Robust design optimization and robust optimal control
Introduction
Practical application of the numerical optimization results is complicated
by the fact that any intricate technical system is a stochastic system
and characteristics of this system have a probabilistic nature. We would
like to emphasize the point that, speaking of stochastic properties of
a technical system within the frame of optimization tasks, we imply a
system's essential parameters spread which occurs during the production
stage despite the up-to-date level of technology. Random deviations of
the system's parameters lead to a random change in system's efficiency.
An efficiency extreme value, obtained during the optimization problem
solving in a traditional (deterministic) setting, is simply a maximum
attainable value and can be considered as just conventional optimum from
the point of view of its practical realization. Thus, one can consider
two different types of optimization criteria. One of them is an ideal
efficiency which can be achieved under the conditions of absolutely precise
practical replication of the preset parameters of the system under consideration.
Other optimization criteria are of probabilistic nature. For example:
mathematical expectation of the efficiency; total probability of assuring
the preset constraints; variance of the efficiency and so on
It is evident that the extreme of one of these criteria doesn't guarantee
the assurance of the high level of another one. Even more, these criteria
may be contradicting each other. Thus, in this case we have a multicriteria
optimization problem.
Our concept
Our concept of robust design optimization and robust optimal control
allows determining the optimal practical technical solution that could
be implemented with the high probability for the given technology level
of the production plants. Many current probabilistic approaches either
employ estimation of probabilistic efficiency criteria only at the stage
of analysis of obtaining deterministic solution, or use significantly
simplified estimates of probabilistic criteria during optimization process.
The distinctive feature of our approach is that during robust design optimization
we solve the optimization problem using direct stochastic formulation,
when estimation of probabilistic criteria is accomplished at each iteration.
This procedure reliably produces truly robust optimal solution. High efficiency
of the robust design optimization is provided by the capabilities of IOSO
algorithms to solve stochastic optimization problems with large level
of noise.
Our robust design optimization concept provides considerable (several
orders of magnitude) reduction in cost and time during development of
the new highly efficient systems. This concept also considerably (several
times) reduces risk associated with the new technical solutions. For example,
during optimal calibration of the automotive engine the development time
was reduced 5 times, during development of the new axial compressor
more than 200 times. The most important feature of IOSO technology is
the ability to solve robust design optimization problems with the large
number of variables (hundreds) and efficiency criteria (dozens).
Real-life example
MULTICRITERIA ROBUST DESIGN OF THE MULTISTAGE AXIAL FLOW COMPRESSOR
Problem features: 140 design variables, 2 criteria.
The trade-off area:
the compressor efficiency from 0.87 up to 0.89 under
the implementation probability from 0.97 down to 0.56 !
The solution N1 could be implemented with a high probability, but it
has low efficiency. Solution N10 has a high efficiency, but has a low
probability of practical implementation. The solutions N4 and N5 have
very similar efficiencies and probability of implementation, but N4 is
better, because it has lower level of efficiency distribution. It is this
solution (N4) that should be implemented in practice.
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