Fuller's problem

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Fuller's problem
State dimension: 1
Differential states: 2
Discrete control functions: 1
Interior point equalities: 4


The first control problem with an optimal chattering solution was given by [Fuller1963]Author: A.T. Fuller
Journal: Journal of Electronics and Control
Pages: 63--71
Title: Study of an optimum nonlinear control system
Volume: 15
Year: 1963
Link to Google Scholar
. An optimal trajectory does exist for all initial and terminal values in a vicinity of the origin. As Fuller showed, this optimal trajectory contains a bang-bang control function that switches infinitely often.

The mathematical equations form a small-scale ODE model. The interior point equality conditions fix initial and terminal values of the differential states.

Mathematical formulation

For t \in [t_0, t_f] almost everywhere the mixed-integer optimal control problem is given by


\begin{array}{llcl}
 \displaystyle \min_{x, w} & \int_{0}^{1} x_0^2 & \; \mathrm{d} t \\[1.5ex]
 \mbox{s.t.} & \dot{x}_0 & = & x_1, \\
 & \dot{x}_1 & = & 1 - 2 \; w, \\[1.5ex]
 & x(0) &=& x_S, \\
 & x(t_f) &=& x_T, \\
 & w(t) &\in&  \{0, 1\}.
\end{array}

Parameters

We use x_S = x_T = (0.01, 0)^T.

Reference Solutions

Solutions obtained with jModelica

The solution found for the relaxed Fuller's problem with jModelica using the solver Ipopt (with the linear solver MA27) is obtained with 12 iterations and the objective is 1.5296058259296967e-05.

a graph with the optimal solution of the Fuller's Problem with Optimica and Ipopt
Solution of the Fuller's Problem with Optimica and Ipopt




Source Code

Miscellaneous and further reading

An extensive analytical investigation of this problem and a discussion of the ubiquity of Fuller's problem can be found in [Zelikin1994]Address: Basel Boston Berlin
Author: Zelikin, M.I.; Borisov, V.F.
Publisher: Birkh\"auser
Title: Theory of chattering control with applications to astronautics, robotics, economics and engineering
Year: 1994
Link to Google Scholar
, a recent investigation of chattering controls in relay feedback systems in [Johansson2002]Author: K.H. Johansson; Barabanov, A.E.; Astr\"om, K.J.
Journal: IEEE Transactions on Automatic Control
Number: 9
Pages: 1414--1423
Title: Limit Cycles with Chattering in Relay Feedback Systems
Volume: 47
Year: 2002
Link to Google Scholar
.

References

[Fuller1963]A.T. Fuller (1963): Study of an optimum nonlinear control system. Journal of Electronics and Control, 15, 63--71Link to Google Scholar
[Johansson2002]K.H. Johansson; Barabanov, A.E.; Astr\"om, K.J. (2002): Limit Cycles with Chattering in Relay Feedback Systems. IEEE Transactions on Automatic Control, 47, 1414--1423Link to Google Scholar
[Zelikin1994]Zelikin, M.I.; Borisov, V.F. (1994): Theory of chattering control with applications to astronautics, robotics, economics and engineering. (%edition%). Birkh\"auser, Basel Boston Berlin, %pages%Link to Google Scholar