59 KiB
+++ title = "Vibration Control of Active Structures - Fourth Edition" author = ["Thomas Dehaeze"] draft = false +++
- Tags
- [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Reference Books]({{< relref "reference_books" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [HAC-HAC]({{< relref "hac_hac" >}})
- Reference
- (Andre Preumont, 2018)
- Author(s)
- Preumont, A.
- Year
- 2018
Introduction
Active Versus Passive
Active structure may be cheaper or lighter than passive structures of comparable performances; or they may offer performances that no passive structure could offer.
Active is not always better, and a control systems cannot compensate for a bad design. Active solution should be considered only after all other passive means have been exhausted.
Feedback control can compensate for external disturbances only in a limited frequency range (the bandwidth), the disturbances are actually amplified by the control system outside this frequency band.
Vibration Suppression
Vibration reduction can be achieved in many different ways:
- stiffening: consists of shifting the resonance frequency of the structure beyond the frequency band of excitation
- damping: consists of reducing the resonance peaks by dissipating the vibration energy
- isolation: consists of preventing the propagation of disturbances to sensitive parts of the system
The design of an active control system involves many issues such as how to configurate the sensors and actuators, how to secure stability and robustness. The power requirements will often determine the size of the actuators and the cost of the project.
Smart Materials and Structures
An active structure consists of a structure provided with a set of actuators and sensors coupled by a controller. If the bandwidth of the controller includes some vibration modes of the structure, its dynamic response must be considered.
If the set of actuators and sensors are located at discrete points of the structure, they can be treated separately. However, for smart structures, the actuators and sensors are often distributed and have a high degree of integration inside the structure, which makes a separate modelling impossible.
Some smart materials are:
- Shape Memory Alloys (SMA): recoverable strain of \(\SI{5}{\percent}\) induced by temperature. They can be used at low frequency and for low precision applications
- Piezoelectric materials: recoverable strain of \(\SI{0.1}{\percent}\) under electric field. They can be used as actuators as well as sensors. Two main classes: ceramics and polymers. Piezopolymers are used mostly as sensors as they require high voltage. The best-known piezoceramic is the Lead-Zirconate-Titanate (PZT).
- Magnetostrictive materials: recoverable strain of \(\SI{0.15}{\percent}\) under magnetic field
- Magneto-Rheological fluids (MR): consists of viscous fluids containing micronsized particules of magnetic material. When the fluid is subjected to a magnetic field, the particules create colunmar structures requiring a minimum shear stress to initiate the flow.
Control Strategies
There are two radically different approached to disturbance rejection: feedback and feedforward.
Feedback
{{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="Figure 1: Principle of feedback control" >}}
The principle of feedback is represented on 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. 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.
In the control of lightly damped structures, feedback control is used for two distinct and complementary purposes: active damping and model-based feedback.
Active Damping:
- The objective of active damping is to reduce the effect of resonant peaks on the response of the structure.
- From \(\frac{y}{d} = \frac{1}{1 + GK}\), this requires \(GK \gg 1\) near the resonances
- It can be generally be achieved without a model of the structure, with guaranteed stability, provided that the actuator and sensor are collocated and have perfect dynamics.
Model based feedback: The objective is to control a variable \(y\) to a desired value \(r\) in spite of the external disturbances \(d\).
- From \(\frac{y}{r} = \frac{GK}{1 + GK}\) we see that this requires large values of \(GK\) in the frequency range where \(y\approx r\) (bandwidth)
- The bandwidth \(\omega_c\) is limited by the accuracy of the model
- The disturbance rejection within the bandwidth of the control system is always compensated by an amplification of the disturbances outside the bandwidth
- When implemented digitally, the sampling frequency \(\omega_s\) must always be two orders of magnitude larger than \(\omega_c\) to preseve reasonably the behavior of the continuous system
Feedforward
{{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="Figure 2: Principle of feedforward control" >}}
The method relies on the availability of a reference signal correlated to the primary disturbance. 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.
The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized.
There is no guarantee that the global response is reduced at other locations. This method is therefor considered as a local one. Because it is less sensitive to phase lag than feedback, it can be used at higher frequencies (\(\omega_c \approx \omega_s/10\)).
The table 1 summarizes the main features of the two approaches.
| Advantages | Disadvantages | |
|---|---|---|
| Active Damping | - Simple to implement | - Effective only near resonance |
| - Does not required accurate model | ||
| - Guaranteed stability (collocated) | ||
| Model Based | - Global method | - Requires accurate model |
| - Attenuate all disturbance within bandwidth | - Limited bandwidth | |
| - Spillover | ||
| - Amplification of disturbances outside bandwidth | ||
| Feedforward Adaptive filtering | - No model is necessary | - Error signal required |
| - Robust to change in plant transfer function | - Local method: may amplify vibration elsewhere | |
| - More effective for narrowband disturbance | - Large amount of real-time computation |
The Various Steps of the Design
{{< figure src="/ox-hugo/preumont18_design_steps.png" caption="Figure 3: The various steps of the design" >}}
The various steps of the design of a controlled structure are shown in figure 3.
The starting point is:
- Mechanical system
- Performance objectives
- Specification of the disturbances
Then the open loop performances can be evaluated:
- The need for active control can be assessed
- The needed bandwidth can be roughly specified
The next step consist of selecting the proper type and location of sensors and actuators:
- The controllability and Observability are important concepts
A model of the structure is developped:
- FEM or identification
- Model reduction to limit the DoF
If the dynamics of the sensors and actuators may significantly affect the behavior of the system, they must be included in the model before the controller design.
Plant Description, Error and Control Budget
From the block diagram of the control system (figure fig:general_plant):
\begin{align*} y &= (I - G_{yu}H)^{-1} G_{yw} w\\\ z &= T_{zw} w = [G_{zw} + G_{zu}H(I - G_{yu}H)^{-1} G_{yw}] w \end{align*}
{{< figure src="/ox-hugo/preumont18_general_plant.png" caption="Figure 4: Block diagram of the control System" >}}
The frequency content of the disturbance \(w\) is usually described by its power spectral density \(\Phi_w (\omega)\) which describes the frequency distribution of the meas-square value.
\[\sigma_w = \sqrt{\int_0^\infty \Phi_w(\omega) d\omega}\]
Even more interesting for the design is the Cumulative Mean Square response defined by the integral of the PSD in the frequency range \([\omega, \infty[\).
\[\sigma_z^2(\omega) = \int_\omega^\infty \Phi_z(\nu) d\nu = \int_\omega^\infty |T_{zw}|^2 \Phi_w(\nu) d\nu \]
It is a monotonously decreasing function of frequency and describes the contribution of all frequencies above \(\omega\) to the mean-square value of \(z\). \(\sigma_z(0)\) is then the global RMS response.
A typical plot of \(\sigma_z(\omega)\) is shown figure fig:cas_plot. It is useful to identify the critical modes in a design, at which the effort should be targeted.
The diagram can also be used to assess the control laws and compare different actuator and sensor configuration.
{{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="Figure 5: Error budget distribution in OL and CL for increasing gains" >}}
Pseudo-inverse
Under-actuated System
Consider the linear system of equation: \[w = J v\] With:
- \(w\) a vector with \(m\) components (measurements)
- \(v\) a vector with \(n\) components (inputs)
- We assume \(m>n\) (under-actuated)
We seek the pseudo-inverse of \(J\) such that \(v = J^+ w\)
The columns of \(J\) are the influence function of the actuators. If the columns of \(J\) are independant, the Jacobian is full rang (\(r=n\)) and the Moore-Penrose pseudo inverse is: \[J^+ = (J^T J)^{-1} J^T\]
Over-actuated System
If there are more actuator than sensor (\(m<n\)), we obtain: \[J^+ = J^T(J J^T)^{-1}\]
Note that the Singular Value Decomposition offers a practical way to compute the pseudo-inverse, both for \(m>n\) and \(n>m\).
Singular Value Decomposition
The Singular Value Decomposition (SVD) is a generalization of the eigenvalue decomposition of a rectangular matrix: \[ J = U \Sigma V^T = \sum_{i=1}^r \sigma_i u_i v_i^T \] With:
- \(U\) and \(V\) orthogonal matrices. The columns \(u_i\) and \(v_i\) of \(U\) and \(V\) are the eigenvectors of the square matrices \(JJ^T\) and \(J^TJ\) respectively
- \(\Sigma\) a rectangular diagonal matrix of dimension \(m \times n\) containing the square root of the common non-zero eigenvalues of \(JJ^T\) and \(J^TJ\)
- \(r\) is the number of non-zero singular values of \(J\)
The pseudo-inverse of \(J\) is: \[ J^+ = V\Sigma^+U^T = \sum_{i=1}^r \frac{1}{\sigma_i} v_i u_i^T \]
The conditioning of the Jacobian is measured by the condition number: \[ c(J) = \frac{\sigma_{max}}{\sigma_{min}} \]
When \(c(J)\) becomes large, the most straightforward way to handle the ill-conditioning is to truncate the smallest singular value out of the sum. This will have usually little impact of the fitting error while reducing considerably the actuator inputs \(v\).
Some Concepts in Structural Dynamics
Equation of Motion of a Discrete System
The general form of the equation of motion governing the dynamic equilibrium between the external, elastic, inertia and damping forces acting on a discrete, flexible structure with a finite number \(n\) of degrees of freedom is
\begin{equation} M \ddot{x} + C \dot{x} + K x = f \end{equation}
With:
- \(x\) is the vector of generalized displacements (translations and rotations)
- \(f\) is the vector of generalized forces (point forces and torques)
- \(M\), \(C\) and \(K\) are respectively the mass, damping and stiffness matrices; they are symmetric and semi-positive definite
The damping matrix \(C\) represents the various dissipation mechanisms in the structure, which are usually poorly known. One of the popular hypotheses is the Rayleigh damping.
\begin{equation} C = \alpha M + \beta K \end{equation}
\(\alpha\) and \(\beta\) are selected to fit the structure under consideration.
Vibration Modes
Consider the free response of an undamped system of order \(n\): \[ M\ddot{x} + K x = 0 \]
If one tries a solution of the form \(x = \phi_i e^{j\omega_i t}\), \(\phi_i\) and \(\omega_i\) must statisfy the eigenvalue problem \[ (K - \omega_i^2 M)\phi_i = 0 \] with:
- \(\omega_i\): the natural frequency
- \(\phi_i\): the corresponding mode shape
The number of mode shapes is equal to the number of degrees of freedom \(n\).
The mode shapes are orthogonal with respect to the stiffness and mass matrices:
\begin{align} \phi_i^T M \phi_j &= \mu_i \delta_{ij} \\\ \phi_i^T K \phi_j &= \mu_i \omega_i^2 \delta_{ij} \end{align}
With \(\mu_i\) the modal mass (also called the generalized mass) of mode \(i\).
Modal Decomposition
Structure Without Rigid Body Modes
Let perform a change of variable from physical coordinates \(x\) to modal coordinates \(z\).
\begin{equation} x = \Phi z \end{equation}
With:
- \(\Phi = [\phi_1, \phi_2, ..., \phi_n]\) the matrix of the mode shapes
- \(z\) the vector of modal amplitudes
The dynamic equation of the system becomes: \[ M \Phi \ddot{z} + C \Phi \dot{z} + K \Phi z = f \]
If we left multiply the equation by \(\Phi^T\) and we use the orthogonalily relationships: \[ diag(\mu_i) \ddot{z} + \Phi^T C \Phi + diag(\mu_i \omega_i^2) z = \Phi^T f \]
If \(\Phi^T C \Phi\) is diagonal, the damping is said classical or normal. In this case: \[ \Phi^T C \Phi = diag(2 \xi_i \mu_i \omega_i) \]
One can verify that the Rayleigh damping \eqref{eq:rayleigh_damping} complies with this condition with modal damping ratios \(\xi_i = \frac{1}{2} ( \frac{\alpha}{\omega_i} + \beta\omega_i )\).
And we obtain decoupled modal equations \eqref{eq:modal_eom}.
\begin{equation} \ddot{z} + 2 \xi \Omega \dot{z} + \Omega^2 z = z^{-1} \Phi^T f \end{equation}
with:
- \(\xi = diag(\xi_i)\)
- \(\Omega = diag(\omega_i)\)
- \(\mu = diag(\mu_i)\)
Typical values of the modal damping ratio are summarized on table tab:damping_ratio.
| Damping Ratio | Application |
|---|---|
| \(\xi \simeq 0.001 - 0.005\) | Space structures |
| \(\xi \simeq 0.01 - 0.02\) | Mechanical engineering |
| \(\xi \simeq 0.05\) | Civil engineering |
| \(\xi \simeq 0.2\) | When ground is involved |
The assumption of classical damping is often justified for light damping, but it is questionable when the damping is large.
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. However, in physical coordinates, the number of degrees of freedom is usually very large. 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. Therefore, the number of degrees of freedom contribution effectively to the response is reduced drastically in modal coordinates.
Dynamic Flexibility Matrix
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: \[ X = G(\omega) F \]
Where \(G(\omega)\) is called the Dynamic flexibility Matrix: \[ G(\omega) = (-\omega^2 M + j\omega C + K)^{-1} F \]
From the modal expansion of the dynamic flexibility matrix can be obtained by coordinate transformation \(x = \phi z\) and we obtain:
\begin{equation} G(\omega) = \sum_{i=1}^n \frac{\phi_i \phi_i^T}{\mu_i \omega_i^2} D_i(\omega) \end{equation}
With:
- \(D_i(\omega)\) is the dynamic amplification factor of mode \(i\) given by
\begin{equation} D_i(\omega) = \frac{1}{1 - \omega^2/\omega_i^2 + 2 j \xi_i \omega/\omega_i} \end{equation}
{{< 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 \(\omega_k \gg \omega_b\)" >}}
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 fig:neglected_modes).
And \(G(\omega)\) can be rewritten on terms of the low frequency modes only: \[ G(\omega) \approx \sum_{i=1}^m \frac{\phi_i \phi_i^T}{\mu_i \omega_i^2} D_i(\omega) + R \]
The quasi-static correction of the high frequency modes \(R\) is called the residual mode. This introduces a feedthrough component in the transfer matrix.
Structure with Rigid Body Modes
Collocated Control System
A collocated control system is a control system where:
- the actuator and the sensor are attached to the same degree of freedom
- they are dual: the product of the actuator signal and the sensor signal represents the energy exchange between the structure and the control system
| Actuator | Sensor |
|---|---|
| Force | Translation |
| Torque | Rotation |
The open-loop FRF of a collocated system corresponds to a diagonal component of the dynamic flexibility matrix.
If we assumes that the collocated system is undamped and is attached to the DoF \(k\), the open-loop FRF is purely real: \[ G_{kk}(\omega) = \sum_{i=1}^m \frac{\phi_i^2(k)}{\mu_i (\omega_i^2 - \omega^2)} + R_{kk} \]
\(G_{kk}\) is a monotonously increasing function of \(\omega\) (figure fig:collocated_control_frf).
{{< figure src="/ox-hugo/preumont18_collocated_control_frf.png" caption="Figure 7: Open-Loop FRF of an undamped structure with collocated actuator/sensor pair" >}}
The amplitude of the FRF goes from \(-\infty\) at the resonance frequencies \(\omega_i\) to \(+\infty\) at the next resonance frequency \(\omega_{i+1}\). Therefore, in every interval, there is a frequency \(z_i\) such that \(\omega_i < z_i < \omega_{i+1}\) where the amplitude of the FRF vanishes. The frequencies \(z_i\) are called anti-resonances.
Undamped collocated control systems have alternating poles and zeros on the imaginary axis. For lightly damped structure, the poles and zeros are just moved a little bit in the left-half plane, but they are still interlacing.
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.
{{< figure src="/ox-hugo/preumont18_collocated_zero.png" caption="Figure 8: Structure with collocated actuator and sensor" >}}
The frequency of the transmission zero \(z_i\) and the mode shape associated are the natural frequency and the mode shape of the system obtained by constraining the d.o.f. on which the control systems acts.
The open-loop zeros are asymptotic values of the closed-loop poles when the feedback gain goes to infinity.
The open-loop poles are independant of the actuator and sensor configuration while the open-loop zeros do depend on it.
By looking at 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.
{{< figure src="/ox-hugo/preumont18_alternating_p_z.png" caption="Figure 9: Bode plot of a lighly damped structure with collocated actuator and sensor" >}}
The open-loop transfer function of a lighly damped structure with a collocated actuator/sensor pair can be written:
\begin{equation} 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)} \end{equation}
The corresponding Bode plot is represented in figure 9. 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 \(\SI{180}{\degree}\). In this way, the phase diagram is always contained between \(\SI{0}{\degree}\) and \(\SI{-180}{\degree}\) as a consequence of the interlacing property.
Electromagnetic and Piezoelectric Transducers
Introduction
Transducers are critical in active structures technology. In many applications, the actuators are the most critical part of the system; however, the sensors become very important in precision engineering where submicron amplitudes must be detected.
Two broad categories of actuators can be distinguish:
- grounded actuator: react on a fixed support. They include torque motors, force motors (shakers), tendons
- structure borne actuator: includes jets, reaction wheels, proof-mass actuators, piezo strips, ...
Voice Coil Transducer
A voice coil transducer is an energy transformer which converts electrical power into mechanical power and vice versa.
The system consists of (see figure fig:voice_coil_schematic):
- A permanent magnet which produces a uniform flux density \(B\) normal to the gap
- A coil which is free to move axially
{{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="Figure 10: Physical principle of a voice coil transducer" >}}
We note:
- \(v\) the velocity of the coil
- \(f\) the external force acting to maintain the coil in equilibrium againt the electromagnetic forces
- \(e\) the voltage difference across the coil
- \(i\) the current into the coil
Faraday's law:
\begin{equation} e = 2\pi n r B v = T v \end{equation}
With \(T = 2\pi n r B\) is the transducer constant.
Lorentz force law:
\begin{equation} f = -i 2\pi n r B = - T i \end{equation}
The total power delivered to the moving coil transducer is equal to the sum of the electric power and the mechanical power: \[ ei + fv = 0 \]
Thus, at any time, there is an equilibrium between the electrical power absorbed by the device and the mechanical power delivered.
Proof-Mass Actuator
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).
{{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="Figure 11: Proof-mass actuator" >}}
If we apply the second law of Newton on the mass: \[ m\ddot{x} + c\dot{x} + kx = f = Ti \]
In the Laplace domain: \[ x = \frac{Ti}{ms^2 + cs + k} \]
The total force applied on the support is: \[ F = -f + cs + k = -m s^2 x = \frac{-ms^2Ti}{ms^2 + cs + k} \]
The transfer function between the total force and the current \(i\) applied to the coil is :
\begin{equation} \frac{F}{i} = \frac{-s^2 T}{s^2 + 2\xi_p \omega_p s + \omega_p^2} \end{equation}
with:
- \(T\) is the transducer constant
- \(\omega_p = \frac{k}{m}\) is the natural frequency of the spring-mass system
- \(\xi_p\) is the damping ratio
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).
{{< figure src="/ox-hugo/preumont18_proof_mass_tf.png" caption="Figure 12: Bode plot \(F/i\) of the proof-mass actuator" >}}
Geophone
The geophone is a transducer which behaves like an absolute velocity sensor above some cutoff frequency. The voltage \(e\) of the coil is used as the sensor output.
If \(x_0\) is the displacement of the support and if the voice coil is open (\(i=0\)), the governing equations are:
\begin{align*} m\ddot{x} + c(\dot{x}-\dot{x_0}) + k(x-x_0) &= 0\\\ T(\dot{x}-\dot{x_0}) &= e \end{align*}
By using the two equations, we obtain:
\begin{equation} \frac{e}{\dot{x_0}} = \frac{-s^2 T}{s^2 + 2\xi_p\omega_p s + \omega_p^2} \end{equation}
Above the corner frequency, the gain of the geophone is equal to the transducer constant \(T\).
{{< figure src="/ox-hugo/preumont18_geophone.png" caption="Figure 13: Model of a geophone based on a voice coil transducer" >}}
Designing geophones with very low corner frequency is in general difficult. Active geophones where the frequency is lowered electronically may constitute a good alternative option.
General Electromechanical Transducer
The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure fig:electro_mechanical_transducer.
{{< figure src="/ox-hugo/preumont18_electro_mechanical_transducer.png" caption="Figure 14: Electrical analog representation of an electromechanical transducer" >}}
In Laplace form the constitutive equations read:
\begin{align} e & = Z_e i + T_{em} v \label{eq:gen_trans_e} \\\ f & = T_{em} i + Z_m v \label{eq:gen_trans_f} \end{align}
With:
- \(e\) is the Laplace transform of the input voltage across the electrical terminals
- \(i\) is the input current
- \(f\) is the force applied to the mechanical terminals
- \(v\) is the velocity of the mechanical part
- \(Z_e\) is the blocked electrical impedance (for \(v=0\))
- \(T_{em}\) is the transduction coefficient representing the electromotive force (in \(\si{\volt\second\per\meter}\))
- \(T_{me}\) is the transduction coefficient representing the force acting on the mechanical terminals to balance the electromagnetic force induced per unit current input (in \(\si{\newton\per\ampere}\))
- \(Z_m\) is the mechanical impedance measured when \(i=0\)
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. Thus, if \(Z_ei\) can be measured and substracted from \(e\), a signal proportional to the velocity is obtained.
To do so, the bridge circuit as shown on figure fig:bridge_circuit can be used.
We can show that
\begin{equation} V_4 - V_2 = \frac{-Z_b T_{em}}{Z_e + Z_b} v \end{equation}
which is indeed a linear function of the velocity \(v\) at the mechanical terminals.
{{< figure src="/ox-hugo/preumont18_bridge_circuit.png" caption="Figure 15: Bridge circuit for self-sensing actuation" >}}
Smart Materials
Smart materials have the ability to respond significantly to stimuli of different physical nature. Figure fig:smart_materials lists various effects that are observed in materials in response to various inputs.
{{< 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" >}}
Piezoelectric Transducer
Piezoelectric materials exhibits two effects described below.
Ability to generate an electrical charge in proportion to an external applied force.
An electric filed parallel to the direction of polarization induces an expansion of the material.
The most popular piezoelectric materials are Lead-Zirconate-Titanate (PZT) which is a ceramic, and Polyvinylidene fluoride (PVDF) which is a polymer.
We here consider a transducer made of one-dimensional piezoelectric material.
\begin{subequations} \begin{align} D & = \epsilon^T E + d_{33} T\\\ S & = d_{33} E + s^E T \end{align} \end{subequations}
With:
- \(D\) is the electric displacement \([C/m^2]\)
- \(E\) is the electric field \([V/m]\)
- \(T\) is the stress \([N/m^2]\)
- \(S\) is the strain
- \(\epsilon^T\) is the dielectric constant under constant stress
- \(s^E\) is the compliance when the eletric field is constant (inverse of Young modulus)
- \(d_{33}\) is the piezoelectric constant \([m/V]\) or \([C/N]\) in the poling direction of the material (convention)
Constitutive Relations of a Discrete Transducer
The set of equations \eqref{eq:piezo_eq} can be written in a matrix form:
\begin{equation} \begin{bmatrix}D\S\end{bmatrix}
\begin{bmatrix} \epsilon^T & d_{33}\\\ d_{33} & s^E \end{bmatrix} \begin{bmatrix}E\T\end{bmatrix} \end{equation}
Where \((E, T)\) are the independent variables and \((D, S)\) are the dependent variable.
If \((E, S)\) are taken as independant variables:
\begin{equation} \begin{bmatrix}D\T\end{bmatrix}
\begin{bmatrix} \epsilon^T(1-k^2) & e_{33}\\\ -e_{33} & c^E \end{bmatrix} \begin{bmatrix}E\S\end{bmatrix} \end{equation}
With:
- \(c^E = \frac{1}{s^E}\) is the Young modulus under short circuited electrodes (\(E = 0\)) in \([N/m^2]\)
- \(e_{33} = \frac{d_{33}}{s^E}\) is the constant relating the electric displacement to the strain for short-circuited electrodes \([C/m^2]\)
\begin{equation} k^2 = \frac{{d_{33}}^2}{s^E \epsilon^T} = \frac{{e_{33}}^2}{c^E \epsilon^T} \end{equation}
\(k\) is called the electromechanical coupling factor of the material. It measures the efficiency of the conversion of the mechanical energy into electrical energy, and vice versa.
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) 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:
\begin{equation} \begin{bmatrix}Q\\Delta\end{bmatrix}
\begin{bmatrix} C & nd_{33}\\\ nd_{33} & 1/K_a \end{bmatrix} \begin{bmatrix}V\f\end{bmatrix} \end{equation}
where
- \(Q = n A D\) is the total electric charge on the electrodes of the transducer
- \(\Delta = S l\) is the total extension (\(l = nt\) is the length of the transducer)
- \(f = AT\) is the total force
- \(V\) is the voltage applied between the electrodes of the transducer
- \(C = \epsilon^T A n^2/l\) is the capacitance of the transducer with no external load (\(f = 0\))
- \(K_a = A/s^El\) is the stiffness with short-circuited electrodes (\(V = 0\))
{{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="Figure 17: Piezoelectric linear transducer" >}}
Equation \eqref{eq:piezo_stack_eq} can be inverted to obtain
\begin{equation} \begin{bmatrix}V\f\end{bmatrix}
\frac{K_a}{C(1-k^2)} \begin{bmatrix} 1/K_a & -nd_{33}\\\ -nd_{33} & C \end{bmatrix} \begin{bmatrix}Q\\Delta\end{bmatrix} \end{equation}
Energy Stored in the Piezoelectric Transducer
Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure fig:piezo_discrete.
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
\begin{equation} dW = V i dt + f \dot{\Delta} dt = V dQ + f d\Delta \end{equation}
{{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="Figure 18: Discrete Piezoelectric Transducer" >}}
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:
\begin{equation} 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} \end{equation}
- The first term is the electrical energy stored in the capacitance \(C(1-k^2)\) (corresponding to fixed geometry \(\Delta = 0\))
- The second term is the piezoelectric energy
- The third term is the elastic strain energy stored in a spring stiffness \(K_a/(1-k^2)\) (corresponding to open electrodes \(Q=0\))
The constitutive equations can be recovered by differentiate the stored energy: \[ f = \frac{\partial W_e}{\partial \Delta}, \quad V = \frac{\partial W_e}{\partial Q} \]
Interpretation of \(k^2\)
Consider a piezoelectric transducer subjected to the following mechanical cycle: first, it is loaded with a force \(F\) with short-circuited electrodes; the resulting extension is \(\Delta_1 = F/K_a\) where \(K_a = A/(s^El)\) is the stiffness with short-circuited electrodes. The energy stored in the system is: \[ W_1 = \int_0^{\Delta_1} f dx = \int_0^{\Delta_1} K_a x dx = \frac{F^2}{2 K_a} \]
At this point, the electrodes are open and the transducer is unloaded according to a path of slope \(K_a/(1-k^2)\), the resulting extension is \(\Delta_2 = \frac{F(1-k^2)}{K_a}\). The energy recovered is \[ W_1 = \int_0^{\Delta_2} f dx = \frac{F \Delta_2}{2} = \frac{F^2(1-k^2)}{2 K_a} \]
The ratio between the remaining stored energy and the initial stored energy is \[ \frac{W_1 - W_2}{W_1} = k^2 \]
Admittance of the Piezoelectric Transducer
Consider the system of figure fig:piezo_stack_admittance, where the piezoelectric transducer is assumed massless and is connected to a mass \(M\). The force acting on the mass is negative of that acting on the transducer, \(f = -M \ddot{x}\).
{{< figure src="/ox-hugo/preumont18_piezo_stack_admittance.png" caption="Figure 19: Elementary dynamical model of the piezoelectric transducer" >}}
From the constitutive equations, one finds
\begin{equation} \frac{I}{V} = s C (1-k^2) \frac{s^2 + z^2}{s^2 + p^2} \end{equation}
where the poles and zeros are respectively \[ p^2 = \frac{K_a}{M},\quad z^2 = \frac{K_a/(1-k^2)}{M} \]
And one can see that
\begin{equation} \frac{z^2 - p^2}{z^2} = k^2 \end{equation}
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).
{{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="Figure 20: Typical admittance FRF of the transducer" >}}
Piezoelectric Beam, Plate and Truss
Piezoelectric Material
Constitutive Relations
Coenergy Density Function
Hamilton's Principle
Piezoelectric Beam Actuator
Hamilton's Principle
Piezoelectric Loads
Laminar Sensor
Current and Charge Amplifiers
Distributed Sensor Output
Charge Amplifier Dynamics
Spatial Modal Filters
Modal Actuator
Modal Sensor
Active Beam with Collocated Actuator/Sensor
Frequency Response Function
Pole-Zero Pattern
Modal Truncation
Admittance of a Beam with a Piezoelectric Patch
Piezoelectric Laminate
Two-Dimensional Constitutive Equations
Kirchhoff Theory
Stiffness Matrix of a Multilayer Elastic Laminate
Multilayer Laminate with a Piezoelectric Layer
Equivalent Piezoelectric Loads
Sensor Output
Beam Model Versus Plate Model
Additional Remarks
Active Truss
Open-Loop Transfer Function
Admittance Function
Finite Element Formulation
Problems
References
Passive Damping with Piezoelectric Transducers
Introduction
Resistive Shunting
Inductive Shunting
Equal Peak Design
Robustness of the Equal Peak Design
Switched Shunt
Equivalent Damping Ratio
Collocated Versus Non-collocated Control
Pole-Zero Flipping
The Root Locus shows, in a graphical form, the evolution of the poles of the closed-loop system as a function of the scalar gain \(g\) applied to the compensator. The Root Locus is the locus of the solution \(s\) of the closed loop characteristic equation \(1 + gG(s)H(s) = 0\) when \(g\) goes from zero to infinity.
If the open-loop transfer function is written \[ G(s)H(s) = k \frac{\Pi_{i=1}^{m} (s - z_i)}{\Pi_{i=1}^{n} (s - p_i)} \] The locus goes from the poles \(p_i\) (for \(g=0\)) to the zeros \(z_i\) (as \(g \rightarrow \infty\)).
The Two-Mass Problem
Collocated Control
Non-collocated Control
Notch Filter
Effect of Pole-Zero Flipping on the Bode Plots
Nearly Collocated Control System
Non-collocated Control Systems
The Role of Damping
Active Damping with Collocated System
Introduction
The role of active damping is to increase the negative real parts of system poles wile maintaining the natural frequencies essentially unchanged.
Active damping requires relatively little control effort; this is why it is also called Low Authority Control (LAC). Other control strategies which fully relocate the closed loop poles are called High Autority Control (HAC).
Lead Control
\[H(s) = g \frac{s+z}{z+p} \quad p \gg z \]
It produces a phase lead in the frequency band between \(z\) and \(p\), bringing active damping to all the modes belonging to \(z < \omega_i < p\).
The closed-loop poles start at the open-llop poles for \(g=0\) and go to the open-loop zeros for \(g\rightarrow\infty\).
The controller does not have any roll-off, but the roll-off of the structure is enough to guarantee gain stability at high frequency.
Direct Velocity Feedback (DVF)
This is a particular case of the Lead controller as \(z\rightarrow 0\) and \(p\rightarrow\infty\).
Structure: \[M \ddot{x} + K x = b u\]
Output is a velocity sensor: \[y = b^T \dot{x}\]
Control: \[u = -g y\]
Positive Position Feedback (PPF)
Sometimes the plant does not have a roll-off of \(-40dB/\text{decade}\), then we can use a second-order PPF: \[H(s) = \frac{-g}{s^2 + 2 \xi_f \omega_f s + {\omega_f}^2}\]
Integral Force Feedback (IFF)
Duality Between the Lead and the IFF Controllers
Root Locus of a Single Mode
Open-Loop Poles and Zeros
Actuator and Sensor Dynamics
Decentralized Control with Collocated Pairs
Cross talk
Force Actuator and Displacement Sensor
Displacement Actuator and Force Sensor
Proof of Equation (7.18)–(7.32)
Vibration Isolation
Introduction
Relaxation Isolator
Electromagnetic Realization
Active Isolation
Sky-Hook Damper
Integral Force Feedback
Flexible Body
Free-Free Beam with Isolator
Payload Isolation in Spacecraft
Interaction Isolator/Attitude Control
Gough–Stewart Platform
Six-Axis Isolator
Relaxation Isolator
Integral Force Feedback
Spherical Joints, Modal Spread
Active Versus Passive
Car Suspension
State Space Approach
Introduction
State Space Description
Single Degree of Freedom Oscillator
Flexible Structure
Inverted Pendulum
System Transfer Function
Poles and Zeros
Pole Placement by State Feedback
Example: Oscillator
Linear Quadratic Regulator
Symmetric Root Locus
Inverted Pendulum
Observer Design
Kalman Filter
Inverted Pendulum
Reduced-Order Observer
Oscillator
Inverted Pendulum
Separation Principle
Transfer Function of the Compensator
The Two-Mass Problem
Analysis and Synthesis in the Frequency Domain
Gain and Phase Margins
Nyquist Criterion
Cauchy's Principle
Nyquist Stability Criterion
Nichols Chart
Feedback Specification for SISO Systems
Sensitivity
Tracking Error
Performance Specification
Unstructured Uncertainty
Robust Performance and Robust Stability
Bode Gain–Phase Relationships
The Bode Ideal Cutoff
Non-minimum Phase Systems
Usual Compensators
System Type
Lead Compensator
PI Compensator
Lag Compensator
PID Compensator
Multivariable Systems
Performance Specification
Small Gain Theorem
Stability Robustness Tests
Residual Dynamics
Optimal Control
Introduction
Quadratic Integral
Deterministic LQR
Stochastic Response to a White Noise
Remark
Stochastic LQR
Asymptotic Behavior of the Closed Loop
Prescribed Degree of Stability
Gain and Phase Margins of the LQR
Full State Observer
Covariance of the Reconstruction Error
Kalman Filter (KF)
Linear Quadratic Gaussian (LQG)
Duality
Spillover
Spillover Reduction
Loop Transfer Recovery (LTR)
Integral Control with State Feedback
Frequency Shaping
Weakness of LQG:
- use frequency independant cost function
- use noise statistics with uniform distribution
To overcome the weakness => frequency shaping either by:
- considering a frequency dependant cost function
- using colored noise statistics
Frequency-Shaped Cost Functionals
Noise Model
Controllability and Observability
Introduction
Definitions
Controllability and Observability Matrices
Examples
Cart with Two Inverted Pendulums
Double Inverted Pendulum
Two d.o.f. Oscillator
State Transformation
Control Canonical Form
Left and Right Eigenvectors
Diagonal Form
PBH Test
Residues
Example
Sensitivity
Controllability and Observability Gramians
Internally Balanced Coordinates
Model Reduction
Transfer Equivalent Realization
Internally Balanced Realization
Example
Stability
Introduction
Phase Portrait
Linear Systems
Routh--Hurwitz Criterion
Lyapunov's Direct Method
Introductory Example
Stability Theorem
Asymptotic Stability Theorem
Lasalle's Theorem
Geometric Interpretation
Instability Theorem
Lyapunov Functions for Linear Systems
Lyapunov's Indirect Method
An Application to Controller Design
Energy Absorbing Controls
Applications
Digital Implementation
Sampling, Aliasing, and Prefiltering
Zero-Order Hold, Computational Delay
Quantization
Discretization of a Continuous Controller
Active Damping of a Truss Structure
Actuator Placement
Implementation, Experimental Results
Active Damping Generic Interface
Active Damping
Experiment
Pointing and Position Control
Active Damping of a Plate
Control Design
Active Damping of a Stiff Beam
System Design
The HAC/LAC Strategy
In active structures for precision engineering applications, the control system is used to reduce the effect of transient and steady-state disturbances on the controlled variables. Active damping is very effective in reducing the settling time of transient disturbances and the effect of steady state disturbances near the resonance frequencies of the system; however, away from the resonances, the active damping is completely ineffective and leaves the closed-loop response essentially unchanged. Such low-gain controllers are often called Low Authority Controllers (LAC), because they modify the poles of the system only slightly.
To attenuate wide-band disturbances, the controller needs larger gains, in order to cause more substantial modifications to the poles of the open-loop system; this is the reason why they are often called High Authority Controllers (HAC). Their design requires a model of the structure, and there is usually a trade-off between the conflicting requirements of performance-bandwidth and stability in the face of parametric uncertainty and unmodelled dynamics.
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.
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure fig:hac_lac_control. 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)
{{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="Figure 21: Principle of the dual-loop HAC/LAC control" >}}
Wide-Band Position Control
Compensator Design
Results
Vibroacoustics: Volume Displacement Sensors
QWSIS Sensor
Discrete Array Sensor
Spatial Aliasing
Distributed Sensor
Tendon Control of Cable Structures
Introduction
Tendon Control of Strings and Cables
Active Damping Strategy
Basic Experiment
Linear Theory of Decentralized Active Damping
Guyed Truss Experiment
Microprecision Interferometer Testbed
Free-Floating Truss Experiment
Application to Cable-Stayed Bridges
Laboratory Experiment
Control of Parametric Resonance
Large Scale Experiment
Application to Suspension Bridges
Footbridge
Laboratory Experiment
Active Control of Large Telescopes: Adaptive Optics
Introduction
Wavefront Sensor
Zernike Modes
Fried Length, Seeing
Kolmogorov Turbulence Model
Strehl Ratio
Power Spectral Density of the Zernike Modes
Deformable Mirror for Adaptive Optics
Stoney Formula
Stroke Versus Natural Frequency
Feedback Control of an AO Mirror
Quasi-static Control
Control of the Mirror Based on the Jacobian
Control of Zernike Modes
Dynamic Response of the AO Mirror
Dynamic Model of the Mirror
Control-Structure Interaction
Passive Damping
Active Damping
Miscellaneous
Segmented AO Mirror
Initial Curvature of the AO Mirror
Active Control of Large Telescopes: Active Optics
Introduction
Monolithic Primary Mirror
Segmented Primary Mirror
SVD Controller
Loop Shaping of the SVD Controller
Dynamics of a Segmented Mirror
Control-Structure Interaction
SISO System
MIMO System
Spillover Alleviation
Scaling Rules
Static Deflection Under Gravity
First Resonance Frequency
Control Bandwidth
Adaptive Thin Shell Space Reflectors
Introduction
Adaptive Plates Versus Adaptive Shells
Adaptive Spherical Shell
Quasi-static Control: Hierarchical Approach
Petal Configuration
MATS Demonstrator
Manufacturing of the Demonstrator
Semi-active Control
Introduction
Magneto-Rheological Fluids
MR Devices
Semi-active Suspension
Semi-active Devices
Narrow-Band Disturbance
Quarter-Car Semi-active Suspension
Problems
Bibliography
Preumont, A., Vibration control of active structures - fourth edition (2018), : Springer International Publishing. ↩