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The state space formulation replaces an
th-order ODE by a
vector first-order ODE.
Review of discrete-time case:
where
-
state vector at time
-
vector of inputs
-
output vector
-
state transition matrix
-
input coefficient matrix
-
output coefficient matrix
-
direct path coefficient matrix
The state-space representation is especially powerful for
- multi-input, multi-output (MIMO) linear systems
- time-varying linear systems
(every matrix can have a time subscript
)
Continuous-Time State Space Models:
In continuous time, we obtain a first-order vector ODE in which a
vector of state time-derivatives is driven by linear
combinations of state variables:
State-Space Advantages:
- State-space models are used extensively in advanced modeling
applications
- Extensive support in Matlab, with many numerically excellent
associated tools and techniques (such as the singular value
decomposition, to name one)
- Analytically powerful for theory work
- Example: Solution of
is
, where
the matrix exponential is defined as
- We won't do much with state-space modeling in this class, but
you should know it exists and that it should be considered for larger,
more complex systems than we will be dealing with
Subsections
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Download DigitizingNewton.pdf
Download DigitizingNewton_2up.pdf
Download DigitizingNewton_4up.pdf