### Inhalt des Dokuments

# Abstraction-based Hybrid Control Synthesis

### From Fachgebiet Regelungssysteme TU Berlin

## Contents |

### Abstract

Hybrid systems are characterised by the interaction of continuous and discrete-event components. Abstraction-based control synthesis approaches aim at "replacing" continuous dynamics by discrete abstractions. The resulting problem is pureley discrete and therefore within the realm of established DES control synthesis methods. An important feature is that safety and nonblocking properties can be guaranteed to carry over from the abstraction level to the underlying hybrid control problem.

### People involved

- Jörg Raisch
- Anne-Kathrin Schmuck
- Dmitry Gromov (now with McGill University)

### Cooperation

- Thomas Moor, Universität Erlangen
- Jen Davoren, University of Melbourne

### Description

The control of physical or chemical processes by digital
computer programs often leads to heterogeneous systems which include both
continuous and discrete-event dynamics. Such *hybrid control systems*
generally exhibit highly complex behaviour. From an engineering point of view,
the systematic design of hybrid control systems is of particular
importance. This represents a mathematically challenging task, primarily
because of the nature of hybrid state sets: purely continuous
systems usually exhibit a nice (vector space) structure. This implies that a rich set
of analysis tools can be applied to investigate continuous system
dynamics. Purely discrete systems can be described by discrete, and in most
cases finite, state sets. Hence, the dynamical behaviour of finite discrete systems
can, at least in principle, be completely investigated by finite enumeration
type methods. The state set of a hybrid system is the product of the state sets of
its constituent components. In general, it is therefore neither finite nor
does it exhibit vector space structure.

A natural approach to avoiding this
problem is to resort to abstraction-based control synthesis methods: roughly
speaking, the external behaviour of the continuous component is approximated
by a discrete-event system (DES);
if the specifications are also discrete, this turns the hybrid control problem
into a purely discrete one, which, in a subsequent step, can be addressed using
established methods from the field of DES theory. Abstraction-based
synthesis of hybrid control systems has been an active area of research for a
number of years, with important contributions from, among others, P. Antsaklis', B. Krogh's and J. Lunze's groups. All of these approaches require the approximation to be *safe*, meaning that any
controller enforcing the specifications for the discrete approximation must be
guaranteed to do the same for the underlying continuous model.
Failure of successful controller synthesis on the approximation level,
however, does not imply that the hybrid control problem cannot be solved, as
increasing approximation accuracy may still allow determination of an adequate
controller. We have therefore suggested a method that provides a set of
discrete abstractions (all of them realisable by finite
automata) which are strictly ordered with respect to
approximation accuracy, e.g.
^{[1]}
^{[2]}. These "l-complete approximations" exactly represent the external behaviour of the continuous system under consideration over an interval of l+1 sampling instants
^{[3]}
^{[4]}, where sampling may either be equidistant, i.e., clock-driven
^{[5]}, or event-triggered
^{[6]}
Clearly, increasing l will increase approximation accuracy, but will also
(exponentially) increase complexity.

In cooperation with T. Moor from Universitaet Erlangen and J. Davoren from the University of Melbourne, we have explored a number of promising approaches to alleviate this problem.

Increasing the integer parameter l increases approximation accuracy *uniformly* --- even though the given specifications may only require a refinement of certain aspects of the discrete approximation. Hence, in ^{[7]}
^{[8]}, we
developed a procedure that, in case of failure during the controller synthesis
step, locates the potential reason for failure in the currently used
approximation. The refinement procedure then focuses its efforts on those aspects of the approximation that have caused the failure instead of doing an unspecific global refinement.

Another approach to counter the increase of complexity is the use of modular
controllers. In ^{[9]}, we identified conditions under which two discrete controllers, each enforcing a particular specification for a continuous plant model, will have an admissible parallel composition that enforces both specifications simultaneously.

Hierarchical control can be interpreted as an attempt to handle complex problems by decomposing them into smaller subproblems and reassembling their solutions in a hierarchical structure. We have investigated an approach that is based on a hierarchy of models describing a given plant at various levels of abstraction. It captures intuitive concepts like information aggregation between different levels of control and is general enough to encompass both continuous and discrete-event levels. Details can be found in the description of our project on hierarchical control theory.

It is interesting to note that l-complete approximations can be used to provide
set-valued estimates for the unknown hybrid system state on the basis of the discrete-valued input and output signals. Estimates are conservative in the sense that
the true state can be guaranteed to be contained in the set-valued estimate. Estimation accuracy can be improved by increasing the parameter l, albeit at the cost of complexity. In ^{[10]}
we have shown that for a class of hybrid systems the same estimate can be obtained via a distributed, or
decentralised, approach involving several less complex approximations, which are run in parallel.
For a larger class of systems, it can be shown that this approach provides an outer approximation
of the estimate provided by a monolithic l-complete estimator.

To compute l-complete or other safe approximations, one basically needs to
propagate bounded subsets of the plant state space under the flow
corresponding to the plant dynamics, and to intersect the results with other
bounded sets. This clearly represents a major problem for nonlinear flows. In
practice, one often resorts to exhaustive simulation type methods, where
instead of a set, a large number of single points is propagated over time. This
not only interferes with the aim of finding a *safe* approximation,
but also drastically increases computational requirements, especially for
high-dimensional systems. We have investigated a class of nonlinear systems where safe approximations can be
computed very efficiently: monotone dynamical systems, which are fairly common
in chemical engineering applications, are characterised by the
fact that there exists a partial order in the state space which is preserved under the
progress of time ^{[11]}.

| |

In cooperation with the PSD group at MPI Magdeburg, we successfully used our results to
synthesise a discrete-event controller for the automatic start-up
of a distillation column. Controller synthesis was based
on a nonlinear 42nd order plant model; the specification was to drive the
plant state into a well-defined vicinity of the desired
operating point within 20 minutes. The figures show a comparison between the open-loop
case, where the control inputs corresponding to the desired operating point were
applied to the plant model, and the closed-loop case consisting of continuous
plant model and discrete controller ^{[11]}. In the former case, it takes many hours to converge to the desired target region (indicated by horizontal lines), in
the latter case, this is achieved, as required, within 20 minutes.
Using our hierarchical hybrid approach, we have also been able to successfully
address the start-up problem for a more complex distillation plant ^{[12]}.

We have have also investigated the combination of the abstraction-based approach described above and an optimal switching strategy developed by Alessandro Giua and Carla Seatzu from the University of Cagliari. In the overall control scheme, the abstraction-based part guarantees safety properties while the remaining part makes optimal use of the remaining degrees of freedom ^{[13]} ^{[14]}.

### Publications

- ↑
- Jörg Raisch, Siu O'Young.
**Discrete Approximation and Supervisory Control of Continuous Systems**.*IEEE Transactions on Automatic Control, Special Issue on Hybrid Systems*, 43 (4):569–573, 1998. - Bibtex
**Author :**Jörg Raisch, Siu O'Young**Title :**Discrete Approximation and Supervisory Control of Continuous Systems**In :***IEEE Transactions on Automatic Control, Special Issue on Hybrid Systems*,**Date :**1998

- Jörg Raisch, Siu O'Young.
- ↑
- Jörg Raisch.
**A Hierarchy of Discrete Abstractions for a Hybrid Plant**.*JESA — European Journal of Automation, Special Issue on Hybrid Dynamical Systems*, 32 (9–10):1073–1095, 1999. - Bibtex
**Author :**Jörg Raisch**Title :**A Hierarchy of Discrete Abstractions for a Hybrid Plant**In :***JESA — European Journal of Automation, Special Issue on Hybrid Dynamical Systems*,**Date :**1999

- Jörg Raisch.
- ↑
- Thomas Moor, Jörg Raisch.
**Supervisory Control of Hybrid Systems within a Behavioural Framework**.*Systems and Control Letters, Special issue on Hybrid Control Systems*, 38 pages 157–166, 1999. - Bibtex
**Author :**Thomas Moor, Jörg Raisch**Title :**Supervisory Control of Hybrid Systems within a Behavioural Framework**In :***Systems and Control Letters, Special issue on Hybrid Control Systems*,**Date :**1999

- Thomas Moor, Jörg Raisch.
- ↑
- Thomas Moor, Jörg Raisch, Siu O'Young.
**Discrete Supervisory Control of Hybrid Systems by l-Complete Approximations**.*Journal of Discrete Event Dynamic Systems*, 12 (1):83–107, 2002. - Bibtex
**Author :**Thomas Moor, Jörg Raisch, Siu O'Young**Title :**Discrete Supervisory Control of Hybrid Systems by l-Complete Approximations**In :***Journal of Discrete Event Dynamic Systems*,**Date :**2002

- Thomas Moor, Jörg Raisch, Siu O'Young.
- ↑
- Jörg Raisch.
**Discrete Abstractions of Continuous Systems — an Input/Output Point of View**.*Mathematical and Computer Modelling of Dynamical Systems, Special issue on Discrete Event Models of Continuous Systems*, 6 (1):6–29, 2000. - Bibtex
**Author :**Jörg Raisch**Title :**Discrete Abstractions of Continuous Systems — an Input/Output Point of View**In :***Mathematical and Computer Modelling of Dynamical Systems, Special issue on Discrete Event Models of Continuous Systems*,**Date :**2000

- Jörg Raisch.
- ↑
- Dieter Franke, Thomas Moor, Jörg Raisch.
**Supervisory control of switched linear systems**.*at–Automatisierungstechnik, Special Issue on Hybrid Systems I: Analysis and Control*, 48 (9):460–469, 2000. - Bibtex
**Author :**Dieter Franke, Thomas Moor, Jörg Raisch**Title :**Supervisory control of switched linear systems**In :***at–Automatisierungstechnik, Special Issue on Hybrid Systems I: Analysis and Control*,**Date :**2000

- Dieter Franke, Thomas Moor, Jörg Raisch.
- ↑
- Thomas Moor, Jen M. Davoren, Jörg Raisch.
**Strategic refinements in abstraction based supervisory control of hybrid systems**. In*Proc. 6th Int. Workshop on Discrete Event Systems*, pages 329–334, Zaragoza, Spain, 2002. - Bibtex
**Author :**Thomas Moor, Jen M. Davoren, Jörg Raisch**Title :**Strategic refinements in abstraction based supervisory control of hybrid systems**In :**In*Proc. 6th Int. Workshop on Discrete Event Systems*,**Date :**2002

- Thomas Moor, Jen M. Davoren, Jörg Raisch.
- ↑
- Thomas Moor, Jen M. Davoren, Jörg Raisch.
**Learning by Doing — Systematic Abstraction Refinement for Hybrid Control Synthesis**. In*IEE Proc. Control Theory & Applications, Special issue on hybrid systems*, volume 153 pages 591-599, 2006. - Bibtex
**Author :**Thomas Moor, Jen M. Davoren, Jörg Raisch**Title :**Learning by Doing — Systematic Abstraction Refinement for Hybrid Control Synthesis**In :**In*IEE Proc. Control Theory & Applications, Special issue on hybrid systems*,**Date :**2006

- Thomas Moor, Jen M. Davoren, Jörg Raisch.
- ↑
- Thomas Moor, Jen M. Davoren, Jörg Raisch.
**Modular Supervisory Control of a Class of Hybrid Systems in a Behavioural Framework**. In*Proc. European Control Conference ECC2001*, pages 870–875, Porto, Portugal, 2001. - Bibtex
**Author :**Thomas Moor, Jen M. Davoren, Jörg Raisch**Title :**Modular Supervisory Control of a Class of Hybrid Systems in a Behavioural Framework**In :**In*Proc. European Control Conference ECC2001*,**Date :**2001

- Thomas Moor, Jen M. Davoren, Jörg Raisch.
- ↑
- J. Raisch, T. Moor, N. Bajcinca, S. Geist, V. Nenchev.
**Distributed State Estimation for Hybrid and Discrete Event Systems Using l-Complete Approximations**. In*Proceedings of the WODES 2010 - 10th International Workshop on Discrete Event Systems*, pages 139 - 144, 2010. - Bibtex
**Author :**J. Raisch, T. Moor, N. Bajcinca, S. Geist, V. Nenchev**Title :**Distributed State Estimation for Hybrid and Discrete Event SystemsUsing l-Complete Approximations

**In :**In*Proceedings of the WODES 2010 - 10th International Workshop on Discrete**Event Systems*,**Date :**2010

- J. Raisch, T. Moor, N. Bajcinca, S. Geist, V. Nenchev.
- ↑
^{11.0}^{11.1}- Thomas Moor, Jörg Raisch.
, volume 279 of Lecture Notes in Control and Information Sciences, pages 247–265. Springer–Verlag, Berlin, Germany, 2002.**Abstraction based supervisory controller synthesis for high order monotone continuous systems** - Bibtex
**Author :**Thomas Moor, Jörg Raisch**In :**, volume 279 of Lecture Notes in Control and Information Sciences, pages 247–265. Springer–Verlag, Berlin, Germany, 2002.**Abstraction based supervisory controller synthesis for high order monotone continuous systems****Date :**2002

- Thomas Moor, Jörg Raisch.
- ↑
- Alexander Itigin, Jörg Raisch, Thomas Moor, Achim Kienle.
**A Two-Level Hybrid Control Strategy for the Start-up of a Coupled Distillation Plant**. In*Proc. ECC2003 — European Control Conference 2003*, Cambridge, United Kingdom, - Bibtex
**Author :**Alexander Itigin, Jörg Raisch, Thomas Moor, Achim Kienle**Title :**A Two-Level Hybrid Control Strategy for the Start-up of a Coupled Distillation Plant**In :**In*Proc. ECC2003 — European Control Conference 2003*,**Date :**

- Alexander Itigin, Jörg Raisch, Thomas Moor, Achim Kienle.
- ↑
- Daniele Corona, Carla Seatzu, Alessandro Giua, Dmitry Gromov, Eckart Mayer, Jörg Raisch.
**Optimal hybrid control for switched affine systems under safety and liveness constraints**. In*Proc. CACSD — IEEE Int. Conf. on Computer Aided Control Systems Design*, pages 35–40, Taipei, Taiwan, 2004. - Bibtex
**Author :**Daniele Corona, Carla Seatzu, Alessandro Giua, Dmitry Gromov, Eckart Mayer, Jörg Raisch**Title :**Optimal hybrid control for switched affine systems under safety and liveness constraints**In :**In*Proc. CACSD — IEEE Int. Conf. on Computer Aided Control Systems Design*,**Date :**2004

- Daniele Corona, Carla Seatzu, Alessandro Giua, Dmitry Gromov, Eckart Mayer, Jörg Raisch.
- ↑
- Carla Seatzu, Dmitry Gromov, Jörg Raisch, Daniele Corona, Alessandro Giua.
**Optimal Control of Discrete-Time Hybrid Automata under Safety and Liveness Constraints**.*Nonlinear Analysis, Special issue on Hybrid Systems and Applications*, 65 (6):1188–1210, 2006. - Bibtex
**Author :**Carla Seatzu, Dmitry Gromov, Jörg Raisch, Daniele Corona, Alessandro Giua**Title :**Optimal Control of Discrete-Time Hybrid Automata under Safety and Liveness Constraints**In :***Nonlinear Analysis, Special issue on Hybrid Systems and Applications*,**Date :**2006

- Carla Seatzu, Dmitry Gromov, Jörg Raisch, Daniele Corona, Alessandro Giua.