Order reduction in optimal event-triggered control design for linear stochastic systems

Abstract

Abstract-This paper considers the optimal design of eventtriggered controllers under a non-traditional average-cost criterion with costly observations. Determining the optimal eventtriggering law can be cast in the dynamic programming framework. Due to the lack of a closed form solution for the value function associated with the dynamic program, the methods for calculating the optimal solution suffer from the curse of dimensionality. Based on structural properties of the optimal solution, we develop a novel approximative method to reduce the dimensionality of the underlying optimization problem from the state dimension of the regulated process to the number of control inputs. As processes often consist of only few inputs compared to the number of state variables, such approach reduces the computational complexity significantly. It is shown that the proposed approximative event-trigger preserves the asymptotic behavior of the closed-loop system. A conditions is derived, when the reduced event-triggering law equals the optimal solution. We propose a measure to evaluate the approximation accuracy of the developed order reduction method. Numerical simulations illustrate the obtained results and validate the effectiveness of the proposed model reduction method compared to the optimal solution.

Suboptimal event‐triggered control for networked control systems

“…Although, the above optimization problem differs only slightly from the dynamic program for solving the optimal waiting strategy given by (14), there is a major burden in solving problem (16), as the variable e k is generally not perfectly known at the event-trigger. Remark 3.6.…”

“…T 2 = 1, constitute special situations, where e k can be fully recovered at the event-trigger. Due to Proposition 3.3 and 3.4, respectively, the optimal event-triggered controller (f * , γ * )is given by γ * = γ CE with a state estimator given by (15) and f * is the solution of the dynamic program stated in (16).…”

“…, the event-trigger obtains additional information Q k−T1+T2 that enables us to determine e k−T1−T2+1 from the difference equation of e k defined in the optimization problem (16), where the noise sequence W k−T1−T2 can be recovered from knowledge of X k−T1−T2+1 and the control inputs U k−T1−T2 . As the sequence δ k−1 is known at the event-trigger, the state e k conditioned on δ k−1 has the Markov property , i.e.…”

“…In contrast to the initial problem stated by (16), the information state I red k does not increase in time, but is bounded by the round-trip time T 1 + T 2 . Therefore, finding the optimal event-trigger f is feasible for practical implementation in the case of infinite horizon problems with a moderate round-trip time.…”

“…When Γ is not positive definite, but only non-negative definite, optimal solutions of (14) generally violate assumption (18) for anyc as is shown in [16]. But asc can be arbitrarily large, there always exists an -optimal event-triggering law taking assumption (18) into account.…”