Demo: Stabilization Technique in INTO-CPS Cláudio Gomes, Casper Thule, Kenneth Lausdahl, and Peter G. Larsen
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Demo: Stabilization Technique in INTO-CPS
Cláudio Gomes, Casper Thule, Kenneth Lausdahl, and
Peter G. Larsen
Example IVP
with
Example:
with
June, 2018
Stability of ODEs
Do all solutions of given IVP tend to an equilibrium?Formally, for any solution , does ?
Application: Does the cruise control drive the car to a stable velocity?
with
Stable velocity:
Let and so that
Then, introduce new variable so that
Solution is
Since , as
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Stability of Ordinary Differential Equations
with
For any solution , does ?
Scalar linear ODEs:
Stable if
Vector linear ODEs: with
Stable if
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Scalar Initial Value Problems -Approximation
with
with
Explicit Euler Method:
v dv_dt
0.0 25.380
5.076 22.158
9.508 19.345
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Numerical Stability of Euler MethodScalar linear ODEs:
Explicit Euler Method:
if
Vector linear ODEs:
if
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Stability Analysis Summary
1. Check if original system is stable.
2. Apply numerical method equation to original system
equation
3. Check if it is stable.
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Application to Co-simulation
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Application to Co-sim.
Cosim unit:
Target form:
Zero order hold:
Cosim unit:
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Cosim unit:
Target form:
Cosim unit with internal iteration:
e.g., Fw. Euler:
Application to Co-sim.
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Application to Co-sim.
Cosim unit:
Target form:
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Application to Co-sim.
Cosim unit:
Ideal coupling cosim:
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Jacobi Coupling
Cosim unit:
Ideal coupling cosim:
Actual cosim coupling:
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Jacobi Coupling
Cosim unit:
Actual cosim coupling:
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Jacobi Coupling
Cosim unit:
Actual cosim coupling:
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Jacobi Coupling - MSD
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Iterative Coupling
Cosim unit:
Ideal coupling cosim:
Actual cosim coupling:
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Iterative Coupling
Cosim unit:
Ideal coupling cosim:
Actual cosim coupling:
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Iterative Coupling
Cosim unit:
Ideal coupling cosim:
Actual cosim coupling:
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Iterative Coupling
Cosim unit:
Actual cosim coupling:
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Iterative Coupling
Cosim unit:
Actual cosim coupling:
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Iterative Coupling
Cosim unit:
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Iterative Coupling
Cosim unit:
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Iterative Coupling - MSD
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