188 lines
7.8 KiB
Markdown
188 lines
7.8 KiB
Markdown
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title = "Guidelines for the selection of weighting functions for h-infinity control"
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author = ["Thomas Dehaeze"]
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draft = false
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Tags
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: [H Infinity Control]({{< relref "h_infinity_control" >}})
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Reference
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: ([Bibel and Malyevac 1992](#org50b9640))
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Author(s)
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: Bibel, J. E., & Malyevac, D. S.
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Year
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: 1992
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## Properties of feedback control {#properties-of-feedback-control}
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<a id="org1554fcc"></a>
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{{< figure src="/ox-hugo/bibel92_control_diag.png" caption="Figure 1: Control System Diagram" >}}
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From the figure [1](#org1554fcc), we have:
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\begin{align\*}
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y(s) &= T(s) r(s) + S(s) d(s) - T(s) n(s)\\\\\\
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e(s) &= S(s) r(s) - S(s) d(s) - S(s) n(s)\\\\\\
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u(s) &= S(s)K(s) r(s) - S(s)K(s) d(s) - S(s)K(s) n(s)
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\end{align\*}
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With the following definitions
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- \\(L(s) = G(s)K(s)\\) is the **loop transfer matrix**
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- \\(S(s) = [I+G(s)K(s)]^{-1}\\) is the **Sensitivity** function matrix
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- \\(T(s) = [I+G(s)K(s)]^{-1}G(s)K(s)\\) is the **Transmissibility** function matrix
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<div class="cbox">
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<div></div>
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\\[ S(s) + T(s) = 1 \\]
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</div>
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<div class="cbox">
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<div></div>
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- **Command following**: \\(S=0\\) and \\(T=1\\) => large gains
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- **Disturbance rejection**: \\(S=0\\) => large gains
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- **Sensor noise attenuation**: \\(T\\) small where the noise is concentrated
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- **Control Sensitivity minimization**: \\(K S\\) small
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- **Robustness to modeling errors**: \\(T\\) small in the frequency range of the expected model undertainties
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</div>
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## SISO tradeoff {#siso-tradeoff}
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We want \\(S\\) small for command following and disturbance rejection.
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We want \\(T\\) small to remain insensitive to sensor noise and modeling errors and to reduce control sensitivity.
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However we cannot keep both \\(S\\) and \\(T\\) small as \\(S(s)+T(s)=1\\).
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We must determine some **tradeoff** between the sensitivity and the complementary sensitivity functions.
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Usually, reference signals and disturbances occur at low frequencies, while noise and modeling errors are concentrated at high frequencies. The tradeoff, in a SISO sense, is to make \\(|S(j\omega)|\\) small as low frequencies and \\(|T(j\omega)|\\) small at high frequencies.
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## \\(H\_\infty\\) and weighting functions {#h-infty--and-weighting-functions}
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<div class="cbox">
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<div></div>
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\\(\mathcal{H}\_\infty\\) control is a design technique with a state-space computation solution that utilizes frequency-dependent weighting functions to tune the controller's performance and robustness characteristics.
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</div>
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<a id="orgce70b5f"></a>
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{{< figure src="/ox-hugo/bibel92_general_plant.png" caption="Figure 2: \\(\mathcal{H}\_\infty\\) control framework" >}}
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New design framework (figure [2](#orgce70b5f)): \\(P(s)\\) is the **generalized plant** transfer function matrix:
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- \\(w\\): exogenous inputs
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- \\(z\\): regulated performance output
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- \\(u\\): control inputs
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- \\(y\\): measured output variables
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The plant \\(P\\) has two inputs and two outputs, it can be decomposed into four sub-transfer function matrices:
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\\[P = \begin{bmatrix}P\_{11} & P\_{12} \\ P\_{21} & P\_{22} \end{bmatrix}\\]
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## Lower Linear Fractional Transformation {#lower-linear-fractional-transformation}
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The transformation from the input \\(w\\) to the output \\(z\\), \\(T\_{zw}\\) is called the **Lower Linear Fractional Transformation** \\(F\_l (P, K)\\).
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<div class="cbox">
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<div></div>
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\\[T\_{zw} = F\_l (P, K) = P\_{11} + P\_{12}K (I-P\_{22})^{-1} P\_{21}\\]
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</div>
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The \\(H\_\infty\\) control problem is to find a controller that minimizes \\(\\| T\_{zw} \\|\_\infty\\) over the space of all realizable controllers \\(K(s)\\) that stabilize the closed-loop system.
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## Weights for inputs/outputs signals {#weights-for-inputs-outputs-signals}
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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)).
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<a id="orgca469d2"></a>
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{{< figure src="/ox-hugo/bibel92_hinf_weights.png" caption="Figure 3: Input and Output weights in \\(\mathcal{H}\_\infty\\) framework" >}}
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The weights on the input and output variables are selected to reflect the spatial and **frequency dependence** of the respective signals and performance specifications.
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These inputs and output weighting functions are defined as rational, stable and **minimum-phase transfer function** (no poles or zero in the right half plane).
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## General Guidelines for Weight Selection: \\(W\_S\\) {#general-guidelines-for-weight-selection--w-s}
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\\(W\_S\\) is selected to reflect the desired **performance characteristics**.
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The sensitivity function \\(S\\) should have low gain at low frequency for good tracking performance and high gain at high frequencies to limit overshoot.
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We have to select \\(W\_S\\) such that \\({W\_S}^-1\\) reflects the desired shape of \\(S\\).
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<div class="cbox">
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<div></div>
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- **Low frequency gain**: set to the inverse of the desired steady state tracking error
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- **High frequency gain**: set to limit overshoot (\\(0.1\\) to \\(0.5\\) is a good compromise between overshoot and response speed)
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- **Crossover frequency**: chosen to limit the maximum closed-loop time constant (\\(\omega\_c \approx 1/\tau\\))
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</div>
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## General Guidelines for Weight Selection: \\(W\_T\\) {#general-guidelines-for-weight-selection--w-t}
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We want \\(T\\) near unity for good tracking of reference and near zero for noise suppresion.
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<div class="cbox">
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<div></div>
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A high pass weight is usualy used on \\(T\\) because the noise energy is mostly concentrated at high frequencies. It should have the following characteristics:
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- The **crossover frequency** is chosen to **limit the closed-loop bandwidth**
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- The **high frequency gain** is set high to proide **sensor noise rejection** and high frequency gain attenuation
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</div>
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When using both \\(W\_S\\) and \\(W\_T\\), it is important to make sure that the magnitude of theise weights at the crossover frequency is less that one to not violate \\(S+T=1\\).
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## Unmodeled dynamics weighting function {#unmodeled-dynamics-weighting-function}
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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)).
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<a id="org96cc166"></a>
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{{< figure src="/ox-hugo/bibel92_unmodeled_dynamics.png" caption="Figure 4: Unmodeled dynamics model" >}}
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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).
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<a id="orgab966f3"></a>
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{{< figure src="/ox-hugo/bibel92_weight_dynamics.png" caption="Figure 5: Example of unmodeled dynamics weight" >}}
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## Inputs and Output weighting function {#inputs-and-output-weighting-function}
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It is possible to **weight the control input and actuator rate**.
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This is used to **prevent actuator saturation** and **limit amplification of sensor noise signals** on the control input signal.
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Typically actuator input weights are constant over frequency and set at the inverse of the saturation limit.
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## Order of the weighting functions {#order-of-the-weighting-functions}
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**The order of the optimal controller is equal to the order of the nominal plant model plus the order of the weights**. The complexity of the controller is increase as the order of the weights increases.
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**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.
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## Bibliography {#bibliography}
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<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.
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