Chapter IV: Control of Dynamic-Topology Networks
4.2 Mode Process Identification
Recall from Section 3.3.1, the Mode Process ID component estimates the current mode ˆ
ϕ(t)N[t] and the TPM ˆP(t) based on state and control trajectories x[0 : t], u[0 : t]. In the context of this dynamic-topology network, ˆϕ(t)N[t] is estimated usingconsistent set narrowing, a variation of nested convex body chasing [70] extended specifically to dynamic-topology network systems. Second, ˆP(t) is estimated using empirical counts based on ˆϕ(t)N[t] and on estimates of the previous modes {ϕˆ(s)N[s]}t−1s=0.
4.2.1 Consistent Set Narrowing
Because the distribution of the external noise process w[t] is unknown other than its norm bound, we employ a consistent set narrowing approach, which checks the set of modes that are “consistent” with the state/control observations. This method is similar to the nested convex body chasing approach described in [70], which was used for model approximation and selection for designing robust controls.
Denote the current mode-index as n,N[t]∈N. By Assumption7, there are at most ∆T −1 state and control observations, x[Tn:t] and u[Tn:t], associated with a single mode ϕn. We thus construct a consistent set as follows.
1 1 2 2 2 ... 2 4 4 ...
1 1 [2,3,5] [2,3] 2 ... 2 [4,5] 4 ...
1 1 1 1 2 ... 2 2 4 ...
Figure 4.1: A visual diagram depicting Mode Process ID with consistent set narrowing. Here,
∆T= 10 andM= 5. Withn,N[t], the upper row of gray circles denotes a realization{ϕn} of the original unobservable mode process{ξn}. The middle row denotes the evolution of the consistent set, updated via (4.2) using the state and control observations x[Tn:t], u[Tn:t].
The estimate{ϕˆ(t)N[t]}of the mode process is at the bottom row of white circles.
Definition 32 (Consistent Sets). Over time, we construct a sequence of consistent sets {C[t]}t∈N in the following way. For each n∈N, we initially set C[Tn],X because no ob- servations about the current mode ϕn have been made yet. Then for each t∈(Tn, Tn+1), if C[t −1]6=∅, a new consistent set is formed by retaining all modes m∈ C[t −1] from the previous iteration where each one-step value of state and control (x[t],x[t+ 1],u[t]) satisfies the norm-boundedness condition of the noisew[t]:
C[t] =
m∈ C[t−1]|
t−1
^
r=Tn
1{kx[r+1]−A(m)x[r]−Bu[r]k∞≤w}
. (4.2)
As (4.2) is being performed for eacht∈N, we update the estimate ˆϕ(t)N[t]of the current mode.
We use ˆϕ(t)N[t]∈argmaxζ∈XPˆ(t−1)[ ˆϕ(t−1)N[t−1], ζ] if|C[t]|>1; otherwise, we update ˆϕ(t)N[t]∈ C[t].
Remark 19. One property of consistent set narrowing, also observed in nested convex body chasing approaches [70], is that at each time t∈Z≥0, the consistent set C[t] always contains the true mode ϕN[t]. This is by definition of the consistent set, and the deterministic nature of the condition (4.2) which defines the narrowing process (equivalent to verifying a simple linear inequality). In the MJS literature, there have been notions of consistency similar to (4.2) according to which unknown modes of MJS are estimated. For example, [136] verifies consistency under the assumption that imperfect measurementsy[t]6=x[t] of the statex[t] are collected. Thus, instead of using the state history x[0 :t] directly, the consistency condition
is designed around the collected measurements y[0 :t] and propagated estimates of x[0 :t]
based on the initial condition x0 and the measurement equation.
Mode detectability is also a concept that has been studied; for instance, in [37], the mode variable (analogous to ξt∈ X in our notation) emits its own signal (analogous to ˆϕ(t)t in our notation) independently of the system dynamics and the previous modes. We note the subscripts of t instead of n because the mode process in [37] is assumed to operate on the same timescale as the system dynamics. In the consistent set narrowing approach, we obtain
ˆ
ϕ(t)n from the state and control trajectories (x[t],x[t + 1],u[t]); this estimate also changes with time as we collect more data about the trajectories.
4.2.2 Empirical Estimation of the TPM
For any n∈N, the estimate of ϕn is most accurate when the maximum possible amount of data from the system has been obtained to create the estimate, i.e., among all t∈[Tn, Tn+1), the value of ˆϕ(t)n is most accurate at time t=Tn+1 −1. For general TN[t]< t < TN[t]+1, ˆP(t) is estimated based on ˆϕ(t)N[t] and only the most accurate estimates of the previous modes {ϕˆ(TN[s]N[s]−1)}t−1s=0. Thus, in the TPM estimation procedure, there is only one estimate associated each true modeϕn. For simplicity of notation in this section only, we fix n,N[t] and denote shorthand ˆϕn0≡ϕˆ(Tn0n0−1) for n0< n and ˆϕn≡ϕˆ(t)n .
Ift=Tnfor somen∈N, estimating ˆP(t)given{ϕˆn0}nn0=1 is straightforward. By Assumption8, it is known which entries of the TPM are nonzero. Thus, we initialize ˆP(t) to be an M×M matrix with a 1 in the nonzero entries; when normalized, this corresponds to a stochastic matrix which has uniform distribution over the feasible transitions (e.g., 1/3 probability each for a row with three nonzero entries) but for estimation purposes, we keep the estimate of the TPM unnormalized until the end of the simulation duration. For each consecutive pair of transitions ( ˆϕn0,ϕˆn0+1) forn0∈ {0,· · · , n−1}, we take ˆP(t)[ ˆϕn0,ϕˆn0+1] = ˆP(t)[ ˆϕn0,ϕˆn0+1] + 1.
IfTn< t < Tn+1 for somen∈N, we have two separate subcases. If ˆϕ(t−1)n = ˆϕ(t)n , then we sim- ply follow the approach above and compute ˆP(t) using the sequence{ϕˆn0}nn0=1. Otherwise, if
ˆ
ϕ(t−1)n 6= ˆϕ(t)N[t], then we again follow the approach above and compute ˆP(t), but using the se- quence{ϕˆn0}n−1n0=1 instead. To incorporate the mode estimate at current mode-indexn, we first need to reset the TPM estimate of the last transition via ˆP(t)[ ˆϕn−1,ϕˆ(t−1)n ] = ˆP(t)[ ˆϕn−1,ϕˆ(t−1)n ]−
1; then we update as usual ˆP(t)[ ˆϕn−1,ϕˆ(t)n ] = ˆP(t)[ ˆϕn−1,ϕˆ(t)n ] + 1. Once the mode sequence es- timates have been processed until current time t, we update ˆP(t) such that each row is
normalized to sum to 1.
Remark 20. The need for including Mode Process ID in the controller architecture Fig- ure 3.2 is closely related to the notion of mode observability, which has been studied ex- tensively in the literature [150, 2, 11, 136]. One common setup is that the measurements come from a (linear) noisy measurement equation such that y[t]6=x[t], and derive mode ob- servability conditions from the imperfect observations y[t] of the state x[t]. Also, the mode process is assumed to operate on the same timescale as the system dynamics. Compared to these methods, the algorithms we chose for implementing Mode Process ID hinge upon assumptions that simplify the mode observability problem. For example, in Assumption 8, the state x[t] is observable and in Assumption 7, we fix the mode switching times to be constant and deterministic rather than stochastic.
We again emphasize that this is because the focus of our paper is on the impact of PLP on control design rather than mode observability, and we aimed to set up a simple scenario to show that our approach can be used when the system has uncertainties. Thus, not all of our assumptions are limiting; for example, compared to our approach, [150] explicitly imposes that the external noise processes{w[t]}t,{v[t]}tare Gaussian white and neither [150] nor [2]
consider the impact of control.
Remark 21. We qualitatively discuss some conditions for mode observability in our specific implementation of Mode Process ID. First, the modes {A(1),· · · , A(M)} cannot be too
“similar” to each other with respect to a certain metric d, (e.g., if d(A(m1), A(m2))< for some threshold >0 and two distinct modes m16=m2 and m1, m2∈ X). Second, when ∆T is too short, the consistent set may not converge to a single mode even if d(A(m1), A(m2))≥ for all pairs (m1, m2)∈ X such that m16=m2. Rigorous derivation of these conditions for our specific case are deferred to future work. This includes designing d and for the consistent set narrowing approach, and deriving conditions on ∆T and the set {A(1),· · · , A(M)} for guaranteed convergence towards a singleton consistent set. Although these conditions are contingent upon our simplifying assumptions, they are expected to be similar to those derived in the aforementioned literature.