Information about Separation Principle
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. As an example of such a principle, it has been proved that if a stable observer and stable state feedback are designed for a 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).
We can design the an observer of the form
And state feedback
Define the error
Then
Now we can write the closed-loop dynamics as
Since this is triangular, the eigenvalues are just those of together with those of . Thus the stability of the observer and feedback are independent.
Control theory is an interdisciplinary branch of engineering and mathematics, that deals with the behavior of dynamical systems. The desired output of a system is called the reference.
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Proof of Separation principle for LTI systems
Consider the system
We can design the an observer of the form
And state feedback
Define the error
Then
Now we can write the closed-loop dynamics as
Since this is triangular, the eigenvalues are just those of together with those of . Thus the stability of the observer and feedback are independent.
References
Brezinski, Claude. Computational Aspects of Linear Control (Numerical Methods and Algorithms). Springer, 2002.For control theory in psychology and sociology, see .
Control theory is an interdisciplinary branch of engineering and mathematics, that deals with the behavior of dynamical systems. The desired output of a system is called the reference.
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In control theory, a state observer is a system that models a real system in order to provide an estimate of its internal state, given measurements of the input and output of the real system. It is typically a computer-implemented mathematical model.
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In control theory, a controller is a device which monitors and affects the operational conditions of a given dynamical system. The operational conditions are typically referred to as output variables of the system which can be affected by adjusting certain input variables.
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BIBO Stability is a form of stability for signals and systems. BIBO stands for Bounded-Input Bounded-Output. If a system is BIBO stable then the output will be bounded for every input to the system that is bounded.
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In control theory, a state observer is a system that models a real system in order to provide an estimate of its internal state, given measurements of the input and output of the real system. It is typically a computer-implemented mathematical model.
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LTI system theory investigates the response of a linear, time-invariant system to an arbitrary input signal. Though the standard independent variable is time, it could just as easily be space (as in image processing and field theory) or some other coordinate.
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In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where the entries below or above the main diagonal are zero. Because matrix equations with triangular matrices are easy to solve they are very important in numerical analysis.
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eigenvector of the transformation and the blue vector is not. Since the red vector was neither stretched nor compressed, its eigenvalue is 1. All vectors with the same vertical direction - i.e., parallel to this vector - are also eigenvectors, with the same eigenvalue.
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In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent.
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