Update Content - 2024-12-17
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@@ -67,7 +67,7 @@ There are two radically different approached to disturbance rejection: feedback
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{{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="<span class=\"figure-number\">Figure 1: </span>Principle of feedback control" >}}
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The principle of feedback is represented on figure [1](#figure--fig:classical-feedback-small). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance.
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The principle of feedback is represented on [Figure 1](#figure--fig:classical-feedback-small). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance.
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The design problem consists of finding the appropriate compensator \\(K(s)\\) such that the closed-loop system is stable and behaves in the appropriate manner.
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In the control of lightly damped structures, feedback control is used for two distinct and complementary purposes: **active damping** and **model-based feedback**.
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@@ -94,14 +94,14 @@ The objective is to control a variable \\(y\\) to a desired value \\(r\\) in spi
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{{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="<span class=\"figure-number\">Figure 2: </span>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](#figure--fig:feedforward-adaptative).
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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](#figure--fig:feedforward-adaptative).
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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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There is no guarantee that the global response is reduced at other locations. This method is therefor considered as a local one.
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Because it is less sensitive to phase lag than feedback, it can be used at higher frequencies (\\(\omega\_c \approx \omega\_s/10\\)).
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The table [1](#table--tab:adv-dis-type-control) summarizes the main features of the two approaches.
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The [Table 1](#table--tab:adv-dis-type-control) summarizes the main features of the two approaches.
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<a id="table--tab:adv-dis-type-control"></a>
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<div class="table-caption">
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@@ -129,7 +129,7 @@ The table [1](#table--tab:adv-dis-type-control) summarizes the main features of
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{{< figure src="/ox-hugo/preumont18_design_steps.png" caption="<span class=\"figure-number\">Figure 3: </span>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](#figure--fig:design-steps).
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The various steps of the design of a controlled structure are shown in [Figure 3](#figure--fig:design-steps).
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The **starting point** is:
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@@ -156,7 +156,7 @@ If the dynamics of the sensors and actuators may significantly affect the behavi
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### Plant Description, Error and Control Budget {#plant-description-error-and-control-budget}
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From the block diagram of the control system (figure [4](#figure--fig:general-plant)):
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From the block diagram of the control system ([Figure 4](#figure--fig:general-plant)):
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\begin{align\*}
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y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\
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@@ -186,7 +186,7 @@ Even more interesting for the design is the **Cumulative Mean Square** response
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It is a monotonously decreasing function of frequency and describes the contribution of all frequencies above \\(\omega\\) to the mean-square value of \\(z\\).
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\\(\sigma\_z(0)\\) is then the global RMS response.
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A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#figure--fig:cas-plot).
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A typical plot of \\(\sigma\_z(\omega)\\) is shown [Figure 5](#figure--fig:cas-plot).
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It is useful to **identify the critical modes** in a design, at which the effort should be targeted.
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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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@@ -398,7 +398,7 @@ With:
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{{< figure src="/ox-hugo/preumont18_neglected_modes.png" caption="<span class=\"figure-number\">Figure 6: </span>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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If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on figure [6](#figure--fig:neglected-modes)).
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If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on [Figure 6](#figure--fig:neglected-modes)).
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And \\(G(\omega)\\) can be rewritten on terms of the **low frequency modes only**:
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\\[ G(\omega) \approx \sum\_{i=1}^m \frac{\phi\_i \phi\_i^T}{\mu\_i \omega\_i^2} D\_i(\omega) + R \\]
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@@ -436,7 +436,7 @@ The open-loop FRF of a collocated system corresponds to a diagonal component of
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If we assumes that the collocated system is undamped and is attached to the DoF \\(k\\), the open-loop FRF is purely real:
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\\[ G\_{kk}(\omega) = \sum\_{i=1}^m \frac{\phi\_i^2(k)}{\mu\_i (\omega\_i^2 - \omega^2)} + R\_{kk} \\]
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\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#figure--fig:collocated-control-frf)).
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\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) ([Figure 7](#figure--fig:collocated-control-frf)).
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<a id="figure--fig:collocated-control-frf"></a>
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@@ -451,7 +451,7 @@ For lightly damped structure, the poles and zeros are just moved a little bit in
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</div>
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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 [8](#figure--fig:collocated-zero).
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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 8](#figure--fig:collocated-zero).
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<a id="figure--fig:collocated-zero"></a>
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@@ -467,7 +467,7 @@ The open-loop poles are independant of the actuator and sensor configuration whi
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</div>
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By looking at figure [7](#figure--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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By looking at [Figure 7](#figure--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="figure--fig:alternating-p-z"></a>
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@@ -479,7 +479,7 @@ 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](#figure--fig:alternating-p-z). 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](#figure--fig:alternating-p-z). 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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@@ -501,7 +501,7 @@ Two broad categories of actuators can be distinguish:
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A voice coil transducer is an energy transformer which converts electrical power into mechanical power and vice versa.
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The system consists of (see figure [10](#figure--fig:voice-coil-schematic)):
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The system consists of (see [Figure 10](#figure--fig:voice-coil-schematic)):
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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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@@ -543,7 +543,7 @@ Thus, at any time, there is an equilibrium between the electrical power absorbed
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#### Proof-Mass Actuator {#proof-mass-actuator}
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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 [11](#figure--fig:proof-mass-actuator)).
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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 11](#figure--fig:proof-mass-actuator)).
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<a id="figure--fig:proof-mass-actuator"></a>
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@@ -574,7 +574,7 @@ with:
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</div>
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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 [12](#figure--fig:proof-mass-tf)).
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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 12](#figure--fig:proof-mass-tf)).
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<a id="figure--fig:proof-mass-tf"></a>
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@@ -610,7 +610,7 @@ Designing geophones with very low corner frequency is in general difficult. Acti
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### General Electromechanical Transducer {#general-electromechanical-transducer}
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The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#figure--fig:electro-mechanical-transducer).
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The consitutive behavior of a wide class of electromechanical transducers can be modelled as in [Figure 14](#figure--fig:electro-mechanical-transducer).
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<a id="figure--fig:electro-mechanical-transducer"></a>
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@@ -637,7 +637,7 @@ With:
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Equation <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 [15](#figure--fig:bridge-circuit) can be used.
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To do so, the bridge circuit as shown on [Figure 15](#figure--fig:bridge-circuit) can be used.
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We can show that
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@@ -655,7 +655,7 @@ which is indeed a linear function of the velocity \\(v\\) at the mechanical term
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### Smart Materials {#smart-materials}
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Smart materials have the ability to respond significantly to stimuli of different physical nature.
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Figure [16](#figure--fig:smart-materials) lists various effects that are observed in materials in response to various inputs.
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[Figure 16](#figure--fig:smart-materials) lists various effects that are observed in materials in response to various inputs.
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<a id="figure--fig:smart-materials"></a>
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@@ -748,7 +748,7 @@ 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 [17](#figure--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> 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 17](#figure--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> 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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@@ -789,7 +789,7 @@ Equation <eq:piezo_stack_eq> can be inverted to obtain
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#### Energy Stored in the Piezoelectric Transducer {#energy-stored-in-the-piezoelectric-transducer}
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Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#figure--fig:piezo-discrete).
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Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on [Figure 18](#figure--fig:piezo-discrete).
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The total power delivered to the transducer is the sum of electric power \\(V i\\) and the mechanical power \\(f \dot{\Delta}\\). The net work of the transducer is
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@@ -831,7 +831,7 @@ The ratio between the remaining stored energy and the initial stored energy is
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#### Admittance of the Piezoelectric Transducer {#admittance-of-the-piezoelectric-transducer}
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Consider the system of figure [19](#figure--fig:piezo-stack-admittance), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
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Consider the system of [Figure 19](#figure--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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<a id="figure--fig:piezo-stack-admittance"></a>
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@@ -853,7 +853,7 @@ 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> constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [20](#figure--fig:piezo-admittance-curve)).
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Equation <eq:distance_p_z> constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement ([Figure 20](#figure--fig:piezo-admittance-curve)).
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<a id="figure--fig:piezo-admittance-curve"></a>
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@@ -1552,7 +1552,7 @@ Their design requires a model of the structure, and there is usually a trade-off
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When collocated actuator/sensor pairs can be used, stability can be achieved using positivity concepts, but in many situations, collocated pairs are not feasible for HAC.
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The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#figure--fig:hac-lac-control).
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The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in [Figure 21](#figure--fig:hac-lac-control).
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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.
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This approach has the following advantages:
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