LOCALIZED LINEAR QUADRATIC REGULATOR
5.5 Adaptive Constraint Update with Performance Guarantee
5.5.1 Lower Bound of Centralized LQR Cost We first assume that the cost matrix
Now we specialize our discussion to the case where the non-localizability measure
∆RM comes from the bias or the uncertainty of the plant model. Specifically, we assume that the localized controller is designed based on the system model(A,B2), but is applied on a different system model(A+∆A,B2+∆B). In this case, we have
∆RM =−∆ARc−∆BMc. (5.27) From (5.27), when the plant has a higher uncertainty on its A matrix, we should penalize more on the system response Rc to make the non-localizability measure
∆RM small. On the other hand, if the system has a higher uncertainty on the gain matrixB2, we should penalize more on the system responseMc.
The LLQR problem can be formulated by incorporating additional constraints (5.10d) - (5.10f) into (5.28), as shown in (5.10). Let Ψ∗j be the optimal value of problem (5.28), and Ψupperj ≥ Ψ∗j be the LLQR cost, i.e., the optimal value of (5.28) when additional constraints (5.10d) - (5.10f) are included. For a large-scale system, the valueΨ∗j cannot be computed in a scalable manner. In order to quantify the degradation ofΨupperj , we propose a scalable algorithm to provide a non-trivial lower boundΨlowerj ≤ Ψ∗j of the centralized optimal cost. In this way, the degrada- tionΨupperj /Ψ∗
jof the LLQR controller is upper bounded by the ratioΨupperj /Ψlower
j .
In other words, we can compute a sub-optimality guarantee Ψlowerj /Ψupper
j for the LLQR controller with a given initial condition x[1] = ej using only local plant model information.
From the discussion of Section 5.2, we can analyze the solution of (5.10), which is an upper bound of the optimal value of (5.28), within the localized region Outj(d+1) for each disturbance wj. Similar to the upper bound problem, here we would like to compute a lower bound of the optimal value of (5.28) based on the network information contained within the set Outj(d+1). Consider the following example.
……
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Figure 5.5: Localized region for the initial conditionx[1]= e1in a large network Example 7. Figure 5.5 shows the interconnection of subsystem x1to other subsys- tems in a large network. The global network in this example can be arbitrary large.
The goal is to compute a lower bound of (5.28)for a given initial conditionx[1]= e1 within a given localized regionOut1(2)= {x1,x2,x3,x4,x5}. In particular, we have no information about the system model outside the localized regionOut1(2). The intuition of our approach is as follows. First, note that the matrices (Dx,Du) are diagonal and positive semidefinite. Therefore, a lower bound of the objective function in (5.28) can be computed by setting xi = 0 andui = 0 for all xi andui
outside the localized region. Then, we restrict the system dynamics (5.28c) within
the localized region, and match the inconsistent part by introducing additional zero penalty control action at the boundary of the localized region. We formalize this idea by the partition of the state and control vectors as follows.
Given a localized region Outj(d +1), we partition the global state vector x into three components: the internal state vector xin, the boundary state vector xb, and the external state vector xout. We partition the global control vector u into two components: the internal control vectoruinand the external control vectoruout. The state vector is partitioned in the following way.
Definition 13. The external state vector xout of a localized regionOutj(d +1) is defined by the set
xout = {xi|xi <Outj(d+1)}.
The boundary state vector xbofOutj(d+1)is a subset ofOutj(d+1)that contains the states coupled from the external state vectorxout. Specifically, we have
xb= {xi|xi ∈Outj(d+1), ∃xk <Outj(d+1),Aikxk ,0}.
Finally, the internal state vector is given by
xin = {xi|xi ∈Outj(d+1), ∀xk <Outj(d+1),Aikxk =0}.
Example 8. Consider the example in Figure 5.5 with localized region Out1(2)
= {x1,x2,x3,x4,x5}. The external state vector of this localized region contains all the states except those inOut1(2), i.e., xout includesx6, . . . ,x8and all the states not shown in the figure. The boundary state vector contains the states that are coupled from the external region, which is xb = {x2,x4}. The internal state vector is given by xin = {x1,x3,x5}. Note that x5is coupled to, but not coupled from the external region, and thus is a component of the internal state vector.
Remark 12. It should be noted that the boundary region defined here for the lower bound problem isdifferentfrom the boundary region defined for the LLQR problem (5.10). For the LLQR problem (an upper bound of (5.28)), the boundary states are those that couple to the external region (e.g., {x5} in Figure 5.5). For the lower bound of (5.28), the boundary states are those that couplefromthe external region (e.g.,{x2,x4}in Figure 5.5).
The external control vectoruout for a localized region Outj(d+1)is defined by the set of control actions that only directly affect the states in xout. Let bi be theith
column of matrix B2. For the scalar subsystem model, the external control vector uout is the set ofui where the nonzero locations ofbiuiare contained within the set xout. The internal control vectoruinis the complement of the external control vector uout.
Example 9. For Figure 5.5, we haveuin= {u1,u2,u3}. The external control vector uout containsu4andu5and all the control actions not shown in the figure.
With the partition of the state and control vectors, we can rewrite the objective function in (5.28a) by
∞
Õ
k=1
xin[k]
xb[k] xout[k]
>
Dx,in 0 0
0 Dx,b 0
0 0 Dx,out
xin[k]
xb[k] xout[k]
+
"
uin[k]
uout[k]
#>"
Du,in 0
0 Du,out
# "
uin[k]
uout[k]
#
. (5.29)
As Dx and Du are diagonal and positive semidefinite, a lower bound for (5.29) is given by
∞
Õ
k=1
"
xin[k]
xb[k]
#> "
Dx,in 0 0 Dx,b
# "
xin[k]
xb[k]
#
+uin[k]>Du,inuin[k]. (5.30)
Note that (5.30) can be computed using the information contained within the local- ized region Outj(d+1).
The system dynamics in (5.28c) can be rewritten as
xin[k +1] xb[k +1] xout[k+1]
=
Ain,in Ain,b 0
Ab,in Ab,b Ab,out
Aout,in Aout,b Aout,out
xin[k]
xb[k] xout[k]
+
Bin,in 0
Bb,in 0 Bout,in Bout,out
"
uin[k] uout[k]
# . (5.31) The zero in equation (5.31) is from the definition — the external state vector xout
does not couple to the internal state vector xin directly, and the external control vectoruout cannot adjust the internal state vector xinand the boundary state vector xb directly. The only possibility that connects the external region to the localized region is through the term Ab,out. This coupling can be captured by introducing an additional control action uadd[k] = Ab,outxout[k]into the system. In this case, the
system dynamics within the localized region is given by
"
xin[k+1] xb[k +1]
#
=
"
Ain,in Ain,b
Ab,in Ab,b
# "
xin[k] xb[k]
# +
"
Bin,in 0 Bb,in I
# "
uin[k] uadd[k]
#
(5.32)
uadd[k]= Ab,outxout[k]. (5.33)
Since we are only interested in the lower bound of (5.28), we can discard the constraint (5.33) to increase the feasible set of the optimization problem. Combining with the objective function (5.30), we can compute a lower bound of (5.28) by solving the following optimization problem
minimize (5.30) (5.34a)
subject to
"
xin[1] xb[1]
#
= er(j) and (5.32), (5.34b)
whereer(j)is the initial conditionej in the reduced dimension. Note that (5.34) is a smaller centralized LQR problem that can be computed using only the plant model information contained within the localized region Outj(d+1). Similar to the LLQR problem, the computational complexity of solving (5.34) depends on the number of nodes in the localized region. If the size of the localized region is significantly smaller than the size of the global network, then (5.34) can be solved efficiently.
Remark 13. To make the LQR problem(5.34)well-posed, we can add a termÍ∞ k=1
u>add[k]uadd[k] in the objective function for some small. As long as is small enough, problem(5.34)still gives a lower bound for the optimal value in(5.28). Example 10. Consider the example in Figure 5.5 with localized region Out1(2)
= {x1,x2,x3,x4,x5}. To compute a lower bound of the optimal cost, we introduce additional control actions on the boundary states x2 andx4. The lower bound can then be solved by the LQR problem shown in Figure 5.6.
Figure 5.6: Lower bound problem for Figure 5.5
Remark 14. For comparison, we include the upper bound problem (LLQR problem) for Figure 5.5 in Figure 5.7. For the upper bound problem, we introduce additional constraint x5 = 0into the system. As discussed before, the boundary states for the upper bound problem is different from that for the lower bound problem.
Figure 5.7: Upper bound problem for Figure 5.5
Remark 15. The upper and lower bounds for distributed optimal control problem (2.8)satisfying a QI subspace constraint can be computed in a similar fashion. First, we reformulate the QI subspace constraint imposed on the distributed controller into a subspace SLC, as shown in(4.15). With a given localized region, the lower bound of the distributed optimal control problem is then given by(5.34)with an additional subspace constraint. This becomes another distributed optimal control problem, but only requires the plant model information contained within the pre-specified localized region. The distribute optimal control problem with reduced dimension can then be solved by the algorithm presented in [29].