3.3 Problem Formulation and Preliminary Results
3.3.1 Problem Setup
Figure 3.7: The set-up of the control across communication networks problem. Later in the chapter we also look at a channel present between the controller and the actuator.
Consider the arrangement in Figure 3.7. Let the discrete-time linear process evolve according to the equation
x(k+ 1) =Ax(k) +Bu(k) +w(k), (3.1)
wherex(k)∈Rn is the process state,u(k)∈Rmis the control input andw(k) is the process noise
assumed to be white, Gaussian, and zero mean with covariance matrixRw3. The initial condition
x(0) is assumed to be independent ofw(k) and to have mean zero and covariance matrixR(0). The state of the plant is measured byN sensors with thei-th sensor generating measurements according to the equation
yi(k) =Cix(k) +vi(k). (3.2)
The measurement noises vi(k)’s are assumed white, zero-mean, Gaussian (with covariance matrix
Rv,i) and independent of the plant noise w(k) and of each other. Every sensor communicates its
own measurements (or some function of the measurements) to the controller. For the moment, we ignore random delays and packet reordering in the channels and model them solely as packet erasure
3
The results we present continue to hold for time-varying systems, but we consider the time-invariant case to simplify the presentation.
links with a fixed delay of one time step. For ease of presentation, we will denote the encoder as a node separate from the sensor and term it the source node.
The controller at every time step calculates a control inputu(k) and transmits it to the actuator. As shown in Figure 3.7, for the time being we ignore the channel between the controller and the actuator. We will revisit both the issue of delay as well as the presence of a controller-actuator channel later in the chapter and show how simple modifications to our design can take care of them. The controller aims at minimizing the quadratic cost function
JT =E " T X k=0 xT(k)Qx(k) +uT(k)Ru(k)+xT(T+ 1)PTc+1x(T+ 1) # , (3.3) whereQ,RandPc
T+1are all positive definite matrices. We make the usual assumptions of the pairs
(A, B) and (A, Q12) being stabilizable. The expectation in the cost function is taken over the initial
conditionx(0) and the noise processes{w(k)} and {vi(k)}’s. Without the channels being present,
this is the classical LQG control synthesis problem. The presence of communication channels that erase packets stochastically, however, alters the problem drastically.
The time-line of the operation of the channels is as follows. At every time step k,
1. The source node at each sensor computes a function of all the information it has access to at that time.
2. The source nodes transmit the functions on the communication links. The controller calculates the controlu(k) based on the information it possesses.
3. The controller observes the messages, if any, received on the links and updates its information set for the next time step. The source nodes update their information sets with the observations yi(k)’s.
This time-line means that the there are two sources of delay even if the channel is not dropping any packets. First, the source nodes at time step k can only transmit a function of measurements till time stepk−1. Further, there is a delay of one time step for transmission over the link. Thus, at time stepk the controller can, at best, have access to measurements till time stepk−2. Removal of any of these sources of delay will lead to only minor adjustments in the results given in the rest of the chapter.
The presence of packet erasure links warrants a discussion on the type of controllers that are allowed. The absolutely optimal LQG performance achievable is given by the classical LQR con- troller/Kalman estimator pair. However, this design does not respect the packetized nature of the communication. Specifically, the controller requires continual access to the Kalman filter output, which, in turn, requires continual access to the measurements from all the sensors. This access might not always be possible because of data loss in the communication links. In order to make
the class of controllers that are allowed more precise, we introduce the following terminology. De- note by si(k) the finite vector transmitted from the sensor i to the controller at time step k. By
causality, si(k) can depend (possibly in a time-varying manner) onyi(0),yi(1),· · ·, yi(k−1), i.e.,
si(k) =fi(k) (yi(0), yi(1),· · · , yi(k−1)).Denote by the variableλi(k) the (random) event whether
or not a transmission was successful on the linkiat time stepkin the realization of the packet loss sequencePi that is occurring in the linki. Theinformation set,I(k) available to the controller at
timekis the union ofN setsIi(k)’s defined by
Ii(k) ={si(j)|λi(j) = ‘received’}.
Also, denote byti(k)< kthe last time-step at which a packet was delivered over linkiprior to time
k. That is
ti(k) = max{j < k|λi(j) = ‘received’}.
Themaximal information set,Imax(k) at time-stepkis then the union of sets Imax
i (k) defined by Iimax(k) ={yi(j)|j < ti(k)}.
The maximal information set is the largest set of output measurements from sensor i on which the control at time-step k can depend. In general, the set of output measurements on which the control depends will be less than this set, since earlier packets, and hence measurements, may have been dropped. As stated earlier, we impose the restriction that the vectors si(k) remain finite
and thus, e.g., do not increase in size as k increases. We will call the set of fi(k)’s which fulfill
this requirement as F. Without loss of generality, we will only consider information-set feedback controllers, i.e., controllers of the formu(k) =u(I(k), k).We denote the set of control laws allowed by U. We shall assume perfect knowledge of the system parameters A, B, C, Rw and Rv,i’s at
the controller. Moreover, we shall assume that the controller (and the decoder) have access to the previous control signalsu(0), u(1),· · ·,u(k−1) while calculating the controlu(k) at timek.
We can thus pose the packetized LQG problem as:
min u∈U,fi∈F JT(u, fi,Pi) =E " T X k=0 uT(k)Ru(k) +xT(k)Qx(k)+xT(T+ 1)Pc(T+ 1)x(T+ 1) # . (3.4) Note that the cost functionalJT above depends on the random packet-drop sequencesPi’s. However,
we donot average across packet-drop processes;the solution we will present is optimal for arbitrary realizations of the packet dropping processes. That is, the controller, encoder and decoder we propose will minimizeJT(u, fi,Pi) over the set of allowable controllersU and allowable functionsF for any
dependence in the cost functional and merely writeJT(u, fi) or justJT.
Our goal, then, is to solve the standard LQG problem with the additional complication of the packet-dropping links. While this may appear a small modification, it is uncleara priori, what the structure of the optimal control algorithm should be, and in what way the packetized links should be used through the design of the encoder and the decoder. We begin by presenting a separation principle.