Signal reconstruction in the presence of finite‐rate measurements: finite‐horizon control applications

Abstract

SUMMARYIn this paper, we study finite-length signal reconstruction over a finite-rate noiseless channel. We allow the class of signals to belong to a bounded ellipsoid and derive a universal lower bound on a worstcase reconstruction error. We then compute upper bounds on the error that arise from different coding schemes and under different causality assumptions. When the encoder and decoder are noncausal, we derive an upper bound that either achieves the universal lower bound or is comparable to it. When the decoder and encoder are both causal operators, we show that within a very broad class of causal coding schemes, memoryless coding prevails as optimal, imposing a hard limitation on reconstruction.Finally, we map our general reconstruction problem into two important control problems in which the plant and controller are local to each other, but are together driven by a remote reference signal that is transmitted through a finite-rate noiseless channel. The first problem is to minimize a finite-horizon weighted tracking error between the remote system output and a reference command. The second * Correspondence to: sree@mit.edu problem is to navigate the state of the remote system from a nonzero initial condition to as close to the origin as possible in finite-time. Our analysis enables us to quantify the tradeoff between time horizon and performance accuracy which is not well-studied in the area of control with limited information as most works address infinite-horizon control objectives (eg. stability, disturbance rejection).

Coding scheme based on spherical polar coordinate for control over packet erasure channel

“…With regard to quantized feedback control, advances have been made in guaranteeing stabilizability or observability [1][2][3][4][5][6][7][8], characterizing general performance trade-offs [9][10][11][12][13], designing schemes (with guaranteed performance) [14][15][16][17][18][19][20][21][22][23][24][25], and so on. The conditions guaranteeing the stabilizability or the observability of quantized feedback control system over deterministic channels have been studied extensively [2][3][4]8].…”