The Optimal Boundary and Regulator Design Problem for Event-Driven Controllers

Event-driven control systems provide interesting benefits such as reducing resource utilization. This paper formulates the optimal boundary and regulator design problem that minimizes the resource utilization of an event-driven controller that achieves a cost equal to the case of periodic controllers. 1 Event-Driven Control System Model We consider the control system ẋ(t) = Ax(t) +B u(t) y(t) = C x(t) (1) with x ∈ R, A ∈ R, B ∈ R, u ∈ R, and C ∈ R. Let u(t) = uk = Lx(ak) = Lxk ∀t ∈ [ak, ak+1[ (2) be the control updates given by a linear feedback controller designed in the continuous-time domain but using only samples of the state at discrete instants a0, a1, . . . , ak, . . . Between two consecutive control updates, u(t) is held constant. In periodic sampling we have ak+1 = ak + h, where h is the period of the controller. Let ek(t) = x(t)−xk be the error evolution between consecutive samples with t ∈ [ak, ak+1[. For several types of event-driven control approaches [1, 2], event conditions can be generalized by introducing a function f(·, ·, Υ ) : R × R → R that defines a boundary measuring the tolerated error with to respect the sampled state [3]. The condition that must be ensured is f(ek(t), xk, Υ ) ≤ η (3) ⋆ This work was supported in part by ArtistDesign NoE IST-2008-214373, and by CICYT DIP-2007-61527. 2 where η is the error tolerance and Υ = {υ1, υ2, . . . , υp}, υi ∈ R is a set of free parameters. Hence, we can define the complete dynamics of the event-driven system by the n+ 1 order non linear discrete-time system ak+1 = ak + Λ(xk, Υ, η) xk+1 = (Φ(Λ(xk, Υ, η)) + Γ (Λ(xk, Υ, η))L)xk (4) where Λ(xk, Υ, η) denotes the time separation between two consecutive activations ak+1 and ak, that solves (1), (2), and (3), assuming that xk = x(ak) is the state sampled at ak, Υ is the set of free parameters of f , and η is the tolerance to the error. We also define Φ(t) = e and Γ (t) = ∫ t 0 edsB. We highlight that we have been able to find an expression for Λ(xk, Υ, η) only by approximating Φ and Γ by Taylor expansion [3]. In all the other cases Λ can only be computed numerically. 2 Optimal Problem Formulation The optimal problem for event-driven controllers can be formulated in two complementary ways: to minimize the cost while using the same amount of resources than the periodic controller, or to minimize the computational demand while achieving the same cost as in the case of the periodic controller. Here we describe the resource usage minimization given a cost constraint. The other formulation simply requires to exchange the goal function and one constraint, as it will be indicated later. Let be a standard quadratic cost function in continuous time defined as J(L, Υ, η) = ∫ al 0 x(t)Qcx(t) + u(t) Rcu(t)dt+ x(al) Ncx(al) (5) The optimal boundary and regulator design problem for resource minimization can be formulated as maximize ∑l−1 k=0 Λ(xk, Υ, η) k w.r.t. L, Υ, η (6) subject to xk+1 = (Φ(Λ(xk, Υ, η)) + Γ (Λ(xk, Υ, η))L)xk (7) ak+1 = ak + Λ(xk, Υ, η) (8) J(L, Υ, η) ≤ Jh (9) where (6) sets the maximization goal equal to the average of the first l sampling intervals, (7) enforces the relationship between two consecutive sampled states, (8) describes the constraint among the activations, and Jh is the cost of an optimal h-periodic controller. Notice that by exchanging (6) with (9) we obtain the complementary problem that minimizes the cost given an upper bound on the period. The problem (6)–(9) can be numerically solved by constrained minimization techniques such as Lagrange multipliers, or by standards procedures for time varying discrete-time systems. 3 3 Example Consider the double integrator system ẋ = [