chirpt0T+ωt,δu2t=0chirptT+ωt,where chirpt is a swept-frequency cosine signal with amplitude being 1% udesigned. As stated in chap. 3.1 of [31], frequencies of the dominant dynamics are generally not higher than 2 Hz for aeroengines. Therefore, the frequency of chirpt is set to increase linearly from 0 to 2 Hz within 20 s, which is adequate to extract the spectral characteristics of rotors. Moreover, zero-mean white Gaussian noises ωt are also added to the excitation signals such that the signal-to-noise ratio (SNR) satisfies
(4)SNR=σsignal2σnoise2=100,in which σ denotes the standard deviation.The excitation signalsδul=u1lt,u2ltTl=1,2 and the deviations δx1l, δx2l, and δy2ll=1,2 generated from the aeroengine simulator are depicted in Figure 2, which have already been normalized by their designed value. With these two samples, the system matrices A,B,C,D under the “full-health” status are identified by the method proposed in [32] as
(5)A=−5.7924.8180.612−4.390,ℬ=0.6280.3630.7340.579,C=10−0.5781.900,D=000.0330.707,Figure 2
Data generated for the system identification.For simplicity, it is assumed that all health parameterswii=1,…,8 degrade simultaneously. Then by varying wi from 1 to 0.99 evenly and still maintaining y at its designed value ydesigned given in Table 3, a group of matrices Al, Bl, Cl, and Dll=1,2,…,11 is obtained via the system identification method above, which constitutes the set L below
(6)ℒ≜ΛΛ=AlℬlClDl,l=1,2,…,11.Table 3
Delay/dropout margin of each disk regionr=0.6q.
−q
−6
−7
−8
−9
−10
−11
−12
−13
ηmax
0.143
0.121
0.104
0.090
0.079
0.070
0.062
0.056
−q
−14
−15
−16
−17
−18
−19
−20
ηmax
0.050
0.046
0.041
0.038
0.034
0.031
0.028Due to the limited space, the values of these system matrices are not listed here.


## 2.3. Derivation of the Norm-Bounded Uncertainty

In this paper, degradations of the health parametersw (representing aging and deterioration) are viewed as the source of uncertainties for the identified state-space models. The degradation has two adverse effects on the identified linear models: On the one hand, the degradation is reflected in the drift of the steady-state value, and on the other hand, it causes the perturbation of the system matrices.Since the identified system matricesA, B, C, and D vary with w, they are supposed to be decomposed as
(7)A=A+ΔA,ℬ=B+ΔB,C=C+ΔC,D=D+ΔD,in which ΔA, ΔB, ΔC and ΔD are norm-bounded uncertainties satisfying
(8)ΔA=GaFaEa,ΔB=GbFbEb,ΔC=GcFcEc,ΔD=GdFdEd,where Gi and Eii∈a,b,c,d are constant matrices of appropriate dimensions. Fi is the uncertain matrix with Lebesgue measurable elements that satisfy FiTFi≤I.When obtaining the uncertain state-space representation (7), the difficulty lies in how to decompose a group of matrices Al further into a uniform norm-bounded description A+GaFaEa, namely, transforming L to the set M defined below.
(9)ℳ≜Λ∣Λ=ABCD+GaFaEaGbFbEbGcFcEcGdFdEd,FiTFi≤I,i∈a,b,c,d.To the best of our knowledge, there exists no transformation method such thatL equals M.Therefore, efforts are made to find a setM which could encompass L, namely, L⊆M. This is handled via the concept of interval matrix introduced below.Definition 1 [33].
With the given matricesP≜pijn×n, Q≜qijn×n, and A≜aijn×n, notation NP,Q is used to denote the set of matrix A which satisfies pij≤aij≤qij, namely,
(10)NP,Q≜A∈ℝn×n∣pij≤aij≤qij,i,j=1,2,…,n.
ThenNP,Q is called interval matrix, in which P and Q are the lower- and upper-bound matrices, respectively.
Based on the lemma given below, the interval matrixNP,Qcan be converted equivalently to a matrix with norm-bounded uncertainty. Therefore, the uncertain representation is obtained by seeking a lower-bound matrix P and an upper-bound matrix Q such that Al∈NP,Ql=1,2,…,11.Lemma 1 [34].
The interval matrixNP,Q is equivalent to the set Ma defined below.
(11)ℳa≜A∈ℝn×nA=A+GaFaEa,FaTFa≤I,in which matrices Ga∈ℝn×n2, Fa∈ℝn2×n2, and Ea∈ℝn2×n are taken as
(12)A=P+Q2,H≜hijn×n=Q−P2,Ga=h11e1⋯h1ne1⋯hn1en⋯hnnen,Ea=h11e1⋯h1nen⋯hn1e1⋯hnnenT,ei=0,…,0,1,⏟i0,…,0T∈ℝn,i=1,2,…,n,Fa=diagε11,…,ε1n,…,εn1,…,εnn,εij≤1.Remark 1.
The equivalence ofNP,Q and Ma has been proved in [34] and thus is omitted here. Readers could verify equalities P=A−GaIn2Ea and Q=A+GaIn2Ea for ease of understanding.
With the given matricesAll=1,2,…,11, by selecting “tight” bound matrices P and Q such that Al∈NP,Q, matrices A, Ga, and Ea are calculated according to Lemma 1 as
(13)A=−5.2484.0801.012−4.933,Ga=0.7380.85900000.6320.737,Ea=0.73800.632000.85900.737T.
Similarly, matricesB, C, D, Gi, and Eii∈b,c,d in (7) are calculated as
(14)B=0.6310.3890.7430.576,Gb=0.0610.16000000.0910.060,Eb=0.06100.091000.16000.060T,C=10−0.7592.107,Gc=0000000.4250.455,Ec=000.42500000.455T,D=000.0340.700,Gd=0000000.0420.080,Ed=000.04200000.080T.
Therefore, the setM encompassing the set L is obtained to redescribe a group of linear models (6) as a nominal matrix (9) with norm-bounded uncertainties.


## 3. Networked Controller Design for the Aeroengine

In this section, the mathematical expression of the NEC system is presented which takes the transmission delay and the packet dropout into account simultaneously. Then a robust networked engine controller is designed with requirements of robustness against delay/dropout and the satisfactory dynamic performance being met with the NCS stability theory and the regional pole placement, respectively. The former will be presented in Section3.2 while the latter in Section 3.3. Then by combining linear matrix inequalities (LMIs) from these two theories, the controller design is synthesized in Section 3.4.


### 3.1. Networked Engine Control System

A schematic of the NEC system architecture is shown in Figure3, where τSCii=1,2,3, τCAii=1,2, and τC denote the transmission delay between the sensors and the controller, the transmission delay between the controller and the actuators, and the calculation time of the controller, respectively.Figure 3
Schematic of the NEC system architecture.To eliminate the potential tracking error due to reasons like modeling biases or plant disturbances, integrators are incorporated in the controller. Based on [35], by putting integrators at the output y, format of the controller is taken as
(15)δut=Kx⋅δxt+Kz⋅zt,where z is the state of the integrator satisfying
(16)zt=∫etdt=∫δycmd−δytdt.Supposed that the sensors are clock driven, the controller and actuators are event driven and the data are transmitted with a single packet and the integrated network-induced delayτ can be approximated as τ=maxiτSCi+τC+maxjτCAj. When packet dropout occurs, the latest packet is utilized for computation and actuation. Then combining (2) with (15) and following the modeling process in [13], the closed-loop system is expressed as
(17)ẋt=A⋅δxt+ℬ⋅δut,δyt=C⋅δxt+D⋅δut,t∈ikh+τk,ik+1h+τk+1,δut=Kx⋅δxt−τk+Kz⋅zt−τk,t=ikh+τk,ik∈1,2,3…,k=1,2,…,in which h is the sampling period. τk is the transmission delay from the sampling instant ikh to the actuating instant ikh+τk. ik is a nondecreasing sequence satisfying ik+1≥ik if the data packet disorder is ignored.More specifically,ik+1=ik+1 means that no packet dropout occurs while ik+1=ik means a packet loss in the transmission. By the definition of the symbol ik, it can be seen that (17) models the transmission delay and the packet dropout simultaneously.Furthermore, by defining an augmented state vector asx¯≜xzT and considering the output reference δy to be zero here, the closed-loop system (17) can be expressed compactly as
(18)x¯̇t=A¯⋅δx¯t+ℬ¯⋅δut,t∈ikh+τk,ik+1h+τk+1,δut=K⋅δx¯t−τk,t=ikh+τk,ik∈1,2,3,…,,k=1,2,…,in which K≜KxKz. A¯ and B¯ are system matrices that can still be decomposed into a norm-bounded form as
(19)with
(20)A¯=A0−C0,B¯=B−D,G¯=Ga0Gb00Gc0Gd,E¯a=EaEc000000T,F¯=diagFa,−Fc,Fb,−Fd,E¯b=00EbEdT.The matricesA¯, B¯, G¯, E¯a, and E¯b can be calculated based on (13) and (14), but they are not listed here due to their high dimensions.


### 3.2. Stability Theory under Delay/Dropout Condition

In the following, the state feedback controllerδu=K·δx¯ will be designed. Before we start, a frequently used lemma is introduced below which eliminates the uncertain term in LMIs effectively without introducing conservatism. Thus, by this lemma, existing theorems can be extended for uncertain systems straightforwardly.Lemma 2 [36].
With the given matricesM, N with appropriate dimensions, and a symmetric matrix W, the matrix inequality
(21)W+NTFTMT+MFN<0,holds for all F satisfying FTF≤I if and only if there exists a scalar ε>0 such that
(22)W+εMMT+ε−1NTN<0.
By Lemma2, the theorem introduced below guarantees the robustness against delay/dropout for the closed-loop system (18).Theorem 1
For the given scalarsη≥ik+1−ikh+τk+1 and ρjj=2,3, the closed-loop system (18) is exponentially asymptotically stable with K=YX−1 if there exist a scalar ε>0 and matrices P>0, T>0, X>0, Y, and Nii=1,2,3 with appropriate dimensions such that (23) holds.
(23)Θη,ρ2,ρ3,P,T,X,Y,ε≜Ω11Ω12Ω13ηN1E¯aXT∗Ω22Ω23ηN2E¯bYT∗∗Ω33ηN30∗∗∗−ηT0∗∗∗∗−εI<0,where
(24)Ω11=symN1−symA¯X+εG¯G¯T,Ω12=N2T−N1−B¯Y−ρ2XA¯T+εG¯G¯Tρ2,Ω13=P+N3T+X−ρ3XA¯T+εG¯G¯Tρ3,Ω22=−symN2−symρ2B¯Y+ερ2G¯G¯Tρ2,Ω23=−N3T+ρ2X−ρ3YTB¯T+ερ2G¯G¯Tρ3,Ω33=ηT+2ρ3X+ερ3G¯G¯Tρ3.Proof.
It can be known from [14] that for the given scalars η and ρjj=2,3, the closed-loop system (18) is exponentially asymptotically stable with K=YX−1 if there exist matrices P>0, T>0, X>0, Y, and Nii=1,2,3 with appropriate dimensions such that (25) holds.
(25)symN1−symA¯XN2T−N1−ℬ¯Y−ρ2XA¯TP+N3T+X−ρ3XA¯TηN1∗−symN2−symρ2ℬ¯Y−N3T+ρ2X−ρ3YTℬ¯TηN2∗∗ηT+2ρ3XηN3∗∗∗−ηT<0.
Then by substituting into (25) and using Lemma 2 to matrix F¯, it can be found that (25) holds if there exists a scalar ε>0 satisying (26).
(26)Ω~11Ω~12Ω~13ηN1∗Ω~22Ω~23ηN2∗∗Ω~33ηN3∗∗∗−ηT+ε−G¯−ρ2G¯−ρ3G¯0−G¯−ρ2G¯−ρ3G¯0T+ε−1E¯aXTE¯bYT00E¯aXTE¯bYT00T<0,where
(27)Ω~11=symN1−symA¯X,Ω~12=N2T−N1−B¯Y−ρ2XA¯T,Ω~13=P+N3T+X−ρ3XA¯T,Ω~22=−symN2−symρ2B¯Y,Ω~23=−N3T+ρ2X−ρ3YTB¯T,Ω~33=ηT+2ρ3X.
Then by Schur complements, (26) is equivalent to (23). The theorem is proved.Remark 2.
In Theorem1, inequality η≥ik+1−ikh+τk+1 describes the effect of the transmission delay and packet dropout simultaneously; hence, η can be viewed as a delay/dropout margin.
(i)
If no packet drops in the transmissionik+1=ik+1, inequality τk+1≤ηmax−h holds. The maximum allowable transmission delay τmax is calculated as
(28)τmax=ηmax−h
(ii)
Assuming a constant delayτ in the transmission, inequality ik+1−ik≤ηmax−τ/h holds. Thus, the maximum allowable number of consecutive packet dropout nmax is calculated as
(29)nmax=ηmax−τh−1


### 3.3.D-Stability Theory

Though Theorem1 guarantees that the closed-loop system is exponentially stable, the dynamic performance can still be poor since no constraints are imposed on the rate of decay. To obtain a satisfactory dynamic performance, a simple way is to assign the closed-loop poles of the augmented system (18), namely, the eigenvalues of A¯+B¯K, to some suitable regions D on the left-half complex plane. To achieve this, the below is introduced first which is usually referred as the D-stability theory.Lemma 3 [37].
With the given LMI regionD≜s∈ℂ:L+sM+s¯MT<0 and matrix A, the eigenvalues of A lie in D if and only if there exists a symmetric matrix X>0 such that
(30)MDA,X≜L⊗X+M⊗AX+MT⊗AXT<0.
Different types of regions, including the half-open plane, vertical strip area, and conic sector, can be covered by the above LMI description. Other complex regions can also be covered as long as they are convex. To guarantee a minimum decay rate and a minimum damping ratio, in this paper, the disk regionC−q,r is chosen as the expected eigenvalue region of which the center is −q and the radius is r. This disk region is described by taking
(31)L=−rqq−r,M=0100.
Then based on Lemma3, the theorem below guarantees all the eigenvalues of A¯+B¯K in the disk region C−q,rTheorem 2.
Given a disk regionC−q,r defined by (31), all eigenvalues of A¯+B¯K lie in C−q,r if and only if there exist a symmetric matrix X>0, a matrix Y with appropriate dimension, and a scalar ε>0 such that
(32)Ξq,r,X,Y,ε≜−rX+εG¯G¯TqX+A¯X+B¯Y0∗−rXE¯aX+E¯bYT∗∗−εI<0,with the control gain matrix being calculated as K=YX−1.Proof.
Being similar to the proof of Theorem1, by substituting  into MDA¯+B¯K,X<0 and using Lemma 2 to eliminate the uncertain matrix F¯, MDA¯+B¯K,X<0 is equivalent to
(33)MDA¯+B¯K,X+εG¯T0TG¯T0+ε−10E¯aX+E¯bKXT0E¯aX+E¯bKX<0.
Then by Schur complements and takingK=YX−1, (33) is equivalent to (32). The theorem is proved.


### 3.4. Controller Design Synthesis

Based on the aforementioned work, the controller with strong delay/dropout robustness and satisfactory dynamic performance can be obtained through solving LMIs in Theorem1 and Theorem 2 simultaneously. These LMIs are written together as
(34)Θη,ρ2,ρ3,P,T,X,Y,ε<0,Ξq,r,X,Y,ε1<0,P>0,T>0,X>0,ε>0,ε1>0,in which Θ and Ξ are matrices defined in (23) and (32), respectively, and q, r, η, and ρjj=2,3 are predesignated scalars.Then for the given scalarsq, r, and ρjj=2,3, the maximum delay/dropout margin ηmax can be calculated by a linear search. Correspondingly, the control gain matrix is calculated as K=YX−1.


## 3.1. Networked Engine Control System

A schematic of the NEC system architecture is shown in Figure3, where τSCii=1,2,3, τCAii=1,2, and τC denote the transmission delay between the sensors and the controller, the transmission delay between the controller and the actuators, and the calculation time of the controller, respectively.Figure 3
Schematic of the NEC system architecture.To eliminate the potential tracking error due to reasons like modeling biases or plant disturbances, integrators are incorporated in the controller. Based on [35], by putting integrators at the output y, format of the controller is taken as
(15)δut=Kx⋅δxt+Kz⋅zt,where z is the state of the integrator satisfying
(16)zt=∫etdt=∫δycmd−δytdt.Supposed that the sensors are clock driven, the controller and actuators are event driven and the data are transmitted with a single packet and the integrated network-induced delayτ can be approximated as τ=maxiτSCi+τC+maxjτCAj. When packet dropout occurs, the latest packet is utilized for computation and actuation. Then combining (2) with (15) and following the modeling process in [13], the closed-loop system is expressed as
(17)ẋt=A⋅δxt+ℬ⋅δut,δyt=C⋅δxt+D⋅δut,t∈ikh+τk,ik+1h+τk+1,δut=Kx⋅δxt−τk+Kz⋅zt−τk,t=ikh+τk,ik∈1,2,3…,k=1,2,…,in which h is the sampling period. τk is the transmission delay from the sampling instant ikh to the actuating instant ikh+τk. ik is a nondecreasing sequence satisfying ik+1≥ik if the data packet disorder is ignored.More specifically,ik+1=ik+1 means that no packet dropout occurs while ik+1=ik means a packet loss in the transmission. By the definition of the symbol ik, it can be seen that (17) models the transmission delay and the packet dropout simultaneously.Furthermore, by defining an augmented state vector asx¯≜xzT and considering the output reference δy to be zero here, the closed-loop system (17) can be expressed compactly as
(18)x¯̇t=A¯⋅δx¯t+ℬ¯⋅δut,t∈ikh+τk,ik+1h+τk+1,δut=K⋅δx¯t−τk,t=ikh+τk,ik∈1,2,3,…,,k=1,2,…,in which K≜KxKz. A¯ and B¯ are system matrices that can still be decomposed into a norm-bounded form as
(19)with
(20)A¯=A0−C0,B¯=B−D,G¯=Ga0Gb00Gc0Gd,E¯a=EaEc000000T,F¯=diagFa,−Fc,Fb,−Fd,E¯b=00EbEdT.The matricesA¯, B¯, G¯, E¯a, and E¯b can be calculated based on (13) and (14), but they are not listed here due to their high dimensions.


## 3.2. Stability Theory under Delay/Dropout Condition

In the following, the state feedback controllerδu=K·δx¯ will be designed. Before we start, a frequently used lemma is introduced below which eliminates the uncertain term in LMIs effectively without introducing conservatism. Thus, by this lemma, existing theorems can be extended for uncertain systems straightforwardly.Lemma 2 [36].
With the given matricesM, N with appropriate dimensions, and a symmetric matrix W, the matrix inequality
(21)W+NTFTMT+MFN<0,holds for all F satisfying FTF≤I if and only if there exists a scalar ε>0 such that
(22)W+εMMT+ε−1NTN<0.
By Lemma2, the theorem introduced below guarantees the robustness against delay/dropout for the closed-loop system (18).Theorem 1
For the given scalarsη≥ik+1−ikh+τk+1 and ρjj=2,3, the closed-loop system (18) is exponentially asymptotically stable with K=YX−1 if there exist a scalar ε>0 and matrices P>0, T>0, X>0, Y, and Nii=1,2,3 with appropriate dimensions such that (23) holds.
(23)Θη,ρ2,ρ3,P,T,X,Y,ε≜Ω11Ω12Ω13ηN1E¯aXT∗Ω22Ω23ηN2E¯bYT∗∗Ω33ηN30∗∗∗−ηT0∗∗∗∗−εI<0,where
(24)Ω11=symN1−symA¯X+εG¯G¯T,Ω12=N2T−N1−B¯Y−ρ2XA¯T+εG¯G¯Tρ2,Ω13=P+N3T+X−ρ3XA¯T+εG¯G¯Tρ3,Ω22=−symN2−symρ2B¯Y+ερ2G¯G¯Tρ2,Ω23=−N3T+ρ2X−ρ3YTB¯T+ερ2G¯G¯Tρ3,Ω33=ηT+2ρ3X+ερ3G¯G¯Tρ3.Proof.
It can be known from [14] that for the given scalars η and ρjj=2,3, the closed-loop system (18) is exponentially asymptotically stable with K=YX−1 if there exist matrices P>0, T>0, X>0, Y, and Nii=1,2,3 with appropriate dimensions such that (25) holds.
(25)symN1−symA¯XN2T−N1−ℬ¯Y−ρ2XA¯TP+N3T+X−ρ3XA¯TηN1∗−symN2−symρ2ℬ¯Y−N3T+ρ2X−ρ3YTℬ¯TηN2∗∗ηT+2ρ3XηN3∗∗∗−ηT<0.
Then by substituting into (25) and using Lemma 2 to matrix F¯, it can be found that (25) holds if there exists a scalar ε>0 satisying (26).
(26)Ω~11Ω~12Ω~13ηN1∗Ω~22Ω~23ηN2∗∗Ω~33ηN3∗∗∗−ηT+ε−G¯−ρ2G¯−ρ3G¯0−G¯−ρ2G¯−ρ3G¯0T+ε−1E¯aXTE¯bYT00E¯aXTE¯bYT00T<0,where
(27)Ω~11=symN1−symA¯X,Ω~12=N2T−N1−B¯Y−ρ2XA¯T,Ω~13=P+N3T+X−ρ3XA¯T,Ω~22=−symN2−symρ2B¯Y,Ω~23=−N3T+ρ2X−ρ3YTB¯T,Ω~33=ηT+2ρ3X.
Then by Schur complements, (26) is equivalent to (23). The theorem is proved.Remark 2.
In Theorem1, inequality η≥ik+1−ikh+τk+1 describes the effect of the transmission delay and packet dropout simultaneously; hence, η can be viewed as a delay/dropout margin.
(i)
If no packet drops in the transmissionik+1=ik+1, inequality τk+1≤ηmax−h holds. The maximum allowable transmission delay τmax is calculated as
(28)τmax=ηmax−h
(ii)
Assuming a constant delayτ in the transmission, inequality ik+1−ik≤ηmax−τ/h holds. Thus, the maximum allowable number of consecutive packet dropout nmax is calculated as
(29)nmax=ηmax−τh−1


## 3.3.D-Stability Theory

Though Theorem1 guarantees that the closed-loop system is exponentially stable, the dynamic performance can still be poor since no constraints are imposed on the rate of decay. To obtain a satisfactory dynamic performance, a simple way is to assign the closed-loop poles of the augmented system (18), namely, the eigenvalues of A¯+B¯K, to some suitable regions D on the left-half complex plane. To achieve this, the below is introduced first which is usually referred as the D-stability theory.Lemma 3 [37].
With the given LMI regionD≜s∈ℂ:L+sM+s¯MT<0 and matrix A, the eigenvalues of A lie in D if and only if there exists a symmetric matrix X>0 such that
(30)MDA,X≜L⊗X+M⊗AX+MT⊗AXT<0.
Different types of regions, including the half-open plane, vertical strip area, and conic sector, can be covered by the above LMI description. Other complex regions can also be covered as long as they are convex. To guarantee a minimum decay rate and a minimum damping ratio, in this paper, the disk regionC−q,r is chosen as the expected eigenvalue region of which the center is −q and the radius is r. This disk region is described by taking
(31)L=−rqq−r,M=0100.
Then based on Lemma3, the theorem below guarantees all the eigenvalues of A¯+B¯K in the disk region C−q,rTheorem 2.
Given a disk regionC−q,r defined by (31), all eigenvalues of A¯+B¯K lie in C−q,r if and only if there exist a symmetric matrix X>0, a matrix Y with appropriate dimension, and a scalar ε>0 such that
(32)Ξq,r,X,Y,ε≜−rX+εG¯G¯TqX+A¯X+B¯Y0∗−rXE¯aX+E¯bYT∗∗−εI<0,with the control gain matrix being calculated as K=YX−1.Proof.
Being similar to the proof of Theorem1, by substituting  into MDA¯+B¯K,X<0 and using Lemma 2 to eliminate the uncertain matrix F¯, MDA¯+B¯K,X<0 is equivalent to
(33)MDA¯+B¯K,X+εG¯T0TG¯T0+ε−10E¯aX+E¯bKXT0E¯aX+E¯bKX<0.
Then by Schur complements and takingK=YX−1, (33) is equivalent to (32). The theorem is proved.


## 3.4. Controller Design Synthesis

Based on the aforementioned work, the controller with strong delay/dropout robustness and satisfactory dynamic performance can be obtained through solving LMIs in Theorem1 and Theorem 2 simultaneously. These LMIs are written together as
(34)Θη,ρ2,ρ3,P,T,X,Y,ε<0,Ξq,r,X,Y,ε1<0,P>0,T>0,X>0,ε>0,ε1>0,in which Θ and Ξ are matrices defined in (23) and (32), respectively, and q, r, η, and ρjj=2,3 are predesignated scalars.Then for the given scalarsq, r, and ρjj=2,3, the maximum delay/dropout margin ηmax can be calculated by a linear search. Correspondingly, the control gain matrix is calculated as K=YX−1.


## 4. Application on the Turbofan Aeroengine Simulator

In this section, the controller design proposed above is applied on the turbofan aeroengine simulator by selecting appropriate closed-loop poles region. The controller is designed based on the uncertain model established in Section2 and then verified on the simulator itself to validate the applicability of the proposed method.


### 4.1. Controller Design

Once the uncertain state-space representation is ready, the center−q and the radii r of the disk pole region C−q,r are determined to calculate the control gain matrix. In practical turbofan applications, the percentage overshoot (PO) and the setting time Tsetting of the engine control system are usually required to satisfy
(35)PO≤2%,Tsetting≤2s.By taking the damping ratioξ=0.8 (dampling angle θ=arccos0.8) and the center q=6, PO and Tsetting are estimated by the formulae of the second-order system as
(36)PO=e−πξ/1−ξ2×100%=1.52%,Tsetting=−lnΔ×1−ξ2q−rΔ=0.02=1.84s,in which r=qsinθ. This fact indicts that the dynamic performance requirements above can be met if q≥6 and r≤0.6q. Based on this criterion, by taking q=8 and r=4.8 and searching ρjj=2,3 between −10,10, the control gain matrix K is calculated for (18) by solving LMIs in (34) as
(37)K=−8.829−3.24441.298−4.6781.378−1.625−3.2596.965.The corresponding delay/dropout margin is calculated asηmax=0.104. Then according to (18), the control gain matrix K is implemented in the form of
(38)Kx=−8.829−3.2441.378−1.625,Kz=41.298−4.678−3.2596.965.In the NEC system, the transmission delayτ is bounded in one sampling period h if a time-triggered data bus is utilized [23]. Therefore, taking τ as its worst value h=0.02s, ηmax=0.104 indicates a maximum allowable number of consecutive packet dropout nmax=3 based on Remark 2, namely, the NEC system is stable under a consecutive packet dropout being not severer than 3 in every 4 sampling cycles when the time delay τ=0.02s.


### 4.2. Simulation

For comparison, another controllerKwithoutΔ is calculated under the same condition but without considering the uncertainties. Comparison is performed between these two controllers to verify the effectiveness of our design.By applying them on the component-level aeroengine simulator and varying the degree of deteriorationw, unit step response simulations are conducted on the output y1 as shown in Figures 4 and 5. In each simulation, the health parameter wii=1,…,8 is taken as 1,0.997,0.994,0.991, while the network condition is taken as the extreme situation τ=0.02s and nmax=3 as discussed above. The control input u and the output y in these figures have already been normalized by their designed values udesigned and ydesigned.Figure 4
Step response under various deteriorations withK.Figure 5
Step response under various deteriorations withKwithoutΔComparing Figure4 (with uncertainties) with Figure 5 (without uncertainties), one can see that the settling time in Figure 4 is smaller than that in Figure 5. This shows that the controller design that is taking uncertainties into consideration is more robust and has a smaller settling time than the one without considering the uncertainties. Besides, it can be observed from Figure 4 that
(1)
The NEC system remains stable under the transmission delay and packet dropout described by the delay/dropout marginηmax,C−8,4.8=0.104
(2)
The outputy1 presents a satisfactory as well as a consistent dynamic performance under different deteriorations. Besides, the response curves do not overshoot significantly since the disk region has a small imaginary part (damping angle θ=arcsin0.6)
(3)
The output could track the reference command without the steady-state errorFrom the points listed above, the robust networked controller designed in this paper has achieved the expected goal, namely, the three control requirements proposed in Introduction.


### 4.3. Discussions

In this section, discussions are extended on the relationship between the delay/dropout margin and the disk regionC−q,r. Besides, the conservatism which is introduced when obtaining the norm-bounded uncertainties is also discussed.


#### 4.3.1. Relationship between the Delay/Dropout Margin and the Disk Region

As mentioned above, the expected disk regionC−q,r should satisfy q≥6 and r≤0.6q. Therefore, by varying the center q∈6,20 and taking the radii r as r=0.6q for a constant damping ratio, different gain matrices KC−q,r are and their delay/dropout margins ηmax are calculated in Table 3.From Table3, it also can be observed that the delay/dropout margins decrease rapidly while the pole region is being far away from the imaginary axis. Therefore, there should be a trade-off between the robustness against delay/dropout and the dynamical performance when designing the controller.


#### 4.3.2. Conservatism Introduced in the Modeling Process

In this paper, a data-based method is applied to obtain the norm-bounded uncertainties in Section2.3. By this method, a group of system matrices (L in (6)) is redescribed as a nominal matrix with norm-bounded uncertainties (M in (9)). However, conservatism is introduced in this modeling process since L is only a subset of M. To demonstrate the conservatism graphically, integrators are incorporated as mentioned in Section 3.1 to augment L and M to L¯ and M¯ defined below as
(39)and
(40)in which M¯ is exactly the augmented system matrices in (18).Based on the gain matrices of various disk regions obtained in the previous section, the closed-loop poles (namely, the eigenvalues ofA¯+B¯KC−q,r) are calculated for L¯ and M¯. As a result, the convex hulls of these poles are drawn in Figure 6, in which the pole region of L¯ is denoted in red while the counterpart of M¯ is denoted in blue. It can be seen that certain conservatism is introduced but is still at an acceptable level.Figure 6
Convex hull of the closed-loop poles under different control gain matricesKC−q,r


#### 4.3.3. Conservatism Introduced in the Controller Design Process

The conservatism is not just introduced by the modeling but also introduced by the controller. In this subsection, we combine the networked controller design theory [13] with the D-stability theorem to design a controller for the uncertain system (18) as a comparison.By using the same disk region selected in this paper and taking the lower bound of time delayτm as 0, the maximum allowable value of ηmax is calculated as 0.112 based on [13]. By contrast, ηmax calculated in this paper is 0.104. Considering a dropout-free transmission, the gap means a larger maximum allowable delay τmax by 8 ms which is small enough. Therefore, only a little conservatism is introduced in the controller design process.However, it should be pointed out that the theorem in [13] is in the form of bilinear matrix inequality (BMI), which has to be solved by a complex iterative algorithm. In contrast, the method proposed in this paper is numerically tractable. This makes the controller much easier to be calculated with existing LMI toolbox.


## 4.1. Controller Design

Once the uncertain state-space representation is ready, the center−q and the radii r of the disk pole region C−q,r are determined to calculate the control gain matrix. In practical turbofan applications, the percentage overshoot (PO) and the setting time Tsetting of the engine control system are usually required to satisfy
(35)PO≤2%,Tsetting≤2s.By taking the damping ratioξ=0.8 (dampling angle θ=arccos0.8) and the center q=6, PO and Tsetting are estimated by the formulae of the second-order system as
(36)PO=e−πξ/1−ξ2×100%=1.52%,Tsetting=−lnΔ×1−ξ2q−rΔ=0.02=1.84s,in which r=qsinθ. This fact indicts that the dynamic performance requirements above can be met if q≥6 and r≤0.6q. Based on this criterion, by taking q=8 and r=4.8 and searching ρjj=2,3 between −10,10, the control gain matrix K is calculated for (18) by solving LMIs in (34) as
(37)K=−8.829−3.24441.298−4.6781.378−1.625−3.2596.965.The corresponding delay/dropout margin is calculated asηmax=0.104. Then according to (18), the control gain matrix K is implemented in the form of
(38)Kx=−8.829−3.2441.378−1.625,Kz=41.298−4.678−3.2596.965.In the NEC system, the transmission delayτ is bounded in one sampling period h if a time-triggered data bus is utilized [23]. Therefore, taking τ as its worst value h=0.02s, ηmax=0.104 indicates a maximum allowable number of consecutive packet dropout nmax=3 based on Remark 2, namely, the NEC system is stable under a consecutive packet dropout being not severer than 3 in every 4 sampling cycles when the time delay τ=0.02s.


## 4.2. Simulation

For comparison, another controllerKwithoutΔ is calculated under the same condition but without considering the uncertainties. Comparison is performed between these two controllers to verify the effectiveness of our design.By applying them on the component-level aeroengine simulator and varying the degree of deteriorationw, unit step response simulations are conducted on the output y1 as shown in Figures 4 and 5. In each simulation, the health parameter wii=1,…,8 is taken as 1,0.997,0.994,0.991, while the network condition is taken as the extreme situation τ=0.02s and nmax=3 as discussed above. The control input u and the output y in these figures have already been normalized by their designed values udesigned and ydesigned.Figure 4
Step response under various deteriorations withK.Figure 5
Step response under various deteriorations withKwithoutΔComparing Figure4 (with uncertainties) with Figure 5 (without uncertainties), one can see that the settling time in Figure 4 is smaller than that in Figure 5. This shows that the controller design that is taking uncertainties into consideration is more robust and has a smaller settling time than the one without considering the uncertainties. Besides, it can be observed from Figure 4 that
(1)
The NEC system remains stable under the transmission delay and packet dropout described by the delay/dropout marginηmax,C−8,4.8=0.104
(2)
The outputy1 presents a satisfactory as well as a consistent dynamic performance under different deteriorations. Besides, the response curves do not overshoot significantly since the disk region has a small imaginary part (damping angle θ=arcsin0.6)
(3)
The output could track the reference command without the steady-state errorFrom the points listed above, the robust networked controller designed in this paper has achieved the expected goal, namely, the three control requirements proposed in Introduction.


## 4.3. Discussions

In this section, discussions are extended on the relationship between the delay/dropout margin and the disk regionC−q,r. Besides, the conservatism which is introduced when obtaining the norm-bounded uncertainties is also discussed.


### 4.3.1. Relationship between the Delay/Dropout Margin and the Disk Region

As mentioned above, the expected disk regionC−q,r should satisfy q≥6 and r≤0.6q. Therefore, by varying the center q∈6,20 and taking the radii r as r=0.6q for a constant damping ratio, different gain matrices KC−q,r are and their delay/dropout margins ηmax are calculated in Table 3.From Table3, it also can be observed that the delay/dropout margins decrease rapidly while the pole region is being far away from the imaginary axis. Therefore, there should be a trade-off between the robustness against delay/dropout and the dynamical performance when designing the controller.


### 4.3.2. Conservatism Introduced in the Modeling Process

In this paper, a data-based method is applied to obtain the norm-bounded uncertainties in Section2.3. By this method, a group of system matrices (L in (6)) is redescribed as a nominal matrix with norm-bounded uncertainties (M in (9)). However, conservatism is introduced in this modeling process since L is only a subset of M. To demonstrate the conservatism graphically, integrators are incorporated as mentioned in Section 3.1 to augment L and M to L¯ and M¯ defined below as
(39)and
(40)in which M¯ is exactly the augmented system matrices in (18).Based on the gain matrices of various disk regions obtained in the previous section, the closed-loop poles (namely, the eigenvalues ofA¯+B¯KC−q,r) are calculated for L¯ and M¯. As a result, the convex hulls of these poles are drawn in Figure 6, in which the pole region of L¯ is denoted in red while the counterpart of M¯ is denoted in blue. It can be seen that certain conservatism is introduced but is still at an acceptable level.Figure 6
Convex hull of the closed-loop poles under different control gain matricesKC−q,r


### 4.3.3. Conservatism Introduced in the Controller Design Process

The conservatism is not just introduced by the modeling but also introduced by the controller. In this subsection, we combine the networked controller design theory [13] with the D-stability theorem to design a controller for the uncertain system (18) as a comparison.By using the same disk region selected in this paper and taking the lower bound of time delayτm as 0, the maximum allowable value of ηmax is calculated as 0.112 based on [13]. By contrast, ηmax calculated in this paper is 0.104. Considering a dropout-free transmission, the gap means a larger maximum allowable delay τmax by 8 ms which is small enough. Therefore, only a little conservatism is introduced in the controller design process.However, it should be pointed out that the theorem in [13] is in the form of bilinear matrix inequality (BMI), which has to be solved by a complex iterative algorithm. In contrast, the method proposed in this paper is numerically tractable. This makes the controller much easier to be calculated with existing LMI toolbox.


## 4.3.1. Relationship between the Delay/Dropout Margin and the Disk Region

As mentioned above, the expected disk regionC−q,r should satisfy q≥6 and r≤0.6q. Therefore, by varying the center q∈6,20 and taking the radii r as r=0.6q for a constant damping ratio, different gain matrices KC−q,r are and their delay/dropout margins ηmax are calculated in Table 3.From Table3, it also can be observed that the delay/dropout margins decrease rapidly while the pole region is being far away from the imaginary axis. Therefore, there should be a trade-off between the robustness against delay/dropout and the dynamical performance when designing the controller.


## 4.3.2. Conservatism Introduced in the Modeling Process

In this paper, a data-based method is applied to obtain the norm-bounded uncertainties in Section2.3. By this method, a group of system matrices (L in (6)) is redescribed as a nominal matrix with norm-bounded uncertainties (M in (9)). However, conservatism is introduced in this modeling process since L is only a subset of M. To demonstrate the conservatism graphically, integrators are incorporated as mentioned in Section 3.1 to augment L and M to L¯ and M¯ defined below as
(39)and
(40)in which M¯ is exactly the augmented system matrices in (18).Based on the gain matrices of various disk regions obtained in the previous section, the closed-loop poles (namely, the eigenvalues ofA¯+B¯KC−q,r) are calculated for L¯ and M¯. As a result, the convex hulls of these poles are drawn in Figure 6, in which the pole region of L¯ is denoted in red while the counterpart of M¯ is denoted in blue. It can be seen that certain conservatism is introduced but is still at an acceptable level.Figure 6
Convex hull of the closed-loop poles under different control gain matricesKC−q,r


## 4.3.3. Conservatism Introduced in the Controller Design Process

The conservatism is not just introduced by the modeling but also introduced by the controller. In this subsection, we combine the networked controller design theory [13] with the D-stability theorem to design a controller for the uncertain system (18) as a comparison.By using the same disk region selected in this paper and taking the lower bound of time delayτm as 0, the maximum allowable value of ηmax is calculated as 0.112 based on [13]. By contrast, ηmax calculated in this paper is 0.104. Considering a dropout-free transmission, the gap means a larger maximum allowable delay τmax by 8 ms which is small enough. Therefore, only a little conservatism is introduced in the controller design process.However, it should be pointed out that the theorem in [13] is in the form of bilinear matrix inequality (BMI), which has to be solved by a complex iterative algorithm. In contrast, the method proposed in this paper is numerically tractable. This makes the controller much easier to be calculated with existing LMI toolbox.


## 5. Conclusion

In this paper, efforts are made to establish the uncertain description for aeroengines with aging and deterioration. Based on this model, the controller with both strong delay/dropout robustness and satisfactory dynamic performance is designed through solving linear matrix inequalities (LMIs). The validity and applicability of this approach are elaborated through simulations under the deteriorated and networked conditions. The corresponding delay/dropout margin is also calculated, which provides a reference for future development of the aeroengine-distributed control system. Admittedly, with some degree of conservatism, the controller design approach presented in this paper offered a feasible and practical mean to obtain a robust networked controller



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*Source: 6010216-2018-10-11.xml*