Abstract

A novel estimation-based and dropout-dependent control design for distributed control systems of aeroengines with packet dropout is proposed. The packet dropout is described as an independent identically distributed (i.i.d.) Bernoulli process with known probability. The objective of the control system is to effectively and stably impel aeroengines. An inverse system is proposed to reconstruct the missing measurements, based on which, a hybrid adaptive Kalman filter (HAKF) is proposed to provide the estimated states for control design. A new state-feedback control design is developed with the estimated states and the actual states. For guaranteeing the stability of aeroengines, both the stability of HAKF and the new controlled system are analyzed and proved. The new design is applied to an aeroengine, HAKF has superior estimation accuracy which ensures the effectiveness, and the desired stability performance is achieved by controlled system in the presence of packet dropout.

1. Introduction

As propulsion, aeroengines are a typical kind of nonlinear system with high-performance requirements. In order to meet the requirements of engine performance and efficiency, it is warranted to ameliorate the control architecture in terms of commonality, expandability, flexibility, and reduced weight [1]. At present, a very effective solution is to convert the engine centralized control system into a distributed control system (DCS). Despite the performance superiority, the distributed control system may cause packet dropout due to the characteristics of network transmission [2]. The dropout of data packets in the control system not only undermines the quality of control but also destroys the stability of the control system. It is noted that the packet dropout will lead to some technical difficulties in the stability-guaranteed control design with packet dropout. Hence, the problem of packet dropout has attracted the attention of many scholars.

Generally, the packet dropout is modelled as a random process with random Bernoulli distribution [3, 4] or time-homogeneous binary Markov distribution [5, 6] in the previous research. Both these two descriptions define the arrival of packet at instant as a binary random variable . In the Bernoulli process, the variable was modelled with probability distribution: where the variable is the packet dropout rate. In the Markov process, the arrival of a packet is featured by a transition probability matrix: where is the state space of the Markov process and the positive parameters and are the failure rate and recovery rate, respectively. Based on these mathematical descriptions of the packet dropout process, the compensation or estimation for the lost information is developed simultaneously.

Considering the information compensation for packet dropout, researchers have proposed numerous and outstanding results of estimation approaches with intermittent observations. Among these approaches, the Kalman filter (KF) has received a great of attention because it is a minimum mean-square error (MMSE) estimation under the packet dropout conditions. Sinopoli proposed the Kalman state estimator by introducing the random binary variable of a Bernoulli packet dropout into the algorithm iteration and proved the stability of the Kalman estimator [7]. Based on the Kalman iteration structure proposed in reference [7], a series of research on the stability of new Kalman filters is developed and published. Reference [8] introduces the peak covariance to describe the stability of the Kalman filter with the stochastic packet dropout and obtains the stability conditions. An improved stability condition for the Kalman filter with bounded packet dropout was presented in reference [9]; a less conservative and sufficient condition for peak-covariance stability is established by the linear matrix inequality (LMI) method. Furthermore, the peak-covariance stability of Kalman filters is proven to be the mean-square stability for a random packet dropout [10, 11]. These theories extend to the analysis of Kalman filters with complicated systems [12–16]. The design of Kalman filters with state-dependent packet dropout for the hybrid measurement system is investigated in reference [12]. Reference [13] proposed a specific estimation design with Markov packet dropout for nonuniform sampling network systems. Marelli analyzed the stability for Kalman filters with random measurement matrices, the necessary and sufficient conditions are given in reference [14]. An extension of the fruits of [14] is made in reference [15] to demonstrate the stability of multisensor Kalman filter over a lossy network. Similarly, the stability analysis of multisensor Kalman filter over lossy networks is also discussed in reference [16]. From all the above references, the Kalman filter supplies a solution for the packet dropout problem, which can compensate for the losing information effectively. In this way, the construction of an estimator can serve as the basis of information compensation with packet dropout.

Due to the control system performance affecting the safety of aeroengine directly, the stability is the primary and most important requirement of the controller design for aeroengines. To guarantee the safety of aeroengine, efforts should be devoted to the controller stability analysis with packet dropout. In the stability with network failure literature, the classical approach is to consider network failure as a disturbance and utilize the Lyapunov theorem to analyze the controller stability [17–21], and then, these authors take advantage of the controller robustness to cover the poor effect. Moreover, the other arguably popular problem is the controller design with unmeasurable states and packet dropout. To identify the unmeasurable states, observers are constructed and the observer-based controllers come after [22–24]. Similarly, the observer-based control system framework is also used for the packet dropout and the relative system stability analysis is researched. Wang et al. designed an observer for the unmeasurable states and achieve the exponentially mean-square stability condition for the observer-based control system [25]. Li et al. proposed a stability analysis approach for the network nonlinear system with global Lipschitz nonlinearities and random packet dropout [26]. Similarly, the interesting publications of the observer-based control system with packet dropout and other problems (such as consecutive packet delays, multiplicative noises, and uncertainties) are investigated in references [19–21, 27]. Furthermore, some new control designs for the system with network failure are proposed [28–30]. The novel control design is attracted to the event-triggered mechanism, which can guarantee the control system stability and reduce the network burden. These studies supply a new direction for solving the network failure problems by the codesign of the control and scheduling approach.

The abovementioned outstanding researches mainly focus on the control design for the system with network failure, robust control, observer-based control, and codesign of control and scheduling approaches are investigated. Actually, the network failure in aeroengine DCS would cause the loss of controller inputs which is the crucial factor for the aeroengine control system. Hence, it is necessary to compensate for the missing controller inputs for the aeroengine. To guarantee the performances of aeroengine DCS, the packet dropout for the specific aeroengine DCS is handled by a separated design framework which consists of the estimation and estimation-based control design, which is quite different from the above research in theory [17–21, 27–30].

The traditional Kalman filters are widely used for estimation. And numerous researches on the design and stability analysis of the Kalman filter with packet dropout were carried out. It is noted that the Kalman structure in the above references gives up the measurement update, an important correct step, due to the measurement dropout. And this way would directly decrease the estimation accuracy. Besides, the stability analysis of both the Kalman filters and the control system are separated; there is a gap in the connection between information compensation and control system stability. Considering aeroengines are specific time-varying, a complex controlled plant with strong nonlinearity requires the high control performance and quality. It is necessary to combine the information compensation and control system stability analysis for the high requirements of the aeroengines.

For the purpose to shorten the gap mentioned above, unlike the popular state estimation methods (traditional KF, see, e.g., references [7, 8]), we proposed a hybrid adaptive Kalman filter (HAKF), an approach with hybrid measurement correct based on the Kalman structure, to increase the estimation accuracy with packet dropout. The main difference between HAKF and the above Kalman filters is that there is a measurement reconstruction step in ours which improves the accuracy. Moreover, the stability of the aeroengine control system with HAKF which incorporates information compensation into the control system stability is investigated. The corresponding stability conditions of this control system were presented in the formulation of linear matrix inequalities (LMIs).

The salient novelties and contributions in this paper are as follows: (1)The new approach deals with packet dropout and the control design separately, which significantly improves the flexibility of the control system design and boosts the performance of the control system. In theory, the novel estimator effectively compensates the missing states, and the controller guarantees the desired stability performance(2)The inverse system combined with references reconstructs the missing measurable information. The reconstructed information serves as the basis of the measurement update in state estimation and improves the accuracy of the new estimation(3)By taking advantage of the reconstructed information, the measurement, and the packet dropout flag , the HAKF improves the estimation accuracy obviously, which is distinctly different from the traditional Kalman filters [7, 8]. Moreover, the stability of the HAKF is guaranteed in theory(4)The proposed control design employs the hybrid information, namely, actual and estimated information. Furthermore, by a new mathematical expectation-based Lyapunov function, the stability of this hybrid control system is guaranteed(5)The new approach is applied to a turbofan engine. The results demonstrate that the HAKF decreases the RMSEs of the estimation by over 57% compared to the traditional Kalman filters, and the control system has superior performances with the proposed control

This paper is organized as follows. In Section 2, the mathematical description of the aeroengine distributed control system with packet dropout is formulated. Section 3 proposes the design and stability analysis of the HAKF. Section 4 presents the dropout-dependent stability analysis of the hybrid signal closed-loop control system. In Section 5, the numerical examples are given to illustrate the comparison of novel HAKF and the traditional filter in references [7, 8]; besides, the stability simulations on the aeroengine are also described. Some conclusions of this paper are drawn finally.

2. Problem Formulation

The aeroengine system is described as where and are the state and control inputs, is the output, is the measurement, and and are the system and measurement noises. The matrices , , , , and are known. The noises and are Gaussian noise with

To improve the aeroengine controller performance and design flexibility, an estimation-based control framework for aeroengine DCS is shown in Figure 1. Data packets are transmitted from sensors to the controller and then to the actuators via a shared network in the DCS. Inevitably, a sophisticated packet may dropout in the data transmission.

Assumption 1. For the aeroengine DCS with stochastic packet dropout, the following assumptions are introduced: (1)Sensors are time-triggered, and the controller is event-triggered(2)The sensor data is sent following a sequence of sampling(3)The initial states transmitted from the controller to the actuator are consistent at the initial moment(4)The actuators take advantage of the data received in the latest previous sample when the packet dropout

Remark 2. The data packets may dropout in the transmission from sensors to controller or/and from the controller to actuators. Assumption 1 provides a strategy to compensate for the lost information between the controller and actuators based on the time-triggered network transmission protocol. While the packet dropout between the sensors and controller, it would cause the performance deterioration of the controlled system (3). Hence, this paper focuses on compensation and control with problem of the missing data packet.

Figure 1 

Framework of aeroengine DCS with estimation-based control approach.

Due to the uncertainties of the external environment, network characteristics, etc., the packet dropout is commonly designed as a random process. The packet dropout in the network channel is considered a statistic random process with multicycle information missing because of the stochastic and disordered uncertainties. Define a random variable to describe whether the controller receives the observations at the sample instant [31–33], and it is formulated as follows:

Here, is a binary random variable subject to independent identically distributed (i.i.d.) process. The variable indicates that the controller accepts the sensor signals successfully and there are no data dropout at instant . Using demonstrates that the controller fails to accept the sensor signals and the packet lost. Therefore, the network transmission process in an aeroengine DCS can be shown as Figure 2, in which is actual information that the controller can receive at instant .

Figure 2 

Network transmission process.

From (7), when , the information for controller was lost, which immediately destroy the controller quality and performance. Considering the information compensation, an estimator is designed as a guard for the controller and the novel controller framework with signal transmission is shown in Figure 3.

Figure 3 

Network transmission with estimator.

3. Hybrid Adaptive Kalman Filter

3.1. Structure of the HAKF

In view of the system (3) with packet dropout, we propose the hybrid adaptive Kalman filter (HAKF) to compensate for the missing information.

The HAKF time update is and the posteriori estimated state is where is the estimated state at instant , and are the priori and posteriori estimated states, and and are the priori and posteriori estimation errors with

and are the priori estimation error covariance and the posteriori estimation error covariance, is the gain of HAKF, and is the hybrid measurement.

Remark 3. The HAKF time update (8) and (9) supply the priori estimated state, and then, the correct module (10) was designed to update the priori estimated state. In view of the time update, equations (8) and (9) are just model based which are irrelevant to the measurements; the priori estimated state and priori error covariance matrix of HAKF are regular as the LKF as shown in [34].

Remark 4. According to the LKF theory, the adjustment of the KF estimation state was motivated by the measurement errors . Equation (7) shows that no measurement is included when a packet dropout and the traditional KF has no correct module to update the priori estimation [7, 8]. When a continuous package dropout occurs, the accuracy of the KF decreases significantly. In practice, the packet dropout comes up occasionally, which means the controller can receive the measurements when the network works. Hence, the HAKF adaptively adopts measurements or reconstructed measurements when the network works or package dropout occurs.

Remark 5. The key and primary difficulty is to design an approach to reconstruct the missing measurements for the HAKF. In Section 3.2, this difficult is addressed by the measurement reconstruction with an inverse system.

3.2. Measurement Reconstruction

The meaningless measurements need to be reconstructed when , inspired by the intelligent inverse control [35], we reconstruct the measurement by combination of inverse system and references. The inverse system connects with a previous system in a series is shown in Figure 4, which can distinctly reveal the relationship between inverse system and the original system .

Figure 4 

Inverse system of an aeroengine system.

Considering system (3) at the instant , its input-output system is

Here, , , and . The subscript means one step after .

The inverse system of (13) is defined as here, ,, is the reference, and satisfying .

Remark 6. In an aeroengine closed-loop tracking control system such as Figure 1, the ideal system response performances are outputs tracking for the reference signals without steady errors, and this performance can be achieved by controller design, a tracking error-driven feedback approach [35]. Based on this characteristic, we reconstruct the sensor signals by the aeroengine inverse system combined with the references , and would be equal to the actual with the assumption of no tracking errors.

Based on the inverse system (14), we reconstruct the output as

Finally, the receiving measurements can be written as

Remark 7. By introducing the packet dropout variable , equation (16) merges the measurement and its reconstructed value to . When the network transmission works, , while the packet dropout occurs, , which can effectively rectify the abandoned correct module in references [7, 8].

It is noted that the HAKF gain in (10) will affect the posteriori estimation error defined in (12). So, in the next subsection, we present the optimization of to minimize the error .

3.3. Design of HAKF Gain Matrix

Define the performance function as here, is the ith entry of . Therefore, the performance function can be regarded as where is the trajectory of a matrix [34, 36].

Introducing equations (10) and (16) into (12), it yields where is the residual of HAKF with

By formulas (3) and (15) and Remark 6, it follows:

By formula (21), the error in (19) can be rewritten as and then, in equation (11) is

Since the measurement noises are uncorrelated to the priori estimated error [37], equation (23) can be simplified as

By equations (5) and (9), it follows

To solve (18), the partial differential to is

Let [38, 39], and it yields

Finally, considering system (3) with the reconstructed measurements (16), by equations (8)–(10), (25), and (27), the HAKF state estimation approach is summarized as

3.4. HAKF Design Procedure

The design procedure of HAKF is summarized as follows, and the iteration steps of the new estimation approach are shown in Figure 5.

Step 1. Set up the initial state values for and the noise and system matrix and

Step 2. Calculate the prior estimated state and prior estimation error covariance at instant based on (28) and (29) with the state variable and input variable at instant

Step 3. Determine the situation of network channel and reconstruct the measurement signals based on (15) and (16) by the binary variable

Step 4. Calculate the Kalman gain based on (32)

Step 5. Update the posterior state with (30) by measurement information and calculate the posterior estimation error covariance based on (31)

Step 6. Output , and update the time instant.

Figure 5 

Iteration steps of HAKF.

3.5. Stability Analysis of HAKF with Packet Dropout

The proposed HAKF (28)–(32) takes advantage of the reconstructed measurement (16), which results in the additional estimated deviation. In this subsection, we present the following main result to investigate the stability of the new estimation algorithm.

Theorem 8. Consider the system (3) with packet dropout (4). If the matrices and , the HAKF (28)–(32) is stable.

Proof. Consider the Lyapunov function candidate for any arbitrary , . Let By (3), (10), and (12), we have Define By formulas (34)–(36), it follows Since the noise always exists, cannot decay to 0 at the end; the best hope is the homogeneous part of (35) must decay with time. Considering the dynamics of (35) is explicitly irrelevant to the noises [40, 41], the stability condition (37) can be simplified as Define Premultiply and postmultiply equation (39) by and , and we have by introducing equations (9) and (25) into (41), we have As . is a binary random variable subject to the i.i.d. process, and , and . Hence, the Lyapunov function is negative.
Theorem 8 is proved.☐

Remark 9. Because the package dropout rate is not 100%, the binary random is 1 with the probability of . Hence, and are negative with the same probability and will damp with the time.

Remark 10. Taking the advantage of , the HAKF renders the state estimation to adapt to the network environment, which characterize the new approach distinctly different from the LKF.

Considering the existence estimation error of the HAKF algorithm and for the purpose of improving system performance, two states are in the aeroengine closed-loop control system. First, when the controller received the measurement information successfully, the actual system states are contributed to design the controller. Otherwise, the inputs of the controller are estimated state by the HAKF with measurement reconstruction. It is worth noting that the state estimator iteratively computes at each sample time, while the inputs of controller vary with the random variable .

4. Dropout-Dependent Stability Analysis for Control System

In Section 3, the stability of HAKF is discussed. In this section, we further investigate the dropout-dependent stability of this controlled system.

We consider the controlled system (3) with the state estimation of HAKF. As shown in Figure 1, to reduce the influence of the estimation error, the controller receives the actual state when and utilizes the estimated state while . Therefore, the closed-loop control system (3) with hybrid signals can be described as where the system hybrid state and input are

where is the control variable calculated by and and are matrices with appropriate dimension. State equation (43) simply reveals the relationship of state at time and , but it is worth paying attention that, because of the estimated state in the system (43), the system process noise variable is not as same as in (3).

Remark 11. Since the two states switch in the system (43) at each sample instant , there are challenges to the stability of the control design. The next stability analysis will respond to the challenges.

In system (43), the four cases of state at instant come after.

Case 1. The system information is missing at both instants and , and there is

Case 2. The system information was sent successfully at instant while it is missing at instant . There is

Case 3. The system information is missing at instant , while it was sent successfully at instant . There is

Case 4. The system information is sent successfully at both instants and , there is

Considering these four cases, the state in (44) can be further written as

The matrices and the noise in the system (43) can be explicated as