Uncertain AoI in stochastic optimal control of constrained LTI systems

Proof of Lemma 1.

First, consider the system dynamics (3) and the control law (7), for the case of available state information:

with general functions:

The recursive state equation can be reformulated to a series of functions f k and κ k , where control laws are rewritten to κ ˜ k , e. g., for k ∈ { 0 , 1 }:

or for arbitrary k ∈ N ≥ 0:

or for linear functions:

with a i and b i [8].

Secondly, if a k ≥ 0, (7) is used in combination with (8). Thus, with the last certain information x l → l : = k − a k , it holds that all states x l + 1 up to x k are not available to the controller:

With (7) and (8), it holds that:

with functions κ k defined in (17) and:

In (19), the states with index up to l can be expressed by the functions given in (18). For the first of the remaining states/inputs in the function arguments, it follows with (18) that:

Recursively for k > l, the states:

and the inputs:

are obtained. Again, with linear functions f k , l and κ k , the control law κ ˜ k in (20) is a linear function of its arguments, thus (again with a set of parameters a i ) one can write:

Eventually, (21) feeds back all disturbances w r with r ≤ l − 1 = k − a k − 1 ⇔ r < k − a k . With V k : = a 0 ( x ) and M k , r : = a r ( w ) , (21) equals the disturbance feedback in Lemma 1 for time k.  □

Proof of Proposition 1.

According to Proposition 1 the co-domain of λ k is given by:

such that

hold. Now recall (14), and let P ( > β ) ⊆ P ( Ω ) denotes a subset of indicator matrices P ( i ) , for which all entries satisfy the inequality 𝟙 k , r ( i ) ≥ 𝟙 k , r ( β ) ∀ k , r ∈ N H . Then the following holds true with θ denoting the behavior of the communication network according to P ( θ ) ∈ P ( Ω ) :

if all control laws u k ( i ) , for which P ( i ) ∈ P ( > β ) holds, are truncated to u k ( β ) . In here, i denotes one run of the Markov process over the whole trajectory (though i is independent of k), which justifies the independence of probabilities. Therefore, the necessary condition results with (15) to:

This is at least satisfied, if the γ k ( x ) -confidence ellipsoid of the state x k ( β ) lies within the admissible state set, i. e.:

Following the same steps of the proof to [8, Proposition 5], condition (22) is satisfied if the LMI L x k + 1 [ j ] ≽ 0 given in (16) is satisfied with the tailored likelihood γ k ( x ) for each half-space j ∈ { 1 , … , n X k + 1 } of X k + 1 , where n X k + 1 denotes the number of half-spaces.

The reasoning for the input follows analogously, but with λ k instead of its product.  □