194 lines
8.3 KiB
Markdown
194 lines
8.3 KiB
Markdown
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title = "Systems and Signals Norms"
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author = ["Thomas Dehaeze"]
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draft = false
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\\[ \SI{1}{\meter\per\second} \\]
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Resources:
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- ([Skogestad and Postlethwaite 2007](#orga6846b3))
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- ([Toivonen 2002](#org300cd1c))
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- ([Zhang 2011](#org037ea69))
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## Definition {#definition}
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<div class="definition">
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<div></div>
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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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</div>
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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="definition">
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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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Consider a system \\(G\\) with input \\(d\\) and output \\(e\\), such that:
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\\[ e = G d \\]
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For performance, we may want the output signal \\(e\\) to be "small" for any allowed input signals \\(d\\).
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We therefore need to specify:
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1. What \\(d\\) are allowed. (Which set does \\(d\\) belong to?)
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Some possible inputs signal sets are:
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- \\(d(t)\\) consists of impulses \\(\delta(t)\\).
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- These generate step changes in the states.
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- \\(d(t) = \sin(\omega t)\\) with fixed frequency
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- \\(d(t)\\) is bounded in energy \\(\\|d(t)\\|\_2 \le 1\\)
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- \\(d(t)\\) is bounded in power \\(\\|d(t)\\|\_\text{pow} \le 1\\)
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- \\(d(t)\\) is bounded in magnitude \\(\\|d(t)\\|\_\infty \le 1\\)
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2. What we mean by "small". (Which norm should be use for \\(e\\)?)
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To measure the output signal, we may consider the following norms:
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- 2-norm (energy): \\(\\|e(t)\\|\_2\\)
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- \\(\infty\text{-norm}\\) (peak magnitude): \\(\\|e(t)\\|\_\infty\\)
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- Power: \\(\\|e(t)\\|\_\text{pow}\\)
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We now consider which system norms result from the definition of input classes and output norms (Table [1](#table--tab:system-norms)).
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<a id="table--tab:system-norms"></a>
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<div class="table-caption">
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<span class="table-number"><a href="#table--tab:system-norms">Table 1</a></span>:
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System norms for sets of inputs signals and three different output norms
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</div>
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| | \\(\delta(t)\\) | \\(\sin(\omega t)\\) | \\(\vert\vert d \vert\vert\_2\\) | \\(\vert\vert d \vert\vert\_\infty\\) | \\(\vert\vert d \vert\vert\_\text{pow}\\) |
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|-------------------------------------------|------------------------------------------|--------------------------------------------------------|------------------------------------------|----------------------------------------------|-------------------------------------------|
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| \\(\vert\vert e \vert\vert\_2\\) | \\(\vert\vert G(s) \vert\vert\_2\\) | \\(\infty\\) | \\(\vert\vert G(s) \vert\vert\_\infty\\) | \\(\infty\\) | \\(\infty\\) |
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| \\(\vert\vert e \vert\vert\_\infty\\) | \\(\vert\vert g(t) \vert\vert\_\infty\\) | \\(\overline{\sigma}(G(j\omega))\\) | \\(\vert\vert G(s) \vert\vert\_2\\) | \\(\vert\vert g(t) \vert\vert\_1\\) | \\(\infty\\) |
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| \\(\vert\vert e \vert\vert\_\text{pow}\\) | 0 | \\(\frac{1}{\sqrt{2}} \overline{\sigma}(G(j\omega))\\) | 0 | \\(\le \vert\vert G(s) \vert\vert\_\infty\\) | \\(\vert\vert G(s) \vert\vert\_\infty\\) |
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## System Norms {#system-norms}
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### \\(\mathcal{H}\_\infty\\) Norm {#mathcal-h-infty--norm}
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<div class="exampl">
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<div></div>
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Consider a proper linear stable system \\(G(s)\\).
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The \\(\mathcal{H}\_\infty\\) norm is the peak value of its maximum singular value:
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\\[ \\|G(s)\\|\_\infty \triangleq \max\_{\omega} \overline{\sigma}(G(j\omega)) \\]
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</div>
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In terms of signals, the \\(\mathcal{H}\_\infty\\) norm can be interpreted as follows:
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- it is the worst case steady-state gain for sinusoidal inputs at any frequency
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- it is equal to the 2-norm in the time domain:
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\\[ \\|G(s)\\|\_\infty = \max\_{d(t)} \frac{\\|e(t)\\|\_2 \neq 0}{\\|d(t)\\|\_2} = \max\_{\\|d(t)\\|\_2 = 1} \\|e(t)\\|\_2 \\]
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### \\(\mathcal{H}\_2\\) Norm {#mathcal-h-2--norm}
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<div class="exampl">
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<div></div>
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Consider a strictly proper system \\(G(s)\\).
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The \\(\mathcal{H}\_2\\) norm is:
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\begin{align\*}
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\\|G(s)\\|\_2 &\triangleq \sqrt{\frac{1}{2\pi} \int\_{-\infty}^{\infty} \text{tr}\left(G(j\omega)^HG(j\omega)\right) d\omega} \\\\\\
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&= \sqrt{\frac{1}{2\pi} \int\_{-\infty}^{\infty} \sum\_i {\sigma\_i}^2(G(j\omega)) d\omega}
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\end{align\*}
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</div>
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In terms of signals, the \\(\mathcal{H}\_\infty\\) norm can be interpreted as follows:
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- it is a measure of the expected RMS value of the output to white noise excitation
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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](#org16354b5)), 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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## Bibliography {#bibliography}
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<a id="org16354b5"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
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<a id="orga6846b3"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.
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<a id="org300cd1c"></a>Toivonen, Hannu T. 2002. “Robust Control Methods.” Abo Akademi University.
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<a id="org037ea69"></a>Zhang, Weidong. 2011. _Quantitative Process Control Theory_. CRC Press.
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