SLIDE 2 5/16/2018 2
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Robust and non-fragile control systems
5
In [1] optimum and robust controllers, designed by using the H2, H1, l1,
and 𝜈 formulations, can produce extremely fragile controllers
Badly chosen optimization criteria => controller parameters that are
m mathematically ill-posed
The stability regions in the parameter space of higher order systems have
“instability holes” and the optimization algorithm can stuff the controller parameter into tight spots close to these holes
Good gain and phase margins are not necessarily reliable indicators of
robustness.
However, poor gain and/or phase margins are accurate indicators of fragileness!
Controller sensitivity which may be important in other non-optimal design
techniques as well.
Example results
5/7/2018
Robust and non-fragile control systems
6 Being situated away from the boundaries of the stability region in the controller parameter space, controllers designed based on the centroid method are both robust and non-fragile.
Non-fragility criterion=minimum distance to the boundary of stability region Non-convex stability region:Finding the center of the largest convex region Optimization problem : 𝐧𝐛𝐲 (𝒔ℂ) =
𝒍 𝒏𝒃𝒚
𝟐 𝑯 (𝒜) Example[2]: IPDT process or a FOPDT Non-fragile PID controller in the viewpoint of the center of mass ~ Ziegler–Nichols
Non-fragile controller design based on centroid of admissible regions
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Robust and non-fragile control systems
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Non-fragile controller design based on pole sensitivity minimization
A new measure: The measure is a weighted sum of a 2-norm of the sensitivity of the individual closed loop system pole/eigenvalues to perturbations in the controller parameters. 𝑦: 𝑗 = 1, … , 𝑜 the controllers parameters
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Robust and non-fragile control systems
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Non-fragile controller design based on pole sensitivity minimization
► Controller fragility will depend upon the particular realization
the controller. ► Handles the fragility problem by minimization of the eigenvalue sensitivity to controller parameter perturbations. ► The eigenvalues pairs closest to the imaginary axis can be weighted more heavily ► Numerical method for obtaining the solution ► Parameter uncertainty is small so that first-order perturbation equations can be obtained
