Update Content - 2020-09-23
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@@ -13,13 +13,16 @@ Tags
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Resources:
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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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- ([Skogestad and Postlethwaite 2007](#org352385f))
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- ([Toivonen 2002](#orga84ee63))
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- ([Zhang 2011](#org74fd92c))
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## Definition {#definition}
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<div class="bblue">
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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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@@ -27,6 +30,8 @@ A norm of \\(e\\) (which may be a vector, matrix, signal of system) is a real nu
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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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@@ -42,7 +47,7 @@ A norm of \\(e\\) (which may be a vector, matrix, signal of system) is a real nu
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## Matrix Norms {#matrix-norms}
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<div class="examp">
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<div class="bgreen">
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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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@@ -96,37 +101,94 @@ We normally use the same p-norm both for the vector and the signal.
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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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SISO Systems => absolute value => bode plot
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MIMO Systems => singular value
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Signal => maximum value
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<div class="bgreen">
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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="bgreen">
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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](#orgd47c42b)), the \\(\mathcal{H}\_2\\) norm has a stochastic interpretation:
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As explained in ([Monkhorst 2004](#org50a92a2)), 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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Minimizing the \\(\mathcal{H}\_2\\) norm can be equivalent as minimizing the RMS value of some signals in the system.
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## Link between signal and system norms {#link-between-signal-and-system-norms}
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## Bibliography {#bibliography}
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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="org50a92a2"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
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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="org352385f"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.
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<a id="orgb2755d2"></a>Toivonen, Hannu T. 2002. “Robust Control Methods.” Abo Akademi University.
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<a id="orga84ee63"></a>Toivonen, Hannu T. 2002. “Robust Control Methods.” Abo Akademi University.
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<a id="org3e1b2ef"></a>Zhang, Weidong. 2011. _Quantitative Process Control Theory_. CRC Press.
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<a id="org74fd92c"></a>Zhang, Weidong. 2011. _Quantitative Process Control Theory_. CRC Press.
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