Intersection form (4-manifold): Difference between revisions

Latest revision as of 16:21, 5 July 2013

In control theory, the state-transition matrix is a matrix whose product with the state vector $x$ at an initial time $t_{0}$ gives $x$ at a later time $t$ . The state-transition matrix can be used to obtain the general solution of linear dynamical systems. It is also known as the matrix exponential.

Overview

Consider the general linear state space model

\dot{x} (t) = A (t) x (t) + B (t) u (t)

y (t) = C (t) x (t) + D (t) u (t)

The general solution is given by

x (t) = Φ (t, t_{0}) x (t_{0}) + \int_{t_{0}}^{t} Φ (t, τ) B (τ) u (τ) d τ

The state-transition matrix $Φ (t, τ)$ , given by

Φ (t, τ) \equiv U (t) U^{- 1} (τ)

where $U (t)$ is the fundamental solution matrix that satisfies

\dot{U} (t) = A (t) U (t)

is a $n \times n$ matrix that is a linear mapping onto itself, i.e., with $u (t) = 0$ , given the state $x (τ)$ at any time $τ$ , the state at any other time $t$ is given by the mapping

x (t) = Φ (t, τ) x (τ)

While the state transition matrix φ is not completely unknown, it must always satisfy the following relationships:

\frac{\partial ϕ (t, t_{0})}{\partial t} = A (t) ϕ (t, t_{0})

and

ϕ (τ, τ) = I

for all

τ

and where

I

is the identity matrix.^[1]

And φ also must have the following properties:

1.	$ϕ (t_{2}, t_{1}) ϕ (t_{1}, t_{0}) = ϕ (t_{2}, t_{0})$
2.	$ϕ^{- 1} (t, τ) = ϕ (τ, t)$
3.	$ϕ^{- 1} (t, τ) ϕ (t, τ) = I$
4.	$\frac{d ϕ (t, t_{0})}{d t} = A (t) ϕ (t, t_{0})$

If the system is time-invariant, we can define φ as:

ϕ (t, t_{0}) = e^{A (t - t_{0})}

In the time-variant case, there are many different functions that may satisfy these requirements, and the solution is dependent on the structure of the system. The state-transition matrix must be determined before analysis on the time-varying solution can continue.

References

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[1] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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[1]

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+In [[control theory]], the '''state-transition matrix''' is a matrix whose product with the state vector <math>x</math> at an initial time <math>t_0</math> gives <math>x</math> at a later time <math>t</math>.  The state-transition matrix can be used to obtain the general solution of linear dynamical systems. It is also known as the [[matrix exponential]].
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+==Overview==
+Consider the general linear [[state space (controls)|state space]] model
+: <math>\dot{\mathbf{x}}(t) = \mathbf{A}(t) \mathbf{x}(t) + \mathbf{B}(t) \mathbf{u}(t)</math>
+: <math>\mathbf{y}(t) = \mathbf{C}(t) \mathbf{x}(t) + \mathbf{D}(t) \mathbf{u}(t)</math>
+The general solution is given by
+: <math>\mathbf{x}(t)= \mathbf{\Phi} (t, t_0)\mathbf{x}(t_0)+\int_{t_0}^t \mathbf{\Phi}(t, \tau)\mathbf{B}(\tau)\mathbf{u}(\tau)d\tau</math>
+The state-transition matrix <math>\mathbf{\Phi}(t, \tau)</math>, given by
+: <math>\mathbf{\Phi}(t, \tau)\equiv\mathbf{U}(t)\mathbf{U}^{-1}(\tau)</math>
+where <math>\mathbf{U}(t)</math> is the fundamental solution matrix that satisfies
+: <math>\dot{\mathbf{U}}(t)=\mathbf{A}(t)\mathbf{U}(t)</math>
+is a <math>n \times n</math> matrix that is a linear mapping onto itself, i.e., with <math>\mathbf{u}(t)=0</math>, given the state <math>\mathbf{x}(\tau)</math> at any time <math>\tau</math>, the state at any other time <math>t</math> is given by the mapping
+:<math>\mathbf{x}(t)=\mathbf{\Phi}(t, \tau)\mathbf{x}(\tau)</math>
+While the state transition matrix &phi; is not completely unknown, it must always satisfy the following relationships:
+:<math>\frac{\partial \phi(t, t_0)}{\partial t} = A(t)\phi(t, t_0)</math> and
+:<math>\phi(\tau, \tau) = I</math> for all <math>\tau</math> and where <math>I</math> is the [[identity matrix]].<ref>{{cite book|first=Roger W.|last=Brockett|title=Finite Dimensional Linear Systems|publisher=John Wiley & Sons|year=1970|isbn=978-0-471-10585-5}}</ref>
+And &phi; also must have the following properties:
+:{| class="wikitable"
+|-
+|1.||<math>\phi(t_2, t_1)\phi(t_1, t_0) = \phi(t_2, t_0)</math>
+|-
+|2.||<math>\phi^{-1}(t, \tau) = \phi(\tau, t)</math>
+|-
+|3.||<math>\phi^{-1}(t, \tau)\phi(t, \tau) = I</math>
+|-
+|4.||<math>\frac{d\phi(t, t_0)}{dt} = A(t)\phi(t, t_0)</math>
+|}
+If the system is [[time-invariant]], we can define &phi; as:
+:<math>\phi(t, t_0) = e^{A(t - t_0)}</math>
+In the time-variant case, there are many different functions that may satisfy these requirements, and the solution is dependent on the structure of the system. The state-transition matrix must be determined before analysis on the time-varying solution can continue.
+==References==
+* {{cite book
+ | author = Brogan, W.L.
+ | year = 1991
+ | title = Modern Control Theory
+ | publisher = Prentice Hall
+ | isbn = 0-13-589763-7
+}}
+{{Reflist}}
+{{Wikibooks|Control Systems/Time Variant System Solutions}}
+[[Category:Control theory]]

Intersection form (4-manifold): Difference between revisions

Latest revision as of 16:21, 5 July 2013

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