Modelling and optimal controller design of networked control systems with multiple delays
FENG-LI LIAN{*, JAMES MOYNE{ and DAWN TILBURY}In this paper we discuss the modelling and control of networked control systems (NCS) where sensors, actuators and controllers are distributed and interconnected by a common communication network. Multiple distributed communica-tion delays as well as multiple inputs and multiple outputs (MIMO) are considered in the modelling algorithm. In addition, the asynchronous sampling mechanisms of distributed sensors are characterized to obtain the actual time delays between sensors and the controller. Due to the characteristics of a network architecture, piecewise constant plant inputs are assumed and discrete-time models of plant and controller dynamics are adopted to analyse the stability and performance of a closed-loop NCS. The analysis result is used to verify the stability and performance of an NCS without considering the impact of multiple time delays in the controller design. In addition, the proposed NCS model is used as a foundation for optimal controller design. The proposed control algorithm utilizes the information of delayed signals and improves the control performance of a control system encountering distributed communication delays. Several simulation studies are provided to verify the control performance of the proposed controller design.
1. Introduction
A major trend in modern industrial and commercial systems is to integrate computing, communication, and control into different levels of machine/factory opera-tions and information processes. The traditional com-munication architecture for control systems, which has been successfully implemented in industry for decades, is point-to-point; that is, a wire connects the central con-trol 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 newrequirements, such as modularity, decentralization of control, inte-grated diagnostics, quick and easy maintenance and lowcost. The introduction of common-bus network architectures can improve the efficiency, flexibility and reliability of these integrated applications through reduced wiring and distributed intelligence, and reduce installation, reconfiguration and maintenance time and costs.
These types of distributed control systems are called networked control systems (NCS): sensors, actuators, and controllers are interconnected by one communica-tion network. The change of communicacommunica-tion
architec-ture from point-to-point to common-bus, however, introduces different forms of time delay uncertainty between sensors, actuators and controllers. These time delays come from the time sharing of the communica-tion medium as well as the computacommunica-tion time required for physical signal coding and communication pro-cessing. The characteristics of time delays can be con-stant, bounded or even random, depending on the network protocols adopted and the chosen hardware. It is well known in control systems that time delays can degrade a system’s performance and even cause system instability. However, though the analysis and modelling of time-delay systems has been a progressive research area, existing methodologies cannot be directly applied due to the discrete and distributed nature of the many different time delays in NCSs.
Most NCS research has focused on two areas: com-munication protocols and controller design. A proper message transmission protocol is necessary to guarantee the network quality of service, whereas advanced con-troller design is desirable to guarantee the control qual-ity of performance. In this paper we consider the modelling of an NCS with multiple communication delays and formulate and solve the optimal control problem based on the proposed discrete-time delay model. In } 2, we survey existing results on time-delay systems and networked control systems. In } 3, we discuss the problem formulation and assumptions used in the discrete-time modelling algorithm. In } 4, we address the delay impact on plant and controller dynamics. In } 5, we provide an example to illustrate the stability analysis of the proposed modelling approach. Based on the proposed modelling framework, we formulate an optimal controller in } 6. We present simulation result of the proposed controller design in } 7. In } 8, we summarize the results of the proposed model-ling and controller design of networked control systems.
International Journal of ControlISSN 0020–7179 print/ISSN 1366–5820 online # 2003 Taylor & Francis Ltd http://www.tandf.co.uk/journals
DOI: 10.1080/0020717031000098426 Received 27 August 2001. Revised 15 January 2003.
* Author for correspondence. e-mail: [email protected] { Electrical Engineering 537, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, 106, Taiwan.
{ Department of Electrical Engineering and Computer Science, University of Michigan, 1124 EECS Building, 1301 Beal Avenue, Ann Arbor, MI 48109-2122, USA. e-mail: [email protected]
} Department of Mechanical Engineering, University of Michigan, 2250 G. G. Brown Building, 2350 Hayward Street, Ann Arbor, MI 48109-2125, USA. e-mail: [email protected]
2. Related work
In this section we discuss related research on analysis of time-delay systems and modelling of control of net-worked control systems.
2.1. Analysis of time-delay systems
Modelling and control of NCSs is based on the time-delay systems analysis framework which has been studied for several decades. In general, delays occur in the transmission of signals or materials between differ-ent subsystems. Large-scale systems such as communi-cation systems, manufacturing systems, transportation systems, power systems and teleoperation systems are typical examples of time-delay systems (Malek-Zavarei and Jamshidi 1987). There have been two approaches used to analyse the stability of time-delay systems: clas-sical (frequency domain) and functional (time domain) approaches (Ogˇuzto¨reli 1966, Marshall 1979, Go´recki et al. 1989) The classical approach utilizes analytical or graphical methods to find the roots of the character-istic equation of a dynamic system. Since delays appear in the characteristic equation as exponential functions, analytical techniques are developed to find solutions of a quasi-polynomial. Alternatively, the Pade´ approxima-tion to the time delay can be used to obtain a pure poly-nomial. In addition, standard graphical methods such as the root locus, Bode plot and Nyquist diagram can be further modified to analyse delayed polynomials or transfer functions. For a discrete-time system with time delays, the system stability can also be easily ver-ified by the root locus analysis in the z-domain (Franklin et al. 1998).
The functional approach, on the other hand, uses delay-differential equations to characterize systems with delays in the state, input or both. Multiple delays (Hsiao and Hwang 1997, Gu 1999) and time-varying delays (Goubet-Bartholome´u¨s et al. 1997, Gu et al. 1998) can be modelled in a similar setting. Based on standard algorithms from functional analysis, solutions of delay-differential equations can be constructed, and their stability can be analysed using Lyapunov’s second methods such as Lyapunov–Krasovskii and/or Lyapunov–Razumikhin stability theorems (Dugard and Verrest 1998).
Since time delays in most applications are considered non-deterministic parameters, controller design for time-delay systems typically uses either a robust or stochastic control approach. Robust H1 controllers
can be designed based on the known system structure or delay uncertainty. Uncertainties in both the system matrix and the input matrix can be included in the delay-differential dynamic equation. Moreover, using a linear matrix inequality (LMI) approach, system stab-ility and performance can be analysed and guaranteed
(Jeung et al. 1996, Li and de Souza 1997, Luo et al. 1998).
These modelling and analysis approaches developed for time-delay systems can only be used as a foundation for analysing the time-delay effects in NCSs. Since all devices in an NCS are distributed on one common-bus network, sample-and-hold and time skew among samples are typical and result in an inherent asynchro-nization in the system model. Also, due to the distri-bution of actuators and the network communication mechanism, the system inputs are piecewise constant with delays, rather than continuous. These properties have not been discussed in previous time-delay analyses; thus further research is needed to model and analyse NCSs.
2.2. Modelling and control of networked control systems
Research in NCSs is different from that in tradi-tional time-delay systems. Because of the variability of network-induced time delays, the NCSs may be time-varying systems, making analysis and design more chal-lenging. Wittenmark et al. (1995) discussed several tim-ing issues such as communication and computation delays, processor jitter and transient errors existing in NCSs. These timing issues must be addressed when deriving a discrete-time state-space model. In their sub-sequent work, a discrete-time model with a single sen-sor-controller delay and a single controller-actuator delay was studied and a stochastic controller design was designed (Nilsson and Bernhardsson 1997, Nilsson et al. 1998). Nilsson (1998) studied the case with mul-tiple sensor-controller and controller-actuator delays; only the case where the total maximum network delay is less than one controller sampling period was consid-ered.
Recently research on the analysis and modelling of NCSs has been conducted using continuous-time and discrete-time models. It is more natural to analyse an NCS from the discrete-time point of viewsince in a typical NCS operation, physical signals (from sensors or to actuators) are sampled and then transmitted on the network medium after a short delay. For discrete-time models, most researchers assume that the network is synchronized and the sampling rates of sensors, con-trollers and actuators are the same. Halevi and Ray (1988) considered the case of a single time delay for sensor-controller and controller-actuator and a single time skewbetween sensor and controller sampling instants. They used the augmented state to include the past delayed signals and derived a closed-loop model for NCSs.
Krtolica et al. (1994) derived a discrete-time time-varying state-space representation of NCSs with F.-L. Lian
random network delays by using the augmented states to include past plant and controller states. The total number of states used depends on the possible range of sensor-controller and controller-actuator delays. Since the maximum possible number of network delays is bounded, and the closed-loop system is a discrete-time model, the system matrix can be viewed as a finite auto-maton with finite states. The stability analysis of such a system can be further described by a Markov chain with finite state transitions.
Branicky et al. (2000) considered a simplified NCS model where the sensor-controller and controller-actua-tor delays are lumped together. Their study only con-sidered the case where the lumped time delay is less than one sampling period; the stability regions for the NCS model were investigated based on the lumped delay. In practical applications, however, sensor-controller and controller-actuator delays are different and time-varying at different networked devices due to the network trans-mission mechanism. Hence, a proper time delay profile in NCSs should be characterized based on the network transmission bandwidth and control system bandwidth. Continuous-time NCS models were also considered by several researchers. Go¨ktas et al. (1996, 1997) used a modified Pade´ approximation and considered the net-work delay as an uncertainty. They designed a robust controller to compensate for the uncertain delay in an ATM network. Kim et al. (1996) used a Lyapunov approach to obtain the maximum allowable delay bound for the stability of a network delayed system. A scheduling algorithm for determining the sampling rate and allocating bandwidth was also provided. Walsh et al. (1999 a, b) also adopted the Lyapunov approach on a continuous-time model to obtain the maximum allow-able transfer interval and to analyse the stability of the closed-loop system. They further analysed the impact of different scheduling algorithms on the maximum allow-able transfer interval. However, only a conservative delay bound was obtained. The impact of delay variance on control performance is discussed in these works, but is not formally characterized.
In practical applications, however, sensor-controller and controller-actuator delays are different and time-varying at different networked devices due to the net-work transmission mechanism. An NCS should be modelled based on the characteristics of network-induced delays and the consideration of network and control parameters. Furthermore, controllers should be designed based on the NCS model, taking into account the delay information. Therefore, in this paper, a discrete-time model of NCSs with multiple inputs and multiple outputs (MIMO) and multiple dis-tributed communication delays is derived. In addition, the asynchronous sampling mechanisms of distributed sensors are characterized to obtain the actual time
delays between sensors and the controller. Based on the proposed NCS model, the stability and performance of a closed-loop system with a standard controller are analysed, and a linear quadratic regulator (LQR) opti-mal control is formulated to compensate for the multiple time delays. A preliminary version of this work was presented in Lian et al. (2001 b, 2002 b).
3. Problem formulation and assumptions
Consider the block diagram of a networked control system with a single controller, but multiple sensors and multiple actuators as shown in figure 1. There are N states (x), M inputs (u), and R outputs (y) in the plant dynamics model, and Q states (z), R inputs (w), and M outputs (v), in the controller dynamics model, i.e. M actuators, R sensors and one controller, where N; M; R and Q are positive constant integers. We use sr; r¼ 1; 2; . . . ; R and am; m¼ 1; 2; . . . ; M to represent
the sensor-controller and controller-actuator delays, re-spectively. The variables wr and um are the delayed yr
and vm signals, respectively. The relationships between
these variables will be addressed later. In the following discussion, we present the system models in continuous time and discrete time. In this paper, time is denoted by t for the continuous-time domain and k for the discrete-time domain.
In figure 1, the continuous-time, state-space model of the linear time-invariant plant dynamics Gp can be
described by the standard form _x
xðtÞ ¼ ApxðtÞ þ BpuðtÞ
yðtÞ ¼ CpxðtÞ
)
ð1Þ
where xðtÞ 2 RN; uðtÞ 2 RM; yðtÞ 2 RRand the constant matrices Ap; Bp and Cp are of compatible dimensions.
Since the controller is implemented at one digital com-puter, the controller is designed in discrete time with a sampling time T and the state-space model of the con-troller dynamics Gccan be expressed as
zkþ1¼ Fzkþ Gwk vk¼ Hzkþ Jwk ) ð2Þ where zk¼4zðkÞ ¼ zðkTÞ 2 RQ, wk¼4wðkÞ ¼ wðkTÞ 2 RR, v k¼ 4
vðkÞ ¼ vðkT Þ 2 RM and the matrices F; G; H and J are of compatible dimensions. Note that wk is a delayed version of the sensor output yðtÞ at some sam-pling instant, and similarly, uðtÞ is a delayed version of the controller output vk.
In practical applications of networked control systems, devices are distributed and have their own pro-cessing units and timing functions. Hence, synchroniza-tion of all devices is extremely difficult. In this paper, we assume that the network is not synchronized; each device may have a different time skewwhen related to Networked control systems with multiple delays
the controller sampling instants. We also assume that the sensor and controller sampling times are the same, and that actuators respond to actuation commands im-mediately after receiving the information from the con-troller. The detailed assumptions and notations used in this paper are described as follows and are illustrated in figure 2.
1. The buffer length at the controller for each sensor and each actuator is equal to one. That is, the controller only uses the newest sensor messages and never sends a stale actuator command. 2. As shown in figure 2, the periods of all R sensors
and one controller are identical and equal to T, but there may be R different time skews, denoted as r; r¼ 1; . . . ; R, among these sensor sampling
instants. The definition of ris the time difference between the sampling instant of the rth sensor and the sampling instant of the controller. We assume that rs are constant.
3. There are two types of time delays: processing delayand communication delay. Processing delays occur at the controller, sensors and actuators, and are denoted as pcðkÞ; psrðkÞ and pamðkÞ,
respect-ively. Communication delays between the sensors and the controller and between the controller and the actuators are denoted as csrðkÞ and camðkÞ,
respectively. In this paper, we add the processing delays to the communication delays. The combined time delays are defined as follows. The combined sensor processing-communication
delay is csrðkÞ þ psrðkÞ and the combined
actuator-controller processing-communication delay is amðkÞ ¼ pcðkÞ þ camðkÞ þ pamðkÞ. Hence, in the
fol-lowing discussion, we only consider the combined time delays. We also assume that these delays are bounded by one sampling period, that is, the cases of vacant sampling and message rejection are not considered in this paper. Note that amðkÞs shown in figure 2 are the combined delays.
Detailed discussions of characterizing communi-cation delays and choosing a proper sampling period, guaranteeing control performance can be found in Lian et al. (2001 a, 2002 a), respectively. 4. As shown in figure 2, the sensor-controller delay srðkÞ depends on both the time skew r and the
sampling period T. If csrðkÞ þ psrðkÞ r, then
srðkÞ ¼ r. Otherwise, srðkÞ ¼ T þ r. Here,
we also assume that the time delay is bounded by the sampling period; that is, maxfsrðkÞ; r ¼
1; . . . ; Rg 2T. Note that this assumption is true for deterministic network protocols such as token passing or priority-based under normal traffic load. However, for those networks with a stochastic medium access control mechanism, this assumption might not be true.
5. Since the controller-actuator delay is amðkÞ, the kth controller sampling interval ½ðk 1ÞT; kT Þ,
can be formulated as ½0; TÞ ¼
½0; amðkÞÞS½amðkÞ; TÞ: Note that if the
actua-tion delay is longer than one sampling period, then amðkÞ ¼ T. This longer delay may be due
F.-L. Lian Controller z(k) uM network channel network channel
...
...
reference input sensor output actuator inputG
pG
c u1 u2 yR y1 y2 a2 a2 aMaM a1 a1 ssRR ss22 ss11 w1 wR w2 vM v1 v2 actuator delay sensor delay ... ... ... ... Plant x(t)Figure 1. The block diagram of a networked control system. Sensors, actuators and controllers in an NCS are distributed and interconnected by communication networks. NCSs are flexible, reconfigurable and efficient. The information of these devices can be easily shared by other subsystems. However, time-delays between sensor-controller and controller-actuator are unavoidable because of the sharing of the communication medium.
to the blocking of message transmission or using too small of a sampling period. The first situation can be improved by further considering network parameters in system design. For the second situation, a larger T should be chosen. However, the control performance may degrade due to a long sampling period. On the other hand, a smaller sampling period increases the control performance as well as the system complexity due to the high network traffic load.
6. In deriving time-delay models, the sensor-control-ler and controlsensor-control-ler-actuator delays are assumed different and time-varying. However, for some cases of control networks, the message transmis-sion times could be viewed as constant parameters as shown in figure 4 of Lian et al. (2002 a). Hence, for the controller design and illustrative examples
in this paper, constant delays are further assumed.
7. For any variable , we will use ¼ =T to denote its value in terms of sampling period. In addition, we split into its integer and fractional parts as
¼ ^þ ~, where ^2 Zþ and 0 ~ <1.
4. Delay impact on plant and controller dynamics One of the main characteristics of NCSs is the dif-ferent communication delays between the plant and con-troller, as shown in figure 1. That is, in general, uðtÞjt¼kT6¼ vk and wk6¼ yðtÞjt¼kT, because of existing
time delays among these signals. The actual values of these communication delays depend on the network pro-tocol adopted as well as the network traffic load. In this section, standard linear systems theory is used to derive the relation between each pair of variables based on the assumptions described in } 3.
4.1. Plant input and controller output
We first study the relation between the controller outputs vkand the plant inputs uðtÞ. Since the actuators receive controller commands discontinuously, we assume that the actuator inputs are piecewise constant as shown in the actuator timing diagram in figure 2. Hence, for the mth actuator and kT t < ðk þ 1ÞT, the input signal umðtÞ can be described as
umðtÞ ¼ vmðk 1Þ1ð0;amðkÞÞðtÞ þ vmðkÞ1ðamðkÞ;TÞðtÞ ð3Þ where 1ðaðkÞ;bðkÞÞðtÞ ¼ 1 if aðkÞ þ kT t < bðkÞ þ kT 0 otherwise ð4Þ That is, umðtÞ is the combination of piecewise constant
functions. For example, consider the first actuator at the kth sampling instant shown in figure 2. The following relation holds: u1ðtÞ ¼ v1ðk 1Þ1ð0;a1ðkÞÞþ v1ðkÞ1ða1ðkÞ;TÞ,
for kT t < ðk þ 1ÞT. Then, uðtÞ ¼ ½u1ðtÞ; . . . ; uMðtÞT
and vðkÞ ¼ ½v1ðkÞ; . . . ; vMðkÞT.
4.2. Plant model in discrete-time domain
In order to analyse the closed-loop system in dis-crete-time, we use the following state-space solution of a first-order matrix differential equation to discretize the continuous-time plant dynamics model (Franklin et al. 1998) xðtÞ ¼ expðApðt t0ÞÞxðt0Þ þ ðt t0 expðApðt q0ÞÞBpuðq0Þ dq0 ð5Þ
We first discretize the plant model at the controller sam-pling instants by applying (5) with t0¼ kT and
t¼ ðk þ 1ÞT. For simplicity, we use the notation Networked control systems with multiple delays
S1S2 SR ∆R … ∆2 ∆1 ... (k-1)T kT (k+1)T R Sensors ( -1st, -- 2nd, … Rth) 1 Controller time skews sampling instants S1S2 SR … ... M Actuators ( -1st, -- 2nd, … Mth)
actuator action instants
a1(k) a1(k-1) a2(k) aM(k) aM(k-1) a2(k-1) ...
Figure 2. The timing diagram for sensors, controller, and actuators in an NCS.
xk¼4xðkÞ ¼ xðkTÞ, A ¼4 expðApTÞ, q ¼ q 0 kT and !ðT; qÞ ¼4 expðApðT qÞÞBp2 R NM :Then xkþ1 ¼ Axkþ ðT 0 expðApðT qÞÞBpuðkT þ qÞ dq ¼ Axkþ XM m¼1 ðT 0 !mðT; qÞumðkT þ qÞ dq ð6Þ
where !m2 RN1 and um2 R, m ¼ 1; . . . ; M are the components of ! and u, respectively. Now, because the actuator commands arrive at different times within the sample interval, um is not constant over
½kT; ðk þ 1ÞT. From (3), we can compute ÐT 0 !mðT; qÞumðkT þ qÞ dq as ðT 0 !mðT; qÞumðkT þ qÞ dq ¼ ðamðkÞ 0 !mðT; qÞvmðk 1Þ dq þ ðT amðkÞ !mðT; qÞvmðkÞ dq ¼4B1mðkÞvmðk 1Þ þ B0mðkÞvmðkÞ ð7Þ
where we also use the timing relations in Assumption 5 and the following definitions: B1mðkÞ ¼
4 ÐamðkÞ 0 !mðT; qÞ dq and B0mðkÞ ¼ 4 ÐT amðkÞ!mðT; qÞ dq. Therefore, by applying (7) to (6), we have xkþ1¼ Axkþ XM m¼1 X1 j¼0 BmjðkÞvmðk jÞ ð8Þ
By further exchanging the order of summations and letting Bjk¼ B 1jðkÞ B2jðkÞ BMj ðkÞ and vk j¼
v1ðk jÞ v2ðk jÞ vMðk jÞ
½ T, we have the
deri-vation xkþ1¼ Axkþ X1 j¼0 XM m¼1 BmjðkÞvmðk jÞ ¼ Axkþ B0kvkþ B1kvk 1 ð9Þ
In (9), A is time-invariant because it is independent of the delay variables amðkÞ, but the Bkjs will depend on the
values of amðkÞs. Therefore, if these amðkÞs depend on k,
the NCS model will be time-varying.
4.3. State value between sampling instants
Because the sampling of sensors happens between the sampling instants of the controller, we need to derive the formula for the state values between sampling instants, i.e. xððk þ 1ÞT TÞ, where 0 < 1. By using (5) again with t0¼ kT and t ¼ ðk þ 1 ÞT
xððk þ 1 ÞTÞ ¼ Axkþ B0kvkþ B1kvk 1 ð10Þ where A¼ expðApðT TÞÞ, B j ðkÞ 2 R NM and vk j 2 RM1:Note that A is not constant but depends
on , in contrast to A in (6), and BjðkÞ can be formu-lated similarly to Bkj. If 0 T < T amðkÞ, then the
mth elements of B0ðkÞ and B1ðkÞ are B0mðkÞ ¼ 4 ÐT amðkÞ!mðT; qÞ dq, and B 1 mðkÞ ¼ 4 ÐamðkÞ 0 !mðT; qÞ dq,
respectively. If T amðkÞ T < T, then the mth
ele-ments of B0ðkÞ and B1ðkÞ are B0mðkÞ ¼ 4
0, and B1mðkÞ ¼
4 ÐT
0 !mðT; qÞ dq, respectively.
4.4. Plant output and controller input
Similarly, for plant outputs yðtÞ and controller inputs wk, we have the following relation due to the
communication delays between the sensors and the con-troller. By Assumption 6 and for simplicity, we have the following equation for the rth sensor-controller delay: srðkÞ ¼ ½ ssrðkÞ T ¼ ½ ^ssrðkÞ þ ~ssrðkÞ T, where ^ssris an
inte-ger and 0 ~ssr<1. Therefore, for any value of srðkÞ and
by (10) xðkT srðkÞÞ ¼ xððk ^ssrðkÞÞT ~ssrðkÞTÞ ¼ A~ssrðkÞxk 1 ^ssrðkÞ þX 1 j¼0 Bj~ss rðkÞðk 1 ^ssrðkÞÞ vðk 1 ^ssrðkÞ jÞ ð11Þ where we let ¼ ~ssrðkÞ in (10). Equation (11) can then be
used to compute the value of the plant output. At the kth sampling instant, the values of the plant output received by the controller are (see figure 1)
wk¼ w1ðkTÞÞ .. . wRðkTÞ 2 6 6 6 4 3 7 7 7 5¼ y1ðkT s1ðkÞÞ .. . yRðkT sRðkÞÞ 2 6 6 6 4 3 7 7 7 5 ¼ C1xðkT s1ðkÞÞ .. . CRxðkT sRðkÞÞ 2 6 6 6 4 3 7 7 7 5¼ C1A~ss1ðkÞxk 1 ^ss1ðkÞ .. . CRA~ssRðkÞxk 1 ^ssRðkÞ 2 6 6 6 4 3 7 7 7 5 þ C1P1j¼0B j ~ ss1ðkÞðk 1 ^ss1ðkÞÞvðk 1 ^ss1ðkÞ jÞ .. . CR P1 j¼0B j ~ ssRðkÞðk 1 ^ssRðkÞÞvðk 1 ^ssRðkÞ jÞ 2 6 6 6 6 4 3 7 7 7 7 5 ¼X 2 i¼1 ?ikxk iþ X3 i¼1 (ikvk i ð12Þ where F.-L. Lian
?ik¼ ?i1k .. . ?iRk 2 6 6 4 3 7 7 5; ?irk ¼ CrA~ssrðkÞ; if ^ssrðkÞ ¼ i 1 0; otherwise 8 > < > : and (ik¼ (i1k .. . (iRk 2 6 6 4 3 7 7 5; (irk¼ CrB0~ssrðkÞðk 1Þ if i¼ 1 Cr½B1~ssrðkÞðk 1Þ þ B 0 ~ ssrðkÞðk 2Þ; if i ¼ 2 CrB1~ssrðkÞðk 2Þ if i¼ 3 8 > > > > > > < > > > > > > : 4.5. Closed-loop model of NCSs
In order to analyse the system property and provide guidelines for controller design, we will derive the closed-loop model that combines the discrete-time plant model (9), controller model (2), and (12).
xkþ1 ¼ Axkþ B0k½Hzkþ Jwk þ B1kvk 1 ¼ Axkþ B0kHzk þ B0kJ X2 i¼1 ?ikxk iþX 3 i¼1 (ikvk i " # þ B1kvk 1 ð13Þ
Also, the controller dynamics (2) can be further expressed as zkþ1¼ Fzkþ G X2 i¼1 ?ikxk iþ X3 i¼1 (ikvk i " # ð14Þ vk¼ Hzkþ J X2 i¼1 ?ikxk iþ X3 i¼1 (ikvk i " # ð15Þ
By further combining (13)–(15) and defining Xk¼ xTk xTk 1xTk 2j zTk j vTk 1 vTk 2vTk 3
T
2 R3NþQþ3R we obtain the closed-loop dynamics as
Xkþ1¼ .kXk ð16Þ where .k¼ .kð1; 1Þ .kð1; 2Þ .kð1; 3Þ .kð2; 1Þ .kð2; 2Þ .kð2; 3Þ .kð3; 1Þ .kð3; 2Þ .kð3; 3Þ 2 6 6 6 4 3 7 7 7 5 2 Rð3NþQþ3RÞð3NþQþ3RÞ ð17Þ
The variables .ði; jÞ are defined as
.kð1; 1Þ ¼ A B 0 kJ?1k B0kJ?2k I2N2N 02NN " # 2 R3N3N .kð1; 2Þ ¼ B 0 kH 02NQ " # 2 R3NQ .kð1; 3Þ ¼ B 0 kJ(1kþ B1k Bk0J(2k B0kJ(3k 02N3R " # 2 R3N3R .kð2; 1Þ ¼ 0QN G?1k G?2k h i 2 RQ3N .kð2; 2Þ ¼ F 2 RQQ .kð2; 3Þ ¼ G(1 k G(2k G(3k 2 RQ3R .kð3; 1Þ ¼ 0RN J? 1 k J?2k 02R3N " # 2 R3R3N .kð3; 2Þ ¼ H 02RQ 2 R3RQ .kð3; 3Þ ¼ J( 1 k J(2k J(3k I2R2R 02RR " # 2 R3R3R
The closed-loop system could be time-varying since .k
will depend on the network delay characteristics. If the network delays are constant, then the closed-loop system will be time-invariant.
At the first step of analysis procedures, we assume the controller has been designed. Then the bound for different network delays can be found based on the stab-ility criterion that all the eigenvalues of .kare less than
1. However, in the MIMO case, there are Mþ R differ-ent time delays and it is very difficult to determine the upper bound of delay values. The models (13)–(15) pro-vide the system structure under network delays. For the controller design, although the exact value of the system matrices are unknown, an estimation algorithm can be developed to identify on-line the constant system par-ameters when the system has constant network delays. For random time delays due to a network protocol or random processing times, a stochastic controller may be used to guarantee stability and performance (Krtolica et al.1994, Tsai and Ray 1997, Nilsson et al. 1998).
5. Illustrative example of NCS stability analysis In this section, we consider a two-axis example of a three-axis milling machine tool. Each axis moves on a linear slide and is driven through a ball screwby a DC motor with a tachometer which provides an angular vel-ocity measurement. The DC motor is driven by a PWM drive. Each axis also has a linear encoder that provides linear position measurement. Therefore, both position and velocity feedback are available. The two axes oper-ate independently. The time constants (sec) for each axis are 0:055 (X) and 0:056 (Y) and the overall gains K ((mm/s)/PWM) are 28.346 (X) and 28.956 (Y), respect-ively.
Then, we define x1¼ Px; x2¼ Vx, x3¼ Py, x4¼ Vy,
u1¼ ux and u2¼ uy, where Pi and Vi are the position
and velocity variables of the i-axis. The state space form of the two-axis system can be expressed as
_ x x1 _ x x2 _ x x3 _ x x4 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 ¼ 0 1 0 0 0 18:18 0 0 0 0 0 1 0 0 0 17:86 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 x1 x2 x3 x4 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 þ 0 0 515:38 0 0 0 0 517:07 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 u1 u2 " #
We further assume that csrðkÞ þ psrðkÞ r, amðkÞ < T
and maxðamÞ < T maxðsrÞ, for r¼ 1; . . . ; 4 and
m¼ 1; 2, respectively. These assumptions can be achieved by properly selecting the sampling period T given sensing and actuation delays. Based on Assumptions 4 and 5 in } 3, we have the following relations: sr¼ r, for r¼ 1; . . . ; 4. Hence, from (3),
the plant input signals u1ðtÞ and u2ðtÞ can be described
as follows, for kT t < ðk þ 1ÞT: uiðtÞ ¼
viðk 1Þ1ð0;aiÞðtÞ þ viðkÞ1ðai;TÞðtÞ, i ¼ 1; 2. Although the
model presented in } 4 is valid for time-varying delays, in this illustrative example, we assume that these actua-tor delays am are constant and known in advance, says
a1¼ 1 ms and a2¼ 2 ms. By applying (9), we can obtain
the discrete-time plant model (at T¼ 10 ms) as
xkþ1 ¼ 1 0:0091 0 0 0 0:8338 0 0 0 0 1 0:0092 0 0 0 0:8365 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 xk þ 0:0198 0 4:2788 0 0 0:0158 0 3:8547 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 vkþ 0:0045 0 0:4336 0 0 0:0086 0 0:8807 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 vk 1
We can further calculate the signal xrðkT srÞ by
obtaining the sensing delays of sr, say s1¼ 3, s2¼ 4,
s3 ¼ 5 and s4¼ 6 (ms). For example, by applying
(11), x1ðkT s1Þ can be described as x1ðkT s1Þ ¼ 1 0:0066½ 0 0xk 1 þ 0:0198 0½ vk 1þ 0:0045½ 0vk 2 Therefore, (12) becomes wk¼ 1 0:0066 0 0 0 0:8966 0 0 0 0 1 0:0048 0 0 0 0:9311 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 xk 1 þ 0:0089 0 2:4632 0 0 0:0023 0 1:0159 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 vk 1þ 0:0032 0 0:4663 0 0 0:0040 0 0:9803 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 vk 2
In order to validate the stability and performance of standard controller design, we first consider a memory-less state feedback controller, i.e. uðtÞ ¼ KxðtÞ, where Kis designed based on the pole placement in continuous time domain. In this case, F; G and H are zero matrices of compatible dimension, but J¼ K. Therefore, Xk¼ xTk xTk 1 zTk vTk 1 vTk 2
T
. In this example, the system dimension is 13. However, if there are no time delays, then the system dimension becomes 7, i.e. Xk¼ xTk zTk vTk 1
T. The 13 eigenvalues (‘’) of . are
plotted in figure 3 along with the 7 eigenvalues (‘’) of the closed-loop system without delays. Figures 3(a) and (b) showtwo different feedback gains K. The values of time delays (sr, r¼ 1; . . . ; 4 and am, m¼ 1; 2) are
iden-tical in figures 3(a) and (b). In each plot, the dotted lines are the real and imaginary axes and the solid line is the unit circle. The ‘’ symbols are the locations of the eigenvalues of the closed-loop system with delays and the ‘’ symbols are those without delays. However, only the four right-most points map to the eigenvalues of original closed-loop system, i.e. eig(Ap BpK). From
this comparison, we find that the closed-loop systems with multiple time delays will perform differently if the controller designer does not consider these time delays at the first design stage. In fact, at some combination of different time delays and sampling periods, the closed-loop systems could be unstable.
6. Formulation for optimal controller design
In this section, we formulate and solve the optimal controller design problem for the NCS model presented in } 4. The standard control algorithm utilizes the state values at the sampling instants as controller inputs. In an NCS, the sensor signals are not all sampled at the same instants. Hence, in order to apply the linear quad-ratic regulator (LQR) optimal controller design pro-cedure, we develop a delay transformation which maps an NCS model into a standard control model with delayed states as state variables. Although the model F.-L. Lian
presented in } 4 is valid for time-varying delays, for the ease of presentation, we only consider the case where sensor and actuator delays are constant and y¼ x, i.e. N¼ R.
The optimal control derivation proceeds as follows. First, we incorporate the sensing and actuation delays into the system model as described in } 4. This results in a delayed state-variable model with the same dimension, (x2 RN) as the original system; however, the A and B matrices have changed to include the delays. If the delays are constant, as is assumed here, the delayed state-variable model is time-invariant. Standard optimal control techniques are then applied to this delayed state-variable model.
Recall that xðkTÞ represents the actual values of the system states at the controller sample times; this value is not measured. xrðkT srÞ, r ¼ 1; . . . ; R are the values of
the rth state received at the controller. Suppose that the sensor delays are all less than one sampling period, i.e. sr< T, r¼ 1; . . . ; R. As stated in (11), we use xðk ~ssrÞ,
where ~ssr¼ sr=T, to denote the true value of the
contin-uous state at time ðk ~ssrÞT. Using (9), we can express
the value of the state at the sensor sampling instants, xðk þ 1 ~ssrÞ as
xðk þ 1 ~ssrÞ ¼ Axðk ~ssrÞ þ
X2 j¼0
Brjvðk jÞ ð18Þ
where Brj¼ ½ B1rj ; . . . ; Bmrj ; . . . ; BMrj and Bmrj ðkÞs are
defined as If T sr am, then B0mrðkÞ ¼4 ðT amþsr !mðT; qÞ dq B1mrðkÞ ¼ 4 ðamþsr 0 !mðT; qÞ dq and B2mrðkÞ ¼ 4 0 If T sr< am, then B0mrðkÞ ¼40; B1mrðkÞ ¼4 ðT amþsr T !mðT; qÞ dq and B2mrðkÞ ¼4 ðamþsr T 0 !mðT; qÞ dq
Note that the actuator delays are taken care of in the B matrix terms. The dependence of Brj on the time k is
dropped because all delays are assumed constant in this section. Since the sensor delays are different for each measurement, we need to carefully extract the cor-rect dynamics for each state. The rth element of the vector xðÞ, i.e. the individual state xrðÞ, can be
described as
xrðk þ 1 ~ssrÞ ¼ Ahr;rixrðk ~ssrÞ þ Ahr; rixh riðk ~ssrÞ
þX
2
j¼0
Brjhr;ivðk jÞ ð19Þ
where Ahr;ri2 R is the (r,r)th element of matrix A, Ahr; ri2 R1ðR 1Þ is the rth rowwith the rth element deleted, Bhr;i2 R1M is the rth rowof matrix B and xh ri is the x vector with the rth element deleted.
xrðk ~ssrÞ are the sensed values of the rth state that
are sampled and sent over the network.
In the meantime, by using (10) and letting ¼ ~ssr, w e
can also express the sensed state xðk þ 1 ~ssrÞ,
Networked control systems with multiple delays
0 0.2 0.4 0.6 0.8 1 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4
Eigenvalues: x(red): without delay, o(blue): with delays
Real Imaginar y (a) 0.2 0 0.2 0.4 0.6 0.8 1 1.2 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5
Eigenvalues: x(red): without delay, o(blue): with delays
Real
Imaginar
y
(b)
Figure 3. The location of eigenvalues of closed-loop systems. (a) Design 1, (b) Design 2.
r¼ 1; . . . ; R, in terms of the state of the system at the controller sample instant, xðkÞ, as
xðk þ 1 ~ssrÞ ¼ A~ssrxðkÞ þ
X1 j¼0
B~ssj
rvðk jÞ ð20Þ
Then, we can separate the rth element from the rest of the vector xðÞ as xrðk þ 1 ~ssrÞ ¼ A hr;i ~ ssr xðkÞ þ X1 j¼0 Bss~jhr;i r vðk jÞ; ð21Þ and xh riðk þ 1 ~ssrÞ ¼ A h r;i ~ ssr xðkÞ þ X1 j¼0 B~ssjh r;i r vðk jÞ ð22Þ In order to formulate the system dynamics in terms of measured states, we further define the new variables
xsðkÞ ¼ x1ðk s1Þ x2ðk s2Þ .. . xRðk sRÞ 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 ; As ¼ Ah1;i~ss 1 Ah2;i~ss 2 .. . AhR;i~ss R 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 and Bjs¼ B~ssjh1;i 1 B~ssjh2;i 2 .. . BjhR;i~ss R 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5
xsðkÞ will be the state variables of the new delayed state-variable model. By applying (21) for r¼ 1; . . . ; R, w e have the equation
xsðk þ 1Þ ¼ AsxðkÞ þ
X1 j¼0
Bsjvðk jÞ ð23Þ
Since As is the combination of A~ssr, i.e. expðAp~ssrTÞ,
which is non-singular, we can multiply both sides of (23) by A 1s and obtain the actual value of the state at
the controller sample instant as a function of the sensed (delayed) values of the states and the inputs
xðkÞ ¼ A 1sxsðk þ 1Þ A 1s
X1 j¼0
Bjsvðk jÞ ð24Þ
We further apply (24) to (22) at time k 1 to find the values of the other system states at the instant when the rth state is sampled xh riðk ~ssrÞ ¼ A h r;i ~ ssr A 1 sxsðkÞ A 1s X1 j¼0 Bjsvðk 1 jÞ " # þX 1 j¼0 Bj< r;>~ss r vðk 1 jÞ ¼ Axr ~ ssrxsðkÞ þ X1 j¼0 Bxrj ~ ssr vðk 1 jÞ ð25Þ
where the first equality is from (24) at time k 1, and, in the second equality, we use the definitions Axr
~ ssr ¼ 4 Ah r;i~ss r A 1 s, and Bx~ssrrj¼ 4 B~ssjh r;i r A h r;i ~ ssr A 1 sBsj . Hence,
by further plugging (25) into (19), we have xrðk þ 1 ~ssrÞ ¼ Ahr;rixrðk ~ssrÞ þ X2 j¼0 Brjhr;ivðk jÞ þ Ahr; ri ( Ah r;i~ss r A 1 s h i xsðkÞ þX 1 j¼0 B~ssjh r;i r A h r;i ~ ssr A 1 sBsj h i vðk 1 jÞ ) ð26Þ
Therefore, by using the definition of xsðkÞ, we can
obtain the newdelayed state-variable model as
xsðk þ 1Þ ¼ AxsxsðkÞ þ X2 j¼0 Bxjsvðk jÞ ð27Þ where Axs ¼ Að1;1Þ 0 0 0 .. . 0 0 0 AðR;RÞ 2 6 6 6 4 3 7 7 7 5þ Ah1; 1iAh 1;i~ss 1 A 1 s .. . AhR; RiAh R;i~ss R A 1 s 2 6 6 6 6 4 3 7 7 7 7 5 and Bxjs¼ B0h1;i1 .. . B0hR;iR 2 6 6 4 3 7 7 5 j¼ 0
B1jh1;iþ Ah1; 1i Bj 1h 1;i ~ ss1 A h 1;i ~ ss1 A 1 sBsj 1 h i .. .
BRjhR;iþ AhR; Ri B~ssj 1h R;iR Ah R;i~ssR A
1 sBsj 1 h i 2 6 6 6 6 4 3 7 7 7 7 5; j¼ 1; 2 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : We further define zðkÞ ¼ ½ xsðkÞT; vðk 1ÞT;
vðk 2ÞTT2 RNþ2R. zðkÞ will be the new state vari-ables in the standard optimal control design technique. Then, we have the state-space model for controller design
zðk þ 1Þ ¼ AzzðkÞ þ BzvðkÞ ð28Þ
where Az¼ Axs B 1 xs B 2 xs 0 0 0 0 I 0 2 6 6 4 3 7 7 5 2RðNþ2RÞðNþ2RÞ and Bz¼ B0xs I 0 2 6 4 3 7 5 2 RðNþ2RÞR
For the controller design of tracking problem, we first let the desired trajectory dðtÞ or dðkÞ be described by the equation _ddðtÞ ¼ Ac ddðtÞ; in continuous time dðk þ 1Þ ¼ AdddðkÞ; in discrete time 9 = ; ð29Þ or where dðÞ ¼ d½ 1ðÞ; . . . ; dRðÞ T
. Furthermore, since the newstate variables are measured at time k ~ssr, w e define the desired trajectory at each individual time instant k ~ssr as dsðkÞ ¼
4
d1ðk ~ss1Þ; . . . ; dRðk ~ssRÞ
½ T.
For the optimal controller design, we then choose the cost function V V¼X N k¼0 zðkÞ zdðkÞ ½ TQ zðkÞ z½ dðkÞ þ vðkÞTRvðkÞ ð30Þ where zdðkÞ ¼ 4
½dsðkÞT; 0TT, i.e. the combined vector of
desired trajectory and zero input, and Q and R are defined as R¼ R0; and Q¼ Qs 0 0 0 R1 0 0 0 R2 2 6 6 4 3 7 7 5
and Qsand Riare the weighting matrices for the original
system states and inputs, respectively. Based on the stan-dard LQR algorithm (Lewis 1986), the optimal control-ler law vðkÞ can be obtained as
vðkÞ ¼ KðkÞ zðkÞ z½ dðkÞ ¼ KsðkÞ x½ sðkÞ dsðkÞ X2 i¼1 KivðkÞvðk iÞ ð31Þ where K¼ Ks; K1v; K2v
. The newoptimal controller uti-lizes the available measured states as well as past inputs as the controller inputs. The number of past inputs depends on the sensor and actuator delays as described in (14) and (15). An NCS-LQR design example for the illustrative example presented in } 5 will be presented in the next section.
7. Numerical examples of NCS-LQR
In this section, we revisit the two-axis example in } 5. Three cases of the optimal controller design will be com-pared. The first case applies the standard LQR control-ler for the original delay-free system. In the second case, the LQR controller that is designed based on delay-free system is used to control the delayed system. The final case considers the LQR design based on the delayed state-variable model discussed in } 6.
The actuation and sensing delays are assumed as fol-lows: a1 ¼ 1, a2¼ 2, s1¼ 3, s2¼ 4, s3¼ 5 and s4 ¼ 6
(ms) and the sampling time is 10 ms. The choice of this delay characteristics is based on the performance analy-sis of one type of priority-based control networks such as DeviceNetTM{ (Lian et al. 2001 a). Also, based on the assumptions used in } 5, equation (27) has the form
xsðk þ 1Þ ¼ 1 0:0090 0 0 0 0:8338 0 0 0 0 1 0:0090 0 0 0 0:8365 2 6 6 6 6 4 3 7 7 7 7 5xsðkÞ þ 0:0323 0 3:9787 0 0 0:0342 0 3:4630 2 6 6 6 6 4 3 7 7 7 7 5vðkÞ þ 0:0104 0 0:4032 0 0 0:0175 0:0088 0:7912 2 6 6 6 6 4 3 7 7 7 7 5vðk 1Þ þ 0 0 0 0 0 0 0 0 2 6 6 6 6 4 3 7 7 7 7 5vðk 2Þ Furthermore, the newdelayed state-variable model (28) becomes zðk þ 1Þ ¼ 1 0:0090 0 0 0:0104 0 0 0 0 0:8338 0 0 0:4032 0 0 0 0 0 1 0:0090 0 0:0175 0 0 0 0 0 0:8365 0:0088 0:7912 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 2 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 5 zðkÞ þ 0:0323 0 3:9787 0 0 0:0342 0 3:4630 1 0 0 1 0 0 0 0 2 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 5 vðkÞ
Networked control systems with multiple delays
{ DeviceNet is a sensor bus communication protocol commonly used as in an NCS. DeviceNet utilizes a non-destructive collision resolution scheme through message priority.
For the LQR optimal controller design, we first choose the weighting matrices
Qs¼ diagð20; 0:05; 20; 0:05Þ
R0¼ diagð0:1; 0:1Þ; R1¼ diagð0:1; 0:1Þ
and
R2¼ diagð0:001; 0:001Þ
Most of the cost function is weighted upon position tracking. Since vðkÞ; vðk 1Þ and vðk 2Þ are not inde-pendent, we discount the weights on previous inputs. By using the discrete-time system matrix A and input matrix B, and the weighting matrices Qs and R0, the
LQR state feedback gain, called Klqr, is computed as
Klqr¼ 3:6038 0:1829 0 0
0 0 3:5879 0:1827
" #
The gain is the same as the standard LQR design for the delay-free systems. On the other hand, by using the NCS system matrix Az, input matrix Bz and the weighting
matrices Q and R, the NCS-LQR state feedback gain, called Kncslqr, is obtained as
Kncslqr¼
3:7917 0:1802 0 0 0:1102 0 0 0
0:0002 0 4:0722 0:1881 0:0016 0:2145 0 0
" #
Kncslqr is computed based on the proposed NCS delayed state-variable model discussed in } 6.
Simulation results of the first LQR design are shown in figures 4–6. Figure 4 shows the location of eigenvalues of three different closed-loop systems: (1) the discrete-time LQR (DT-LQR) controller for the delay-free system, (2) the DT-LQR controller for the delayed F.-L. Lian 1 0.5 0 0.5 1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Eigenvalue Location Real Image Delayed Plant w/ DTLQR Delayed Plant w/ NCSLQR DelayFree Plant w/ DTLQR
Figure 4. Eigenvalue location of three different closed-loop systems: Klqr for delay-free system, Klqr for delayed system and Kncslqr for delayed system.
0 5 0 50 100 150 200 X axis Time (sec) P osition (mm) 0 5 100 50 0 50 X axis Time (sec) V elocity (mm/sec) 0 5 0 50 100 150 200 Y axis Time (sec) P osition (mm) 0 5 100 50 0 50 Y axis Time (sec) V elocity (mm/sec) 0 50 100 150 200 0 50 100 150 200 X Axis (mm) Y Axis (mm)
2D XY Plane of DTLQR w/o Delay
system, and (3) the NCS-LQR controller for the delayed system. The eigenvalue locations for the NCS-LQR case are closer to those of the DT-LQR for the delay-free system than those of the DT-LQR for the delayed system.
Figure 5 shows the simulation results of the tracking performance of the two-axis system, i.e. X and Y axes, with the DT-LQR controller for the delay-free system. Here, two constant feed-rates for forward and backward motions are considered as the desired trajectory for both X and Y axes. For this standard LQR case where time delays are not considered, vðkÞ ¼ Klqr½xðkÞ dðkÞ is
used as controller input. The plot on the left-hand side is the top-viewtrajectory, and the four plots on the right-hand side are the positions and velocities of X and Y axes. This case can be used as a basis for compar-ison. Figure 6 shows the summary of the tracking errors
with different indexes. The left-hand plot is the root-mean-square tracking errors of (1) the entire simulation time (0–7 s), (2) the first second (0–1 s), and (3) the last two seconds (5–7 s). The right-hand plot is the result of the integral of the time multiplied by the absolute values of the error (ITAE). The NCS-LQR controller improves the control performance over the DT-LQR controller when considering the time delays.
The simulation results of two other desired trajec-tories, a circle with a radius of 100 mm, and constant trajectories for both position and velocity, are also shown in figures 7–10. The circular trajectory can be viewed as a time-varying trajectory; the constant trajec-tory is a regulation case for both position and velocity. For both types of trajectories, the proposed NCS-LQR guarantees better control performance than the DT-LQR. Especially, the 2-D top-viewplots in figures 7 Networked control systems with multiple delays
Delayed Plant w/ DTLQR to DelayFree case Delayed Plant w/ NCSLQR to DelayFree case DelayFree Plant w/ DTLQR to itself
0 1 2 3 4 0 10 20 30 40
Ratio of RMS of Tracking Error
1: 07 (sec), 2: 01 (sec), 3: 57 (sec)
0 1 2 3 4 0 10 20 30 40
Ratio of ITAE of Tracking Error
1: 07 (sec), 2: 01 (sec), 3: 57 (sec)
Figure 6. The summary of tracking errors: Klqrfor delay-free system, Klqrfor delayed system and Kncslqr for delayed system. Left:
RMS, right: ITAE. 100 50 0 50 100 50 0 50 100 X Axis (mm) Y Axis (mm) 2D XY Plane of DTLQR w Delay 50 0 50 50 0 50 X Axis (mm) Y Axis (mm) 2D XY Plane of NCSLQR w Delay
F.-L. Lian
Delayed Plant w/ DTLQR to DelayFree case Delayed Plant w/ NCSLQR to DelayFree case DelayFree Plant w/ DTLQR to itself
0 1 2 3 4
0 5 10 15
Ratio of RMS of Tracking Error
1: 07 (sec), 2: 01 (sec), 3: 57 (sec)
0 1 2 3 4 0 5 10 15 20 25 30
Ratio of ITAE of Tracking Error
1: 07 (sec), 2: 01 (sec), 3: 57 (sec)
Figure 8. The summary of ‘circular’ tracking errors: Klqrfor delay-free system, Klqrfor delayed system and Kncslqrfor delayed system. Left: RMS, right: ITAE.
5 0 5 6 4 2 0 2 4 6 X Axis (mm) Y Axis (mm) 2D XY Plane of DTLQR w Delay 5 0 5 5 0 5 X Axis (mm) Y Axis (mm) 2D XY Plane of NCSLQR w Delay
Figure 9. The 2-D top viewof ‘constant’ X–Y axis system: left: Klqrfor delayed system; right: Kncslqr for delayed system.
Delayed Plant w/ DTLQR to DelayFree case Delayed Plant w/ NCSLQR to DelayFree case DelayFree Plant w/ DTLQR to itself
0 1 2 3 4 0 5 10 15 20 25 30
Ratio of RMS of Tracking Error
1: 07 (sec), 2: 01 (sec), 3: 57 (sec)
0 1 2 3 4 0 5 10 15 20 25 30
Ratio of ITAE of Tracking Error
1: 07 (sec), 2: 01 (sec), 3: 57 (sec)
Figure 10. The summary of ‘constant’ tracking errors: Klqrfor delay-free system, Klqrfor delayed system and Kncslqr for delayed system. Left: RMS, right: ITAE.
and 9 showthe dramatic improvement of the proposed NCS-LQR controller design.
8. Summary and future work
In this paper we analysed and modelled a MIMO networked control system with multiple time delays. The time delays between sensor-controller and control-ler-actuator and the time skews at different devices’ sam-pling instants were considered and included in the derivation of a discrete-time MIMO model. By includ-ing these time delay parameters, both the control system and network system designers can utilize this model to design networked control systems and optimize their overall performance.
The closed-loop NCS model discussed in } 4 only included a standard controller designed without consid-ering the time delay effect a priori. Therefore, this closed-loop stability analysis was used as a verification tool for the stability and performance of an NCS. Based on the time-delay modelling algorithm and considering the case of constant communication delays, a delayed state-variable model was formulated for the standard LQR controller design in } 6. This NCS-LQR controller improved the control performance over the DT-LQR controller when considering the time delays. Simulation studies provided in }} 5 and 7 demonstrated the utility of the proposed design algorithms for stability analysis and performance improvement.
The advantage of having an NCS architecture is that it provides the flexibility to quickly reconfigure the system architecture, and to easily share information with other subsystems. The change of system configura-tion may also change the time delay of a networked device even in a deterministic NCS. Hence, the advanced networked controller should be able to maintain a proper level of network traffic load and adaptively mod-ify the controller algorithm. This slowly changing situa-tion can be modelled as an adaptive control problem which can adjust on-line to the changing system par-ameters. Also, before a truly deterministic network is available for control applications, the time-varying delays should be included in the modelling algorithm, and a stochastic controller design could be adopted. For the uncertainties existing in an NCS such as the variance of time delays and system uncertainty, a robust control-ler design could then be applied. In addition, an intelli-gent controller design that incorporates both network and control parameters to further improve both net-work and control performance should be studied in the future. This intelligent networked controller can identify/estimate network performance and operate among multiple modes of control action, depending on network traffic or control performance.
Acknowledgements
The authors are grateful to the anonymous referees for their careful reading of the paper and for their pertinent suggestions. This research was supported in part by the NSF Engineering Research Center for Reconfigurable Machining Systems, University of Michigan, under the grant EEC95-92125, the DARPA MICA Program, the NewFaculty Research Fund at National Taiwan University and the National Science Council, Taiwan, ROC, under the grant NSC 91-2213-E-002-133.
References
Branicky, M. S., Phillips, S. M., and Zhang, W., 2000, Stability of networked control systems: explicit analysis of delay. Proceedings of 2000 American Control Conference, Chicago, IL, USA, pp. 2352–2357.
Dugard, L., and Verrest, E. I., 1998, Stability and Control of Time-delay Systems (Berlin–Heidelberg–NewYork: Springer-Verlag).
Franklin, G. F., Po well, J. D., and Workman, M. L., 1998, Digital Control of Dynamic Systems, third edition (Menlo Park, CA: Addison-Wesley).
Go« ktas, F., Smith, J. M., and Bajcsy, R., 1996, -Synthesis for distributed control systems with network-induced delays. Proceedings of the 35th Conference on Decision and Control, Kobe, Japan, pp. 813–814.
Go« ktas, F., Smith, J. M., and Bajcsy, R., 1997, Telerobotics over communication networks. Proceedings of the 36th Conference on Decision and Control, San Diego, CA, USA, pp. 2399–2404.
Go¤ recki, H., Fuksa, S., Grabowski, P., and Korytowski, A., 1989, Analysis and Synthesis of Time Delay Systems(NewYork: Wiley).
Goubet-Bartholome¤u« s, A., Dambrine, M., and Richard, J. P.,1997, Stability of perturbed systems with time-varying delays. Systems and Control Letters, 31, 155– 163.
Gu, K.,1999, Discretized Lyapunov functional for uncertain systems with multiple time-delay. International Journal of Control, 72, 1436–1445.
Gu, Y., Wang, S., Li, Q., Cheng, Z., and Qian, J., 1998, On delay-dependent stability and decay estimate for uncertain systems with time-varying delay. Automatica, 34, 1035–1039.
Halevi, Y., and Ray, A., 1988, Integrated communication and control systems: Part I—analysis. ASME Journal of Dynamic Systems, Measurement, and Control, 110, 367– 373.
Hsiao, F.-H., and Hwang, J.-D., 1997, Stability analysis of uncertain feedback systems with multiple time delays and series nonlinearities. Journal of Franklin Institute, 334, 491–505.
Jeung, E. T., Oh, D. C., Kim, J. H., and Park, H. B., 1996, Robust controller design for uncertain systems with time delays: LMI approach. Automatica, 32, 1229–1331. Kim, Y. H., Kwon, W. H., and Park, H. S., 1996, Stability
and a scheduling method for network-based control systems. IECON Proceedings, Taipei, Taiwan, pp. 934–939.
Krtolica, R., O« zgu«ner, U« , Chan, H., Go« ktas, H., Winkelman, J., and Liubakka, M., 1994, Stability of linear Networked control systems with multiple delays
feedback systems with random communication delays. International Journal of Control, 59, 925–953.
Lewis, F. L.,1986, Optimal Control (NewYork: Wiley). Li, X., and de Souza, C. E., 1997, Delay-dependent robust
stability and stabilization of uncertain linear delay systems: a linear matrix inequality approach. IEEE Transactions on Automatic Control, 42, 1144–1148.
Lian, F.-L., Moyne, J. R., and Tilbury, D. M., 2001 a, Performance evaluation of control networks: Ethernet, ControlNet, and DeviceNet. IEEE Control Systems Magazine, 21, 66–83.
Lian, F.-L., Moyne, J. R., and Tilbury, D. M., 2001 b, Analysis and modeling of networked control systems: MIMO case with multiple time delays. Proceedings of 2001 American Control Conference, Arlington, VA, USA, pp. 4306–4312.
Lian, F.-L., Moyne, J. R., and Tilbury, D. M., 2002 a, Network design consideration for distributed control sys-tems. IEEE Transactions on Control Systems Technology, 10, 297–307.
Lian, F.-L., Moyne, J. R., and Tilbury, D. M., 2002 b, Optimal controller design and evaluation for a class of net-worked control systems with distributed constant delays. Proceedings of 2002 American Control Conference, Anchorage, AL, USA, pp. 3009–3014.
Luo, J. S., van den Bosch, P. P. J., Weiland, S., and Goldenberge, A.,1998, Design of performance robustness for uncertain linear systems with state and control delays. IEEE Transactions on Automatic Control, 43, 1593– 1596.
Malek-Zavarei, M., and Jamshidi, M., 1987, Time-Delay Systems: Analysis, Optimization and Applications (North-Holland: Amsterdam).
Marshall, J. E.,1979, Control of Time-Delay Systems (Petter Peregrinus: Stevenage UK).
Nilsson, J.,1998, Real-time control systems with delays. PhD thesis, Lund Institute of Technology, Lund, Sweden. Nilsson, J., and Bernhardsson, B., 1997, LQG control over
a Markov communication network. Proceedings of the 36th Conference on Decision and Control, San Diego, CA, USA, pp. 4586–4591.
Nilsson, J., Bernhardsson, B., and Wittenmark, B., 1998, Stochastic analysis and control of real-time systems with random time delays, Automatica, 34, 57–64.
OGˇuzto« reli, M. N.,1966, Time-lag Control Systems (New York: Academic Press).
Tsai, N.-C., and Ray, A., 1997, Stochastic optimal control under randomly varying distributed delays. International Journal of Control, 68, 1179–1202.
Walsh, G. C., Beldiman, O., and Bushnell, L., 1999 a, Asymptotic behavior of networked control systems. Proceedings of the International Conference on Control Applications, Hawaii, USA, pp. 1448–1453.
Walsh, G. C., Ye, H., and Bushnell, L., 1999 b, Stability analysis of networked control systems. In Proceedings of the American Control Conference, San Diego, CA, USA, pp. 2876–2889.
Wittenmark, B., Nilsson, J., and Torngren, M., 1995, Timing problems in real-time control systems. Proceedings of American Control Conference, Seattle, WA, USA, pp. 2000–2004.
