zeros to the right, and the last one even moves to infinity as the feedthrough ...... into electrical energy and vice-versa; its constitutive equations are e = Tv. (2.14) ..... Figure 2.16: (a) Design # 1: 2 membranes, 2 flexible joints, magnet in the leg.
Chapter 1
Active damping 1.1 1.1.1
Introduction Why suppress vibrations ?
Mechanical vibrations span amplitudes from meters (civil engineering) to nanometers (precision engineering). Their detrimental effect on systems may be of various natures: Failure: vibration-induced structural failure may occur by excessive strain during transient events (e.g. building response to earthquake), instability due to particular operating conditions (flutter of bridges under wind excitation), or simply fatigue (mechanical parts in machines). Comfort: Examples where vibrations are detrimental to comfort are numerous: noise in helicopters, car suspensions, wind-induced sway of buildings. Operation of precision devices: Numerous systems in precision engineering, especially optical systems, put severe restrictions on mechanical vibrations. Precision machine tools, DVD readers, telescopes are typical examples. Moore’s law on the number of transistors on an integrated circuit could not hold without a constant improvement of the wafer stepper accuracy. The performances of large interferometer such as the VLTI are limited by microvibrations affecting the various parts of the optical path. Lightweight segmented telescopes (space as well as earth-based) will be impossible to build in their final shape with an accuracy of a fraction of the wavelength, because of the various disturbances sources such as the thermal gradients (which dominates the space environment). Such systems will not exist without the capability to control actively the reflector shape.
1
1.1.2
How to reduce vibrations ?
Vibration reduction can be achieved in many different ways, depending on the problem; the most common are stiffening, damping and isolation. 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 systems. Damping may be achieved passively, with fluid dampers, eddy currents, elastomers or hysteretic elements, or by transferring kinetic energy to Dynamic Vibration Absorbers (DVA). One can also use transducers as energy converters, to transform vibration energy into electrical energy that is dissipated in electrical networks, or stored (energy harvesting). Recently, semiactive devices (also called semi-passive) have become available; they consist of passive devices with controllable properties. The Magneto-Rheological (MR) fluid damper is a famous example; piezoelectric transducers with switched electrical networks is another one. When high performance is needed, active control can be used; this involves a set of sensors (strain, acceleration, velocity, force,. . .), a set of actuators (force, inertial, strain,...) and a control algorithm (feedback or feedforward). Active damping is the main focus of this chapter. The design of an active control system involves many issues such as how to configurate the sensors and actuators (map of strain energy or kinetic energy), how to secure stability and robustness (collocated actuator/sensor pairs); the power requirement will often determine the size of the actuators, and the cost. An alternative which will be discussed later in this text is the so-called Hybrid control: It combines active and passive features to achieve performance at reduced cost.
1.2
Structural control
As compared to other control problems, structural control has a number of specific features. Firstly, the systems involved have in general a large number of degrees of freedom (d.o.f.) and a large number of modes (in fact infinite, but in most cases, it will be more than enough to consider a discrete approximation of the distributed system). In most cases, Finite Elements (F.E.) will tend to produce models with too many d.o.f.. This problem is usually solved by working in modal coordinates and truncating the model beyond the frequency band of interest. Attention must be paid, however, that the high frequency modes outside the frequency band of interest influence the position of the open-loop zeros of the system and call for 2
a quasi-static correction. Secondly, many structures involved in structural control are lightly damped (ξ ∼ 0.001 to 0.05). This means that the stability margin of the uncontrolled modes is small, sometimes very small, and that they are subject to spillover, which means that the control system always tends to destabilize the flexible modes just outside the control bandwidth and this has to be handled adequately; the only margin against spillover instability is provided by damping of the residual modes. The combination of a large number of modes with a small stability margin calls for specific control strategies emphasizing robustness, with respect to the residual dynamics (high frequency modes) and also with respect to the changes in the system parameters. Control systems with collocated (dual) actuator/sensor pairs exhibit special properties which are especially attractive in this respect. They are analyzed in detail here. In this chapter, we try to relate familiar concepts of structural dynamics and classical concepts of linear systems and control theory; an elementary knowledge of both fields is assumed.
1.3
Plant description
Consider the block diagram of Fig.1.1, in which the plant consists of the structure and its actuator and sensor. w is the disturbance applied to the structure, z is the controlled variable or performance metric (that one wants to keep as close as possible to 0), u is the control input and y is the sensor output (they are all assumed scalar for simplicity). H(s) is the control feedback law, expressed in the Laplace domain (s is the Laplace variable). We define the open-loop transfer functions : Gzw (s): Open-loop transfer function between w and z Gzu (s): Open-loop transfer function between u and z Gyw (s): Open-loop transfer function between w and y Gyu (s): Open-loop transfer function between u and y From the definition of the open-loop transfer functions, y = Gyw w + Gyu Hy
(1.1)
y = (I − Gyu H)−1 Gyw w
(1.2)
or
3
Disturbance
w
Control input
u
Plant
z
Performance metric
y
Output measurement
H(s) Figure 1.1: Block diagram of the control system. It follows that u = Hy = H(I − Gyu H)−1 Gyw w = Tuw w
(1.3)
On the other hand z = Gzw w + Gzu u
(1.4)
Combining the two foregoing equations, one finds the closed-loop transmissibility between the disturbance w and the control metric z : z = Tzw w = [Gzw + Gzu H(I − Gyu H)−1 Gyw ]w
1.3.1
(1.5)
Error budget
The frequency content of the disturbance w is usually described by its Power Spectral Density (PSD), Φw (ω) which describes the frequency distribution of the mean-square (MS) value Z 2 σw
∞
=
Φw (ω)dω
0
(1.6)
[the units of Φw is readily obtained from this equation; it is expressed in units of w squared per (rad/s)]. From(1.5), the PSD of the performance metric z is given by : Φz (ω) = |Tzw |2 Φw (ω)
4
(1.7)
Φz (ω) gives the frequency distribution of the mean-square value of the performance metric. Even more interesting for design is the cumulative PSD, defined by the integral of the PSD in the frequency range [0, ω] Z ω Z ω 2 σz (ω) = Φz (ν)dν = |Tzw |2 Φw (ν)dν (1.8) 0
0
It is a monotonously increasing function of frequency and describes the contribution of all the frequencies below ω to the mean-square value of the error budget. σz (ω) is expressed in the same units as the performance metric z; a typical plot is shown in Fig.1.2 for an hypothetical system with 4 modes.
Figure 1.2: Error budget distribution for various control configurations. For lightly damped structures, the diagram exhibits steps at the natural frequencies of the modes and the magnitude of the steps gives the contribution of each mode to the error budget, in the same units as the performance metric. It is very helpful to identify the critical modes in a design, at which the effort should be targeted. This diagram can be used to assess the control laws as well as actuator and sensor configurations. In a similar way, the control budget can be assessed from Z σu2 (ω)
= 0
Z
ω
Φu (ν)dν =
0
ω
|Tuw |2 Φw (ν)dν
(1.9)
σu (ω) describes how the RMS control input is distributed over the various modes of the structure and plays a critical role in the actuator design. As one sees, the frequency content of the disturbance w, described by Φw (ω), is essential in the evaluation of the error and control budgets and it 5
is very difficult, even risky, to attempt to design a controller without prior information on the disturbance.
1.4 1.4.1
Equations of structural dynamics Equation of motion including seismic excitation
In this section, we recall the equations governing the dynamics of a linear structure subjected to a point force f and a single-axis seismic excitation of acceleration x ¨0 (Fig.1.3.a). The seismic response has some special features which need to be treated with care.
Figure 1.3: (a) Structure subjected to a single-axis support excitation (b) Fictitious shaking table representation highlighting the reaction force f0 (c) Partition of the global displacement into rigid body motion and flexible motion relative to the base. We start from Hamilton’s principle, which states that the variational indicator : Z t2 Z t2 (δL + δWnc )dt = [δ(T ∗ − V ) + δWnc ]dt = 0 (1.10) t1
t1
vanishes for arbitrary variations of the path between two instants t1 and t2 compatible with the kinematics, and such that the configuration is fixed 6
at t1 and t2 . In this equation, T ∗ is the kinetic coenergy (often simply called kinetic energy), V the strain energy and δWnc is the virtual work associated with the non-conservative forces (the external force f and the support reaction f0 in this case). If one decomposes the global displacements x (generalized displacements including translational as well as rotational degrees of freedom, d.o.f.) into the rigid body mode and the flexible motion relative to the base (Fig.1.3.c). x = 1x0 + y
(1.11)
where x0 is the support motion, 1 is the unit rigid body mode (all translational d.o.f. along the axis of excitation are equal to 1 and the rotational d.o.f. equal to 0). The virtual displacements satisfy δx = 1δx0 + δy
(1.12)
where δx0 is arbitrary and δy satisfies the clamped boundary conditions at the base. Let b be the influence vector of the external loading f (for a point force, b contains all 0 except 1 at the d.o.f. where the load is applied), so that bT x is the generalized displacement of the d.o.f. where f is applied. The various energy terms involved in Hamilton’s principle are respectively 1 V = y T Ky 2
1 T ∗ = x˙ T M x˙ 2
δWnc = f0 δx0 + f bT δx = (f0 + f bT 1)δx0 + f bT δy
(1.13) (1.14)
One notices that the kinetic coenergy depends on the absolute velocity while the strain energy depends on the flexible motion alone, that is the motion relative to the base. For a point force, bT 1 = 1. Substituting into the variational indicator, one finds Z t2 [x˙ T M δ x˙ − y T Kδy + (f0 + f )δx0 + f bT δy]dt = 0 (1.15) t1
and, upon integrating the first term by parts, taking into account that δx(t1 ) = δx(t2 ) = 0, Z t2 [(−¨ xT M 1 + f0 + f )δx0 − (¨ xT M + y T K − f bT )δy]dt = 0 (1.16) t1
for arbitrary δx0 and δy. It follows that Mx ¨ + Ky = bf 7
(1.17)
f0 = 1T M x ¨−f
(1.18)
The presence of f in the right side of this equation is necessary to achieve static equilibrium; it is often ignored in the formulation. Combining (1.11) with (1.17), one gets the classical equation expressed in relative coordinates M y¨ + Ky = −M 1¨ x0 + bf
(1.19)
Also combining with (1.17), (1.18) can be rewritten alternatively f0 = −1T Ky
(1.20)
The first term in the right hand side of (1.19) is simply the inertia forces associated with a rigid body acceleration x ¨0 of the support. The foregoing analysis can be generalized to a single support multi-axis excitation rather easily (x0 becomes a vector in this case); the extension to several supports with differential motion is more difficult and beyond the scope of this text (e.g. Clough and Penzien, 1975, or Preumont, 1994). The structural damping has been ignored for simplicity; if a viscous damping is assumed on the relative coordinates y, a contribution C y˙ is added to the left hand side of (1.19), and (1.20) must be changed accordingly.
1.4.2
Modal coordinates
Consider the homogenous equation governing the free response (no base motion x0 nor external force f ) of the conservative (undamped) system: M y¨ + Ky = 0
(1.21)
If one tries a solution y = φi ejωi t , the mode shape φi and the natural frequency ωi must satisfy the eigenvalue problem (K − ωi2 M )φi = 0
(1.22)
Because M and K are non-negative definite, the mode shapes are real and the eigenvalues ωi2 ≥ 0 . The number of modes is equal to the number of d.o.f. of the system, but the structural response is often dominated by the first ones. We will address the modal truncation in detail a little later. 8
Equation (1.22) being homogeneous, the amplitude of the mode shape φi can be scaled arbitrarily; φi satisfy the orthogonality conditions φTi M φj = µi δij
(1.23)
φTi Kφj = µi ωi2 δij
(1.24)
where δij is the Kronecker delta index (δij = 1 if i = j, δij = 0 if i 6= j). µi is the modal mass of mode i; it may be chosen arbitrarily, to normalize the mode shapes, µi = 1 is often used for simplicity. The matrix of mode shapes is defined as Φ = (φ1 , φ2 , ..., φn ); it satisfies the orthogonality conditions ΦT M Φ = diag(µi )
(1.25)
ΦT KΦ = diag(µi ωi2 )
(1.26)
In what follows, we will assume normal (or modal ) damping; this means that the matrix ΦT CΦ is diagonal. By analogy with the single d.o.f. oscillator, we define the modal damping ratio ξi by ΦT CΦ = diag(2ξi µi ωi )
(1.27)
Now, let us consider equation (1.19) again (with damping) M y¨ + C y˙ + Ky = −M 1¨ x0 + bf
(1.28)
and let us perform a change of variables from physical coordinates y (motion relative to the base) to modal coordinates z according to y = Φz
(1.29)
z is the vector of modal amplitudes. Substituting into the foregoing equation, left multiplying by ΦT and using the orthogonality relationships, one gets a set of decoupled equations µi z¨i + 2ξi µi ωi z˙i + µi ωi2 zi = −φTi M 1¨ x0 + φTi bf
(1.30)
Γi = −φTi M 1
(1.31)
is known as the modal participation factor of mode i, it is simply the work on mode i of the inertia forces associated with a unit acceleration of the support. φTi b is the modal displacement of the point where the force is applied. We define the vector of modal participation factors Γ = (Γ1 , Γ2 , ..., Γi , ...)T as Γ = −ΦT M 1 9
(1.32)
1.4.3
Support reaction, dynamic mass
Now let us consider the reaction force f0 due to the seismic motion x ¨0 (we assume f = 0 in what follows). From (1.18) and (1.11) f0 = 1T M x ¨ = 1T M (1¨ x0 + y¨) = 1T M 1¨ x0 + 1T M Φ¨ z f0 = mT x ¨0 − ΓT z¨
(1.33)
where (1.32) has been used and mT = 1T M 1 is the total mass of the system. The fact that mT is the total mass of the system can be seen from the expression of the kinetic energy (1.13): if a rigid body velocity x˙ = 1x˙ 0 is applied, the total kinetic energy is 1 T 1 1 x˙ M x˙ = x˙ T0 1T M 1x˙ 0 = mT x˙ 20 2 2 2
(1.34)
The dynamic mass of the system is defined as the ratio between the complex amplitude of the harmonic force applied to the shaker, F0 , and the amplitude ¨ 0 , for every excitation frequency : of the acceleration of the shaking table, X ¨ 0 ejωt x ¨0 = X
f0 = F0 ejωt
z = Zejωt
(1.35)
Assuming no damping for simplicity, the relationship between the amplitude ¨ 0 and Z follows from (1.30): X
Z = diag[
Γi ¨0 ]X − ω2)
µi (ωi2
(1.36)
Combining with (1.33), the dynamic mass reads n
X Γ2 F0 ω2 i = mT + ( 2 ) ¨0 µ ωi − ω 2 X i=1 i
(1.37)
Before discussing this result, let us consider the alternative formulation based on (1.20); using (1.22) one gets f0 = −1T KΦz = −1T M Φ diag(ωi2 ) z = ΓT diag(ωi2 ) z
10
(1.38)
and combining with (1.36), one finds n
X Γ2 F0 ω2 i ( 2 i 2) = ¨0 µ ωi − ω X i=1 i
(1.39)
Comparing (1.37) and (1.39) at ω = 0, one finds that mT =
n X Γ2 i
i=1
(1.40)
µi
where the sum extends to all the modes. Γ2i /µi is called the effective modal mass of mode i; it represents the part of the total mass which is associated with mode i for this particular type of excitation (defined by the vector 1). Equation (1.37) and (1.39) are equivalent if all the modes are included in the modal expansion. However, if equation (1.39) is truncated after m < n modes, the modal mass of the high frequency mode is simply ignored, making this result statically incorrect. A quasi-static correction can be applied, by assuming that the high frequency modes respond in a quasi-static manner (i.e. as for ω = 0) ; this leads to m n X X Γ2i ω2 Γ2i F0 = ( 2 i 2) + ¨0 µ ωi − ω µ X i=1 i i=m+1 i
(1.41)
and upon using (1.40), m
m
m
i=1
i=1
i=1
X Γ2 X Γ2 X Γ2 ω2 ω2 F0 i i i = ( 2 i 2 ) + mT − = mT + ( 2 ) ¨0 µi ωi − ω µi µi ωi − ω 2 X
(1.42)
Thus, one recovers the truncated form of (1.37), which is statically correct. If the damping in included, equation (1.37) becomes m
X Γ2 F ω2 i = mT + ( 2 ) ¨0 µ ωi + 2jξi ωi ω − ω 2 X i=1 i
(1.43)
It can be truncated after the m modes which belong to the bandwidth of the excitation without any error on the static mass.
1.4.4
Dynamic flexibility matrix
This is all for the seismic excitation. Now, consider the steady state harmonic response to a vector excitation f = F ejωt . The governing equation is Mx ¨ + C x˙ + Kx = f 11
(1.44)
the steady state response is also harmonic, x = Xejωt , and the amplitude of F and X are related by X = [−ω 2 M + jωC + K]−1 F = G(ω)F
(1.45)
Where the matrix G(ω) is called the dynamic flexibility matrix ; it is a dynamic generalization of the static flexibility matrix, G(0) = K −1 . The modal expansion of G(ω) can be obtained by transforming (1.44) into modal coordinates x = Φz as we did earlier. The modal response is also harmonic, z = Zejωt and one finds easily that Z = diag{
µi (ωi2
1 } ΦT F + 2jξi ωi ω − ω 2 )
(1.46)
leading to X = ΦZ = Φ diag{
1 } ΦT F + 2jξi ωi ω − ω 2 )
µi (ωi2
(1.47)
Comparing with (1.45), one finds the modal expansion of the dynamic flexibility matrix: 2
−1
G(ω) = [−ω M + jωC + K]
=
n X i=1
φi φTi µi (ωi2 + 2jξi ωi ω − ω 2 )
(1.48)
where the sum extends to all the modes. Glk (ω) expresses the complex amplitude of the structural response of d.o.f. l when a unit harmonic force ejωt is applied at d.o.f. k. G(ω) can be rewritten G(ω) =
n X φi φT i
i=1
where Di (ω) =
µi ωi2
Di (ω)
1 1 − ω 2 /ωi2 + 2jξi ω/ωi
(1.49)
(1.50)
is the dynamic amplification factor of mode i. Di (ω) is equal to 1 at ω = 0, it exhibits large values in the vicinity of ωi , |Di (ωi )| = (2ξi )−1 , and then decreases beyond ωi (Fig.1.4). According to the definition of G(ω) the Fourier transform of the response X(ω) is related to the Fourier transform of the excitation F (ω) by X(ω) = G(ω)F (ω) 12
(1.51)
F
Excitation bandwidth
!b
!
Di
1 2øi
Mode outside the bandwidth
1
0
!i
!b
!k
!
Figure 1.4: (a) Fourier spectrum of the excitation F with a limited frequency content ω < ωb and (b) dynamic amplification of mode i such that ωi < ωb and ωk À ωb . This equation means that all the frequency components work independently, and if the excitation has no energy at one frequency, there is no energy in the response at that frequency. From Fig.1.4, one sees that when the excitation has a limited bandwidth, ω < ωb , the contribution of all the high frequency modes (i.e. such that ωk À ωb ) to G(ω) can be evaluated by assuming Dk (ω) ' 1. As a result, G(ω) =
m X φi φT
i 2 µ ω i=1 i i
Di (ω) +
n X φi φTi µ ω2 i=m+1 i i
(1.52)
The first term in the right hand side is the contribution of all the modes which respond dynamically and the second term is a quasi-static correction 13
for the high frequency modes, taking into account that G(0) = K −1 =
n X φi φT i
i=1
(1.53)
µi ωi2
(1.52) can be rewritten in terms of the low frequency modes only: G(ω) =
m X φi φT
i 2 µ ω i=1 i i
Di (ω) + K −1 −
m X φi φT i
i=1
µi ωi2
(1.54)
The quasi-static correction of the high frequency modes is often called the residual mode, denoted by R. Unlike all the terms involving Di (ω) which reduce to 0 as ω → ∞, R is independent of the frequency and introduces a feedthrough (constant) component in the transfer matrix. We will shortly see that R has a strong influence on the location of the transmission zeros and that neglecting it may lead to substantial errors in the prediction of the performance of the control system.
1.5
Collocated control system
A collocated control system is a control system where the actuator and the sensor are attached to the same d.o.f.. It is not sufficient to be attached to same location, but they must also be dual, that is a force actuator must be associated with a displacement (or velocity or acceleration) sensor, and a torque actuator with an angular (or angular velocity) sensor, in such a way that the product of the actuator signal by the sensor signal represents the energy (power) exchange between the structure and the control system. Such systems enjoy very interesting properties. The open-loop Frequency Response Function (FRF) of a collocated control system corresponds to a diagonal component of the dynamic flexibility matrix. If the actuator and sensor are attached to d.o.f. k, the open-loop FRF reads Gkk (ω) =
m X φ2 (k) i
i=1
µi ωi2
Di (ω) + Rkk
(1.55)
If one assumes that the system is undamped, the FRF is purely real Gkk (ω) =
m X i=1
φ2i (k) + Rkk µi (ωi2 − ω 2 )
(1.56)
All the residues are positive (square of the modal amplitude) and, as a 14
Gkk(!)
resonance
Gkk(0) = Kà1 kk static response
Rkk
!i+1
!i
residual
zi
! mode
antiresonance
Figure 1.5: Open-loop FRF of an undamped structure with a collocated actuator/sensor pair (no rigid body modes). result, Gkk (ω) is a monotonously increasing function of ω, which behaves as illustrated in Fig.1.5. The amplitude of the FRF goes from −∞ at the resonance frequencies ωi (corresponding to a pair of imaginary poles at s = ±jωi in the open-loop transfer function) to +∞ at the next resonance frequency ωi+1 . Since the function is continuous, in every interval, there is a frequency zi such that ωi < zi < ωi+1 where the amplitude of the FRF vanishes. In structural dynamics, such frequencies are called anti-resonances; they correspond to purely imaginary zeros at ±jzi , in the open-loop transfer function. Thus, undamped collocated control systems have alternating poles and zeros on the imaginary axis (Martin, 1978). The pole / zero pattern is that of Fig.1.6.a. For a lightly damped structure, the poles and zeros are just moved a little in the left-half plane, but they are still interlacing (Fig.1.6.b). If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero, zi , the amplitude of the response of the collocated sensor vanishes. This means that the structure oscillates at zi according to the shape shown in dotted line on Fig.1.7.b. We will establish in the next section that this shape, and the frequency zi , are actually a mode shape and a natural frequency of the system obtained by constraining the d.o.f. on which the control system acts. We know from control theory that the open-loop zeros are asymptotic values of the closed-loop poles, when the feedback gain goes to infinity (Franklin et al., 1986, Kailath, 1980). The natural frequencies of the constrained system depend on the d.o.f. where the constraint has been added (this is indeed well known in control 15
Im(s)
(a)
jzi j!i
Im(s)
x
x
x
x
x
Re(s)
x
(b)
Re(s)
Figure 1.6: Pole/Zero pattern of a structure with collocated (dual) actuator and sensor. (a) undamped. (b) Lightly damped. (Only the upper half of the complex plane is shown, the diagram is symmetrical with respect to the real axis). theory that the open-loop poles are independent of the actuator and sensor configuration while the open-loop zeros do depend on it). However, from the foregoing discussion, for every actuator/sensor configuration, there will be one and only one zero between two consecutive poles, and the interlacing property applies for any location of the collocated pair. Referring once again to Fig.1.5, one easily sees that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically in such a way that its high frequency asymptote becomes tangent to the frequency axis. This produces a shift in the location of the transmission zeros to the right, and the last one even moves to infinity as the feedthrough component disappears from the FRF. Thus, neglecting the residual modes tends to overestimate the frequency of the transmission zeros. As we shall see shortly, the closed-loop poles which remain at finite distance move on loops joining the open-loop poles to the open-loop zeros ; therefore, altering the open-loop pole/zero pattern has a direct impact on the closed-loop poles. The open-loop transfer function of a undamped structure with a collocated actuator/sensor pair (Fig 1.6.a) can be written Q 2 2 (s /z + 1) (ωi < zi < ωi+1 ) (1.57) G(s) = G0 Q i 2 i2 (s /ωj + 1) j For a lightly damped structure (Fig.1.6.b), it reads
16
u (a)
y (b)
(c)
g
Figure 1.7: (a) Structure with collocated actuator and sensor; (b) structure with additional constraint; (c) structure with additional stiffness along the controlled d.o.f.
Q 2 2 (s /z + 2ξi s/zi + 1) G(s) = G0 Q i 2 2i j (s /ωj + 2ξj s/ωj + 1)
(1.58)
The corresponding Bode and Nyquist plots are represented in Fig 1.8. Every imaginary pole at ±jωi introduces a 1800 phase lag and every imaginary zero at ±jzi a 1800 phase lead. In this way, the phase diagram is always contained between 0 and −1800 , as a consequence of the interlacing property. For the same reason, the Nyquist diagram consists of a set of nearly circles (one per mode), all contained in the third and fourth quadrants. Thus, the entire curve G(ω) is below the real axis (the diameter of every circle is proportional to ξi−1 ).
1.5.1
Transmission zeros and constrained system
We now establish that the transmission zeros of the undamped system are the poles (natural frequencies) of the constrained system. Consider the undamped structure of Fig.1.7.a (a displacement sensor is assumed for simplicity). The governing equations are 17
Im(G)
w =0
Re(G)
G dB
! = zi
wi
w
zi
f 0°
w
-90°
! = !i
-180°
Figure 1.8: Nyquist diagram and Bode plots of a lightly damped structure with collocated actuator and sensor. Structure: Mx ¨ + Kx = b u
(1.59)
y = bT x
(1.60)
Output sensor : u is the actuator input (scalar) and y is the sensor output (also scalar). The fact that the same vector b appears in the two equations is due to collocation. For a stationary harmonic input at the actuator, u = u0 ejω0 t ; the response is harmonic, x = x0 ejω0 t , and the amplitude vector x0 is solution of (K − ω02 M )x0 = b u0
(1.61)
The sensor output is also harmonic, y = y0 ejω0 t , and the output amplitude is given by y0 = bT x0 = bT (K − ω02 M )−1 b u0 (1.62) Thus, the transmission zeros (anti-resonance frequencies) ω0 are solutions of bT (K − ω02 M )−1 b = 0
(1.63)
Now, consider the system with the additional stiffness g along the same d.o.f. as the actuator/sensor (Fig 1.7.c). The stiffness matrix of the modified system is K + gbbT . The natural frequencies of the modified system are solutions of the eigenvalue problem [K + gbbT − ω 2 M ]φ = 0 18
(1.64)
For all g the solution (ω, φ) of the eigenvalue problem is such that (K − ω 2 M )φ + gbbT φ = 0
(1.65)
bT φ = −bT (K − ω 2 M )−1 gbbT φ
(1.66)
or
Since bT φ is a scalar, this implies that bT (K − ω 2 M )−1 b = −
1 g
(1.67)
Taking the limit for g → ∞, one sees that the eigenvalues ω satisfy bT (K − ω 2 M )−1 b = 0
(1.68)
which is identical to (1.63). Thus, we have established that ω = ω0 ; the imaginary zeros of the undamped collocated system, solutions of (1.63), are the poles of the constrained system (1.64) at the limit, when the stiffness g added along the actuation d.o.f. increases to ∞: lim [(K + gbbT ) − ω02 M ]x0 = 0
g→∞
1.5.2
(1.69)
Nearly collocated control system
In many cases, the actuator and sensor pair are close to each other without being strictly collocated. This situation is examined here. u s a
y
Figure 1.9: Structure with nearly collocated actuator/sensor pair. Consider the undamped system of Fig.1.9 where the actuator input u is applied at a and the sensor y is located at s. Returning to the modal expansion of the dynamic flexibility matrix (1.48), the open-loop FRF of the system is no longer given by a diagonal term, but rather by n
G(ω) =
X φi (a)φi (s) y = u µ (ω 2 − ω 2 ) i=1 i i 19
(1.70)
where φi (a) and φi (s) are the modal amplitudes at the actuator and the sensor locations, respectively (the sum includes all the normal modes in this case). Comparing with (1.56), the residues of (1.70) are no longer guaranteed to be positive; however, if the actuator location a is close to the sensor location s, the modal amplitudes φi (a) and φi (s) will be close to each other, at least for the low frequency modes, and the corresponding residues will again be positive. The following result can be established in this case : If two neighboring modes are such that their residues φi (a)φi (s) and φi+1 (a)φi+1 (s) have the same sign, there is always an imaginary zero between the two poles (Martin, 1978). Since G(ω) is continuous between ωi and ωi+1 , this result will be established if one proves that the sign of G(ω) near ωi is opposite to that near ωi+1 . At ω = ωi + δω, G(ω) is dominated by the contribution of mode i and its sign is sign[
φi (a)φi (s) ] = −sign[φi (a)φi (s)] ωi2 − ω 2
(1.71)
At ω = ωi+1 − δω, G(ω) is dominated by the contribution of mode i + 1 and its sign is sign[
φi+1 (a)φi+1 (s) ] = sign[φi+1 (a)φi+1 (s)] 2 ωi+1 − ω2
(1.72)
− Thus, if the two residues have the same sign, the sign of G(ω) near ωi+1 is + opposite to that near ωi . By continuity, G(ω) must vanishes somewhere in − between, at zi such that ωi+ < zi < ωi+1 . Note, however, that when the residues of the expansion (1.70) are not all positive, there is no guarantee that G(ω) is an increasing function of ω, and one can find situations where there are more than one zero between two neighboring poles.
1.5.3
Non-collocated control systems
Since the low frequency modes vary slowly in space, the sign of φi (a)φi (s) tend to be positive for low frequency modes when the actuator and sensor are close to each other, and the interlacing of the poles and zeros is maintained at low frequency. This is illustrated in the following example: Consider a simply supported uniform beam of mass per unit length m and bending stiffness EI. The natural frequencies and mode shapes are respectively ωi2 = (iπ)4 20
EI ml4
(1.73)
Sensor motion
u
s>a
y
a = l=10
Mode 1 sin(ùx=l)
Mode 2 sin(2ùx=l)
l=2
2l=3
l=3
l=4
Mode 3 sin(3ùx=l)
3l=4
l=2
Mode 4 sin(4ùx=l)
Figure 1.10: Uniform beam with non-collocated actuator/sensor pair. Mode shapes 1 to 4. and
iπx (1.74) l (the generalized mass is µi = ml/2). Note that the natural frequency increases as the square of the mode order. We assume that a force actuator is placed at a = 0.1 l and we examine the evolution of the open-loop zeros as a displacement sensor is moved to the right from s = a (collocated), towards the end of the beam (Fig 1.10). The evolution of the open-loop zeros with the sensor location along the beam is shown in Fig 1.11; the plot shows the ratio zi /ω1 , so that the openloop poles (independent of the actuator/sensor configuration) are at 1, 4, 9, 25, etc.... For s = a = 0.1 l, the open-loop zeros are represented by ◦; they alternate with the poles. Another position of the actuator/sensor pair along the beam would lead to a different position of the zeros, but always alternating with the poles. As the sensor is displaced from the actuator, s > a, the zeros tend to φi (x) = sin
21
! !1
Actuator
z4
25
Mode 5
s 0.2= l
z3 Mode 4
16
z2
0.25
z1 Mode 3
9 0.33
Mode 2
4 0.5 1 0.1
0.2 0.3 0.4 0.5
0.6
Mode 1 0.7 0.8 0.9 1.0
s l
Figure 1.11: Evolution of the imaginary zeros when the sensor moves away from the actuator along a simply supported beam (the actuator is at 0.1 l). The abscissa is the sensor location, the ordinate is the frequency of the transmission zero. increase in magnitude as shown in Fig.1.11, but the low frequency ones still alternate. When s = 0.2 l, z4 becomes equal to ω5 and there is no zero any longer between ω4 and ω5 when s exceeds 0.2 l. Such a situation, where an imaginary zero comes from below an imaginary pole to a larger value is called a pole/zero flipping. Similarly, z3 flips with ω4 for s = l/4, z2 flips with ω3 for s = l/3 and z1 flips with ω2 for s = l/2. Examining the mode shapes, one notices that the pole-zero flipping always occurs at a node of the mode shapes, and this corresponds to a change of sign in φi (a)φi (s), as discussed above. This simple example illustrates the behavior of the pole/zero pattern for nearly collocated control systems: the poles and zeros are still interlacing at low frequency, but not at higher frequency, and the frequency where the interlacing stops decreases as the distance between the actuator and sensor increases. A more accurate analysis (Spector and Flashner, 1989, Miu, 1993) shows that: For structures such as bars in extension, shafts in torsion or simply connected spring-mass systems (non dispersive), when the sensor is displaced from the actuator, the zeros migrate along the imaginary axis towards infin-
22
ity. The imaginary zeros are the resonance frequencies of the two substructures formed by constraining the structure at the actuator and sensor (this generalizes the result of section 1.5.1).
Imaginary zeros migrate towards
j1
Real zeros come from 1
Figure 1.12: Evolution of the zeros of a beam when the sensor moves away from the actuator. Every pair of imaginary zeros which disappears at infinity reappears on the real axis. For beams with specific boundary conditions, the imaginary zeros still migrate along the imaginary axis, but every pair of zeros that disappears at infinity reappears symmetrically at infinity on the real axis and moves towards the origin (Fig 1.12). Systems with right half plane zeros are called non-minimum phase. Thus, non collocated control systems are always nonminimum phase, but this does not cause difficulties if the right half plane zeros lie well outside the desired bandwidth of the closed-loop system. When they interfere with the bandwidth, they put severe restrictions on the control system, by reducing significantly the phase margin.
1.6
Active damping with collocated system
In this section, we will use the interlacing properties of collocated (dual) systems to develop Single Input-Single Output (SISO) active damping schemes with guaranteed stability. By active damping, we mean that the primary objective of the controller is simply to increase the negative real part of the system poles, while maintaining the natural frequencies essentially un23
Im(s)
(a)
Dynamic Amplification (dB)
(b)
Open-loop
þ
Damping
0
sinþ = ø
!
Re(s)
Figure 1.13: Role of damping (a) system poles (b) dynamic amplification. changed. This will simply attenuate the resonance peak in the dynamic amplification (Fig 1.13). Recall that the relationship between the damping ratio ξ and the angle φ with respect to the imaginary axis (Fig.1.13.a) is sin φ = ξ, and that the dynamic amplification at resonance (Fig.1.13.b) is 1/2ξ. By analogy with a swing which can be driven fairly easily at its natural frequency, this strategy will often require relatively little control effort. This is why it is also called Low Authority Control (LAC), by contrast with other control strategies which fully relocate the closed-loop poles (natural frequency and damping) and are called High Authority control (HAC). A remarkable feature of the LAC controllers discussed here is that the control law requires very little knowledge of the system (at most the knowledge of the natural frequencies). However, guaranteed stability does not mean guaranteed performances; good performance does require information on the system as well as on the disturbance applied to it, for appropriate actuator/sensor placement, actuator sizing, sensor selection and controller tuning. Actuator placement means good controllability of the dominant modes; this will be reflected by well separated poles and zeros, leading to wide loops in the root-locus plots. In order to keep the formal complexity to a minimum, we will assume no structural damping and perfect actuator and sensor dynamics throughout most of this section. The impact of the actuator and sensor dynamics on stability, and the beneficial effect of passive damping is discussed in (Preumont, 2002).
24
1.6.1
Lead control
Consider a undamped structure with a collocated, dual actuator/sensor pair. We assume that the open-loop FRF G(ω) does not have any feedthrough (constant) component, so that G(ω) decays at high frequency as s−2 (the high frequency decay is called roll-off ; the roll-off rate is −40 dB/decade in this case). The open-loop transfer function of such a system reads
G(s) =
n X i=1
(bT φi )2 µi (s2 + ωi2 )
(1.75)
where bT φi is the modal amplitude at the actuator/sensor location. This corresponds, typically, to a point force actuator collocated with a displacement sensor, or a torque actuator collocated with an angular sensor. The pole-zero pattern is that of Fig 1.14 (where 3 modes have been assumed); there are two structural poles in excess of zeros, to provide a roll-off rate s−2 ( a feedthrough component would introduce an additional pair of zeros). This system can be damped with a lead compensator : H(s) = g
s+z s+p
(p À z)
(1.76)
The block diagram of the control system is shown in Fig 1.15. This controller takes its name from the fact that it produces a phase lead in the frequency band between z and p, bringing active damping to all the modes belonging to z < ωi < p. Figure 1.14 also shows the root locus of the closed-loop poles when the gain g is varied from 0 to −∞. The closed-loop poles which remain at finite distance start at the open-loop poles for g = 0 and eventually go to the open-loop zeros for g → ∞. Since there are two poles more than zeros, two branches go to infinity. The controller does not have any roll-off, but the roll-off of the structure is enough to guarantee gain stability at high frequency.1 Note that the asymptotic values of the closed-loop poles for large gains being the open-loop zeros zi , which are the natural frequencies of the constrained system, they are therefore independent of the lead controller parameters z and p. For a structure with well separated modes, the individual loops in the root-locus (Fig 1.14) are to a large extent independent of each 1
For clarity purposes, throughout this chapter, the root-locus plots often use different scales on the real and imaginary axes, so that the angles are not relevant.
25
Im(s)
jz i
j! i
àp
Re(s)
àz
Lead
Structure
Figure 1.14: Open-loop pole/zero pattern and root locus of the lead compensator applied to a structure with collocated actuator/sensor (open-loop transfer function with two poles in excess of zeros). Different scales are used on the real and imaginary axes.
+ à
s+z g s+p
u
G(s) =
P i
(b Tþ i) 2 ö i(s 2+! 2i )
y
Figure 1.15: Block diagram of the lead compensator applied to a structure with collocated actuator/sensor (open-loop transfer function G(s) with two poles in excess of zeros).
26
other, and the root-locus for a single mode can be drawn from the lead controller and the asymptotic values ωi and zi of that mode only (Fig 1.16). The characteristic equation for this simplified system can be written from the pole-zero pattern:
Im(s)
jz i ømax
j!i
Re(s)
àp
àz
Figure 1.16: Structure with well separated modes and lead compensator, root-locus of a single mode.
1+α
(s2 + zi2 )(s + z) =0 (s2 + ωi2 )(s + p)
(1.77)
where α is the variable parameter going from α = 0 (open-loop) to infinity. This can be written alternatively 1+
1 (s2 + ωi2 )(s + p) =0 α (s2 + zi2 )(s + z)
If z and p have been chosen in such a way that z ¿ ωi < zi ¿ p, this can be approximated in the vicinity of jωi by 1+
p (s2 + ωi2 ) =0 α s(s2 + zi2 )
(1.78)
This characteristic equation turns out to be the same as that of the Integral Force Feedback (IFF) controller discussed in section 1.6.4. It can be shown that the maximum achievable modal damping is given by (Preumont, 2002)
27
2
ξmax =
1.6.2
zi − ωi 2ωi
(ωi > zi /3)
(1.79)
Direct velocity feedback
The Direct Velocity Feedback (DVF) is the particular case of the lead controller as z → 0 and p → ∞. Returning to the the basic equations: Structure: Mx ¨ + Kx = bu (1.80) Output (velocity sensor) : y = bT x˙
(1.81)
u = −gy
(1.82)
Control : one finds easily the closed-loop equation Mx ¨ + gbbT x˙ + Kx = 0
(1.83)
Upon transforming into modal coordinates, x = Φx and taking into account the orthogonality condition (1.25) and (1.26), one gets diag(µi )¨ z + gΦT bbT Φ z˙ + diag(µi ωi2 )z = 0
(1.84)
where z is the vector of modal amplitudes. The matrix ΦT bbT Φ is in general fully populated. For small gains, one may assume that it is diagonally dominant, ' diag(bT φi )2 . This assumption leads to a set of decoupled equations. Mode i is governed by µi z¨i + g(bT φi )2 z˙i + µi ωi2 zi = 0
(1.85)
By analogy with single a d.o.f. oscillator, one finds that the active modal damping ξi is given by 2ξi µi ωi = g(bT φi )2 (1.86) or ξi =
g(bT φi )2 2µi ωi
(1.87)
Thus, for small gains, the closed-loop poles sensitivity to the gain (i.e. the departure rate from the open-loop poles) is controlled by (bT φi )2 , the square of the modal amplitude at the actuator/sensor location. 2
This results from (1.126), the pole in (1.78) playing the same role as the zero in (1.125), and vice versa.
28
Now, let us examine the asymptotic behavior for large gains. For all g, the closed-loop eigenvalue problem (1.83) is (M s2 + gbbT s + K)x = 0
(1.88)
it follows that x = −(K + M s2 )−1 gsbbT x or bT x = −gsbT (K + M s2 )−1 bbT x Since
bT x
(1.89)
is a scalar, one must have sbT (K + M s2 )−1 b = −
1 g
(1.90)
and taking the limit for g → ∞ sbT (K + M s2 )−1 b = 0
(1.91)
The solutions of this equation are s = 0 and the solutions of (1.63), that is, the eigenvalues of the constrained system. The fact that the eigenvalues are purely imaginary, s = ±jω0 stems from the fact that K and M are symmetric and semi-positive definite. Typical root locus plots for a lead controller and a DVF controller are compared in Fig.1.17. Im(s)
(a)
g
Im(s)
(b)
g
Structure
Re(s)
Structure
Re(s)
Lead
DVF
Figure 1.17: Collocated control system. (a) Root locus for a lead controller. (b) DVF controller. As for the lead controller, for well separated modes, those which are far enough from the origin can be analyzed independently of each other. In this way, the characteristic equation for mode i is approximated by 1+g
s(s2 + zi2 ) =0 (s2 + ω12 )(s2 + ωi2 ) 29
(besides the poles at ±jωi and the zeros at ±jzi , we include the zero at s = 0 and the poles at ±jω1 ) which in turn, if ωi > zi À ω1 , may be approximated by s2 + zi2 1+g 2 =0 (1.92) s(s + ωi2 ) in the vicinity of mode i. This root locus is again identical to that of the IFF, (1.125), and the formula for the maximum modal damping (1.126) applies ξmax =
1.6.3
ωi − zi 2zi
(1.93)
Positive Position Feedback (PPF)
There are frequent situations where the open-loop FRF does not exhibit a roll-off of −40 dB/decade as in the previous section. In fact, a feedthrough component may arise from the truncation of the high frequency dynamics, as in (1.54), or because of the physical nature of the system (e.g. beams or plates covered with collocated piezoelectric patches, piezoelectric truss). In these situations, the degree of the numerator of G(s) is the same as that of the denominator and the open-loop pole-zero pattern has an additional pair of zeros at high frequency. Since the overall degree of the denominator of H(s)G(s) must exceed the degree of the numerator, the controller H(s) must have more poles than zeros. The Positive Position Feedback was proposed to solve this problem (Goh and Caughey, 1985, Fanson and Caughey, 1990). The second-order PPF controller consists of a second order filter H(s) =
s2
−g + 2ξf ωf s + ωf2
(1.94)
where the damping ξf is usually rather high (0.5 to 0.7), and the filter frequency ωf is adapted to target a specific mode. The block diagram of the control system is shown in Fig.1.18; the negative sign in H(s), which produces a positive feedback, is the origin of the name of this controller. Figure 1.19 shows typical root loci when the PPF poles are targeted to mode 1 and mode 2, respectively (i.e. ωf close to ω1 or ω2 , respectively). One sees that the whole locus is contained in the left half plane, except one branch on the positive real axis, but this part of the locus is reached only for large values of g, which are not used in practice. The stability condition can be established as follows: the characteristic equation of the closed-loop system reads 30
+ à
u
àg s 2+2ø f! f+! 2f
G(s) =
P i
(b Tþ i) 2 ö i(s 2+! 2i )
y
Figure 1.18: Block diagram of the second-order PPF controller applied to a structure with collocated actuator and sensor (the open-loop transfer function has the same number of poles and zeros).
ψ(s) = 1 + gH(s)G(s) = 1 − or
n X g bT φi φTi b { }=0 s2 + 2ξf ωf s + ωf2 i=1 µi (s2 + ωi 2 )
n X ψ(s) = s2 + 2ξf ωf s + ωf2 − g{ i=1
bT φi φTi b }=0 µi (s2 + ωi 2 )
According to the Routh-Hurwitz criterion for stability, if one of the coefficients of the power expansion of the characteristic equation becomes negative, the system is unstable. It is not possible to write the power expansion ψ(s) explicitly for an arbitrary value of n, however, one can see easily that the constant term (in s0 ) is an = ψ(0) = ωf2 − g
n X bT φi φT b i
i=1
µi ωi 2
In this case, an becomes negative when the static loop gain becomes larger than 1. The stability condition is therefore gG(0)H(0) =
n g X bT φi φTi b { } ω1 ) travel on nice stabilizing loops as in the case of a pure AMD actuator (Fig.3.10), as a result of the interlacing property of poles and zeros. Figure 3.13.a compares the transmissibility curves x ¨n /¨ x0 between the 5
This is just an example; in practice, some trial and error may be necessary to achieve the appropriate closed-loop behavior.
123
Figure 3.13: (a) Transmissibility x ¨n /¨ x0 between the ground acceleration and the acceleration of the top floor for the DVA and the Hybrib control (dB). (b) Cumulative RMS of the top acceleration. ground acceleration and the acceleration of the top floor for the DVA and the Hybrid control. One sees that the magnitude near mode 1 is lower with the hybrid controller than with the DVA alone; on the other hand, the hybrid control provides active damping to the higher modes as well, unlike the DVA. Fig.3.13.b displays the cumulative RMS value of the top acceleration, σx¨n , defined in a way similar to (3.14). Figure 3.14 compares the hybrid control with the purely active control (AMD); in both cases, the gains have been selected to achieve the same magnitude of the transmissibility for mode 1 (Fig.3.14.a). One observes that the hybrid control has slightly inferior performances on the higher frequency modes. The cumulative RMS shear force f0 is shown in Fig.3.14.b and the cumulative RMS control effort u is shown in Fig.3.14.c; the control effort required by the hybrid control is substantially lower than of the AMD.
124
Figure 3.14: (a) Transmissibility x ¨n /¨ x0 between the ground acceleration and the acceleration of the top floor for the DVA and the Hybrib control (dB). (b) Cumulative RMS of the shear force f0 . (c) Cumulative RMS control effort u.
125
3.6
Shear control
In the sequel of this chapter, we replace the inertial device on the top floor by a device acting between the ground and the first floor (we call it shear control for lack of another name, because the control system tends to modify the shear behavior of the first floor). Two control configurations are considered: (i) A force actuator combined with a relative displacement (or velocity) sensor and (ii) A displacement actuator combined with a collocated force sensor. These are very classical, dual configurations which can be realized with various actuation technologies going from electromagnetic to hydraulic for the first one, ball screw or piezoelectric for the second one, depending on the application. The difference between the two architectures stems from the fact that the force actuator brings no stiffness in open-loop while the displacement actuator brings the extra stiffness Ka of the actuator in the first floor.
3.7
Force actuator, displacement sensor
Figure 3.15: (a) Shear frame with a force actuator u and collocated relative displacement sensor y = x1 − x0 . (b) Configuration corresponding to the transmission zeros zi . The first control configuration consists of a force actuator attached be126
tween the ground and the first floor, with a collocated displacement (or velocity) sensor measuring the relative motion y = x1 − x0 (Fig.3.15.a). The open-loop poles are the natural frequencies of the system when the actuator is removed. In the case of the shear frame example, r π 2i − 1 k pi = ωi = 2 sin[ ] (i = 1, . . . , n) (3.20) m 2 2n + 1 According to chapter 1, the transmission zeros are the natural frequencies of the structure when the sensor output is zero, that is when the relative motion between the ground and the first floor is blocked (Fig.3.15.b); these are the natural frequencies of the shear frame with one storey less: r k π 2i − 1 zi = 2 sin[ ] (i = 1, . . . , n − 1) (3.21) m 2 2n − 1
3.7.1
Direct Velocity Feedback
For this actuator/sensor configuration, the most obvious control laws are the Lead or the Direct Velocity Feedback (DVF); they are very similar and only the DVF will be considered here, u = −gs (x1 − x0 )
(3.22)
The pole/zero pattern and the root-locus are shown in Fig.3.16; the string of interlacing poles and zeros near the imaginary axis are the open-loop features of the structure (a small uniform modal damping of ξi = 0.01 has been assumed), while the zero at the origin belongs to the DVF controller. There is only one pole in excess of zeros and the root locus has a single asymptote on the negative real axis. The rest of the root-locus consists of loops going from the open-loop poles to the nearby transmission zeros; the entire root locus is contained in the left half plane, meaning an infinite gain margin. Note that the loops leave the open-loop poles orthogonal to the imaginary axis as expected for pure viscous damping. We will discuss the results a little later. Note also that the open-loop pole/zero pattern is obtain directly from modal analysis and that the entire root-locus can be drawn from the open-loop pole/zero pattern and the control law. This makes the design straightforward.
127
Figure 3.16: Shear control with force actuator. Root locus of the DVF, u = −gs (x1 − x0 ).
3.7.2
First-order Positive Position Feedback
An alternative to the DVF is the first-order6 Positive Position Feedback (PPF) (Baz, Poh and Fedor, 1992, Høgsberg and Krenk, 2006), which consists of g (x1 − x0 ) (3.23) u= 1 + τs As the name says, it is a positive feedback; it behaves statically like a negative stiffness of constant g. The root-locus is shown in Fig.3.17. The string of interlacing open-loop poles and zeros near the imaginary axis belong to the structure and are the same as in the previous case (DVF), because the two controllers use the same actuator and sensor configuration. The pole on the real axis belong to the controller; there are 3 poles in excess of zeros, leading to 3 asymptotes, 2 at ±1200 and the positive real axis, which means that the controller is conditionally stable. The stability limit is reached when the pole travelling on the real axis reaches the origin. The stability 6
PPF controllers have been examined in section 1.6.3; unlike the second-order PPF, it is a wide-band controller which does not need to be tuned on a targeted mode.
128
Figure 3.17: Shear control with a force actuator. Root locus of the PPF, g u = 1+τ s (x1 − x0 ). condition is given by (1.100): gG(0) < 1
(3.24)
where G(s) = y/u is the open-loop transfer function. Since G(0) is in fact the static compliance of the system seen from the actuator, the stability condition simply means that the negative stiffness of the controller should not exceed the static stiffness of the system; this would in fact lead to the static collapse of the first floor. Comparing the two root-locus plots of the DVF and PPF, one observes that the loop corresponding to the low frequency mode is much wider for the PPF than for the DVF, which might suggest better performance; however, a closer examination reveals that only the initial part of the loop is available, because of the stability condition. Note also that in the PPF case, the loops do not leave the open-loop poles orthogonal to the imaginary axis as in the DVF case (as a result of the negative stiffness which softens the system), which suggests that the control
129
effort may be larger, as we will see shortly7 .
3.7.3
Comparison of the DVF and the PPF
A comparison of the DVF and the PPF has been conducted with the shear frame example of Fig.3.15; it is reported in Fig.3.18. The gain of the two controllers have been selected to produce similar performances on the first mode of the transmissibility x ¨n /¨ x0 between the ground acceleration and that of the top floor (Fig.3.18.a). Observe that the DVF tends to bring more damping to the higher modes, and that the peaks in the transmissibility for the PPF occur at lower frequencies (softening). The RMS top acceleration is compared in Fig.3.18.b; the results are extremely close in this problem which is dominated by the first mode. Fig.3.18.c compares the shear force, f0 = k(x1 − x0 ); the results obtained with the DVF are substantially better than those of the PPF; this seems to be associated with the negative stiffness inherent to the PPF. This effect is even more apparent in Fig.3.18.d which compares the control effort (3.18); the control effort of the PPF is more than doubled with respect to that of the DVF. This is not surprising, when you think about it, since part of the control is used to soften the structure rather than damping vibration. This will have a direct impact on the actuator design and cost.
3.8
Displacement actuator, force sensor
To conclude, consider an active strut consisting of a displacement actuator and a collocated force sensor (Fig.3.19); the control input is the unconstrained expansion u = δ and the actuator stiffness is Ka , so that the constitutive equation of the actuator is ∆ = u + f /Ka
(3.25)
where ∆ is the total expansion of the strut and f is positive in traction8 . This control system being collocated, it has interlacing open-loop poles and zeros. The open-loop poles are obtained by setting u = 0, which means 7
There is an interesting analogy with the motion of a swing: everybody knows from personal experience that increasing or reducing the amplitude a swing which vibrates at its natural frequency is easy to do, with little effort; this amounts to moving the poles a little to the right, or to the left, i.e. changing the real part. On the other hand, moving the swing at a frequency different from its natural one requires a lot of effort. 8 This equation simply states that the total extension of the active strut is the sum of the unconstrained expansion and the elastic deformation.
130
Figure 3.18: Comparison of the DVF and PPF for the shear frame example. (a) Transmissibility x ¨n /¨ x0 (dB); the gains have been selected to achieve similar performances for the first mode. (b) Cumulative RMS top acceleration x ¨n . (c) Cumulative RMS shear force at the base f0 . (d) Cumulative RMS control effort u.
131
Figure 3.19: (a) Shear frame with a displacement actuator u = δ and collocated force sensor y = f , and IFF controller. (b) Configuration corresponding to the open-loop poles pi ; Ka is the axial stiffness of the actuator. (c) Configuration corresponding to the transmission zeros zi (actuator removed). that the actuator will appear as an additional static stiffness Ka between the ground and the first floor (Fig.3.19.b); defining the reduced stiffness of the actuator κa = Ka /k (3.26) the stiffness matrix of the open-loop system is 2 + κa −1 0 . . . 0 −1 2 −1 . . . 0 0 −1 2 ... 0 K = k ... 0 0 −1 2 −1 0 . . . 0 −1 1
(3.27)
the natural frequencies of this system are the open-loop poles pi of the control system. The transmission zeros are obtained by cancelling the sensor output, y = f , which amounts to removing the active strut (Fig.3.19.c); the natural frequencies of this system are the transmission zeros zi of the control system. Comparing with the dual case discussed in the previous section (DVF), which involves a force actuator and a displacement sensor, 132
Im(s)
Re(s)
Figure 3.20: Root locus of the IFF u = δ = (g/s)(f /Ka ) (the scale on the real axis has been magnified as compared to Fig.3.16 and 3.17). we observe that the transmission zeros of this problem are the open-loop poles of the previous problem, given by (3.20), and that the open-loop poles of this problem would become asymptotically identical to the transmission zeros of the previous problem for an infinitely stiff control element, Ka → ∞. The control law adopted in this case is the Integral Force Feedback (IFF): u=δ=
g f s Ka
(3.28)
It is a positive feedback, based on the force f measured in the strut (Ka at the denominator is for normalization purposes: f /Ka is the elastic elongation of the strut). The open-loop pole/zero pattern is quite similar to that of the DVF, in reverse order: the string of transmission zeros are at lower frequencies than the open-loop poles. The controller consists of a pole at the origin, instead of a zero in the previous case. For well separated modes, the maximum
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achievable modal damping is given by (1.126) ξimax =
pi − zi 2zi
(3.29)
where pi is the open-loop pole and zi is the corresponding transmission zero. This formula may be used to calculate the minimum stiffness of the actuator to achieve a given value of the damping in the targeted mode (mode 1 in this case). With z1 = ω1 given by (3.2) and ξimax fixed by the designer, one gets the minimum value of pi which is the first natural frequency corresponding to the stiffness matrix (3.27); this allows to compute Ka .
3.8.1
Comparison of the IFF with the DVF
A comparison of the IFF and the DVF has been conducted on the shear frame example. The active member stiffness has been selected according to κa = Ka /k = 5. Figure 3.20 shows the root-locus of the IFF; the control gain has been selected to match the performance of the DVF on the first mode of the transmissibility x ¨n /¨ x0 . The results of the comparison between the IFF and the DVF are summarized in Fig.3.21: Figure 3.21.a compares the transmissibility Tx¨n w with that in open-loop; as expected, the peaks in the IFF curve have slightly higher frequencies, due to the stiffening from the active member, and the active damping of the higher modes is less pronounced than for the DVF, due to the nature of the control law (the control amplitude decreases with frequency). Figure 3.21.b compares the cumulative RMS of the top acceleration. Figure 3.21.c compares the cumulative RMS value of the force sensor output of the IFF, y = f and the input force u of the DVF.9 Finally, Fig.3.21.d compares the power requirements of the two control strategies, ˙ for the DVF and E[f δ˙ ] for the IFF. Taking into account respectively E[u ∆] ˙ for the DVF and δ˙ = g f /Ka for the the control laws, respectively u = −g ∆ IFF, the MS power requirements read Z ∞ 2 2 ˙ ˙ σDV F = E[u ∆] = gE[∆ ] ∼ gω 2 |T∆w |2 dω (3.30) 0
2 ˙ σIF F = E[f δ ] = E[
g 2 f ]∼ Ka
Z
0
∞
g |Tf w |2 dω Ka
(3.31)
(the seismic excitation is assumed to be a white noise). Figure 3.21.d compares the cumulative MS values for the two control laws; they are very close to each other, and mainly concentrated on the first mode. 9
both represent the effort in the control element.
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Figure 3.21: Comparison of the IFF and the DVF for the shear frame example. (a) Transmissibility x ¨n /¨ x0 (dB); the gains have been selected to achieve similar performances in the first mode. (b) Cumulative RMS value of the top acceleration x ¨n . (c) Cumulative RMS value of the force sensor output, y = f , of the IFF, and of the input force u of the DVF. (d) Cumulative MS of the power requirements. 135
3.9
References
Asami, T, Nishihara, O., Baz, A., Analytical solutions to H∞ and H2 optimization of dynamic vibration absorbers attached to damped linear systems, Trans. ASME, J. of Vibration and Acoustics, Vol.124, 284-295, April 2002. Baz, A., Poh, S., Fedor, J., Independent Modal Space Control with Positive Position Feedback, Trans. ASME, J. of Dynamic Systems, Measurement and Control, Vol.114, No 1, 96-103, March 1992. Crandall, S.H., Mark, W.D., Random Vibration in Mechanical Systems, Academic Press, 1963. Den Hartog, J.P., Mechanical Vibrations, McGraw-Hill, 1956. Geradin, M., Rixen, D., Mechanical Vibrations, 2nd Edition, Wiley, 1997. Høgsberg, J.R., Krenk, S., Linear control strategies for damping of flexible structures, J. of Sound and Vibration, 293, 59-77, 2006. Ormondroyd, J., Den Hartog, J.P., The theory of the damped vibration absorber, Trans. ASME, J. of Applied Mechanics, 50:7, 1928. Preumont, A., Vibration Control of Active Structures, An Introduction, 2nd Edition, Kluwer, 2002. Preumont, A., Dufour, J.P., Melekian, Ch., Active Damping by Local Force Feedback with Piezoelectric Actuators, AIAA J. of Guidance, Vol.15, No 2, 390-395, March-April 1992.
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Index active damping, 23 force feedback, 63, 68 Active Mass Damper (AMD), 109, 117 Fr¨ obenius norm, 79, 83 active truss, 34 fraction of modal strain energy, 37 anti-resonance, 15, 18 gain stability, 25 Bingham model, 100 Gough-Stewart platform, 66 Bode plots, 17 Hamilton’s principle, 6, 35 collocated control system, 14, 23 hexapod, 67 constrained system, 17, 29, 47 High Authority Control (HAC), 24 cross talk, 44 hybrid control, 122 cubic architecture, 66 Integral Force Feedback (IFF), 27, 34, cumulative PSD, 5 37, 41, 51, 133 cumulative RMS value, 94, 117, 120 decentralized control, 44, 68 Direct Velocity Feedback (DVF), 28, 48, 127 dual (pair), 14, 64, 126 duality DVF-IFF, 41 dynamic amplification factor, 12 dynamic flexibility matrix, 12 dynamic mass, 10 Dynamic Vibration Absorber (DVA), 2, 95, 109, 112 effective modal mass, 11 energy converter, 59 equal peak design, 99, 113 error budget, 4 feedforward, 55 feedthrough, 14
jerk, 84 joints (flexible), 69, 74 joints (spherical), 69 lag compensator, 43 lead compensator, 25, 41, 127 Linear Quadratic Regulator (LQR), 89 Low Authority Control (LAC), 24 Magneto-Rheological fluid (MR), 2, 100 Maxwell unit, 57 membrane, 73 modal coordinate, 8 modal damping, 9, 28 modal participation factor, 9 moving coil transducer, 59
137
non-minimum phase, 23 Nyquist plot, 17 performance index, 89, 91 pole/zero flipping, 22 Positive Position Feedback (PPF), 30, 48 power requirements, 134 Power Spectral Density (PSD), 4 quarter-car model, 85, 101 relaxation isolator, 57 residual mode, 14 road profile, 83 roll-off, 25 Routh-Hurwitz criterion, 31 seismic response, 6 semi-active device, 100 semi-active sky-hook, 102 semi-active suspension, 101 shear control, 126 shear frame, 110 six-axis isolator, 65 sky-hook damper, 60, 62, 68, 87, 91 spillover, 3 suspension (active), 85 suspension (adaptive), 85, 101 suspension (semi-active), 85, 99, 101 thermal analogy, 36 transmissibility, 4, 55, 76, 88, 116 transmission zeros, 17, 39, 45, 48, 127, 132 Tuned Mass Damper (TMD), 109, 112 vibration isolation, 54 voice coil, 59, 71, 74 wheel-hop mode, 86 zeros, 17, 39, 45, 48, 64, 70, 127, 132 138
