### Active Versus Passive {#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 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 {#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 {#control-strategies}
There are two radically different approached to disturbance rejection: feedback and feedforward.
The principle of feedback is represented on figure [1](#orge1596ba). 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
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](#org8128933).
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 {#plant-description-error-and-control-budget}
{{<figuresrc="/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.
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[\\).
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.
- \\(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\\)
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 {#some-concepts-in-structural-dynamics}
### Equation of Motion of a Discrete System {#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
<divclass="cbox">
<div></div>
\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
</div>
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.
<divclass="cbox">
<div></div>
\begin{equation}
C = \alpha M + \beta K
\end{equation}
</div>
\\(\alpha\\) and \\(\beta\\) are selected to fit the structure under consideration.
### Vibration Modes {#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\\).
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 )\\).
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.
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.
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:
{{<figuresrc="/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 [6](#org8a88959)).
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 {#structure-with-rigid-body-modes}
### Collocated Control System {#collocated-control-system}
<divclass="cbox">
<div></div>
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
{{<figuresrc="/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}\\)wheretheamplitudeoftheFRFvanishes.Thefrequencies \\(z\_i\\)arecalled**anti-resonances**.
<divclass="cbox">
<div></div>
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 [8](#org6c053d5).
{{<figuresrc="/ox-hugo/preumont18_collocated_zero.png"caption="Figure 8: Structure with collocated actuator and sensor">}}
<divclass="cbox">
<div></div>
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 [7](#orgbd3bc07), 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.
{{<figuresrc="/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:
The corresponding Bode plot is represented in figure [9](#org63fc16c). 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 {#electromagnetic-and-piezoelectric-transducers}
### Introduction {#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
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](#orgc64500b)).
Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#org1f0c996)).
{{<figuresrc="/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 {#general-electromechanical-transducer}
{{<figuresrc="/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.
{{<figuresrc="/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">}}
If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#orgf820772)) 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:
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
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:
- 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\\) {#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}\\).
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 [20](#orgff1c070)).
#### Beam Model Versus Plate Model {#beam-model-versus-plate-model}
#### Additional Remarks {#additional-remarks}
### Active Truss {#active-truss}
#### Open-Loop Transfer Function {#open-loop-transfer-function}
#### Admittance Function {#admittance-function}
### Finite Element Formulation {#finite-element-formulation}
### Problems {#problems}
### References {#references}
## Passive Damping with Piezoelectric Transducers {#passive-damping-with-piezoelectric-transducers}
### Introduction {#introduction}
### Resistive Shunting {#resistive-shunting}
### Inductive Shunting {#inductive-shunting}
#### Equal Peak Design {#equal-peak-design}
#### Robustness of the Equal Peak Design {#robustness-of-the-equal-peak-design}
### Switched Shunt {#switched-shunt}
#### Equivalent Damping Ratio {#equivalent-damping-ratio}
## Collocated Versus Non-collocated Control {#collocated-versus-non-collocated-control}
### Pole-Zero Flipping {#pole-zero-flipping}
<divclass="cbox">
<div></div>
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.
</div>
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 {#the-two-mass-problem}
#### Collocated Control {#collocated-control}
#### Non-collocated Control {#non-collocated-control}
### Notch Filter {#notch-filter}
### Effect of Pole-Zero Flipping on the Bode Plots {#effect-of-pole-zero-flipping-on-the-bode-plots}
### Nearly Collocated Control System {#nearly-collocated-control-system}
### Non-collocated Control Systems {#non-collocated-control-systems}
### The Role of Damping {#the-role-of-damping}
## Active Damping with Collocated System {#active-damping-with-collocated-system}
### Introduction {#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 {#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) {#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) {#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) {#integral-force-feedback--iff}
### Duality Between the Lead and the IFF Controllers {#duality-between-the-lead-and-the-iff-controllers}
#### Root Locus of a Single Mode {#root-locus-of-a-single-mode}
#### Open-Loop Poles and Zeros {#open-loop-poles-and-zeros}
### Actuator and Sensor Dynamics {#actuator-and-sensor-dynamics}
### Decentralized Control with Collocated Pairs {#decentralized-control-with-collocated-pairs}
#### Cross talk {#cross-talk}
#### Force Actuator and Displacement Sensor {#force-actuator-and-displacement-sensor}
#### Displacement Actuator and Force Sensor {#displacement-actuator-and-force-sensor}
### Proof of Equation (7.18)–(7.32) {#proof-of-equation--7-dot-18----7-dot-32}
### Active Damping Generic Interface {#active-damping-generic-interface}
#### Active Damping {#active-damping}
#### Experiment {#experiment}
#### Pointing and Position Control {#pointing-and-position-control}
### Active Damping of a Plate {#active-damping-of-a-plate}
#### Control Design {#control-design}
### Active Damping of a Stiff Beam {#active-damping-of-a-stiff-beam}
#### System Design {#system-design}
### The HAC/LAC Strategy {#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 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)
<aid="org6703487"></a>Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. <https://doi.org/10.1007/978-3-319-72296-2>.
