Update Content - 2020-09-22
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Resources:
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- ([Skogestad and Postlethwaite 2007](#org140f9cc))
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- ([Toivonen 2002](#orgc1385a9))
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- ([Zhang 2011](#org8471dd8))
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- ([Skogestad and Postlethwaite 2007](#orgf3c8b69))
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- ([Toivonen 2002](#orgb2755d2))
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- ([Zhang 2011](#org3e1b2ef))
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## \\(\mathcal{H}\_\infty\\) Norm {#mathcal-h-infty--norm}
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## Definition {#definition}
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A norm of \\(e\\) (which may be a vector, matrix, signal of system) is a real number, denoted \\(\\|e\\|\\), that satisfies the following properties:
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1. Non-negative: \\(\\|e\\| \ge 0\\)
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2. Positive: \\(\\|e\\| = 0 \Longleftrightarrow e = 0\\)
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3. Homogeneous: \\(\\|\alpha \cdot e\\| = |\alpha| \cdot \\|e\\|\\) for all complex scalars \\(\alpha\\)
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4. Triangle inequality: \\(\\|e\_1 + e\_2\\| \le \\|e\_1\\| + \\|e\_2\\|\\)
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## Vector Norms {#vector-norms}
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- **Vector 1-norm (Sum Norm)**:
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\\[ \\|a\\|\_1 \triangleq \sum\_i |a\_i| \\]
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- **Vector 2-norm (Euclidean Norm)**:
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\\[ \\|a\\|\_2 \triangleq \sqrt{\sum\_i |a\_i|^2} \\]
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- **Vector p-norm**:
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\\[ \\|a\\|\_p \triangleq \left( \sum\_i |a\_i|^p \right)^{1/p} \\]
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- **Vector \\(\infty\text{-norm}\\) (Max Norm)**: ()
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\\[ \\|a\\|\_\infty \triangleq \max\_i |a\_i| \\]
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## Matrix Norms {#matrix-norms}
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<div class="examp">
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<div></div>
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A norm on a matrix \\(\\|A\\|\\) is a matrix norm if, in addition to the four norm properties, it also satisfies the multiplicative property:
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\\[ \\|AB\\| \le \\|A\\| \cdot \\|B\\| \\]
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</div>
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- **Sum matrix norm**:
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\\[ \\|A\\|\_\text{sum} \triangleq \sum\_{i,j} |a\_{ij}| \\]
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- **Frobenius matrix norm (Euclidean Norm)**:
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\\[ \\|A\\|\_F \triangleq \sqrt{\sum\_{i,j} |a\_{ij}|^2} = \sqrt{\text{tr}(A^H A)} \\]
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- **Max element norm**: (which is not a _matrix_ norm)
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\\[ \\|A\\|\_\text{max} \triangleq \max\_{i,j} |a\_{ij}| \\]
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## Induced Matrix Norms {#induced-matrix-norms}
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Induced matrix norms are important because of their close relationship to signal amplification in systems.
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Consider the figure below where \\(w\\) is the input vector, \\(z\\) the output vector and where the "amplification" or "gain" of the matrix \\(A\\) is defined by the ration \\(\\|z\\|/\\|w\\|\\).
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{{< figure src="/ox-hugo/induced_matrix_norm.png" >}}
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The maximum gain for all possible input directions is given by the **induced norm**:
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\\[ \\|A\\|\_{ip} \triangleq \max\_{w \neq 0} \frac{\\|Aw\\|\_p}{\\|w\\|\_p} \\]
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Thus, the induced norm gives the largest possible "amplification" of the matrix.
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The following equivalent definition is also used:
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\\[ \\|A\\|\_{ip} = \max\_{\\|w\\|\_p \le 1} \\|Aw\\|\_p \\]
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## Signal Norms {#signal-norms}
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For signals, we may compute the norm in two steps:
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1. "Sum up" the channels at a given time using a vector norm.
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For a scalar, we simply take the absolute value.
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2. "Sum up" in time using a temporal norm.
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We normally use the same p-norm both for the vector and the signal.
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- **1-norm in time (Integral Absolute Error)**:
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\\[ \\|e(t)\\|\_1 = \int\_{-\infty}^{\infty} \sum\_i |e\_i(\tau)| d\tau \\]
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- **2-norm in time (Quadratic Norm)**:
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\\[ \\|e(t)\\|\_2 = \sqrt{\int\_{-\infty}^{\infty} \sum\_i |e\_i(\tau)|^2 d\tau} \\]
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- **\\(\infty\text{-norm}\\) in time (Peak value in time)**:
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\\[ \\|e(t)\\|\_\infty = \max\_\tau \left( \max\_i |e\_i(\tau)| \right) \\]
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- **Power-Norm or RMS-Norm**:
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\\[ \\|e(t)\\|\_\text{pow} = \lim\_{T\to \infty} \sqrt{\frac{1}{2T} \int\_{-T}^T \sum\_i |e\_i(\tau)|^2 d\tau} \\]
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## Signal Interpretation of Various System Norms {#signal-interpretation-of-various-system-norms}
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## System Norms {#system-norms}
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### \\(\mathcal{H}\_\infty\\) Norm {#mathcal-h-infty--norm}
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SISO Systems => absolute value => bode plot
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MIMO Systems => singular value
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Signal
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Signal => maximum value
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## \\(\mathcal{H}\_2\\) Norm {#mathcal-h-2--norm}
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### \\(\mathcal{H}\_2\\) Norm {#mathcal-h-2--norm}
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The \\(\mathcal{H}\_2\\) is very useful when combined to [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}}).
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As explained in ([Monkhorst 2004](#orgafef987)), the \\(\mathcal{H}\_2\\) norm has a stochastic interpretation:
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As explained in ([Monkhorst 2004](#orgd47c42b)), the \\(\mathcal{H}\_2\\) norm has a stochastic interpretation:
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> The squared \\(\mathcal{H}\_2\\) norm can be interpreted as the output variance of a system with zero mean white noise input.
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@@ -41,10 +123,10 @@ Minimizing the \\(\mathcal{H}\_2\\) norm can be equivalent as minimizing the RMS
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## Bibliography {#bibliography}
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<a id="orgafef987"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
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<a id="orgd47c42b"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
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<a id="org140f9cc"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.
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<a id="orgf3c8b69"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.
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<a id="orgc1385a9"></a>Toivonen, Hannu T. 2002. “Robust Control Methods.” Abo Akademi University.
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<a id="orgb2755d2"></a>Toivonen, Hannu T. 2002. “Robust Control Methods.” Abo Akademi University.
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<a id="org8471dd8"></a>Zhang, Weidong. 2011. _Quantitative Process Control Theory_. CRC Press.
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<a id="org3e1b2ef"></a>Zhang, Weidong. 2011. _Quantitative Process Control Theory_. CRC Press.
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