Simple delay-based implementation of continuous-time controllers

Abstract

Abstract-The objective of this work is to study the benefits that delay can provide in simplifying the control process of large-scale systems, motivated by the availability of different types of delays in man-made and biological systems. We show that a continuous-time linear time-invariant (LTI) controller can be approximated by a simple controller that mainly uses delay blocks instead of integrators. More specifically, three methods are proposed to approximate a pre-designed stabilizing LTI controller arbitrarily precisely by a simple delay-based controller composed of delay blocks, a few integrators and possibly a unity feedback. Different problems associated with the developed approximation procedures, such as finding the optimal number of delay blocks or studying the robustness of the designed controller with respect to delay values, are then addressed.

Synthesis of embedded control systems with high sampling frequencies

“…• The approximating signalĝ(t) can be contrived in such a way that the infinity norm of the error G(s) −Ĝ(s) becomes less than any prescribed tolerance. Regarding the second point made above, it is shown in [10] that ifĝ(t) overlaps with g(t) at its breakpoints, then the approximation error ∥G(s) −Ĝ(s)∥ ∞ satisfies the following inequality:…”

“…So far, the controller G was assumed to be single-input single-output. The generalization to the multi-input multioutput case can be easily carried out in line with the discussion provided in [10]. The details of this important generalization can be found in [11].…”

“…Given a prescribed maximum error, we discussed in [10] how to find a permissibleĝ(t) based on either g(t) directly or its discretized counterpart with the discretization step h (the complexity of findinĝ g(t) using the latter method is linear with respect to the number of discrete samples). An analog implementation of the controllerĜ(s) is visualized in Figure 3(a), which consists of three blocks as follows:…”

“…More precisely, increasing the sampling frequency makes the digital controller extremely sensitive to measurement noise and computational round-off errors. Based on our recent result in [10], the present work aims to propose a robust digital-control scheme for continuoustime systems that can be used in two important scenarios: (i) having a high sampling frequency with limited computational power (ii) having a slow processor with jitter and irregular sampling times. Note that the second scenario occurs when the sampling frequency is relatively faster than the slow processing rate and, in addition, the sampling times are prone to delays and irregularities [12].…”

Spectrum design using distributed delay

“…Here, apart from designing the system parameters, one may also tune the delays in order to place the eigenvalues appropriately [13,14], though limitations may arise due to the achievable minimal value of the delay. Such limitations may be compensated for in systems with distributed delays where apart from system parameters one may design the delay distribution [15,16].…”