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Separation principle
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<p>In <a href="/facts/Control_theory/aIuERpn6">control theory</a>, 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 <a href="/facts/State_observer/HugNt2hg">observer</a> for the state of the system, which feeds into an optimal deterministic <a href="/facts/Controller_(control_theory)/aIuERpn6">controller</a> for the system. Thus the problem can be broken into two separate parts, which facilitates the design. </p><p>The first instance of such a principle is in the setting of deterministic linear systems, namely that if a stable <a href="/facts/State_observer/HugNt2hg">observer</a> and a stable state <a href="/facts/Feedback/XnuWH3zp">feedback</a> are designed for a <a href="/facts/LTI_system_theory/kEmj0QDC">linear time-invariant system</a> (LTI system hereafter), then the combined observer and feedback is <a href="/facts/BIBO_stability/RxD8h2f5">stable</a>. The separation principle does not hold in general for nonlinear systems. </p><p>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 <a href="/facts/Kalman_filter/wNK7rnbk">Kalman filter</a> and a <a href="/facts/Linear-quadratic_regulator/8YybEFpU">linear-quadratic regulator</a>. This is known as <a href="/facts/Linear-quadratic-Gaussian_control/3sI4R7zA">linear-quadratic-Gaussian control</a>. 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 <a href="/facts/Separation_principle_in_stochastic_control/6QUrIBAn">separation principle in stochastic control</a>. </p><p>The separation principle also holds for high gain observers used for state estimation of a class of nonlinear systems and control of quantum systems. </p>