Fuller's problem

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 <bibref>Fuller1963</bibref>. 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

$\begin{array}{ll}\underset{x,w}{\min }& \underset{0}{\overset{1}{\int }}{x}_0^2\mathrm{d}t\\ \text{s.t.}& {\dot{x}}_0(t)& =& x_1(t),\\ & {\dot{x}}_1(t)& =& 1-2w(t),\\ & x(0)& =& x_S,\\ & x(t_f)& =& x_T,\\ & w(t)& \in & \{0,1\}.\end{array}$

Parameters

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

Reference Solutions

Solutions obtained with optimica

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

Solution of the Fuller's Problem with Optimica and Ipopt

Source Code

• C code at Fuller's Problem (C)
• optimica at Fuller's Problem (optimica)

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 <bibref>Zelikin1994</bibref>, a recent investigation of chattering controls in relay feedback systems in <bibref>Johansson2002</bibref>.

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%