Rate Sufficient Conditions for Closed-loop Control over Half-duplex ...

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Abstract—The problem of remotely controlling an unstable noiseless linear time invariant system over noisy half-duplex relay channels with average power ...
Rate Sufficient Conditions for Closed-loop Control over Half-duplex AWGN Relay Channels Ali A. Zaidi∗ , Tobias Oechtering∗ , Mikael Skoglund∗ and Serdar Y¨uksel† ∗ School

Abstract—The problem of remotely controlling an unstable noiseless linear time invariant system over noisy half-duplex relay channels with average power constraints is considered. For information transmission, we propose a coding scheme based on the Schalkwijk-Kailath scheme. Therefore, we derive conditions on rate which are sufficient for mean square stability of the linearly controlled LTI system over non-orthogonal, orthogonal, and two-hop AWGN relay channels.

I. I NTRODUCTION The problem of remotely controlling dynamical systems over communication channels has gained significant attention in recent years. Such problems ask for interaction between stochastic control theory and information theory [1]. The minimum data rate below which the stability of an LTI system is impossible has been derived in stochastic and deterministic settings in [2–4], where they considered quantization errors and noise-free rate-limited channels. In [5, 6] are necessary rate conditions required to stabilize an LTI plant almost surely. However, from [7] we know that the characterization by Shannon capacity is not enough for sufficient conditions for moment stability in closed-loop control. In [8] a simple coding scheme is proposed to mean square stabilize an LTI plant over noise-free rate-limited channels. The mean square stability of discrete plant over signal-to-noise ratio constraint channels is addressed in [9, 10]. In [11] the authors considered noisy communication links between both observer–controller and controller–plant. In this paper we study conditions on the rate which are sufficient for stability of a discrete time LTI plant over noisy relay channels. The achievable information rate over the relay channel depends on the processing strategy of the relay. The most well known relaying strategies are amplify-and-forward (AF), compress-and-forward, and decode-and-forward [12]. AF strategy is well suited for delay sensitive closed-loop control applications and is therefore addressed in this paper. We focus our study on half-duplex AWGN relay channels with one time unit delayed noiseless feedback channel. We consider the half-duplex relay channel as a part of the feedback control loop. For coding we propose to use a Schalkwijk-Kailath scheme [13] based coding strategy which is suitable for feedback channels [14, 15] since it implements the RobbinsMonro stochastic approximation algorithm. The objective of this work is to derive sufficient rates for stability of an LTI

Plant

of Electrical Engineering and the ACCESS Linnaeus Center Royal Institute of Technology (KTH), Stockholm, Sweden † Mathematics and Engineering, Department of Mathematics and Statistics Queen’s University, Kingston, Ontario, Canada

Yt

O/E

St

AWGN Relay Channel

Rt

D/C Ut

Fig. 1. The unstable plant has to be controlled by the actions of observer/encoder (O/E) and decoder/controller (D/C) over the AWGN relay channel.

plant in mean square sense [2, 3, 7–9] over the given relay channels. II. P ROBLEM F ORMULATION We consider a scalar discrete-time LTI system, whose state equation is given by Xt+1 Yt

= λXt + Ut = Xt

(1)

where {Xt } ⊆ R, {Ut } ⊆ R and {Yt } ⊆ R are state, control and observation processes. We assume that the open-loop system is unstable (λ > 1) and the initial state X0 is a random variable with variance σx2 and an arbitrary probability distribution. We consider a remote control setup, where the observed state value is transmitted to the controller over an AWGN relay channel as shown in Fig. 1. In order to communicate the observed state value Yt over the noisy channel, an encoder E is lumped with the observer O and a decoder D is lumped with the controller C. In addition there is an intermediate relay node R within the channel to support communication from E to D. At any time instant t, St and Rt are the input and the output of the AWGN relay channel and Ut is the control action. Let ft denote the observer/encoder policy, then we have St = ft (Y0 , Y1 , ..., Yt , R1 , R2 , ..., Rt−1 ) which PT −1 must satisfy an average power constraint limT →∞ T1 t=0 E[St2 ] ≤ PS . Further let γt denote the decoder/controller policy, then Ut = γt (R1 , R2 , ..., Rt ) which must satisfy a cost constraint, P ∞ 2 t=0 E[Ut ] < ∞ . The objective in this paper is to find a sufficient condition on the system parameters so that the system in (1) can be mean square stabilized.

Definition 1: A system is said to be mean square stable if and only if lim E[Xt ] = 0,

t→∞

lim E[Xt2 ] = 0,

t→∞

t = 1, 3, 5, . . . t = 2, 4, 6, . . .

where h ∈ R denotes the gain of E − D link. III. S TABILITY R ESULTS We will first present our results in a comprehensive fashion and then provide the proofs in the next section. Theorem 2: The scalar linear time invariant system in (1) can be mean square stabilized over the half-duplex AWGN relay channel if !!   ˜ (β) M 2h2 βPS 1 log 1 + + log 1 + log (λ) < , ˜ (β) 4 N N (4) ˜ (β) = ˜ (β) = PR NR + N , and M where β ∈ [0, 1], N 2βPS +NR q 2 p 2βPS PR N . 2h2 (1 − β)PS + (2βPS +N 2 R )(2h βPS +N )

R Zt

(2)

regardless of the initial state X0 . There can be various configurations of the relay channel. In this work we focus on a half-duplex relay channel where the relay cannot receive and transmit signals simultaneously. Moreover, we consider the linear relaying strategy which is in particular suitable for delay sensitive closed-loop control applications. Although we know that linear strategies are not optimal in general for multi-sensor settings [16, 17], we consider AF relaying for the sake of simplicity. That is the relay amplifies the received signal under an average power constraint PR and forwards it to the decoder/controller. A general half-duplex AWGN relay channel is depicted in Fig. 2, where the variables Se,t and Sr,t denote the transmitted signals from the encoder E and the relay R at any discrete time step t. The variables Zr,t and Zt denote the mutually independent white noise components at the relay and at the decoder respectively with Zr,t ∼ N (0, Nr ) and Zt ∼ N (0, N ). The information transmission from the encoder consists of two phases as shown in Fig. 2. In the first transmission phase, the encoder E transmits message V1 with an average power 2βPS where 0 < β ≤ 1. The relay R listens but remains silent. In the second transmission phase both the encoder E and the relay R transmit with average powers 2(1−β)PS and PR respectively. The relay transmits an amplified version of the noisy signal received during the first transmission. The amplification at the relay is done under an average power constraint PR . Therefore the relay transmit signal at the discrete time step t is given by r PR (Se,t-1 + Zr,t-1 ) (3) Sr,t = a(Se,t-1 + Zr,t-1 ) = P S + NR q R where the amplification factor a is chosen equal to PSP+N R 2 in order to satisfy the average power constraint i.e., E[Sr,t ] = PR . Accordingly the relay channel output at the decoder is Rt which is given by Rt = hSe,t + Zt Rt = hSe,t + Sr,t + Zt

Zr,t

Se,t

E

Rt

D

(a) First transmission phase.

R

Sr,t Zt

Se,t

E

Rt

D

(b) Second transmission phase. Fig. 2.

Half-duplex AWGN relay channel.

An optimal choice of the power allocation parameter β depends on the quality (i.e., SNR) of E − D, E − R, and R − D links. Remark 3: The term on the right hand side of (4) is an achievable information rate over the half-duplex AWGN relay channel with noiseless feedback. This is shown in Appendix A. Remark 4: The following infinite horizon quadratic cost can be achieved:  ∞ X  2  λ4 σǫ20 1 + λ2 b1 + q b2 + λ2 b1 c2 2 E Xt + qUt = , (1 − λ4 b1 c1 ) t=1 (5) σ2 N 2h2 βPS N , b = where q > 0, σǫ20 = h2xPS , b1 = 2h2 βP 2 2 2h βPS +N , S +N ˜

˜

M (β) N c1 = M˜ (β)+ ˜ , and c2 = M ˜ (β)+N ˜ . This is shown in N Appendix B. A relay channel is said to be orthogonal if the signal spaces of the encoder and the relay are orthogonal. The given halfduplex relay channel is orthogonal if β = 1 that is in the second transmission phase the encoder stays quiet and only the relay transmits. Corollary 5: The scalar linear time invariant system in (1) can be mean square stabilized over an orthogonal half-duplex AWGN relay channel if !!   ˜ (1) M 1 2h2 PS , (6) log 1 + + log 1 + log (λ) < ˜ (1) 4 N N

where term on the right hand side of the inequality is an achievable information rate over the orthogonal half-duplex AWGN relay channel. A relay channel is said to be two-hop when there is no direct communication link from the encoder to the decoder and the information can be communicated only via the relay. The given half-duplex relay channel becomes two-hop if h = 0. Naturally for this case we chose β = 1. Corollary 6: The scalar linear time invariant system in (1) can be mean square stabilized over a two-hop half-duplex AWGN relay channel if   1 2PS PR log (λ) < log 1 + , (7) 4 PR NR + N (2PS + NR )

where the term on the right hand side of the inequality is an achievable information rate over the two-hop half-duplex AWGN relay channel. Remark 7: For a setup which is equivalent to the two-hop relay channel, we find a necessary condition in [11, Theorem 4.1] which reads as      1 PR 2PS log (λ) < min log 1 + , log 1 + . 4 N NR The condition in (7) becomes both necessary and sufficient if either the E − R link is noiseless (NR = 0) or the R − D link is noiseless (N = 0). IV. P ROOF Observe that if at any time instant t, the controller C gets to know the value of initial state X0 , then it can perfectly control the LTI system in (1) by bringing its state value Xt to zero. Thus, X0 is the message, that needs to be conveyed to the controller C. In order to stabilize the system, the {E, D}–pair works such that with every use of the communication channels (noisy feedforward and noiseless feedback), the controller C gets better information about X0 , which results into an improved control action. ˆ 0,t be the estimate of the initial state in Lemma 8: Let X ˆ 0,t − X0 ) be the time step t at the controller C, and let ǫt , (X associated estimation error. Then the scalar linear system in (1) can be moment stabilized over a communication channel if mǫt = 0,

lim λ2t σǫ2t = 0,

t→∞

for which mǫt = 0 and the quantity λ2t σǫ2t converges to zero as t → ∞. The proposed coding scheme is as follows. At time step t = 0: q PS The encoder E observes X0 and inputs Se,0 = 2 X0 to σx the relay channel. • The relay R neither listens nor it transmits. • The decoder D observes R0 = hSe,0 + Z0 . It then estimates q ˆ 0,0 = 1 σx2 R0 . X0 as X



h

The variance of ǫ0 is given by   σ2 N σǫ20 , E ǫ20 = 2x . h PS

The estimation error ǫ0 is a scalar multiple of a zero mean Gaussian variable, therefore ǫ0 ∼ N (0, σǫ20 ). First transmission phase, time steps t = 1, 3, 5, . . . : • The encoder q E observes Xt = λt ǫt-1 according to (9). It then inputs Se,t = 2βPS /σǫ2t−1 ǫt−1 to the relay channel. • The relay R listens but remains silent. • The decoder D observes Rt = hSe,t + Zt and computes the MMSE estimate of ǫt−1 , which is given by ǫˆt−1 = E[ǫt−1 |R1 , R2 , ..., Rt ] (a) (b) E[ǫt−1 Rt ] = E[ǫt−1 |Rt ] = Rt , E[Rt2 ]

(8)

σǫ2t

where mǫt and denote the first and the second moment of the initial state estimation error. Proof: In order to stabilize the given system, the controller C takes control action using the estimate of the initial ˆ 0,t provided by D. The control action in the system state X ˆ 0,0 . For t ≥ 1, the control actions first time step is U0 = −λX t+1 ˆ ˆ 0,t−1 ). With these control are given by Ut = −λ (X0,t − X actions, the system state equation in (1) can be rewritten as   ˆ 0,t − X0 . (9) Xt+1 = λt+1 X The mean value of the state at any t is given by h i ˆ 0,t − X0 = λt+1 mǫ . E[Xt+1 ] = λt+1 E X t

If mǫt = 0 for all t, then we have E[Xt+1 ] = 0 for all t. Further, the mean squared value of the state at any t is given by  2  2 ˆ 0,t − X0 = λ2(t+1) σǫ2t . E[Xt+1 ] = λ2(t+1) E X

If limt→∞ λ2t σǫ2t = 0 then the given system can be mean square stabilized. In order to prove Theorem 4, we propose a coding scheme for the general half-duplex relay channel. This scheme is based on the Schalkwijk–Kailath coding scheme [13]. By employing the proposed coding scheme, we then find a condition on λ

PS

The corresponding estimation error is given by s σx2 1 ˆ 0,0 − X0 = ǫ0 , X Z0 . h PS

where (a) follows from the orthogonality principle of MMSE estimation (that is E[ǫt Rt ] = 0) [18]; and (b) follows from the fact that the optimum MMSE estimator for a Gaussian variable is linear [18]. The variance of ǫˆt−1 is given by   E2 [ǫt-1 Rt ] = σ ˆǫ2t-1 , E ǫˆ2t = E[Rt2 ]



2h2 βPS 2h2 βPS + N



σǫ2t-1 . (10)

The decoder then updates its estimate of the initial state using ǫˆt−1 . The updated estimate is given by ˆ 0,t = X ˆ 0,t−1 − ǫˆt−1 . X

(11)

The MMSE estimation error is then given by ˆ 0,t − X0 (a) ǫt = X = ǫt−1 − ǫˆt−1 ,

(12)

where (a) follows from (11). The variance of ǫt is given by σǫ2t

    E2 [ǫt-1 Rt ] = , E ǫ2t = E ǫ2t-1 − E[Rt2 ]



N 2h2 βPS + N



σǫ2t-1 .

(13)

The estimation error ǫt is a linear combination of zero mean Gaussian variables, therefore we have ǫt ∼ N (0, σǫ2t ).

Second transmission phase, time steps t = 2, 4, 6, . . . : t • The encoder E observes Xt = λ ǫt-1 according to (9). The encoder output is given by s 2 (1 − β) PS Se,t = ǫt−1 . σǫ2t−1 q PR • The relay transmits Sr,t = (2βPS +NR ) (Se,t−1 + ZR,t−1 ). • The decoder D observes Rt = hSe,t + Sr,t + Zt = L1 ǫt−1 + L2 ǫt−2 + Z˜t , (14) r r 2(1−β)h2 PS 2βPS PR where L1 = , L = , and 2 σǫ2 (2βPS +NR )σǫ2 t−1 t−2 q ˜ (β)). Z˜t = Zt + 2βPPSR+NR Zr,t−1 with Z˜t ∼ N (0, N The decoder then computes the MMSE estimate of ǫt−1 given all previous channel outputs {R0 , R1 , ..., Rt } in the following three steps: 1) Compute MMSE prediction of Rt given {R1 , R2 , ..., Rt−1 } ˆ t = L2 ǫˆt−2 , R where ǫˆt−2 is the MMSE estimate of ǫt−2 . 2) Compute the innovation ˆ t = L1 ǫt−1 + L2 (ǫt−2 − ǫˆt−2 ) + Z˜t It = Rt − R (a)

(b)

= (L1 + L2 )ǫt−1 + Z˜t = M (β)ǫt−1 + Z˜t

(15)

where (a) follows from (12); and (b) follows by M (β) , L1 + L2 . 3)Compute MMSE estimate of ǫt given {R1 , R2 , ..., Rt−1 , It }. The estimation error ǫt−1 is independent of {R1 , R2 , ..., Rt−1 }, therefore compute the estimate ǫˆt−1 given It only (a)

ǫˆt−1 = E[ǫt−1 |It ] =

E[ǫt−1 It ] It , E[It2 ]

where (a) follows from an MMSE estimation of a Gaussian variable. The variance of ǫˆt−1 is given by ! M 2 (β)σǫ2t−1  2  E2 [ǫt-1 It ] (a) σǫ2t-1 E ǫˆt-1 = = ˜ E [It2 ] M 2 (β)σǫ2t−1 + N ! ˜ (β) M (b) σǫ2t-1 , (16) = ˜ (β) + N ˜ M   where (a) follows from E [ǫt−1 It ] = M (β)σǫ2t−1 , E It2 = ˜ ; and (b) follows from M 2 (β)σǫ2t−1 + N ˜ (β) , M 2 (β)σ 2 = M ǫt−1 s 2(1 − β)h2 PS σǫ2t−1 σǫ2t−2

σǫ2

+

s

2βPS PR σǫ2t−1 (2βPS + NR )σǫ2t−2

!2

,

N from (13), yields where the substitution of σ2t−1 = 2h2 βP S +N ǫt−2 q p  2βPS PR N 2 ˜ (β) = 2(1−β)h2 PS + M 2 (2βPS+NR )(2h βPS+N ) .

The decoder then updates its estimate of the initial state according to (11) and the associated estimation error is given by (12). The variance of the estimation error is given by   (a)     E ǫ2t = E ǫ2t−1 + E ǫˆ2t−1 − 2E [ǫt−1 ǫˆt−1 ] ! ˜  2   2  (c) N (b) = E ǫt−1 − E ǫˆt−1 = σǫ2t−1 , (17) ˜ (β) + N ˜ M   where (a) follows from (12); (b) follows from E ǫˆ2t−1 = 2 t−1 It ] E [ǫt−1 ǫˆt−1 ] = E [ǫ ; and (c) follows from (16). E[It2 ] By recursively using (13) and (17) we can write the estimation error variance σǫ2t at any time step t. The estimation error variance at the odd time steps (t = 1, 3, 5, . . . ) is given by ! t−21   t+21 ˜ N N 2 σǫ20 σ ǫt = ˜ (β)+ N ˜ 2h2 βPS +N M ! 2t  2t  ˜ N N Kσǫ20 , (18) = ˜ (β)+ N ˜ 2h2 βPS +N M   21 ˜ (β)+N ˜) N (M where K = (2h . Similarly the estimation error 2 βP +N )N ˜ S variance at the even time steps (t = 2, 4, 6, . . . ) is given by ! 2t  2t  ˜ N N 2 σǫ20 . (19) σ ǫt = ˜ (β)+ N ˜ 2h2 βPS +N M We use Lemma 8 to find a sufficient condition on λ for which the system in (1) can be mean square stabilized over the general half-duplex AWGN relay channel. We observe from (18) and (19) that λ2t σǫ2t → 0 as t → ∞ if   ˜ N N λ 4   < 1,  (20) 2 ˜ ˜ (2h βPS + N ) M (β) + N

from which (4) follows. The policy satisfies the required cost constraint P∞ controller 2 ] < ∞ which is shown in Appendix. B.  E[U t t=0 V. C ONCLUSION

We study the problem of mean square stabilizing a discrete time LTI system over a half-duplex AWGN relay channel. We propose to use a Schalkwijk-Kailath based coding strategy to remotely control a system over the relay channel. In every transmission the observer/encoder unit sends an innovation with respect to the current information available at the decoder/controller over the relay channel. We find rate sufficient conditions for mean square stability of the system over the given channel. Our results reveal a relationship between the communication channel parameters (i.e., signal-to-noise ratios of the encoder–decoder, encoder–relay and relay–decoder links) and the possibility of stabilizing the plant. An interesting extension of this work would be to consider instantaneous non-linear relaying strategies which can potentially increase the achievable rate and thus extend the class of stablizable systems over the considered relay channel.

A PPENDIX A ACHIEVABILITY The given scheme can be seen as a point to point communication channel, where R2t-1 is the channel output corresponding to the input Se,2t-1 and I2t is the channel output corresponding to the input Se,2t for t = 1, 2, 3, . . . . Since P (I2t , R2t-1 |Se,2t , Se,2t-1 ) = P (I2t |Se,2t )P (R2t-1 |Se,2t-1 ), the channel is memoryless. The information rate is given by  1  2T 2T 2T 2T lim I {Se,2t-1 }t=1 , {Se,2t }t=1 ; {R2t-1 }t=1 , {I2t }t=1 T →∞ 2T    1 2T 2T h {R2t-1 }t=1 , {I2t }t=1 − = lim T →∞ 2T   2T 2T 2T 2T h {R2t-1 }t=1 , {I2t }t=1 | {Se,2t-1 }t=1 , {Se,2t }t=1 2T 1 X (h (R2t-1 ) + h (I2t )) − T →∞ 2T t=1

(a)

2T 1 X (h (R2t-1 |Se,2t-1 ) + h (I2t |Se,2t )) T →∞ 2T t=1  1 (b) T h (R2t-1 ) + h (I2t ) − = lim T →∞ 2T   h (R2t-1 |Se,2t-1 ) − h (I2t |Se,2t )

1 (I (Se,2t-1 ; R2t-1 ) + I (Se,2t ; I2t )) , (21) 2 where (a) follows from the fact the fact that the channel is memoryless, E[R2l-1 R2k-1 ] = E[I2l I2k ] = 0 for k 6= l, and E[R2l-1 I2k ] = 0 for all l, k = 1, 2, 3, ..; and (b) follows from the fact that R2t-1 and I2t are i.i.d. variables. For the first transmission phase the mutual information between the transmitted variable and the received variable is given by =

(22)

where (a) follows from R2t-1 ∼ N (0, 2h2 βPS + N ) and Z2t-1 ∼ N (0, N ). In the second phase the decoder computes the innovation It according to (15). The mutual information between the transmitted variable and the innovation variable is then given by I (Se,2t ; I2t ) = h(I2t ) − h(I2t |Se,2t ) = h(I2t ) − h(Z˜2t ) ! ˜ (β) M (a) 1 , (23) = log 1 + ˜ (β) 2 N ˜ (β) + N ˜ ) and Z˜2t ∼ where (a) follows from I2t ∼ N (0, M ˜ (β)). N (0, N A PPENDIX B Q UADRATIC COST

t=1

(a)

E[Ut2 ] =

2T X t=1

(b)

λ2(t+1) σ ˆǫ2t-1 =

T X t=1

(λ4t b2 σǫ22t-2 + λ4t+2 c2 σǫ22t-1 )

t=1

(d)

=

T

σǫ20 (b2 + λ2 c2 b1 ) X 4 (λ b1 c1 )t , b1 c1 t=1

(24)

ˆ 0,t-1 − X ˆ 0,t ) and (11); where (a) follows from Ut = λt+1 (X (b) follows by separating even and odd indexed terms of the sequence {ˆ σk2 }; (c) follows from (10), (16), and by defining 2 ˜ (β) M βPS b2 = 2h2h 2 βP +N , c2 = ˜ (β)+N ; and (d) follows from (13), S M N , c1 = (17), and by defining b1 = 2h2 βP S +N same way we can compute the following 2T X t=1

E[Xt2 ] =

2T X t=1

˜ N ˜ (β)+N . M

In the

T

λ2t σǫ2t-1 =

σǫ20 (1+λ2 b1 ) X 4 (λ b1 c1 )t (25) b1 c1 t=1

R EFERENCES

lim

2T X

=

T X

According to (20) the term (λ4 b1 c1 ) < 1 in (24) and (25), PT λ4 b1 c1 therefore limT →∞ t=1 (λ4 b1 c1 )t = 1−λ and we get (5). 4b c 1 1

= lim

I (Se,2t-1 ; R2t-1 ) = h(R2t-1 ) − h(R2t-1 |Se,2t-1 )   2h2 βPS (a) 1 , = h(R2t-1 ) − h(Z2t-1 ) = log 1 + 2 N

(c)

(λ4t σ ˆǫ22t-2 + λ4t+2 σ ˆǫ22t-1 )

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