3.2 KiB
| title | allDay | date | endDate | completed | type |
|---|---|---|---|---|---|
| State of the Art | true | 2024-10-03 | 2024-10-10 | null | single |
1.25 Pages TARGET: ~700 words
Outline
- Robust Control
- How is robust control validation done?
- How do people generate unstructured perturbations?
Take 1
Attempt
Robust control as a field determines how resilient a control system is to a difference in plant dynamics for a given characteristic. In a real system, there will always be some inaccuracy in the model of plant dynamics, disturbances, or other noise. These unmodeled features will affect plant behavior if they are not anticipated. Robust control gives us tools to design for these perturbations proactively. We can design characteristics such as performance and stability to guarantee as 'robust'.
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:
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:
$$ \mathcal{P} = \left{ \frac{1}{(m+e_m)s^2 + (b+e_b)s + (k + e_k)} \right} \text{ : }
\matrix{m_{min} \leq m+e_m \leq m_{max} \
b_{min} \leq b +e_b \leq b_{max} \
k_{min} \leq k +e_k \leq k_{max}} $$
where e_m is the difference between the nominal mass and the actual as-built mass. e_b and e_k follow similar logic. Structured perturbations are easy to use to create perturbed plants: simply pick values for e_m, e_b, and e_k that are within the allowable bounds and plug them in to create a new, perturbed transfer function.
Structured perturbations also require a lot of leg work to create
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 disk multiplicative perturbation)
(Explain how actually getting to W_2 isn't really trivial).