Update few notes
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@ -33,9 +33,6 @@ Instrument principles:
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- autocollimators with a flat mirror
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### How that thing with two autocollimators can work? {#how-that-thing-with-two-autocollimators-can-work}
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## Sources of error in displacement interferometry {#sources-of-error-in-displacement-interferometry}
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Two error sources:
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@ -5,7 +5,7 @@ draft = false
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+++
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Tags
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: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Reference Books]({{< relref "reference_books" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}})
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: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Reference Books]({{< relref "reference_books" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [HAC-HAC]({{< relref "hac_hac" >}})
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Reference
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: <sup id="454500a3af67ef66a7a754d1f2e1bd4a"><a href="#preumont18_vibrat_contr_activ_struc_fourt_edition" title="Andre Preumont, Vibration Control of Active Structures - Fourth Edition, Springer International Publishing (2018).">(Andre Preumont, 2018)</a></sup>
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@ -61,7 +61,7 @@ There are two radically different approached to disturbance rejection: feedback
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#### Feedback {#feedback}
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<a id="orgd64232a"></a>
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<a id="orgcd0067e"></a>
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{{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="Figure 1: Principle of feedback control" >}}
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@ -87,12 +87,12 @@ The objective is to control a variable \\(y\\) to a desired value \\(r\\) in spi
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#### Feedforward {#feedforward}
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<a id="orga8d6c2f"></a>
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<a id="orgf21f883"></a>
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{{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="Figure 2: Principle of feedforward control" >}}
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The method relies on the availability of a **reference signal correlated to the primary disturbance**.
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The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in figure [2](#orga8d6c2f).
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The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in figure [2](#orgf21f883).
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The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized.
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@ -123,11 +123,11 @@ The table [1](#table--tab:adv-dis-type-control) summarizes the main features of
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### The Various Steps of the Design {#the-various-steps-of-the-design}
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<a id="orgb7c5b7f"></a>
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<a id="orgca19f4b"></a>
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{{< figure src="/ox-hugo/preumont18_design_steps.png" caption="Figure 3: The various steps of the design" >}}
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The various steps of the design of a controlled structure are shown in figure [3](#orgb7c5b7f).
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The various steps of the design of a controlled structure are shown in figure [3](#orgca19f4b).
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The **starting point** is:
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@ -161,7 +161,7 @@ y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\\\
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z &= T\_{zw} w = [G\_{zw} + G\_{zu}H(I - G\_{yu}H)^{-1} G\_{yw}] w
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\end{align\*}
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<a id="orgc176272"></a>
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<a id="org06b8843"></a>
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{{< figure src="/ox-hugo/preumont18_general_plant.png" caption="Figure 4: Block diagram of the control System" >}}
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@ -191,7 +191,7 @@ It is useful to **identify the critical modes** in a design, at which the effort
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The diagram can also be used to **assess the control laws** and compare different actuator and sensor configuration.
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<a id="orge3c2efe"></a>
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<a id="orgefc00fd"></a>
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{{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="Figure 5: Error budget distribution in OL and CL for increasing gains" >}}
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@ -334,9 +334,9 @@ If we left multiply the equation by \\(\Phi^T\\) and we use the orthogonalily re
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If \\(\Phi^T C \Phi\\) is diagonal, the **damping is said classical or normal**. In this case:
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\\[ \Phi^T C \Phi = diag(2 \xi\_i \mu\_i \omega\_i) \\]
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One can verify that the Rayleigh damping [eq:rayleigh_damping](#eq:rayleigh_damping) complies with this condition with modal damping ratios \\(\xi\_i = \frac{1}{2} ( \frac{\alpha}{\omega\_i} + \beta\omega\_i )\\).
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One can verify that the Rayleigh damping \eqref{eq:rayleigh_damping} complies with this condition with modal damping ratios \\(\xi\_i = \frac{1}{2} ( \frac{\alpha}{\omega\_i} + \beta\omega\_i )\\).
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And we obtain decoupled modal equations [eq:modal_eom](#eq:modal_eom).
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And we obtain decoupled modal equations \eqref{eq:modal_eom}.
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<div class="cbox">
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<div></div>
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@ -370,15 +370,15 @@ Typical values of the modal damping ratio are summarized on table [tab:damping_r
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The assumption of classical damping is often justified for light damping, but it is questionable when the damping is large.
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If one accepts the assumption of classical damping, the only difference between equation [eq:general_eom](#eq:general_eom) and [eq:modal_eom](#eq:modal_eom) lies in the change of coordinates.
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If one accepts the assumption of classical damping, the only difference between equation \eqref{eq:general_eom} and \eqref{eq:modal_eom} lies in the change of coordinates.
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However, in physical coordinates, the number of degrees of freedom is usually very large.
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If a structure is excited in by a band limited excitation, its response is dominated by the modes whose natural frequencies are inside the bandwidth of the excitation and the equation [eq:modal_eom](#eq:modal_eom) can often be restricted to theses modes.
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If a structure is excited in by a band limited excitation, its response is dominated by the modes whose natural frequencies are inside the bandwidth of the excitation and the equation \eqref{eq:modal_eom} can often be restricted to theses modes.
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Therefore, the number of degrees of freedom contribution effectively to the response is **reduced drastically** in modal coordinates.
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#### Dynamic Flexibility Matrix {#dynamic-flexibility-matrix}
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If we consider the steady-state response of equation [eq:general_eom](#eq:general_eom) to harmonic excitation \\(f=F e^{j\omega t}\\), the response is also harmonic \\(x = Xe^{j\omega t}\\). The amplitude of \\(F\\) and \\(X\\) is related by:
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If we consider the steady-state response of equation \eqref{eq:general_eom} to harmonic excitation \\(f=F e^{j\omega t}\\), the response is also harmonic \\(x = Xe^{j\omega t}\\). The amplitude of \\(F\\) and \\(X\\) is related by:
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\\[ X = G(\omega) F \\]
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Where \\(G(\omega)\\) is called the **Dynamic flexibility Matrix**:
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@ -398,7 +398,7 @@ With:
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D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \xi\_i \omega/\omega\_i}
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\end{equation}
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<a id="org44f39ce"></a>
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<a id="orgde5a280"></a>
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{{< figure src="/ox-hugo/preumont18_neglected_modes.png" caption="Figure 6: Fourier spectrum of the excitation \\(F\\) and dynamic amplitification \\(D\_i\\) of mode \\(i\\) and \\(k\\) such that \\(\omega\_i < \omega\_b\\) and \\(\omega\_k \gg \omega\_b\\)" >}}
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@ -443,7 +443,7 @@ If we assumes that the collocated system is undamped and is attached to the DoF
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\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [fig:collocated_control_frf](#fig:collocated_control_frf)).
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<a id="org384a73f"></a>
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<a id="orgc840265"></a>
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{{< figure src="/ox-hugo/preumont18_collocated_control_frf.png" caption="Figure 7: Open-Loop FRF of an undamped structure with collocated actuator/sensor pair" >}}
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@ -459,7 +459,7 @@ For lightly damped structure, the poles and zeros are just moved a little bit in
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If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [fig:collocated_zero](#fig:collocated_zero).
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<a id="org0af6db0"></a>
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<a id="orgd860ef7"></a>
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{{< figure src="/ox-hugo/preumont18_collocated_zero.png" caption="Figure 8: Structure with collocated actuator and sensor" >}}
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@ -476,7 +476,7 @@ The open-loop poles are independant of the actuator and sensor configuration whi
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By looking at figure [fig:collocated_control_frf](#fig:collocated_control_frf), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right.
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<a id="orgadf3ccb"></a>
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<a id="org245f75f"></a>
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{{< figure src="/ox-hugo/preumont18_alternating_p_z.png" caption="Figure 9: Bode plot of a lighly damped structure with collocated actuator and sensor" >}}
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@ -486,7 +486,7 @@ The open-loop transfer function of a lighly damped structure with a collocated a
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G(s) = G\_0 \frac{\Pi\_i(s^2/z\_i^2 + 2 \xi\_i s/z\_i + 1)}{\Pi\_j(s^2/\omega\_j^2 + 2 \xi\_j s /\omega\_j + 1)}
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\end{equation}
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The corresponding Bode plot is represented in figure [9](#orgadf3ccb). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\).
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The corresponding Bode plot is represented in figure [9](#org245f75f). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\).
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In this way, the phase diagram is always contained between \\(\SI{0}{\degree}\\) and \\(\SI{-180}{\degree}\\) as a consequence of the interlacing property.
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@ -513,7 +513,7 @@ The system consists of (see figure [fig:voice_coil_schematic](#fig:voice_coil_sc
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- A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap
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- A coil which is free to move axially
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<a id="orgc2d5dd7"></a>
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<a id="org605d681"></a>
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{{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="Figure 10: Physical principle of a voice coil transducer" >}}
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@ -553,7 +553,7 @@ Thus, at any time, there is an equilibrium between the electrical power absorbed
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A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) (figure [fig:proof_mass_actuator](#fig:proof_mass_actuator)).
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<a id="orgd3962b1"></a>
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<a id="org5fa45df"></a>
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{{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="Figure 11: Proof-mass actuator" >}}
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@ -585,7 +585,7 @@ with:
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Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [fig:proof_mass_tf](#fig:proof_mass_tf)).
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<a id="orgc1aa3b5"></a>
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<a id="org0b93a83"></a>
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{{< figure src="/ox-hugo/preumont18_proof_mass_tf.png" caption="Figure 12: Bode plot \\(F/i\\) of the proof-mass actuator" >}}
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@ -610,7 +610,7 @@ By using the two equations, we obtain:
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Above the corner frequency, the gain of the geophone is equal to the transducer constant \\(T\\).
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<a id="orgedd24f2"></a>
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<a id="org45de7ce"></a>
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{{< figure src="/ox-hugo/preumont18_geophone.png" caption="Figure 13: Model of a geophone based on a voice coil transducer" >}}
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@ -621,7 +621,7 @@ Designing geophones with very low corner frequency is in general difficult. Acti
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The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [fig:electro_mechanical_transducer](#fig:electro_mechanical_transducer).
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<a id="org281a432"></a>
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<a id="orgdc255dc"></a>
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{{< figure src="/ox-hugo/preumont18_electro_mechanical_transducer.png" caption="Figure 14: Electrical analog representation of an electromechanical transducer" >}}
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@ -643,7 +643,7 @@ With:
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- \\(T\_{me}\\) is the transduction coefficient representing the force acting on the mechanical terminals to balance the electromagnetic force induced per unit current input (in \\(\si{\newton\per\ampere}\\))
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- \\(Z\_m\\) is the mechanical impedance measured when \\(i=0\\)
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Equation [eq:gen_trans_e](#eq:gen_trans_e) shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals.
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Equation \eqref{eq:gen_trans_e} shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals.
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Thus, if \\(Z\_ei\\) can be measured and substracted from \\(e\\), a signal proportional to the velocity is obtained.
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To do so, the bridge circuit as shown on figure [fig:bridge_circuit](#fig:bridge_circuit) can be used.
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@ -656,7 +656,7 @@ We can show that
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which is indeed a linear function of the velocity \\(v\\) at the mechanical terminals.
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<a id="org54c14d5"></a>
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<a id="orgab771fe"></a>
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{{< figure src="/ox-hugo/preumont18_bridge_circuit.png" caption="Figure 15: Bridge circuit for self-sensing actuation" >}}
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@ -666,7 +666,7 @@ which is indeed a linear function of the velocity \\(v\\) at the mechanical term
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Smart materials have the ability to respond significantly to stimuli of different physical nature.
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Figure [fig:smart_materials](#fig:smart_materials) lists various effects that are observed in materials in response to various inputs.
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<a id="orgeb59953"></a>
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{{< figure src="/ox-hugo/preumont18_smart_materials.png" caption="Figure 16: Stimulus response relations indicating various effects in materials. The smart materials corresponds to the non-diagonal cells" >}}
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@ -718,7 +718,7 @@ With:
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#### Constitutive Relations of a Discrete Transducer {#constitutive-relations-of-a-discrete-transducer}
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The set of equations [eq:piezo_eq](#eq:piezo_eq) can be written in a matrix form:
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The set of equations \eqref{eq:piezo_eq} can be written in a matrix form:
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\begin{equation}
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\begin{bmatrix}D\\S\end{bmatrix}
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@ -761,7 +761,7 @@ It measures the efficiency of the conversion of the mechanical energy into elect
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</div>
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If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [fig:piezo_stack](#fig:piezo_stack)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating [eq:piezo_eq_matrix_bis](#eq:piezo_eq_matrix_bis) over the volume of the transducer:
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If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [fig:piezo_stack](#fig:piezo_stack)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating \eqref{eq:piezo_eq_matrix_bis} over the volume of the transducer:
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\begin{equation}
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\begin{bmatrix}Q\\\Delta\end{bmatrix}
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@ -782,11 +782,11 @@ where
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- \\(C = \epsilon^T A n^2/l\\) is the capacitance of the transducer with no external load (\\(f = 0\\))
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- \\(K\_a = A/s^El\\) is the stiffness with short-circuited electrodes (\\(V = 0\\))
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{{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="Figure 17: Piezoelectric linear transducer" >}}
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Equation [eq:piezo_stack_eq](#eq:piezo_stack_eq) can be inverted to obtain
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Equation \eqref{eq:piezo_stack_eq} can be inverted to obtain
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\begin{equation}
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\begin{bmatrix}V\\f\end{bmatrix}
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@ -810,11 +810,11 @@ The total power delivered to the transducer is the sum of electric power \\(V i\
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dW = V i dt + f \dot{\Delta} dt = V dQ + f d\Delta
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\end{equation}
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<a id="orge625816"></a>
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{{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="Figure 18: Discrete Piezoelectric Transducer" >}}
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By integrating equation [eq:piezo_work](#eq:piezo_work) and using the constitutive equations [eq:piezo_stack_eq_inv](#eq:piezo_stack_eq_inv), we obtain the analytical expression of the stored electromechanical energy for the discrete transducer:
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By integrating equation \eqref{eq:piezo_work} and using the constitutive equations \eqref{eq:piezo_stack_eq_inv}, we obtain the analytical expression of the stored electromechanical energy for the discrete transducer:
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\begin{equation}
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W\_e(\Delta, Q) = \frac{Q^2}{2 C (1 - k^2)} - \frac{n d\_{33} K\_a}{C(1-k^2)} Q\Delta + \frac{K\_a}{1-k^2}\frac{\Delta^2}{2}
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@ -847,7 +847,7 @@ The ratio between the remaining stored energy and the initial stored energy is
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Consider the system of figure [fig:piezo_stack_admittance](#fig:piezo_stack_admittance), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
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The force acting on the mass is negative of that acting on the transducer, \\(f = -M \ddot{x}\\).
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{{< figure src="/ox-hugo/preumont18_piezo_stack_admittance.png" caption="Figure 19: Elementary dynamical model of the piezoelectric transducer" >}}
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@ -866,9 +866,9 @@ And one can see that
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\frac{z^2 - p^2}{z^2} = k^2
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\end{equation}
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Equation [eq:distance_p_z](#eq:distance_p_z) constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [fig:piezo_admittance_curve](#fig:piezo_admittance_curve)).
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Equation \eqref{eq:distance_p_z} constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [fig:piezo_admittance_curve](#fig:piezo_admittance_curve)).
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{{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="Figure 20: Typical admittance FRF of the transducer" >}}
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@ -1574,7 +1574,7 @@ This approach has the following advantages:
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- The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin)
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- The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin)
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{{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="Figure 21: Principle of the dual-loop HAC/LAC control" >}}
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@ -29,6 +29,14 @@ Section 2.2.2 Force and Motion
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> One should however be aware that another non-destructive source of non-linearity is found in a tried important field of mechanics, called _kinematics_.
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> The relation between angles and positions is often non-linear in such a mechanism, because of the changing angles, and controlling these often requires special precautions to overcome the inherent non-linearities by linearisation around actual position and adapting the optimal settings of the controller to each position.
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{{< figure src="/ox-hugo/schmidt14_high_low_freq_regions.png" caption="Figure 1: Stabiliby condition and robustness of a feedback controlled system. The desired shape of these curves guide the control design by optimising the lvels and sloppes of the amplitude Bode-plot at low and high frequencies for suppression of the disturbances and of the base Bode-plot in the cross-over frequency region. This is called **loop shaping design**" >}}
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Section 4.3.3
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|
||||
> On might say that a high value of the unity-gain crossover frequency and corresponding high-frequency bandwidth limit is rather an unwanted side-effect of the required high loop-gain at lower frequencies, than a target for the design of a control system as such.
|
||||
|
||||
Section 9.3: Mass Dilemma
|
||||
|
||||
> A reduced mass requires improved system dynamics that enable a higher control bandwidth to compensate for the increase sensitivity for external vibrations.
|
||||
|
||||
@ -16,10 +16,54 @@ Author(s)
|
||||
Year
|
||||
: 2007
|
||||
|
||||
<div style="display: none"> \(
|
||||
% H Infini
|
||||
\newcommand{\hinf}{\mathcal{H}_\infty}
|
||||
% H 2
|
||||
\newcommand{\htwo}{\mathcal{H}_2}
|
||||
% Omega
|
||||
\newcommand{\w}{\omega}
|
||||
% H-Infinity Norm
|
||||
\newcommand{\hnorm}[1]{\left\|#1\right\|_{\infty}}
|
||||
% H-2 Norm
|
||||
\newcommand{\normtwo}[1]{\left\|#1\right\|_{2}}
|
||||
% Norm
|
||||
\newcommand{\norm}[1]{\left\|#1\right\|}
|
||||
% Absolute value
|
||||
\newcommand{\abs}[1]{\left\lvert#1\right\lvert}
|
||||
% Maximum for all omega
|
||||
\newcommand{\maxw}{\text{max}_{\omega}}
|
||||
% Maximum singular value
|
||||
\newcommand{\maxsv}{\overline{\sigma}}
|
||||
% Minimum singular value
|
||||
\newcommand{\minsv}{\underline{\sigma}}
|
||||
% Under bar
|
||||
\newcommand{\ubar}[1]{\text{\b{$#1$}}}
|
||||
% Diag keyword
|
||||
\newcommand{\diag}[1]{\text{diag}\{{#1}\}}
|
||||
% Vector
|
||||
\newcommand{\colvec}[1]{\begin{bmatrix}#1\end{bmatrix}}
|
||||
\)</div>
|
||||
|
||||
<div style="display: none"> \(
|
||||
\newcommand{\tcmbox}[1]{\boxed{#1}}
|
||||
% Simulate SIunitx
|
||||
\newcommand{\SI}[2]{#1\,#2}
|
||||
\newcommand{\ang}[1]{#1^{\circ}}
|
||||
\newcommand{\degree}{^{\circ}}
|
||||
\newcommand{\radian}{\text{rad}}
|
||||
\newcommand{\percent}{\%}
|
||||
\newcommand{\decibel}{\text{dB}}
|
||||
\newcommand{\per}{/}
|
||||
% Bug with subequations
|
||||
\newcommand{\eatLabel}[2]{}
|
||||
\newenvironment{subequations}{\eatLabel}{}
|
||||
\)</div>
|
||||
|
||||
|
||||
## Introduction {#introduction}
|
||||
|
||||
<a id="org43a1dd8"></a>
|
||||
<a id="orga7066d6"></a>
|
||||
|
||||
|
||||
### The Process of Control System Design {#the-process-of-control-system-design}
|
||||
@ -190,7 +234,7 @@ Notations used throughout this note are summarized in tables [table:notatio
|
||||
|
||||
## Classical Feedback Control {#classical-feedback-control}
|
||||
|
||||
<a id="org49cc073"></a>
|
||||
<a id="orgaabb1e5"></a>
|
||||
|
||||
|
||||
### Frequency Response {#frequency-response}
|
||||
@ -239,7 +283,7 @@ Thus, the input to the plant is \\(u = K(s) (r-y-n)\\).
|
||||
The objective of control is to manipulate \\(u\\) (design \\(K\\)) such that the control error \\(e\\) remains small in spite of disturbances \\(d\\).
|
||||
The control error is defined as \\(e = y-r\\).
|
||||
|
||||
<a id="org594420f"></a>
|
||||
<a id="org3b71acb"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_classical_feedback_alt.png" caption="Figure 1: Configuration for one degree-of-freedom control" >}}
|
||||
|
||||
@ -551,7 +595,7 @@ We cannot achieve both of these simultaneously with a single feedback controller
|
||||
|
||||
The solution is to use a **two degrees of freedom controller** where the reference signal \\(r\\) and output measurement \\(y\_m\\) are independently treated by the controller (Fig. [fig:classical_feedback_2dof_alt](#fig:classical_feedback_2dof_alt)), rather than operating on their difference \\(r - y\_m\\).
|
||||
|
||||
<a id="org658969c"></a>
|
||||
<a id="org9265d45"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_alt.png" caption="Figure 2: 2 degrees-of-freedom control architecture" >}}
|
||||
|
||||
@ -560,7 +604,7 @@ The controller can be slit into two separate blocks (Fig. [fig:classical_fe
|
||||
- the **feedback controller** \\(K\_y\\) that is used to **reduce the effect of uncertainty** (disturbances and model errors)
|
||||
- the **prefilter** \\(K\_r\\) that **shapes the commands** \\(r\\) to improve tracking performance
|
||||
|
||||
<a id="org8799861"></a>
|
||||
<a id="org0e3d8d7"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_classical_feedback_sep.png" caption="Figure 3: 2 degrees-of-freedom control architecture with two separate blocs" >}}
|
||||
|
||||
@ -629,7 +673,7 @@ With (see Fig. [fig:performance_weigth](#fig:performance_weigth)):
|
||||
|
||||
</div>
|
||||
|
||||
<a id="org20f2d1e"></a>
|
||||
<a id="org0656ee4"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_weight_first_order.png" caption="Figure 4: Inverse of performance weight" >}}
|
||||
|
||||
@ -653,7 +697,7 @@ After selecting the form of \\(N\\) and the weights, the \\(\hinf\\) optimal con
|
||||
|
||||
## Introduction to Multivariable Control {#introduction-to-multivariable-control}
|
||||
|
||||
<a id="org3b68fc1"></a>
|
||||
<a id="org25e187e"></a>
|
||||
|
||||
|
||||
### Introduction {#introduction}
|
||||
@ -696,7 +740,7 @@ For negative feedback system (Fig. [fig:classical_feedback_bis](#fig:classi
|
||||
- \\(S \triangleq (I + L)^{-1}\\) is the transfer function from \\(d\_1\\) to \\(y\\)
|
||||
- \\(T \triangleq L(I + L)^{-1}\\) is the transfer function from \\(r\\) to \\(y\\)
|
||||
|
||||
<a id="orgb1e90db"></a>
|
||||
<a id="org10be303"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_classical_feedback_bis.png" caption="Figure 5: Conventional negative feedback control system" >}}
|
||||
|
||||
@ -1011,7 +1055,7 @@ The **structured singular value** \\(\mu\\) is a tool for analyzing the effects
|
||||
|
||||
The general control problem formulation is represented in Fig. [fig:general_control_names](#fig:general_control_names).
|
||||
|
||||
<a id="orge52557f"></a>
|
||||
<a id="org410e618"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_general_control_names.png" caption="Figure 6: General control configuration" >}}
|
||||
|
||||
@ -1041,7 +1085,7 @@ We consider:
|
||||
- The weighted or normalized exogenous inputs \\(w\\) (where \\(\tilde{w} = W\_w w\\) consists of the "physical" signals entering the system)
|
||||
- The weighted or normalized controlled outputs \\(z = W\_z \tilde{z}\\) (where \\(\tilde{z}\\) often consists of the control error \\(y-r\\) and the manipulated input \\(u\\))
|
||||
|
||||
<a id="orga94c007"></a>
|
||||
<a id="org98354a0"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_general_plant_weights.png" caption="Figure 7: General Weighted Plant" >}}
|
||||
|
||||
@ -1084,7 +1128,7 @@ where \\(F\_l(P, K)\\) denotes a **lower linear fractional transformation** (LFT
|
||||
|
||||
The general control configuration may be extended to include model uncertainty as shown in Fig. [fig:general_config_model_uncertainty](#fig:general_config_model_uncertainty).
|
||||
|
||||
<a id="org4adeb70"></a>
|
||||
<a id="orgaee1f77"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_general_control_Mdelta.png" caption="Figure 8: General control configuration for the case with model uncertainty" >}}
|
||||
|
||||
@ -1112,7 +1156,7 @@ MIMO systems are often **more sensitive to uncertainty** than SISO systems.
|
||||
|
||||
## Elements of Linear System Theory {#elements-of-linear-system-theory}
|
||||
|
||||
<a id="orgad24d7e"></a>
|
||||
<a id="orgb820714"></a>
|
||||
|
||||
|
||||
### System Descriptions {#system-descriptions}
|
||||
@ -1398,7 +1442,7 @@ RHP-zeros therefore imply high gain instability.
|
||||
|
||||
### Internal Stability of Feedback Systems {#internal-stability-of-feedback-systems}
|
||||
|
||||
<a id="org0711897"></a>
|
||||
<a id="orgb1e4209"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_classical_feedback_stability.png" caption="Figure 9: Block diagram used to check internal stability" >}}
|
||||
|
||||
@ -1545,7 +1589,7 @@ It may be shown that the Hankel norm is equal to \\(\left\\|G(s)\right\\|\_H = \
|
||||
|
||||
## Limitations on Performance in SISO Systems {#limitations-on-performance-in-siso-systems}
|
||||
|
||||
<a id="orga81403c"></a>
|
||||
<a id="org76a7a2f"></a>
|
||||
|
||||
|
||||
### Input-Output Controllability {#input-output-controllability}
|
||||
@ -1937,7 +1981,7 @@ Uncertainty in the crossover frequency region can result in poor performance and
|
||||
|
||||
### Summary: Controllability Analysis with Feedback Control {#summary-controllability-analysis-with-feedback-control}
|
||||
|
||||
<a id="org4107db4"></a>
|
||||
<a id="org4ab6880"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_classical_feedback_meas.png" caption="Figure 10: Feedback control system" >}}
|
||||
|
||||
@ -1966,7 +2010,7 @@ In summary:
|
||||
Sometimes, the disturbances are so large that we hit input saturation or the required bandwidth is not achievable. To avoid the latter problem, we must at least require that the effect of the disturbance is less than \\(1\\) at frequencies beyond the bandwidth:
|
||||
\\[ \abs{G\_d(j\w)} < 1 \quad \forall \w \geq \w\_c \\]
|
||||
|
||||
<a id="orgd603c07"></a>
|
||||
<a id="orga143a9d"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_margin_requirements.png" caption="Figure 11: Illustration of controllability requirements" >}}
|
||||
|
||||
@ -1988,7 +2032,7 @@ The rules may be used to **determine whether or not a given plant is controllabl
|
||||
|
||||
## Limitations on Performance in MIMO Systems {#limitations-on-performance-in-mimo-systems}
|
||||
|
||||
<a id="org2ba9bd9"></a>
|
||||
<a id="org6b25e5b"></a>
|
||||
|
||||
|
||||
### Introduction {#introduction}
|
||||
@ -2299,7 +2343,7 @@ We here focus on input and output uncertainty.
|
||||
In multiplicative form, the input and output uncertainties are given by (see Fig. [fig:input_output_uncertainty](#fig:input_output_uncertainty)):
|
||||
\\[ G^\prime = (I + E\_O) G (I + E\_I) \\]
|
||||
|
||||
<a id="orgcf627e9"></a>
|
||||
<a id="org367c804"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_input_output_uncertainty.png" caption="Figure 12: Plant with multiplicative input and output uncertainty" >}}
|
||||
|
||||
@ -2435,7 +2479,7 @@ However, the situation is usually the opposite with model uncertainty because fo
|
||||
|
||||
## Uncertainty and Robustness for SISO Systems {#uncertainty-and-robustness-for-siso-systems}
|
||||
|
||||
<a id="org83d3f33"></a>
|
||||
<a id="org80d55a0"></a>
|
||||
|
||||
|
||||
### Introduction to Robustness {#introduction-to-robustness}
|
||||
@ -2509,7 +2553,7 @@ which may be represented by the diagram in Fig. [fig:input_uncertainty_set]
|
||||
|
||||
</div>
|
||||
|
||||
<a id="orgc4dec07"></a>
|
||||
<a id="org865770b"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_set.png" caption="Figure 13: Plant with multiplicative uncertainty" >}}
|
||||
|
||||
@ -2563,7 +2607,7 @@ To illustrate how parametric uncertainty translate into frequency domain uncerta
|
||||
In general, these uncertain regions have complicated shapes and complex mathematical descriptions
|
||||
- **Step 2**. We therefore approximate such complex regions as discs, resulting in a **complex additive uncertainty description**
|
||||
|
||||
<a id="org6f945a2"></a>
|
||||
<a id="org168b9ff"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_uncertainty_region.png" caption="Figure 14: Uncertainty regions of the Nyquist plot at given frequencies" >}}
|
||||
|
||||
@ -2586,7 +2630,7 @@ At each frequency, all possible \\(\Delta(j\w)\\) "generates" a disc-shaped regi
|
||||
|
||||
</div>
|
||||
|
||||
<a id="orgdc40ec3"></a>
|
||||
<a id="org46ced8b"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_uncertainty_disc_generated.png" caption="Figure 15: Disc-shaped uncertainty regions generated by complex additive uncertainty" >}}
|
||||
|
||||
@ -2643,7 +2687,7 @@ To derive \\(w\_I(s)\\), we then try to find a simple weight so that \\(\abs{w\_
|
||||
|
||||
</div>
|
||||
|
||||
<a id="org7fa26f9"></a>
|
||||
<a id="org797da76"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_uncertainty_weight.png" caption="Figure 16: Relative error for 27 combinations of \\(k,\ \tau\\) and \\(\theta\\). Solid and dashed lines: two weights \\(\abs{w\_I}\\)" >}}
|
||||
|
||||
@ -2682,7 +2726,7 @@ The magnitude of the relative uncertainty caused by neglecting the dynamics in \
|
||||
Let \\(f(s) = e^{-\theta\_p s}\\), where \\(0 \le \theta\_p \le \theta\_{\text{max}}\\). We want to represent \\(G\_p(s) = G\_0(s)e^{-\theta\_p s}\\) by a delay-free plant \\(G\_0(s)\\) and multiplicative uncertainty. Let first consider the maximum delay, for which the relative error \\(\abs{1 - e^{-j \w \theta\_{\text{max}}}}\\) is shown as a function of frequency (Fig. [fig:neglected_time_delay](#fig:neglected_time_delay)). If we consider all \\(\theta \in [0, \theta\_{\text{max}}]\\) then:
|
||||
\\[ l\_I(\w) = \begin{cases} \abs{1 - e^{-j\w\theta\_{\text{max}}}} & \w < \pi/\theta\_{\text{max}} \\ 2 & \w \ge \pi/\theta\_{\text{max}} \end{cases} \\]
|
||||
|
||||
<a id="orgcac6bc4"></a>
|
||||
<a id="org8ddf130"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_neglected_time_delay.png" caption="Figure 17: Neglected time delay" >}}
|
||||
|
||||
@ -2692,7 +2736,7 @@ Let \\(f(s) = e^{-\theta\_p s}\\), where \\(0 \le \theta\_p \le \theta\_{\text{m
|
||||
Let \\(f(s) = 1/(\tau\_p s + 1)\\), where \\(0 \le \tau\_p \le \tau\_{\text{max}}\\). In this case the resulting \\(l\_I(\w)\\) (Fig. [fig:neglected_first_order_lag](#fig:neglected_first_order_lag)) can be represented by a rational transfer function with \\(\abs{w\_I(j\w)} = l\_I(\w)\\) where
|
||||
\\[ w\_I(s) = \frac{\tau\_{\text{max}} s}{\tau\_{\text{max}} s + 1} \\]
|
||||
|
||||
<a id="org119416a"></a>
|
||||
<a id="orge3ddb3c"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_neglected_first_order_lag.png" caption="Figure 18: Neglected first-order lag uncertainty" >}}
|
||||
|
||||
@ -2709,7 +2753,7 @@ However, as shown in Fig. [fig:lag_delay_uncertainty](#fig:lag_delay_uncert
|
||||
|
||||
It is suggested to start with the simple weight and then if needed, to try the higher order weight.
|
||||
|
||||
<a id="orge82f57e"></a>
|
||||
<a id="orgb652b95"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_lag_delay_uncertainty.png" caption="Figure 19: Multiplicative weight for gain and delay uncertainty" >}}
|
||||
|
||||
@ -2749,7 +2793,7 @@ We use the Nyquist stability condition to test for robust stability of the close
|
||||
&\Longleftrightarrow \quad L\_p \ \text{should not encircle -1}, \ \forall L\_p
|
||||
\end{align\*}
|
||||
|
||||
<a id="org0430058"></a>
|
||||
<a id="org0fda45b"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback.png" caption="Figure 20: Feedback system with multiplicative uncertainty" >}}
|
||||
|
||||
@ -2765,7 +2809,7 @@ Encirclements are avoided if none of the discs cover \\(-1\\), and we get:
|
||||
&\Leftrightarrow \quad \abs{w\_I T} < 1, \ \forall\w \\\\\\
|
||||
\end{align\*}
|
||||
|
||||
<a id="org40d7367"></a>
|
||||
<a id="org4ead586"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_nyquist_uncertainty.png" caption="Figure 21: Nyquist plot of \\(L\_p\\) for robust stability" >}}
|
||||
|
||||
@ -2803,7 +2847,7 @@ And we obtain the same condition as before.
|
||||
We will derive a corresponding RS-condition for feedback system with inverse multiplicative uncertainty (Fig. [fig:inverse_uncertainty_set](#fig:inverse_uncertainty_set)) in which
|
||||
\\[ G\_p = G(1 + w\_{iI}(s) \Delta\_{iI})^{-1} \\]
|
||||
|
||||
<a id="org0cbbf1c"></a>
|
||||
<a id="orgaad9987"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_inverse_uncertainty_set.png" caption="Figure 22: Feedback system with inverse multiplicative uncertainty" >}}
|
||||
|
||||
@ -2855,7 +2899,7 @@ The condition for nominal performance when considering performance in terms of t
|
||||
Now \\(\abs{1 + L}\\) represents at each frequency the distance of \\(L(j\omega)\\) from the point \\(-1\\) in the Nyquist plot, so \\(L(j\omega)\\) must be at least a distance of \\(\abs{w\_P(j\omega)}\\) from \\(-1\\).
|
||||
This is illustrated graphically in Fig. [fig:nyquist_performance_condition](#fig:nyquist_performance_condition).
|
||||
|
||||
<a id="orga872ba6"></a>
|
||||
<a id="org8e66342"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_nyquist_performance_condition.png" caption="Figure 23: Nyquist plot illustration of the nominal performance condition \\(\abs{w\_P} < \abs{1 + L}\\)" >}}
|
||||
|
||||
@ -2880,7 +2924,7 @@ Let's consider the case of multiplicative uncertainty as shown on Fig. [fig
|
||||
The robust performance corresponds to requiring \\(\abs{\hat{y}/d}<1\ \forall \Delta\_I\\) and the set of possible loop transfer functions is
|
||||
\\[ L\_p = G\_p K = L (1 + w\_I \Delta\_I) = L + w\_I L \Delta\_I \\]
|
||||
|
||||
<a id="orga55c360"></a>
|
||||
<a id="org3ca06cf"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback_weight_bis.png" caption="Figure 24: Diagram for robust performance with multiplicative uncertainty" >}}
|
||||
|
||||
@ -3046,7 +3090,7 @@ with \\(\Phi(s) \triangleq (sI - A)^{-1}\\).
|
||||
|
||||
This is illustrated in the block diagram of Fig. [fig:uncertainty_state_a_matrix](#fig:uncertainty_state_a_matrix), which is in the form of an inverse additive perturbation.
|
||||
|
||||
<a id="org7061c4c"></a>
|
||||
<a id="orgd286b2a"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_uncertainty_state_a_matrix.png" caption="Figure 25: Uncertainty in state space A-matrix" >}}
|
||||
|
||||
@ -3064,7 +3108,7 @@ We also derived a condition for robust performance with multiplicative uncertain
|
||||
|
||||
## Robust Stability and Performance Analysis {#robust-stability-and-performance-analysis}
|
||||
|
||||
<a id="orga4090db"></a>
|
||||
<a id="orgb076a9b"></a>
|
||||
|
||||
|
||||
### General Control Configuration with Uncertainty {#general-control-configuration-with-uncertainty}
|
||||
@ -3075,13 +3119,13 @@ where each \\(\Delta\_i\\) represents a **specific source of uncertainty**, e.g.
|
||||
|
||||
If we also pull out the controller \\(K\\), we get the generalized plant \\(P\\) as shown in Fig. [fig:general_control_delta](#fig:general_control_delta). This form is useful for controller synthesis.
|
||||
|
||||
<a id="orgeddb6de"></a>
|
||||
<a id="org0853688"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_general_control_delta.png" caption="Figure 26: General control configuration used for controller synthesis" >}}
|
||||
|
||||
If the controller is given and we want to analyze the uncertain system, we use the \\(N\Delta\text{-structure}\\) in Fig. [fig:general_control_Ndelta](#fig:general_control_Ndelta).
|
||||
|
||||
<a id="org40e5b3e"></a>
|
||||
<a id="orgc524251"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_general_control_Ndelta.png" caption="Figure 27: \\(N\Delta\text{-structure}\\) for robust performance analysis" >}}
|
||||
|
||||
@ -3101,7 +3145,7 @@ Similarly, the uncertain closed-loop transfer function from \\(w\\) to \\(z\\),
|
||||
|
||||
To analyze robust stability of \\(F\\), we can rearrange the system into the \\(M\Delta\text{-structure}\\) shown in Fig. [fig:general_control_Mdelta_bis](#fig:general_control_Mdelta_bis) where \\(M = N\_{11}\\) is the transfer function from the output to the input of the perturbations.
|
||||
|
||||
<a id="org35c320b"></a>
|
||||
<a id="orge0e68f2"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_general_control_Mdelta_bis.png" caption="Figure 28: \\(M\Delta\text{-structure}\\) for robust stability analysis" >}}
|
||||
|
||||
@ -3153,7 +3197,7 @@ Three common forms of **feedforward unstructured uncertainty** are shown Fig.&nb
|
||||
|
||||
|  |  |  |
|
||||
|----------------------------------------------------|----------------------------------------------------------|-----------------------------------------------------------|
|
||||
| <a id="org44a89ed"></a> Additive uncertainty | <a id="org22a596e"></a> Multiplicative input uncertainty | <a id="org34aa45a"></a> Multiplicative output uncertainty |
|
||||
| <a id="org94556ee"></a> Additive uncertainty | <a id="org205e138"></a> Multiplicative input uncertainty | <a id="org884d99b"></a> Multiplicative output uncertainty |
|
||||
|
||||
In Fig. [fig:feedback_uncertainty](#fig:feedback_uncertainty), three **feedback or inverse unstructured uncertainty** forms are shown: inverse additive uncertainty, inverse multiplicative input uncertainty and inverse multiplicative output uncertainty.
|
||||
|
||||
@ -3176,7 +3220,7 @@ In Fig. [fig:feedback_uncertainty](#fig:feedback_uncertainty), three **feed
|
||||
|
||||
|  |  |  |
|
||||
|--------------------------------------------------------|------------------------------------------------------------------|-------------------------------------------------------------------|
|
||||
| <a id="org1808a4d"></a> Inverse additive uncertainty | <a id="org75e65aa"></a> Inverse multiplicative input uncertainty | <a id="org8c1d406"></a> Inverse multiplicative output uncertainty |
|
||||
| <a id="org17a4e6d"></a> Inverse additive uncertainty | <a id="org2765e1d"></a> Inverse multiplicative input uncertainty | <a id="org33356e1"></a> Inverse multiplicative output uncertainty |
|
||||
|
||||
|
||||
##### Lumping uncertainty into a single perturbation {#lumping-uncertainty-into-a-single-perturbation}
|
||||
@ -3246,7 +3290,7 @@ where \\(r\_0\\) is the relative uncertainty at steady-state, \\(1/\tau\\) is th
|
||||
Let's consider the feedback system with multiplicative input uncertainty \\(\Delta\_I\\) shown Fig. [fig:input_uncertainty_set_feedback_weight](#fig:input_uncertainty_set_feedback_weight).
|
||||
\\(W\_I\\) is a normalization weight for the uncertainty and \\(W\_P\\) is a performance weight.
|
||||
|
||||
<a id="org150edcc"></a>
|
||||
<a id="org2ebb26f"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_set_feedback_weight.png" caption="Figure 29: System with multiplicative input uncertainty and performance measured at the output" >}}
|
||||
|
||||
@ -3406,7 +3450,7 @@ Where \\(G = M\_l^{-1} N\_l\\) is a left coprime factorization of the nominal pl
|
||||
|
||||
This uncertainty description is surprisingly **general**, it allows both zeros and poles to cross into the right-half plane, and has proven to be very useful in applications.
|
||||
|
||||
<a id="org647e6f9"></a>
|
||||
<a id="org71a706b"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty.png" caption="Figure 30: Coprime Uncertainty" >}}
|
||||
|
||||
@ -3438,7 +3482,7 @@ where \\(d\_i\\) is a scalar and \\(I\_i\\) is an identity matrix of the same di
|
||||
Now rescale the inputs and outputs of \\(M\\) and \\(\Delta\\) by inserting the matrices \\(D\\) and \\(D^{-1}\\) on both sides as shown in Fig. [fig:block_diagonal_scalings](#fig:block_diagonal_scalings).
|
||||
This clearly has no effect on stability.
|
||||
|
||||
<a id="orgfd9a9f8"></a>
|
||||
<a id="org949fb62"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_block_diagonal_scalings.png" caption="Figure 31: Use of block-diagonal scalings, \\(\Delta D = D \Delta\\)" >}}
|
||||
|
||||
@ -3754,7 +3798,7 @@ with the decoupling controller we have:
|
||||
\\[ \bar{\sigma}(N\_{22}) = \bar{\sigma}(w\_P S) = \left|\frac{s/2 + 0.05}{s + 0.7}\right| \\]
|
||||
and we see from Fig. [fig:mu_plots_distillation](#fig:mu_plots_distillation) that the NP-condition is satisfied.
|
||||
|
||||
<a id="org573e9d8"></a>
|
||||
<a id="orga318a7a"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_mu_plots_distillation.png" caption="Figure 32: \\(\mu\text{-plots}\\) for distillation process with decoupling controller" >}}
|
||||
|
||||
@ -3779,7 +3823,7 @@ The peak value is close to 6, meaning that even with 6 times less uncertainty, t
|
||||
|
||||
We here consider the relationship between \\(\mu\\) for RP and the condition number of the plant or of the controller.
|
||||
We consider unstructured multiplicative uncertainty (i.e. \\(\Delta\_I\\) is a full matrix) and performance is measured in terms of the weighted sensitivity.
|
||||
With \\(N\\) given by [eq:n_delta_structure_clasic](#eq:n_delta_structure_clasic), we have:
|
||||
With \\(N\\) given by \eqref{eq:n_delta_structure_clasic}, we have:
|
||||
\\[ \overbrace{\mu\_{\tilde{\Delta}}(N)}^{\text{RP}} \le [ \overbrace{\bar{\sigma}(w\_I T\_I)}^{\text{RS}} + \overbrace{\bar{\sigma}(w\_P S)}^{\text{NP}} ] (1 + \sqrt{k}) \\]
|
||||
where \\(k\\) is taken as the smallest value between the condition number of the plant and of the controller:
|
||||
\\[ k = \text{min}(\gamma(G), \gamma(K)) \\]
|
||||
@ -3877,7 +3921,7 @@ The latter is an attempt to "flatten out" \\(\mu\\).
|
||||
For simplicity, we will consider again the case of multiplicative uncertainty and performance defined in terms of weighted sensitivity.
|
||||
The uncertainty weight \\(w\_I I\\) and performance weight \\(w\_P I\\) are shown graphically in Fig. [fig:weights_distillation](#fig:weights_distillation).
|
||||
|
||||
<a id="orgfb7536a"></a>
|
||||
<a id="orgd273607"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_weights_distillation.png" caption="Figure 33: Uncertainty and performance weights" >}}
|
||||
|
||||
@ -3900,11 +3944,11 @@ The scaling matrix \\(D\\) for \\(DND^{-1}\\) then has the structure \\(D = \tex
|
||||
- Iteration No. 3.
|
||||
Step 1: The \\(\mathcal{H}\_\infty\\) norm is only slightly reduced. We thus decide the stop the iterations.
|
||||
|
||||
<a id="org9fe8930"></a>
|
||||
<a id="org10a3970"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_dk_iter_mu.png" caption="Figure 34: Change in \\(\mu\\) during DK-iteration" >}}
|
||||
|
||||
<a id="org7d6de99"></a>
|
||||
<a id="org400285f"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_dk_iter_d_scale.png" caption="Figure 35: Change in D-scale \\(d\_1\\) during DK-iteration" >}}
|
||||
|
||||
@ -3912,13 +3956,13 @@ The final \\(\mu\text{-curves}\\) for NP, RS and RP with the controller \\(K\_3\
|
||||
The objectives of RS and NP are easily satisfied.
|
||||
The peak value of \\(\mu\\) is just slightly over 1, so the performance specification \\(\bar{\sigma}(w\_P S\_p) < 1\\) is almost satisfied for all possible plants.
|
||||
|
||||
<a id="org22cca49"></a>
|
||||
<a id="org519a9ca"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_mu_plot_optimal_k3.png" caption="Figure 36: \\(mu\text{-plots}\\) with \\(\mu\\) \"optimal\" controller \\(K\_3\\)" >}}
|
||||
|
||||
To confirm that, 6 perturbed plants are used to compute the perturbed sensitivity functions shown in Fig. [fig:perturb_s_k3](#fig:perturb_s_k3).
|
||||
|
||||
<a id="orgbb4e9b9"></a>
|
||||
<a id="orgfcb21f2"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_perturb_s_k3.png" caption="Figure 37: Perturbed sensitivity functions \\(\bar{\sigma}(S^\prime)\\) using \\(\mu\\) \"optimal\" controller \\(K\_3\\). Lower solid line: nominal plant. Upper solid line: worst-case plant" >}}
|
||||
|
||||
@ -3973,7 +4017,7 @@ If resulting control performance is not satisfactory, one may switch to the seco
|
||||
|
||||
## Controller Design {#controller-design}
|
||||
|
||||
<a id="org8e7c092"></a>
|
||||
<a id="orga616dec"></a>
|
||||
|
||||
|
||||
### Trade-offs in MIMO Feedback Design {#trade-offs-in-mimo-feedback-design}
|
||||
@ -3993,7 +4037,7 @@ We have the following important relationships:
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
|
||||
<a id="org988fd36"></a>
|
||||
<a id="org8d4f22a"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_classical_feedback_small.png" caption="Figure 38: One degree-of-freedom feedback configuration" >}}
|
||||
|
||||
@ -4035,7 +4079,7 @@ Thus, over specified frequency ranges, it is relatively easy to approximate the
|
||||
|
||||
Typically, the open-loop requirements 1 and 3 are valid and important at low frequencies \\(0 \le \omega \le \omega\_l \le \omega\_B\\), while conditions 2, 4, 5 and 6 are conditions which are valid and important at high frequencies \\(\omega\_B \le \omega\_h \le \omega \le \infty\\), as illustrated in Fig. [fig:design_trade_off_mimo_gk](#fig:design_trade_off_mimo_gk).
|
||||
|
||||
<a id="orgd19569b"></a>
|
||||
<a id="org6e3f117"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_design_trade_off_mimo_gk.png" caption="Figure 39: Design trade-offs for the multivariable loop transfer function \\(GK\\)" >}}
|
||||
|
||||
@ -4092,7 +4136,7 @@ The solution to the LQG problem is then found by replacing \\(x\\) by \\(\hat{x}
|
||||
|
||||
We therefore see that the LQG problem and its solution can be separated into two distinct parts as illustrated in Fig. [fig:lqg_separation](#fig:lqg_separation): the optimal state feedback and the optimal state estimator (the Kalman filter).
|
||||
|
||||
<a id="org9f29f94"></a>
|
||||
<a id="org6f521b9"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_lqg_separation.png" caption="Figure 40: The separation theorem" >}}
|
||||
|
||||
@ -4142,7 +4186,7 @@ Where \\(Y\\) is the unique positive-semi definite solution of the algebraic Ric
|
||||
|
||||
</div>
|
||||
|
||||
<a id="org714f37d"></a>
|
||||
<a id="orgf0f14d9"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_lqg_kalman_filter.png" caption="Figure 41: The LQG controller and noisy plant" >}}
|
||||
|
||||
@ -4163,7 +4207,7 @@ It has the same degree (number of poles) as the plant.<br />
|
||||
|
||||
For the LQG-controller, as shown on Fig. [fig:lqg_kalman_filter](#fig:lqg_kalman_filter), it is not easy to see where to position the reference input \\(r\\) and how integral action may be included, if desired. Indeed, the standard LQG design procedure does not give a controller with integral action. One strategy is illustrated in Fig. [fig:lqg_integral](#fig:lqg_integral). Here, the control error \\(r-y\\) is integrated and the regulator \\(K\_r\\) is designed for the plant augmented with these integral states.
|
||||
|
||||
<a id="org7af49f1"></a>
|
||||
<a id="orgb7cfb99"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_lqg_integral.png" caption="Figure 42: LQG controller with integral action and reference input" >}}
|
||||
|
||||
@ -4176,18 +4220,18 @@ Their main limitation is that they can only be applied to minimum phase plants.
|
||||
|
||||
### \\(\htwo\\) and \\(\hinf\\) Control {#htwo--and--hinf--control}
|
||||
|
||||
<a id="org64e6be0"></a>
|
||||
<a id="org6da7635"></a>
|
||||
|
||||
|
||||
#### General Control Problem Formulation {#general-control-problem-formulation}
|
||||
|
||||
<a id="org4fb3ed6"></a>
|
||||
<a id="org1448cec"></a>
|
||||
There are many ways in which feedback design problems can be cast as \\(\htwo\\) and \\(\hinf\\) optimization problems.
|
||||
It is very useful therefore to have a **standard problem formulation** into which any particular problem may be manipulated.
|
||||
|
||||
Such a general formulation is afforded by the general configuration shown in Fig. [fig:general_control](#fig:general_control).
|
||||
|
||||
<a id="org6b116f5"></a>
|
||||
<a id="org3b91a51"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_general_control.png" caption="Figure 43: General control configuration" >}}
|
||||
|
||||
@ -4438,7 +4482,7 @@ The elements of the generalized plant are
|
||||
\end{array}
|
||||
\end{equation\*}
|
||||
|
||||
<a id="org9710d93"></a>
|
||||
<a id="org35551f8"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_dist_rejection.png" caption="Figure 44: \\(S/KS\\) mixed-sensitivity optimization in standard form (regulation)" >}}
|
||||
|
||||
@ -4447,7 +4491,7 @@ Here we consider a tracking problem.
|
||||
The exogenous input is a reference command \\(r\\), and the error signals are \\(z\_1 = -W\_1 e = W\_1 (r-y)\\) and \\(z\_2 = W\_2 u\\).
|
||||
As the regulation problem of Fig. [fig:mixed_sensitivity_dist_rejection](#fig:mixed_sensitivity_dist_rejection), we have that \\(z\_1 = W\_1 S w\\) and \\(z\_2 = W\_2 KS w\\).
|
||||
|
||||
<a id="org57086a7"></a>
|
||||
<a id="org55460a0"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_ref_tracking.png" caption="Figure 45: \\(S/KS\\) mixed-sensitivity optimization in standard form (tracking)" >}}
|
||||
|
||||
@ -4471,7 +4515,7 @@ The elements of the generalized plant are
|
||||
\end{array}
|
||||
\end{equation\*}
|
||||
|
||||
<a id="org41ecea3"></a>
|
||||
<a id="org007c976"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_mixed_sensitivity_s_t.png" caption="Figure 46: \\(S/T\\) mixed-sensitivity optimization in standard form" >}}
|
||||
|
||||
@ -4499,7 +4543,7 @@ The focus of attention has moved to the size of signals and away from the size a
|
||||
Weights are used to describe the expected or known frequency content of exogenous signals and the desired frequency content of error signals.
|
||||
Weights are also used if a perturbation is used to model uncertainty, as in Fig. [fig:input_uncertainty_hinf](#fig:input_uncertainty_hinf), where \\(G\\) represents the nominal model, \\(W\\) is a weighting function that captures the relative model fidelity over frequency, and \\(\Delta\\) represents unmodelled dynamics usually normalized such that \\(\hnorm{\Delta} < 1\\).
|
||||
|
||||
<a id="org093006a"></a>
|
||||
<a id="orgabff04a"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_input_uncertainty_hinf.png" caption="Figure 47: Multiplicative dynamic uncertainty model" >}}
|
||||
|
||||
@ -4521,7 +4565,7 @@ The problem can be cast as a standard \\(\hinf\\) optimization in the general co
|
||||
w = \begin{bmatrix}d\\r\\n\end{bmatrix},\ z = \begin{bmatrix}z\_1\\z\_2\end{bmatrix}, \ v = \begin{bmatrix}r\_s\\y\_m\end{bmatrix},\ u = u
|
||||
\end{equation\*}
|
||||
|
||||
<a id="org6632f75"></a>
|
||||
<a id="org5056f35"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_hinf_signal_based.png" caption="Figure 48: A signal-based \\(\hinf\\) control problem" >}}
|
||||
|
||||
@ -4532,7 +4576,7 @@ This problem is a non-standard \\(\hinf\\) optimization.
|
||||
It is a robust performance problem for which the \\(\mu\text{-synthesis}\\) procedure can be applied where we require the structured singular value:
|
||||
\\[ \mu(M(j\omega)) < 1, \quad \forall\omega \\]
|
||||
|
||||
<a id="orgfa2cdae"></a>
|
||||
<a id="org7befd92"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_hinf_signal_based_uncertainty.png" caption="Figure 49: A signal-based \\(\hinf\\) control problem with input multiplicative uncertainty" >}}
|
||||
|
||||
@ -4590,7 +4634,7 @@ For the perturbed feedback system of Fig. [fig:coprime_uncertainty_bis](#fi
|
||||
|
||||
Notice that \\(\gamma\\) is the \\(\hinf\\) norm from \\(\phi\\) to \\(\begin{bmatrix}u\\y\end{bmatrix}\\) and \\((I-GK)^{-1}\\) is the sensitivity function for this positive feedback arrangement.
|
||||
|
||||
<a id="org79cab4a"></a>
|
||||
<a id="org4f3b2f4"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_bis.png" caption="Figure 50: \\(\hinf\\) robust stabilization problem" >}}
|
||||
|
||||
@ -4631,13 +4675,13 @@ for a specified \\(\gamma > \gamma\_\text{min}\\), is given by
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
|
||||
The Matlab function `coprimeunc` can be used to generate the controller in [eq:control_coprime_factor](#eq:control_coprime_factor).
|
||||
It is important to emphasize that since we can compute \\(\gamma\_\text{min}\\) from [eq:gamma_min_coprime](#eq:gamma_min_coprime) we get an explicit solution by solving just two Riccati equations and avoid the \\(\gamma\text{-iteration}\\) needed to solve the general \\(\mathcal{H}\_\infty\\) problem.
|
||||
The Matlab function `coprimeunc` can be used to generate the controller in \eqref{eq:control_coprime_factor}.
|
||||
It is important to emphasize that since we can compute \\(\gamma\_\text{min}\\) from \eqref{eq:gamma_min_coprime} we get an explicit solution by solving just two Riccati equations and avoid the \\(\gamma\text{-iteration}\\) needed to solve the general \\(\mathcal{H}\_\infty\\) problem.
|
||||
|
||||
|
||||
#### A Systematic \\(\hinf\\) Loop-Shaping Design Procedure {#a-systematic--hinf--loop-shaping-design-procedure}
|
||||
|
||||
<a id="org556e0fc"></a>
|
||||
<a id="org929fa3b"></a>
|
||||
Robust stabilization alone is not much used in practice because the designer is not able to specify any performance requirements.
|
||||
|
||||
To do so, **pre and post compensators** are used to **shape the open-loop singular values** prior to robust stabilization of the "shaped" plant.
|
||||
@ -4650,7 +4694,7 @@ If \\(W\_1\\) and \\(W\_2\\) are the pre and post compensators respectively, the
|
||||
|
||||
as shown in Fig. [fig:shaped_plant](#fig:shaped_plant).
|
||||
|
||||
<a id="orga1726e5"></a>
|
||||
<a id="orgbc1e59e"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_shaped_plant.png" caption="Figure 51: The shaped plant and controller" >}}
|
||||
|
||||
@ -4687,7 +4731,7 @@ Systematic procedure for \\(\hinf\\) loop-shaping design:
|
||||
This is because the references do not directly excite the dynamics of \\(K\_s\\), which can result in large amounts of overshoot.
|
||||
The constant prefilter ensure a steady-state gain of \\(1\\) between \\(r\\) and \\(y\\), assuming integral action in \\(W\_1\\) or \\(G\\)
|
||||
|
||||
<a id="org8641bb9"></a>
|
||||
<a id="org83cf8d8"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_shapping_practical_implementation.png" caption="Figure 52: A practical implementation of the loop-shaping controller" >}}
|
||||
|
||||
@ -4713,7 +4757,7 @@ But in cases where stringent time-domain specifications are set on the output re
|
||||
A general two degrees-of-freedom feedback control scheme is depicted in Fig. [fig:classical_feedback_2dof_simple](#fig:classical_feedback_2dof_simple).
|
||||
The commands and feedbacks enter the controller separately and are independently processed.
|
||||
|
||||
<a id="org58fc63d"></a>
|
||||
<a id="org8f1d974"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_classical_feedback_2dof_simple.png" caption="Figure 53: General two degrees-of-freedom feedback control scheme" >}}
|
||||
|
||||
@ -4724,7 +4768,7 @@ The design problem is illustrated in Fig. [fig:coprime_uncertainty_hinf](#f
|
||||
The feedback part of the controller \\(K\_2\\) is designed to meet robust stability and disturbance rejection requirements.
|
||||
A prefilter is introduced to force the response of the closed-loop system to follow that of a specified model \\(T\_{\text{ref}}\\), often called the **reference model**.
|
||||
|
||||
<a id="org6a86c67"></a>
|
||||
<a id="orgd00d786"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_coprime_uncertainty_hinf.png" caption="Figure 54: Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping design problem" >}}
|
||||
|
||||
@ -4749,7 +4793,7 @@ The main steps required to synthesize a two degrees-of-freedom \\(\mathcal{H}\_\
|
||||
|
||||
The final two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller is illustrated in Fig. [fig:hinf_synthesis_2dof](#fig:hinf_synthesis_2dof).
|
||||
|
||||
<a id="org18530c5"></a>
|
||||
<a id="org3d681ec"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_hinf_synthesis_2dof.png" caption="Figure 55: Two degrees-of-freedom \\(\mathcal{H}\_\infty\\) loop-shaping controller" >}}
|
||||
|
||||
@ -4821,7 +4865,7 @@ where \\(u\_a\\) is the **actual plant input**, that is the measurement at the *
|
||||
|
||||
The situation is illustrated in Fig. [fig:weight_anti_windup](#fig:weight_anti_windup), where the actuators are each modeled by a unit gain and a saturation.
|
||||
|
||||
<a id="org0e606b3"></a>
|
||||
<a id="org3867b27"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_weight_anti_windup.png" caption="Figure 56: Self-conditioned weight \\(W\_1\\)" >}}
|
||||
|
||||
@ -4869,14 +4913,14 @@ Moreover, one should be careful about combining controller synthesis and analysi
|
||||
|
||||
## Controller Structure Design {#controller-structure-design}
|
||||
|
||||
<a id="org3d2d0b9"></a>
|
||||
<a id="org6fc0469"></a>
|
||||
|
||||
|
||||
### Introduction {#introduction}
|
||||
|
||||
In previous sections, we considered the general problem formulation in Fig. [fig:general_control_names_bis](#fig:general_control_names_bis) and stated that the controller design problem is to find a controller \\(K\\) which based on the information in \\(v\\), generates a control signal \\(u\\) which counteracts the influence of \\(w\\) on \\(z\\), thereby minimizing the closed loop norm from \\(w\\) to \\(z\\).
|
||||
|
||||
<a id="org3079cf1"></a>
|
||||
<a id="org366605e"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_general_control_names_bis.png" caption="Figure 57: General Control Configuration" >}}
|
||||
|
||||
@ -4911,7 +4955,7 @@ The reference value \\(r\\) is usually set at some higher layer in the control h
|
||||
|
||||
Additional layers are possible, as is illustrated in Fig. [fig:control_system_hierarchy](#fig:control_system_hierarchy) which shows a typical control hierarchy for a chemical plant.
|
||||
|
||||
<a id="org82916a6"></a>
|
||||
<a id="org42e952b"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_system_hierarchy.png" caption="Figure 58: Typical control system hierarchy in a chemical plant" >}}
|
||||
|
||||
@ -4933,7 +4977,7 @@ However, this solution is normally not used for a number a reasons, included the
|
||||
|
||||
|  |  |  |
|
||||
|--------------------------------------------------|--------------------------------------------------------------------------------|-------------------------------------------------------------|
|
||||
| <a id="org94698d3"></a> Open loop optimization | <a id="org0b81e20"></a> Closed-loop implementation with separate control layer | <a id="orgd6b172c"></a> Integrated optimization and control |
|
||||
| <a id="org6986695"></a> Open loop optimization | <a id="orgaae7402"></a> Closed-loop implementation with separate control layer | <a id="orge8ee4d7"></a> Integrated optimization and control |
|
||||
|
||||
|
||||
### Selection of Controlled Outputs {#selection-of-controlled-outputs}
|
||||
@ -5140,7 +5184,7 @@ A cascade control structure results when either of the following two situations
|
||||
|
||||
|  |  |
|
||||
|-------------------------------------------------------|---------------------------------------------------|
|
||||
| <a id="org9cde265"></a> Extra measurements \\(y\_2\\) | <a id="orgd964ccc"></a> Extra inputs \\(u\_2\\) |
|
||||
| <a id="org4e7be08"></a> Extra measurements \\(y\_2\\) | <a id="org1a947e7"></a> Extra inputs \\(u\_2\\) |
|
||||
|
||||
|
||||
#### Cascade Control: Extra Measurements {#cascade-control-extra-measurements}
|
||||
@ -5189,7 +5233,7 @@ With reference to the special (but common) case of cascade control shown in Fig.
|
||||
|
||||
</div>
|
||||
|
||||
<a id="org754439a"></a>
|
||||
<a id="org664489f"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_cascade_control.png" caption="Figure 59: Common case of cascade control where the primary output \\(y\_1\\) depends directly on the extra measurement \\(y\_2\\)" >}}
|
||||
|
||||
@ -5239,7 +5283,7 @@ We would probably tune the three controllers in the order \\(K\_2\\), \\(K\_3\\)
|
||||
|
||||
</div>
|
||||
|
||||
<a id="orga098898"></a>
|
||||
<a id="org12e1e27"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_cascade_control_two_layers.png" caption="Figure 60: Control configuration with two layers of cascade control" >}}
|
||||
|
||||
@ -5354,7 +5398,7 @@ We get:
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
<a id="org043573a"></a>
|
||||
<a id="orgffa343f"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_partial_control.png" caption="Figure 61: Partial Control" >}}
|
||||
|
||||
@ -5413,7 +5457,7 @@ The selection of \\(u\_2\\) and \\(y\_2\\) for use in the lower-layer control sy
|
||||
Consider the conventional cascade control system in Fig. [fig:cascade_extra_meas](#fig:cascade_extra_meas) where we have additional "secondary" measurements \\(y\_2\\) with no associated control objective, and the objective is to improve the control of \\(y\_1\\) by locally controlling \\(y\_2\\).
|
||||
The idea is that this should reduce the effect of disturbances and uncertainty on \\(y\_1\\).
|
||||
|
||||
From [eq:partial_control](#eq:partial_control), it follows that we should select \\(y\_2\\) and \\(u\_2\\) such that \\(\\|P\_d\\|\\) is small and at least smaller than \\(\\|G\_{d1}\\|\\).
|
||||
From \eqref{eq:partial_control}, it follows that we should select \\(y\_2\\) and \\(u\_2\\) such that \\(\\|P\_d\\|\\) is small and at least smaller than \\(\\|G\_{d1}\\|\\).
|
||||
These arguments particularly apply at high frequencies.
|
||||
More precisely, we want the input-output controllability of \\([P\_u\ P\_r]\\) with disturbance model \\(P\_d\\) to be better that of the plant \\([G\_{11}\ G\_{12}]\\) with disturbance model \\(G\_{d1}\\).
|
||||
|
||||
@ -5430,7 +5474,7 @@ A set of outputs \\(y\_1\\) may be left uncontrolled only if the effects of all
|
||||
|
||||
</div>
|
||||
|
||||
To evaluate the feasibility of partial control, one must for each choice of \\(y\_2\\) and \\(u\_2\\), rearrange the system as in [eq:partial_control_partitioning](#eq:partial_control_partitioning) and [eq:partial_control](#eq:partial_control), and compute \\(P\_d\\) using [eq:tight_control_y2](#eq:tight_control_y2).
|
||||
To evaluate the feasibility of partial control, one must for each choice of \\(y\_2\\) and \\(u\_2\\), rearrange the system as in \eqref{eq:partial_control_partitioning} and \eqref{eq:partial_control}, and compute \\(P\_d\\) using \eqref{eq:tight_control_y2}.
|
||||
|
||||
|
||||
#### Measurement Selection for Indirect Control {#measurement-selection-for-indirect-control}
|
||||
@ -5474,7 +5518,7 @@ Then to minimize the control error for the primary output, \\(J = \\|y\_1 - r\_1
|
||||
|
||||
In this section, \\(G(s)\\) is a square plant which is to be controlled using a diagonal controller (Fig. [fig:decentralized_diagonal_control](#fig:decentralized_diagonal_control)).
|
||||
|
||||
<a id="orge8c0a58"></a>
|
||||
<a id="org301990a"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/skogestad07_decentralized_diagonal_control.png" caption="Figure 62: Decentralized diagonal control of a \\(2 \times 2\\) plant" >}}
|
||||
|
||||
@ -5590,7 +5634,7 @@ We then derive **necessary conditions for stability** which may be used to elimi
|
||||
|
||||
For decentralized diagonal control, it is desirable that the system can be tuned and operated one loop at a time.
|
||||
Assume therefore that \\(G\\) is stable and each individual loop is stable by itself (\\(\tilde{S}\\) and \\(\tilde{T}\\) are stable).
|
||||
Using the **spectral radius condition** on the factorized \\(S\\) in [eq:S_factorization](#eq:S_factorization), we have that the overall system is stable (\\(S\\) is stable) if
|
||||
Using the **spectral radius condition** on the factorized \\(S\\) in \eqref{eq:S_factorization}, we have that the overall system is stable (\\(S\\) is stable) if
|
||||
|
||||
\begin{equation}
|
||||
\rho(E\tilde{T}(j\omega)) < 1, \forall\omega
|
||||
@ -5813,7 +5857,7 @@ For performance, we need \\(|1 + L\_i|\\) to be larger than each of these:
|
||||
|1 + L\_i| > \max\_{k,j}\\{|\tilde{g}\_{dik}|, |\gamma\_{ij}|\\}
|
||||
\end{equation}
|
||||
|
||||
To achieve stability of the individual loops, one must analyze \\(g\_{ii}(s)\\) to ensure that the bandwidth required by [eq:decent_contr_one_loop](#eq:decent_contr_one_loop) is achievable.
|
||||
To achieve stability of the individual loops, one must analyze \\(g\_{ii}(s)\\) to ensure that the bandwidth required by \eqref{eq:decent_contr_one_loop} is achievable.
|
||||
Note that RHP-zeros in the diagonal elements may limit achievable decentralized control, whereas they may not pose any problems for a multivariable controller.
|
||||
Since with decentralized control, we usually want to use simple controllers, the achievable bandwidth in each loop will be limited by the frequency where \\(\angle g\_{ii}\\) is \\(\SI{-180}{\degree}\\)</li>
|
||||
<li>Check for constraints by considering the elements of \\(G^{-1} G\_d\\) and make sure that they do not exceed one in magnitude within the frequency range where control is needed.
|
||||
@ -5831,7 +5875,7 @@ If the plant is not controllable, then one may consider another choice of pairin
|
||||
If one still cannot find any pairing which are controllable, then one should consider multivariable control.
|
||||
|
||||
<ol class="org-ol">
|
||||
<li value="7">If the chosen pairing is controllable, then [eq:decent_contr_one_loop](#eq:decent_contr_one_loop) tells us how large \\(|L\_i| = |g\_{ii} k\_i|\\) must be.
|
||||
<li value="7">If the chosen pairing is controllable, then \eqref{eq:decent_contr_one_loop} tells us how large \\(|L\_i| = |g\_{ii} k\_i|\\) must be.
|
||||
This can be used as a basis for designing the controller \\(k\_i(s)\\) for loop \\(i\\)</li>
|
||||
</ol>
|
||||
|
||||
@ -5852,7 +5896,7 @@ Thus sequential design may involve many iterations.
|
||||
|
||||
#### Conclusion on Decentralized Control {#conclusion-on-decentralized-control}
|
||||
|
||||
A number of **conditions for the stability**, e.g. [eq:decent_contr_cond_stability](#eq:decent_contr_cond_stability) and [eq:decent_contr_necessary_cond_stability](#eq:decent_contr_necessary_cond_stability), and **performance**, e.g. [eq:decent_contr_cond_perf_dist](#eq:decent_contr_cond_perf_dist) and [eq:decent_contr_cond_perf_ref](#eq:decent_contr_cond_perf_ref), of decentralized control systems have been derived.
|
||||
A number of **conditions for the stability**, e.g. \eqref{eq:decent_contr_cond_stability} and \eqref{eq:decent_contr_necessary_cond_stability}, and **performance**, e.g. \eqref{eq:decent_contr_cond_perf_dist} and \eqref{eq:decent_contr_cond_perf_ref}, of decentralized control systems have been derived.
|
||||
|
||||
The conditions may be useful in **determining appropriate pairings of inputs and outputs** and the **sequence in which the decentralized controllers should be designed**.
|
||||
|
||||
@ -5861,7 +5905,7 @@ The conditions are also useful in an **input-output controllability analysis** f
|
||||
|
||||
## Model Reduction {#model-reduction}
|
||||
|
||||
<a id="orga673906"></a>
|
||||
<a id="org7648c32"></a>
|
||||
|
||||
|
||||
### Introduction {#introduction}
|
||||
|
||||
51
content/paper/bryson93_contr_spacec_aircr.md
Normal file
51
content/paper/bryson93_contr_spacec_aircr.md
Normal file
@ -0,0 +1,51 @@
|
||||
+++
|
||||
title = "Control of spacecraft and aircraft"
|
||||
author = ["Thomas Dehaeze"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [HAC-HAC]({{< relref "hac_hac" >}})
|
||||
|
||||
Reference
|
||||
: <sup id="4970865a21830fff7b1daeec187bfa68"><a href="#bryson93_contr_spacec_aircr" title="Bryson, Control of Spacecraft and Aircraft, Princeton university press Princeton, New Jersey (1993).">(Bryson, 1993)</a></sup>
|
||||
|
||||
Author(s)
|
||||
: Bryson, A. E.
|
||||
|
||||
Year
|
||||
: 1993
|
||||
|
||||
|
||||
## 9.2.3 Roll-Off Filters {#9-dot-2-dot-3-roll-off-filters}
|
||||
|
||||
> Synthesizing control logic using only one vibration mode means we are consciously **neglecting the higher-order vibration modes**.
|
||||
> When doing this, it is a good idea to insert "roll-off" into the control logic, so that the loop-transfer gain decreases rapidly with frequency beyond the control bandwidth.
|
||||
> This reduces the possibility of destabilizing the unmodelled higher frequency dynamics ("**spillover**").
|
||||
|
||||
|
||||
## 9.5 Robust Compensator Synthesis {#9-dot-5-robust-compensator-synthesis}
|
||||
|
||||
> LQG synthesis using feedback of estimated states will produce almost the same good response as LQR [...] for systems with control system bandwidths that are well below the frequency of the first vibration mode.
|
||||
> However, it may not be true for systems with higher control system bandwidths, even when one or more vibration modes are included in the control design model.
|
||||
|
||||
<!--quoteend-->
|
||||
|
||||
> If a rate sensor is co-located with an actuator on a flexible body, and its signal is fed back to the actuator, all vibration modes are stabilized.
|
||||
> If a rate sensor is not co-located with an actuator on a flexible body, ans its signal is fed back to the actuator, some vibration modes are stabilized and others are destabilized, depending on the location of the sensor relative to the actuator.
|
||||
|
||||
|
||||
## 9.5.2 Low-Authority Control/High-Authority Control {#9-dot-5-dot-2-low-authority-control-high-authority-control}
|
||||
|
||||
> Figure [fig:bryson93_hac_lac](#fig:bryson93_hac_lac) shows the concept of Low-Authority Control/High-Authority Control (LAC/HAC) is the s-plane.
|
||||
> LAC uses a co-located rate sensor to add damping to all the vibratory modes (but not the rigid-body mode).
|
||||
> HAC uses a separated displacement sensor to stabilize the rigid body mode, which slightly decreases the damping of the vibratory modes but not enough to produce instability (called "spillover")
|
||||
|
||||
<a id="orgf5c85db"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/bryson93_hac_lac.png" caption="Figure 1: HAC-LAC control concept" >}}
|
||||
|
||||
> LAC/HAC is usually insensitive to small deviation of the plant dynamics away from the design values, that is, it is **robust** to plant parameter changes.
|
||||
|
||||
# Bibliography
|
||||
<a id="bryson93_contr_spacec_aircr"></a>Bryson, A. E., *Control of spacecraft and aircraft* (1993), : Princeton university press Princeton, New Jersey. [↩](#4970865a21830fff7b1daeec187bfa68)
|
||||
@ -33,7 +33,7 @@ Usually quoted as a percentage of the fill-scale range (FSR):
|
||||
|
||||
With \\(e\_m(v)\\) is the mapping error.
|
||||
|
||||
<a id="org64f54e9"></a>
|
||||
<a id="org18802f9"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/fleming13_mapping_error.png" caption="Figure 1: The actual position versus the output voltage of a position sensor. The calibration function \\(f\_{cal}(v)\\) is an approximation of the sensor mapping function \\(f\_a(v)\\) where \\(v\\) is the voltage resulting from a displacement \\(x\\). \\(e\_m(v)\\) is the residual error." >}}
|
||||
|
||||
@ -42,7 +42,7 @@ With \\(e\_m(v)\\) is the mapping error.
|
||||
|
||||
If the shape of the mapping function actually varies with time, the maximum error due to drift must be evaluated by finding the worst-case mapping error.
|
||||
|
||||
<a id="org81ab6a9"></a>
|
||||
<a id="org65fb6f9"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/fleming13_drift_stability.png" caption="Figure 2: The worst case range of a linear mapping function \\(f\_a(v)\\) for a given error in sensitivity and offset." >}}
|
||||
|
||||
@ -147,9 +147,9 @@ The empirical rule states that there is a \\(99.7\%\\) probability that a sample
|
||||
This if we define the resolution as \\(\delta = 6 \sigma\\), we will referred to as the \\(6\sigma\text{-resolution}\\).
|
||||
|
||||
Another important parameter that must be specified when quoting resolution is the sensor bandwidth.
|
||||
There is usually a trade-off between bandwidth and resolution (figure [3](#orgd8c6776)).
|
||||
There is usually a trade-off between bandwidth and resolution (figure [3](#org954f29f)).
|
||||
|
||||
<a id="orgd8c6776"></a>
|
||||
<a id="org954f29f"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/fleming13_tradeoff_res_bandwidth.png" caption="Figure 3: The resolution versus banwidth of a position sensor." >}}
|
||||
|
||||
|
||||
@ -41,6 +41,8 @@ In this paper, the piezoelectric actuator/electronics adds a time delay which is
|
||||
- **Low Stiffness** actuator is defined as the ones where the transmissibility stays below 0dB at all frequency
|
||||
- **High Stiffness** actuator is defined as the ones where the transmissibility goes above 0dB at some frequency
|
||||
|
||||
<a id="org6e18c94"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ito16_low_high_stiffness_actuators.png" caption="Figure 1: Definition of low-stiffness and high-stiffness actuator" >}}
|
||||
|
||||
|
||||
@ -52,6 +54,8 @@ In this paper, the piezoelectric actuator/electronics adds a time delay which is
|
||||
|
||||
## Controller Design {#controller-design}
|
||||
|
||||
<a id="orgc911fbe"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/ito16_transmissibility.png" caption="Figure 2: Obtained transmissibility" >}}
|
||||
|
||||
|
||||
|
||||
@ -16,5 +16,9 @@ Author(s)
|
||||
Year
|
||||
: 2018
|
||||
|
||||
<a id="org55ab131"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/oomen18_next_gen_loop_gain.png" caption="Figure 1: Envisaged developments in motion systems. In traditional motion systems, the control bandwidth takes place in the rigid-body region. In the next generation systemes, flexible dynamics are foreseen to occur within the control bandwidth." >}}
|
||||
|
||||
# Bibliography
|
||||
<a id="oomen18_advan_motion_contr_precis_mechat"></a>Oomen, T., *Advanced motion control for precision mechatronics: control, identification, and learning of complex systems*, IEEJ Journal of Industry Applications, *7(2)*, 127–140 (2018). http://dx.doi.org/10.1541/ieejjia.7.127 [↩](#73fd325bd20a6ef8972145e535f38198)
|
||||
|
||||
25
content/zettels/hac_hac.md
Normal file
25
content/zettels/hac_hac.md
Normal file
@ -0,0 +1,25 @@
|
||||
+++
|
||||
title = "HAC-HAC"
|
||||
author = ["Thomas Dehaeze"]
|
||||
draft = false
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
High-Authority Control/Low-Authority Control
|
||||
|
||||
From <sup id="454500a3af67ef66a7a754d1f2e1bd4a"><a href="#preumont18_vibrat_contr_activ_struc_fourt_edition" title="Andre Preumont, Vibration Control of Active Structures - Fourth Edition, Springer International Publishing (2018).">(Andre Preumont, 2018)</a></sup>:
|
||||
|
||||
> The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [1](#org2e37874). The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure. This approach has the following advantages:
|
||||
>
|
||||
> - The active damping extends outside the bandwidth of the HAC and reduces the settling time of the modes which are outsite the bandwidth
|
||||
> - The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin)
|
||||
> - The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin)
|
||||
|
||||
<a id="org2e37874"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/hac_lac_control_architecture.png" caption="Figure 1: HAC-LAC Control Architecture" >}}
|
||||
|
||||
# Bibliography
|
||||
<a id="preumont18_vibrat_contr_activ_struc_fourt_edition"></a>Preumont, A., *Vibration control of active structures - fourth edition* (2018), : Springer International Publishing. [↩](#454500a3af67ef66a7a754d1f2e1bd4a)
|
||||
@ -12,8 +12,8 @@ Tags
|
||||
|
||||
## Backlinks {#backlinks}
|
||||
|
||||
- [Multivariable feedback control: analysis and design]({{< relref "skogestad07_multiv_feedb_contr" >}})
|
||||
- [Multivariable control systems: an engineering approach]({{< relref "albertos04_multiv_contr_system" >}})
|
||||
- [Position control in lithographic equipment]({{< relref "butler11_posit_contr_lithog_equip" >}})
|
||||
- [Implementation challenges for multivariable control: what you did not learn in school!]({{< relref "garg07_implem_chall_multiv_contr" >}})
|
||||
- [Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods]({{< relref "li01_simul_fault_vibrat_isolat_point" >}})
|
||||
- [Multivariable control systems: an engineering approach]({{< relref "albertos04_multiv_contr_system" >}})
|
||||
- [Multivariable feedback control: analysis and design]({{< relref "skogestad07_multiv_feedb_contr" >}})
|
||||
|
||||
@ -17,13 +17,37 @@ Tags
|
||||
|
||||
## Relative Position Sensors {#relative-position-sensors}
|
||||
|
||||
<a id="orgf9f8137"></a>
|
||||
<a id="table--tab:characteristics-relative-sensor"></a>
|
||||
<div class="table-caption">
|
||||
<span class="table-number"><a href="#table--tab:characteristics-relative-sensor">Table 1</a></span>:
|
||||
Characteristics of relative measurement sensors <a class='org-ref-reference' href="#collette11_review">collette11_review</a>
|
||||
</div>
|
||||
|
||||
{{< figure src="/ox-hugo/position_sensor_characteristics_relative_sensor.png" caption="Figure 1: Characteristics of relative measurement sensors <sup id=\"642a18d86de4e062c6afb0f5f20501c4\"><a href=\"#collette11_review\" title=\"Collette, Artoos, Guinchard, Janssens, , Carmona Fernandez \& Hauviller, Review of sensors for low frequency seismic vibration measurement, cern, (2011).\">(Collette {\it et al.}, 2011)</a></sup>" >}}
|
||||
| Technology | Frequency (Hz) | Resolution [nm rms] | Range | Temperature Range [\\(^o \degree C\\)] |
|
||||
|----------------|----------------------------|---------------------|--------------------------|----------------------------------------|
|
||||
| LVDT | \\(\text{DC}-200\,[Hz]\\) | 10 | \\(1-10\,[mm]\\) | -50,100 |
|
||||
| Eddy current | \\(5\,[kHz]\\) | 0.1-100 | \\(0.5-55\,[mm]\\) | -50,100 |
|
||||
| Capacitive | \\(\text{DC}-100\,[kHz]\\) | 0.05-50 | \\(50\,[nm] - 1\,[cm]\\) | -40,100 |
|
||||
| Interferometer | \\(300\,[kHz]\\) | 0.1 | \\(10\,[cm]\\) | -250,100 |
|
||||
| Encoder | \\(\text{DC}-1\,[MHz]\\) | 1 | \\(7-27\,[mm]\\) | 0,40 |
|
||||
| Bragg Fibers | \\(\text{DC}-150\,[Hz]\\) | 0.3 | \\(3.5\,[cm]\\) | -30,80 |
|
||||
|
||||
<a id="org4ac51f1"></a>
|
||||
<a id="table--tab:summary-position-sensors"></a>
|
||||
<div class="table-caption">
|
||||
<span class="table-number"><a href="#table--tab:summary-position-sensors">Table 2</a></span>:
|
||||
Summary of position sensor characteristics. The dynamic range (DNR) and resolution are approximations based on a full-scale range of \(100\,\mu m\) and a first order bandwidth of \(1\,kHz\) <a class='org-ref-reference' href="#fleming13_review_nanom_resol_posit_sensor">fleming13_review_nanom_resol_posit_sensor</a> (<a href="fleming13_review_nanom_resol_posit_sensor.html">notes</a>)
|
||||
</div>
|
||||
|
||||
{{< figure src="/ox-hugo/position_sensor_characteristics.png" caption="Figure 2: Position sensor characteristics <sup id=\"3fb5b61524290e36d639a4fac65703d0\"><a href=\"#fleming13_review_nanom_resol_posit_sensor\" title=\"Andrew Fleming, A Review of Nanometer Resolution Position Sensors: Operation and Performance, {Sensors and Actuators A: Physical}, v(nil), 106-126 (2013).\">(Andrew Fleming, 2013)</a></sup>" >}}
|
||||
| Sensor Type | Range | DNR | Resolution | Max. BW | Accuracy |
|
||||
|----------------|----------------------------------|---------|------------|----------|-----------|
|
||||
| Metal foil | \\(10-500\,\mu m\\) | 230 ppm | 23 nm | 1-10 kHz | 1% FSR |
|
||||
| Piezoresistive | \\(1-500\,\mu m\\) | 5 ppm | 0.5 nm | >100 kHz | 1% FSR |
|
||||
| Capacitive | \\(10\,\mu m\\) to \\(10\,mm\\) | 24 ppm | 2.4 nm | 100 kHz | 0.1% FSR |
|
||||
| Electrothermal | \\(10\,\mu m\\) to \\(1\,mm\\) | 100 ppm | 10 nm | 10 kHz | 1% FSR |
|
||||
| Eddy current | \\(100\,\mu m\\) to \\(80\,mm\\) | 10 ppm | 1 nm | 40 kHz | 0.1% FSR |
|
||||
| LVDT | \\(0.5-500\,mm\\) | 10 ppm | 5 nm | 1 kHz | 0.25% FSR |
|
||||
| Interferometer | Meters | | 0.5 nm | >100kHz | 1 ppm FSR |
|
||||
| Encoder | Meters | | 6 nm | >100kHz | 5 ppm FSR |
|
||||
|
||||
|
||||
### Strain Gauge {#strain-gauge}
|
||||
@ -74,7 +98,7 @@ Description:
|
||||
| Keysight | [link](https://www.keysight.com/en/pc-1000000393%3Aepsg%3Apgr/laser-heads?nid=-536900395.0&cc=FR&lc=fre) |
|
||||
|
||||
<div class="table-caption">
|
||||
<span class="table-number">Table 1</span>:
|
||||
<span class="table-number">Table 3</span>:
|
||||
Characteristics of Environmental Units
|
||||
</div>
|
||||
|
||||
@ -84,13 +108,13 @@ Description:
|
||||
| Renishaw | 0.2 | 1 | 6 | 1 |
|
||||
| Picoscale | 0.2 | 2 | 2 | 1 |
|
||||
|
||||
Figure [3](#orgce716a4) is taken from
|
||||
Figure [1](#org1c8180d) is taken from
|
||||
<sup id="7658b1219a4458a62ae8c6f51b767542"><a href="#jang17_compen_refrac_index_air_laser" title="Yoon-Soo Jang \& Seung-Woo Kim, Compensation of the Refractive Index of Air in Laser Interferometer for Distance Measurement: a Review, {International Journal of Precision Engineering and
|
||||
Manufacturing}, v(12), 1881-1890 (2017).">(Yoon-Soo Jang \& Seung-Woo Kim, 2017)</a></sup>.
|
||||
|
||||
<a id="orgce716a4"></a>
|
||||
<a id="org1c8180d"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/position_sensor_interferometer_precision.png" caption="Figure 3: Expected precision of interferometer as a function of measured distance" >}}
|
||||
{{< figure src="/ox-hugo/position_sensor_interferometer_precision.png" caption="Figure 1: Expected precision of interferometer as a function of measured distance" >}}
|
||||
|
||||
|
||||
### Fiber Optic Displacement Sensor {#fiber-optic-displacement-sensor}
|
||||
@ -111,5 +135,5 @@ Figure [3](#orgce716a4) is taken from
|
||||
|
||||
## Backlinks {#backlinks}
|
||||
|
||||
- [A review of nanometer resolution position sensors: operation and performance]({{< relref "fleming13_review_nanom_resol_posit_sensor" >}})
|
||||
- [Measurement technologies for precision positioning]({{< relref "gao15_measur_techn_precis_posit" >}})
|
||||
- [A review of nanometer resolution position sensors: operation and performance]({{< relref "fleming13_review_nanom_resol_posit_sensor" >}})
|
||||
|
||||
@ -12,9 +12,9 @@ Tags
|
||||
|
||||
## Backlinks {#backlinks}
|
||||
|
||||
- [Multivariable feedback control: analysis and design]({{< relref "skogestad07_multiv_feedb_contr" >}})
|
||||
- [The design of high performance mechatronics - 2nd revised edition]({{< relref "schmidt14_desig_high_perfor_mechat_revis_edition" >}})
|
||||
- [Parallel robots : mechanics and control]({{< relref "taghirad13_paral" >}})
|
||||
- [Modal testing: theory, practice and application]({{< relref "ewins00_modal" >}})
|
||||
- [The art of electronics - third edition]({{< relref "horowitz15_art_of_elect_third_edition" >}})
|
||||
- [Vibration Control of Active Structures - Fourth Edition]({{< relref "preumont18_vibrat_contr_activ_struc_fourt_edition" >}})
|
||||
- [Parallel robots : mechanics and control]({{< relref "taghirad13_paral" >}})
|
||||
- [The design of high performance mechatronics - 2nd revised edition]({{< relref "schmidt14_desig_high_perfor_mechat_revis_edition" >}})
|
||||
- [Multivariable feedback control: analysis and design]({{< relref "skogestad07_multiv_feedb_contr" >}})
|
||||
|
||||
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