Localized LQR as a SLS Problem

LOCALIZED LINEAR QUADRATIC REGULATOR

5.1 Problem Statement

5.1.3 Localized LQR as a SLS Problem

we consider the control action taken at subsystemi, which is specified byui =Kix, then if K_i is completely dense then subsystem i must collect measurements from every other subsystem j ∈ V. In order to design a controller that is scalable to implement, a natural solution is to impose sparsity constraints on the controllerK such that each row only has a small number of nonzero terms — in this way each subsystemi only needs to collect a small number of measurements to compute its control action. Unfortunately, this naive approach fails if the underlying topology of G of the networked system is strongly connected. Specifically, when the topology ofGis strongly connected, any sparse constraint setC is not QI (cf. Example 1 in Section 2.4), which means that the constrained optimal control problem is always non-convex. Although recent methods based on convex relaxations [19] can be used to solve certain cases of the non-convex optimal control problem (5.5) with sparse constraint set C, the underlying synthesis optimization problem is itself still large- scale and does not admit a scalable reformulation. The need to address scalability, both in the synthesis and implementation of a controller, is the driving motivation of the LLQR framework.

localized SLC L_d (this will be formally defined later) and a FIR SLC F_T on the system response in (5.6), which leads to the LLQR problem given by

minimize

{R,M} k h

C₁ D₁₂ i

"

R M

# k_H²

2 (5.7a)

subject to h

zI− A −B₂ i

"

R M

#

= I (5.7b)

"

R M

#

∈ L_d∩ F_T ∩1

zRH_∞. (5.7c)

The goal of this chapter is to show that (i) the LLQR problem (5.7) can besolved in a localized and scalable way if thed-localized SLC and the FIR SLC are prop- erly specified, and (ii) the LLQR controller achieving the desired localized system response can be implemented in a localized and scalable way. At this point, we assume that the LLQR problem (5.7) is feasible. The feasibility of (5.7), which is called the(d,T)state feedback localizability of the system(A,B₂), will be formally defined in the end of this section.

d-localized SLC

The notion of d-localized SLC L_d is defined based on the interaction graph G of the interconnected system. We first recall some standard terminology from graph theory.

Definition 6. In an unweighted graphG, thelengthof a path is the number of edges it uses.

Definition 7. For an interconnected system with an unweighted interaction graphG, thedistance from subsystem j to subsystemiis defined by the length of the shortest path from node j to nodei in the graphG, and is denoted bydist(j →i).

With the definition of the distance function on the interaction graph, we define the d-incoming and outgoing sets of subsystem j as follows.

Definition 8. Thed-outgoing set of subsystemjis defined asOutj(d):= {i|dist(j → i) ≤ d}, and the d-incoming set of subsystem j is defined as Inj(d) := {i|dist(i →

j) ≤ d}.

Example 4. For a system(5.1)with interaction graph illustrated in Figure 5.2, the 2-incoming and2-outgoing sets of subsystem5are given byIn₅(2)= {2,3,4,5}and Out₅(2)= {5,6,7,8,9,10}, respectively.

In5(2)

Out₅(2)

Figure 5.2: Illustration of the 2-incoming and 2-outgoing sets of subsystem 5.

Our approach to making the controller synthesis task specified in optimization problem (5.7) scalable is to confine, orlocalizethe effects of each process disturbance wj to a d-outgoing set at each subsystem j, for a d much smaller than the radius of the interaction graph G. As we make precise in the sequel, this implies that each sub-controller j can be synthesized using the localized plant model contained within its(d+1)-outgoing set Outj(d+1), and implemented by collecting data from subsystems contained within its(d+1)-incoming set Inj(d+1).

With this approach in mind, we say that the system responseR mapping the state disturbancewto the statexisd-localized if its impulse response can be appropriately covered byd-outgoing sets. The formal definition of ad-localized system response is described as follows.

Definition 9. For the system model(5.2), let Ri j denote the transfer function from the perturbationwjat sub-system jto the statexiat sub-systemi. The mapRis said to be d-localized if and only if for every subsystem j, Ri j = 0 for alli < Outj(d).

Similarly, letMi jdenote the transfer function from the perturbationwjat sub-system j to the controluiat sub-systemi. The mapMis said to bed-localized if and only if for every subsystem j,Mi j =0for alli <Outj(d).

Remark 8. Alternatively, we can also say thatR is d-localized if and only if for every subsystemi,Ri j = 0for all j <Ini(d).

Remark 9. Let supp(·) : R^m×n → {0,1}^m×n be the support operator, where {supp(A)}i j = 1 if Ai j , 0 and {supp(A)}i j = 0 otherwise. Let A = supp(A) ∪ supp(I), where∪is the OR operator on binary matrices. For the scalar subsystem model, a convenient way to impose d-localized SLC on the system response R is given by

supp(R) ⊆ supp

A^d

. (5.8)

In words, this says that the transfer matrixRisd-localized if and only if the effect of each disturbance is contained, or localized, to within a region of radius d from the source of the disturbance. In general, we can specify a localized constraint with different values ofdfor the system responseRandM, respectively. As will be shown later in this chapter, a common approach is to enforce ad-localized constraint onR, and a(d+1)-localized constraint onM. We call this special type of localized constraint ad-localized SLC, and is denoted byL_d (cf. (5.7)).

Definition 10. The subspace L_d is called a d-localized SLC if it constrains the system responseRto bed-localized, andMto be(d+1)-localized.

(d,T)state feedback localizability

With the definition of thed-localized SLC, we formally define the notion of(d,T) state feedback localizability of a system(A,B₂)as follows.2

Definition 11. The system (5.2) with system matrices (A,B₂) and an underlying graphGis said to be(d,T)state feedback localizable if (5.7)is feasible.

Recall that a system is said to be controllable if there exists a FIR closed loop response. Here, we say that a system is(d,T)localizable if there exists a localized FIR closed loop response (cf., Section 4.2.9). In this sense, localizability can be considered as a stricter notion of the controllability of a system.

The relations between the localizability of the system, the delay pattern of the communication network, and the actuation architecture of the controllers will be discussed in details in Section 5.3.