www.ietdl.org Published in IET Control Theory and Applications Received on 17th December 2008 Revised on 21st May 2009 doi: 10.1049/iet-cta.2008.0571
ISSN 1751-8644
New study of controller design for networked control systems T.C. Yang1 C. Peng2 D. Yue3 M.R. Fei4 1
Department of Engineering and Design, University of Sussex, Sussex BN1 9QT, UK School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing, People’s Republic of China 3 Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, People’s Republic of China 4 Shanghai Key Laboratory of Power Station Automation Technology, School of Mechatronics Engineering and Automation, Shanghai University, Shanghai, People’s Republic of China E-mail:
[email protected] 2
Abstract: Recently there is a great research interest in networked control systems (NCSs). The intended contributions of this study, in the order of they are presented, are (a) to point out a major limitation of current studies on NCSs: many theoretical results – although important and have some significant technical merit – cannot be applied to distributed control systems and therefore have little application value; (b) in contrast to that simple numerical examples – often linear systems – are widely used to demonstrate some theoretical results. The authors study the NCS controller design for a non-linear unstable system, which has been historically used for the study of large-scale system control; (c) application of a new design approach: it is suggested that the NCS controller design problem, where data packet time delay, loss or out-of-order need to be addressed, is equivalent to a design problem where the system is point-to-point connected but has delayed and unreliable measurement subsystems; (d) a theorem for the asymptotic stability of a class of non-linear time-delay systems and (e) the stability theorem and the proposed new design approach are applied to the controller design for a model of a network-connected non-linear unstable system stated in (b). Extensive simulation results presented here have demonstrated the effectiveness of the proposed new design approach and the stability theorem.
1
Introduction
Recently, networked control systems (NCSs) have attracted a lot of research interests, thereby leading to a significant number of publications, to name a few see [1 – 10] and references therein. In this paper, we first point out two limitations in the existing NCS research: (i) most studies are limited to linear systems and examples used to demonstrate the applications of theoretical results in most papers are very simple numerical examples and (ii) the main applications of NCSs are for distributed control systems. However, some theoretical results, including some recent works – for example see [1 – 4], although important and have some significant technical merit – are based on some assumptions which are not valid for distributed control systems. This issue was raised before [11] and in this paper, we will re-address this in detail in Section 2. IET Control Theory Appl., 2010, Vol. 4, Iss. 7, pp. 1109 – 1121 doi: 10.1049/iet-cta.2008.0571
In Section 3, we present a model of two pendulums coupled by a spring (TPCS) studied in this paper for networked control. This non-linear unstable system has been used as a classical test bed for the study of decentralised control of large systems [12 – 16]. Each pendulum of the two is treated as a subsystem and they are coupled by a spring between them. In this paper, we extend the study of controller design by introducing networked signal transfer to replace the traditional point-to-point connections used in all the previous studies. We also suggest a different approach to deal with the uncertainties in networked signal transfer: the NCS design problem is equivalent to a problem of designing a robust controller for a time-delay system where all the signals are point-to-point connected. Our main theoretical contribution is presented in Section 4. For a class of non-linear time-delay systems, we derive a 1109
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www.ietdl.org sufficient stability condition based on a Lyapunov – Krasovskii functional and this is applied to design a controller for the TPCS system. Extensive simulation results are presented in Section 5, followed by some conclusions in Section 6.
2 NCS for distributed control systems A networked control structure – Fig. 1 [17], where solid lines represent physical links and broken lines for signal flows – has many advantages, such as modularity, decentralisation 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 integrated applications through reduced wiring and distributed intelligence; and reduce installation, reconfiguration and maintenance time and costs. These advantages are for distributed control systems where there are multi-sensing sources and multi-actuator nodes. In Fig. 1 each labelled ‘sensor’, such as ‘sensor 1’, actually represents a sensing source of a group of sensors which can send several measured signals from one location in one data packet. However, for different sensor nodes in the figure, since they are located in different places, the measurements taken there have to be sent via different packets. The same also applies to the labelled ‘actuator’ in the figure.
In contrast to a typical NCS structure of Fig. 1, if there is only one sensing source and only one actuator node, this becomes Fig. 2 and this is termed as one-sensor one-actuator (OSOA) structure in this paper. Under this structure, one can assume that the controller is ‘event-driven’ and when a data packet is sent at t ¼ t0 from the sensor node, it will arrive at the actuator node t ¼ t3 (Fig. 3a). Without losing generality,
Figure 3 Typical timing diagrams for event-driven and time-driven controllers a Event-driven controller b Time-driven controller
Figure 1 Typical NCS set-up for distributed control
Figure 2 Remotely-controlled system through a communication network 1110 & The Institution of Engineering and Technology 2010
IET Control Theory Appl., 2010, Vol. 4, Iss. 7, pp. 1109 – 1121 doi: 10.1049/iet-cta.2008.0571
www.ietdl.org there are three main delays in this networked signal transfer and control: (i) sensor-controller delay tsc ¼ t1 2 t0 , (ii) controller computational delay tc ¼ t2 2 t1 (it is acceptable to assume tc ≃ 0 and to neglect this delay) and (iii) controller-actuator delay tca ¼ t3 2 t2 . tsc and tca are also called networkinduced delays. The overall delay t is
t = tsc + tc + tca
(1)
Packet out-of-order and loss in transmissions can also be included in the modelling. This will be further discussed in Section 3. Under a general NCS structure of Fig. 1, however, the assumption that the controller is ‘event-driven’ as made in [1 – 4] and other papers is not valid. Equation (1) which is based on ‘event-driven’ controllers is also not valid. For example, in Fig. 3a if a packet from ‘sensor 1’ arrives at the controller node at t ¼ t1 and another packet from ‘sensor 2’ arrives at the controller node at t ¼ t1 + 1, where 1 is a small random number, how can an ‘event-driven’ controller response to these packets? Therefore for a typical NCS, the controller must be ‘time-driven’: it calculates the controller output periodically based on the received signals stored in the memory (see Fig. 3b). In a typical NCS application, there are many sensing nodes (the above diagram (Fig. 3b) only shows two). A time-driven controller calculates a control signal at each regular time instance according to its own CPU clock, that is, at . . . (k 2 1)T, kT, (k + 1)T, . . . . An event-driven controller, however, starts a calculation whenever it receives a packet from a sending source. Compare the two Figs. 3a and b; it is clear that an event-driven controller cannot work for a typical distributed NCS application.
Remark 1: The first systematic modelling for general NCSs of Fig. 1 can be found in [18], where the controller is timedriven. The limitation of the model in [18] is that all delays are assumed to be bound by one sampling period. Some further study on this can be found in [19] and other literatures.
Figure 4 Two pendulums (inverted rods) coupled by a spring (TPCS)
is still used in this paper), each pendulum of the two is treated as a subsystem and they are coupled by a spring between them. This two-subsystem model is widely used in the large-scale system decentralised control literatures, to name a few, see [12– 16]. The variables in Fig. 4 are
ui: angular displacement of pendulum i (i ¼ 1, 2); ti: torque input generated by the actuator for pendulum i (i ¼ 1, 2), without causing any unnecessary confusion, the symbol ‘t’ is reused here for torque; F: spring force; ls: spring length;
f: slope of the spring to the earth;
Some researchers also realise the fundamental limitation of the OSOA structure of Fig. 2. In their paper, instead of using the term of an NCS, they call the system a ‘remote control system’ [20].
and the constants are
Remark 2: An inverted pendulum on a cart and a DC motor
mi: mass of pendulum i (i ¼ 1, 2);
have been used for NCS experimental studies in some publications. However, each of these ‘remote control’ experiment sets has only one sensor node and one actuator node, and therefore does not represent a typical distributed NCS.
3
System of TPCS
For a system of TPCS, Fig. 4 (here instead of pendulums, there are two inverted rods. Without losing generality and in line with other researchers, the term ‘inverted pendulum’ IET Control Theory Appl., 2010, Vol. 4, Iss. 7, pp. 1109 – 1121 doi: 10.1049/iet-cta.2008.0571
li: length of pendulum i (i ¼ 1, 2);
L: distance of two pendulums;
k: spring constant. The mass of each pendulum is uniformly distributed. The length of the spring is chosen so that F ¼ 0 when u1 ¼ u2 ¼ 0, which implies that (u1 u˙ 1 u2 u˙ 2 )T = 0 is an (unstable) equilibrium of the system if t1 ¼ t2 ¼ 0. For simplicity, the mass of the spring is neglected. 1111
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www.ietdl.org The dynamic equations of the coupled pendulums are ..
[m1 (l1 )2 /3] u 1 = t1 + m1 g(l1 /2)sin u1 + l1 F cos(u1 − f) (2) ..
[m2 (l2 )2 /3] u 2 = t2 + m2 g(l2 /2)sin u2 − l2 F cos(u2 − f) (3) where g ¼ 9.8 m/s2 is the constant of gravity, and F = k (ls − [L2 + (l1 − l2 )2 ]1/2 )
(4)
ls = [(L + l2 sin u2 − l1 sin u1 )2 + (l2 cos u2 − l1 cos u1 )2 ]1/2 (5)
f = tan−1
l1 cos u1 − l2 cos u2 L + l2 sin u2 − l1 sin u1
(6)
The following numerical values are used for the plant l1 = 1 m l2 = 0.8 m m1 = 1 kg m2 = 0.8 kg L = 1.2 m k = 4 N/m and the initial conditions of the two pendulums are noted as: u1 (0) = x01 u2 (0) = x02.
Remark 3: There is a modelling error in [14 – 16]. The momentum of the inertia of a pendulum (rod) about its pivot point is m(l )2/3, but in [14 – 16] this was wrongly given as ml/2. To include the dynamics associated with networked communications, in this paper it is assumed that (a) The measurements of the angular position and speed of pendulum 1, u1 and u˙ 1 , are sampled at a rate with a period of T ¼ 0.01 s. These two signals are put into one data packet at the sensor node 1 and transmitted through communication channel 1. In each data packet, there is a sequential integer number, an index, for the order of sampling at the plant end. (Note that one byte in the data segment of a packet will be sufficient for this purpose.) Similarly, u2 and u˙ 2 are sampled at the same rate and the packets, with their own index numbers, are sent from the sensor node 2 and transmitted through communication channel 2 of the same network. Each data packet arriving at the receiving end, the controller node, is subject to random time delay, packet out-of-order and packet loss. The characteristics and statistics of the packet transfer in these two channels can be different. (b) The controller is time-driven with a sampling (update) rate of T ¼ 0.01 s. The received measurement signals are stored in controller memory (zero-order-hold) to be used for the control signal calculation. Once a new packet arrives, if its index indicates that it contains the latest sampled signals at the plant, then these signals will be used to update the signals stored in the memory. In this way, although the arrived data packets maybe out-of-order due 1112 & The Institution of Engineering and Technology 2010
to variations of random delays in network transmissions, the controller always uses the latest plant measurements available to it. In addition, packet loss in transmission can be treated in the same way as long-delayed packets [19]. A detailed example of this is given in Section 5.1 (Table 2). (c) For the TPCS system studied in this paper and for a class of applications, the signal transfer between the controller and the actuators can be assumed to be ‘point-to-point fixed’. This is in line with the current technical trend of ‘intelligent actuators’ [7] where a digital controller can be implemented on a microprocessor embedded in each actuator. Basically, in order to implement an NCS, each actuator must at least have a network card and some processing power to handle the receiving of signals from a network. It is easy and cheap to increase this processing power to enable an additional function of a digital controller, if the computational demand of the control algorithm is not very high (a controller embedded in a network-enabled actuator needs to calculate only the control signal for the local actuator itself ). This can also eliminate the need of signal transfer between the controller and the actuators, and there is no associated time delay which affects the system stability and performance. It is worth noting that in an early research on NCSs [21], which has been widely quoted in later NCS studies, the word ‘smart actuator’ was used. For the same reasons given here, in theoretical analysis and simulations presented in [21], it also assumed that controllers are embedded in actuators, that is, controllers and actuators are physically combined as an intergraded unit. The TPCS system with the networked communications is termed as TPCS-NCS in this paper. The characteristics of the NCS signal transfer described in (a) and (b) above represents the main features in many applications and can be readily implemented in computer simulations. An analytical model – includes the all aspects of time delay, packet outof-order and loss – and the design based on such a model are, however, very challenging and are the main focus of many NCS researches. In this paper, we propose and demonstrate that a different approach to deal with the uncertainties in networked signal transfer can also be used. Consider, for example, a TPCS system where all signals are point-to-point connected but with imperfect and unreliable measurement subsystems to measure the pendulum angles and angle speeds. The measurement process itself is subject to random time delay – equivalent to network-induced delay in NCSs – and occasionally, it fails to take the required measurement – equivalent to data packet loss, or packet outof-order which makes late-arrived packets useless in NCSs. This system is termed as TPCS-P2P system in this paper. Here P2P stands for point-to-point and is an abbreviation used by some researchers. Similar to the TPSC-NCS controller, the TPCS-P2P controller always uses the latest measurement signals available to it. In addition to wellknown measurement noise applicable to all systems, a TPCS-P2P system has measurement delays and IET Control Theory Appl., 2010, Vol. 4, Iss. 7, pp. 1109 – 1121 doi: 10.1049/iet-cta.2008.0571
www.ietdl.org uncertainties – when a measurement (or a few consecutive measurements) is failed, the controller will use the previous value, which is the same as what will happen in an NCS if there is packet loss or out-of-order. The value of the signal error viewed from the controllers of the both TPCS-NCS and TPCS-P2P systems is the same – the difference between the current process output and its previous value. Therefore the controller design for the TPCS-P2P system with point-to-point connections and imperfect and unreliable measurement subsystems is equivalent to that for a TPCS-NCS system with perfect measurements but having uncertainties in networked signal transfer – time delay, packet loss and out-of-order.
Remark 4: If a network is used for the signal transfer between the controller and actuators, this can also be equivalently treated as point-to-point connected but with delayed and unreliable actuators. In the next section, Section 4, we first use a Lyapunov– Krasovskii functional to derive a sufficient stability condition, Theorem 1, for a class of non-linear time-delay systems. Next, this theorem is re-stated as Theorem 2 for the controller design and is applied to design a controller for the point-to-point connected TPCS system: TPCSP2P. The controller designed, as suggested, is applied to the network-connected system: TPCS-NCS which is the model used in the simulations. The simulation results presented in Section 5 have demonstrated the effectiveness of the new design approach proposed here.
that x˙ (t) = Ax(t) + Bu(t) + h1 (t, x(t)) + h2 (t, x(t))
(7)
u(t) = −Kx(t − h(t))
(8)
where h(t) is a random time delay bounded by h1 and h2: h1 ≤ h(t) ≤ h2; and hi (t, x(t)) i ¼ 1, 2 satisfies hTi (t, x(t))hi (t, x(t)) ≤ xT (t)HiT Hi x(t)
∀t ≥ 0
(9)
where Hi is a constant matrix.
Remark 5: Note that for any Hi the inequality (9) defines a class of piecewise-continuous functions Ha = {h : Rn+1 Rn |hTi hi ≤ xT HiT Hi x in the domain of continuity}
(10)
The class Ha consists of functions that satisfy hi (t, 0) ¼ 0 in their domains of continuity, and x ¼ 0 is an equilibrium of the system. The closed-loop system of (7) and (8) is x˙ (t) = Ax(t) − BKx(t − h(t)) + h1 (t, x(t)) + h2 (t, x(t)) (11)
Theorem 1: For the given h1 , h2 ≥ 0 and K, and hi (t, x(t)) ¼ 1, 2 satisfying (9), if there exist real matrices P, Q, R, S and T . 0 with appropriate dimensions to make the following linear matrix inequality (LMI) holds
V11 V12 V= ,0 ∗ V22
4 Stabilising controller design for a class of non-linear systems
(12)
4.1 Class of non-linear systems and a new theorem for the stability
then the system (11) is asymptotically stable, where (see equation at the bottom of the page)
Consider a class of non-linear systems where the delayed state variables are used for the feedback control, and the non-linear dynamics can be separated from its linear part in such a way
and I is an identity matrix with an appropriate dimension; 11 and 12 are positive numbers; d ¼ (h1 + h2)/2 and ∗ denotes a symmetrical entry.
⎡
V11
V12
PA + A T P − R + Q − T PBK + R T P ⎢ ⎢ ∗ −R − S S 0 ⎢ =⎢ ∗ ∗ −S − Q − T 0 ⎢ ⎢ ⎣ ∗ ∗ ∗ −11 I ∗ ∗ ∗ ∗ ⎡ ⎤ h2 A T R (d − h1 )A T S dA T T 11 H1T 12 H2T ⎢ ⎥ ⎢ h2 K T BT R (d − h1 )K T BT S dK T BT T 0 0 ⎥ ⎢ ⎥ =⎢ 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ h2 R (d − h1 )S dT 0 0 ⎦ (d − h1 )S
h2 R V22 = diag{−R,
−S,
−T ,
−11 I ,
IET Control Theory Appl., 2010, Vol. 4, Iss. 7, pp. 1109 – 1121 doi: 10.1049/iet-cta.2008.0571
dT
0
⎤ P ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎦ −12 I
0
−12 I } 1113
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www.ietdl.org Before we present our proof, it is worth pointing out that if one wants to achieve the global stability of the closed-loop system, then inequality (9) must be satisfied in the entire state-space. In many applications, due to various physical limitations and practical considerations, the system is required only to operate in a subset of the state-space around the system equilibrium point. Therefore very often one only needs to satisfy inequality (9) within the defined subset. Furthermore, as can be seen from the proof, this subset is invariant – any state starts within this subset will be remained there and finally converged to the equilibrium point.
Proof: Construct a positive-definite Lyapunov – Krasovskii functional V (t) = V1 (t) + V2 (t) + V3 (t) + V4 (t) + V5 (t)
V3 (t) = V4 (t) = V5 (t) =
t
−h2
−d
−d
T −R R x(t) x(t) ≤ R −R x(t − h(t)) x(t − h(t))
t−h(t) − x˙ T (v)(d − h1 )S x˙ (v) dv t−d
x(t − h(t)) T −S S x(t − h(t)) ≤ x(t − d) S −S x(t − d)
t x˙ T (v)d T x˙ (v) dv −
x(t) x(t − d)
T
−T T T −T
x(t) x(t − d)
(19)
jT (t) = [xT (t) xT (t − h(t)) xT (t − d) hT1 (t, x(t)) hT2 (t, x(t))] equation (13), for V (t), and all the equations and inequalities of (14) – (19) leads to
t+s
t
V˙ (t) ≤ jT (t) V11 j(t) + x˙ T (t)[d2 T + (d − h1 )2 S + h22 R] x˙ (t)
x˙ T (v)(d − h1 )S x˙ (v) dv ds
+ 11 hT1 (t, x(t))h1 (t, x(t)) + 12 hT2 (t, x(t))h2 (t, x(t))
t+s
(20)
x˙ T (v) d T x˙ (v) dv ds
t+s
A sufficient condition for asymptotic stability of system (11) is that there exist real matrices P, Q, R, S and T . 0 and a feedback gain K such that V˙ (t) , 0 for all j (t) = 0, that is
Taking the time derivative of V (t) with respect to t along the trajectory of (11) leads to V˙ 1 (t) = 2xT (t) P x˙ (t)
(14)
V˙ 2 (t) = xT (t) Q x(t) − xT (t − d) Q x(t − d)
(15)
V˙ 3 (t) = x˙ T (t)h22 R˙x(t) − ≤ x˙
x˙ T (v)h2 R˙x(v) dv
Using the notation
and P, Q, R, S, T . 0.
T
t−h(t)
≤
x˙ T (v)h2 R˙x(v) dv ds
−h1 t
0
t
t−d
V1 (t) = xT (t)Px(t)
t V2 (t) = xT (s)Qx(s) ds t−d
−
(13)
where
0
Applying Jessen’s inequality [22] for R, S, T . 0 leads to
(t)h22 R˙x(t)
−
t t−h2
x˙ T (v)h2 R˙x(v) dv
t
t−h(t)
V˙ 4 (t) = x˙ T (t)(d − h1 )2 S x˙ (t) − ≤ x˙ T (t)(d − h1 )2 S x˙ (t) −
V˙ 5 (t) = x˙ T (t) d2 T x˙ (t) −
x˙ T (v)h2 R˙x(v) dv
t−h1 t−d
t−d
t
(16)
T
x˙ T (v)(d − h1 )S x˙ (v) dv (17)
x˙ T (v)d T x˙ (v) dv
t−d
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+ 11 hT1 (t, x(t))h1 (t, x(t)) + 12 hT2 (t, x(t))h2 (t, x(t)) (21) , 0 ∀j(t) = 0 Since inequality (9) is hTi (t, x(t))hi (t, x(t)) ≤ xT (t)HiT Hi x(t) ∀t ≥ 0, then if
jT (t) V11 j(t) + x˙ T (t)[d2 T + (d − h1 )2 S + h22 R] x˙ (t)
x˙ (v)(d − h1 )S x˙ (v) dv
t−h(t)
V˙ (t) ≤ jT (t) V11 j(t) + x˙ T (t)[d2 T + (d − h1 )2 S + h22 R] x˙ (t)
(18)
+ 11 xT (t)H1T H1 x(t) + 12 xT (t)H2T H2 x(t) , 0
(22)
it will make inequality (21) satisfied. Furthermore, by the Schur complement, inequalities (22) and (9) are equivalent to the existence of P, Q, R, S, T . 0 and K such that (12) holds. V=
V11 V12 ,0 ∗ V22
(12)
which is required by the theorem. This finally completes the proof of V˙ (t) , 0, and therefore the proof of Theorem 1. A IET Control Theory Appl., 2010, Vol. 4, Iss. 7, pp. 1109 – 1121 doi: 10.1049/iet-cta.2008.0571
www.ietdl.org Theorem 1 can be restated as Theorem 2 for controller design.
Theorem 2: For h1 , h2 ≥ 0; h1(t, x(t)) and h2(t, x(t)) in (7) satisfying (9), if there exist a real matrix Y and symmetric positive matrices P, R, Q, S, T and X with appropriated dimensions to make the following matrix inequality holds
C11 ∗
C12 C22
,0
(23)
problem of optimisation subject to some LMIs (To limit the length of the paper, the details are not presented here.) Then this optimisation problem can be solved by applying the well-known cone complementarity linearisation algorithms [24]. In summary, if (i) inequality (9) can be hold and (ii) matrices Y, X and other matrices can be found to satisfy (23), then the feedback control of (8) with K ¼ YX 2T guarantees the asymptotic stability of a class of timedelayed non-linear systems of (7) and (8).
then the system (7) and (8) is asymptotically stable with a state feedback control gain of K ¼ YX 2T, where (see equation at the bottom of the page) and I is an identity matrix with an appropriate dimension; 11 and 12 are positive numbers and d ¼ (h1 + h2)/2.
Proof: Define X = P −1
= XRX T Y = KX T P = XPX T R
(24)
= XQX T S = XSX T T = XTX T Q
then pre- and post-multiplying both sides of the V matrix in the LMI of (12) with diag(X,X,X,I,I,I,I,I,I,I ) and its transpose, respectively, leads to
C11 ∗
C12 C
,0
(25)
22
22 ¼ diag(2R 21, 2S 21, 2T 21, 2I, 2I ). From (24), where C 22 = C22 , and therefore if the LMI of one can find that C (12) holds, (23) also holds. This completes the proof. A Here some non-linear terms, such as −1 X , − X T S−1 X and − X T T −1 X in C22 of (23), −X T R are involved. Follow a very similar procedure as given in [23]; solving (23) can be transformed to an equivalent
⎡
C11
⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣ ⎡
C12
−T +Q AX T + XA T − R
h2 XA T ⎢ ⎢ h2 Y T B T ⎢ =⎢ 0 ⎢ ⎢ ⎣ h2 I h2 I
∗ ∗
(7)
u(t) = −Kx(t − h(t))
(8)
Theorem 1 gives a sufficient condition for the system stability when the measurement delay is bounded by h1 ≤ h(t) ≤ h2 . A system satisfying this sufficient condition will also have some degree of robustness which can tolerate some measurement errors. As discussed in the previous section, a signal error may be caused by different reasons in the TPCS-NCS and TPCS-P2P systems, but viewed from the controllers of the both systems its value is the same. This is the difference between the current process output and its previous value. In real applications, these errors caused by imperfect and unreliable measurements are equivalent to the errors caused by data packet loss or out-of-order in NCSs. This paper, however, does not give an analytical study on the robustness of the closed-loop system designed. This is our planned future research. Another topic under study is a decentralised controller design for a class of non-linear time-delay systems consisting of a number of subsystems.
4.2 Stabilising controller design for the TPCS-P2P system Applying Theorem 2 to the controller design for the TPCSP2P system involves three steps: (i) to separate the linear part and the non-linear part of the TPCS dynamics of (2) and (3)
BY + R − S −R ∗
T S −T −S − Q
0 0 −11 I
∗
∗
∗
∗
∗
∗
(d − h1 )XA T T T
(d − h1 )Y B 0 (d − h1 )I (d − h1 )I −1
x˙ (t) = Ax(t) + Bu(t) + h1 (t, x(t)) + h2 (t, x(t))
−1
dXA T dY T BT 0 dI dI
X , −X T S X , −X T T C22 = diag{−X T R IET Control Theory Appl., 2010, Vol. 4, Iss. 7, pp. 1109 – 1121 doi: 10.1049/iet-cta.2008.0571
XH1T 0 0 0 0 −1
I
⎤ T
∗
I
⎤
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎦ 0 0
−12 I
XH2
⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎦ 0
X , −11 I , −12 I } 1115
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www.ietdl.org to obtain a state-space equation in a form of (7); (ii) for h1(t, x(t)) and h2(t, x(t)) in (7) to find constant matrices H1 and H2 to satisfy inequality (9) and (iii) to find all the required constants and matrices to satisfy (23) and to obtain the matrices X, Y and K ¼ YX 2T used in (8).
d z(t) = A z(t) + Bu(t) + h1 (t, z(t)) + h2 (t, z(t)) dt
Step (ii): If we limit the angles of the two pendulums to |u| ≤ p/6, then p |g(sin u − u) ≤ 0.45|u|, |u| ≤ 6
and
2 3 F cos2 (u1 − f) m1 l 1 2 3 F cos2 (u2 − f) + m2 l2 2 2 3 3 ≤ F + F ≤ zT H2T H2 z m1 l1 m2 l 2
hT2 (t, z(t))h2 (t, z(t)) =
where z = [ u1 u˙ 1 u2 u˙ 2 ]T ; u = [ t1 t2 ]T ; ⎡ ⎤ ⎡ 0 1 0 0⎤ 0 0 ⎢ 3 ⎥ ⎢ 3 g 0 0 0⎥ ⎢ 0 ⎥ ⎢ 2l ⎥ 2 ⎢ ⎥ m l ⎢ ⎥ 1 1 ⎥ A=⎢ 1 ⎥, B = ⎢ ⎢ ⎥ 0 0 0 1 0 0 ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ 3 3 ⎦ g 0 0 0 0 2l2 m2 l22 ⎡ ⎡ ⎤ ⎤ 0 0 3 3 ⎢ g(sin u − u ) ⎥ ⎢ ⎥ ⎢ 2l ⎢ m l F cos (u1 − f) ⎥ 1 1 ⎥ ⎢ 1 ⎢ 11 ⎥ ⎥ h1 = ⎢ ⎥, h2 = ⎢ ⎥ 0 0 ⎢ ⎢ ⎥ ⎥ ⎣ 3 ⎣ ⎦ ⎦ 3 g(sin u2 − u2 ) F cos (u2 − f) − 2l2 m2 l2
hT1 (t,
|F | ≤ k(1.49|u1 | + 0.18|u2 |) This leads to
Step (i): From (2) and (3), one can have
Therefore
one can draw a three-dimensional figure of /F/ against /u1/ and /u2/ and have
2 3 z(t))h1 (t, z(t)) = g (sin u1 − u1 )2 2l1 2 3 + g (sin u2 − u2 )2 2l2 3 × 0.45 2 ≤ (u1 )2 2l1 3 × 0.45 2 + (u2 )2 = zT H1T H1 z 2l2
where ⎡
1.49k ⎢ 0 H2 = b⎢ ⎣ 0 0
3 × 0.45 ⎢ 2l1 ⎢ ⎢ 0 H1 = ⎢ ⎢ ⎢ 0 ⎣
Step (iii): With all the equations and matrices obtained in the previous steps, we use the MATLAB LMI toolbox to solve the transformed optimisation problem to obtain a maximum allowable delay bound h2 ¼ 0.06 s (h1 is set to h1 ¼ 0 in our design). The other main matrices obtained are ⎡
1.7782 −4.5698 ⎢ −4.5698 17.6371 ⎢ X =⎣ −0.0001 −0.0015 −0.0015 0.0091 Y =
0
0
K =
0
0 3 × 0.45 0 2l2 0 0
0⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎦
5
0
Next, from (4) and (5) F = k (ls − [L2 + (l1 − l2 )2 ]1/2 )
(4)
ls = [(L + l2 sin u2 − l1 sin u1)2 + (l2 cos u2 − l1 cos u1)2 ]1/2 (5) 1116 & The Institution of Engineering and Technology 2010
−4.7700 −2.0430 0.0000 0.0003
⎤ −0.0001 −0.0015 −0.0015 0.0091 ⎥ ⎥ 1.7789 −4.5574 ⎦ −4.5574 17.5621 0.0000 0.0007 −2.4421 −1.0486
and
⎤ 0
⎤ 0 0⎥ ⎥ 0⎦ 0
3 2 3 2 b= + m1 l1 m2 l2
⎡
0 0.18k 0 0 0 0 0 0
−8.919 −0.0018
−2.427 −0.0034 −0.0002 −4.5521
−0.0004 −1.241
(26)
Simulation studies
The controller designed for the TPCS-P2P system, (26) above, now is used to control the TPCS-NCS system with the communication details specified in Section 3. Since the controller design is based on the sufficient stability condition of Theorem 1, in simulation tests we will make some time delay . h2 ¼ 0.06 s. It is assumed that all random time delay has Gaussian distribution and the standard deviation is 0.01. System responses to different mean values of delay are simulated. IET Control Theory Appl., 2010, Vol. 4, Iss. 7, pp. 1109 – 1121 doi: 10.1049/iet-cta.2008.0571
www.ietdl.org The system sampling period is 0.01 s, and the simulations stop at t ¼ 3 s. Therefore there are 301 simulation steps and 301 random data for the amount of time delay in each channel of the two.
5.1 TPCS-NCS system when data packet loss is not explicitly considered in simulations The cases with data packet drop-out explicitly considered will be studied in the next subsection. In applications when a packet arrived with a long delay – normally it will be outof-order – it will be discarded by the controller since the data in the packet is not the latest signal available to the controller. The effect of this is the same as that of a packet lost in transmissions. Therefore the event of a data packet drop-out has been implicitly included in the time-delay cases simulated in this subsection, where a few longdelayed packets are out-of-order and are treated in the same way as that of lost packets. This detail is shown in Table 2 and will be explained later. As described in Section 3, the measurements of u1 and u˙ 1 are put into a data packet and transmitted through communication channel 1 (u1 and u˙ 1 have the same time delay). The mean delay in communication channel 1 is noted as E1 . In this channel, the number of delays .0.06 s during the simulation is noted as N1_d .0.06 . Similarly, u2 and u˙2 are transmitted through communication channel 2. The mean delay in this channel is noted as E2 , and the number of delays .0.06 s in 301 delay data is noted as N2_d .0.06 . Notice that since in the simulations the initial seeds to generate the random numbers for delays in the two channels are different, 0 and 1000, respectively, therefore at the same sampling instance the delays in the two channels are different even when E1 ¼ E2 . Two kinds of system time-responses are simulated:
For different values of E1 and E2 , the system time responses to the non-zero initial condition are summarised in Table 1. The plots of the time responses for cases A-5 and A-10 are given in Figs. 5 and 6, respectively. Fig. 7 is a plot of the delays in the two channels. Notice that, for case A-10 another simulation has been run for 10 s. The result, not shown here to limit the length of the paper, does confirm that the system response shown in Fig. 6 (ended at t ¼ 3 s.) is stable. For the period between 3 and 10 s, the same as that shown in Fig. 6, it is a decayed oscillatory response. Although it takes a while to settle, it is indeed stable. These results show that, even a significant number of delays . h2 ¼ 0.06 s, the system remains stable. Table 1 Non-zero initial condition responses (u1(0) ¼ 0.4 rad, u2(0) ¼ 0.2 rad) Case no.
E1
N1_d.0.06
E2
N2_d.0.06
Jinit
A-1
0.04
7
0.04
2
4.85
A-2
0.04
7
0.05
38
4.87
A-3
0.04
7
0.06
146
4.89
A-4
0.05
51
0.06
146
4.78
A-5
0.06
161
0.06
146
5.06 Fig. 5
A-6
0.06
161
0.08
291
5.24
A-7
0.08
292
0.10
all
5.59
A-8
0.08
292
0.12
all
6.36 Fig. 7, delay data
A-9
0.10
all
0.12
all
6.84
A-10
0.12
all
0.12
all
8.17 Fig. 6
1. System responses to a non-zero initial condition
u1 (0) = 0.4 rad u2 (0) = −0.2 rad u˙ 1 (0) = 0 u˙ 2 (0) = 0 2. At a zero initial condition, a step disturbance of 2 Nm is applied to pendulum 1. For the ease of making comparison, we also introduce two performance indexes: 301 2 [u1(i) + u22(i)] (P-1): Jinit ¼ Si¼1 condition responses, and
for
non-zero
initial
2 (P-2): Jstep ¼ S301 i¼1[(u1_r2 steady 2 u1(i)) + (u2_steady 2 2 u2(i)) ] for response to a step disturbance of 2 Nm applied to pendulum 1, where u1_steady and u2_steady are the steadystate values of u1(t ¼ 6) ¼ 0.3607 and u2(t ¼ 6) ¼ 0.2854.
IET Control Theory Appl., 2010, Vol. 4, Iss. 7, pp. 1109 – 1121 doi: 10.1049/iet-cta.2008.0571
Figure 5 Response to the non-zero initial conditio, (case A-5) 1117
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www.ietdl.org Table 3 Response to a step disturbance of 2 Nm applied to pendulum 1 Case no.
Figure 6 Response to the non-zero initial condition (case A-10)
E1
N1_d.0.06
E2
N2_d.0.06
Jstep
B-1
0.04
7
0.04
2
10.10
B-2
0.04
7
0.05
38
10.00
B-3
0.04
7
0.06
146
9.89
B-4
0.05
51
0.06
146
9.68
B-5
0.06
161
0.06
146
9.46
B-6
0.06
161
0.08
291
9.26
B-7
0.08
292
0.10
all
8.67 Fig. 8
B-8
0.08
292
0.12
all
8.52
B-9
0.10
all
0.12
all
8.20
B-10
0.12
all
0.12
all
8.08
subsection, the events of these happening are actually implicitly included in the simulations. For different values of E1 and E2 , the system time responses to a step disturbance are summarised in Table 3, and a plot of the time response for case B-7 is given in Fig. 8. From Table 3, one may notice that when the delay increases, the performance index Jstep actually decreases. This can be explained as follows. The delay is usually considered undesirable since it has the tendency to deteriorate the system performance or even destabilise the system. However, in some applications–for example in a delayed resonator–delay is deliberately introduced in the control to improve the stability and performance [25]. How the delay affects the performance of this particular example here needs to be further investigated, but this is beyond the scope of this paper.
Figure 7 Random delays, E1 ¼ 0.08 s; E2 ¼ 0.12 s (case A-8) Now consider the first seven data packets in communication channel 2 for case A-1, where E2 ¼ 0.04 s and N2_d .0.06 ¼ 2. The sending times, delays and arriving times are given in Table 2. It can be seen that packets 1 and 3 are out-of-order when they arrived, and therefore the signals in these packets are not to be used for control calculation. This is the same result if packets 1 and 3 were lost in the transmission. One can therefore conclude that, although the data packet loss and out-of-order is not explicitly considered in ‘delay-only-studies’ of this
5.2 TPCS-NCS system with data packet loss explicitly considered in simulations The characteristics of time delays in case A-5, E1 ¼ 0.06 s and E2 ¼ 0.06 s is used in the all simulations in this subsection. In addition, different cases of explicit packet loss in channel 1 are simulated for the non-zero initial condition responses. The results are summarised in Table 4 and a plot for case C-9 is given in Fig. 9. In Table 4,
Table 2 Data packets transmission times and if they are out-of-order Packet no. sending time
1
2
3
4
5
6
7
0.0
0.01
0.02
0.03
0.04
0.05
0.06
delay
0.0489 0.0388 0.0474 0.0264 0.0301 0.0440 0.0381
arriving time
0.0489 0.0488 0.0674 0.0564 0.0701 0.0940 0.0981
if the data is used
N
1118 & The Institution of Engineering and Technology 2010
Y
N
Y
Y
Y
Y
IET Control Theory Appl., 2010, Vol. 4, Iss. 7, pp. 1109 – 1121 doi: 10.1049/iet-cta.2008.0571
www.ietdl.org
Figure 8 Response to a step disturbance (case B-7)
Figure 9 Response to non-zero initial condition, case C-9 (184 packets in channel 1 are lost)
Table 4 Non-zero initial condition responses with data-packet loss in channel 1 Case no.
Nloss_1
Nloss_2
The first eight sequence numbers of the Packets lost (no. 3 is for t ¼ 0.02 s . . . )
A-5
0
0
no explicit packet loss
C-1
2
21
3
7
21
25
39
44
74
78
5.34
C-2
3
42
3
7
17
21
25
39
44
45
5.50
C-3
4
61
3
7
9
17
21
25
35
39
5.77
C-4
6
91
3
4
7
9
16
17
21
25
6.02
C-5
6
110
3
4
7
9
16
17
21
25
6.01
C-6
8
130
3
4
7
9
15
16
17
21
6.53
C-7
11
147
2
3
4
5
7
9
14
15
7.27
C-8
12
169
2
3
4
5
7
9
12
14
11.19
C-9
13
184
2
3
4
5
7
9
10
12
19.78
Nloss_1 is the total number of the packet lost during the initial response period of 0 , t ≤ 0.2 s, which is often critical to the system stability. Nloss_2 is the total number of the packet lost during the whole simulation period 0 , t ≤ 3 s.
Remark 6: For the purpose of demonstrating the proposed new design approach, most simulations here are based on very poor network quality of service (QoS) – long delays and/or a significant number of packets are lost. In real applications, there is a minimum standard of QoS to be achieved by communication engineers before a control engineer can consider using the network for NCSs. Even for wireless networks, the error rate– this includes data packet arriving out-of-order and lost in transmission – should be less than 10% in the worst circumstances [26]. In summary, all simulation results in this section show that, for this particular non-linear unstable system with a IET Control Theory Appl., 2010, Vol. 4, Iss. 7, pp. 1109 – 1121 doi: 10.1049/iet-cta.2008.0571
Jinit 5.06
Fig. 5
Fig. 9
networked sensing signal transmission, the controller designed in Section 4.2 – in addition to the stability when all delays are no more than 0.06 s guaranteed by a sufficient condition (Theorem 1) – is ‘sufficiently robust’ leading to all stable responses for the cases simulated in this section. Nevertheless, an analytical study on this robustness is yet to be researched.
6
Conclusions
To address the two issues in the current study of NCSs – (i) examples used to demonstrate some theoretical results are too simple and often linear systems and (ii) some research outcomes are not valid for the main NCS applications of distributed control systems – a non-linear unstable system, which has been historically used for the study of large-scale system control, is studied in this paper. From the application point of view, it is suggested that the NCS design problem, where data packet time delay, loss or 1119
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7
Acknowledgment
The authors would like to acknowledge the Natural Science Foundation of China (NSFC) for its support under grant numbers 60704024 and 60774059.
8
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