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In control theory, we may need to find out whether or not a system such as ˙ = + () = + is controllable, where , , and are, respectively, , , and matrices for a system with inputs, state variables and outputs.
For example, if matrix D = 0 and matrix C does not have full row rank, then some positions of the output are masked by the limiting structure of the output matrix, and therefore unachievable. Moreover, even though the system can be moved to any state in finite time, there may be some outputs that are inaccessible by all states.
In control theory, a Kalman decomposition provides a mathematical means to convert a representation of any linear time-invariant (LTI) control system to a form in which the system can be decomposed into a standard form which makes clear the observable and controllable components of the system.
In control theory, Ackermann's formula is a control system design method for solving the pole allocation problem for invariant-time systems by Jürgen Ackermann. [1] One of the primary problems in control system design is the creation of controllers that will change the dynamics of a system by changing the eigenvalues of the matrix representing the dynamics of the closed-loop system. [2]
The Observability 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: + = + = = | = = =
The observability and controllability of a system are mathematical duals (i.e., as controllability provides that an input is available that brings any initial state to any desired final state, observability provides that knowing an output trajectory provides enough information to predict the initial state of the 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.
The phrase H ∞ control comes from the name of the mathematical space over which the optimization takes place: H ∞ is the Hardy space of matrix-valued functions that are analytic and bounded in the open right-half of the complex plane defined by Re(s) > 0; the H ∞ norm is the supremum singular value of the matrix over that