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Architectures and coder design for networked control systems

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Architectures and coder design for networked control systems
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  Automatica 44 (2008) 248–257www.elsevier.com/locate/automatica Brief paper Architecturesandcoderdesignfornetworkedcontrolsystems  Graham C. Goodwin ∗ , Daniel E. Quevedo, Eduardo I. Silva School of Electrical Engineering and Computer Science, The University of Newcastle, NSW 2308, Australia Received 28 August 2006; received in revised form 20 May 2007; accepted 21 May 2007Available online 14 September 2007 Abstract In networked control systems (NCSs) achievable performance is limited by the communication links employed to transmit signals in theloop. In the present work, we characterise LTI coding systems which optimise performance for various NCS architectures. We study NCSswhere the communication link is situated between plant output and controller, and NCSs where the communication link is located betweencontroller and actuator. Furthermore, we present a novel NCS architecture, which is based upon theYoula parameterisation. We show that, whichof these architectures gives best performance depends,  inter alia , upon characteristics of a related non-networked design, plant disturbancesand reference signal. A key aspect of our work, resides in the utilisation of fixed signal-to-noise ratio channel models which give rise toparsimonious designs, where channel utilisation is kept low. The results are verified with simulations utilising bit-rate limited channels.   2007 Elsevier Ltd. All rights reserved. Keywords:  Networked control systems; Noisy channels; Coding schemes; Minimum variance control; Quantisation 1. Introduction In traditional control systems, one commonly assumes thatthe interconnection of plant and controller is  transparent  ,i.e., transmitted signals are equal to received signals. Thisparadigm is often appropriate and underlies many successfulcontrol design methods, especially for linear time invari-ant (LTI) systems; see, e.g., Goodwin, Graebe, and Salgado(2001). However, in some situations, the assumption of trans-parent communication is not justified. Control systems wherethe communication link constitutes a bottleneck in achievableperformance are commonly termed  networked control systems (NCSs); see, for example, articles contained in the special issue(Baillieul &Antsaklis, 2007), and the survey papers (Hespanha, Naghshtabrizi, & Xu, 2007; Tipsuwan & Chow, 2003). Thecommunication link can either be dedicated or consist of anetwork which is shared between several users. Novel aspects  This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor TongwenChen under the direction of Editor Ian Petersen. ∗ Corresponding author. Tel. +61249217072; fax +61249601712.  E-mail addresses:  graham.goodwin@newcastle.edu.au(G.C. Goodwin), dquevedo@ieee.org (D.E. Quevedo),eduardo.silva@studentmail.newcastle.edu.au (E.I. Silva).0005-1098/$-see front matter    2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2007.05.015 introduced by the presence of non-transparent communicationlinks in control include time delays, data-dropouts and quanti-sation, see also Ishii and Francis (2002); Hristu-Varsakelis andLevine (2005); Nair, Fagnani, Zampieri, and Evans (2007);Schenato, Sinopoli, Franceschetti, Poolla, and Sastry (2007).From an analysis perspective, even basic system theoreticnotions, such as closed loop stability are far from trivial inthe networked control context; see, e.g., Elia (2005); Fagnaniand Zampieri (2004); Freudenberg, Braslavsky, and Middleton(2005); Goodwin, Haimovich, Quevedo, and Welsh (2004);Li and Baillieul (2004); Nair et al. (2007); Tang and de Silva(2006); Wong and Brocket (1999).When designing NCSs, the characteristics of the communi-cation system should be explicitly taken account of to ensureacceptableperformancelevels.Thisraisesnewchallenges.Akeyobservation is that, in NCSs, there exist additional degrees of freedom in the design process, when compared to traditionalcontrol loops. As a consequence, to optimise performance, itis useful to investigate architectural issues and signal codingmethods, see also Canudas de Wit, Rodríguez, Fornés, andGómez-Estern (2006); De Perisi and Isidori (2004); Tatikondaand Mitter (2004b); Quevedo and Goodwin (2005).Several NCS architectures have been studied. One can distin-guish configurations where the channel is located in the  up-link  ,  G.C. Goodwin et al. / Automatica 44 (2008) 248–257   249 i.e., between sensors and controller input (De Perisi & Isidori,2004; Wong & Brocket, 1999), and where it lies in the  down-link  , i.e., between controller output and actuators (Freudenberget al., 2005). More general architectures, where the process-ing power is distributed, have also been examined, for exam-ple, in Ling and Lemmon (2004); Tang and de Silva (2006);Tatikonda and Mitter (2004b); Goodwin et al. (2004).The goal of the present work is to compare various NCS ar-chitecturesforthecontrolofasingle-inputsingle-output(SISO)plant. We will consider a design situation, where an LTI con-troller is to be implemented in an NCS which employs a signal-to-noise ratio constrained communication channel. This type of channel model is commonly utilised in the signal processingliterature, see, e.g., Jayant and Noll (1984). The controller is assumed to have already been designed for a given plant so asto achieve desired control objectives when the communicationlink is assumed to be transparent. However, when implement-ing this controller in an NCS, unavoidably performance lossesoccur. We will show how performance degradation, as mea-sured by the variance of the tracking error, can be minimisedthrough appropriate signal coding. As in other contemporaryapproaches to NCS design, see Elia (2005); Lian, Moyne, and Tilbury (2003); Xiao, Johansson, Hindi, Boyd, and Goldsmith(2003), we will employ design methodologies that utilise LTIsystemtheoreticideas.Thiswillallowustoobtainresultswhichquantify the achievable performance of various NCS architec-tures. The present paper extends work described in Goodwin,Quevedo, and Silva (2006); Quevedo, Welsh, Goodwin, andMcLeod (2006).The remainder of this work is organised as follows: InSection 2, we present the nominal control design which is tobe implemented via an NCS and give details about the specificcommunication link to be utilised. Optimal coding for twodifferent standard NCS architectures is studied in Section 3. InSection 4, we present a novel NCS architecture and show howto equip it with optimal coding systems. Section 5 documentsa design study. Finally, conclusions are drawn in Section 6. 2. NCS elements 2.1. Nominal design We will examine a networked scenario, where the controlloop is closed through a communication link. Specifically, wewill consider the situation where an LTI SISO discrete timecontroller, say  C(z) , has already been designed for a discretetime SISO LTI plant  G(z) . This design has been carried outassuming that the plant is affected by output disturbances, 1 d  ,that the output measurement is corrupted by noise,  n , and that agiven reference signal,  r  , should be tracked; see Fig. 1. We willrefer to this design as the  nominal design  and we will assumethat it gives satisfactory performance when the communicationlinks are transparent. 1 We use vector space notation to denote signals. For example,  y  denotes { y() }  ∈ N . The symbol  z  is used to refer to both the argument of thez-transform and to the forward shift operator.Fig. 1. Standard non-networked control system.Fig. 2. Standard non-networked control system for stable plant: Youla form. In the closed loop of  Fig. 1, the tracking error, e  r  − y , (1)is given by e  = S(z)(r  − d) + T(z)n , (2)where  S(z)  ( 1  +  G(z)C(z)) − 1 and  T(z)  1  −  S(z)  are theloop sensitivity functions; see, e.g., Goodwin et al. (2001).The signals  r  ,  d   and  n  are assumed to be independent station-ary zero mean processes, with power spectral densities (PSDs) | R( e  j  ) | 2 ,  | D( e  j  ) | 2 and  | N( e  j  ) | 2 , respectively.In the particular case when  G(z)  is stable, the nominal designcan alternatively be realised in the form of the affine parame-terisation of all stabilising controllers ( Youla parameterisation )depicted in Fig. 2; see, e.g., Goodwin et al. (2001); Morari and Zafiriou (1989). In that figure,  ˆ G(z)  is an explicit model for theplant. The  Youla controller   C Y  (z)  in Fig. 2 is related to  C(z) in Fig. 1 via  C(z) = C Y  (z)( 1 − C Y  (z)  ˆ G(z)) − 1 .Wewillignoredifferencesbetweentheplanttransferfunction G(z)  and the model  ˆ G(z) . As a consequence, the signal   0 in Fig. 2 satisfies   0  =  d   +  n . Thus,   0  summarises essentialinformation about the system to be controlled. Heuristically,this signal is a useful candidate to be transmitted through acommunication link in an NCS, where channel utilisation is tobe kept low. We will return to this Youla configuration later inSection 4.  250  G.C. Goodwin et al. / Automatica 44 (2008) 248–257  2.2. The communication link  The novel ingredient in an NCS, when compared to a tra-ditional control loop, is the communication link. From a de-sign perspective, this opens the possibility of coding signalsprior to transmission and also plays a key role in the achievableperformance. Accordingly, we will consider a communicationlink consisting of a communication channel together with anencoder–decoder pair, as shown in Fig. 3.We start by describing the channel model. There exist sev-eral ways of characterising a communication channel; see, e.g.,Tatikonda and Mitter (2004a); Tse and Viswanath (2005). Wewill focus on an additive signal-to-noise ratio constrained chan-nel model, where the channel output  w  is related to the channelinput  v  via w  = v  + q , (3)and  q  is zero mean stationary discrete time random process,having PSD  | Q( e  j  ) | 2 and variance  q  =  12      −  | Q( e  j  ) | 2 d  . (4)The signal-to-noise ratio constraint implies that   q  is relatedto the variance of   v ,   v , by  =   v  q , (5)where    is the (finite) channel signal-to-noise ratio. Therefore,we can write | Q( e  j  ) | 2 =   v   | Q 0 ( e  j  ) | 2 , (6)where  Q 0 ( e  j  )  is such that  ( 1 / 2  )     −  | Q 0 ( e  j  ) | 2 d  = 1.Many (source) coding schemes have been studied in the NCSliterature; see, e.g., Canudas de Wit et al. (2006); Matveev andSavkin (2005); De Perisi and Isidori (2004); Quevedo,Goodwin, and Welsh (2004); Tatikonda and Mitter (2004b);Wong and Brocket (1999). These methods vary in complex-ity and on the assumptions made regarding the informationavailable to the encoder and decoder. In the present work, weconcentrate on LTI encoder–decoders pairs that have accessonly to local information; see, e.g., Jayant and Noll (1984). Inorder not to modify the nominal design relations (as discussedin Section 2.1), we restrict the encoder–decoder pair to achieve  perfect reconstruction , i.e., we assume that F(z)F  − 1 (z) = 1 (7) Fig. 3. Communication link. (see Fig. 3). To avoid unstable pole zero cancellations and non- causality, we restrict the encoder  F(z)  to be stable, minimumphase and biproper. The design of the encoder–decoder pairs,i.e., of   F(z)  and  F  − 1 (z)  in Fig. 3, will form a central theme in the subsequent analysis. 3. Optimal coding in standard NCS architectures This section examines the two most common single-channelnetworked control architectures, namely, where the channel islocated in the down-link and where it is located in the up-link.These architectures are depicted in Figs. 4 and 5, respectively. 3.1. Optimal coding for the down-link case In the architecture of  Fig. 4 it holds that the tracking error(see (1)) is given by e  = S(z)(r  − d) + T(z)n − S(z)F  D (z) − 1 G(z)q ,where we have used the additive noise channel in (3). Using(2), we note that the variance of the component of the trackingerror that arises from the communication link is given by J   =  12      −  | S( e  j  )F  − 1 D  ( e  j  )G( e  j  )Q( e  j  ) | 2 d  . (8)The above expression serves to quantify the impact of thecommunication link on the performance of the NCS. It dependsupon the encoder–decoder pair used.Basedon(8),onemightbetemptedtosimplyset F  − 1 D  ( e  j  )  ≈ 0, for all frequencies   . This, however, would lead to a decodersuch that  F  D ( e  j  )  → ∞ ,  ∀   and hence, to an unboundedsignal  v . This is clearly unacceptable. Thus, encoder–decoderdesign has to be carried out more carefully.A key point is that   q  is related to   v  via (5). Indeed, fromFig. 4, it follows that  v  =  12      −  | T( e  j  )Q( e  j  ) | 2 d  +  12      −  | S( e  j  )C( e  j  )F  D ( e  j  )  D ( e  j  ) | 2 d  , (9)where |  D ( e  j  ) | 2  | D( e  j  ) | 2 +| N( e  j  ) | 2 +| R( e  j  ) | 2 (10)is the PSD of   d   + n + r .Substituting (6) into (9) gives  v  =   2      −  | S( e  j  )C( e  j  )F  D ( e  j  )  D ( e  j  ) | 2 d  , (11)where     −  12      −  | T( e  j  )Q 0 ( e  j  ) | 2 d   − 1 . (12)  G.C. Goodwin et al. / Automatica 44 (2008) 248–257   251Fig. 4. Channel in the down-link.Fig. 5. Channel in the up-link. Using (6), (8) and (11), it follows that  J   in (8) satisfies J   =   2      −  | S( e  j  )F  − 1 D  ( e  j  )G( e  j  )Q 0 ( e  j  ) | 2 d  ×  12      −  | S( e  j  )C( e  j  )F  D ( e  j  )  D ( e  j  ) | 2 d  . (13) Remark 1  (  Bound on   ). In (13),  J   is the variance of the effectthat the communication link has on the tracking error. There-fore,    in (12) has to be positive. This imposes a constrainton   , namely   >( 2  ) − 1      −  | T( e  j  )Q 0 ( e  j  ) | 2 d  . If     doesnot satisfy this constraint, then the fixed signal-to-noise ratiomodel will not hold, and instability may occur. This observa-tion is consistent with the results reported in Freudenberg et al.(2005). (Note that the results in Freudenberg et al. (2005) are based on a channel model with fixed noise variance, rather thanfixed signal-to-noise ratio, as considered here.)Thefollowingtheoremcharacterisesoptimalencoder–decoderpairs for the NCS in Fig. 4. Theorem 2  (  Down-link Channel (Quevedo et al., 2006 ) ). Consider the NCS depicted in Fig . 4  and the loss function J in (13).  Then  J   J  opt D  ,  where J  opt D      12      −  | S( e  j  )T( e  j  )  D ( e  j  )Q 0 ( e  j  ) | d   2 (14) and     is defined in  (12).  This performance bound is tight and can be achieved by all encoders which satisfy | F  D ( e  j  ) | 2 = k D  G( e  j  )Q 0 ( e  j  )C( e  j  )  D ( e  j  )  ,  ∀   ∈ [−  ,   ] , (15) where  k D  > 0  is any positive  (  fixed  )  real number  . Proof.  Using the Cauchy–Schwartz inequality in (13) it fol-lows that  J   J  opt D  , and that equality is achieved if and only if there exists  k D  constant and positive such that | S( e  j  )F  − 1 D  ( e  j  )G( e  j  )Q 0 ( e  j  ) |= k D | S( e  j  )C( e  j  )F  D ( e  j  )  D ( e  j  ) | ,for all   , i.e., if and only if (15) holds.   The class of coding systems specified via (15) achieves thebesttrade-offbetweenchannelutilisationandtheeffectofchan-nel noise on loop performance, as measured by the variance of the tracking error.We will denote any coder  F  D (z)  that satisfies(15) by  F  opt D  (z) .Since (15) is a restriction on the  magnitude  of   F  opt D  ( e  j  ) ,one can always resort to appropriate all-pass filters to guaran-tee that  F  opt D  (z)  is biproper, stable and minimum phase (recallSection 2.2). Remark 3  ( Parsimony of the channel model ). We can see thatby using a fixed signal-to-noise ratio channel model, optimalcoders do not have an infinite gain over all frequencies (seealso Lu & Skelton, 1999). In contrast, if the channel noise variance (rather than the signal-to-noise ratio) were to be fixed(as in Freudenberg et al., 2005), then setting  F  − 1 D  ( e  j  )  ≈  0,  ∀  (so that  v  becomes unbounded) would minimise the channelinduced plant output variance. In such cases, for a parsimoniousdesign which only incurs limited channel utilisation a differentloss function should be used (see, e.g., Quevedo et al., 2004). 3.2. Optimal coding for the up-link case We next investigate optimal encoder–decoder pairs for thealternative architecture depicted in Fig. 5, where the commu-nication system is located in the up-link.  252  G.C. Goodwin et al. / Automatica 44 (2008) 248–257  As before, we define the tracking error variance due to chan-nel effects as  J  . Proceeding as in the down-link case analysedin Section 3.1, we conclude that in the NCS of  Fig. 5 J   =   2      −  | F  − 1 U   ( e  j  )T( e  j  )Q 0 ( e  j  ) | 2 d  ×  12      −  | S( e  j  )F  U  ( e  j  )  U  ( e  j  ) | 2 d  , (16)where    is as in (12) and |  U  ( e  j  ) | 2  | D( e  j  ) | 2 +| N( e  j  ) | 2 +| G( e  j  )C( e  j  )R( e  j  ) | 2 (17)is the PSD of   d   + n + G(z)C(z)r ; compare to (10).As in the down-link architecture, when designing the cod-ing system, there exists a trade-off between channel noise at-tenuation and channel utilisation. Optimal performance can beattained by means of the following theorem: Theorem 4  ( Up-link channel ).  Consider the NCS depicted inFig . 5  and the loss function J in  (16).  Then  J   J  opt U   ,  where J  opt U       12      −  | S( e  j  )T( e  j  )  U  ( e  j  )Q 0 ( e  j  ) | d   2 (18) and     is defined in  (12).  This bound is tight and is attained byall encoders which satisfy | F  U  ( e  j  ) | 2 = k U   G( e  j  )C( e  j  )Q 0 ( e  j  )  U  ( e  j  )  , ∀   ∈ [−  ,   ] ,(19) where  k U   is any  (  fixed  )  positive real number  . Proof.  Similar to the proof of Theorem 2.   We will denote any coder that satisfies (19) by  F  opt U   (z) . Asin the down-link case, all-pass filters can be used to imposeadditional properties on  F  opt U   (z) . 3.3. Performance comparison For the two architectures examined so far, the only differencein the optimal performance resides in the terms   D ( e  j  )  and  U  ( e  j  )  (see (14) and (18)). Further insight can be gained byanalysing three different scenarios: 3.3.1. Disturbance rejection Assume that the reference is zero. Comparison of (10) and(17) shows that, if   r  =  0, then   D ( e  j  )  =   U  ( e  j  )  ∀   ∈[−  ,   ] . Thus, in a regulation loop, both architectures,  whenequipped with optimal coder–decoder pairs , attain the sameperformance. It is interesting to note that, in this situation, (15)and (19) suggest choosing  F  opt U   (z)  =  C(z)F  opt D  (z) . With thischoice,  placing the channel in the down-link is algebraicallyequivalent to placing it in the up-link  . 3.3.2. Cancelling nominal design Here we consider a special case where the controller cancels(stable) poles and zeros of the plant model. 2 We also assumethat  n  =  0,  r  =  0, and that the channel noise is white, i.e., | Q 0 ( e  j  ) | =  1,  ∀   ∈ [−  ,  ] . It then follows from (19) that,if there exists    ∈  R  such that  | D( e  j  ) | =   | G( e  j  )C( e  j  ) | , ∀   ∈ [−  ,   ] , then the optimal encoder in the architecturerepresented in Fig. 5 can be chosen as  F  opt U   (z)  =  1 (see alsoQuevedoetal.,2006).Thus,inthiscase,itisoptimaltosendthemeasured plant output without any coding. Stated differently,for the simple case under study, if uncoded signals are used,and if the physical constraints allow this to be done, then itis preferable to  place the communication link in the up-link rather than in the down-link  . (See Quevedo et al., 2006 for anexperimental validation of this observation). 3.3.3. Non-zero references With non-zero reference signals, the two NCS architecturesstudied so far will, in general, give different performance. Inparticular, if   d   =  n  =  0, then the achievable loss functions in(14) and (18) are lower bounded by J  opt D  =    12      −  | S( e  j  )T( e  j  )Q 0 ( e  j  )R( e  j  ) | d   2 , J  opt U   =    12      −  | (T( e  j  )) 2 Q 0 ( e  j  )R( e  j  ) | d   2 . (20)To investigate this situation further, it is worthwhile recallingthat common design strategies will ensure that  T( e  j  )  ≈  1  ( ⇔ S( e  j  )  ≈  0 )  at those frequencies where  R( e  j  )  is significant,and that  | T( e  j  ) |  is small at other frequencies (Goodwin et al.,2001). As a consequence, we would anticipate, by examining(20), that  J  opt D  <J  opt U   . We conclude that, for non-zero referencesignals(andoptimalcoding),thearchitectureof Fig.4generallyoutperforms that of  Fig. 5, i.e., if possible  the channel should be placed in the down-link  .In the following section we will propose a novel NCS ar-chitecture, which, when equipped with optimal coder–decoderpairs, will often give enhanced performance relative to the twoarchitectures analysed so far. Remark5 ( Twochannelarchitectures ). Tosomeextent,ourre-sults can be applied to NCS architectures with communicationchannels in both the up-link and down-link. Indeed, if signal-to-noise ratios of these channels are sufficiently large, then thedesign of each coder–decoder pair can be based on the chan-nel noise model in each respective link and expressions (15)and (19). 4. Networked Youla architecture for stable plants We will use the form of theYoula parameterisation describedin Section 2.1 to develop an alternative parsimonious NCS 2 This is the case of, for example, simple internal model based controllers,see Morari and Zafiriou (1989).
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