Information Sciences 235 (2013) 45–54
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Data-driven predictive control for networked control systems Yuanqing Xia ⇑, Wen Xie, Bo Liu, Xiaoyun Wang School of Automation, Beijing Institute of Technology, Beijing 100081, China Key Laboratory of Intelligent Control and Decision of Complex Systems, Beijing 100081, China
a r t i c l e
i n f o
Article history: Available online 16 February 2012 Keywords: Networked control system Data-driven predictive control Random network delay
a b s t r a c t This paper is concerned with the problem of data-driven predictive control for networked control systems (NCSs), which is designed by applying the subspace matrices technique, obtained directly from the input/output data transferred from networks. The networked predictive control consists of the control prediction generator and network delay compensator. The control prediction generator provides a set of future control predictions to make the closed-loop system achieve the desired control performance and the network delay compensator eliminates the effects of the network transmission delay. The effectiveness and superiority of the proposed method is demonstrated in simulation as well as experiment study. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction In the last decade, network technology has dramatically been developed. Recently, more and more network technologies have been applied to control systems [21,18,12,24,13,22]. This kind of control systems in which a control loop is closed via communication channel is called networked control systems. Now, networked control is a new area in control systems [4,11,15,16,9]. Particularly, Internet based control systems allow remote monitoring and adjustment of plants over the Internet, which makes the control systems benefit from the ways of retrieving data and reacting to plant fluctuations from anywhere around the world at any time. The networked control has also opened up a complete new range of real-world applications, namely tele-manufacturing, tele-surgery, museum guidance, traffic control, space exploration, disaster rescue, and health care. In networked control system (NCS), the plant, controller, sensor, actuator and reference command are connected through a network. Most attention in this area has been paid to the design and analysis of NCSs [10,25]. The stability problem of closed-loop NCS in the presence of network delays and data packet drops has been addressed in [23,19,20]. In [1,2], the quantized feedback control and H1 output tracking control are analyzed respectively. To reduce the network traffic load, a sampled-data NCS scheme has been presented and some necessary and sufficient conditions for global exponential stability of the closedloop systems via state/output feedback, without/with network delays have been established in [7]. The random network delays in the controller to actuator channel in NCSs have been studied in [5] and the fixed network delay and the random network delays in both forward and feedback channels have been considered in paper [6], but the random network delays are not in the form of a Markov process. In [8,17], the problems of stochastic stability of networked control systems with random time-delays have been discussed, in which the random time-delays are modeled as a Markov process. In recent years, the techniques of Internet of Things are developed rapidly, the research of NCSs plays a key role in Internet of Things. The Internet of Things will draw on the functionality offered by all of these technologies to realize the vision of
⇑ Corresponding author at: School of Automation, Beijing Institute of Technology, Beijing 100081, China. E-mail addresses:
[email protected] (Y. Xia),
[email protected] (W. Xie),
[email protected] (B. Liu),
[email protected] (X. Wang). 0020-0255/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2012.01.047
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Y. Xia et al. / Information Sciences 235 (2013) 45–54
Fig. 1. Internet of things.
a fully interactive and responsive network environment. Fig. 1 shows the future applications of the Internet of Things. During the research of Internet of Things, we find data collection and processing are very important. First, it is difficult even impossible for us to get the accurate physical models of every objects in Internet of Things. The only way we know the objects in Internet of Things is data we can get. Secondly, due to the sensor technology, we can detect changes in the physical status of things. Thus, mass data about things are collected and stored. Thanks to the advances in computer science, especially in the aspects of computing ability and storage together with high quality and reliable measurements from process instruments, make it possible to collect and process the data efficiently. Finally, all the objects and devices are connected to large databases and networks—and indeed to the network of networks (the Internet). Information and commands are transmitted through network. Under traditional design frameworks, the data from the plant are used to build the model, since dynamic models are the prerequisite of control and monitoring. Once the design of a controller or a monitor is completed, the model often ceases to exist. However, the use of models also introduces unavoidable modeling error and complexity in building the model. With emergence of the Internet of Things, mass data have to be processed. The data-driven subspace approach has been proposed for industrial process control field. The data-driven control method also can be developed for complex systems. Especially, data-driven method is particularly suitable for NCSs since only digital data can be transferred through network and received by controller and actuator. However, the use of the network will lead to intermittent losses or delays of the communicated information. These losses could deteriorate the performance and may even cause the system to go unstable. Thus, in this paper, a data-driven predictive networked controller is proposed to solve these problems without modeling. The paper is organized as follows. In Section 2, a model based networked control system is introduced briefly. Section 3 describes the method of data-driven predictive control. In Section 4, the data-driven predictive networked control scheme is designed to the ball and beam system. The simulation and experiment examples show the effectiveness and superiority of the proposed scheme, respectively. Conclusions are provided in Section 5.
2. Brief introduction of the model based networked control systems Consider a discrete, linear time-invariant (LTI) dynamic system S with unknown process disturbances and measurement noises, which is described in state-space form as:
xðk þ 1Þ ¼ AxðkÞ þ BuðkÞ þ wðkÞ
ð1Þ
yðkÞ ¼ CxðkÞ þ DuðkÞ þ v ðkÞ
ð2Þ
where x(k) 2 Rn is the system state, u(k) 2 Rl is the system input, y(k) 2 Rm is the system output, w(k) N(0, Q) are the unknown process disturbances and v(k) N(0, R) are the unknown measurement noises; A, B, C, D are constant system matrices of suitable finite dimensions.
Y. Xia et al. / Information Sciences 235 (2013) 45–54
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Fig. 2. Model based networked control systems.
The feedback control scheme of system (1) and (2) developed in the current work can be shown in Fig. 2. Here, a buffer is set at the controller node to satisfy that the measurements are processed in sequence. Kalman filter (KF) is adopted to estimate the state and then produce the predictive states with finite horizon N:
^xðkjkÞ ¼ KFðS; u ^ ðk 1jk 1Þ; yðkÞÞ ^xðk þ ijkÞ ¼ KFðS; u ^ ðkjkÞ; yðkÞÞ; i ¼ 1; 2; . . . ; N
ð3Þ
^ ðk þ ijkÞ ¼ Kðk þ iÞ^xðk þ ijkÞ; u
ð5Þ
ð4Þ
i ¼ 1; 2; . . . ; N
where KF represents the compact form of Kalman filter expression and K(k + i) is time-varying Kalman filter gain [16]. To overcome unknown network transmission delays, a networked predictive control scheme is proposed. It mainly consists of a control prediction generator and a network delay compensator. The control prediction generator is designed to generate a set of future control predictions. The network delay compensator is used to compensate for the unknown random network delays. This paper considers the case where the network delays are in the forward (from controller to actuator) channel (CAC) and feedback (from sensor to controller) channel (SCC). A very important characteristic of the network is that it can transmit a set of data at the same time. Thus, it is assumed that predictive control sequence at time k is packed and sent to the plant side through a network. The network delay compensator chooses the latest control value from the control prediction sequences available on the plant side. For example, when there is no time delay in the channel from sensor to controller and the time delay from controller to actuator is ki, if the following predictive control sequences are received on the plant side:
h h .. . h
uTtk1 jtk1 ; uTtk1 þ1jtk1 ; . . . ; uTtjtk1 ; . . . ; uTtþNk1 jtk1 uTtk2 jtk2 ; uTtk2 þ1jtk2 ; . . . ; uTtjtk2 ; . . . ; uTtþNk2 jtk2
uTtkt jtkt ; uTtkt þ1jtkt ; . . . ; uTtjtkt ; . . . ; uTtþNkt jtkt
iT iT ð6Þ
iT
where the control values utjtki for i = 1, 2, . . . , t, are available to be chosen as the control input of the plant at time t, the output of the network delay compensator, i.e., the input to the actuator will be
ut ¼ utjtminfk1 ;k2 ;...;kt g
ð7Þ
In fact, by using the networked predictive control scheme presented in this section, the control performance of the closedloop system with network delay is very similar to that of the closed-loop system without network delay. The controller sends packets to the plant node:
fuðk þ ijkÞj i ¼ 0; 1; . . . ; Ng
ð8Þ
At time instant k, the actuator chooses a preferable control signal as the actual input of the controlled dynamic system:
uðkÞ ¼ uðkjk iÞ where i ¼ arg mini fuðkjk iÞg is available. The stability proof and idea can be found in [15,16].
ð9Þ
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3. Data driven predictive control In this section, we present a data driven predictive control scheme, which is proposed in [3]. Suppose the measurements of the inputs and the outputs u(n), y(n) for n 2 {1, 2, . . . , 2N + j 1} are available. The data block Hankel matrices for u(n), represented as Up and Uf, with N-block rows and j-block columns are defined as
0 B B Up ¼ B @
uð0Þ
uð1Þ
uðj 1Þ
uð1Þ
uð2Þ
uðjÞ
1 C C C A
ð10Þ
uðN 1Þ uðNÞ uðN þ j 2Þ 0
uðNÞ
B uðN þ 1Þ B Uf ¼ B @ uð2N 1Þ
uðN þ 1Þ
uðN þ j 1Þ
1
uðN þ 2Þ
uðN þ jÞ
C C C A
uð2NÞ
ð11Þ
uð2N þ j 2Þ
Each block element in the above data Hankel matrices is a column vector of inputs. Similar data Hankel matrices for y(n), represented as Yp and Yf, can be written. The past and future state sequences are defined as
X p ¼ ð xð0Þ xð1Þ xðj 1Þ Þ
ð12Þ
X f ¼ ð xðNÞ xðN þ 1Þ xðN þ j 1Þ Þ
ð13Þ
Following the above subspace notations, data column vectors over a predictive horizon for the output and input, respectively, are similarly defined as
2
3 yðkÞ 6 7 .. 6 7 . 7; yf ¼ 6 6 7 4 yðk þ N 2Þ 5
2
3 uðkÞ 6 7 .. 6 7 . 7 uf ¼ 6 6 7 4 uðk þ N 2Þ 5
yðk þ N 1Þ 2
wp1
h wp ¼ wTp1
uðk þ N 1Þ
3
yðk NÞ 6 7 .. 6 7; ¼4 . 5 yðk 1Þ wTp2
ð14Þ
2
wp2
3 uðk NÞ 6 7 .. 7 ¼6 . 4 5 uðk 1Þ
iT
ð15Þ
ð16Þ
Using regression analysis approach, we have
yf ¼ Lw wp þ Lu uf þ Le ef It follows that yf consists of predicted output over a horizon. Thus
y ¼ yf Similarly
u ¼ uf Considering ef consists of white noise, the optimal prediction is
^f ¼ Lw wp þ Lu uf y Thus
^¼y ^f y ^ f can be obtained by solving a least squares problem: The prediction Y
W p 2 min Y ðL L Þ w u f Lw ;Lu U f F where Yf, Uf, Wp are defined in [3].
ð17Þ
Y. Xia et al. / Information Sciences 235 (2013) 45–54
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The solution is given by
ðLw Lu Þ ¼ Y f
Wp Uf
y
1 W p ðW p U f Þ ¼ Y f W Tp U Tf Uf
ð18Þ
where represents the Moore–Penrose pseudo-inverse. Substitute to the MPC objective function
^f ÞT ðr f y ^f Þ þ uTf ðkIÞuf J ¼ ðr f y By taking derivative, the optimal future control can be computed as
uf ¼ ðkI þ LTu Lu Þ1 LTu ðr f Lw wp Þ Note that this procedure only involves the projection step of subspace methods. No model has been extracted or used to derive this predictive controller. In other words, the data-driven predictive controller is obtained directly from subspaces built by measured data, skipping the procedure of modeling. 4. Data driven networked control systems design In this section, data-driven networked control systems are introduced. The subspace projection method is applied to generate the predictive control signals. The only difference between data-driven networked control systems and model-based control systems is the controller, which is shown in Fig. 3. When we get data including past input and past output from the sensors over network, we apply the data-driven predictive control algorithm described in Section 3. Thus, we can obtain a sequence of predictive control inputs, which will be
Fig. 3. Data driven networked control systems.
Fig. 4. Ball and beam system.
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Y. Xia et al. / Information Sciences 235 (2013) 45–54 Table 1 Symbol description. J Jb M R g
Moment of inertia of the beam Moment of inertia of the ball Mass of the ball Radius of the ball Acceleration of gravity
The network−induced delay in both forward and feedback channel 6 Time delay in the forward channel 4
2
0
0
50
100 time instant
150
200
8 Time delay in the feedback channel 6 4 2 0
0
50
100 time instant
150
200
Fig. 5. Simulation results: time-delay in feedback and forward channels.
transmitted to a buffer over network. The compensator will choose the right control input according to the predictive networked control scheme described in Section 2. From the description we can see that the proposed data driven networked control scheme is different from the model based networked control scheme in [14], by which the control input can be obtained directly without modeling. Next, the data-driven networked control scheme is applied to ball and beam system, as is shown in Fig. 4 and the variables used in the ball and beam system are listed in Table 1. By choosing the beam angle h and the ball position c as generalized position coordinates for the system, the Lagrangian equations of motion are:
þ M c€ þ Mg sin h Mch_ 2 R2 s ¼ Mc2 þ J þ Jb €h þ 2Mcc_ h_ þ Mg c cos h
0¼
Jb
ð19Þ ð20Þ
where s is the torque applied to the beam. 2 The system parameters used are M ¼ 0:11 kg; R ¼ 0:015 m; J b ¼ 9:9 106 kg m and g ¼ 9:81 m=s2 (thus, for simulation b ¼ 0:7143). Defining b ¼ M= J b =R2 þ M and changing the coordinates in the input space using the invertible nonlinear transformation
s ¼ 2Mcc_ h_ þ Mg c cos h þ Mc2 þ J þ Jb u
ð21Þ
_ T , then the system can be written in state-space form as to define a new input u, and let x ¼ ½x1 ; x2 ; x3 ; x4 T ¼ ½c; c_ ; h; h
3 2 3 3 2 0 x2 x_ 1 6 x_ 7 6 bðx x2 g sin x Þ 7 6 0 7 1 4 3 7 6 7 6 27 6 7 þ 6 7u 6 7¼6 5 405 4 x_ 3 5 4 x4 2
x_ 4
0
1
ð22Þ
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4.1. Numerical example Here, we modeled the ball and beam system in MATLAB and adopted the data-driven prediction networked control scheme to the simulative system. The time-delays are produced by the computer simulation as Fig. 5 and the output simulation results are shown as Fig. 6. The simulation results shows the validity of the proposed scheme. 4.2. Experiment results First, for the sake of comparison, we apply LQR predictive control strategy based on Kalman filtering to the actual networked ball and beam system. The algorithm is described as [14] and is apparently model based. In this experiment, the time-delays produced in feedback and forward network channels are as Fig. 7. Fig. 8 shows the ball position output in the Kalman-LQR predictive control experiment, which indicates that the ball position output becomes violent while the networked ball and beam system is influenced by the transmission delay. Thus, the system cannot be stable.
The outputs and the actual inputs of Ballbeam system 3 y(k) 2 1 0 −1
0
50
100 time instant
150
200
50 u(k)
0
−50
0
50
100 time instant
150
200
Fig. 6. Simulation results: position output and control input.
6
Time delay in the network
5 4 3 2 1 0 0
100
200
300
400
500 Time(s)
600
700
800
Fig. 7. Time-delay in the Kalman-LQR predictive networked control experiment.
900
1000
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Y. Xia et al. / Information Sciences 235 (2013) 45–54
0.4 0.35
Position of the ball(m)
0.3 0.25 0.2 0.15 0.1 0.05 0
0
100
200
300
400
500
600
700
800
900
1000
Time(s) Fig. 8. Position output in the Kalman-LQR predictive networked control experiment.
6
Time delay in the network
5
4
3
2
1
0 0
500
1000
1500
Time(s) Fig. 9. Time-delay in the data-driven predictive networked control experiment.
Next, we will adopt the proposed data-driven predictive control scheme into the networked ball and beam system. The detailed algorithm is described in Fig. 3, that is, data-driven predictive control algorithm described in Section 3 combined with predictive networked control scheme described in Section 2. The time-delays in feedback and forward channels are obtained as Fig. 9. The ball position output is obtained as Fig. 10. It can be seen that the ball position output tracks the reference signal rapidly, although there exist transmission delays in the networked system. Thus, the data-driven predictive networked control scheme is proved to be effective and preponderant than the Kalman-LQR predictive algorithm described in [14].
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Y. Xia et al. / Information Sciences 235 (2013) 45–54
0.4 0.35
Position of the ball(m)
0.3 0.25 0.2 0.15 0.1 0.05 0 0
500
1000
1500
Time(s) Fig. 10. Position output in the data-driven predictive networked control experiment.
5. Conclusions and discussion A data-driven networked predictive control scheme has been proposed in this paper for MIMO networked control systems with random network delays in CAC and SCC. Based on the network feature of transmitting a data packet at each time, the proposed data-driven networked predictive controller consists of the control prediction generator and the network delay compensator. The former provides a set of future control predictions to satisfy the system performance requirements. The latter generates a particular set of predictive manipulated variables at each sampling time instant. The predictive manipulated variables are encapsulated into data packets then delivered to the plant node through the unreliable channel with network-induced time delays. Simulation results and experiment results show the proposed control scheme is effective. The data-driven control scheme in this paper is obtained based on linear systems, however, the ball and beam systems are nonlinear systems. It is very interesting that the data-driven control scheme works better than the ball and beam systems controlled by other methods. This leaves many question about data-driven control method: for example, how to distinguish linear systems and nonlinear systems under the data-driven control scheme; how to evaluate the performance? As for fast controlled plants, how to generate the initial control signals? If the data drop occurs, how to compute the subspace objection with intermittent observations? How to analyze the stability of data-driven based nonlinear systems? Acknowledgments The authors thank the referees for their valuable and helpful comments which have improved the presentation. The work of Yuanqing Xia was supported by the National Basic Research Program of China (973 Program) (2012CB720000), the National Natural Science Foundation of China (60974011), Program for New Century Excellent Talents in University of China (NCET-08-0047), the Ph.D. Programs Foundation of Ministry of Education of China (20091101110023 and 20111101110012), and Program for Changjiang Scholars and Innovative Research Team in University, and Beijing Municipal Natural Science Foundation (4102053 and 4101001). References [1] [2] [3] [4] [5] [6] [7] [8]
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