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Subband coding for networked control systems

Subband coding for networked control systems
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  See discussions, stats, and author profiles for this publication at: Subband coding for networked control systems  Article   in  International Journal of Robust and Nonlinear Control · November 2009 DOI: 10.1002/rnc.1393 CITATIONS 11 READS 10 3 authors , including: Some of the authors of this publication are also working on these related projects: Stochastic Model Predictive Control   View projectDaniel QuevedoUniversität Paderborn 214   PUBLICATIONS   3,383   CITATIONS   SEE PROFILE All content following this page was uploaded by Daniel Quevedo on 22 September 2014. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the srcinal documentand are linked to publications on ResearchGate, letting you access and read them immediately.  INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL  Int. J. Robust Nonlinear Control  2009;  19 :1817–1836Published online 6 November 2008 in Wiley InterScience ( DOI: 10.1002/rnc.1393 Subband coding for networked control systems ‡ Daniel E. Quevedo ∗ , † , Eduardo I. Silva and Graham C. Goodwin School of Electrical Engineering and Computer Science ,  The University of Newcastle ,  Australia SUMMARYWe study a source coding method for networked control systems (NCSs) for SISO LTI plant models. Tomake efficient use of limited communication resources between controller and plant actuator, we developa non-uniformly sampled subband coding NCS architecture, which we design to minimize a bound on thetracking error variance caused by quantization effects. Our main contribution, in relation to earlier work on subband coding NCSs, lies in that we allow each subband to be equipped with pre- and post-filters. Tocharacterize optimal filters and to solve the subband bit-allocation problem, we model quantization effectsvia signal-to-noise ratio constraints and explicitly take into account the closed-loop nature of signals.Simulation results verify that the resultant designs perform exceptionally well, even in non-idealizedsituations. Copyright  q  2008 John Wiley & Sons, Ltd. Received 22 January 2008; Revised 1 August 2008; Accepted 9 September 2008KEY WORDS : networked control systems; subband coding; quantization; signal-to-noise ratio channels 1. INTRODUCTIONThere has been an increasing trend toward using practical communication channels in low-levelcontrol loops, see, e.g.  [ 1–3 ]  and the references therein. Performance of these, so-called  networked control systems  (NCSs), is often limited by available communication resources. For example, due tobit-rate limitations, all signals to be transmitted need to be quantized  [ 4 ] . Furthermore, data may beaffected by time-delays and data-dropouts  [ 5,6 ] . While traditional control system design generallyneglects communication aspects, see, e.g.  [ 7 ] , in NCSs these should be taken into account.Owing to the presence of communication channels, in NCSs there exist additional degreesof freedom in the design process when compared with traditional control loops. Thus, to make ∗ Correspondence to: Daniel E. Quevedo, School of Electrical Engineering and Computer Science, The University of Newcastle, Callaghan, NSW 2308, Australia. † E-mail: ‡ A preliminary version of this paper was presented at the 3rd IFAC Symposium on System, Structure and Control,Foz do Iguac¸u, Brazil, October 2007.Copyright  q  2008 John Wiley & Sons, Ltd.  1818  D. E. QUEVEDO, E. I. SILVA AND G. C. GOODWIN efficient use of the available communication resources, it is useful to re-examine architecturalissues and to develop appropriate coding methods  [ 8–10 ] .If the communication constraints imposed by the network are very severe, then a completecontrol re-design for NCSs may be necessary, see also  [ 10,11 ] . In more lenient situations, e.g.if the network has sufficient throughput and delays incurred are approximately constant, then emulation-based approaches , where the impact of the network on a given nominal control designis examined, become interesting alternatives  [ 12,13 ] . For example,  [ 14 ]  focuses on networks,which do not introduce delays, and shows,  inter-alia , that source coding schemes that use a scalarquantizer and appropriately designed LTI pre- and post-filters can be used to mitigate the effectof quantization effects on the closed loop. The work in  [ 14 ]  also illustrates a key feature of quantized control systems, namely, that at least for open-loop stable plants, the approximation of quantization effects via a signal-to-noise ratio constrained channel model is often accurate, even if rather coarse quantizers with a rate of, say, 3 bits / sample are used (see also experimental resultsdocumented in  [ 15 ] ). These simple quantization models are widespread for the design of signalprocessing systems, see, e.g.  [ 16 ]  and can also be used for emulation-based design of NCSs withmore complex feedback coders  [ 17 ] . Alternative approaches for the design of feedback coders foruse in closed-loop control can be found in  [ 18,19 ] .An important property of controllers in most feedback loops is that they serve various purposes,including stabilization, reference tracking and disturbance compensation. Controller output signalscan, thus, be regarded as consisting of several partial signals, each having a distinct characteristic.For example, in PI-controllers the P-part is fast and is affected by noise, whereas the I-part is muchslower and serves essentially for regulation purposes. If in an NCS communication resources areto be used parsimoniously, then it makes sense to use source coding methods where each of thesepartial signals is treated differently.The above observation led us to investigate in  [ 20 ]  an NCS architecture, where the controlleroutput is composed of parallel sub-signals. Each sub-signal is sent over a different channel to theplant input. These channels can be either separate physical media or consist of so-called ‘virtualchannels’, where only a single physical link is used. § By formulating a finite-set constrainedpredictive controller, which explicitly takes into account the associated communication constraintsresulting from individual sampling rates and quantization sets, it was illustrated in  [ 20 ]  that theuse of virtual channels may give performance gains.The use of parallel (virtual) channels in NCSs has also been advocated in  [ 22 ] . In that work, itis shown that to stabilize an open-loop unstable scalar system, it is useful to complement a noisycommunication channel with an additional noiseless channel that operates at low sampling andbit rates. An alternative setting is described in  [ 23 ] . There, the authors focus on time-delays andillustrate the performance benefits, which result from switching between channels and associatedcontrollers. In  [ 24 ]  the advantage of using multiple description coding for state estimation overparallel channels with erasures was illustrated.Clearly, the use of parallel channels in NCSs is related to the use of filter bank and subbandcoding methods, which underlie many signal and image processing applications (e.g. MP3 audiocompression), see, e.g.  [ 25–27 ] . The use of filter banks in NCSs was addressed in  [ 28–30 ] . In § The differentiation between virtual channels results from the use of different communication schemes, see, e.g.  [ 21 ] .For example, a single physical channel can be used as various virtual channels by trading off competing channelcharacteristics such as, time-delays and sampling rates versus occurrence of data dropouts. If Ethernet with IPv6is used, then separate virtual channels are obtained if packets are assigned individual priorities.Copyright  q  2008 John Wiley & Sons, Ltd.  Int. J. Robust Nonlinear Control  2009;  19 :1817–1836DOI: 10.1002/rnc  SUBBAND CODING FOR NCS  1819particular, Ishii and Hara  [ 29,30 ]  advocate the use of an NCS architecture where the partial fractionexpansion of a given LTI controller yields subband signals of the corresponding control signal.Each subband is sampled at a rate, which is sufficiently high to avoid aliasing, and is assigned anindividual bit rate. At the plant input side, a filter bank for reconstruction of the subband signalscompensates for random data loss and can be designed by using  H ∞  methods  [ 29,30 ] .In the present work, we study an alternative emulation-based design procedure for subbandcoding NCSs, where channels do not introduce time-delays. As in  [ 29,30 ] , our approach uses agiven nominal LTI controller, whose decomposition yields subband signals of the control input,each having a sampling rate such that aliasing effects can be neglected. Our main contribution liesin the fact that we incorporate LTI pre- and post-filters in each subband. We show how these filterscan be designed to minimize the variance of the component of the control tracking error causedby channel effects. This allows us to provide bounds on the performance of the NCS architecturestudied. For that purpose, we build upon  [ 14 ]  and focus on quantization effects, which we modelvia signal-to-noise ratio constraints. Here, we explicitly take into account the fact that channelnoises are fed back through the loop. The pre- and post-filters are restricted to satisfy a perfectreconstruction constraint, which ensures that, in the absence of channel effects, the nominal loopsensitivities are recovered. The filters obtained depend upon the power spectral densities (PSDs) of closed-loop signals and, thus, upon the nominal control design. They are, however, independent of the channel signal-to-noise ratios. Consequently, we can solve the problem of allocating bit rates tothe quantizers employed in the subband coding NCS architecture after the filters have already beendesigned. Simulation studies presented in the sequel document that even in situations where signalsare not strictly bandlimited and, thus, the expressions developed are only approximate, subbandcoding NCS performance can be significantly improved through use of the pre- and post-filtersproposed.An outline of the remainder of the paper is as follows: In Section 2 we present the subbandcoding NCS architecture of interest here. The subband channel model is described in Section 3.Section 4 shows how to design the pre- and post- filters. The optimal bit-allocation problem isaddressed in Section 5. Section 6 documents a simulation study. Section 7 draws conclusions.The present paper is roughly based upon our recent conference contribution  [ 31 ] . It also extendsthe approach in  [ 14 ]  to subband coding NCSs.2. SUBBAND CODING NCS ARCHITECTUREAs foreshadowed in the introduction, the NCS design approach presented in this work comprises anembellishment of a given nominal LTI control design for use in NCSs with parallel communicationchannels. In this section, we will first present the nominal design, and then characterize theassociated subband signals. 2.1. Nominal design We will consider an LTI controller, say  C  (  z ) , which has already been designed so that, in theabsence of network effects, the output of an SISO discrete-time plant  G (  z )  tracks a zero meanstochastic reference signal  r  , see Figure 1.The plant is assumed to be perturbed by a zero mean output disturbance process  d   and the plantoutput  y  is assumed to be affected by a zero mean measurement noise process   . We will assume Copyright  q  2008 John Wiley & Sons, Ltd.  Int. J. Robust Nonlinear Control  2009;  19 :1817–1836DOI: 10.1002/rnc  1820  D. E. QUEVEDO, E. I. SILVA AND G. C. GOODWIN Figure 1. Non-networked LTI control loop. that the three processes  r  ,  d   and    are uncorrelated. In this non-networked situation, the trackingerror e  r  −  y satisfies e = S  (  z )( r  − d  ) + T  (  z )   (1)where S  (  z )   11 + G (  z ) C  (  z ),  T  (  z )   G (  z ) C  (  z ) 1 + G (  z ) C  (  z ) are the nominal closed-loop sensitivities, see, e.g.  [ 7 ] .The PSD of the tracking error in this nominal case is given by |  E  nom ( e  j  ) | 2 =| S  ( e  j  ) | 2 ( |  R ( e  j  ) | 2 +|  D ( e  j  ) | 2 ) +| T  ( e  j  ) | 2 |  ( e  j  ) | 2 (2)where  |  R ( e  j  ) | 2 ,  |  D ( e  j  ) | 2 and  |  ( e  j  ) | 2 are the PSDs of   r  ,  d   and   , respectively. The associatedtracking error variance is given by:  nom e  = 12      −  |  E  nom ( e  j  ) | 2 d   (3)In the sequel, we will assume that  S  (  z )  is stable and that the nominal design characterized by (1)and (2) gives satisfactory performance  in the absence of channel effects . The main theme of thepresent work lies in designing a subband coding NCS architecture, which emulates this nominaldesign, i.e. which minimizes network effects. 2.2. Controller decomposition The distinguishing characteristic of NCSs is that communication constraints limit achievable perfor-mance. As can be concluded from  [ 20,22,29 ] , performance gains achieved by multi-channelNCS architectures, when compared with one-channel configurations, are intimately related to thepossibility of allocating channel resources for each channel separately. Copyright  q  2008 John Wiley & Sons, Ltd.  Int. J. Robust Nonlinear Control  2009;  19 :1817–1836DOI: 10.1002/rnc
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