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@@ -8,7 +8,7 @@ Tags
: [H Infinity Control]({{< relref "h_infinity_control" >}})
Reference
: ([Bibel and Malyevac 1992](#org50b9640))
: ([Bibel and Malyevac 1992](#org395ccd3))
Author(s)
: Bibel, J. E., & Malyevac, D. S.
@@ -19,11 +19,11 @@ Year
## Properties of feedback control {#properties-of-feedback-control}
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{{< figure src="/ox-hugo/bibel92_control_diag.png" caption="Figure 1: Control System Diagram" >}}
From the figure [1](#org1554fcc), we have:
From the figure [1](#orgd464a3c), we have:
\begin{align\*}
y(s) &= T(s) r(s) + S(s) d(s) - T(s) n(s)\\\\\\
@@ -77,11 +77,11 @@ Usually, reference signals and disturbances occur at low frequencies, while nois
</div>
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{{< figure src="/ox-hugo/bibel92_general_plant.png" caption="Figure 2: \\(\mathcal{H}\_\infty\\) control framework" >}}
New design framework (figure [2](#orgce70b5f)): \\(P(s)\\) is the **generalized plant** transfer function matrix:
New design framework (figure [2](#orgf088f75)): \\(P(s)\\) is the **generalized plant** transfer function matrix:
- \\(w\\): exogenous inputs
- \\(z\\): regulated performance output
@@ -108,9 +108,9 @@ The \\(H\_\infty\\) control problem is to find a controller that minimizes \\(\\
## Weights for inputs/outputs signals {#weights-for-inputs-outputs-signals}
Since \\(S\\) and \\(T\\) cannot be minimized together at all frequency, **weights are introduced to shape the solutions**. Not only can \\(S\\) and \\(T\\) be weighted, but other regulated performance variables and inputs (figure [3](#orgca469d2)).
Since \\(S\\) and \\(T\\) cannot be minimized together at all frequency, **weights are introduced to shape the solutions**. Not only can \\(S\\) and \\(T\\) be weighted, but other regulated performance variables and inputs (figure [3](#orgff0b295)).
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{{< figure src="/ox-hugo/bibel92_hinf_weights.png" caption="Figure 3: Input and Output weights in \\(\mathcal{H}\_\infty\\) framework" >}}
@@ -154,15 +154,15 @@ When using both \\(W\_S\\) and \\(W\_T\\), it is important to make sure that the
## Unmodeled dynamics weighting function {#unmodeled-dynamics-weighting-function}
Another method of limiting the controller bandwidth and providing high frequency gain attenuation is to use a high pass weight on an **unmodeled dynamics uncertainty block** that may be added from the plant input to the plant output (figure [4](#org96cc166)).
Another method of limiting the controller bandwidth and providing high frequency gain attenuation is to use a high pass weight on an **unmodeled dynamics uncertainty block** that may be added from the plant input to the plant output (figure [4](#orgc150230)).
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{{< figure src="/ox-hugo/bibel92_unmodeled_dynamics.png" caption="Figure 4: Unmodeled dynamics model" >}}
The weight is chosen to cover the expected worst case magnitude of the unmodeled dynamics. A typical unmodeled dynamics weighting function is shown figure [5](#orgab966f3).
The weight is chosen to cover the expected worst case magnitude of the unmodeled dynamics. A typical unmodeled dynamics weighting function is shown figure [5](#org42e3b7d).
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{{< figure src="/ox-hugo/bibel92_weight_dynamics.png" caption="Figure 5: Example of unmodeled dynamics weight" >}}
@@ -182,6 +182,7 @@ Typically actuator input weights are constant over frequency and set at the inve
**The order of the weights should be kept reasonably low** to reduce the order of th resulting optimal compensator and avoid potential convergence problems in the DK interactions.
## Bibliography {#bibliography}
<a id="org50b9640"></a>Bibel, John E, and D Stephen Malyevac. 1992. “Guidelines for the Selection of Weighting Functions for H-Infinity Control.” NAVAL SURFACE WARFARE CENTER DAHLGREN DIV VA.
<a id="org395ccd3"></a>Bibel, John E, and D Stephen Malyevac. 1992. “Guidelines for the Selection of Weighting Functions for H-Infinity Control.” NAVAL SURFACE WARFARE CENTER DAHLGREN DIV VA.