Separation principle

Summary

In control theory, a separation principle, more formally known as a principle of separation of estimation and control, states that under some assumptions the problem of designing an optimal feedback controller for a stochastic system can be solved by designing an optimal observer for the state of the system, which feeds into an optimal deterministic controller for the system. Thus the problem can be broken into two separate parts, which facilitates the design. The first instance of such a principle is in the setting of deterministic linear systems, namely that if a stable observer and a stable state feedback are designed for a linear time-invariant system (LTI system hereafter), then the combined observer and feedback is stable. The separation principle does not hold in general for nonlinear systems. Another instance of the separation principle arises in the setting of linear stochastic systems, namely that state estimation (possibly nonlinear) together with an optimal state feedback controller designed to minimize a quadratic cost, is optimal for the stochastic control problem with output measurements. When process and observation noise are Gaussian, the optimal solution separates into a Kalman filter and a linear-quadratic regulator. This is known as linear-quadratic-Gaussian control. More generally, under suitable conditions and when the noise is a martingale (with possible jumps), again a separation principle applies and is known as the separation principle in stochastic control. The separation principle also holds for high gain observers used for state estimation of a class of nonlinear systems and control of quantum systems. Consider a deterministic LTI system: where represents the input signal, represents the output signal, and represents the internal state of the system. We can design an observer of the form and state feedback Define the error e: Then Now we can write the closed-loop dynamics as Since this is a triangular matrix, the eigenvalues are just those of A − BK together with those of A − LC.

Official source

https://en.wikipedia.org/wiki/Separation_principle

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Related publications (1)

A Distributed Luenberger Observer for Linear State Feedback Systems With Quantized and Rate-Limited Communications

Colin Neil Jones, Ye Pu, Andrea Alessandretti, Francisco Fernandes Castro Rego

This article addresses the problem of simultaneous distributed state estimation, and control of linear systems with linear state feedback, subjected to process, and measurement noise, under the constraints of quantized, and rate-limited network data transmission. In the set-up adopted, sensors and actuators communicate through a network with a strongly connected topology. Unlike the case of centralized linear systems, for which the separation principle holds, the above practical assumption prevents the separate design of observers, and controller because each of the nodes does not necessarily have access to the control inputs generated at all the other nodes. We derive a linear distributed Luenberger observer, and a set of sufficient conditions that guarantee ultimate boundedness of the estimation error, and system state vectors, with bounds that depend on the L-infinity norm of the noise signals, and the number of bits used in the transmissions. A numerical example illustrates the performance and effectiveness of the proposed algorithm in controlling a network of open-loop unstable systems.

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2021