### Inhalt des Dokuments

# Optimal Control of Hybrid Systems

### From Fachgebiet Regelungssysteme TU Berlin

## Contents |

### Abstract

Hybrid optimal control problems are highly nontrivial, as one has to deal not only with the infinite dimensional optimisation problems related to the continuous dynamics, but also with a potential combinatorial explosion related to the discrete part. Because of the large number of potential applications, there has been considerable interest in optimal hybrid control problems. We have focused on some specific, but practically important, classes of hybrid systems and derived necessary conditions of optimality and efficient conceptual algorithms to solve the related problems

### People involved

- Vadim Azhmyakov (now with CINVESTAV)
- Sid Ahmed Attia (now with GE)
- Dmitry Gromov (now with McGill University)
- Jörg Raisch

### Cooperation

- Vadim Azhmyakov (CINVESTAV)

### Description

Through V. Azhmyakov and S.A. Attia, our group became interested in the theory of optimal hybrid control. Hybrid optimal control problems are highly nontrivial, as one has to deal not only with the infinite dimensional optimization problems related to the continuous dynamics, but also with a potential combinatorial explosion related to the discrete part. Because of the large number of potential applications, there has been considerable interest in optimal hybrid control problems, with important contributions from, e.g., Clarke, Antsaklis, Caines, Egerstadt, Piccoli and their coworkers. One of the most convenient ways to deal with the problem is to formulate it as a sequential problem, i.e., for a particular execution the time axis is partitioned into subintervals. In each interval, the discrete state remains constant, and the continuous dynamics is characterised by a set of ODEs. Transitions between intervals/ discrete states are either triggered internally (typically by the continuous state "hitting" some manifold) or externally (by a discrete control signal). The former is often referred to as autonomous switching, the latter as controlled switching.

We have focused on some specific, but practically important, classes of hybrid systems and derived
necessary conditions of optimality and efficient conceptual algorithms
to solve the related problems: first, we have investigated hybrid systems
with autonomous switching and continuous control inputs. In contrast
to the work by Caines and coworkers, we derived necessary
optimality conditions without recourse to the technique of needle
variations. Instead we apply a generalized Lagrange multiplier rule^{[1]}.
This allows us to obtain necessary conditions for a
weak minimum as opposed to the Maximum Principle, which gives
necessary conditions for a strong minimum. The difference
between the two types of minima is the norm used to compare two
feasible trajectories. The weak necessary conditions of optimality are
said to hold if the continuous trajectories associated with the same
discrete state are compared in the sense of the infinity norm in
contrast to a strong minimum, where the 1-norm is usually
employed. The problem is first formulated as an abstract optimisation
problem in an appropriate Sobolev space. The differential equations
are considered as operators acting on Sobolev spaces, and
the switching surfaces are embedded into the operator as equality
constraints. A generalised Lagrange multiplier is then applied to
extract the necessary conditions of optimality.

As a second class, we have investigated hybrid systems with autonomous
switching where discrete transitions are accompanied by instantaneous
changes (jumps) in the continuous states and where these state jumps
(i.e., the differences between "new" and "old" values of the
continuous states) can be considered as the sole control
variables. Hybrid systems with jumps in the continuous states are
often referred to as impulsive hybrid systems. In a first step,
necessary conditions of optimality are established based on a
variational approach. For this, a smooth variation preserving the
switching sequence for the discrete state is introduced around the
optimal trajectory. Applying the Lagrange principle gives a sequence
of boundary-value problems that need to be solved and an equality
condition on the gradient of the cost functional with respect to the
jump parameters. Closed form expressions of the gradient are then
obtained using a parameter variation where the effects of parametric
variation are propagated on the whole trajectory. An algorithm based
on gradient descent techniques is then proposed together with some
convergence results. The algorithm uses forward-backward integration
of the system dynamics and the adjoint equations together with a
pointwise update of the jump parameters. Details are provided in
^{[2]}
^{[3]}
.

As a third class, we have considered hybrid systems with autonomous
switching, continuous control inputs *and* controlled state
jumps. Using a simple transformation, the problem under study can be
formulated as a hybrid systems with autonomous switching where jump
parameters are considered as a part of the control. Based on the
results in ^{[1]}, we develop a new set of
necessary conditions of optimality ^{[4]}.
A combination of
the algorithm developed for the class of impulsive autonomous hybrid
systems ^{[2]}^{[3]} together with a gradient based approach^{[5]}
for updating the
control can be used to extract both the continuous control signals and the controlled jump
parameters.

Finally, ^{[6]}
discusses how a class of optimal control problems for switched systems that are affine in the control input can be treated as convex problems.

### Publications

- ↑
^{1.0}^{1.1}- V. Azhmyakov, S. A. Attia, D. Gromov, J. Raisch.
, volume 4416 of Lecture Notes in Computer Science (LNCS), pages 637-640. Springer-Verlag, 2007.**Necessary optimality conditions for a class of hybrid optimal control problems** - Bibtex
**Author :**V. Azhmyakov, S. A. Attia, D. Gromov, J. Raisch**In :**, volume 4416 of Lecture Notes in Computer Science (LNCS), pages 637-640. Springer-Verlag, 2007.**Necessary optimality conditions for a class of hybrid optimal control problems****Date :**2007

- V. Azhmyakov, S. A. Attia, D. Gromov, J. Raisch.
- ↑
^{2.0}^{2.1}- S.A. Attia, V. Azhmyakov, J. Raisch.
**State jump optimization for a class of hybrid autonomous systems**. In*Proc. 2007 IEEE Multi-conference on Systems and Control*, pages 1408-1413, 2007. - Bibtex
**Author :**S.A. Attia, V. Azhmyakov, J. Raisch**Title :**State jump optimization for a class of hybrid autonomous systems**In :**In*Proc. 2007 IEEE Multi-conference on Systems and Control*,**Date :**2007

- S.A. Attia, V. Azhmyakov, J. Raisch.
- ↑
^{3.0}^{3.1}- S. A. Attia, V. Azhmyakov, J. Raisch.
**On an Optimization Problem for a Class of Impulsive Hybrid Systems.**.*Discrete Event Dynamic Systems: Theory and Applications – Special issue on Hybrid Systems Optimization*, 20 pages 215 - 231, 2010. - Bibtex
**Author :**S. A. Attia, V. Azhmyakov, J. Raisch**Title :**On an Optimization Problem for a Class of Impulsive Hybrid Systems.**In :***Discrete Event Dynamic Systems: Theory and Applications – Special issue on Hybrid Systems Optimization*,**Date :**2010

- S. A. Attia, V. Azhmyakov, J. Raisch.
- ↑
- V. Azhmyakov, S. A. Attia, J. Raisch.
, volume 4981 of Lecture Notes in Computer Science (LNCS), pages 30-42. Springer-Verlag, 2008.**On the Maximum Principle for Impulsive Hybrid System** - Bibtex
**Author :**V. Azhmyakov, S. A. Attia, J. Raisch**In :**, volume 4981 of Lecture Notes in Computer Science (LNCS), pages 30-42. Springer-Verlag, 2008.**On the Maximum Principle for Impulsive Hybrid System****Date :**2008

- V. Azhmyakov, S. A. Attia, J. Raisch.
- ↑
- V. Azhmyakov, J. Raisch.
**A gradient-based approach to a class of hybrid optimal control problems**. In*Proc. 2nd IFAC Conference on Analysis and Design of Hybrid Systems2*, pages 89-94, 2006. - Bibtex
**Author :**V. Azhmyakov, J. Raisch**Title :**A gradient-based approach to a class of hybrid optimal control problems**In :**In*Proc. 2nd IFAC Conference on Analysis and Design of Hybrid Systems2*,**Date :**2006

- V. Azhmyakov, J. Raisch.
- ↑
- V. Azhmyakov, M.V. Basin, J. Raisch.
**A Proximal Point Based Approach to Optimal Control of Affine Switched Systems**.*Discrete Event Dynamic Systems*, 2011. - Bibtex| PDF | DOI
**Author :**V. Azhmyakov, M.V. Basin, J. Raisch**Title :**A Proximal Point Based Approach to Optimal Control of Affine Switched Systems**In :***Discrete Event Dynamic Systems*,**Date :**2011

- V. Azhmyakov, M.V. Basin, J. Raisch.