# H ∞ Networked Cascade Control System Design for Turboshaft Engines with Random Packet Dropouts

Authors: Xiaofeng Liu; Xu Sun
Journal: International Journal of Aerospace Engineering
 (2017)

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## Abstract

The distributed control architecture becomes more and more important in future gas turbine engine control systems, in which the sensors and actuators will be connected to the controllers via a network. Therefore, the control problem of network-enabled high-performance distributed engine control (DEC) has come to play an important role in modern gas turbine control systems, while, due to the properties of the network, the packet dropouts must be considered. This study introduces a distributed control system architecture based on a networked cascade control system (NCCS). Typical turboshaft engine distributed controllers are designed based on the NCCS framework withH∞ state feedback under random packet dropouts. The sufficient robust stable conditions are derived via the Lyapunov stability theory and linear matrix inequality approach. Simulations illustrate the effectiveness of the presented method.


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## Body

## 1. Introduction

The distributed control system (DCS) is a control system wherein control elements are distributed throughout the system. This is in contrast to the centralized ones, which use a single controller at a central location. In a DCS, a hierarchy of controllers is connected by communication networks for information/data transmission. The advantages of the DCS architecture, such as reduction of system weight, higher reliability, modularity, and less low life cost, merit increasing attention from industrial companies and engineers.Conventional gas turbine engine control systems are designed as a centralized architecture (which called as Full Authority Digital Engine Control, FADEC) to protect the control elements from the extreme environment [1]. While, with the increasingly development of sophisticated electronics with higher reliability in high temperature environment, the requirements of increased performance, more convenient operation, reduction of design, and maintenance cost make the control system to use a more effective architecture. Thus, the distributed engine control (DEC) architecture came into being [2, 3].Due to the distributed architecture, the sensors and controllers are connected by the communication networks, as well as between the controllers and the actuators. DEC seeks to advance the state of the art in gas turbine engine control systems by using a digital communication network with a more robust network. This will lead to the development of gas turbine engine control systems with greater extensibility and higher capacity for upgrades. DEC is extensively studied in [2, 4–7] and the references therein.The DEC architecture can be viewed as an NCCS. For example, the GE T700 turboshaft engine is a two-spool gas turbine engine consisting of a gas generator and a free power turbine [8, 9], and the power turbine is connected to the rotor system by a shaft and a gear box. Conventionally, the power turbine can be considered as a part of the rotor system [10]. The input of the rotor system is the gas generator’s output and shaft torque; therefore, the whole turboshaft engine system combined with control systems can be reviewed as a cascade control system (CCS) [11].As for the DEC using the communication networks to close the control loop, there are fundamental factors to affect the DEC system. They include network-induced time delay, packet dropouts, and bandwidth constraints [12, 13]. Therefore, to guarantee the desired performance and to ensure stability, the control system should be robust to these factors. The network-induced time delay in NCCSs occurs when the sensors, controllers and actuators transfer information/data through the networks, and it can degrade the performance of the control systems and even can destabilize the system [14]. Since the network-induced time delay is unavoidable in the NCCSs, the existing literature, such as [15–18] and the references therein, has discussed the time delay, and many useful approaches have been proposed and even applied to the industrial systems see [19–21] and the references therein.However, few papers have discussed the DEC robust control in gas turbine engine control systems. For example, Belapurkar et al. [22] analyzed the stability of set-point controller for partially DEC systems with time delays by using LQR method. Yedavalli et al. [13] discussed the stability of DEC systems under communication packet dropouts. Merrill et al. [2] provided a DEC design approach based on quadratic invariance optimal control theory to the control performance of various types of decentralized network configurations.This paper is concerned with the problem ofH∞ controller design for gas turbine engine distributed control by using state feedback control in the form of NCCSs with packet dropouts. The rest of the paper is organized as follows. In Section 2, the architecture of distributed engine control system is thoroughly described, and the state feedback control problem is formed. H∞ state feedback controllers are designed based on Lyapunov stability theory and LMI approach in Section 3. A numerical simulation example is presented in Section 4 to illustrate the effectiveness of the approach. Conclusion will be found in Section 5.Notation. Throughout the paper, the superscripts “T” and “(-1)” represent matrix transposition and matrix inverse, respectively. · refers to the Euclidean vector norm, and E· stands for the mathematical expectation operator with respect to the given probability measure P. In symmetric block matrices or long matrix expressions, an asterisk (∗) represented a term that is induced by symmetry. diag⁡{·} stands for a block-diagonal matrix.


## 2. Problem Formulation

### 2.1. DEC System Architecture Description

This study utilized a GE T700 turboshaft engine. Figure1 shows the simplified diagram. The inputs to the gas generator were the power turbine speed set value, NP, and the fuel flow rate, WF. The outputs were the gas generator speed, NG, engine torque transmitted by the power turbine shaft, QS, compressor static discharge pressure, PS3, and power turbine inlet temperature, T45. The controller design process begins with a linearized, state-space model of the system. Figure 2 shows the simplified model in this case.Figure 1
Block diagram of the open-loop gas generator/rotor system.Figure 2
Block diagram of the simplified linearized gas generator and rotor system.Control laws essentially work to maintainNP, constant at the set point by modulating WF. The control accomplishes this by scheduling a nominal NG speed as a function of XCPC, T1, and P1. The control trims this NG demand to isochronously adjust NP to NP set input. PLA position limits the maximum permissible NG, while the control further limits the maximum T45. The control limits the NG acceleration/deceleration rate as a function of NG scheduled WF/PS3 limit. The DEC discussed herein has one network, which is inserted in the gas generator controller and the gas generator. Figure 3 shows the architecture. The abovementioned description illustrates that the GE T700 control structure is a cascade control structure, wherein the desired primary process output can only be controlled by controlling the secondary control process output.Figure 3
Block diagram of the NCCS model.Primary Plant. The state-space representation of the rotor system is provided by the following equation [9, 23]:(1)
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1
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=
1
0
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1
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.Secondary Plant. The continuous-time linear model of the gas generator is shown as follows:(2)
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,where w(t) is exogenous process white noise signal belonging to l2[0,∞) and the noise parameters matrix B3 should be used as design parameters to achieve desirable system frequency response characteristics [9].The following assumptions are partially taken from [24, 25]:(a)
The controllers are event-driven. The primary controller computes the values and sends them to the secondary controller after obtaining the latest samples of the primary plant outputs. The secondary controller then computes the control command and sends it to the actuator as soon as it receives the latest samples of the secondary plant and the control output of the primary plant controller through a common network.(b)
The actuator is time-driven. In other words, the actuator actuates the plants once it receives the control command. The actuator will then use the previous value by zero-order-hold to precede the secondary process in case of packet loss.(c)
The sensors are time-driven; that is, they periodically sample the outputs and send them to the corresponding controllers.(d)
The data packet transmitted from the controller to the plant may be delayed. The delay is assumed to be a fixed one and less than a sampling periodh (i.e., τk∈[0,h]).(e)
The data packet is assumed to be transmitted between the primary and secondary controllers in a single packet without any loss. However, the data packet transmitted between the secondary controller and the actuator may be delayed or may meet a possible failure in a random manner.


### 2.2. State Feedback Control of DEC System

By considering the network-induced delayτk, the controllers are event-driven, the actuator is time-driven, and the engine receives the piece-wised control input is given by(3)ut=u~2k-1,kh≤t<kh+τk,u~2k,kh+τk≤t<k+1h,(4)u~2k=u~2k-1,ifu2klost  during  transmission,u2k,ifu2ktransmitted  successfully;that is, the actuator receives the signal u2(k) if the data is transmitted successfully; otherwise, the previous value will be used in the actuator by zero-order-hold, where u2(k) is the control output of the secondary controller.Since the actuator is time-driven, the packet loss may happen in a random manner. Then,u~2(k) can be rewritten by [25, 26](5)u~2k=λku2k+1-λku~2k-1,where λ(k) is a Bernoulli distributed stochastic variable taking the value 0 or 1. λ(k)=1 represents the successful state transmission of the delayed packet and λ(k)=0 describes the complete packet loss. It assumed that λ(k) satisfies the Bernoulli distribution [27]:(6)Prob⁡λk=1=α,Prob⁡λk=0=1-α,where α is a positive scalar, E[λ(k)-α]=0, and E[(λ(k)-α)2]=α(1-α).Considering the system reference inputNPr=0, the static state feedback controller is utilized by a discrete-time form:(7)u2k=u1k+K2αx2k,where K2(α) is the probability-dependent (packet dropouts probability) state feedback gain matrix given by K2(α)=K21+αK22 and K1 is(8)u1k=K1x1k,where x1(k) is the state vector of rotor system in discrete-time form and K1 is the state feedback gain.By using (3), the rotor system and engine with sampling period, kh,(k+1)h, are discretized to(9)x1k+1=A1x1k+B1y2k,y1k=C1x1k,where A1=eA1h,  B1=∫0heA1sdsB1, and(10)x2k+1=A2x2k+B21u~2k-1+B22u~2k+B3wk,y2k=C2x2k,where A2=eA2h, B21=∫h-τkheA2sdsB2,  B22=∫0h-τkeA2sdsB2, and B3=∫0heA2sdsB3. Then, considering the random packet loss, combined (5) and (7) with (10), (10) becomes(11)x2k+1=A2x2k+B21-1-λkB22u~2k-1+λkB22K1x1k+K2αx2k+B3wk,y2k=C2kx2k.Thus, the discretized system can be further expanded as(12)x2k+1=A2+αB22K2αx2k+λk-αB22K1x1k+K2αx2k+αB22K1x1k+B21+1-αB22u~2k-1+α-λkB22u~2k-1+B3wk,y2k=C2kx2k.Since the goal of this paper is to design the state controllers to regulate the power turbine speed in presence of disturbances, the output of the closed-loop is determined byy1(k), and the input is exogenous disturbance w(k). Observing (9) and (12), x1(k), x2(k), and u~2(k-1) are chosen as the closed-loop state vectors. Therefore, the closed-loop state-space form is given by(13)xk+1=Aα+λk-αBαxk+Dwk,yk=Cxk,where A(α), B(α), C, and D can be seen in(14)Aα=A1B1C20αB22K1A2+αB22K2αB21+1-αB22αK1αK2α1-αI,Bα=000B22K1B22K2α-B22K1K2α-I,C=C1T00T,D=0B30.


## 2.1. DEC System Architecture Description

This study utilized a GE T700 turboshaft engine. Figure1 shows the simplified diagram. The inputs to the gas generator were the power turbine speed set value, NP, and the fuel flow rate, WF. The outputs were the gas generator speed, NG, engine torque transmitted by the power turbine shaft, QS, compressor static discharge pressure, PS3, and power turbine inlet temperature, T45. The controller design process begins with a linearized, state-space model of the system. Figure 2 shows the simplified model in this case.Figure 1
Block diagram of the open-loop gas generator/rotor system.Figure 2
Block diagram of the simplified linearized gas generator and rotor system.Control laws essentially work to maintainNP, constant at the set point by modulating WF. The control accomplishes this by scheduling a nominal NG speed as a function of XCPC, T1, and P1. The control trims this NG demand to isochronously adjust NP to NP set input. PLA position limits the maximum permissible NG, while the control further limits the maximum T45. The control limits the NG acceleration/deceleration rate as a function of NG scheduled WF/PS3 limit. The DEC discussed herein has one network, which is inserted in the gas generator controller and the gas generator. Figure 3 shows the architecture. The abovementioned description illustrates that the GE T700 control structure is a cascade control structure, wherein the desired primary process output can only be controlled by controlling the secondary control process output.Figure 3
Block diagram of the NCCS model.Primary Plant. The state-space representation of the rotor system is provided by the following equation [9, 23]:(1)
N
˙
P
N
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M
R
Q
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M
R
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x
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1
t
=
0
0
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1
J
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-
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A
M
J
M
R
1
J
M
R
K
M
R
D
M
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D
A
M
J
M
R
-
K
M
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M
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T
-
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M
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M
R
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A
1
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P
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R
Q
M
R
︸
x
1
t
+
2
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0
2
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R
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T
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B
1
Q
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y
2
t
,
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y
1
t
=
1
0
0
︸
C
1
N
P
N
M
R
Q
M
R
.Secondary Plant. The continuous-time linear model of the gas generator is shown as follows:(2)
N
˙
G
Q
˙
S
T
˙
45
P
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S
3
N
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P
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x
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2
t
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F
2
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3
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2
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2
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t
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t
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y
2
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1
0
0
0
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C
2
N
G
Q
S
T
45
P
S
3
N
P
,where w(t) is exogenous process white noise signal belonging to l2[0,∞) and the noise parameters matrix B3 should be used as design parameters to achieve desirable system frequency response characteristics [9].The following assumptions are partially taken from [24, 25]:(a)
The controllers are event-driven. The primary controller computes the values and sends them to the secondary controller after obtaining the latest samples of the primary plant outputs. The secondary controller then computes the control command and sends it to the actuator as soon as it receives the latest samples of the secondary plant and the control output of the primary plant controller through a common network.(b)
The actuator is time-driven. In other words, the actuator actuates the plants once it receives the control command. The actuator will then use the previous value by zero-order-hold to precede the secondary process in case of packet loss.(c)
The sensors are time-driven; that is, they periodically sample the outputs and send them to the corresponding controllers.(d)
The data packet transmitted from the controller to the plant may be delayed. The delay is assumed to be a fixed one and less than a sampling periodh (i.e., τk∈[0,h]).(e)
The data packet is assumed to be transmitted between the primary and secondary controllers in a single packet without any loss. However, the data packet transmitted between the secondary controller and the actuator may be delayed or may meet a possible failure in a random manner.


## 2.2. State Feedback Control of DEC System

By considering the network-induced delayτk, the controllers are event-driven, the actuator is time-driven, and the engine receives the piece-wised control input is given by(3)ut=u~2k-1,kh≤t<kh+τk,u~2k,kh+τk≤t<k+1h,(4)u~2k=u~2k-1,ifu2klost  during  transmission,u2k,ifu2ktransmitted  successfully;that is, the actuator receives the signal u2(k) if the data is transmitted successfully; otherwise, the previous value will be used in the actuator by zero-order-hold, where u2(k) is the control output of the secondary controller.Since the actuator is time-driven, the packet loss may happen in a random manner. Then,u~2(k) can be rewritten by [25, 26](5)u~2k=λku2k+1-λku~2k-1,where λ(k) is a Bernoulli distributed stochastic variable taking the value 0 or 1. λ(k)=1 represents the successful state transmission of the delayed packet and λ(k)=0 describes the complete packet loss. It assumed that λ(k) satisfies the Bernoulli distribution [27]:(6)Prob⁡λk=1=α,Prob⁡λk=0=1-α,where α is a positive scalar, E[λ(k)-α]=0, and E[(λ(k)-α)2]=α(1-α).Considering the system reference inputNPr=0, the static state feedback controller is utilized by a discrete-time form:(7)u2k=u1k+K2αx2k,where K2(α) is the probability-dependent (packet dropouts probability) state feedback gain matrix given by K2(α)=K21+αK22 and K1 is(8)u1k=K1x1k,where x1(k) is the state vector of rotor system in discrete-time form and K1 is the state feedback gain.By using (3), the rotor system and engine with sampling period, kh,(k+1)h, are discretized to(9)x1k+1=A1x1k+B1y2k,y1k=C1x1k,where A1=eA1h,  B1=∫0heA1sdsB1, and(10)x2k+1=A2x2k+B21u~2k-1+B22u~2k+B3wk,y2k=C2x2k,where A2=eA2h, B21=∫h-τkheA2sdsB2,  B22=∫0h-τkeA2sdsB2, and B3=∫0heA2sdsB3. Then, considering the random packet loss, combined (5) and (7) with (10), (10) becomes(11)x2k+1=A2x2k+B21-1-λkB22u~2k-1+λkB22K1x1k+K2αx2k+B3wk,y2k=C2kx2k.Thus, the discretized system can be further expanded as(12)x2k+1=A2+αB22K2αx2k+λk-αB22K1x1k+K2αx2k+αB22K1x1k+B21+1-αB22u~2k-1+α-λkB22u~2k-1+B3wk,y2k=C2kx2k.Since the goal of this paper is to design the state controllers to regulate the power turbine speed in presence of disturbances, the output of the closed-loop is determined byy1(k), and the input is exogenous disturbance w(k). Observing (9) and (12), x1(k), x2(k), and u~2(k-1) are chosen as the closed-loop state vectors. Therefore, the closed-loop state-space form is given by(13)xk+1=Aα+λk-αBαxk+Dwk,yk=Cxk,where A(α), B(α), C, and D can be seen in(14)Aα=A1B1C20αB22K1A2+αB22K2αB21+1-αB22αK1αK2α1-αI,Bα=000B22K1B22K2α-B22K1K2α-I,C=C1T00T,D=0B30.


## 3. Main Results

### 3.1. System Performance Requirement

In this paper, the goal is to design controllers (7) and (8) for the turboshaft engine NCCS, such that, in the presence of random packet losses, the closed-loop system (13) is stable, and the H∞ performance constraint is satisfied [28](15)∑k=0∞Eyk2<γ2∑k=0∞Ewk2for all nonzero w(k), where γ>0 is a prescribed scalar.


### 3.2. Controller Design

Lemma 1 (Schur complement).
Given constant matricesΩ1, Ω2, and Ω3, where Ω1=Ω1T and Ω2=Ω2T>0, then Ω1+Ω3TΩ2-1Ω3<0 if and only if(16)Ω1Ω3TΩ3-Ω2<0,or-Ω2Ω3Ω3TΩ1<0.Theorem 2.
Let the positive scalarα be given. The closed-loop system (13) is stable with an H∞ performance index γ, if there exist symmetric positive definite matrices P and matrices R1, R2, R3, K1, K21, and K22 such that the following LMI holds:(17)O10O2O3O4∗-γ2IO500∗∗O600∗∗∗O60∗∗∗∗-I<0,where(18)O1=P-diag⁡R1+R1T,R2+R2T,R3+R3T,O2=R1A1TαK1TB22TαK1TR2TC2TB1TR2TA2T+αK2TαB22TK2Tα0R3TB21T+1-αR3TB22T1-αR3T,O3=0K1TB22TK1T0K2TαB22TK2Tα0-R3TB22T-R3T,O4=R1TC1T00,O5=0B30T,O6=-P,K2α=K21+αK22,and the state feedback gain matrices can be gained by(19)K1=K1R1-1,K21=K21R2-1,K22=K22R3-1.Proof.
In order to conclude the controller design conditions, the following Lyapunov function can be defined:(20)Vk=xTkP-1xk.
Now, for any nonzerow(k),(21)EVk+1-EVk+EykTyk-γ2EwkTwk=EAα+λk-αBαxk+1+DwkTP-1Aα+λk-αBαxk+1+Dwk-xkTP-1xk+ykTyk-γ2wkTwk≤E(AαxkTP-1(Aαxk+α1-αxTkBTαP-1Bαxk-xkTP-1xk+ykTyk-γ2wkTwk=ExkwkT·Oα·xkwk,where O(α) can be seen in(22)Oα=AαTP-1Aα+α1-αBαTP-1Bα+CTC-P-1AαTP-1DDTP-1AαDTP-1D-γ2I.and O(α) can be rewritten in(23)Oα=-P-10∗-γ2I+AαTDT·P-1·AαD+α1-αBαT0·P-1·Bα0+CT0·C0.Thus, by applying Lemma 1, (24) can be obtained(24)Oα=-P-10AαTα1-αBαTCT∗-γ2IDT00∗∗-P00∗∗∗-P0∗∗∗∗-I.
Now, the goal is to proveO(α)<0. By means of the partition matrices R=diag⁡{R1,R2,R3}, P, the values of A(α), B(α), C, D, and K1, K21, and K22 in LMI (17), the following inequality (25) can be gotten:(25)P-R+RT0RTAαTα1-αRTBαTRTCT∗-γ2IDT00∗∗-P00∗∗∗-P0∗∗∗∗-I<0.
If (25) holds, then by using P-2R≥-RTP-1R, the following inequality (26) can be gained:(26)-RTP-1R0RTAαTα1-αRTBαTRTCT∗-γ2IDT00∗∗-P00∗∗∗-P0∗∗∗∗-I<0.
Equation (17) can then be obtained by pre- and postmultiplying (26) by diag⁡(R-1,I,I,I,I). Therefore, for zero to ∞ with respect to k, it yields:(27)∑k=0∞Eyk2<γ2∑k=0∞Ewk2+EV0-EV∞.Since x(0)w(0)=00, the closed-loop system (13) is stable, and it satisfies (15).Algorithm for the Controllers Design (a)
The continuous closed-loop system parameters are derived based on Figure2. (b)
The continuous system parameters are discretized. (c)
The convex optimization problem (17) is solved to obtain the feasible solutions in terms of positive definite matrices P, nonsingular slack matrices Ri,  (i=1,2,3), and matrices K1, K21, K22, and γ. (d)
The controller parametersK1, K21, and K22 are derived based on Theorem 2. (e)
Stop.


## 3.1. System Performance Requirement

In this paper, the goal is to design controllers (7) and (8) for the turboshaft engine NCCS, such that, in the presence of random packet losses, the closed-loop system (13) is stable, and the H∞ performance constraint is satisfied [28](15)∑k=0∞Eyk2<γ2∑k=0∞Ewk2for all nonzero w(k), where γ>0 is a prescribed scalar.


## 3.2. Controller Design

Lemma 1 (Schur complement).
Given constant matricesΩ1, Ω2, and Ω3, where Ω1=Ω1T and Ω2=Ω2T>0, then Ω1+Ω3TΩ2-1Ω3<0 if and only if(16)Ω1Ω3TΩ3-Ω2<0,or-Ω2Ω3Ω3TΩ1<0.Theorem 2.
Let the positive scalarα be given. The closed-loop system (13) is stable with an H∞ performance index γ, if there exist symmetric positive definite matrices P and matrices R1, R2, R3, K1, K21, and K22 such that the following LMI holds:(17)O10O2O3O4∗-γ2IO500∗∗O600∗∗∗O60∗∗∗∗-I<0,where(18)O1=P-diag⁡R1+R1T,R2+R2T,R3+R3T,O2=R1A1TαK1TB22TαK1TR2TC2TB1TR2TA2T+αK2TαB22TK2Tα0R3TB21T+1-αR3TB22T1-αR3T,O3=0K1TB22TK1T0K2TαB22TK2Tα0-R3TB22T-R3T,O4=R1TC1T00,O5=0B30T,O6=-P,K2α=K21+αK22,and the state feedback gain matrices can be gained by(19)K1=K1R1-1,K21=K21R2-1,K22=K22R3-1.Proof.
In order to conclude the controller design conditions, the following Lyapunov function can be defined:(20)Vk=xTkP-1xk.
Now, for any nonzerow(k),(21)EVk+1-EVk+EykTyk-γ2EwkTwk=EAα+λk-αBαxk+1+DwkTP-1Aα+λk-αBαxk+1+Dwk-xkTP-1xk+ykTyk-γ2wkTwk≤E(AαxkTP-1(Aαxk+α1-αxTkBTαP-1Bαxk-xkTP-1xk+ykTyk-γ2wkTwk=ExkwkT·Oα·xkwk,where O(α) can be seen in(22)Oα=AαTP-1Aα+α1-αBαTP-1Bα+CTC-P-1AαTP-1DDTP-1AαDTP-1D-γ2I.and O(α) can be rewritten in(23)Oα=-P-10∗-γ2I+AαTDT·P-1·AαD+α1-αBαT0·P-1·Bα0+CT0·C0.Thus, by applying Lemma 1, (24) can be obtained(24)Oα=-P-10AαTα1-αBαTCT∗-γ2IDT00∗∗-P00∗∗∗-P0∗∗∗∗-I.
Now, the goal is to proveO(α)<0. By means of the partition matrices R=diag⁡{R1,R2,R3}, P, the values of A(α), B(α), C, D, and K1, K21, and K22 in LMI (17), the following inequality (25) can be gotten:(25)P-R+RT0RTAαTα1-αRTBαTRTCT∗-γ2IDT00∗∗-P00∗∗∗-P0∗∗∗∗-I<0.
If (25) holds, then by using P-2R≥-RTP-1R, the following inequality (26) can be gained:(26)-RTP-1R0RTAαTα1-αRTBαTRTCT∗-γ2IDT00∗∗-P00∗∗∗-P0∗∗∗∗-I<0.
Equation (17) can then be obtained by pre- and postmultiplying (26) by diag⁡(R-1,I,I,I,I). Therefore, for zero to ∞ with respect to k, it yields:(27)∑k=0∞Eyk2<γ2∑k=0∞Ewk2+EV0-EV∞.Since x(0)w(0)=00, the closed-loop system (13) is stable, and it satisfies (15).Algorithm for the Controllers Design (a)
The continuous closed-loop system parameters are derived based on Figure2. (b)
The continuous system parameters are discretized. (c)
The convex optimization problem (17) is solved to obtain the feasible solutions in terms of positive definite matrices P, nonsingular slack matrices Ri,  (i=1,2,3), and matrices K1, K21, K22, and γ. (d)
The controller parametersK1, K21, and K22 are derived based on Theorem 2. (e)
Stop.


## 4. Simulation Examples

This section presents the effectiveness evaluation of the proposed method under simulations in the GE T700 turboshaft gas turbine engine DEC control systems. The model of the engine is based on partial derivatives calculated from an accurate nonlinear model [1]. The rotor system and the gas generator models in continuous time form are provided:(28)N˙PN˙MRQ˙MR=00-285.71430-0.45339.06625.2650-5.2131-42.5958NPNMRQMR+571.4286082.5714QS,NP=100NPNMRQMR,N˙GQ˙ST˙45P˙S3N˙P=-126.827.0412.3622.1716.7254.6757.21-77.02-76.2150.81-336.6223.3-130.7-83.32172.1161.22.459-21.8-63.091.79962.42-73.55-104.2-91.44-102.3NGQST45PS3NP+-11.744.2453.5617.4559.35WF+0.020.020.020.020.02w,QS=01000NGQST45PS3NP.The coefficients after the discretization are provided as follows:(29)A1=0.93520.0640-2.26700.00210.99340.07180.0418-0.04130.5952,B1=4.57030.03620.7847,C1=100,A2=0.39270.1572-0.05240.00100.07720.69851.0228-0.4391-0.40800.0411-0.59330.5866-0.1759-0.83110.34840.80050.0546-0.06650.63670.01870.1142-0.83500.02420.07690.0953,B21=-0.01530.19080.35240.05020.1099,B22=0.02060.09200.25530.0138-0.1128,B3=0.12130.0715-0.12140.10190.0328,C2=01000.Simulation 1.
Let the packet loss probability valueα=0.3, and given the initial conditions as x1(0)=10.20.2T,x2(0)=0.90000.41890.78430.64981.0000T, the simulation time is T=20s, sampling time is h=0.01s, and assuming that the two network-induced delays are both equivalent to τk, which is not longer than the sampling period, τk=0.005s. The goal of this simulation is to design the controller gains such that the closed-loop system is robustly stable with a disturbance attenuation level γ>0. The optimized solution of (17) can be calculated by using the LMI toolbox in MATLAB:(30)P=10-7×-0.4640-0.00370.0128-0.03980.06840.0015-0.0110-0.0418-0.0011-0.0037-0.5633-0.00050.00010.0003-0.0002-0.0006-0.00050.00010.0128-0.0005-0.6465-0.01800.02330.00380.0120-0.0035-0.0015-0.03980.0001-0.0180-0.6070-0.02630.01300.0106-0.0424-0.01420.06840.00030.0233-0.0263-0.7761-0.09930.01610.1228-0.03200.0015-0.00020.00380.0130-0.0993-0.5875-0.07240.0404-0.0347-0.0110-0.00060.01200.01060.0161-0.0724-0.58740.00410.0364-0.0418-0.0005-0.0035-0.04240.12280.04040.0041-0.5022-0.0119-0.00110.0001-0.0015-0.0142-0.0320-0.03470.0364-0.0119-0.5462,R1=10-5×0.2236-0.00520.0866-0.00520.0172-0.00170.0866-0.00170.0363,R2=10-4×0.01880.00010.0485-0.02170.00070.00010.00010.0004-0.00000.00010.04850.00040.1304-0.05840.0030-0.0217-0.0000-0.05840.02780.00400.00070.00010.00300.00400.0237,R3=10-8×6.8719,K1=-0.0016-0.00010.0035,K21=-0.0380-0.27800.06350.1037-0.0242,K22=0.0287-0.00080.0847-0.0323-0.0080,γopt=0.3035.
Figures4 and 5 show the responses of the state variables in the closed-loop system under packet dropouts, and the system states converge to zero. Meanwhile, Figure 6 illustrates that the gas generator control loop (inner loop) is much faster than the rotor system control loop (outer loop). Therefore, by Theorem 2, the closed-loop system (13) is robust stable with H∞ disturbance-rejection-attenuation level γ, and it is noted that the obtained controllers make sure the fast response of inner loop to eliminate disturbances.Figure 4
Rotor system state responsex1.Figure 5
Gas generator state responsex2.Figure 6
Rotor system controller outputu.Simulation 2.
In order to show the effectiveness evaluation of the proposed method under different values of packet loss probabilityα, the obtained state feedback controller parameters and disturbance attenuation level γ are presented in Table 1. In Table 1, the case α=0 represents the successful transmission, α≠1 representing the packet dropout, and α=1 denotes the complete loss of transmission case.
Figures7 and 8 show that the closed-loop system (13) can be stabilized with or without packet dropouts. Figure 9 illustrates that the dynamical behavior of the closed-loop system takes longer to converge to zero. Figures 10 and 11 show the network-induced packet loss responses. Thus, the designed controllers are well suited for the considered turboshaft engine model and work well over the network-induced imperfections and input disturbances.Table 1
Optimizedγopt and control parameters for various values of α.
α K 1 K 21   &   K 22 γ o p t 0 [ 0.0009 - 0.0000 - 0.0016 ] [ 0.0421 0.2644 - 0.0754 - 0.1307 0.0297 ] 0.2999 [ 0 0 0 0 0 ] 0.3 [ - 0.0016 - 0.0001 0.0035 ] [ - 0.0380 - 0.2780 0.0635 0.1037 - 0.0242 ] 0.3035 [ 0.0287 - 0.0008 0.0847 - 0.0323 - 0.0080 ] 0.7 [ - 0.0047 - 0.0003 0.0102 ] [ - 0.1013 - 0.6667 0.1705 0.2835 - 0.0654 ] 0.2970 [ 0.0741 - 0.0013 0.2336 - 0.0698 - 0.0287 ] 1.0 [ - 0.0075 - 0.0004 0.0162 ] [ - 0.1611 - 0.9143 0.2504 0.4084 - 0.0926 ] 0.3014 [ 0.0972 - 0.0003 0.3161 - 0.0869 - 0.0415 ]Figure 7
Rotor system statex1 under different α.Figure 8
Gas generator statex2 under different α.Figure 9
Rotor system controller outputu1 under different α.Figure 10
Gas generator controller outputu2 under different α.Figure 11
Gas generator controller outputu2 under different α (partial view).


## 5. Conclusions

This study considered the novel robustH∞ distributed engine control problem to guarantee the engine performance with random packet dropouts and disturbances. A distributed control system architecture of a typical turboshaft engine was also described accordingly. This distributed architecture can be transformed into a networked cascade control system. The state feedback controllers were designed to robustly stabilize the closed-loop system under packet loss and disturbances. The sufficient conditions for stability were derived based on the Lyapunov stability and the LMI approach. The controller design problem under consideration is solvable if the LMI was feasible. Simulation examples were provided to show the effectiveness of the approach.



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*Source: 5435090-2017-02-16.xml*