RESEARCH ARTICLE
Design of Distributed Engine Control Systems with Uncertain Delay Xiaofeng Liu1,2,3*, Yanxi Li4, Xu Sun1 1 School of Transportation Science and Engineering, Beihang University, Beijing, PR China, 2 Collaborative Innovation Center for Advanced Aero-Engine, Beijing, PR China, 3 Aircraft/Engine Integrated System Safety Beijing Key Laboratory, Beijing, PR China, 4 School of Energy and Power Engineering, Beihang University, Beijing, PR China *
[email protected]
Abstract a11111
OPEN ACCESS Citation: Liu X, Li Y, Sun X (2016) Design of Distributed Engine Control Systems with Uncertain Delay. PLoS ONE 11(9): e0163545. doi:10.1371/ journal.pone.0163545
Future gas turbine engine control systems will be based on distributed architecture, in which, the sensors and actuators will be connected to the controllers via a communication network. The performance of the distributed engine control (DEC) is dependent on the network performance. 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 with a H1 output feedback under network-induced time delays and uncertain disturbances. The sufficient conditions for robust stability are derived via the Lyapunov stability theory and linear matrix inequality approach. Both numerical and hardware-in-loop simulations illustrate the effectiveness of the presented method.
Editor: Houbing Song, West Virginia University, UNITED STATES Received: July 24, 2016 Accepted: September 9, 2016 Published: September 26, 2016 Copyright: © 2016 Liu et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: All relevant data are within the paper. Funding: This work was supported by the National Natural Science Foundation of China (NSFC) under grant nos. 61573035 and 61104146 and the China Scholarship Council (CSC) under grant no. 201506025135. Competing Interests: The authors have declared that no competing interests exist.
Introduction A distributed control system (DCS) is a control system, wherein the control elements are distributed throughout the system, unlike centralized ones, where only a single controller at a central location is used. In a DCS, a hierarchy of controllers is connected by communication networks for information transmission. The advantages of the DCS architecture, such as system weight reduction, higher reliability, modularity, and less maintenance costs, merit increasing attention from industrial companies and engineers. Conventional gas turbine engine control systems are designed as a centralized architecture called the full authority digital engine control (FADEC)to protect the control elements from extreme environment [1]. With the increasing development of sophisticated electronics in gas turbine engine control systems, increased performance, more convenient operation, and reduction of design and maintenance costs require a more effective architecture for the control systems; hence, the development of the DEC architecture [2]. The sensors and controllers are connected by communication networks and between the controllers and the actuators because of the distributed architecture [3]. The DEC architecture can be viewed as an NCCS. For example, the GE T700 turboshaft engine is a two-spool engine
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Design of Distributed Engine Control Systems with Uncertain Delay
consisting of a gas generator and a free power turbine [4] [5]. The power turbine is connected to the rotor system by a shaft and a gearbox. The power turbine can be conventionally considered as a part of the rotor system [6]. The rotor system input is the gas generator’s output, which is the shaft torque. Therefore, the whole turboshaft engine system, combined with the control systems, can be reviewed as a cascade control system (CCS) [7]. Fundamental factors affect the DEC system that uses the communication network to close the control loop. They include network-induced time delay, packet dropouts, and bandwidth constraints [8] [9]. Hence, the control system should be robust to these factors to guarantee the desired performance and ensure stability. The network-induced time delay in the NCCSs occurs when the sensors, controllers, and actuators transfer information/data through the networks, which can degrade the performance of the control systems and even destabilize the system [10]. The network-induced time delay is unavoidable in the NCCSs. Hence, existing literature, such as [11] [12] [13] [14] and the references therein, discuss the time delay. Moreover, many useful approaches were proposed [15] [16] [17] [18] and applied to the industrial systems (see [19] [20] [21] [22] [23] and the references therein). However, only a few studies discussed the DEC robust control in gas turbine engine control systems. Accordingly, Belapurkar et al. [24] analyzed the stability of a set-point controller for partial DEC systems with time delays by using the linear quadratic regulator (LQR) method. Yedavalli et al. [9] discussed the DEC system stability 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. The present study is concerned with the problem of the H1 controller design for the gas turbine engine-distributed control by using an output feedback control in the form of NCCSs with uncertain delays. The rest of the paper is organized as follows: the architecture of the distributed engine control system is thoroughly described in the next section. An NCCS model of a GE T700 turboshaft engine is established and the H1 output feedback controllers are designed based on Lyapunov stability theory and LMI approach in the following section. Simulation examples are presented in the simulation results section to illustrate the effectiveness of the approach. The conclusion is found in the last section.
DEC System Architecture of the GE T700 Turboshaft Engine This study utilized a GE T700 turboshaft engine. Fig 1 shows the simplified diagram. Table 1 presents the abbreviations of the engine parameters. 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. Control laws essentially work to maintain the power turbine speed, NP, constant at the set point by modulating the fuel flow, WF. The control accomplishes this by scheduling a nominal NG speed as a function of the XCPC, T1 and P1. The control trims this NG demand to isochronously adjust NP to the NP set input. The power lever angle (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 an NG scheduled WF/PS3 limit. The DEC discussed herein has one network, which is inserted in the gas generator controller and the gas generator. Fig 2 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.
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Design of Distributed Engine Control Systems with Uncertain Delay
Fig 1. Block diagram of the open-loop gas generator/rotor system. doi:10.1371/journal.pone.0163545.g001
The following assumptions are partially taken from [25] [26]: • 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 Table 1. Symbols of the GE T700 turboshaft engine. Symbols and meaning PLA
Power lever angle (throttle)
NG
Gas generator speed
NP
Power turbine speed
NMR
Main rotor blade velocity
QMR
Rotor torque state
QS
Engine shaft torque
XCPC
Collective pitch
P1
Inlet pressure
PS3
Static pressure at Station 3
T1
Inlet temperature
T45
Inter-turbine gas temperature
WF
Fuel flow
JG
Power turbine inertia
JT
Lumped power turbine/dynamometer inertia
JMR
Main rotor blade inertia
KMR
Stiffness of the centrifugal restoring springs
DMR
Lag hinge damping
DAM
Aero damping
r
Reference input
x
Model state vector
y
Model output vector
u
Model input vector
doi:10.1371/journal.pone.0163545.t001
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Design of Distributed Engine Control Systems with Uncertain Delay
Fig 2. Block diagram of the NCCS model. doi:10.1371/journal.pone.0163545.g002
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. • 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. • The sensors are time driven, that is, they periodically sample the outputs and send them to the corresponding controllers. • 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 period h (i.e., τk 2 [0, h]). • 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.
Robust H1 Output Control for GE T700 Model description The controller design process begins with a linearized, state-space model of the system. Fig 3 shows the simplified model in this case. Primary plant: The state-space representation of the rotor system is provided by the following equation: x_ 1 ðtÞ ¼ A1 x1 ðtÞ þ B1 y2 ðtÞ ð1Þ y1 ðtÞ ¼ C1 x1 ðtÞ where, x1 = [NP NMR QMR]T, and y1 = NP are the state vector and the output of the rotor system, respectively. y2 = QS is the gas generator output. The matrices A1 , B1 , and C1 are provided as follows: 2 3 1 2 3 0 0 2 6 7 JT 6 J 7 6 7 T 6 7 6 7 DAM 1 6 0 7 6 7 A1 ¼ 6 7; C1 ¼ ½ 1 0 0 : 7; B 1 ¼ 6 0 JMR JMR 6 7 6 7 4 2 DMR 5 6 7 4 DMR DAM DMR DMR 5 JT KMR KMR JMR JT JMR Secondary plant: The continuous-time linear model of the gas generator is shown as follows:
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Design of Distributed Engine Control Systems with Uncertain Delay
Fig 3. Block diagram of the simplified linearized gas generator and rotor system. doi:10.1371/journal.pone.0163545.g003
x_ 2 ðtÞ ¼ A2 x2 ðtÞ þ B2 uðtÞ þ B3 wðtÞ ð2Þ y2 ðtÞ ¼ C2 x2 ðtÞ where, x2 = [NG QS T45 PS3 NP]T, y2 = QS are the state and output vectors; u(t) = WF is the control input; and w(t) is the exogenous disturbance signal belonging to l2[0, 1). The matrices A2 , B2 , and C2, are presented as follows: 2 3 2 3 1 dQG 1 dQG 0 0 0 0 6 J dN 7 6 J dW 7 6 7 6 7 G G G F 6 6 7 7 6 2 DMR dQP 6 7 7 2 DMR dQ 2 DMR dQ P7 P 7 6 6 0 0 0 6 J 7 6 JT dNG dNP 7 dWF 7 T 6 6 JT 7 6 7 6 7 6 7 6 7 dT45 dT45 6 7 6 7; C2 ¼ ½ 0 1 0 0 0 0 0 0 0 A2 ¼ 6 ; B2 ¼ 6 7 7 dNG dWF 6 6 7 7 6 7 6 7 6 6 7 7 dPS3 dPS3 6 6 7 7 0 0 0 0 6 7 6 7 dN dW G F 6 7 6 7 6 6 7 7 4 4 5 5 2 dQP 2 dQP 1 2 dQP 0 0 JT dWF JT dNG JT JT dNP and B3 is a real constant matrix with an appropriate dimension.
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Design of Distributed Engine Control Systems with Uncertain Delay
Output feedback control The output feedback controller is utilized in K1 considering the system reference input NPr = 0. The controller uses a discrete-time form in practical applications. u1 ðkÞ ¼ K1 y1 ðkÞ
ð3Þ
where, y1(k) is the output vector of the rotor system in discrete-time form, and K1 is the output feedback gain. The K2 also uses the output feedback form as follows: u2 ðkÞ ¼ u1 ðkÞ þ K2 y2 ðkÞ
ð4Þ
where, y2(k) is the output vector of the gas generator in discrete-time form, and K2 is the output feedback gain. The gas generator receives the piecewise control input provided as follows by considering the network-induced delay τk: ( u2 ðk 1Þ kh t < kh þ tk uðtÞ ¼ ð5Þ u2 ðkÞ kh þ tk t < ðk þ 1Þh The rotor system and the gas generator with the sampling period, [kh, (k+1)h], are discretized as follows by using Eq (5): x1 ðk þ 1Þ ¼ A1 x1 ðkÞ þ B1 y2 ðkÞ ð6Þ y1 ðkÞ ¼ C1 x1 ðkÞ where, A1 ¼ eA1 h , B1 ¼
Rh 0
eA1 s dsB1 :
x2 ðk þ 1Þ ¼ A2 x2 ðkÞ þ Bk21 u2 ðkÞ þ Bk22 u2 ðk
1Þ þ B3 wðkÞ
ð7Þ
y2 ðkÞ ¼ C2 x2 ðkÞ R khþt Rh eA2 ðkhþh sÞ dsB2 , Bk22 ¼ kh k eA2 ðkhþh sÞ dsB2 , B3 ¼ 0 eA2 s dsB3 . Rh t Rh Let m = kh + h − s, then Bk21 ¼ 0 k eA2 m dmB2 , Bk22 ¼ h tk eA2 m dmB2 . Correspondingly,
where, A2 ¼ eA2 h , Bk21 ¼
R khþh
khþtk
τk 2 [0, h]. Therefore, let tk ¼ ð1 þ dk Þh2, dk 2 [−1, 1]: Z Z h2 R h2 dk h2 A m k A2 m A2 h2 2 B21 ¼ 0 e dmℬ2 ¼ e dmℬ2 þ e 0
k Þ k2 ¼ k Let Fm ¼ max k Fðd dk 2½ 1;1
R h2 0
dk h2
eA2 m dmℬ2
0
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflffl {zfflfflfflfflfflfflfflfflfflfflffl }
B21
F ðdk Þ
h k Þ=Fm , then eA2 m dm k2 , D ¼ Fm eA2 2 , F ¼ Fðd
Bk21 ¼ B21 þ DFB2 : Let B22 ¼
Rh h 2
eA2 m dmB2 , then Bk22 ¼ B22
DFB2 :
x2 ðk þ 1Þ ¼ A2 x2 ðkÞ þ ðB21 þ DFB2 Þu2 ðkÞ þ ðB22
DFB2 Þu2 ðk
1Þ þ B3 wðkÞ ð8Þ
y2 ðkÞ ¼ C2 x2 ðkÞ
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Design of Distributed Engine Control Systems with Uncertain Delay
In general, the network-induced delay in an NCCS is usually uncertain because of the network transmission and network load uncertainty. A robust H1 control is an efficient tool to analyze these problems and deal with the uncertainty. This study aims to design the output controllers to regulate the power turbine speed in the presence of disturbances. Hence, the closed-loop output is determined by y1(k), and the input is an exogenous disturbance w(k). Observing Eqs (6) and (7), x1(k), x2(k), and u2(k − 1) are chosen as the closed-loop state vectors. Therefore, the closed-loop state-space form is provided as follows:
2
x1 ðk þ 1Þ
3
2
A1
6 7 6 6 7 6 7 6 6 6 x2 ðk þ 1Þ 7 ¼ 6 O1 6 7 6 5 4 4 u2 ðkÞ K1 C1
B1 C2 O2 K2 C2
0
32
x1 ðkÞ
3
2
0
3
76 7 6 7 76 7 6 7 76 7 6 7 O3 76 x2 ðkÞ 7 þ 6 B3 7w 76 7 6 7 54 5 4 5 u2 ðk 1Þ 0 0
ð9Þ
y1 ðkÞ ¼ C1 x1 ðkÞ
where, O1 ¼ ðB21 þ DFB2 ÞK1 C1 , O2 ¼ A2 þ ðB21 þ DFB2 ÞK2 C2 , O3 ¼ B22 DFB2 . Definition 1 Given a certain constant γ > 0, the γ-suboptimal output feedback H1 stabilization control laws exist for the closed-loop DEC Eq (9), and the disturbance attenuation degree of the system is γ, if there exist output feedback control laws Eqs (3) and (4), which make the DEC system Eq (9) as robust asymptotically stable, and the closed-loop output y1(k) and the disturbance w(k) are subject to the H1 norm-bounded constraint ky1(k)k2 γkw(k)k2 under zero initial conditions. The following lemmas are required to derive the sufficient conditions for robust asymptotic stability: Lemma 1 (Schur Complement) Given constant matrices O1, O2, and O3, where O1 ¼ OT1 and O2 ¼ OT2 > 0, then O1 þ OT3 O2 1 O3 < 0 if and only if:
2 4
O1 O3
OT3 O2
3 5 < 0 or
"
O2 OT3
O3
# 0 exists satisfying: T k
O1 þ a 1 OT2 O2 þ aO3 OT3 < 0
Theorem 1 For the closed-loop NCCS shown in Eq (9) with disturbances, given a constant γ > 0, the optimization problem Eq (10) has the feasible solution min ρ if symmetric positive-
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Design of Distributed Engine Control Systems with Uncertain Delay
definite matrices, X, Y, Z, W1, and W2, exist with corresponding dimensions and a constant λ > 0, 2 X
6 6 6 0 6 6 6 6 0 6 6 6 0 6 6 6 6B W 6 21 1 6 6 6 A1 X 6 6 6 6 W1 6 6 6B W 6 2 1 4 C1 X
0 Y 0
0 0 Z
0
ðB21 W1 Þ
T
0
ðA2 þ B21 W2 Þ
0
ðB22 ZÞ
ðA1 XÞ T
ðB1 C2 YÞ
T
BT3
T
ðW1 Þ
T
ðW2 Þ
T
ðB2 W1 Þ
T
ðB2 W2 Þ
T
0
T
0
0
ðB2 ZÞ
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
A2 þ B21 W2
B22 Z
B3
lDDT
B1 C2 Y
0
0
0
X
W2
0
0
0
0
B2 Z
0
0
0
0
lI
0
0
0
0
0
0
0
Y
ðC1 XÞ
0
0
B2 W2
rI
T
Z
T
3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 < 0 ð10Þ 7 7 7 7 7 7 7 7 7 7 7 7 5
I
The γ-optimal output feedback H1 control laws then exist, and the gain matrices are provided as follows: K1 ¼
W1 W ; K2 ¼ 2 C1 X C2 Y
ð11Þ
pffiffiffiffi The optimal solution ρ becomes available, and min g ¼ r . Proof: A quadratic Lyapunov function in discrete-time form is defined as follows to derive the sufficient conditions for robust asymptotic stability of Eq (9): Vk ¼ x1T ðkÞPx1 ðkÞ þ x2T ðkÞQx2 ðkÞ þ uT2 ðk
1ÞSu2 ðk
1Þ
ð12Þ
where, P, Q, and S are the symmetric positive-definite matrices with corresponding dimensions. The ΔVk can be obtained by the following equation by taking the Lyapunov function derivative: DVk
¼ Vkþ1
Vk
¼ x1T ðk þ 1ÞPx1 ðk þ 1Þ þ x2T ðk þ 1ÞQx2 ðk þ 1Þ þ uT2 ðkÞSu2 ðkÞ x1T ðkÞPx1 ðkÞ
x2T ðkÞQx2 ðkÞ
uT2 ðk
1ÞSu2 ðk 2
¼ ½ x1T ðkÞ x2T ðkÞ uT2 ðk
1Þ
1Þ x1 ðkÞ
3
6 7 6 7 6 x ðkÞ 7 6 7 2 6 7 wT ðkÞ X 6 7 6 7 6 u2 ðk 1Þ 7 6 7 4 5
ð13Þ
wðkÞ
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Design of Distributed Engine Control Systems with Uncertain Delay
where, 2
X
T
AT1 PA 1 þ ðK1 C1 Þ SK1 C1
T
P þ OT1 QO1
6 6 6 ðB C ÞT PA þ ðK C ÞT SK C þ OT QO 6 1 2 1 2 2 1 1 1 2 6 ¼6 6 6 OT3 QO1 6 4
AT1 PB1 C2 þ ðK1 C1 Þ SK2 C2 þ OT1 QO2 T
T
T 2
ðB1 C2 Þ PB1 C2 þ ðK2 C2 Þ SK2 C2
Q þ O O2
T
S þ OT3 QO3
6 6 6 ðB C ÞT PA þ ðK C ÞT SK C 6 1 2 1 2 2 1 1 6 ¼6 6 6 0 6 4
BT3 QO3
T
AT1 PB1 C2 þ ðK1 C1 Þ SK2 C2
P
T
T
ðB1 C2 Þ PB1 C2 þ ðK2 C2 Þ SK2 C2
0
3
2
OT1
Q
0
0
7 7 O QB3 7 7 7 7 7 T O3 QB3 7 7 5 BT3 QB3
3
7 6 7 7 6 7 6 T7 0 07 7 6 O2 7 7 6 7 7 þ 6 7 Q ½ O1 7 6 T7 6O 7 S 07 7 6 37 5 4 5
0
0
0
3
T 2
O QO3
BT3 QO2
AT1 PA 1 þ ðK1 C1 Þ SK1 C1
OT1 QB3
T 2
OT3 QO2
BT3 QO1 2
OT1 QO3
O2
O3
B3
BT3
0
The performance index function can be defined as follows considering the closed-loop robust stability with disturbance: 1 X
J¼
y1T ðkÞy1 ðkÞ
g2 wT ðkÞwðkÞ
ð14Þ
k¼0
The following condition must be satisfied if the disturbance attenuation degree is γ > 0: k y1 ðkÞ k2 gk wðkÞ k2
ð15Þ
(i.e., J 0). The following equation is obtained for the system with a non-zero disturbance satisfying w(t) 2 [0, 1) and zero initial conditions: 1 X
ðy1T ðkÞy1 ðkÞ
J
g2 wT ðkÞwðkÞ þ DVk Þ
k¼0
3 1 2 x1 ðkÞ 7 B6 7 C 6 7 B6 7 C 6 6 7 B6 7 C 6 7 B6 0 7 C 6 0 0 0 7 B6 C 6 x ðkÞ 7 7 ð16Þ 2 B6 7 C 6 7 T B 6 7 C 7 w ðkÞ B6 7 þ X C 6 6 7 B6 7 C 6 7 B6 0 7 C 0 0 0 6 B6 7 C 6 u2 ðk 1Þ 7 7 B6 7 C 6 7 @4 5 A 4 5 0 0 0 g2 I wðkÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 02
¼ ½ x1 ðkÞT
T
x2 ðkÞ
u2 ðk
T
1Þ
C1T C1
0
0
0
3
Γ
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Design of Distributed Engine Control Systems with Uncertain Delay
where, 2
Γ
6 6 6 6 6 ¼6 6 6 6 4 2 6 6 6 6 6 6 6 ¼6 6 6 6 6 6 6 4 2 6 6 6 6 6 6 6 6 6 6 6 6 ¼6 6 6 6 6 6 6 6 6 6 6 4 2 6 6 6 6 6 6 6 6 6 6 6 6 ¼6 6 6 6 6 6 6 6 6 6 6 4
T
AT1 PA 1 þ ðK1 C1 Þ SK1 C1 T
T
P þ C1T C1
AT1 PB1 C2 þ ðK1 C1 Þ SK2 C2
T
T
ðB1 C2 Þ PA 1 þ ðK2 C2 Þ SK1 C1
ðB1 C2 Þ PB1 C2 þ ðK2 C2 Þ SK2 C2
0
0
0
0
T
AT1 PA 1 þ ðK1 C1 Þ SK1 C1 T
P
T
T
T
ðB1 C2 Þ PB1 C2 þ ðK2 C2 Þ SK2 C2
Q
0
g2 I
0
0
S
2
0
OT2
0
OT3
0
0
g2 I
BT3
O1
O2
O3
B3
Q
T
ðK2 C2 Þ
T
0
OT1
0
0
T 2
O
0
0
0
0
0
0
0
OT3
0
A1
B1 C2
P
K1 C1
K2 C2
0
S
0
0
0
0
0
0
0
0
0
g2 I
BT3
O1
O2
0
0
O3
B3
Q
0
ðB1 C2 Þ
T
ðK1 C1 Þ
0
1
0 1
S
0
OT1
AT1
0
T 2
Q
ðB1 C2 Þ
0
QT3
0 0
0
Q
0
0
0
0
0
g2 I
BT3
O1
O2
O3
B3
Q
A1
B1 C2
0
0
0
P
K1 C1
K2 C2
0
0
0
0
0 S
1
ðK1 C1 Þ
0 1
T
O2
O3
B3
3
OT1
0
AT1
3
BT3
0
0
OT1
7 6 7 7 6 7 6 T7 0 7 7 6 O2 7 7 6 7 7 þ 6 7 Q ½ O1 7 6 T7 6O 7 0 7 7 6 37 5 4 5
0
0
3
0
Q
P
0 S
AT1 PB1 C2 þ ðK1 C1 Þ SK2 C2
0
C1T C1
Q
T
P þ C1T C1
ðB1 C2 Þ PA 1 þ ðK2 C2 Þ SK1 C1
C1T C1
T
0
7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 1
3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 1
T
3
7 7 7 ðK2 C2 Þ 7 7 7 7 7 0 7 7 7 7 7 0 7 7 7 7 0 7 7 7 7 7 0 7 5
PLOS ONE | DOI:10.1371/journal.pone.0163545 September 26, 2016
T
S
1
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Design of Distributed Engine Control Systems with Uncertain Delay
Substituting O1, O2, and O3 into Γ,
2
Γ
C1T C1
P
0
6 6 6 0 6 6 6 6 6 0 6 6 6 6 ¼6 0 6 6 6 6 ðB þ DFB ÞK C 6 21 2 1 1 6 6 6 6 A1 6 4
Q
C1T C1
P
A2 þ ðB21 þ DFB2 ÞK2 C2
6 6 6 0 6 6 6 6 6 0 6 6 6 6 ¼6 0 6 6 6 6 B KC 6 21 1 1 6 6 6 6 A1 6 4
0 Q 0
0
0 B21
DFB2
DFB2 Þ
ðB21
g2 I
BT3
B3
Q
AT1 T
T
0 0
1
0
0
0
0
P
K2 C2
0
0
0
0
0
0
0 S
ðB21 K1 C1 Þ
T
AT1 T
0
ðA2 þ ðB21 K2 C2 Þ
0
BT22
0
g2 I
BT3
0
ðB1 C2 Þ
0
A2 þ B21 K2 C2
B22
B3
Q
B1 C2
0
0
0
P
0
0
0
0
1
0
2
B2
0
0 0
ðK1 C1 Þ
ðB2 K1 C1 Þ
T
1
T
1
T
T
3
7 7 7 ðK2 C2 Þ 7 7 7 7 7 0 7 7 7 7 7 0 7 7 7 7 0 7 7 7 7 7 0 7 5 T
S
1
3
7 7 7 ðK2 C2 Þ 7 7 7 7 7 0 7 7 7 7 7 0 7 7 7 7 0 7 7 7 7 7 0 7 5 T
S
1
3
6 7 6 7 6 T 7 6 ðB2 K2 C2 Þ 7 6 7 6 7 6 7 T 6 B2 7 6 7 6 7 6 7 6 7 T 0 þ 6 7F ½ 0 0 6 7 6 7 6 7 6 7 0 6 7 6 7 6 7 6 7 6 7 0 6 7 4 5
PLOS ONE | DOI:10.1371/journal.pone.0163545 September 26, 2016
T
ðK1 C1 Þ
ðB1 C2 Þ
B1 C2
0
K1 C1 K2 C2 2 3 0 6 7 6 7 607 6 7 6 7 6 7 6 7 607 6 7 6 7 6 7 6 7 þ 6 0 7F ½ B2 K1 C1 B2 K2 C2 6 7 6 7 6 7 6D7 6 7 6 7 6 7 6 7 607 6 7 4 5 0
T
ðA2 þ ðB21 þ DFB2 ÞK2 C2 Þ
0
S
0
ððB21 þ DFB2 ÞK1 C1 Þ
0
0
0
K1 C1 2
0
0
0 0
DT
0 0
0
11 / 24
Design of Distributed Engine Control Systems with Uncertain Delay
Using Lemma 2, a constant λ > 0 exists to satisfy:
2
Γ
C1T C1
P
0
6 6 6 0 6 6 6 6 6 0 6 6 6 6 ¼6 0 6 6 6 6 B KC 6 21 1 1 6 6 6 6 A1 6 4
Q 0
0 0 S
ðB21 K1 C1 Þ
0
T
AT1 T
0
ðA2 þ ðB21 K2 C2 Þ
0
BT22
0
g2 I
BT3
0
ðB1 C2 Þ
0
0
A2 þ B21 K2 C2
B22
B3
Q
B1 C2
0
0
0
P
K1 C1 K2 C2 0 0 2 T 3 ðB2 K1 C1 Þ 6 7 6 7 6 T 7 6 ðB2 K2 C2 Þ 7 6 7 6 7 6 7 T 6 7 B 2 6 7 6 7 6 7 6 7 þ l 16 7½ B2 K1 C1 B2 K2 C2 0 6 7 6 7 6 7 6 7 0 6 7 6 7 6 7 6 7 6 7 0 6 7 4 5
0
0
B2
0
0
1
ðK1 C1 Þ
0 1
T
T
3
2
0
3
7 6 7 7 6 7 7 607 ðK2 C2 Þ 7 6 7 7 6 7 7 6 7 7 6 7 7 607 0 7 6 7 7 6 7 7 6 7 7 6 7 7 þ l 6 0 7½ 0 0 7 6 7 7 6 7 7 6 7 7 6D7 0 7 6 7 7 6 7 7 6 7 7 6 7 7 607 0 7 6 7 5 4 5 1 0 S T
0
0
0 DT
0
0
0 0 < 0
0
PLOS ONE | DOI:10.1371/journal.pone.0163545 September 26, 2016
12 / 24
Design of Distributed Engine Control Systems with Uncertain Delay
Therefore, J 0. Using Lemma 1, 2
Γ
C1T C1
P
6 6 6 6 0 6 6 6 6 0 6 6 6 6 6 0 6 ¼6 6 6 B KC 6 21 1 1 6 6 6 6 A1 6 6 6 6 6 K1 C1 6 4 B2 K1 C1 2
P
6 6 6 0 6 6 6 6 6 0 6 6 6 6 0 6 6 6 6 ¼6 6 B21 K1 C1 6 6 6 6 A1 6 6 6 6 KC 6 1 1 6 6 6 6 B2 K1 C1 6 4 C1
0
0
Q
0
0 S
0
ðB21 K1 C1 Þ
T
AT1 T
ðB1 C2 Þ
T
ðK1 C1 Þ
T
ðK2 C2 Þ
T
0
ðA2 þ ðB21 K2 C2 Þ
0
BT22
0
0
g2 I
BT3
0
0
0
0
0
0
A2 þ B21 K2 C2
B22
B3
B1 C2
0
0
0
P
K2 C2
0
0
0
0
S
B2
0
0
0
0
B2 K2 C2 0 Q 0
0 0 S
0
lDDT
Q
ðB21 K1 C1 Þ
1
T
AT1 T
ðB1 C2 Þ
T
1
0
ðK1 C1 Þ
T
ðK2 C2 Þ
T
1
ðB2 K1 C1 Þ
7 7 7 ðB2 K2 C2 Þ 7 7 7 7 T B2 7 7 7 7 7 7 0 7 7 7 7 0 7 7 7 7 7 0 7 7 7 7 0 7 7 5 lI ðB2 K1 C1 Þ
T
ðB2 K2 C2 Þ
T
C1T
0
ðA2 þ B21 K2 C2 Þ
0
BT22
0
0
BT2
0
g2 I
BT3
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
A2 þ B21 K2 C2
B22
B3
B1 C2
0
0
0
P
K2 C2
0
0
0
0
S
B2
0
0
0
0
lI
0
0
0
0
0
0
0
3
T
0
B2 K2 C2
T
lDDT
Q
1
1
1
0
3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
