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Abstraction-based Hybrid Control Synthesis

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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.

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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].

Distillation column start-up: open-loop. Profiles shown every 5hours.
Distillation column start-up: closed-loop. Profiles shown every 10min.

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].


  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. 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.
  8. 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.
  9. 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.
  10. 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.
  11. 11.0 11.1
    Thomas Moor, Jörg Raisch. Abstraction based supervisory controller synthesis for high order monotone continuous systems, volume 279 of Lecture Notes in Control and Information Sciences, pages 247–265. Springer–Verlag, Berlin, Germany, 2002.
  12. 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,
  13. 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.
  14. 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.

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