CAPÍTULO IV: ASPECTOS DE LA LECTURA
IV.4. EL DESARROLLO DE LA SENSIBILIDAD A TRAVÉS DE LA LECTURA
We propose the following design for the encoder and the decoder. At every time stepk
• The encoder (at the sensor end) has access to measurements till time stepk−1 in its information set. It runs the modified Kalman filter according to equations (3.9), (3.10) and transmits the output ¯x(k|k−1) of this filter across the link.
• The decoder (at the controller end) maintains two variables: a variable ψ(k) that takes into account the effect of the control inputs, and a local variable ˆxdec(k
|k−1) (with initial condition ˆ
xdec(0
| −1) = 0). These variables are updated as follows.
– The decoder can calculate the termsP(k|k−1) andP(k−1|k−1) as in the Kalman filter equations. It then calculates
ψ(k) =Bu(k−1) +γ(k−1)ψ(k−1),
where
and the initial condition is given byψ(0) = 0.
– If λ(k−1) = ‘received’, the link successfully transmitted the packet containing the estimate ¯x(k−1|k−2) at time stepk−1. The decoder sets
ˆ
xdec(k−1|k−2) = ¯x(k−1|k−2) +ψ(k−1).
– If λ(k−1) = ‘dropped’, the packet was dropped. The decoder implements the linear predictor
ˆ
xdec(k−1|k−2) =Aˆxdec(k−2|k−3) +Bu(k−2). (3.12)
– Finally, the decoder outputs the estimate
ˆ xdec(k
|k−2) =Aˆxdec(k
−1|k−2) +Bu(k−1).
The algorithm given above is optimal in the following sense.
Proposition 3.2 In the algorithm described above, xˆdec(k
|k−2) = ˆx(k|Imax(k),
{u(t)}kt=0−1).
Proof The proof is obvious given the relation (3.11). The information needed from the sensor at time stepkfor the calculation of ˆx(k−1|k−2) is precisely ¯x(k−1|k−2). The impact of the control input can be taken care of by the term Ψ(k−1) which is the same as the termψ(k−1) calculated in the algorithm by the decoder. Now, for the case whenλ(k−1) = ‘received’, the decoder in the algorithm has access to ¯x(k−1|k−2) and Ψ(k−1). Thus, it can calculate the centralized Kalman filter output ˆx(k−1|k−2). Upon executing the time update step, it calculates ˆx(k|k−2) which is ˆx(k|Imax(k),{u(t)}k−1
t=0). For the case when λ(k) = ‘dropped’, the decoder propagates the best
Kalman filter estimate ˆxdec(k−1|k−2) with the control inputsu(k−2) andu(k−1). Thus, in this
case too, ˆxdec(k|k−2) = ˆx(Imax(k),
{u(t)}kt=0−1)
Proposition 3.2 also solves the pure estimation problem in which the state of a dynamic process needs to be estimated across a packet erasure link. If the decoder sets the term ψ(k) identically to 0, the optimal encoder and decoder structures are given as above. Moreover, taken together, Propositions 3.1 and 3.2 solve the packet-based LQG control problem posed in Section 3.3.1 for the case of a single sensor and a single link.
Proposition 3.3 Consider the packet-based optimal control problem described in Section 3.3.1 for the case of only one sensor being present, For any packet dropping process P, an LQR state feed- back design together with the optimal transmission-estimation algorithm described above achieves the minimum ofJT(u, f,P).
Remarks
1. The information vector ¯x(k|k−1) ‘washes away’ the effect of any previous packet losses. If λ(k−1) = ‘received’, ˆxdec(k
|k−2) is identical to the case when all the previous measurements
{y(0), y(1),· · ·, y(k−2)} were available to the controller.
2. We have made no assumption about the packet dropping behavior. The algorithm provides the optimal estimate for an arbitrary packet drop sequence, irrespective of whether the packet drop can be modeled as an i.i.d. process (or a more sophisticated model like a Markov chain). We also have not made any assumption about the statistics of the packet drops being known. 3. We do not assume knowledge of the cost matrices Q and R at the sensor end. Thus, the cost function (and hence the optimal controller) can be changed at will without affecting the sensor/encoder operation. This is important, e.g., in our MVWT work where the matrices Q andR are user-specified while the encoder code is much harder to change.
Presence of delays
If we assume that there is a provision for time-stamping the packets sent by the encoder, the solution can readily be extended to the case when the channel applies a random delay to the packet so that packets might arrive at the decoder delayed or even out-of-order. At each time step, the decoder will face one of four possibilities, and will update its estimate as described below:
• It has access to ¯x(k−1|k−2). It uses this to calculate the estimate ˆxdec(k|k−2) according
to the algorithm given above.
• It does not receive anything. It uses the predictor equation (3.12) on ˆxdec(k−1|k−3). • It receives ¯x(m|m−1) while at a previous time step, it has already received ¯x(n|n−1), where
n > m. It discards ¯x(m|m−1) and uses (3.12) on ˆxdec(k−1|k−3).
• It receives ¯x(m|m−1) and at no previous time step has it received ¯x(n|n−1), wheren > m. It uses ¯x(m|m−1) to calculate ˆxdec(m|m−2) and obtains ˆxdec(k|k−2) through the repeated
application of (3.12).
Channel between the controller and the actuator
If we look at the proof of the separation principle above, the crucial assumption was that the controller knows what control input is applied at the plant. Thus, if we have a channel between the controller and the plant, the separation principle would still hold, provided there is a provision for an acknowledgment from the receiver to the transmitter for any packet successfully received over that channel4. Since the decoder can now have access to the control input applied at every time
4
step, it is apparent that our algorithm is optimal for this case as well. We can also ask the question of the optimal encoder-decoder design for the controller-actuator channel. The optimal decoding at the actuator end will depend on the information that is assumed to be known to the actuator (e.g. the cost matricesQandRand the measurements from the sensor). Design of the decoder for various information sets is an interesting open problem.