H [infinity] Networked Cascade Control System Design for Turboshaft Engines with Random Packet Dropouts

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 with [subscript]H∞[/subscript] 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.

[ProQuest: [...] denotes non US-ASCII text; see PDF]

Xiaofeng Liu 1 and Xu Sun 1

Academic Editor:Wen Bao

School of Transportation Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing, China

Received 1 December 2016; Accepted 29 January 2017; 16 February 2017

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 of _H∞ 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 ([low *]) 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. Figure 1 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 omitted; refer to PDF]

Figure 2: Block diagram of the simplified linearized gas generator and rotor system.

[figure omitted; refer to PDF]

Control laws essentially work to maintain _NP , 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.

[figure omitted; refer to PDF]

Primary Plant . The state-space representation of the rotor system is provided by the following equation [9, 23]: [figure omitted; refer to PDF]

Secondary Plant . The continuous-time linear model of the gas generator is shown as follows: [figure omitted; refer to PDF] 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 period h (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 delay _τk , the controllers are event-driven, the actuator is time-driven, and the engine receives the piece-wised control input is given by [figure omitted; refer to PDF] 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] [figure omitted; refer to PDF] 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]: [figure omitted; refer to PDF] where α is a positive scalar, E[λ(k)-α]=0, and E[(λ(k)-α⁾² ]=α(1-α).

Considering the system reference input _NPr =0, the static state feedback controller is utilized by a discrete-time form: [figure omitted; refer to PDF] where _K2 (α) is the probability-dependent (packet dropouts probability) state feedback gain matrix given by _K2 (α)=_K21 +α_K22 and _K1 is [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] where _A1 =^{e_A1 h} , _B1 =^{∫0he_A1 s} ds_B1 , and [figure omitted; refer to PDF] where _A2 =^{e_A2 h} , _B21 =^{∫h-_τk he_A2 s} ds_B2 , _B22 =^{∫0h-_τke_A2 s} ds_B2 , and _B3 =^{∫0he_A2 s} ds_B3 . Then, considering the random packet loss, combined (5) and (7) with (10), (10) becomes [figure omitted; refer to PDF]

Thus, the discretized system can be further expanded as [figure omitted; refer to PDF]

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 by _y1 (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 [figure omitted; refer to PDF] where A(α), B(α), C, and D can be seen in [figure omitted; refer to PDF]

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] [figure omitted; refer to PDF] for all nonzero w(k), where γ>0 is a prescribed scalar.

3.2. Controller Design

Lemma 1 (Schur complement).

Given constant matrices _Ω1 , _Ω2 , and _Ω3 , where _Ω1 =^Ω1T and _Ω2 =^Ω2T >0, then _Ω1 +^Ω3TΩ2-1_Ω3 <0 if and only if [figure omitted; refer to PDF]

Theorem 2.

Let the positive scalar α 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: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and the state feedback gain matrices can be gained by [figure omitted; refer to PDF]

Proof.

In order to conclude the controller design conditions, the following Lyapunov function can be defined: [figure omitted; refer to PDF]

Now, for any nonzero w(k), [figure omitted; refer to PDF] where O(α) can be seen in [figure omitted; refer to PDF] and O(α) can be rewritten in [figure omitted; refer to PDF] Thus, by applying Lemma 1, (24) can be obtained [figure omitted; refer to PDF]

Now, the goal is to prove O(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: [figure omitted; refer to PDF]

If (25) holds, then by using P-2R≥-^RTP-1 R, the following inequality (26) can be gained: [figure omitted; refer to PDF]

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: [figure omitted; refer to PDF] 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 Figure 2.

(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 parameters _K1 , _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: [figure omitted; refer to PDF]

The coefficients after the discretization are provided as follows: [figure omitted; refer to PDF]

Simulation 1.

Let the packet loss probability value α=0.3, and given the initial conditions as _x1 (0)=^(10.20.2)T , _x2 (0)=^{(0.90000.41890.78430.64981.0000)T} , the simulation time is T=20 s, sampling time is h=0.01 s, and assuming that the two network-induced delays are both equivalent to _τk , which is not longer than the sampling period, _τk =0.005 s. 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: [figure omitted; refer to PDF]

Figures 4 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 response _x1 .

[figure omitted; refer to PDF]

Figure 5: Gas generator state response _x2 .

[figure omitted; refer to PDF]

Figure 6: Rotor system controller output u.

[figure omitted; refer to PDF]

Simulation 2.

In order to show the effectiveness evaluation of the proposed method under different values of packet loss probability α, 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.

Figures 7 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: Optimized _γopt 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 state _x1 under different α.

[figure omitted; refer to PDF]

Figure 8: Gas generator state _x2 under different α.

[figure omitted; refer to PDF]

Figure 9: Rotor system controller output _u1 under different α.

[figure omitted; refer to PDF]

Figure 10: Gas generator controller output _u2 under different α.

[figure omitted; refer to PDF]

Figure 11: Gas generator controller output _u2 under different α (partial view).

[figure omitted; refer to PDF]

5. Conclusions

This study considered the novel robust _H∞ 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.

Notations

[figure omitted; refer to PDF] :

Power lever angle (throttle)

[figure omitted; refer to PDF] :

Gas generator speed

[figure omitted; refer to PDF] :

Power turbine speed

[figure omitted; refer to PDF] :

Main rotor blade velocity

[figure omitted; refer to PDF] :

Rotor torque state

[figure omitted; refer to PDF] :

Engine shaft torque

[figure omitted; refer to PDF] :

Collective pitch

[figure omitted; refer to PDF] :

Inlet pressure

[figure omitted; refer to PDF] :

Static pressure at Station 3

[figure omitted; refer to PDF] :

Inlet temperature

[figure omitted; refer to PDF] :

Interturbine gas temperature

[figure omitted; refer to PDF] :

Fuel flow

[figure omitted; refer to PDF] :

Power turbine inertia

[figure omitted; refer to PDF] :

Lumped power turbine/dynamometer inertia

[figure omitted; refer to PDF] :

Main rotor blade inertia

[figure omitted; refer to PDF] :

Stiffness of the centrifugal restoring springs

[figure omitted; refer to PDF] :

Lag hinge damping

[figure omitted; refer to PDF] :

Aero damping

[figure omitted; refer to PDF] :

Reference input

[figure omitted; refer to PDF] :

Model state vector

[figure omitted; refer to PDF] :

Model output vector

[figure omitted; refer to PDF] :

Model input vector

[figure omitted; refer to PDF] :

Partial derivatives

[figure omitted; refer to PDF] :

Process white noise.

Acknowledgments

The authors would like to thank Professor Ming Cao, Dr. Xiaodong Chen, and Dr. Qingkai Yang from University of Groningen for their thoughtful remarks that improved the presentation of this paper. This work was supported by National Natural Science Foundation of China (NSFC) [Grant nos. 61573035 and 61104146] and the China Scholarship Council (CSC) [Grant no. 201506025135].

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