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: [H Infinity Control]({{< relref "h_infinity_control" >}}) : [H Infinity Control]({{< relref "h_infinity_control" >}})
Reference Reference
: <sup id="5b41da575e27e6e86f1a1410a0170836"><a class="reference-link" href="#bibel92_guidel_h" title="Bibel \&amp; Malyevac, Guidelines for the selection of weighting functions for H-infinity control, NAVAL SURFACE WARFARE CENTER DAHLGREN DIV VA, (1992).">(Bibel \& Malyevac, 1992)</a></sup> : ([Bibel and Malyevac 1992](#org47391fd))
Author(s) Author(s)
: Bibel, J. E., & Malyevac, D. S. : Bibel, J. E., & Malyevac, D. S.
@ -19,11 +19,11 @@ Year
## Properties of feedback control {#properties-of-feedback-control} ## Properties of feedback control {#properties-of-feedback-control}
<a id="org5999225"></a> <a id="org55b0783"></a>
{{< figure src="/ox-hugo/bibel92_control_diag.png" caption="Figure 1: Control System Diagram" >}} {{< figure src="/ox-hugo/bibel92_control_diag.png" caption="Figure 1: Control System Diagram" >}}
From the figure [1](#org5999225), we have: From the figure [1](#org55b0783), we have:
\begin{align\*} \begin{align\*}
y(s) &= T(s) r(s) + S(s) d(s) - T(s) n(s)\\\\\\ 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> </div>
<a id="org4e0009c"></a> <a id="orgbbca2ea"></a>
{{< figure src="/ox-hugo/bibel92_general_plant.png" caption="Figure 2: \\(\mathcal{H}\_\infty\\) control framework" >}} {{< figure src="/ox-hugo/bibel92_general_plant.png" caption="Figure 2: \\(\mathcal{H}\_\infty\\) control framework" >}}
New design framework (figure [2](#org4e0009c)): \\(P(s)\\) is the **generalized plant** transfer function matrix: New design framework (figure [2](#orgbbca2ea)): \\(P(s)\\) is the **generalized plant** transfer function matrix:
- \\(w\\): exogenous inputs - \\(w\\): exogenous inputs
- \\(z\\): regulated performance output - \\(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} ## 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](#orgdd8fae0)). 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](#org75a0ac3)).
<a id="orgdd8fae0"></a> <a id="org75a0ac3"></a>
{{< figure src="/ox-hugo/bibel92_hinf_weights.png" caption="Figure 3: Input and Output weights in \\(\mathcal{H}\_\infty\\) framework" >}} {{< 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} ## 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](#org0d13a20)). 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](#orgd3e0294)).
<a id="org0d13a20"></a> <a id="orgd3e0294"></a>
{{< figure src="/ox-hugo/bibel92_unmodeled_dynamics.png" caption="Figure 4: Unmodeled dynamics model" >}} {{< 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](#org45b0983). 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](#org6d5884c).
<a id="org45b0983"></a> <a id="org6d5884c"></a>
{{< figure src="/ox-hugo/bibel92_weight_dynamics.png" caption="Figure 5: Example of unmodeled dynamics weight" >}} {{< figure src="/ox-hugo/bibel92_weight_dynamics.png" caption="Figure 5: Example of unmodeled dynamics weight" >}}
@ -181,5 +181,7 @@ 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. **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
<a class="bibtex-entry" id="bibel92_guidel_h">Bibel, J. E., & Malyevac, D. S., *Guidelines for the selection of weighting functions for h-infinity control* (1992).</a> [↩](#5b41da575e27e6e86f1a1410a0170836) ## Bibliography {#bibliography}
<a id="org47391fd"></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.