Problem Description

We consider the wireless control architecture shown in Fig. 5 deployed to control a discrete-time linear time-invariant plant described by the difference equation

xk+1=Axk+Buk+wk, k≥0, (3.1)

where xk ∈ Rn is the plant’s state with x0 given,uk ∈ Rm the driving input, and{wk,k ≥ 0}is

the process noise composed of independent identically distributed (i.i.d)n-dimensional Gaussian random variableswk∼ N0,W with zero mean and covarianceW. We assume the plant is unstable

(λmax(A)>1) but that(A,B)is stabilizable.

The wireless sensor collects state measurementsxkthat it communicates with powerpk∈[0,pmax]

over a wireless fading channel with coefficienthk. At the other side of the channel the receiver/

controller uses the received information to determine a control inputuk that it feedbacks into the

plant.

The channel fading state hk,k ≥ 0 is modeled as an i.i.d. process (block fading cf. 2.2) with

some known probability distributionφh on the positive realsR+, independent of the plant pro-

cess noise {wk,k ≥ 0}. We make the technical assumption that the distribution of the channel

state is absolutely continuous, i.e., has a probability density function onR+. To allow for trans-

missions adapted to the current channel conditions the transmitter has access to the channel state informationhk before transmitting at timek.

Following the wireless channel modeled introduced in 2.2, the probability of successfully decoding the message at the receiver/controller at timek, denoted hereby byqk, is a function of the current

channel statehk as well as the current transmit powerpk,

qk =q(hk,pk). (3.2)

The typical form of this function is depicted in Fig. 2. We assume that q(h,p) is a known increasing function of the producth p. We denote the success of transmission at each time stepk by an indicator variableγk ∈ {0, 1}. Variablesγk ∼Bern(qk)are Bernoulli distributed with time-

varying success probabilities qk given by (3.2). With this communication model the controller

Plant: x_k+1=Ax_k+Bu_k+w_k Sensor/ Transmitter z−1 Channel: qk=q(hk,pk) γk∼Bern(qk) yk=γkxk Receiver/ Controller xk xk,pk yk,γk uk hk wk γk γk−1

Figure 5: Wireless control system architecture. A sensor measures the plant and wireless fading channel statesxk,hkrespectively and transmits with powerpk. Messages are successfully decoded

at the controller with probability qk that depends on the channel statehk and the powerpk. The

sensor receives acknowledgments with a one-step delay.

assume that the controller also getsγk so that it can distinguish between the cases xk = 0 and γk = 0. Packet receipt acknowledgmentγk is also sent to the sensor as provided by 802.11 and

TCP protocols. We assume lossless acknowledgments, so that the sensor knows what information is received at the controller.

The problem addressed in this chapter is the joint design of the control inputsukand the transmit

powerspk. The control inputukis determined by the received informationy0:k,γ0:k. The powerpk

is determined as a function of the plant state measurementsx0:k, the observed channel realizations

h0:k, and the controller acknowledgments γ0:k. Informally, to conserve power at the sensor side

we want to transmit information only when the statexkdeviates from its desired value or when

the channel realizationhkis favorable. In the first case transmission is necessary to keep the plant

under control. In the latter case the transmission cost is minimal. A formal problem specification is presented in the next section after the following remarks.

To formulate the joint design of plant controller and power management we introduce an equiva- lent architecture. In view of (3.2), choosing pkis equivalent to choosing the desired probability of

successful decodingqk at time kand transmitting with the minimum required power to achieve

thisqk, namely

pk =p(hk,qk):=inf{0≤p≤ pmax:q(hk,p)≥qk}. (3.3)

We can therefore interpretqk as our decision variable withp(hk,qk)denoting the cost of selecting

transmission success probabilityqk. This leads to the equivalent control system architecture shown

in Fig. 6 where a scheduler block responsible for decidingqkreplaces the sensor/transmitter block

of Fig 5. Our formulation generalizes the simple transmit-or-not decision as considered in, e.g., Xu and Hespanha (2004).

We note for future reference that the assumed monotonicity of the functionq(h,p)on the product h pimplies that the power functionp(h,q)is increasing inqand decreasing inh. Using maximum

Plant: xk+1=Axk+Buk+wk Scheduler: decidesqk usesp(h_k,q_k) z−1 Channel: γk∼Bern(qk) y_k=γkxk Receiver/ Controller xk xk,qk y_k,γk uk hk wk γk γk−1

Figure 6: Equivalent wireless control system architecture. A scheduler decides the successful decoding probability qk and transmits the state measurement xk with the required power pk =

p(hk,qk). The controller receives the message with probabilityqk.

power pmax, the transmitter can achieve a maximum successful decoding probabilityqmax(h) :=

q(h,pmax)for a given channel state h. Therefore, the decision variablesqk belong in the interval [0,qmax(hk)]. We also make the following assumptions.

Assumption 3.1. The maximum achievable successful decoding probability qmax(h)satisfies

Ehqmax(h)>qcrit:=1−1/λmax(A)2, (3.4)

where expectation is taken over the channel distributionφh.

Assumption 3.2. For any channel realization h, the function p(h,q)in (3.3) is continuous in the successful decoding probability variable q.

Assumption 3.1 is essentially a stability condition, which as we will see in the following section states that transmitter has enough power to keep the plant state bounded in second moment, and it will be used to establish our main Theorems 3.1 and 3.2. Assumption 3.2 is of a technical nature and will be used in Theorem 3.2.

In the architecture of Fig. 6 the communication decisionqk is chosen as a function of the infor-

mation available at the sensor, while the plant control signaluk is a function of the information

available at the controller. These choices are in general allowed to be randomized. The sequence

π :={q0,q1, . . .}, or equivalently the power allocation{p0,p1, . . .}, is termed the communication

policy, whereas the sequenceθ:={u0,u1, . . .}denotes the control policy. With fixed policiesπ,θ,

all random variables are defined on an appropriate product space and have a measure that we denote asPπ,θ_{. We use} Eπ,θ _{to denote integration with respect to}Pπ,θ_{, which we simplify to}E

when not leading to confusion. We remark that sensor and controller know each other’s policy. The policy pair(π,θ)incurs a control cost and a communication cost. As a control cost we adopt

the standard linear quadratic regulator cost J_LQRN (π,θ):=Eπ,θ N−1

∑

for some pair of matricesR>0 andQ≥0, with(A,Q1/2)detectable. The communication cost is given by the expected power consumption

J_PWRN (π,θ):=Eπ,θ N−1

∑

p(hk,qk). (3.6)

To quantify the tradeoff between plant performance and power consumption we combine the LQR cost in (3.5) and the power cost in (3.6) into the limit aggregate cost

J(π,θ):=lim sup N→_∞ 1 N h J_LQRN (π,θ) +λJ_PWRN (π,θ) i , (3.7)

for some positive constantλ > 0. Our goal is to design plant and power control policies θ and π respectively that minimize the joint cost (3.7). These policies depend on what information is

available to the sensor and controller. The specific information structure considered is introduced next.