# Stability optimization of hybrid dynamical systems

**Katja Mombaur, Hans-Georg Bock, Johannes Schlöder** - IWR**Moritz Dieh**l - KU Leuven, Belgium**D. Noll** - Univ P. Sabatier, Toulouse, France

In this research, we are interested to optimize the open-loop stability of periodic solutions of optimal control problems involving hybrid dynamics.

Periodic optimal control problems are of interest in many areas of application, e.g. the flight control of aircrafts and spacecrafts, the study of cyclic biological, ecological or economical dynamical systems,

the simulation of periodic motions in robotics, the control of periodic chemical processes, etc.

Problems involving hybrid dynamics, i.e. multi-phase dynamical problems with a combination of continuous phases and of discrete events - in other words discontinuities in the dynamic variables - represent a particularly challenging class of problems We are looking for periodic solutions of these hybrid systems that are open-loop stable. Open-loop stable systems are stable in the sense of Lyapunov without any feedback corrections and recover automatically from small perturbations.

Such solutions of periodic hybrid dynamic systems are characterized by the fact that all eigenvalues of the monodromy matrix (also called the Jacobian of the Poincaré map) lie inside the unit circle. We produce such stable solutions by minimizing the spectral radius of the monodromy matrix of the optimal control problem solution, in the intention to bring it below one. Such a criterion based on monodromy matrix information is computationally challenging since first order sensitivity information of the trajectory is required for function evaluation. In addition, the spectral radius of a nonsymmetric matrix is nondifferentiable - and in some cases even non-Lipschitz - at points where multiple eigenvalues become equal. In addition to eigenvalue optimization, we have also explored alternative criteria based on other matrix norms, or proposed a smoothed

spectral radius.

### References

K. D. Mombaur, H. G. Bock, J. P. Schlöder, and R. W. Longman *- Open-loop stable solution of periodic optimal control problems*, ZAMM (Journal of Applied Mathematics and Mechanics), Vol. 85, No. 7, July 2005

K. Mombaur, *Using optimization to create self-stable human-like running,* Robotica, Vol. 27, 2009, p. 321-330, published online June 2008

M. Diehl, K. Mombaur, D. Noll: *Stability Optimization of Hybrid Periodic Systems via a Smooth Criterion*, IEEE Transactions on Automatic Control, Vol. 54, No. 8, Aug. 2009