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Control of Hybrid Systems

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Hybrid systems are characterised by the interaction of continuous and discrete-event components. Such systems are ubiquitous, and from an engineering point of view, the systematic design of hybrid control systems is of particular importance. Our research has been mostly concerned with the following specific aspects.

Abstraction-based Hybrid Control Synthesis

The systematic design of hybrid control systems 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 many researchers. The employed approximation needs to be safe, meaning that a 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 improving 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 "strongest 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].

While strongest $l$-complete approximation was originally developped as a refinable abstraction technique for systems that are time invariant in a behavioral sense, this can be extended to a larger class of systems to provide strongest asynchronous $l$-complete approximations as finite-state abstractions [7]. In [8], it was shown that there exist simulation relations between the ``natural realizations of these abstractions and the underlying infinite state model of the continuous component. Bisimulation relations only exist if, in addition to equality of the respective external behaviors, the underlying state space exhibits an additional property. To obtain the latter result, a notion of simulation relations for behavioral systems with multiple time scales from [9] was used.

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 [10] [11], 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 [12], 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.

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 [13] 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 [14].

Finite state symbolic abstractions can also be employed when specifications are not purely logical but contain temporal requirements that can, e.g., be expressed as syntactically co-safe Linear Temporal Logic (scLTL) formulae. As an example, we addressed the time-optimal control of a robot that has to collect and move a finite number of objects to particular spots in space, while maintaining given temporal logic constraints [15]. We also explored the case when the locations of the objects are a-priori unknown and the robot has to discover them with a limited sensing range, pick them up and move them to a designated spot [16].

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. 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. By applying a generalized Lagrange multiplier rule[17], it was possible 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 [18] [19] .

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 system with autonomous switching where jump parameters are considered as a part of the control. For this, a new set of necessary conditions of optimality can be developed [20].

Finally, [21] discusses how a class of optimal control problems for switched systems that are affine in the control input can be treated as convex problems.

Switched Linear Systems

Switched linear systems are an important subclass of hybrid systems and consist of a finite number of linear time-invariant systems with continuous dynamics and some switching mechanism that orchestrates between them. Dynamical systems of this class can be found in various fields of engineering applications, such as aviation technology, power electronics, automotive engineering or power generation. One of the most important system properties is of course stability. We have studied analytical tools for investigating stability of switched linear systems [22] [23] [24]. Quadratic stability of a class of switched linear systems associated with symmetric transfer function matrices was treated in [25], [26]. In the context of control design, in [27] [28] [29], the concept of common left eigenstructure assignment was used to achieve exponential stabilisation of switched linear systems with single and multiple inputs.

People Involved

  • Anne-Kathrin Schmuck (now with (now with Max-Planck-Institut für Softwaresysteme, Kaiserslautern)
  • Dmitry Gromov (now with St. Petersburg State University)
  • Vladislav Nenchev (now with BMW)
  • Naim Bajcinca (now with TU Kaiserlautern)
  • Vadim Azhmyakov (now with Universidad de Medellin)
  • Sid Ahmed Attia (now with GE)
  • Kai Wulff (now with TU Ilmenau)
  • Ishan Pendharkar (now with Bombardier)
  • Yashar Kouhi (now with TU Freiberg)
  • Jörg Raisch


  • Robert Shorten (University College Dublin)
  • Vadim Azhmyakov (Universidad de Medellin)
  • Thomas Moor (Universität Erlangen)
  • Jen Davoren (University of Melbourne)


  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. A.-K. Schmuck, J. Raisch. Asynchronous l-Complete Approximations. Systems and Control Letters, pages 67-75, 2014.
  8. A.-K. Schmuck, J. Raisch. Constructing (Bi)Similar Finite State Abstractions using Asynchronous l-Complete Approximations. pages 6744–6751, 2014. 53rd IEEE Conference on Decision and Control, Los Angeles, USA.
  9. A.-K. Schmuck, J. Raisch. Simulation and Bisimulation over Multiple Time Scales in a Behavioral Setting. In Proceedings of the 22nd Mediterranean Conference on Control and Automation, Palermo, Italy, page 517–524, 2014.
  10. 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.
  11. 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.
  12. 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.
  13. 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.
  14. 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.
  15. V. Nenchev, C. Belta, J. Raisch. Optimal motion planning with temporal logic and switching constraints. In 14th European Control Conf. (ECC'15), Linz, Austria, pages 1135–1140, 2015.
  16. V. Nenchev, J. Raisch. Towards time-optimal exploration and control by an autonomous robot. In 21st Mediterranean Conference on Control and Automation (MED'13), pages 1236-1241, Platanias-Chania, Crete – Greece, June 2013.
  17. V. Azhmyakov, S. A. Attia, D. Gromov, J. Raisch. Necessary optimality conditions for a class of hybrid optimal control problems, volume 4416 of Lecture Notes in Computer Science (LNCS), pages 637-640. Springer-Verlag, 2007.
  18. 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.
  19. 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.
  20. V. Azhmyakov, S. A. Attia, J. Raisch. On the Maximum Principle for Impulsive Hybrid System, volume 4981 of Lecture Notes in Computer Science (LNCS), pages 30-42. Springer-Verlag, 2008.
  21. V. Azhmyakov, M.V. Basin, J. Raisch. A Proximal Point Based Approach to Optimal Control of Affine Switched Systems. Discrete Event Dynamic Systems, 22 (1):61–81, 2012.
  22. Ishan Pendharkar, Kai Wulff, Jörg Raisch. On the construction of quadratically stable switched linear systems with multiple component systems. In Proceedings of the Conference on Decision and Control, 2007, pages 6274 - 6279,
  23. Ishan Pendharkar, Kai Wulff, Jörg Raisch. A behavioral-theoretic approach to quadratic stability of switched linear systems. In 13th IEEE IFAC International Conference on Methods and Models in Automation and Robotics, Szczecin, Poland, 2007.
  24. I. Pendharkar, K. Wulff, J. Raisch. On stability of switched differential algebraic systems – conditions and applications. In Proc. IFAC World Congress, Seoul, 2008.
  25. Kouhi, Y., Bajcinca, N.,, Raisch, J.,, Shorten, R.. A new stability result for switched linear systems. In Proc. of the European Control Conference, pages 2152-2156, 2013.
  26. Yashar Kouhi, Naim Bajcinca, Jörg Raisch, Robert Shorten. On the quadratic stability of switched linear systems associated with symmetric transfer function matrices. Automatica, 50 (11):2872 - 2879, 2014.
  27. Kouhi, Y., Bajcinca, N.. On the left eigenstructure assignment and state feedback design. In Proc. of the American Control Conference, pages 4326-4327, 2011.
  28. Kouhi, Y., Bajcinca, N.. Robust control of switched linear systems. In Proc. Conf. on Decision and Control and European Control Conference, 2011.
  29. Kouhi, Y., Bajcinca, N.. Nonsmooth control design for stabilizing switched linear systems by left eigenstructure assignment. In Proc. IFAC World Congress, pages 380-385, 2011.

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