Stability and stabilization of hybrid systemsMikael Johansson
School of Electrical Engineering
KTH
Torque control and bumpless transfer
Base controller: non-linear PI
Changes in acceleration when shifting gears avoided via bumpless transfer:
for all feasible gear changes ij.
( compatible values of Ki, changes in integral state)
Hybrid system model
Need extended hybrid model that allows for jumps in the continuous state
LMI formulation possible if jump map is affine in x.
Numerical example
Closed loop system is switched linear system
where and
Simulation for
Stability
If affine reset maps
then, is guaranteed by solution to LMI
Can extend discontinuous Lyapunov function computations from Lecture 2
Gear-box example: solution found exponential convergence to vref
Remark: analysis needs to be repeated for each value of vref
(as in bioreactor example)
Part III – Examples
Constrained control via min-max selectors
Substrate feeding control
Automatic gear-box control
A simple relay system
More of a theoretical challenge…
[Hassibi, 2000]
Consider a linear control system under hysteresis relay feedback…
Simulations suggest system is stable, yet no pwq Lyapunov function found.
The challenge
Q: why do piecewise quadratic methods fail, how can they be improved?
The more general challenge:
Put the methods to the test of challenging engineering problems, and
help to contribute to the development to improved analysis tools!
References
M. Johansson, “Piecewise linear control systems – a computational approach”, Springer Lecture Notes in Control and Information Sciences no 284, 2002.
S. Velut, “Probing control – analysis and design with application to fed-batch bioreactors”, PhD thesis, Lund University, Lund, Sweden, June 2005.
S. Pettersson, ”Analysis and design of hybrid systems”, PhD thesis, Chalmers University, Gothenburg, Sweden, 1999.
A. Hassibi, “Lyapunov methods in the analysis of complex dynamical systems”, Stanford University, Stanford, CA, 2000.
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