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4 Qualifying Exam/2 Writing/2. QE State of the Art.md
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@ -20,7 +20,7 @@ Robust control as a field determines how resilient a control system is to a diff
Robustness is dependent on two features: the characteristic to be guaranteed, and the set of reasonably possible perturbed plants $\mathcal{P}$. Usually the characteristic we're interested in is internal stability or performance. The possible set of plants, however, is less straightforward. The set $\mathcal{P}$ can be structured or unstructured. A structured set in this instance can be a discrete number of possible perturbed plants, or possibly a parametric study with a finite number of parameters. Let's consider an example.
Suppose a plant representing a spring-mass-damper system is described as follows @controltutorialsformatlab&simulinkInvertedPendulumSystem:
Suppose a plant representing a spring-mass-damper system is described as follows @controltutorialsformatlab&simulinkInvertedPendulumSystem :
$$ P = \frac{X(s)}{F(s)} = \frac{1}{ms^2 + bs +k}$$
A structured perturbation might take each of these physical parameters $m$, $b$, and $k$ and attribute a likely range or tolerance to their value:
$$ \mathcal{P} = \left\{ \frac{1}{(m+e_m)s^2 + (b+e_b)s + (k + e_k)} \right\} \text{ : }
@ -33,7 +33,10 @@ where $e_m$ is the difference between the nominal mass and the actual as-built m
**Limitation:** *Structured perturbations limit the form of perturbation possible to sample.* Because structured perturbations either are chosen a priori or through a parametric study, the form of possible perturbed plants is limited. Structured perturbations do not allow for unmodelled dynamics to be included as a possible perturbation.
The other type of uncertainty considered is unstructured uncertainty. This type of uncertainty does not assume a form and thus is able to capture unmodelled behavior in its robustness analysis. Unstructured sets are advantageous compared to structured sets for this reason. Robustness with respect to unstructured sets provides a guarantee of resilience to adverse conditions that are unanticipated, or difficult to model. One popular way of implementing unstructured uncertainty is the disk multiplicative perturbation. The disk multiplicative perturbation
The other type of uncertainty considered is unstructured uncertainty. This type of uncertainty does not assume a form and thus is able to capture unmodelled behavior in its robustness analysis. Unstructured sets are advantageous compared to structured sets for this reason. Robustness with respect to unstructured sets provides a guarantee of resilience to adverse conditions that are unanticipated, or difficult to model. One popular way of implementing unstructured uncertainty is the disk multiplicative perturbation. The disk multiplicative perturbation is defined as follows:
$$ \tilde P = (1+\Delta W_2) P $$
Where $\Delta$ is a variable stable transfer function with $||\Delta||_\infty < 1$, and $W_2$ is the uncertainty profile.
(The disk multiplicative perturbation)