Digital and Sampled-Data Output Feedback Control over Signal-to-Noise Ratio Constrained Communication Channels

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1 00 American Control Conference Marriott Waterfront Baltimore MD USA June 30-July 0 00 WeA075 Digital and Sampled-Data Output Feedback Control over Signal-to-Noise Ratio Constrained Communication s AJ Rojas Abstract Communication channels are known to impose a number of constraints on control objectives when explicitly considered in feedback loops One recent line of work considers the problem of feedback stabilisation subject to a channel signal-to-noise ratio SNR constraint he present paper extends such formulation to study the stabilisability of digital output feedback loops and sampled-data output feedback loops We then quantify the difference in channel capacity between the present setting and the discrete-time output feedback loop setting his difference is known to represent a fundamental limitation in control over networks performance I INRODUCION he study of control problems with feedback over communication channels has seen great activity in recent years; see for example [] [] and references therein One recent line of research introduced a framework to study stabilisability of a feedback loop over channels with a signal to noise ratio SNR constraint [3] [4] [5] hese papers obtained the infimal SNR required to stabilise an unstable plant over an additive white Gaussian noise AWGN channel A distinctive characteristic of the SNR approach is that it is a linear formulation For the case of linear time invariant LI controllers and minimum phase plant models with no time delay these conditions match those derived in [6] by application of Shannon s theorem [7 03] Our first contribution in the present paper is to extend on the LI infimal SNR output feedback results of [3] by considering the case of digital output feedback loop he digital characteristic is modeled by means of a quantiser block at the channel output see Figure We approximate the quantiser by means of an additive quantiser error process and then obtain the infimal LI SNR for digital output feedback loop stabilisability in closed-form Our second contribution quantifies the infimal LI SNR for sampled-data output feedback loop stabilisability in closedform In this case the plant model is a discrete-time equivalent of a continuous-time plant model subject to Digital-to- Analog and Analog-to-Digital conversion DAC-ADC see Figure 3 Both the DAC and ADC block contain further quantisers which we also approximate by means of additive quantiser error processes Invoking a spectral factorisation argument we are then able to quantify the infimal LI SNR for sampled-data output feedback loop stabilisability in closed-form AJ Rojas is with the ARC Centre of Excellence for Complex Dynamic Systems and Control he University of Newcastle Australia alejandrorojas@newcastleeduau until April 00 Afterwards AJ Rojas is with the Departamento de Ingeniería Eléctrica Universidad de Concepción Chile Fig plant Cz rk uk nk sk Gz yk Digital output feedback stabilisation of a discrete-time unstable Of the two possible configurations for the location of the idealised communication channel measurement path and control path we consider the case of a memoryless AWGN channel over the measurement path he rest of the paper is organized as follows: in Section II we present the main assumptions for the present paper In Section III we study the infimal LI SNR for digital output feedback loop stabilisability In Section IV we present the solution to the more complex problem of infimal LI SNR for sampled-data output feedback loop stabilisability We conclude with Section V by presenting our final remarks erminology: let C denote the complex plane Let D D D and D denote respectively the open unit-circle closed unit-circle open and closed unit circle complements in the complex plane C with D the unit-circle itself Cz uk II ASSUMIONS rk qk nk sk Gz yk Fig Digital output feedback control stabilisation of a discrete-time unstable plant General assumptions for the discrete-time feedback system depicted in Figure are lant model assumptions: throughout the present work if not stated otherwise it is assumed that the plant model Gz is a real rational function with the following properties: - relative degree n g = - all its zeros have moduli less than - m unstable poles ρ i > each with multiplicity n i i = m /0/$ AACC 59

2 model assumptions: he AWGN channel is characterised by two parameters: the admissible input power level of the channel and the channel additive noise process nk additive noise process: the channel additive noise process is labelled nk and it is a zero-mean iid Gaussian white noise process with variance σ uantisation Error rocess: he quantisation error process q represents the quantisers and it is assumed to be an additive white uniformly distributed noise process independent of nk defined for [ ] he quantiser output is given then by u qk i i M u q k M y q k = M u q k < M M u q k > M y q and u q are respectively the output and input of the quantiser he quantiser level length is defined as = M/ b with b the number of allocated bits he overall range for the quantiser is given by [M M] M R M selected to be such that the probability of overflow is small he value of b is assumed such that the number of quantisation levels is high [8 37] he variances for the proposed quantisation error process is defined as σ q = / Notice that the quantiser s approximation by means of a white uniform additive noise is an accepted methodology in signal processing literature [9] and furthermore it has proved to give good predictions of the overall system behaviour [0] III INFIMAL LI SNR FOR DIGIAL OUU FEEDBACK LOO SABILISABILIY In this section we are concerned about guaranteeing the stabilisability of a digital output feedback loop whenever a memoryless AWGN channel is explicitly considered A roblem Definition We assume that Cz is such that the closed-loop system is stable in the sense that for any distribution of initial conditions the distribution of all signals in the loop will converge exponentially rapidly to a stationary distribution he channel input power defined by s ow lim k E { s k } is required to satisfy an imposed power constraint > E { s } for some predetermined power level E { s } stands for lim k E { s k } and it is introduced to easy the notation Under reasonable stationarity assumptions [ 44] the power in the channel input may be computed as E { s } = sn z σ sq z σ q sn z = CzGz CzGz sqz = CzGz CzGz are the transfer function that relates sk with nk and sk with qk respectively hus the power constraint at the input of the channel translates to the SNR bound σ on the H norm of sn z and sq z σ > snz sqz σq σ From we observe that a fundamental limitation in the SNR of the memoryless AWGN channel will be given then by the simultaneous infimum of sn z and sqz which indeed are at the core of the infimal LI SNR problem definition that follows roblem : Infimal LI SNR for Digital Output Feedback Loop Stabilisability Find a proper rational stabilising controller Cz such that the digital output feedback loop subject to the assumptions presented in Section II is stable and the transfer functions in achieve the infimum on the admissible channel SNR in B Infimal LI SNR for Digital Output Feedback Loop Stabilisability In the present subsection we quantify the infimal LI SNR for Digital Output Feedback Loop Stabilisability heorem : Infimal LI SNR for Digital Output Feedback Loop Stabilisability Consider the digital LI output feedback represented in Figure and that all the elements of the digital LI output feedback satisfy the assumptions listed in Section II then m σ > ρ i ni σ q σ 3 roof: Start by recognising that sq z = z and sn z = z with z = CzGz/ CzGz We then observe that σ > inf Cz stab z σ q σ we have introduced the infimum notation as to make explicit that we are solving roblem On the other hand from [3 heorem III] we have that inf Cz stab z = ρ i ni which concludes the proof Remark : Notice that as the variance of the quantisation error σ q tends to zero we recover the infimal LI SNR for stabilisability result reported in [3 heorem III] IV INFIMAL LI SNR FOR SAMLED-DAA OUU FEEDBACK LOO SABILISABILIY In this section we are concerned about guaranteeing the stabilisability of a digital output feedback loop subject to digital-to-analog and analog-to-digital conversions whenever a memoryless AWGN channel is explicitly considered In Figure 3 we have a representation of the setting of interest As a first reasonable approximation we consider the holder block together with the continuous-time plant model Gs and the sampler block with sampling time to be an equivalent discrete-time plant model Gz see Figure 4 60

3 Fig 3 plant Cz rk uk nk DAC Holder sk Gs ADC yt Output feedback control stabilisation of a sampled-data unstable Cz rk uk nk Gz Holder sk Gs yk Fig 4 Output feedback control stabilisation of an equivalent discrete-time unstable plant for more details Notice that the assumptions presented in Section II are still in place indeed as a result of this hold and sampling process is known that continuous-time plant models of arbitrary relative degree will be mapped to discrete-time plant models with relative degree n g = he change of relative degree is in general due to the introduction of sampling non-minimum phase zeros Nonetheless this unwanted for the present paper characteristics can be avoided if we consider for example results such as [] these zeros are mapped inside the unit circle instead Finally we approximate each quantiser in Figure 4 as an equivalent quantisation error as suggested in Section II see Figure 5 Each quantisation error process is independent of each other and of the channel additive noise nk However for the sake of simplicity we consider each quantisation error to have the same characteristics thus we will only refer to M and b when appropriate without distinguishing between the three quantisation errors Next we briefly restate roblem for the setting in Figure 5 A roblem Definition Fig 5 plant Cz uk ek rk sk q k nk q k Gz qok yk Output feedback control stabilisation of a discrete-time unstable We assume that Cz is such that the closed-loop system is stable he channel input power under reasonable stationarity assumptions [ 44] may be computed as σ > snz soz σo σ s z sn z = CzGz CzGz soz = s z = σ σ sz σ σ 4 CzGz CzGz CzGz Gz sz = CzGz are the transfer function that relates sk with nk and sk with each quantisation error q i k respectively with i = 0 From 4 we observe that a fundamental limitation in the SNR of the memoryless AWGN channel is given by the simultaneous infimum of sn z soz sz and s z which indeed are at the core of the infimal SNR problem definition that follows roblem : Infimal LI SNR for Sampled-Data Output Feedback Loop Stabilisability Find a proper rational stabilising controller Cz such that the feedback control loop subject to the assumptions presented in II is stable and the transfer functions in 5 achieve the infimum on the admissible channel SNR 4 B Spectral Factorisation From Figure 5 we have that when the memoryless AWGN channel is located in the feedback path the augmented system consisting of Gz the channel model and each quantisation error is described by xk =A G xk B G ukb G q k ek=c G xk q o k nk q k We then have that the measurement and process noises are given by 5 6 wk = B G q k vk = q o k q k nk 7 hus the covariance matrix is given by [ ] [ ] wk [wj E { vj vk ] W S } = S δ V kj with δ kj the Kronecker s delta By replacing 7 we obtain W=B G B G σ S = 0 V=σ o σ σ 8 In general the procedure of LG optimisation with recovery involves the solution of two Riccati equations one associated with the design of the observer and another with the design of the regulator he Riccati equations for the associated observer is defined as A G ΣC G Σ= A G ΣA G C G ΣC G V CGΣAG W 9 K p = A G ΣC G C G ΣC G V 6

4 Define the observer sensitivity function as S est z = L est z 0 the open loop transfer function L est z is given by C G zi A G K p A well known fact that stems from the assumed plant model see for example [3] is that the optimal state feedback sensitivity S est z is recovered when using loop transfer recovery LR as the optimal output feedback sensitivity Sz = / CzGz that is Sz = S est z By means of a spectral factorisation argument we now then characterise the optimal S est z and thus Sz that takes part into the infimal LI SNR for Sampled-Data Output Feedback Loop Stabilisability heorem : Induced Spectral Factorisation he Riccati equation in 9 induces the following spectral factorisation S estzc G ΣC G σ o σ σ S est z = σ o σ σ Gzσ Gz S est z is as in 0 roof: From equation 56 in [4 p 85] and adapting the notation to the present paper we have C G zi A G K p C G ΣC G V K p z I A G C G = VC G zi A G Wz I A G C G o obtain replace WV and S as in 8 into whilst recognising S est z as in 0 on the LHS of which ends the proof From equation we have that the plant Gz together with σ σo σ and σ will determine the sensitivity function S est z C Infimal LI SNR for Sampled-Data Output Feedback Loop Stabilisability Consider that Ĉz the controller that solves roblem is in place We specify the plant model to be Gz = qz pz = qz m z ρ i ni pz 3 he polynomial qz is assumed known such that the overall relative degree is n g = and that all its solutions for qz = 0 are in D As we stated before we are ultimately attempting to characterise the particular Ŝz that takes part into infimal LI SNR for sampled-data output feedback loop stabilisability solution From we obtain the location of the poles of Ŝz Ŝ C G ΣC G z σo σ Ŝ z = σ pzpz qz qz σo σ σ pzpz 4 σ From 4 we recognize that the poles of Ŝz labelled z i z i D i = m are the Schur s solutions of σ pzpz qz σo σ σ qz = 0 5 We have then that a characterisation of the optimal Ŝz is given by m Ŝz = z ρ i ni pz r z z 6 i si r is equal to the degree of pz in 3 s i is the multiplicity of each pole z i and the hat notation is to stress that this is the infimal sensitivity function We also stress that although we do not have a closed-form for each z i they can be computed by any of the many currently available algorithms for the purpose of finding the solutions of a polynomial thus for all purposes we consider them as known quantities Remark : Notice that the stable poles of Gz will also play a role in as different from the case of simple stabilisability only the unstable poles of Gz played a role see for example [4] We now quantify the infimal LI SNR for sampled-data output feedback loop stabilisability for the case of a memoryless AWGN channel heorem 3: Infimal LI SNR for Sampled-Data Output Feedback Loop Stabilisability Assume the plant to be as in 3 and the channel model to be a memoryless AWGN channel as in Figure 5 hen for the sampled-data output feedback loop to be stabilisable the channel SNR must satisfy σ > σ o σ σ o σ m σ s r j σ σ l= m s i l= m il l! m il = s i l! g il = s i l! g il l! d sil dz sil d sil dz sil s m j d l dz l ρ i ni mjp z p z z j p ρ i ni d l ḡjp z p p dz l z z j m j= z / ρ j nj pz rj= z z j sj rj= j i j i qz z z j sj 7 8 roof: Introduce a coprime factorisation such that Gz = Nz/Mz qz m Nz = m z ρ i ni pz Mz = ni z ρi z ρ i 6

5 and the parameterisation of all stabilising controllers see [5 pp 64-65] Cz = Xz Mzz/ Y z Nzz Xz and Y z satisfy the Bezout identity NzXz MzY z = and z is the Youla parameter see for example [6] From and we have σ > σ o σ ˆz σ o σ σ σ ŜzGz σ Notice that for the infimal solution Ŝz is given as in 6 thus σ > σ o σ NX NM ˆ σ o σ σ qz r z z i si σ σ 9 We start by analyzing the term NX NM ˆ which can be factorised as NX NM ˆ = M NX N ˆ since Mz is all-pass Introduce now the decomposition M znzxz = Γ z Γz Γ is in H and Γ is in H and replace NX NM ˆ = Γ Γ N ˆ 0 he term Γ is the infimal LI SNR obtained from applying [3 heorem III] We then know that is equal to m ρ i ni Also directly from 0 we can observe that the infimal Youla parameter z for stabilisability with no quantisation error is then given by ΓzN z since this choice would ensure the squared H norm of Γ N to be immediately zero We observe then NX NM ˆ = ρ i ni N N ˆ Reintroduce now Mz again since it is all-pass into the squared H norm term add and subtract the term NzXz and rearrange terms and recognize z = NXNM and ˆz = NX NM ˆ NX NM ˆ m = ρ i ni ˆ Since ˆz = Ŝz z = Sz and Sz = m zρi z/ ρ i ni we have NX NM ˆ = ρ i ni m z ρ i ni pz r z z i si z ρi z / ρ i ni By recognizing and extracting Mz we can then claim that NX NM ˆ m = ρ i ni m ρ i ni z / ρ i ni pz r z z i si r z z i si Applying partial fraction expansion on the term inside the RHS squared H norm and then invoking the Residue heorem see for example [7 pp 69 7] we obtain NX NM ˆ = ρ i ni ρ i ni l= m il l! s j r d l dz l mjp z p z z j p with m il as in 8 Notice that the term r z z i si in the numerator of the squared H norm does not contribute to the residue since it and its derivatives in z are zero at z = z i We now focus on the second norm in 9 for which applying partial fraction expansion and then invoking the Residue heorem gives σ σ l= g il l! r s j d l dz l ḡjp z p z z j p with g il as in 8 Finally we observe that the result in plus the above result gives the expression in 7 which concludes the proof heorem 3 quantifies the infimal LI SNR for sampled-data output feedback loop stabilisability Example : Consider the plant to be Gz = K/z ρ with ρ R ρ > and K R From we have that σ Gz σo σ σ Gz = z ρ ρ K σ ρσ o σ σ z z ρz ρ therefore z the solution for the numerator and pole of Ŝz is given by z = ρ ρ K σ ρσo σ σ ρ ρ K σ ρσo σ 4 σ the only solution that satisfies z D With z known 63

6 applying heorem 3 gives σ > ρ z ρ z σ o σ σ o σ ρ σ z ρ z σ K σ z 3 We observe that as σ o σ σ 0 then z /ρ we recover the infimal LI SNR for stabilisability of ρ D Capacity As in the previous section we can now quantify the channel capacity difference imposed by the sampled-data nature of the output feedback loop Corollary 4: Capacity Difference Consider a plant model as in 3 and a memoryless AWGN channel as in Figure 5 hen the infimal channel capacity Ĉ for stabilisability with input disturbance rejection must satisfy Ĉ γ = m n i log ρ i = log l= l= m il l! s r j γ γ m ρ i ni 4 d l mjp z p p dz l z z j and γ = σ o σ σ o σ m σ ρ i ni ρ i ni s m il r j d l mjp z p l! dz l z z j p σ σ m s i l= g il l! s m j d l ḡjp z p p dz l z z j with m il and g il as in 8 roof: Directly from heorem the definition of the capacity for a memoryless AWGN channel and the fact that it does not increase with feedback Example : In Example we quantified the infimal LI SNR for sampled-data output feedback loop stabilisability In the present example we apply the result from Corollary 4 to obtain the channel capacity difference which is then given by Ĉ log ρ = log σ o σ σ o σ σ z /ρ z ρ ρ z /ρ ρ z σ σ K z Observe that as expected when σ o σ σ 0 the RHS of the above expression is zero and the channel capacity matches the infimal channel capacity for stabilisability of log ρ V FINAL REMARKS In the present paper we have presented the infimal LI SNR for digital output feedback loop stabilisability solution in closed-form for the case of a memoryless AWGN channel on the measurement path and minimum phase unstable plants with relative degree one Future directions for research should include a more general plant model as well as the case of a memoryless AWGN channel located between the controller and the plant model in particular for the sampleddata output feedback loop case VI ACKNOWLEDGEMEN he author wishes to thank everyone at the Australian Research Council Centre of Excellence for Complex Dynamic Systems and Control and the School of Electrical Engineering and Computer Science he University of Newcastle Australia for their invaluable support through the years REFERENCES [] Special Issue on Networked Control Systems IEEE ransactions on Automatic Control 499 September 004 [] GN Nair F Fagnani S Zampieri and RJ Evans Feedback Control under Data Rate Constraints: an Overview In roceedings of the IEEE special issue on he Emerging echnology of Networked Control Systems January 007 [3] JH Braslavsky RH Middleton and JS Freudenberg Feedback Stabilisation over Signal-to-Noise Ratio Constrained s IEEE ransactions on Automatic Control 58: [4] AJ Rojas JH Braslavsky and RH Middleton Output Feedback Control over a Class of Signal to Noise Ratio Constrained Communication s In roceedings of the 006 American Control Conference Minneapolis USA June 006 [5] AJ Rojas Signal-to-noise ratio performance limitations for input disturbance rejection in output feedback control Systems & Control Letters 585: May 009 [6] GN Nair and RJ Evans Stabilizability of stochastic linear systems with finite feedback data rates SIAM J Control and Optimization 43: July 004 [7] M Cover and JA homas Elements of Information heory John Wiley & Sons 99 [8] RG Gallager rinciples of Digital Communication Cambridge University ress 008 [9] N Jayant and Noll Digital coding of waveforms rinciples and approaches to speech and video rentice Hall 984 [0] B Widrow and I Kollár uantization Noise Cambridge University ress 008 [] KJ Åström Introduction to Stochastic Control heory Academic ress 970 [] JI Yuz GC Goodwin and H Garnier Generalised hold functions for fast sampling rates In roceedings of the 43rd IEEE Conference on Decision and Control pages Atlantis aradise Island Bahamas December 004 [3] JM Maciejowski Asymptotic Recovery for Discrete-ime Systems IEEE ransactions on Automatic Control 306: [4] B Anderson and J Moore Optimal Filtering Dover publications NY 005 [5] JC Doyle BA Francis and AR annenbaum Feedback Control heory Macmillan ublishing Company 99 [6] M Vidyasagar Control System Synthesis: A Factorization Approach MI ress 985 [7] RV Churchill and JW Brown Complex Variables and Applications McGraw-Hill International Editions fifth edition