Transcript
EFFECT OF FEEDBACK CONTROL ON THE POWER CONSUMPTION OF INDUCED-STRAIN ACTUATORS Sriram Chandrasekaran Douglas K. Lindner Bradley Department of Electrical and Computer Engineering Virginia Tech Blacksburg, VA 24061
[email protected]
Donald J. Leo Center for Intelligent Materials, Systems and Structures, Mechanical Engineering Department Virginia Tech, Blacksburg, VA 24061-0261
[email protected]
ABSTRACT In this paper we study the closed loop power flow characteristics between a controlled piezoelectric actuator and a current controlled drive amplifier for two different structural control laws. We determine the real and reactive power flow through the structure and actuator into the amplifier when the structure is excited with a sinusoidal disturbance force under both control laws. The dependence of the real and reactive components of the power on the material properties of the actuator, structure and the configuration of the controller is presented. These real and reactive power estimates are useful for sizing the drive amplifier for the actuator. 1. INTRODUCTION It is well known that piezoelectric actuators are effective in suppressing vibrations in active structural control systems. The main thrust of previous work has primarily been to show the time domain or frequency domain behavior of the piezoelectric actuators and structures under closed loop operating conditions, thus demonstrating their control capabilities. In this paper we focus on the power flow characteristics of a piezoelectric actuator functioning in a closed loop control system. Piezoelectric actuators impose some special requirements on the amplifier because their impedance is primarily reactive.
This means that the amplifier must handle significantly higher voltages and
circulating currents than suggested by the real electrical/mechanical power requirements of the actuator. The amplifier must be sized appropriately to accommodate this circulating power. Drive amplifiers for piezoelectric and electrostrictor actuators have been discussed by several authors. Linear amplifiers for piezoelectric actuators are discussed by Warkentin [1994].
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High voltage switching amplifiers for
piezoelectric actuators are discussed by Clingman [1997, 1998].
The switching amplifiers for
electrostrictors are reported by Zvonar, et. al [1996]. It has been noted that charge controlled amplifiers naturally reduce the distortion induced by the nonlinearities of the piezoelectric actuator [Main et. al 1997]. A similar result for electrostrictor actuators has been reported by Zvonar and Lindner [1998]. This reduced distortion can have beneficial effects on the performance of the control system. The analysis of power flow through the amplifier and actuator has been discussed by Warkentin [1994], Leo [2000], and Lindner and Zvonar [1997, 1998]. Warkentin shows that actively damping the structure with a controlled piezoelectric actuator causes the mechanical power injected into the structure by the external disturbance source to be absorbed by the structure and funneled to the electrical source. Similar results were presented by Chandrasekaran and Lindner [2000] that show that the electrical power at the actuator terminals has a negative real component, which indicate that the actuator feeds electrical power back to the source. Leo [2000] analyzed the relationship between energy dissipation and control parameters for a piezoelectric structure driven by a linear amplifier. The results demonstrated that the energy dissipation in a linear amplifier can be substantially greater than the real work performed by the actuator, highlighting the inefficiency of the control system when using linear amplifiers to power a piezoelectric actuator. In this paper we will show how the configuration of the amplifier and of the structural control loop affect the characteristics of the electromechanical power transfer between the actuator and the drive amplifier. The analysis is performed on an electromechanical model of a piezoelectric stack actuator coupled to a mechanical structure. The structure is acted on by an external disturbance force, which is assumed to be sinusoidal at the resonant frequency of the structure. We introduce a generic structural control law that feeds back the displacement of the structure to the current into the actuator. This generic control law, hence, requires the use of a current controlled amplifier to drive the actuator. We assume that actuator is driven by an ideal current source/sink whose value is determined by the control law. The closed loop impedance of the actuator is determined which is turn is useful to study the power flow characteristics between the actuator and the amplifier. The electromechanical power transfer characteristics between the amplifier and the actuator are presented for two specific examples of the structural control law. The first control law is used to actively damp the structure by feeding back the acceleration of the structure to the actuator current. This control law was proposed in Chandrasekaran and Lindner [2000] along with a detailed analysis of the 2
power flow and amplifier requirements. The second example consists of using a conventional positive position controller that feeds back the displacement to the actuator current. Positive position feedback was originally introduced by Goh and Caughey [1985] as an alternative to direct velocity feedback, and has since been shown to be a very practical method of incorporating damping into a resonant structure ([Fanson and Caughey (1990); Dosch, Leo, and Inman [1995]). 2. MODELING OF THE ACTUATOR AND STRUCTURE In this section, the electromechanical model of the piezoelectric actuator attached to a simple mechanical structure is presented.
This model accommodates bidirectional power flow making it
amenable to study the power flow between the actuator and the amplifier. The mechanical model of the actuator-structure is represented by a simple mass-spring-damper system acted on by a sinusoidal disturbance force Fext as shown in Figure 2.1. We assume that this structure is lightly damped. A drive amplifier operating off a fixed DC voltage Vdc provides the power to the piezoelectric actuator. We also assume that there is a structural control feedback loop to the amplifier, which feeds back acceleration of the structure. The mission requirements for this control system are to reduce the effect of the disturbance force on the motion of the structure. We assume that the actuator has a multilayered stack configuration as shown in Figure 2.2. Each layer is rectangular with width w, length l and thickness d. The actuator is formed by stacking n of these layers together. Contiguous layers are polarized in opposite directions and the voltage is applied to the layers as shown in Figure 2.2. A block diagram of the coupled electromechanical model of the actuator and the structure is shown in Figure 2.3. The development of this model is explained in detail in Chandrasekaran and Lindner [2000]. The constant k2 in the model is the electromechanical coupling coefficient given by (2.1)
2 Y11d 33 k = ε 33 2
This constant is defined as the fraction of the input electrical energy that is mechanically deliverable. The constants K1 and K2 depend on the coupling between the actuator and the structure. They are determined by the location of the actuator, configuration of the actuator, bonding layers, modal coupling coefficients of the structure, and other factors. If we interpret the physical system represented by Figure 3
2.1 to be a stacked actuator (Figure 2.2) bonded rigidly to a mass, then the constants K1 and K2 are given by (2.2)
1 nd K 2 = lw K1 =
3. ACTIVE CONTROL OF STRUCTURE In this section, a generic feedback control law relating the displacement of the structure and the actuator current is introduced. We assume that a unity gain current controlled amplifier synthesizes the actuator current according to the control law. Under this assumption, expressions for the displacement of the structure, actuator current, voltage and impedance under closed loop operation are determined. In the next section, these expressions are used to determine and compare the power transfer characteristics between the amplifier and the actuator for two specific controller configurations. From Figure 2.3, the force fa exerted by the actuator on the structure is given by fa =
Y33 K1K 2
(3.1)
d 33 1 Y33 K 2 q ε 33 nlw 1 − k 2 1 1 Kaq nlw K1
x−
1− k2 d = K a x − 33 ε 33
Here, Ka is the equivalent stiffness of the actuator and is given by
Ka =
(3.2)
Y33 K1K 2 . 1− k2
(
)
Substituting Equation (3.1) into Equation (2.1) we obtain M!x! + Bx! + (K + K a )x = f ext +
(3.3)
d 33 1 1 Kaq ε 33 nlw K1
So the addition of the actuator to the system increases the stiffness of the structure as we expect. Using the expressions for K1 and K2 given by Equation (2.2), Equation (3.3) can be rewritten as M!x! + Bx! + (K + K a )x = f ext
(3.4)
1 k2 + q nd 33 1 − k 2
Transforming Equation (3.4) to the Laplace domain we obtain k2 1 X (s ) = G (s ) Fext (s ) + Q ( s ) 2 nd 1 − k 33
(3.5)
4
where, G (s ) =
1 2 2 s + 2ζω n s + ω n M K + Ka M
ωn = ζ =
(3.6)
1
B 2
M K + Ka
Representing the charge Q(s) as the integral of the actuator current Ia(s), Equation (3.5) is transformed to (3.7) k 2 I a (s ) 1 X (s ) = G (s ) Fext (s ) + nd33 1 − k 2 s We define a generic control law relating the current in the actuator and the displacement of the structure given by, I a (s ) = K (s )X (s )
(3.8)
Substituting for actuator current in Equation (3.7), we get
1 k 2 K (s )X (s ) X (s ) = G (s ) Fext (s ) + nd 33 1 − k 2 s
(3.9)
The block diagram of the complete system after introducing this control law is shown in Figure 3.1. This block diagram assumes there is a current controlled amplifier with unity gain in the loop. From Equation (3.9), the displacement can be obtained as
X (s ) =
G (s )
1 − G (s )
1 k K (s ) 2 nd33 1 − k s 2
(3.10)
Fext (s )
From Figure 3.1, the voltage across the actuator according to the control law can be obtained as Va (s ) =
(3.11)
1 1 k2 I a (s ) − X (s ) sCblk nd33 1 − k 2
where, Cblk is the blocked capacitance of the actuator given by, Cblk = ε 33
(
nlw 1− k2 d
)
(3.12)
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The complex impedance of the actuator is then obtained as Z a (s ) =
Va (s ) 1 1 k 2 X (s ) = − I a (s ) sCblk nd 33 1 − k 2 I a (s )
(3.13)
Applying the control law in Equation (3.8), the closed loop impedance Za(s), reduces to Z a (s ) =
(3.14)
1 1 k2 1 − 2 sCblk nd33 1 − k K (s )
From Equations (3.8) and (3.10), the current in the actuator can be obtained as I a (s ) =
G (s )K (s ) 1 − G (s )
1 k K (s ) 2 nd 33 1 − k s 2
(3.15)
Fext (s )
In the next section, these expressions are used to study the electromechanical power transfer between the amplifier and the actuator. 4. POWER TRANSFER CHARACTERISITCS In the previous section we introduced a generic feedback control law relating the displacement of the structure to the actuator current. This feedback control law requires a current controlled amplifier. In this section, we determine the power transfer characteristics between the current controlled amplifier and the actuator for two specific examples of the control law. We assume that the external disturbance force is sinusoidal at the resonant frequency of the structure and use sinusoidal steady state analysis to study the power transfer characteristics. A schematic of a current controlled amplifier driving the actuator is shown in Figure 4.1. The amplifier is driven from a constant voltage source. It is forced by the current controller to synthesize the current to be driven into the actuator in response to the command from the structural controller. Due to the capacitive nature of the actuator the current and voltage at the actuator terminals are oscillatory. In particular, the amplifier must be able to sink current. In the following analysis, we assume that the amplifier acts as an ideal current source/sink to the actuator and the structure. In other words, the power transfer analysis is performed based on the system shown in Figure 4.2. The power amplifier is made to appear as an ideal current source/sink to the actuator. Mechanical power is injected into the actuator/structure from the external disturbance and electrical power is injected from the amplifier. From Equation (3.15), the actuator current at a given frequency ω, is given by
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I a ( jω ) =
G ( jω )K ( jω ) 1 − G ( jω )
1 k K ( jω ) 2 nd 33 1 − k jω 2
Fext ( jω )
(4.1)
The complex apparent electrical power at the terminals of the actuator is given by Pe ( jω ) = Va ( jω )I a∗ ( jω ) = I a ( jω ) Z a ( jω )
(4.2)
2
where we have used Equation (3.13). The mechanical power can be obtained from the external force and the velocity of the structure. The velocity of the structure is given by sX (s ) =
sG (s )
1 − G (s )
1 k K (s ) 2 nd 33 1 − k s 2
(4.3)
Fext (s )
The complex apparent mechanical power is then given by
[
]
Pm ( jω ) = Fext ( jω ) − jωX ∗ ( jω ) =
− jωG (− jω ) 2 Fext ( jω ) 2 1 k K (− jω ) 1 − G (− jω ) nd33 1 − k 2 − jω
(4.4)
These power quantities are complex, indicating that there is a net average power flow through the system and a circulating reactive power component between the amplifier and the actuator/structure. The average and reactive components are obtained from the real and imaginary parts of the apparent power. The average and reactive components of the electrical power are given by Pe, ave ( jω ) = I a ( jω ) Re[Z a ( jω )]
(4.5)
2
Pe, reac ( jω ) = I a ( jω ) Im[Z a ( jω )] 2
It can be seen that the real and imaginary parts of closed loop impedance determine the average and reactive components of the electrical power. Similarly, the average and reactive components of the apparent mechanical power are given by
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Pm, ave ( jω ) − jωG (− jω ) = Re 2 2 ( ) 1 − k K j ω Fext ( jω ) 1 − G (− jω ) 2 − jω nd 33 1 − k Pm, reac ( jω ) − jωG (− jω ) = Im 2 1 k 2 K (− jω ) Fext ( jω ) 1 − G (− jω ) nd 33 1 − k 2 − jω (4.6) We now study the power transfer characteristics between the amplifier and the actuator due to two distinct controller configurations given by K1 (s ) = − K f s 2 K 2 (s ) = g
(4.7) sω 2f
2
s + 2ζ f ω f s 2
1 + ω 2f
1 k2 nd 33 1 − k 2
The controller K1(s) was introduced by Chandrasekaran and Lindner [2000] as a method to actively damp the structure by feeding back the acceleration of the structure to the actuator current. The second controller configuration K2(s) represents the conventional positive position feedback (PPF) control law. From Equation (4.1), the current in the actuator due to active damping controller K1(s) can be obtained as I a1 (s ) =
− s 2G (s )K f G (s )K1 (s ) F ( s ) = Fext (s ) ext 1 k 2 K1 (s ) 1 k2 1 − G (s ) 1 + G (s ) sK f nd33 1 − k 2 s nd33 1 − k 2
(4.8)
Substituting for the G(s) from Equation (3.6), the current is given by I a1 (s ) =
ω 2K f
1 Fext (s ) 2M
1 1 k s + ωn s 2 + 2ζω n + K f 2 M nd 1 − k 33 2
(4.9) It can be seen from Equation (4.9), that this control law adds damping to the structure without changing the resonant frequency of the structure. In a similar fashion, the actuator current due to the PPF control law can be obtained as 8
I a 2 (s ) =
=
G (s )K 2 (s ) Fext (s ) 1 k 2 K 2 (s ) 1 − G (s ) nd33 1 − k 2 s sg 2ω 2f
2 2 2 2 s + 2ζω n s + ω n s + 2ζ f ω f s + ω f
(
)(
)
1 Fext (s ) g 2ω 2f 1 k2 M − 2 M nd33 1 − k (4.10)
We assume that the external force is sinusoidal at the resonant frequency of the structure and that the damping ζ, of the uncontrolled structure is zero in the following analysis. The actuator currents at the resonant frequency ωn, of the structure, from Equations (4.8) and (4.9), are then given by I a1 ( jω n ) ζ = 0 =
I a 2 ( jω n ) ζ = 0
− jω n
(4.11)
Fext ( jω n )
1 k nd 1 − k 2 33 − jω n = Fext ( jω n ) 1 k2 nd 1 − k 2 33 2
It can be seen that the actuator currents for both control laws are identical at the resonant frequency and is only determined by the material properties of the actuator. The current controlled amplifier driving the actuator has to be designed to support the amplitude given by Equation (4.11).
Having the
amplitudes of the actuator current to be identical for both control laws could mean that the design of the amplifier could be performed independent of the control law. To determine the electrical power, we first calculate the closed loop impedance of the actuator due to each of the control laws according to Equation (3.14). The closed loop impedances are then given by 1 1 k2 1 Z a1 (s ) = + 2 2 sCblk nd 33 1 − k s K f 2
2 2 1 k 2 s + 2ζ f ω f s + ω f 1 Z a 2 (s ) = − sCblk nd 33 1 − k 2 sg 2ω 2f
=−
(
s 2 + 2ζ f ω f s + ω 2f 1 − g 2 Cblk sg
)
ω 2f
2
9
(4.12) where,
g=
(4.13)
g 1 k2 nd 1 − k 2 33
The complex impedances from Equation (4.12) are given by Z a1 ( jω ) =
k2 1 1 1 − 2 2 jωCblk nd 33 1 − k ω K f
(
2 2 2 1 ω − ω f 1 − g Cblk Z a 2 ( jω ) = jω g 2ω 2f
) −
2ζ f g 2ω f
(4.14) From Equation (4.14), it can be seen that the real parts of the closed loop impedances for both the control laws are negative. Hence, there is a net flow of power from the actuator back to the amplifier. The origin of this net flow of power back to the amplifier is identified in the following. The average power due to each of the control laws can then be determined according to (4.15) 1 k2 1 2 Pe, ave1 ( jω ) = I a1 ( jω ) − nd 33 1 − k 2 ω 2 K f 2ζ f 2 Pe, ave2 ( jω ) = I a 2 ( jω ) − 2 g ωf Using Equation (4.11) to determine the average electrical powers at the resonant frequency ωn, of the structure with ζ = 0, we get, Pe, ave1 ( jω n )
Pe, ave 2 ( jω n )
ζ =0
ζ =0
=−
1
Fext ( jω n )
1 k2 nd 1 − k 2 K f 33 2ζ f ω n2 − = 2 2 2 1 k g ωf nd 1 − k 2 33
2
F ( jω ) 2 n ext
2ζ f ω n2 F ( jω ) 2 = − 2 n g ω f ext 10
(4.16) where, we have used Equation (4.13). From Equations (4.6) and (4.7), the corresponding expressions for the average mechanical power are given by Pm, ave1 ( jω ) − jω M = Re 2 2 1 1 k Fext ( jω ) ω n2 − ω 2 − jω 2ζω n + K f 2 M nd 1 − k 33 − jω ω 2f − ω 2 − 2 jζ f ω f ω Pm, ave2 ( jω ) M = Re 2 2 g 2ω 2f Fext ( jω ) 2 2 2 − − − − − ω ω 2 j ζω ω ω ω 2 j ζ ω ω n n f f f M
(
)
[(
[(
][(
)
]
)
)
]
(4.17) With ζ = 0 and the frequency set to the resonant frequency ωn, of the structure, the expressions for the average mechanical power reduce to Pm, ave1 ( jω n )
ζ =0
Pm, ave2 ( jω n )
ζ =0
=
1
Fext ( jω n )
2
(4.18)
1 k2 K f nd 1 − k 2 33 2 2ζ f ω n 2 = 2 Fext ( jω n ) g ω f
It can be seen from Equations (4.16) and (4.18), that the magnitudes of the average electrical and mechanical powers are respectively equal at the resonant frequency for both the control laws. Hence, in the absence of structural damping, the actuator absorbs the average mechanical power injected into the structure by the external mechanical force and redirects it to the electrical source. This is however, true over all frequencies, but is not explicitly shown due to the complexity of the expressions for the electrical and mechanical power. In the presence of structural damping, part of the average mechanical power is dissipated in the damping and the remainder is siphoned back to the amplifier by the actuator. Due to the capacitive nature of the actuator, there is a considerable amount of reactive power circulating between the actuator and the amplifier. An estimation of the amount of circulating reactive power and a study of its dependence on the actuator constants and controller configuration is critical to the design of 11
the drive amplifier. Using Equations (4.5) and (4.14), the reactive components of the electrical power for the two control laws are given by 1 2 Pe, reac1 ( jω ) = − I a1 ( jω ) ω C blk ω 2 − ω 2f 1 − g 2 Cblk 2 1 Pe, reac 2 ( jω ) = − I a 2 ( jω ) ω g 2ω 2f
(
)
(4.19) For the first control law, the reactive power is determined only by the blocked capacitance of the actuator. From the expression for the actuator current at the resonant frequency given by Equation (4.11), it can be seen that the reactive power circulating between the actuator and the amplifier, at the resonant frequency, is determined only by the material properties of the actuator and the resonant frequency of the structure and is independent of the controller gain. The reactive power at the resonant frequency of the structure, due to the second control law however, is determined not only by the blocked capacitance of the actuator but also by the parameters of the controller. This is because the controller has a reactive part unlike that in the first control law where the controller is purely real. The reactive components of the mechanical power at ωn, can be determined from Equation (4.6) and are given by Pm, reac1 ( jω ) − jω M = Im 2 2 1 1 k Fext ( jω ) ω n2 − ω 2 − jω 2ζω n + K f 2 M nd k 1 − 33 − jω ω 2f − ω 2 − 2 jζ f ω f ω Pm, reac 2 ( jω ) M = Im 2 2 g 2ω 2f Fext ( jω ) 2 2 2 − − 2 j − − 2 j − ω ω ζω ω ω ω ζ ω ω n n f f f M
(
)
[(
[(
)
]
)
][(
)
]
(4.20)
The reactive components at the resonant frequency with ζ = 0, are then obtained as
12
Pm, reac1 ( jω n ) Pm, reac 2 ( jω n )
ζ =0
ζ =0
=0 =
(4.21)
(
ω n ω 2f − ω n2 g
ω 2f
2
)F
ext
( jω ) 2
If the resonant frequency ωf, of the controller is chosen to be equal to ωn, then the reactive power due to the second control law also reduces to zero. Simulation Results
Simulation results of the closed loop system are presented in this section for both control laws. The controller parameters are chosen such that the same level of damping is added to the structure under both control laws. The magnitude response of the displacement of the structure under open and closed loop is shown in Figure 4.3. The resonant frequency ωf, of the positive position feedback controller is chosen to be equal to the resonant frequency ωn, of the structure. The average electrical power at the terminals of the actuator, plotted as a function of the frequency, is shown in Figure 4.4. It can be seen that the positive position feedback control law changes the resonant frequency of the structure in addition to actively damping it. Hence, in spite of the active damping added to the structure being the same under both control laws, the average electrical power due to the two control laws are slightly different. The reactive electric power at the actuator due to the two control laws is shown in Figure 4.5 as a function of frequency. Since the resonant frequency ωf, of the PPF controller was chosen to be equal ωn, the reactive components of the closed loop impedance of the actuator due to the two control laws (Equation (4.19)) are identical at the resonant frequency. Hence, since the actuator currents are also identical at the resonant frequency, the reactive electrical power at ωn is the same for both control laws as seen in Figure 4.5. The instantaneous electrical and mechanical power for the two control laws are shown in Figures 4.6 and 4.7 for a sinusoidal external disturbance at the resonant frequency ωn, of the structure. It can be seen that the average mechanical power is equal to the negative of the average electrical power for both control laws indicating that the actuator redirects the external mechanical energy back to the amplifier. In addition, it can also be seen that the oscillatory components of the electrical power are
13
identical for the two control laws. This is due to the fact that the reactive components of the electrical power are identical at ωn. The reactive mechanical powers are zero at ωn according to Equation (4.21).
5. BROADBAND POWER FLOW CHARACTERISTICS The expressions for mechanical and electrical power flow illustrate that these quantities are a function of frequency for a specified control law. The simulation in the previous section and Equation (4.21) demonstrate that under certain conditions the average power flow of an active damping controller and a second-order positive position feedback controller will be almost identical. This occurs when the second-order compensator is tuned to the structural resonance and approximately the same amount of closed-loop damping is added to the system. It is has been shown both analytically and experimentally that tuning the second-order compensator to the structural resonance is not the optimal solution for multiple-mode control [McEver and Leo (2000)]. In this section we will investigate the average and reactive power flow characteristics for the case of a second-order compensator that is not tuned exactly to the target frequency. The algorithm developed by McEver and Leo (2000) can be utilized to show that the filter parameters
ωf =ω 2
ζ f = 2/2
g 2 = 1/ 2
(5.1)
will yield a closed-loop damping ratio of ½ in both the compensator pole and the structural pole for a single-mode system.
The feedback parameter of compensator 1, Kf, can be tuned to produce an
equivalent closed-loop damping ratio. Equation (4.5) is then used to compute the average and reactive components of the power for both control laws. Figure 5.1 compares the average and reactive power flows for positive position feedback and active damping for systems with identical closed-loop damping ratios. The average power has a similar form although the peak of the average power is less than 1 for the case of positive position feedback. The significant differences between the two control laws lies in the reactive power flow. The reactive power flow is maximized near the structural resonance for the case of active damping. In the case of positive position feedback, the reactive power flow exhibits a zero crossing near the resonance of the uncontrolled system.
This zero in the reactive power is a consequence of utilizing a resonant
compensator. The compensator resonance produces a zero in the reactive power near the resonance of the uncontrolled system, see Equation (4.19). 14
These results demonstrate that the reactive power flow is a strong function of the type of feedback control law even when the closed-loop damping ratios are equivalent. Figure 5.2 is a comparison of the normalized reactive power flows for a range of closed-loop damping ratios. Figure 5.2a demonstrates that the off-resonance reactive power changes substantially for the active damping control law, particularly in the region above the uncontrolled natural frequency. In contrast, the variation in the reactive power flow of the PPF compensator is localized near the uncontrolled resonance, although some changes do occur in the quasi-static region below resonance. The difference between the two cases is explained by examining the expression for the complex electrical impedances, Equation (4.14). The reactive component of the active damping control law is equivalent to the blocked capacitance of the actuator. As such, it is inversely proportional to the driving frequency, ω. The second-order equation that describes the positive position feedback controller results in a reactive impedance that varies as a function of frequency and the controller parameters. Furthermore, the sign of the reactive impedance can change depending on the relative magnitude of the two components in the numerator of the real part of Za2(jω). Both of these attributes are exhibited by the impedance in Figure 5.2b. Variations in the reactive power flow impact the amplifier design. As discussed in the previous section, the circulating power plays an important role in determining the peak power requirements. This analysis demonstrates that the circulating power varies substantially as a function of the control law. Utilizing a resonant compensator for active damping of structural resonance can minimize the effects of the reactive power. CONCLUSIONS In this paper we have presented a detailed analysis of the power flow characteristics between a current controlled amplifier and a piezoelectric actuator under two different structural control laws. A coupled electromechanical model of the actuator and the structure acted by an external sinusoidal disturbance force is used for the analysis. A generic structural control law that feeds back displacement of the structure to the current into the actuator is introduced. It is shown that closed loop impedance and hence the power flow characteristics are determined by the controller configurations. It is shown that under both control laws, the closed loop impedance of the actuator has a negative real part in addition to a reactive component due to the capacitive nature of the actuator. The negative real part indicates that 15
there is a net flow of power from the actuator to the amplifier. This means that the actuator absorbs the mechanical energy injected into the structure from the external disturbance force and directs it to the electrical source. In addition, due to the capacitive nature of the actuator, there is a considerable amount of reactive energy that needs to be circulated between the actuator and the electrical source. The relative magnitudes of the real and reactive components of the electromechanical power are determined by the control law. The drive amplifier that controls the actuator current in response to the control law needs to be able to accommodate the circulating reactive power and the net flow of average power back from the actuator. ACKNOWLEDGEMENTS The research reported in this paper is supported by the AFOSR under grant numbers F49620-97-1-0254 and made use of ERC Shared Facilities supported by the National Science Foundation under Award Number EEC-9731677. REFERENCES Warkentin. D. J, Crawley. E.F, 1994, “Power flow and amplifier design for piezoelectric actuators in intelligent structures,” Proceedings of the SPIE, The International Society for Optical-Engineering, vol. 2190, pp. 283-94. Chandrasekaran. S. Lindner. D. K., 2000, “Power Flow Through Controlled Piezoelectric Actuators,”submitted to the Journal of Intelligent Material Systems and Structures. Clingman, J. D. 1997, “Drive Electronics for large piezoactuators,” Proceeding of SPIE's 1997 North American Symposium on Smart Structures and Materials: Industrial and Commercial Applications of Smart Structures Technologies, Janet M. Sater; Ed., Vol. 3044, San Diego, CA, pp. 459-467. Clingman. D.J, Gamble. M, 1998, “High voltage switching power amplifiers [for helicopter blade actuator],” Proceedings of the SPIE, The International Society for Optical-Engineering, vol. 3326, pp. 472-8. Goh, C.J., Caughey, T.K., 1985, “On the Stability Problem Caused by Finite Actuator Dynamics in the Collocated Control of Large Space Structures,” International Journal of Control, vol. 41, no. 2, pp. 787—802. Fanson, J.L, Caughey, T.K., 1990, “Positive Position Feedback Control for Large Space Structures,” AIAA Journal, vol. 28, no. 4, pp. 717—724. Dosch, J.J., Leo, D.J., Inman, D.J., 1995, “Modeling and control for vibration suppression of a flexible active structure,” Journal of Guidance, Control, and Dynamics, vol. 18, pp. 340-346. Zvonar, G. A., J. Luan, F. C. Lee, D. K. Lindner, S. Kelly, D. Sable, and T. Schelling, 1996, “High-Frequency Switching Amplifiers For Electrostrictive Actuators”, Proceedings of SPIE's 1996 North American Symposium on Smart Structures and Materials: Industrial and Commercial Applications of Smart Structures Technologies, C. Robert Crowe; Ed., Vol. 2721, San Diego, CA, pp. 465-475. Main JA, Garcia E. “Piezoelectric stack actuators and control system design: strategies and pitfalls”. Journal of Guidance Control & Dynamics, vol.20, no.3, May-June 1997, pp.479-85.
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McEver, M., Leo, D.J., 2000, “Autonomous vibration suppression using on-line pole-zero identification,” Proceedings of the ASME Adaptive Structures and Materials Symposium, to appear. Zvonar, G. A. and D. K. Lindner, 1998 "Power Flow Analysis of Electrostrictive Actuators Driven by Switchmode Amplifiers," Journal on Intelligent Material Systems and Structures, special issue on the 3rd Annual ARO Workshop on Smart Structures, Vol. 9, No 3, pp. 210 - 222. Zvonar G. A, Lindner D. K., 1997, “Nonlinear electronic control of an electrostrictive actuator,” SPIE-Int. Soc. Opt. Eng. Proceedings of Spie - the International Society for Optical Engineering, vol.3044, pp.448-58. USA. Leo, D.J., 2000, “Energy analysis of piezoelectric-actuated structures driven by linear amplifiers,” Journal of Intelligent Material Systems and Structures, vol. 10, no. 1, pp. 36-45.
17
x K B
Fext
M Piezo Actuator va
!x!
Drive Amplifier
Vdc
Structural Controller
Figure 2.1. System Under Consideration
d
nd 3
2 1
l
w
A= lw
ia +
va
_
Figure 2.2. Actuator Configuration
18
Actuator
q
D3
1 nlw
d ε33
d33 ε33 +
va
Σ
d33 ε33
_
S3
+ _
Σ d
Y33 K1
1 1− k 2 T3 K2 fa
fext
+ _
_
x!
!x! 1 s
1 M
Σ_
1 s
x
B K
Structure
Figure 2.3. Complete electromechanical model of actuator-structure Actuator-Structure
ia
1 s
q
d ε33
1 nlw d33 ε33
Unity gain current amplifier
1
va
+ _Σ
d33 ε33
_ +Σ
d
Y33 K1
1
Structural K(s) Controller
1− k 2
K2
fext
B _
_ +
Σ
+
_Σ
1 M
1 s K
Figure 3.1. Block diagram of closed loop system
19
1 s
x
Vdc
ACTUATOR+ STRUCTURE
ia
Power P = v a i a va Amplifier e
x
_
Current Controller
Σ
fex
+
K(s) Structural Controller
Figure 4.1. Schematic of amplifier driving the actuator
Pe = v a ia
ACTUATOR+ STRUCTURE
ia
va
Pm = f ext x!
x
K(s)
Figure 4.2. Simplified representation of the controlled actuator-structure
20
-70
-80
Open loop
-90
X ( jω ) Fext ( jω )
-100
-110
-120
Closed loop
-130
10
0
ω/ωn
Figure 4.3. Magnitude response of displacement of structure under open loop and closed loop (active damping (--) and PPF (-)) operating conditions. 12
10
8
Pe,ave (ω )
6
4
2
0
-2 10
0
ω/ωn
Figure 4.4. Average electrical power at actuator terminals under both control laws (Active damping (--) and PPF (-))
21
10
8
6
Pe,reac (ω ) 4
2
0
10
ω/ωn
0
Figure 4.5. Reactive electrical power at actuator terminals under both control laws (Active damping (--) and PPF (-))
pe1 (t ) = va1ia1
5
Electrical power
0
-5
Pe,ave1
-10
-15 0
0.01
0.015
0.02
0.025
0.03
0.035
0.04
pm1 (t ) = f ext x!1
12
Mechanical power
0.005
10 8 6
Pm,ave1
4 2 0 -2 0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
time in secs
Figure 4.6. Instantaneous electrical and mechanical power at ωn due to active damping control law
22
pe 2 (t ) = va 2ia 2
Electrical power
5
0
-5
Pe,ave2
-10
-15
0
Mechanical power
12 10
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
pm 2 (t ) = f ext x! 2
8
Pm,ave2
6 4 2 0 -2
0
0.005
0.01
0.015
0.02
0.025
0.03
time in secs
0.035
0.04
Figure 4.7. Instantaneous electrical and mechanical power at ωn due to active damping control law
23
Normalized reactive power
Normalized average power
PPF
active damping 0
-
0.1
1
active damping
0 PPF
0.1
10
ω / ωn
1
10
ω / ωn
(a)
(b)
Figure 5.1: Power flow comparison for active damping and positive position feedback for identical closedloop damping ratios: (a) normalized average power, (b) normalized reactive power .
Normalized reactive power
ζ= 0.38, 1.00 0 ζ= 1.00
ζ= 0.50, 0.50
ζ= 0.75 ζ= 0.08, 0.69
ζ= 0.38 ζ= 0.05 0 0.1
1
10
ω / ωn
0.1
1
10
ω / ωn
(a)
(b)
Figure 5.2: Variation in normalized reactive power for active damping (a) and positive position feedback (b).
24
