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| 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 [[state observer|observer]] for the state of the system, which feeds into an optimal deterministic [[controller (control theory)|controller]] for the system. Thus the problem can be broken into two separate parts, which facilitates the design.
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| As an example of such a principle, it has been proved that if a [[BIBO stability|stable]] [[state observer|observer]] and stable state [[feedback]] are designed for a [[LTI system theory|linear time-invariant system]], then the combined observer and feedback will be stable. The separation principle does not hold in general (for example for non-linear systems). Another example is the separation of the [[linear-quadratic-Gaussian control]] solution into the [[Kalman filter]] and optimal controller for a [[linear-quadratic regulator]]. A separation principle also exists for the control of a quantum systems.
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| == Proof of separation principle for LTI systems ==
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| Consider the system
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| :<math> | |
| \begin{align}
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| \dot{x}(t) & = A x(t) + B u(t) \\
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| y(t) & = C x(t)
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| \end{align}
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| </math>
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| where
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| :<math>u(t)</math> represents the input signal,
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| :<math>y(t)</math> represents the output signal, and
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| :<math>x(t)</math> represents the internal state of the system.
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| We can design an observer of the form
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| :<math>\dot{\hat{x}} = ( A - L C ) \hat{x} + B u + L y \, </math>
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| And state feedback
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| :<math>u(t) = - K \hat{x} \, </math>
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| Define the error ''e'':
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| :<math>e = x - \hat{x} \, </math>
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| Then
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| :<math>\dot{e} = (A - L C) e \, </math> | |
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| :<math>u(t) = - K ( x - e ) \, </math>
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| Now we can write the closed-loop dynamics as
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| : <math>\begin{bmatrix}
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| \dot{x} \\
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| \dot{e} \\
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| \end{bmatrix} =
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| \begin{bmatrix}
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| A - B K & L C \\
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| 0 & A - L C \\
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| \end{bmatrix}
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| \begin{bmatrix}
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| x \\
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| e \\
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| \end{bmatrix}</math>
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| Since this is [[triangular matrix|triangular]], the [[eigenvalues]] are just those of ''A'' − ''BK'' together with those of ''A'' − ''LC''. Thus the stability of the observer and feedback are [[Linear independence|independent]].
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| == References ==
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| * Brezinski, Claude. ''Computational Aspects of Linear Control (Numerical Methods and Algorithms)''. Springer, 2002.
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| {{engineering-stub}}
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| [[Category:Control theory]]
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| [[Category:Stochastic control]]
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