Paula Andrea

The idea of periodically invariant sets for finding a control strategy for a periodic linear discrete-time system was proposed by Blanchin and Ukovich (1993). Recently, there has been research work on applying this idea to controlling periodic linear systems (Gondhalekar, 2011; Gondhalekar and Jones, 2011; Zhou et al., 2011), especially Model Predictive Control (MPC) for building systems due to the periodic nature of the disturbances (Ma et al., 2009; Gondhalekar et al., 2010; Ma et al., 2012). In this subsection, we will employ this notion and its computation to determine the sequence{Ct}∞t=τ as well as a safe feedback control law for the invariance phase as discussed above. To emphasize the robustness of the control law with respect to disturbances, we will use the termrobust periodic invariant6_.

We first review the basic definitions and results for robust periodic invariant sequence of sets for constrained systems.

Definition 4.2 A finite sequence _S=_{C0, . . . ,Cδ−1} of sets Ct⊆Safet, ∀t= 0, . . . , δ−1,

6_{The terms}

is a robust periodic invariant sequenceof sets for constrained system (4.30)if for every t= 0, . . . , δ₋1,

∀x∈Ct∃u∈ Ut : f(x, u, d)∈C(t+1) modδ∀d∈ Dt.

A sequence _S? = _{C?0, . . . ,C?δ−1} is said to be the maximal robust periodic invariant

sequenceif it is robust periodic invariant and contains all robust periodic invariant sequences {C0, . . . ,Cδ−1} for the system, meaning that Ct⊆C?t ∀t= 0, . . . , δ−1. 2

Using the robust backward reachability operatorR−1 _{defined in Equation (4.31), the maxi-}

mal robust periodic invariant sequence, if it exists, can be computed by the procedure in Algorithm 4.5 on the following page (cf. Procedure 4.1 in Blanchin and Ukovich, 1993). Starting from the desired safe sets {Safet}δt=0−1 for one period, the algorithm essentially

compute the one-step backward reachable sets repeatedly until it detects one of the two terminating conditions: (1) a fixed point is found (line 5) in which case the obtained sequence

{Ω0, . . . ,Ωδ−1}is maximal; or (2) an empty backward reachable set is found (line 8) in which

case the maximal robust periodic invariant sequence does not exist.

One important question regarding Algorithm 4.5 is its termination, that is whether the algorithm will terminate after a finite number of iterations. In general, there is no guarantee that Algorithm 4.5 will terminate. However, it is shown in (Bertsekas, 1972) and in (Blanchin and Ukovich, 1993) that under certain compactness and continuity conditions, convergence of the sequence{Ω0, . . . ,Ωδ−1} to the maximal one can be guaranteed. Specifically, for the

affine dynamics in Equation (4.30) and assuming that the setsUt, Dt,Safet are convex and compact polytopes, we can guarantee that (Blanchin and Ukovich, 1993)

• IfS? does not exist then Algorithm 4.5 will terminate (by the condition in line 8);

• IfS? exists then the sequence{Ω0, . . . ,Ωδ−1}will converge to it, thus we can obtain an

arbitrarily close over-approximation of S? by executing the algorithm for a sufficiently large number of iterations.

Algorithm 4.5 Computation of Maximal Robust Periodic Invariant Sequence 1: Initialize a sequenceΩt←Safet,t= 0, . . . , δ−1

2: t←0

3: while true do . main iteration

4: Compute R← R−_(t1_{−1) modδ}(Ωtmodδ)TSafe(t−1) modδ 5: if t≤ −δ and Ω_{(t−1) modδ} =R then

6: return FoundS?={Ω0, . . . ,Ωδ−1}

7: end if

8: if R=_∅ then

9: return “Maximal robust periodic invariant sequence does not exist”

10: end if

11: Ω(t−1) modδ←R

12: t_←t₋1

13: end while

Suppose thatS? exists and we can compute it. Then for any initial statex(0)∈C?0a feedback

control lawu(t) =κ(t, x(t)) that can maintain the system safe indefinitely (i.e.,x(t)_∈Safet for allt_∈N) must keepx(t) inS?, as verified by the following result.

Proposition 4.2 (adapted from Blanchin and Ukovich, 1993, Corollary 3.1) A control

strategy u(t) =κ(t, x(t))maintains the system (4.30) safe indefinitely with the initial set _C?₀ if and only if, for x(t)∈C?tmodδ, it satisfies the conditions

f(x(t), κ(t, x(t)), d)_∈C?_{(t+1) modδ}, ∀d∈ Dt κ(t, x(t))∈ Ut

for all t_≥0. ₂

A proof of the result can be found in (Blanchin and Ukovich, 1993).

OnceS? is obtained, it is straightforward to derive a control law that satisfies Proposition 4.2 as, for anyt∈Nand x(t)∈C?_tmodδ,

κ(t, x(t)) =any u_{∈ U}tsuch thatf(x(t), u, d)∈C?(t+1) modδ ∀d∈ Dt. (4.34)

function as suggested by Blanchin and Ukovich (1993), e.g., to reduce the total energy consumption. An MPC strategy can also be formulated for determining the control as

minimize u(t),...,u(t+N−1) t+N−1 X i=t ci(x(i), u(i)) +cf(x(t+N)) subject to u(i)∈ Ui x(i+ 1) =f(x(i), u(i), d(i)) f(x(i), u(i), d)∈C?imodδ, ∀d∈ Di

in which the constraints are satisfied for all i=t, . . . , t+N ₋1 and • t∈Nis the current time step;

• N ∈Nis a given finite horizon,N ≥1;

• di ∈ Di is the nominal disturbance at timei,i=t, . . . , t+N −1;

• ci(x(i), u(i)) is the (scalar) cost at timeiwith respect to state x(i) and control u(i), i=t, . . . , t+N₋1;

• cf(x(t+N))is the (scalar) terminal cost depending on x(t+N).

The optimization is solved for an optimal sequence of controls u(t), . . . , u(t+N ₋1) but onlyu(t) is applied. At the next time step,t+ 1, the optimization is re-formulated and the process is repeated.

Recall that in theinvariance phase of safe green scheduling (cf. Section 4.4.2), we aim to

maintain the system’s state in a periodic sequence of subsets of Safet, ∀t≥τ. Obviously S? is the sequence of sets that we are looking for. Once the state x has been driven toC?_τmodδ at time stepτ (in theconvergence phase), the control law derived above is used to guarantee