Estimation and control over networks using event based transmission methods

Abstract The thesis is concerned with the state estimation and control problem over the network in which an event-based sampling scheme at sensor nodes is proposed.. In the event-driven

공학박사 학위논문

이벤트 기반 전송 방법을 이용한

추정 및 제어 Estimation and Control over Networks using

event-based transmission methods

울 산 대 학 교 대 학 원

전기전자정보시스템공학부

Nguyen Vinh Hao

이벤트 기반 전송 방법을 이용한

추정 및 제어 Estimation and Control over Networks using

event-based transmission methods

Nguyen Vinh Hao 의 공학박사 학위 논문을 인준함

Estimation and Control over Networks using

event-based transmission methods

by Vinh Hao Nguyen

A dissertation submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy (Electrical Engineering)

in The University of Ulsan December, 2008

Acknowledgements

I would like to express my sincere appreciation to my thesis advisor, Prof Soo Suh, who has guided me through my Ph.D research with his patience, vision, and wisdom Prof Suh is never satisfied by mediocre research, he has always encouraged me

Young-to challenge myself with perfectionism and persistence I thank him for helping me understand the essence of scientific research and find the real potential of myself He has been a continual source of fresh ideas in the process of earning my degree

I would also like to thank Prof Hong-Hee Lee and committee members, for taking their time to review my thesis and be on my committee

Many thanks to my friends and roommates, who have dealt with my late nights of thesis work and occasional fits of frustration with good nature A big thank to my labmates, for their friendship and their help during my three years in Korea

Last but not least, I would like to thank my wife, Mrs Do Thi Kim Chung, for being such a good friend and providing me supports on every aspect of my life I would like to thank my parents for their unconditional and endless love, support, encouragement, and for taking care of my son during my Ph.D study

Abstract

The thesis is concerned with the state estimation and control problem over the network in which an event-based sampling scheme at sensor nodes is proposed If the network speed is high and the traffic is sparse, the traditional periodic sampling approach has many merits But when the network bandwidth is limited due to executing tasks of several nodes, time delay becomes large and randomly varying Therefore, to avoid these problems the sensor data transmission rate should be reduced

In the event-driven sampling scheme, sensor data are transmitted to the estimator node only if the difference between the current sensor value and the last transmitted one is greater than a given threshold The research has shown that the event-based sampling scheme is more efficient than the periodic sampling one in some situations, especially in network bandwidth improvement

The main contribution of thesis is to find the optimal threshold value at each sensor node which is a trade-off parameter between the sensor data transmission rate and the control performance Then the modified Kalman filters are formulated to estimate states of the system under conditions of system noises, packet loss, etc At last, the optimal LQG controllers are set up to solve the control problem over the network

The simulation and experimental results have pointed out the feasibility and efficiency of the event-driven sampling scheme in network bandwidth improvement with less degradation of control performance This is very useful in the realistic applications where sensor data transmission rate needs to be lowered due to joining of many sensor nodes or saving power in wireless networks

Contents

Acknowledgements

Abstract

1 Introduction

1.1 Problem overview

1.2 Networked control systems

1.2.1 Network architecture

1.2.2 Network protocols

1.3 Fundamental issues in networked control systems

1.3.1 Network delays

1.3.2 Data rate constraints

1.3.3 Network bandwidth constraints

1.3.4 Sampling and quantization

1.3.5 Data packet dropouts

1.4 Motivation and contributions of thesis

1.4.1 Motivation

1.4.2 Previous works

1.4.3 Contributions

1.5 Thesis outline

2 Event-based sampling and state estimation problem

2.1 Introduction

2.2 Event-based sampling scheme

2.3 State estimation using event-based sampling

2.4 Estimation performance analysis

2.5 Estimation performance of the multirate filter

2.6 Simulation results

2.7 Conclusion

3 State estimation for networked monitoring systems

3.1 Introduction

3.2 Problem formulation

ii iii 1 1 2 3 3 5 5 6 7 7 8 9 9 10 11 11 13 13 13 14 16 17 19 21 22 22 23

3.3 Send-on-delta based state estimation for multi-output systems

3.4 Optimal δ i computing problem

3.5 Numerical and experimental simulation

3.6 Experimental results over ZigBee network

3.7 Conclusion

4 Controller design for networked control systems

4.1 Introduction

4.2 Problem formulation

4.3 Send-on-delta multirate controller design

4.3.1 SOD estimator design

4.3.2 SOD multirate controller design

4.3.3 Optimal δi computing problem

4.3.4 Stability of the SOD multirate controller

4.4 Simulation results

4.5 Conclusion

5 Networked estimation with an area-triggered transmission method

5.1 Introduction

5.2 Area-triggered sampling scheme

5.2.1 Effect of noise on sensor data transmission rate

5.2.2 Π i computation and SOA sampling in discrete time

5.2.3 Effect of noise on signal distortion

5.3 State estimation with SOA transmission method

5.3.1 Bound of Δ i(t, tlast,i)

5.3.2 State estimation

5.4 Simulation results

5.5 Conclusion

6 Networked estimation with packet dropouts .

6.1 Introduction

6.2 Effect of packet dropouts on system performance

6.2.1 Estimation performance of multirate filter with packet dropouts

6.2.2 Estimation performance of the SOD filter with packet dropouts

6.2.3 Evaluation results

6.3 Modified SOD sampling scheme

23 25 27 32 34 35 35 36 38 38 39 39 41 43 46 47 47 48 49 51 52 55 56 57 58 61 62 62 63 63 64 64 65

6.4 State estimation with modified SOD transmission method

6.4.1 Measurement noise increased due to multiple packet dropouts

6.4.2 State estimation

6.5 Optimal δt,i computing problem

6.5.1 Sensor data transmission rate by condition (6.8b)

6.5.2 Estimation error covariance due to packet dropouts

6.5.3 Optimal δt,i computation

6.6 Simulation results

6.6.1 Case 1

6.6.2 Case 2

6.7 Conclusion

7 Conclusions and future work

7.1 Conclusions

7.2 Future work

References

68 69 69 71 71 71 72 73 73 77 79 80 80 81 83

List of Figures

1.1 A control system with a traditional wiring configuration

1.2 A control system with an NCS configuration

1.3 A compact NCS configuration used throughout the thesis

1.4 Configuration of an NCS with delays

2.1 Event-based sampling scheme

2.2 Structure of the event-based Kalman filter

2.3 Error covariance of two filters under the same bandwidth conditions

2.4 Pk value of two filters

2.5 Estimation error of two filters

3.1 Structure of the event-based Kalman filter for the multi-out systems

3.2 Experimental of the state estimation system through a CAN bus

3.3 The relationship between number of sensor data transmissions and si/δi

3.4 Estimation error: standard KF, proposed SOD KF, naive SOD KF

3.5 Experiment of the state estimation system through ZigBee network

3.6 Estimation error: standard KF, proposed SOD KF, naive SOD KF

4.1 Configuration of a networked control system

4.2 Block diagram of a multirate control system

4.3 Estimation error in 3 methods

4.4 Step response with initial position

5.1 a SOD sampling scheme

5.1 b SOA sampling scheme

5.2 Sensor output with noise in discrete time

5.3 Effect of R on data transmission rate and distortion for y1

5.4 Effect of R on data transmission rate and distortion for y2

5.5 Structure of the modified Kalman filter

5.6 Estimation error in case 1

5.7 Estimation error in case 2

6.1 Error covariance without packet loss in two sampling schemes

6.2 Error covariance increased due to packet loss in two sampling schemes

1 2 2 6 14 16 19 20 20 25 28 29 31 32 33 36 37 45 45 49 49 51 53 54 57 60 60 65 65

6.3 a Principle of conventional SOD sampling

6.3 b Principle of modified SOD sampling

6.4 Multiple packet dropouts detection

6.5 Measurement noise increased due to multiple packet dropouts

6.6 Structure of the modified Kalman filter

6.7 δt,1 of (6.17) along with δy,1 and ξ1

6.8 δt,2 of (6.17) along with δy,2 and ξ2

6.9 Estimation error in two filters

6.10 Instants the sensor node transmits data due to condition (6.8b)

6.11 Estimation error in two filters

6.12 Instants the sensor node transmits data due to condition (6.8b)

7.1 The modified Extended Kalman filter for nonlinear systems

66 66 67 69 70 74 74 76 76 78 78 82

List of Tables

2.1 Error covariance in two sampling schemes with different values r and δ

3.1 Numerical results with different estimation performance constraints

3.2 Numerical results in ZigBee network

4.1 Control performance of the standard controller

4.2 Control performance of the proposed controller

4.3 Control performance of the multirate controller

5.1 Estimation performance of 2 methods with different threshold values in case 1 5.2 Estimation performance of 2 methods with different threshold values in case 2 6.1 Estimation error for case 1 in two filters

6.2 Estimation error for case 2 in two filters

19 31 33 44 44 44 59 59 75 77

The rise of NCSs stems in part from necessity Traditionally, control systems have been implemented using point-to-point wiring, i.e., each sensor and actuator is connected

to a centralized controller (often a micro-processor) via a designated wire (Fig.1.1) This configuration ensures real-time communication between components of a control system However, as the complexity and scale of a control system increase, point-to-point wiring becomes cumbersome or impractical: the increased wiring burden brings problems involving weight, cost, maintenance, and reliability At the same time, the microprocessor

in which the controller is implemented provides a limited number of input/output (I/O) ports and limited computing capability, so that point-to-point wiring becomes impossible when the number of sensors and actuators is greater than the number of I/O ports or when the computation load needed exceeds the capability of the processor

PLANT

Sensor

1

Sensor p

Actuator 1

Controller

Actuator q

Figure 1.1 A control system with a traditional wiring configuration

In an NCS (Fig.1.2), the controller is connected to a communication medium that provides access to all the sensors and actuators; the medium is shared by all its users, so that only a limited number of connections can be supported simultaneously The NCS configuration has proved remarkably flexible, enabling many novel features in feedback control systems For example, using wireless communication, sensors and actuators in an NCS can easily change their locations to form different ad hoc groups that are customized for different tasks

PLANT

Sensor node 1

Sensor node p

Actuator node 1

Controller node 1

Actuator node q

In this thesis, the configuration of NCS is limited to a compact system as illustrated

in Fig.1.3, where the plant is an SIMO system and only one controller/estimator node is connected to the network All sensor nodes are connected to the controller/estimator node

by a serial network for state estimation and control

Sensor node p

Actuator node

1 ( )

y t y t p( ) u t( )

Figure 1.3 A compact NCS configuration used throughout the thesis

1.2 Networked control systems

1.2.1 Network architecture

Consider a typical network architecture in a modern manufacturing system shown

in Figure 1.2 The shared communication network architecture has three different levels, namely, the Information/System (IS) network, the Discrete-Event/Cell (DEC) network, and the Continuous-Variable/Device (CVD) network This classification is based on the functionality as well as signal characteristics in typical industrial applications

The top level IS network is used to carry non-time-critical information such as daily or hourly production data, and to communicate with factory-wide databases Messages on an IS network typically have a large data size but low frequency

The middle level is the DEC network, which carries commands or updated working configurations for different cells or subsystems Generally, the messages in a DEC network are discrete and event-based The DEC network messages may be periodic, sporadic, or time-critical When timing is critical, large time delays and lost data at this level may cause coordination problems between different subsystems The analysis and control of DEC network systems such as manufacturing systems and multi-task robotic systems has been studied using discrete-event system techniques such as finite state machines, with or without timing parameters [1], [2]

The bottom level is the CVD network, which communicates physical signals such

as position, velocity, and temperature by the means of network coding and messaging Sensors, actuators, and controllers are the types of devices interconnected by CVD networks Messages are transmitted periodically and in real-time; data sizes are small, but message frequency may be high Time delays and lost data at this level may degrade the system performance and even cause system instability [3], [4] The three levels of networks are separated because they require different information characteristics and functionalities, although they may be connected by gateways or bridges

If the same network is used for multiple levels, the large-size data packets transmitted at the IS-network level could degrade network efficiency in both CVD and DEC networks, and the high-frequency data packets at the CVD-network level might further delay the message transmission in either the DEC or IS networks

1.2.2 Network protocols

Based on the time-delay characteristics of control networks installed in industrial automation systems, we classify these networks into three types: stochastic, bounded, and

constant This classification focuses on the medium access control (MAC) mechanism and the time delay between two devices In this section, we first describe different types of message connections and then discuss how the MAC determines the time delay characteristics for three types of control networks: Ethernet-based (stochastic), token-passing (bounded), and CAN-based (constant)

The three types of control networks commonly implemented in industry are Ethernet based, token-passing, and CAN-based networks Ethernet (IEEE 802.3) is a CSMA/CD (Carrier Sense Multiple Access with Collision Detection) network protocol Simply speaking, CSMA/CD specifies that every node should detect the network availability before sending out messages and, if there are message collisions, a collided node stops transmitting and waits a random length of time to retransmit messages Hence, due to the random binary exponential backoff (BEB) algorithm the transmission delays are non-deterministic Although there exist modifications to Ethernet, for example, switched Ethernet, which seeks to reduce the collision possibility and improve the determinism of the network, the time delay between two devices is inherently stochastic if the BEB algorithm is applied

A token-passing network has bounded time delays As the name suggests, there is one token passing around the network and only the device with the token can transmit messages Transmission bandwidth is thus divided between all devices although there is some overhead associated with passing the token If the network is not saturated, the message time delay will be bounded; the magnitude of these delays depends on the data rate, the total number of devices on the network, message size, and maximum token holding time Typical examples of token-passing control networks are ControlNet [5], MAP (IEEE 802.4), and Profibus [6]

CAN-based network (Control Area Network) protocols are optimized for short messages, and utilize a CSMA/AMP (Carrier Sense Multiple Access with Arbitration on Message Priority) media access method Thus, the protocol is message oriented and each message has a specific priority which is used to determine access to the bus in case of simultaneous transmission The bit-stream of a transmission is synchronized on the start bit and the arbitration is performed on the following message identifier where a logic '0' is dominant over a logic '1' Hence, if two devices want to send messages at the same time, they first continue to send the message frames and then listen to the network If one of them receives a bit different from the one it sends out, it loses the right to send its message

and the other wins the arbitration With this method, an ongoing transmission is never corrupted Any node that wants to transmit a message waits until the bus is free and then starts to send the identifier of its message bit by bit If the message periods and releasing times are known, the time delays between any two devices can be predetermined and may

be constant CAN-based networks include DeviceNet [7] and Smart Distributed System [8]

1.3 Fundamental issues in NCSs

Despite its flexibility and effectiveness, the NCS architecture also introduces new problems which have been beyond the scope of traditional control systems theory until recently In this section, we will briefly analyze some basic problems in NCSs, including network-induced delay, network scheduling, data rate constraints, sampling and quantization, and network packet dropouts

1.3.1 Network delays

The network-induced delay in NCSs occurs when sensors, actuators, and controllers exchange data across the network This delay can degrade the performance of control system designed without considering it and can even destabilize the system

Packet on random access networks are affected by random delays, and the case transmission time of a packet is unbounded Therefore, CSMA networks are generally considered non-deterministic However, if network messages are prioritized, higher-priority messages have a better chance for timely transmission (such as CAN and DeviceNet)

worst-On scheduling networks, packet transmission delays occur while waiting for the token or time slot They can be made both bounded and constant by transmitting packets periodically The effects of all these delay components can be typically captured by the sensor-to-controller delay τsc and the controller-to-actuator delay τca (Fig 1.4)

In state-space models, time delays in the feedback loop of a control system can be effectively captured by introducing additional states to keep track of the delayed information This technique is often termed “state augmentation” (also known as “state lifting” or “state extensification”) For example, it is shown that [9], for a sampled-data control systems with constant feedback and a sample period h, a constant delay τ of (r − 1)h < τ < rh (where r ∈ N) will increase the system order by a factor of r For scalar

linear systems, the relationship between the sampling period and allowable time delay can

be illustrated by a stability region plot [10], which can be obtained via analytical or numerical methods

Actuator

Sensor node

Controller node

Figure 1.4 Configuration of an NCS with delays

Bounded time-varying delays in a control system’s feedback loop can also be modeled by proper state augmentation to include the plant’s state and all the delayed control information; the detail of this technique is illustrated in [10] and [11] Also using the augmented state model, some stability and performance analysis tools are given in [12] for MIMO systems having multiple time delays in different feedback loops In [13], an LQR optimal control problem is formulated and studied based on the same model

1.3.2 Data rate constraints

Communication constraints also manifest themselves in the form of data rate limits

on the communication medium The effects of data rate limits on networked control systems have typically been studied from information theoretic perspectives, and the communication medium is often modeled as a coded channel with a bandwidth limit

The work in [14] investigates state estimation in NCSs where the observations are transmitted to an estimator with a finite data rate The authors introduce a recursive coder-estimator scheme, in which the coding decision can be dependent on the whole past history

of the observation process, and the estimator can be dependent on the whole sequence of past codewords Necessary and sufficient conditions are established for the existence of stable and asymptotically convergent coder-estimator schemes Under the coder-estimator framework, feedback stabilization under data rate constraint is investigated in [15] It is shown that, if the plant is a continuous-time LTI system, then memoryless coding and

control suffice to ensure the containability of the system, meaning that given small enough initial conditions, the trajectory of the system will lie in an n-dimensional sphere of an arbitrary size

The work in [16] investigates the stability of infinite-dimensional linear time plant when the controller receives observation data at a known rate It is shown that, under a finite horizon cost equal to the m-th output moment, the problem reduces to quantizing the initial output As the horizon approaches infinity, asymptotic quantization theory can be applied to directly obtain the limiting coding and control scheme Necessary and sufficient conditions can then be derived for the system to be asymptotically stabilizable in the m-th moment at a given data rate Under some restrictions on the initial condition distribution, a coding-estimation scheme is presented in [17]; this scheme works for finite dimensional, time-varying nonlinear system that satisfies a Lipschitz-type condition

discrete-1.3.3 Network bandwidth constraints

One of the fundamental communication constraints in a communication network is medium access It comes about because a communication bandwidth can only provide limited number of simultaneous medium access channels for its users As a consequence,

in an NCS, only limited number of sensors and actuators are allowed to communicate with the controller at any one time

In modern communication networks, medium access constraints are often resolved via various Medium Access Control (MAC) protocols which define the access scheduling and collision arbitration policies in the network MAC protocols can be roughly divided into two categories, namely sequential MAC protocols and random MAC protocols Under sequential MAC protocols, each user of the network accesses the shared medium according

to a pre-configured sequence Under random MAC protocols, every user attempts to access the media whenever it has a packet to transmit; if there are other users wanting to access the medium at the same moment, an arbitration policy is used to resolve the packet collision

1.3.4 Sampling and quantization

Sampling schemes can either be time-driven or event-driven Astrom and Bernhardsson [18] compared the merits of the Riemann sampling (time-driven) and the

Lebesgue sampling (event-driven) for one-dimensional systems Yook et al [19] proposed that a node should broadcast the true value of the local plant state when it differs from the estimate known to the remote nodes by more than a given threshold They show that this scheme results in a system that is bounded-input bounded-output stable The relation between the threshold level and the message exchange rate is investigated through simulations

A signal has to be quantized before being encoded and sent to a digital channel A quantizer is a device that converts real numbers (an analog signal) into a finite set of integers (a digital signal) Mathematically, a quantizer is a piecewise constant function, mapping a quantization region to a quantization point Usually, a quantization region is a pre-specified rectangular shape More efficient quantization schemes (for controls) have been developed over the years Both [20] and [21] advocate logarithmic-based quantization methods Roughly speaking, quantization region becomes larger logarithmically as the distance from the origin grows Quantization regions are allowed to evolve with time to capture the system dynamics [22]

1.3.5 Data packet dropouts

Dropping network packet occasionally happens on an NCS when there are node failures or message collisions Although most network protocols are equipped with transmission-retry mechanisms, they can only re-transmit for a limited time After this time has expired, the packets are dropped Furthermore, for real-time feedback control data such

as sensor measurement and calculated control signals, it may be advantageous to discard the old, untransmitted message and transmit a new packet if it becomes available In this way, the controller always receives fresh data for control calculation

Normally, feedback-controlled plants can tolerate a certain amount of data loss, but

it is valuable to determine whether the system is stable when only transmitting the packets

at a certain rate, and to compute acceptable lower bounds on the packet transmission rate

In [10], it is shown that an NCS with data packet dropout can be modeled as an asynchronous dynamical system (ADS) with rate constraints on events [23] Using that model, one can calculate the minimum transmission rate that guarantees the stability of an NCS whose closed-loop dynamics are stable without the presence of packet dropout Another possibility for addressing dropped data packets is to model the arrival of data as a random process For example, the work in [24] studies state estimation of SISO NCSs, in

which scalar observations arrive according to a Poisson process; the work in [25] presents

a Kalman Filter scheme for MIMO systems in which the arrival of output information (all outputs as packets) is modeled as a Bernoulli process These works are generalized in [26] where outputs are divided into two parts each of which can be received or lost by the Kalman Filter independently

In a very recent study [27], the optimal H2 filtering problems associated respectively with possible delay of one sampling period, uncertain observations and multiple packet dropouts are studied under a unified framework The H2-norm of systems with stochastic parameters is defined and computed via a Lyapunov equation and a steady-state filter is designed via an LMI approach The authors modeled the multiple packet dropout case, where the random dropout rate is transformed into a stochastic parameter in the system’s representation

1.4 Motivation and contributions of thesis

1.4.1 Motivation

A major trend in modern industrial and commercial systems is to integrate computing, communication, and control into different levels of machine/factory operations and information processes The traditional communication architecture for control systems, which has been successfully implemented in industry for decades, is point-to-point, that is,

a wire connects the central control computer with each sensor or actuator point However, expanding physical setups and functionality are pushing the limits of the point-to-point architecture Hence, a traditional centralized point-to-point control system is no longer suitable to meet new requirements, such as modularity, decentralization of control, integrated diagnostics, quick and easy maintenance, and low cost The introduction of common-bus network architectures can improve the efficiency, flexibility and reliability of these integrated applications, and reduce installation, reconfiguration, maintenance time and costs

The change of communication architecture from point-to-point to common-bus, however, introduces several issues as mentioned in Section 1.3 Most NCS research has dealt with the issues: network-induced delay, network constraints, and packet dropouts When several nodes are connected to the network, network-induced delay is inevitable It

is well-known in control systems that time delays can degrade the system's performance and even cause system instability

This dissertation focuses on the issue of signal sampling scheme and data transmission method in order to reduce data transmission rate on the network If the transmission rate is low, it is clearly that network-induced delay is also small and can be ignored in some certain cases Therefore, the problem of estimation and control over networks becomes easier to deal with when ignoring network delay

1.4.2 Previous works

The traditional way to design networked control systems is to sample the signals equidistant in time A nice feature of this approach is that analysis and design becomes very simple For linear time-invariant processes the closed loop system becomes linear and periodic But, the periodic sampling scheme introduces large data transmission rate over network because the sensor nodes have to send data every sampling time

Event-based sampling scheme, also called level-triggered sampling, level-crossing sampling, send-on-delta, deadband, or Lebesgue sampling, samples the signal when its output has changed with a specified amount A disadvantage of this sampling scheme is that analysis and design are complicated Therefore, it has not received much attention although much work on systems of this type was done in the period 1960-1980

However, recent works have discussed event-driven alternatives to traditional triggered sampling scheme It has been shown to be more efficient than time-triggered one

time-in some situations, especially time-in network bandwidth improvement We will review a number of previous researches here to understand the related work

• In [18], the authors provided a comparison of time-triggered impulse control and level-triggered one where the level-triggered scheme gave lower average error for the same average rate of impulse

• In [28][29], an adjustable deadband was defined on each node to reduce network traffic The node does not broadcast a new message if its signal is within the deadband A method to determine the size of the deadbands was presented that relies

on a performance metric that takes into account system response as well as network traffic

• In [30], the authors introduced the use of a level-crossing sampling scheme based on hysteretic quantization for feedback stabilization under data-rate constraints Hysteresis allows us to implement 1-bit coding feedback communication which has the potential to achieve the most efficient data-rates In addition, under noisy

scenarios, hysteresis would also minimize spurious sampling, further contributing to low data-rate feedback communication This coding sampling scheme becomes highly efficient under data-rate constraints since the nodes only transmit one bit, 0 or

1, per sample

• The work [31][32] solved an optimal level-triggered sampling design problem in which the distortion of a filter is minimized over a finite horizon A method for designing good level-triggered control schemes was obtained by reducing the continuous time problem to discrete time Then, by numerical procedures, the performance of the level-triggered scheme is computed for comparison with that of the periodically switched control scheme

• In [33][34][35], where the event-based sampling has the name of send-on-delta method, the authors presented the analytical method for estimation of the mean sampling rate and sampling effectiveness defined as a ratio of the number of samples taken in periodic and event-based schemes

1.4.3 Contributions

Motivated by the perspective results of the event-based sampling in [28-35], the aim of this thesis is to explore the problem of estimation and control over networks using event-based transmission method We address the following problems:

• Derive formulation for the problem of state estimation and optimal LQG control when the sensor nodes are sampled by the event-based method

• Find the optimal threshold value at each sensor node such that the overall sensor data transmission rate is minimized and the degradation of control performance is small

• Consider the impact of packet dropouts on system performance and propose a novel event-based sampling scheme to improve system performance

1.5 Thesis outline

Chapter 2: we introduce the event-based sampling scheme in which sensor data are transmitted to the estimator node only if the difference between the current sensor value and the last transmitted one is greater than a given threshold We compare the event-based sampling with the multi-rate sampling to evaluate the system performance as well as network traffic

Chapter 3: We formulate the state estimation problem for the linear systems when the sensor nodes are sampled by the event-based method Then we derive the optimization problem to find the optimal threshold value at each sensor node The threshold value is a trade-off parameter between the overall sensor data transmission rate and the estimation performance If the threshold value is large, the sensor data transmission rate is small (improve network bandwidth) but estimation performance degrades much Otherwise, if the threshold value is small, the estimation performance is good but sensor data transmission rate is increased (network traffic is high)

Chapter 4: An optimal LQG controller is designed for the NCS in which sensor data are sent to the controller node with the event-based method, and the controller node send data to the actuator node periodically We prove the stability of the proposed controller and show that control performance of the proposed controller is better than that

of the multirate controller

Chapter 5: We propose a novel even-based sampling scheme to improve system performance The conventional event-based sampling scheme has some flaws that degrade system performance For instance, the transmission rate of the even-based method becomes large when the sensor noise is large Furthermore, it does not detect signal oscillations or steady-state error if the signal variation remains within the threshold range during a long time

Chapter 6: The impact of packet dropouts on system performance is considered and evaluated Then a modified estimator is formulated in the case of multiple packet dropouts

Chapter 7: We summarize the contributions of the thesis and discuss the possibility

of future works

We also introduce a multi-rate sampling scheme, in which the output signal is sampled by the time-triggered scheme, in order to compare system performance as well as data transmission rate in two sampling schemes

2.2 Event-based sampling scheme

Consider a networked monitoring system (Fig 1.3) described by the linear continuous-time model:

x ∈R

( )

R

∈( )

w t Q

In this chapter, we only consider a scalar system for simplicity in problem formulation The multi-output system is considered in the next chapters The output signal

of the system (2.1) is sampled by an event-based scheme as depicted in Fig.2.1, where the slash line presents measurement output value at the sensor node and the solid line presents the sensor data value received at the estimator node

i) Let y be the latest output value that the sensor node sent to the estimator last

node at time t last

ii) The next sensor value of ( )y t will be transmitted to the estimator node only if:

( ) last

y t −y > (2.2) δ

where is a given threshold value The threshold is a design parameter that determines

the resolution of signal observations The smaller threshold value δ , the higher resolution

of signal tracking The digitized samples including the current signal value and the protocol overhead are sent in messages through the network The larger value is, the more network traffic is reduced, but the lower resolution of signal tracking is Therefore, a state estimator is essential in order to reduce the error between the measurement output value and the received value

) y last( )t

t

( )

δ δ δ

last

y t

Figure 2.1 Event-based sampling scheme

2.3 State estimation using event-based sampling

The following assumptions are made on the data transmission over networks (Fig.1.3) using even-based sampling:

i) Measurement output ( )y t is sampled at period T but its data are only

transmitted to the estimator node if condition (2.2) is satisfied (if the difference

between the current value and the previously transmitted one is greater than δ )

ii) For simplicity in problem formulation, transmission delay from the sensor nodes to the estimator node is ignored

The estimator node estimates states of the plant regularly at period T regardless

whether sensor data arrive or not If the sensor data do not arrive, the estimator node knows that the current value of sensor output does not change more than the range ( )

compared to the last arriving one This implicit information is used to estimate the current states in case the sensor data do not arrive

( )

last

y

If there is no sensor data received for , the estimator node considers that

the measurement value of sensor output y t is still equal to but the measurement noise increases from to v t , where is defined by:

> last i,

)( , )

n t t l

(Δ

last

y

( ,t t last

( ) ( )= ( )+ ast Δ )( ,t t last) y t( ) y last

Δ = − , Δ( ,t t last) ≤ (2.3) δ

Assuming that has the uniform distribution with zero mean, covariance of

is computed:

( ,t t last)Δ( ,t t last)

E t t

δ δ

A modified Kalman filter for state estimation at step k , where there is a change

in the measurement update part of the discrete Kalman filter algorithm [41], is given as in

the Fig.2.2 We use the discretized system model sampled at period T , is process noise covariance of the discretized system:

A =e ,

0

T

Ar A r d

In the event-based Kalman filter (Fig.2.2), the state of the plant is estimated

regularly at every period T regardless of whether or not sensor data arrive If sensor data

arrive then , the modified Kalman filter acts like conventional Kalman filter Otherwise, it uses as the measurement value and

( ,t t last) 0

last

y R = R+δ2/ 3 as measurement noise covariance for state estimation

− +

Initialization

sensor data arrive?

0 0 0

ˆSet ,

Figure 2.2 Structure of the event-based Kalman filter

2.4 Estimation performance analysis

The event-based Kalman filter in Fig.2.2 can be represented by updated equations

as follows:

• Time update:

1 1

( )

1 2

where γ = k 1 if the estimator node receives sensor data, otherwise γ = k 0

From (2.6) and (2.7), the prediction error covariance is:

have the steady state Let and be the upper bound of error covariance as sensor data arrive and not arrive respectively If the estimator node does not receive sensor data ( for a sufficiently long time interval, in (2.8) will converge to the upper bound value:

max 0 max 0 2

max 0

0 max 0 2

=+

(2.9)

where C0 = A R d2 +Q C d 2 and R = R+δ2/ 3 The solution to (2.9) is:

2 2 max 0 ( 0 ) ( 0 ) 4 d /2

(2.10)

has a unique positive solution

2.5 Estimation performance of the multirate filter

The principle of multirate sampling scheme is that sensor node samples the output and sends data at period , where r is a multirate ratio The estimator node estimates the state of the system at every period T regardless of whether sensor data

1 1

−

− − + + +

where γ k =1 if the estimator node receives sensor data, otherwise γ k = 0

From (2.12) and (1.13), the prediction error covariance is:

Assuming the estimator node receives sensor data at instant kT After that we have

consecutive instants during which the estimator node does not receive sensor data The next sensor data will arrive at instant kT Thus using (2.14) we have:

(r −1

rT

+

(2.15) 2

1

k d k

P− A P− + = +

2

1 2

2 2( 1) 2

0 2( 1) 2( 1)

2

11

k d d

1

2 2

1 2

− −

+ −

− + −

− + −

2( 1)

0 2 0 2( 1)

2( 1) 2 2

1111

r

r d

d

r d

max 1 k

= = = solution of (2.18) (2.19)

2( 1) 2( 1)

max 0 1 max 1 2

11

T =

To compare the estimation performance of two sampling schemes multirate (MUL)

and send-on-delta (SOD) under the same bandwidth condition, we choose r and δ such

that the number of sensor data transmissions are identical in two sampling schemes Then

we compute Pmaxγ=0, Pmaxγ=1 which are given in (2.10), (2.11) for the SOD filter and

(2.19), (2.20) for the multirate filter The values r and δ are chosen as in Table 2.1 to

compute Pmaxγ=0, Pmaxγ=1

Table 2.1 Error covariance in two sampling schemes with different values r and δ

# of packets Pmaxγ=0 Pmaxγ= 0

For ease in comparison, error covariance values in Table 2.1 are intuitively represented according to number of data transmissions as in Fig.2.3 As we see in Fig.2.3, the values Pmaxγ=0, Pmaxγ=1 in the SOD filter are much smaller than those in the multirate

filter Especially when sensor data transmission rate is small (large δ value), the SOD

filter becomes much better than the multirate filter Therefore we believe that the SOD filter will give better estimation performance than the multirate filter in the same network bandwidth condition

Fig.2.4 shows estimation error covariance , and Fig.2.5 shows estimation error

of two filters in time for a concrete case as We see that estimation error in the SOD filter has smaller upper bound than that in the multirate filter under the same network bandwidth

Figure 2.4 P k value of two filters as r =10, 2.38δ =

Figure 2.5 Estimation error of two filters as r =10, 2.38δ =

2.7 Conclusion

In this chapter, we have formulated a state estimator using event-based sampling scheme Through theoretical analysis and numerical simulation, we have proven that the SOD filter introduces better estimation performance than the multirate filter under the same network bandwidth On the other hand, we can conclude that with the same estimation performance requirement, the sensor data transmission rate in the event-based sampling is smaller than that in the time-triggered sampling Therefore, when the event-based sampling is applied, we get benefit from network bandwidth improvement

Chapter 3

State estimation for networked monitoring systems

employing an event-based transmission method

3.1 Introduction

Recently networks are increasingly used in control applications as mediums to transfer control signals, sensor data, and other monitoring data [1], [42] When sensor data are transmitted through a serial network for the state estimation, it is sometimes required that sensor data traffic is reduced so that the serial network can be used for other traffic The reduction of network traffic can be achieved by reducing either data size or the number of data transmission

The first approach is considered in [14] and [43] In [14], sensor data size reduction

is considered by using the special coding method In [43], the observer is designed using the non-uniform quantizer The second approach is considered in [47] and [46] In [47], sensor data traffic reduction is achieved by using state estimators in each subsystem of a distributed control system The main disadvantage of this method is large computational burden in subsystems to implement state estimators In [46], priority is given based on sensor data value change to reduce estimation performance degradation under high traffic condition

In this chapter, we propose a new estimation method, which can reduce the number

of data transmission with relatively small estimation performance degradation We employ

a send-on-delta (SOD) method in which sensor data are transmitted only if their values change more than the specified ± value [45] This method is often used in the building automation to reduce sensor data traffic When this send-on-traffic method is used even if there is no sensor data transmission, we can know that the sensor value is within §± range from the previously transmitted sensor value We believe that this fact is not fully exploited in the previous research [46]

We also discuss how to choose ± δ value, which is a trade-off parameter between the

sensor data transmission rate and the estimation performance

process noise with covariance Q , and is the measurement noise with covariance R

( )

y t

i) Let p be the latest output value that the i-th sensor node sent to

the estimator node at time t last i,

i last i

y t −y > δ i (3.2) where is a given threshold value δ

The following assumptions are made on the data transmission over networks (Fig.1.3) using even-based sampling:

i) Measurement output ( )y t is sampled at period T but its data are only i

transmitted to the estimator node if condition (3.2) is satisfied

ii) For simplicity in problem formulation, transmission delay from the sensor nodes to the estimator node is ignored

The estimator node estimates states of the plant regularly at period T regardless

whether sensor data arrive or not If the sensor data do not arrive, the estimator node knows that the current value of sensor output does not change more than the range ( )

compared to the last arriving one This implicit information is used to estimate the current states in case the sensor data do not arrive

When the SOD method is used, the estimator board continuously receives information about the sensor values even if sensor data are not received For example, if

the last received i-th sensor value is at time and if there is no i-th sensor data

received for , then the estimator node knows that:

considers that the measurement value of i-th sensor output is still equal to but the measurement noise increases from to , where

for each sensor output is defined as in (2.3):

A modified Kalman filter for state estimation at step k , where there is a change

in the measurement update part of the discrete Kalman filter algorithm [41], is given as in

the Fig.2.2 We use the discretized system model sampled at period T , is process noise covariance of the discretized system:

A =e ,

0

T

Ar A r d

In the event-based Kalman filter (Fig.2.2), the state of the plant is estimated

regularly at every period T regardless of whether or not sensor data arrive If sensor data

arrive then , the modified Kalman filter acts like conventional Kalman filter Otherwise, it uses as the measurement value and

,1 ,2 ,

last last last last p

− +

Initialization

i-th sensor

data arrive?

0 0 0

ˆSet ,

Figure 3.1 Structure of the event-based Kalman filter for the multi-output systems

3.4 Optimal δ i computing problem

An optimization algorithm is proposed for the computing problem Note that plays a trade-off parameter role between the sensor data transmission rate and the estimation performance If is small, the sensor data transmission rate is generally higher, and the estimation performance is generally better

i

i

δ

First we investigate the relationship between and the sensor data transmission

rate For the given i-th sensor, if we choose smaller , the sensor data transmission rate of

the i-th sensor will increase For two sensors (i-th and j-th sensor), does not

necessarily means that the transmission rate of j-th sensor is higher than the transmission rate of i-th sensor To see this, note that Δi( ,t t last i,) in (3.4) is given as follows:

transmission is inversely proportional to Using this observation, we can assume that

is proportional to i-th sensor data transmission rate We introduce a function

relating to overall sensor data transmission rate as follows:

p i p

i i

an upper bound of the proposed filter error covariance by setting measurement error covariance as follows:

i

δ

2 2 1

( / 3, , p/ 3)

The measurement error covariance in (3.8) corresponds to the worst case: when there is no sensor data transmission at all for all sensors The upper bound (the steady-state error covariance when is used as the measurement error covariance) can be computed from the following discrete algebraic Riccati equation:

The solution of (3.9) is denoted by P δ( , , )1 δ p

In the following optimization algorithm to choose , we try to reduce the overall sensor transmission rate subject to the estimation performance constraint

min , ,subject to Diag , ,

p p

f

k P

Note that is the steady-state error covariance when all sensor data are

received at each estimate period T Thus is the best achievable performance and values in (3.10) should be chosen to satisfy the following constraint:

p

k P

3.5 Numerical and experimental simulation

To verify the proposed estimation algorithm, an experiment was done using CAN (Controller Area Network) The experimental setup is given in Fig 2 The plant is emulated in PC using Matlab real time workshop Two Atmel AT90CAN128 microcontroller boards are used as sensor nodes We do not use the real sensors in the sensor nodes but use A/D converters as sensors and sensor data are generated by National Instruments NI-6025E card in PC

Analog Output 2 (Acceleration)

CAN bus

Analog Output 1 (Position)

ADC 16bit ADC 16bit

PC NI-6025E CARD

( AT90CAN128 BOARD )

INTERFACE

Estimator board

Plant (emulated in PC)

Figure 3.2 Experiment of the state estimation system through a CAN bus

The following one-dimensional position estimation problem is emulated in PC, where the system equation is given by:

[position, speed, acceleration]

1 2Diag (0, , 0)