126 lines
6.6 KiB
Markdown
126 lines
6.6 KiB
Markdown
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title = "Respect the unstable"
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author = ["Dehaeze Thomas"]
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draft = false
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Tags
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:
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Reference
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: (<a href="#citeproc_bib_item_1">Stein 2003</a>)
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Author(s)
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: Stein, G.
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Year
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: 2003
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## Introduction {#introduction}
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> The second trend has been evident at our conferences, and certainly in our journal, over the years.
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> This trend is the increasing worship of abstract mathematical results in control at the expense of more specific examinations of their practical, physical consequences.
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<div class="important">
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**Basic facts about unstable plants**:
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- Unstable systems are fundamentally, and quantifiably more difficult to control than stable ones
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- Controllers for unstable systems are operationally critical
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- Closed-loop systems with unstable components are only locally stable
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</div>
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## The Bode Integrals {#the-bode-integrals}
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<div class="important">
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**Bode Integrals**:
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The first integral applies to stable plants and the second to unstable plants.
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They are valid for every stabilizing controller, assuming only that both plan and controller have finite bandwidths.
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In words, the integrals state that the log of magnitude of sensitivity function of a SISO feedback system, integrated over frequency, is constant.
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The constant is zero for stable plants, and it is positive for unstable ones.
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It becomes larger as the number of unstable poles increases and/or as the poles more farther into the right-half plane.
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\begin{align}
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\int\_0^\infty \ln |S(j\omega)| d \omega & = 0 \label{eq:bode\_integral\_stable} \\\\
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\int\_0^\infty \ln |S(j\omega)| d \omega & = \pi \sum\_{p \in P} \text{Re}(p) \label{eq:bode\_integral\_unstable}
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\end{align}
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</div>
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## A Bode Integral Interpretation {#a-bode-integral-interpretation}
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Bode integral can be thought as **conservation laws**.
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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.
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The total amount of this quantity is always the same.
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It is equal to zero for stable plant/compensator pairs, and it is equal to some fixed positive amount for unstable ones.
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Since we are talking about the log of sensitivity magnitude, it follows that negative values are good, and positive values are bad.
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<div class="definition">
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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).
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It is here proposed to call it **dirt**
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</div>
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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](#figure--fig:stein03-serious-design)).
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<a id="figure--fig:stein03-serious-design"></a>
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{{< figure src="/ox-hugo/stein03_serious_design.png" caption="<span class='figure-number'>Figure 1: </span>Sensitivity reduction at low frequency unavoidably leads to sensitivity increase at higher frequencies" >}}
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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](#figure--fig:stein03-formal-design)).
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<a id="figure--fig:stein03-formal-design"></a>
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{{< figure src="/ox-hugo/stein03_formal_design.png" caption="<span class='figure-number'>Figure 2: </span>Sensitivity shaping automated by modern control tools" >}}
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## Available bandwidth {#available-bandwidth}
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An argument is sometimes made that the Bode integrals are not really restrictive because we only seek to dig holes over finite frequency bands.
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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](#figure--fig:stein03-spreading-it-thin)).
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<a id="figure--fig:stein03-spreading-it-thin"></a>
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{{< figure src="/ox-hugo/stein03_spreading_it_thin.png" caption="<span class='figure-number'>Figure 3: </span>It is possible to spead the increase of the sensitivity function over a larger frequency band" >}}
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The weakness of this argument is evident from standard classical theory.
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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.
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This means that the loop cannot simply attenuate at high frequencies but must attenuate in a very precise way.
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The loop must maintain very good frequency response fidelity over wide frequency ranges.
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But a key fact about physical systems is that they do not exhibit good frequency response fidelity beyond a certain bandwidth.
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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.
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Let us call that bandwidth the available bandwidth" \\(\Omega\_a\\), to distinguish it from other bandwidths such as crossover or \\(3-dB\\) magnitude loss.
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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\\))| < δ/ω^2$).
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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.
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Note that the available bandwidth is not a function of the compensator or of the control design process.
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Rather, it is an a priori constraint imposed by the physical hardware we use in the control loop.
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Most importantly, the available bandwidth is always finite.
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Given all this, Bode's integrals really reduce to finite integrals over the range \\(0 \ge \omega \ge \Omega\_a\\):
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\begin{align}
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\int\_0^{\Omega\_a} \ln{|S(j \omega)|} d \omega &= \delta \\\\
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\int\_0^{\Omega\_a} \ln{|S(j \omega)|} d \omega &= \pi \sum\_{p \in P} \text{Re}(p) + \delta
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\end{align}
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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\\).
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Only a small error \\(\delta\\) occurs outside that range, associated with the tail of the complete integrals.
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## Bibliography {#bibliography}
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<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
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<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Stein, Gunter. 2003. “Respect the Unstable.” <i>IEEE Control Systems Magazine</i> 23 (4). IEEE: 12–25.</div>
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</div>
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