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+title = "Respect the unstable"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+:
+
+
+Reference
+: ([Stein 2003](#org50b7ac8))
+
+Author(s)
+: Stein, G.
+
+Year
+: 2003
+
+
+## Introduction {#introduction}
+
+> The second trend has been evident at our conferences, and certainly in our journal, over the years.
+> This trend is the increasing worship of abstract mathematical results in control at the expense of more specific examinations of their practical, physical consequences.
+
+<div class="important">
+  <div></div>
+
+**Basic facts about unstable plants**:
+
+-   Unstable systems are fundamentally, and quantifiably more difficult to control than stable ones
+-   Controllers for unstable systems are operationally critical
+-   Closed-loop systems with unstable components are only locally stable
+
+</div>
+
+
+## The Bode Integrals {#the-bode-integrals}
+
+<div class="important">
+  <div></div>
+
+**Bode Integrals**:
+
+The first integral applies to stable plants and the second to unstable plants.
+They are valid for every stabilizing controller, assuming only that both plan and controller have finite bandwidths.
+In words, the integrals state that the log of magnitude of sensitivity function of a SISO feedback system, integrated over frequency, is constant.
+The constant is zero for stable plants, and it is positive for unstable ones.
+It becomes larger as the number of unstable poles increases and/or as the poles more farther into the right-half plane.
+
+\begin{align}
+  \int\_0^\infty \ln |S(j\omega)| d \omega & = 0 \label{eq:bode\_integral\_stable} \\\\\\
+  \int\_0^\infty \ln |S(j\omega)| d \omega & = \pi \sum\_{p \in P} \text{Re}(p) \label{eq:bode\_integral\_unstable}
+\end{align}
+
+</div>
+
+
+## A Bode Integral Interpretation {#a-bode-integral-interpretation}
+
+Bode integral can be thought as **conservation laws**.
+They state that a certain quantity, the integrated value of the log of the magnitude of the sensitivity function, is conserved under the action of feedback.
+The total amount of this quantity is always the same.
+It is equal to zero for stable plant/compensator pairs, and it is equal to some fixed positive amount for unstable ones.
+
+Since we are talking about the log of sensitivity magnitude, it follows that negative values are good, and positive values are bad.
+
+<div class="definition">
+  <div></div>
+
+It is curious, somehow, that our field has not adopted a name for this quantity being conserved (i.e. the integrated log of sensitivity magnitude).
+It is here proposed to call it **dirt**
+
+</div>
+
+The job of a serious control designer is then to more dirt from one place to another, using appropriate tools, without being able to get rid of any of it (illustrated in Figure [1](#org956e0bf)).
+
+<a id="org956e0bf"></a>
+
+{{< figure src="/ox-hugo/stein03_serious_design.png" caption="Figure 1: Sensitivity reduction at low frequency unavoidably leads to sensitivity increase at higher frequencies" >}}
+
+In the same spirit, the job of a more academic control designer with more abstract tools such as LQG, \\(\mathcal{H}\_\infty\\), is to set parameters (weights) of a synthesis machine to adjust the contours of the machine's digging blades to get just the right shape for the sensitivity function (Figure [2](#org73c6dd3)).
+
+<a id="org73c6dd3"></a>
+
+{{< figure src="/ox-hugo/stein03_formal_design.png" caption="Figure 2: Sensitivity shaping automated by modern control tools" >}}
+
+
+## Available bandwidth {#available-bandwidth}
+
+An argument is sometimes made that the Bode integrals are not really restrictive because we only seek to dig holes over finite frequency bands.
+We then have an infinite frequency range left over into which to dump the dirt, so we can make the layer arbitrarily thin (Figure [3](#orgb2839fe)).
+
+<a id="orgb2839fe"></a>
+
+{{< figure src="/ox-hugo/stein03_spreading_it_thin.png" caption="Figure 3: It is possible to spead the increase of the sensitivity function over a larger frequency band" >}}
+
+The weakness of this argument is evident from standard classical theory.
+A thin layer, say with \\(\ln|S| = \epsilon\\) requires a loop transfer function whose Nyquist diagram falls on a near-unit circle, centered at \\((-1 + j 0)\\) with a radius \\(\approx (1-\epsilon)\\), over a wide frequency range.
+This means that the loop cannot simply attenuate at high frequencies but must attenuate in a very precise way.
+The loop must maintain very good frequency response fidelity over wide frequency ranges.
+
+But a key fact about physical systems is that they do not exhibit good frequency response fidelity beyond a certain bandwidth.
+This is due to uncertain or unmodeled dynamics in the plant, to digital control implementations, to power limits, to nonlinearities, and to many other factors.
+Let us call that bandwidth the available bandwidth" \\(\Omega\_a\\), to distinguish it from other bandwidths such as crossover or \\(3-dB\\) magnitude loss.
+The available bandwidth is the frequency up to which we can keep \\(G(j\omega) K(j\omega)\\) close to a nominal design and beyond which we can only guarantee that the actual loop magnitude will attenuate rapidly enough (e.g. \\(|G(j\omega) K(j\omeg\\))| < &delta;/&omega;^2$).
+In today's popular robust control jargon, the available bandwidth is the frequency range over which the unstructured multiplicative perturbations are substantially less than unity.
+
+Note that the available bandwidth is not a function of the compensator or of the control design process.
+Rather, it is an a priori constraint imposed by the physical hardware we use in the control loop.
+Most importantly, the available bandwidth is always finite.
+
+Given all this, Bode's integrals really reduce to finite integrals over the range \\(0 \ge \omega \ge \Omega\_a\\):
+
+\begin{align}
+  \int\_0^{\Omega\_a} \ln{|S(j \omega)|} d \omega &= \delta \\\\\\
+  \int\_0^{\Omega\_a} \ln{|S(j \omega)|} d \omega &= \pi \sum\_{p \in P} \text{Re}(p) + \delta
+\end{align}
+
+All the action of the feedback design, the sensitivity improvements as well as the sensitivity deterioration, must occur within \\(0 \ge \omega \ge \Omega\_a\\).
+Only a small error \\(\delta\\) occurs outside that range, associated with the tail of the complete integrals.
+
+
+## Bibliography {#bibliography}
+
+<a id="org50b7ac8"></a>Stein, Gunter. 2003. “Respect the Unstable.” _IEEE Control Systems Magazine_ 23 (4). IEEE:12–25.
diff --git a/content/book/smith99_scien_engin_guide_digit_signal.md b/content/book/smith99_scien_engin_guide_digit_signal.md
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+title = "The scientist and engineer's guide to digital signal processing - second edition"
+author = ["Thomas Dehaeze"]
+draft = false
++++
+
+Tags
+: [Digital Signal Processing]({{< relref "digital_signal_processing" >}})
+
+Reference
+: ([Smith 1999](#org18cc45c))
+
+Author(s)
+: Smith, S. W.
+
+Year
+: 1999
+
+
+## Bibliography {#bibliography}
+
+<a id="org18cc45c"></a>Smith, Steven W. 1999. _The Scientist and Engineer’s Guide to Digital Signal Processing - Second Edition_. California Technical Publishing.
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