The aim of this work is to design controllers through explicit minimization of the settling time of a closed-loop response, by using a class of methods adequate for this objective. To the best of our knowledge, all the methods available in the literature do not minimize directly the settling time but only related objective functions. Indeed, the settling time objective function is not only non-smooth but also discontinuous. Therefore we propose to use direct search methods, which do not use any gradient information. An important reason is a recent result that some direct search methods are guaranteed to convergence on such discontinuous objective functions. The proposed approach is self-standing but can also improve the solutions obtained with the alternatives of the literature, which lead to good solutions but suboptimal in terms of the settling time. Note also that this approach is very flexible and can be adapted to a broad range of objectives as well as nonlinear systems or controllers, as long as the time response can be simulated.
arXiv:1109.5966v3 [math.OC] 6 Dec 2011
Minimum settling time control design through
direct search optimization
Emile Simon
November 19, 2018
The aim of this very brief third version is to present succinctly, but clearly
with two animations, two optimization results from the second version of the
paper available on http://arxiv.org/abs/1109.5966v2 which can still be con-
sulted for more details. A future and much improved version of the paper
will be deposited on arXiv at the beginning of 2012.
The objective considered here is to optimize PID coefficients x = [Kp, Ki, Kd]
to minimize the following objective function:
f(x) = the rise time (= time needed for the response z(x, t) to reach
0.98) divided by 100 (the fixed maximum time value for which the response
is evaluated)
+ the maximum deviation of z(x, t) (out of the following settling range:
zmax(t) = 1.02 for t > 0 and zmin(t) = 0.98 for t > rise time).
1
First figure of unit step response optimization, starting from PID coeffi-
cients obtained by Ziegler-Nichols:
Press play to start the animation.
NB/ The response z(x, t) is drawn in green when the corresponding ob-
jective value f(x) is better than best value previously found by the direct
search optimization method, and in red otherwise.
The horizontal black
dashed lines represent the settling range defined above.
2
0
20
40
60
80
100
0
0.5
1
1.5
time t [s]
time response z(t)
Kp = 4.059, Ki = 0.4388, Kd = 9.386 : f(x) = 0.5421
Second figure of unit step response optimization, starting from unstabi-
lizing random PID coefficients:
time [s]
Both optimziations lead to the same solution, with rise time around 11.94s
and maximum deviation almost zero (can be fine-tuned with another objec-
tive function, the settling time, but this will have an insignificant effect on
the solutions here).
3
0
20
40
60
80
100
−6
−4
−2
0
2
4
6
Kp = 0.4395, Ki = 0.5447, Kd = 0.7855 : f(x) = 6.835
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