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| | A '''weighting pattern''' for a [[linear dynamical system]] describes the relationship between an input <math>u</math> and output <math>y</math>. Given the [[time-variant system]] described by |
| | : <math>\dot{x}(t) = A(t)x(t) + B(t)u(t)</math> |
| | : <math>y(t) = C(t)x(t)</math> |
| | then the output can be written as |
| | : <math>y(t) = y(t_0) + \int_{t_0}^t T(t,\sigma)u(\sigma) d\sigma</math> |
| | where <math>T(\cdot,\cdot)</math> is the weighting pattern for the system. For such a system the weighting pattern is <math>T(t,\sigma) = C(t)\phi(t,\sigma)B(\sigma)</math> such that <math>\phi</math> is the [[state transition matrix]]. |
| | |
| | The weighting pattern will determine a system, but note that if there exists a [[Realization (systems)|realization]] for this weighting pattern then there exists many that do so.<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> |
| | |
| | ==Linear time invariant system== |
| | In a [[LTI system]] then the weighting pattern is: |
| | ; Continuous |
| | : <math>T(t,\sigma) = C e^{A(t-\sigma)} B</math> |
| | where <math>e^{A(t-\sigma)}</math> is the [[matrix exponential]]. |
| | |
| | ; Discrete |
| | : <math>T(k,l) = C A^{k-l-1} B</math>. |
| | |
| | == References == |
| | {{Reflist}} |
| | |
| | [[Category:Control theory]] |
Latest revision as of 08:12, 23 January 2013
A weighting pattern for a linear dynamical system describes the relationship between an input 
and output 
. Given the time-variant system described by
then the output can be written as
where 
is the weighting pattern for the system. For such a system the weighting pattern is 
such that 
is the state transition matrix.
The weighting pattern will determine a system, but note that if there exists a realization for this weighting pattern then there exists many that do so.[1]
Linear time invariant system
In a LTI system then the weighting pattern is:
- Continuous
where 
is the matrix exponential.
- Discrete
.
References
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