Difference between revisions of "Fuller's problem"
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\begin{array}{llcl} | \begin{array}{llcl} | ||
| − | \displaystyle \min_{x, w} & \int_{0}^{1} x_0^2 \; \mathrm{d} t \\[1.5ex] | + | \displaystyle \min_{x, w} & \int_{0}^{1} x_0^2 & \; \mathrm{d} t \\[1.5ex] |
\mbox{s.t.} & \dot{x}_0 & = & x_1, \\ | \mbox{s.t.} & \dot{x}_0 & = & x_1, \\ | ||
& \dot{x}_1 & = & 1 - 2 \; w, \\[1.5ex] | & \dot{x}_1 & = & 1 - 2 \; w, \\[1.5ex] | ||
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== Reference Solutions == | == Reference Solutions == | ||
| − | ===Solutions obtained with | + | ===Solutions obtained with jModelica=== |
| − | The solution found for the relaxed Fuller's problem with | + | 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. |
[[File:Fullerspng.png|left|200px|thumb|alt=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]] | [[File:Fullerspng.png|left|200px|thumb|alt=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]] | ||
<br /> | <br /> | ||
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</gallery> | </gallery> | ||
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| + | |||
| + | == Variants == | ||
| + | |||
| + | * Dropped terminal constraints with penalized deviation as additional Mayer term, see [[Fuller's initial value problem]], | ||
| + | * Several binary controls and dropped terminal constraints with penalized deviation as additional Mayer term, see [[Fuller's initial value multimode problem]], | ||
== Source Code == | == Source Code == | ||
* [[:Category:Muscod | Muscod code]] at [[Fuller's Problem (Muscod)]] | * [[:Category:Muscod | Muscod code]] at [[Fuller's Problem (Muscod)]] | ||
| − | * [[:Category: | + | * [[:Category:JModelica | JModelica code]] at [[Fuller's Problem (JModelica)]] |
== Miscellaneous and further reading == | == Miscellaneous and further reading == | ||
Latest revision as of 23:01, 8 January 2018
| 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
. 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.
Contents
Mathematical formulation
For ![t \in [t_0, t_f]](https://mintoc.de/images/math/5/5/8/55823791d9100bcb5461801aff4f6edd.png)
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}](https://mintoc.de/images/math/a/2/6/a26a83d77c0b2eb335b75f9d293664d8.png)
Parameters
We use 
.
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.
Variants
- Dropped terminal constraints with penalized deviation as additional Mayer term, see Fuller's initial value problem,
- Several binary controls and dropped terminal constraints with penalized deviation as additional Mayer term, see Fuller's initial value multimode problem,
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, 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.
References
| [Fuller1963] | A.T. Fuller (1963): Study of an optimum nonlinear control system. Journal of Electronics and Control, 15, 63--71 | |
| [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--1423 | |
| [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% | |
