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The controllability Gramian can be found as the solution of the Lyapunov equation given by + =. In fact, we can see that if we take = as a solution, we are going to find that: + = + = = | = = =
Controllability and observability are dual aspects of the same problem. Roughly, the concept of controllability denotes the ability to move a system around in its entire configuration space using only certain admissible manipulations. The exact definition varies slightly within the framework or the type of models applied.
The Gram matrix is symmetric in the case the inner product is real-valued; it is Hermitian in the general, complex case by definition of an inner product. The Gram matrix is positive semidefinite, and every positive semidefinite matrix is the Gramian matrix for some set of vectors. The fact that the Gramian matrix is positive-semidefinite can ...
where is the state vector and is the control vector. Suppose our goal is to drive the system to an equilibrium from every initial state in some domain .Without loss of generality, suppose the equilibrium is at = (for an equilibrium , it can be translated to the origin by a change of variables).
The square of the Hilbert-Schmidt norm of the Hankel operator associated with a linear system is the sum of squares of the Hankel singular values of this system.
Observability is a measure of how well internal states of a system can be inferred from knowledge of its external outputs. In control theory, the observability and controllability of a linear system are mathematical duals.
In control theory, the Gramian or Gram matrix is an important contribution named after him. The Controllability Gramian and Observability Gramian are both important in the analysis of the stability of control systems. The Gram matrix is also important in deep learning, where it is used to represent the distribution of features in style transfer.
That is called the discrete Observability Gramian. We can easily see the correspondence between discrete time and the continuous time case, that is, if we can check that W d c {\displaystyle {\boldsymbol {W}}_{dc}} is positive definite, and all eigenvalues of A {\displaystyle {\boldsymbol {A}}} have magnitude less than 1 {\displaystyle 1} , the ...