Optimal Control of Hybrid Systems

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.

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. ↑ 1.0 1.1
    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.
    Bibtex
    Author : V. Azhmyakov, S. A. Attia, D. Gromov, J. Raisch
    In : 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.
    Date : 2007
     
  2. ↑ 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,
          pages 1408-1413, 2007.
    Date : 2007
     | PDF 
  3. ↑ 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,
          20 pages 215 - 231, 2010.
    Date : 2010
     | PDF 
  4. ↑
    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.
    Bibtex
    Author : V. Azhmyakov, S. A. Attia, J. Raisch
    In : On the Maximum Principle for Impulsive Hybrid System, volume 4981 of Lecture Notes in Computer Science (LNCS), pages 30-42. Springer-Verlag, 2008.
    Date : 2008
     | PDF 
  5. ↑
    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,
          pages 89-94, 2006.
    Date : 2006
     
  6. ↑
    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
    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,
          2011.
    Date : 2011
     | PDF | DOI 
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