1. Field of the Invention
The invention refers to a controller unit for resetting a deflection of an oscillator excited with a harmonic oscillation, a device including such a controller unit, in particular a rotation rate sensor, as well as to a method for operating and for manufacturing such a controller unit.
2. Description of the Prior Art
Conventional control methods are tailored to control problems with constant or only slowly changing command variables. The value of a controlled process variable affected from a disturbance is kept close to a predetermined set point, or is updated as close as possible to a changing set point. Some applications (e.g. micromechanical rotation rate sensors for analysis of a Coriolis force) provide a control loop for resetting a deflection of an oscillator oscillating with its resonance frequency when stationary. A controller for such a control loop with a harmonic oscillation as command variable is conventionally designed such that a harmonic force signal exciting the oscillator is compensated so that the oscillator—apart form the harmonic oscillation corresponding to the command variable—performs as little movement as possible.
Typically, this feedback control problem is solved as illustrated in
The measurement signal (system output signal) is fed to a controller unit 120 with a demodulator 122. In the demodulator 122, the system output signal is multiplied with a harmonic signal of frequency ω0 which is equal to the resonance angular frequency ω0 of the oscillator 190, wherein a baseband version of the system output signal as well as additional frequency conversion products are formed. A low pass filter 124 damps higher frequency components, in particular at the double resonance angular frequency 2·ω0 of the oscillator 190. The baseband signal is fed to the controller 126, which operates in the baseband, whose design and dimensions can be established by known controller design methods. The controller 126 is, for example, a continuous PI-controller. Due to its integral component, high stationary position can be achieved in case of a constant command variable.
The output of the controller 125 is multiplied (modulated) with a harmonic signal of frequency ω0 equal to the resonance angular frequency □0 of the oscillator 190 in a modulator 128. The modulation product is fed to an actuator 180 as a controller signal, the actuator executing according to the controller signal a force to the oscillator 190 that acts opposite to the deflection of the oscillator 190. With the resonance angular frequency ω0 and the damping s0 of the oscillator as well as with the amplification A and the system dead time TS of the system formed of the actuator 180, the oscillator 190 and the sensor 170, the transfer function of the oscillator 190 to be controlled is given by equation (1):
In what follows it is assumed that the damping s0 of the oscillator 190 is much smaller than its resonance angular frequency (s0<<ω0), and that the oscillator 190 is excited altogether with the harmonic force signal Fe, which has a force amplitude superposing, respectively amplitude modulating an exciting oscillation with the resonance angular frequency ω0 of the oscillator:
Fe=F·cos(ω0·T) (2)
According to
The low pass filter 124 which has to show sufficient damping at the double resonance angular frequency of the oscillator to damp the frequency conversion product sufficiently at 2·ω0, limits the bandwidth of the controller and hence its reaction rate with respect to changes in the force amplitude F.
For a continuous PI-controller with amplification factor KP and the integral action coefficient KI the step response u(t) is produced by a step signal σ(t) as input signal according to equation (3):
u(t)=(KP+KI·t)·σ(t). (3)
By L-transformation of σ(t) and equation (3), the transfer function GR(s) follows:
A pole at s=0 resulting from the integral component is characteristic for the continuous PI-controller. In a PI-controller used in connection with a controlled system of first order with a system function Gs(s), the system parameter Ks, and the boundary angular frequency ω1 is, according to equation (5),
then the controller parameter amplification factor KP and integral action coefficient KI are typically chosen so that the pole in the system function GS(s) (system pole) is compensated by the zero of the transfer function of the controller GR(s) (controller zero). Equating coefficients in the equations (4) and (5) results in a condition for the controller parameter given by the relation according to equation (6):
Equation (6) determines only the ratio of the amplification factor KP to the integral action coefficient KI. The product of the system transfer function GS(s) and controller transfer function GR(s) gives the transfer function of the corrected open loop Gk(s). As the system pole according to equation (5) and the controller zero according to equation (4) cancel, the transfer function of the corrected open loop Gk(s) is given by the relation according to equation (7).
From the corrected open loop frequency response, the stability properties of the closed loop can be deduced via the Nyquist criterion. Because of the integral characteristics of the corrected open loop an absolute value characteristic results which declines with 20 db/decade. The phase always amounts to −90° for positive frequencies, to which application of the Nyquist criterion is typically limited. The phase characteristic is an odd function and has, at frequency 0, a 180° step from +90° for negative frequencies to −90° for positive frequencies. The transfer function Gw(s) for the closed loop generally results from that of the corrected open loop Gk(s) according to equation (8):
From equation (8) it follows that the transfer function Gw(s) for the closed loop is only stable when the locus of the corrected open loop neither encloses nor runs through the point −1 for 0≦ω<∞. One condition equivalent to this is that, at the transition of the absolute value characteristic of the corrected open loop through the 0 dB line, the phase of the corrected open loop is larger than −180°. As the phase is constant at −90° in the above case, the closed loop is thus always stable independent of the choice of amplification factor KP.
The bandwidth of the closed loop can be deduced from the frequency at the transition of the absolute value characteristic through the 0 dB line. The absolute value frequency response can be shifted via the amplification factor KP along the ordinate and, thus, the transition through the 0 dB line, respectively influencing the bandwidth that results from it.
The use of a classical PI-controller assumes a comparatively constant common variable. For this reason applications in which a harmonic common variable of almost constant frequency is to be controlled require a demodulator and a downstream low pass filter, which generate a corresponding baseband signal from the harmonic input signal.
It is therefore the object of the invention to provide an improved controller concept for resetting the deflection of oscillators of the type that oscillate harmonically in the stationary case (e.g. the deflection of one of the movably supported units of a rotation rate sensor) affected by a disturbance.
The present invention addresses the preceding and other objects by providing, in a first aspect, a controller unit. Such controller unit comprises a PI-controller with a proportional transfer element and an integrating transfer element arranged parallel to the proportional transfer element. A controller input of the controller unit is connected with both transfer elements.
A transfer function of the PI-controller has a conjugate complex pole at a controller angular frequency ωr in the s-plane or a pole at e±jω
In a second aspect, the invention provides a device. Such device includes a movably supported oscillator that is excitable to an oscillation with the resonance angular frequency ω0 along a direction of excitation and a controller unit comprising a PI-controller with a proportional transfer element and an integrating transfer element arranged parallel to the proportional transfer element where a controller input of the controller unit is connected with both transfer elements.
An integral action coefficient of the integrating transfer element and an amplification factor of the proportional transfer element are chosen so that the PI-controller is suitable for generating, at admission with a harmonic input signal of the controller angular frequency ωr modulated by the step function at the controller input, a harmonic oscillation of the controller angular frequency ωr with rising amplitude at the controller output. The controller angular frequency ωr is equal to the resonance angular frequency ω0.
In a third aspect, the invention provides a rotation rate sensor. The sensor includes a movably supported oscillator that is excitable in a direction of excitation to an oscillation with resonance angular frequency ω0, and a controller unit comprising a PI-controller with a proportional transfer element and an integrating transfer element arranged parallel to the proportional transfer element controller input of the controller unit is connected with both transfer elements.
The transfer function of the PI-controller has a conjugate complex pole at the resonance angular frequency ω0 in the s-plane or at e±jω
In a fourth aspect, the invention provides a method for operating a rotation rate sensor. Such method includes the steps of generating a measurement signal by a sensor reproducing a deflection of an oscillator and generating a controller signal for an actuator from the measurement signal, wherein the actuator counteracts the deviation of the oscillator from a harmonic oscillation with the resonance angular frequency ω0. The controller signal is derived by means of a controller unit from the measurement signal. The controller unit comprises a PI-controller with a proportional transfer element and an integrating transfer element, arranged parallel to the proportional transfer element. A controller input of the controller unit is connected with both transfer elements.
A transfer function of the PI-controller has a conjugate complex pole at the resonance angular frequency ω0 in the s-plane or a pole at e±jω
In a fifth aspect, the invention provides a method for manufacturing a rotation rate sensor. The method consists of dimensioning a controller unit comprising a PI-controller with a proportional transfer element and an integrating transfer element arranged parallel to the proportional transfer element where a controller input of the controller unit is connected with both transfer elements. The PI-controller is provided with a transfer function that has a conjugate complex pole at a controller angular frequency ωr in the s-plane or a pole at e±jω
The preceding and other features of the invention will become further apparent from the detailed description that follows. Such written description is accompanied by a set of drawing figures in which numerals, corresponding to numerals of the written description, point to the features of the invention. Like numerals point to like features of the invention throughout both the written description and the drawing figures.
Turning to the drawings,
Analogous to equation (3), equation (9) below describes the relation between the controller output signal u(t) and the controller input signal xd(t) for xd(t)=σ(t):
u(t)=(KP+KI·t)·sin(ω0·t)·σ(t). (9)
The Laplace-transform of the controller output signal u(t) and controller input signal xd(t) are derived as in equations (9a) and (9b) below:
The transfer function GR0(s) of the PI-controller 225 for harmonic command variables follows as in equation (10) below:
A conjugate complex pole at s=±jω0 resulting from the generalized integral component is characteristic of the continuous PI-controller 225. With an harmonic oscillation of frequency ω0 at the controller input, the PI-controller 225 generates no phase shift at the controller output. For adjusting an arbitrary phase, the controller unit 220 therefore additionally includes a dead time element 226 with controller dead time TR in series with the PI-controller 225. The transfer function GR(s) of the controller unit 220 thus follows from equation (11) below:
The controller parameters Ki, KP are chosen so that the controller zeros of the controller transfer functions according to equation (11) compensate the conjugate complex system pole in the system transfer functions according to equation (1). Equations (12a) and (12b) result from equating the coefficients of equations (1) and (11) for determination of the controller parameters Ki, Kp:
According to one embodiment, the damping s0 and the resonance angular frequency ω0 of the oscillator 190 are chosen so that s0<<ω0 is satisfied and that, hence, equation (12b) is satisfied in very good approximation. Equation (12c) results from equation (12a) as a dimensioning rule for the ratio of the integral action coefficient KI to the amplification factor KP:
The transfer function Gk(s) of the corrected open loop results from the product of the system transfer function GS(s) and the controller transfer function GR(s). As the expression for the conjugate complex system pole and the conjugate complex controller zeros cancel by appropriate dimensioning according to equations (12b), (12c), the transfer function Gk(s) of the corrected open loop results as equation (13):
By feedback control with a conventional PI-controller, a jump from +90° to −90° occurs in the phase frequency response of the corrected open loop at the frequency ω=0. In contrast, a 180° phase jump occurs at the frequency ω0 in the PI-controller 225 designed for harmonic command variables (not necessarily between +90° and −90°). According to one embodiment, the controller dead time TR is therefore chosen so that the 180° phase jump occurs as close as possible to ω0, for example by dimensioning the controller parameters according to equation (14a) below:
If the phase shift produced by the system dead time TS alone at ω0 is less than 90°, then the phase ratio of 180° can also be generated by an inverting controller. In this case the phases produced by the controller dead time TR and the system dead time TS at ω0, respectively, must merely add to π/2. The dimensioning rule for the controller dead time TR is then:
The stability properties of the closed loop can be deduced via the Nyquist criterion from the frequency response of the corrected open loop. The corrected open loop consists of the generalized integrator and the combination of system dead time TS and controller dead time TR. By appropriate dimensioning of the controller dead time TR according to equations (14a) or (14b), the phase characteristics at the frequency ω0 has a 180° jump between +90° for lower frequencies ω<ω0 to −90° to higher frequencies ω>ω0. The transfer function Gw(s) of the closed loop is related to that of the corrected open loop Gk(s), again, according to equation (8).
When the controller dead time TR is determined according to equation (14a) the closed loop is exactly stable when the locus of the corrected open loop neither encloses nor runs through the point −1 for 0≦ω<ω0. In contrast, when the controller dead time TR is determined according to equation (14b) and the PI-controller 225 generates a 180° phase then the closed loop is exactly stable when the locus of the corrected open loop at a negative real axis starts at a value larger than −1.
In the interval 0≦ω<ω0 the absolute value characteristic intersects the 0 dB line at the gain crossover frequency where the frequency difference from ω0 at the gain crossover frequency determines the bandwidth of the closed loop. The absolute value frequency response and, hence, the gain crossover frequency can be shifted by the amplification factor KP along the ordinate so that the resulting bandwidth of the closed loop is adjustable. According to one embodiment, the amplification factor KP is chosen so that the bandwidth is maximal within the limits given by the stability criteria.
The graphs of the left hand column (from top to bottom) of
amplification A=s02+ω02; and system dead time
The controller zero is chosen so that the system pole is compensated. As the phase (at ω0) which is produced by the system dead time is smaller than 90°, the phase ratio of 180° can be realized by a minus sign in the controller (inverting controller). For an amplification factor KP=− 1/10 the integral action coefficient KI results from equation (12c) and the controller dead time TR results from equation (14b) as TR=π/4·ω0.
The resulting bandwidth of the closed loop amounts to approximately 500 Hz and is clearly larger than in the comparative example of a conventional PI-controller operated in the baseband.
The device of
The controller unit 220 has a PI-controller 225, which includes a proportional transfer element 224 with an amplification factor KP and a integrating transfer element 222 with an integral action coefficient KI, for harmonic command variables. The integral action coefficient KI and the amplification factor KP are chosen so that the zero of the controller transfer function of the PI-controller 225 and the conjugate complex pole of the system transfer function, which describes the oscillator 190, compensate in the s-plane.
According to one embodiment, the damping s0 of the oscillator 190 with respect to deflection in the direction of excitation is much smaller than the resonance angular frequency ω0 of the oscillator 190 and the ratio of the integral action coefficient KI to the amplification factor KP in sec−1 corresponds approximately to the damping s0. Moreover, the amplification factor KP can be chosen so that the resulting bandwidth is as high as possible for respective stability requirements. The integral action coefficient KI is then chosen in relation to the damping s0 and the amplification factor KP according to equation (12c).
According to one embodiment, the system formed from the actuator 180, the oscillator 190 and the sensor 170 has a dead time TS and the controller unit 220 has a dead time element 226 with the controller dead time TR acting serially to the PI-controller 225. The controller dead time TR is chosen in relation to the resonance frequency ω0 of the oscillator 290 and the system dead time TS is chosen so that the phase frequency response of the corrected open loop at the frequency ω0 has a phase jump from +90° to −90° towards higher frequencies.
According to a first variant of this embodiment, the PI-controller for harmonic command variables does not flip the sign and the controller dead time TR is chosen so that the product of the resonance angular frequency ω0 and the sum of system dead time TS and controller dead time TR is 3π/2. According to another variant of this embodiment, the PI-controller 225 inverts the sign, shifts the phase about 180°, and the phase effected by the controller dead time TR and the system dead time TS at the resonance angular frequency ω0 merely adds to π/2 so that the product of the resonance angular frequency ω0 and the sum of system dead time TS and controller dead time TR is π/2.
As the controller unit 220 provides no baseband transformation (which requires a low pass filter for damping higher frequency conversion products), the controller 220 can be formed with a considerable broader band. The controller unit 220 reacts faster to disturbances than comparative controllers that provide a baseband transformation.
According to an embodiment in which the system including the actuator 180, the oscillator 190 and the sensor 170 has a system dead time TS, the controller unit 220 includes a dead time element 326 arranged in series with the discrete PI-controller 325 with a controller dead time TR. The system dead time TS as well as the controller dead time TR are expressed as multiples of the sampling time T according to equations (16a) and (16b) below:
TS=βS·T and (16a)
TRβD·T. (16b)
In this process, the controller dead time TR is determined so that the corrected open loop has a phase jump at the resonance angular frequency ω0 from +90° and −90° towards higher frequencies.
According to one embodiment, the ratio of the integral action coefficient KI to the amplification factor KP is adjusted so that the controller zero of the controller transfer function compensates the conjugate complex system pole of the system transfer function in the s-plane. According to another embodiment, controller parameters are chosen so that the transfer function of the closed loop of an equivalent baseband system has a double real eigenvalue. The controller unit 220 is, for example, realized as a digital circuit (e.g., as ASIC (application specific integrated circuit), DSP (digital signal processor) or FPGA (Field Programmable Gate Array)).
u(k)=(KP+KI·T·k)·sin(ω0·T·k)·σ(k) (17)
The input function Xd(z) and the output function U(z) result from z-transformations according to equations (18a) and (18b) below:
The transfer function GR0(z) of the discrete PI-controller 325 for harmonic command variables is then, according to equation (18c) below:
As the generalized integral portion, such a discrete PI-controller has a pole at z=e±jω
The model of the continuous controlled system according to equation (1) has to be discretized accordingly. To this end in the transfer function G(s) of the controlled system according to equation (1) the system dead time TS is at first expressed as a multiple of the sampling time T according to equation (16a):
Generally a step transfer function G(z) of a discretized
model of a continuous controlled system with the transfer function G(s) can be calculated according to equation (21):
Employing the following abbreviations according to equations (21a) to (21e)
the step transfer function G(z) for the oscillator 190 resulting from equations (20) and (21) is, according to equation (22):
In one embodiment of the invention, the controller dead time TR is determined so that the phase frequency response of the compensated open loop has a phase jump from +90° to −90° towards higher frequencies at the resonance angular frequency ω0. The z-transfer function for the compensated open loop results in analogy to equation (13) from the multiplication of the system transfer function G(z) according to equation (20) with the controller transfer function GR(z) according to equation (19):
GK(z)=G0(z)·GR0(z)·z−((β
Analogous to the equations (14a) and (14b), the controller parameter βD is chosen such that the transfer function of the corrected open loop Gk(z) has a phase jump from +90° to −90° at the resonance angular frequency ω0:
In comparison to equation (14a) one finds an additional part of ½ω0T with respect to the continuous controller, which represents a retardation that can be traced back to the discretizing of an additional half sampling cycle. As in the case of the continuous controller, a phase jump of 180° can be generated by a minus sign in the controller, provided that the phase shift generated by the system dead time βS·T and the discretization, respectively, are smaller than 90° at the resonance angular frequency ω0 so that the phases generated by the discretization, the controller dead time βD·T and the system dead time βS·T, need only add up to π/2. Accordingly, the dimensioning rule for βD results in this case in equation (24b):
The equations (24a) and (24b) normally lead to a non-integral value for βD. Generally, the controller parameter βD has an integral part nD and a remainder 1/aD with aD>1 according to equation (25):
According to one embodiment, the integral part nD can be approximated by a retardation chain in accordance with the length denoted by nD and the fraction 1/aD of a sampling cycle can be approximated by an all-pass filter of first order according to equation (26):
According to one embodiment, the parameter αD of the all-pass filter is chosen such that the phase of the exact transfer function z−α
According to one embodiment αD is determined such that, via nested intervals, the zeros of the function according to equation (28) are determined as follows:
The determination of nD and aD according to equations (25) and (28) is independent of the method for determining the further controller parameters KP and KI.
According to one embodiment of a method for manufacturing a controller unit that includes dimensioning the discrete PI-controller 325 according to
Equating coefficients with respect to z0 provides the dimensioning rule of equation (30):
According to one embodiment, the damping s0 of the oscillator 190 and the sampling time T are chosen such that s0·T<<1 holds whereby approximations according to (31a) and (31b) are sufficiently precise:
e−s
e−2·s
Making the approximations according to equations (31a) and (31b), the two independent dimensioning rules according to equations (29) and (30) can be approximated by the dimensioning rule below:
In one embodiment, the ratio of the integral action coefficient KI to the amplification factor KP is set equal (or nearly equal) to the damping s0 of the oscillator. The dimensioning of the discrete PI-controller 325 according to the described method, which includes the compensation of the system pole by the controller zero, leads to a good reference action of the closed loop.
According to another embodiment of a method for manufacturing a controller unit, which includes the dimensioning of a discrete PI-controller 325, the integral action coefficient KI and the amplification factor KP are determined by suitable eigenvalue specification for a system formed from the discrete PI-controller 325 and a discrete baseband model of the oscillator 190. For this, a baseband model G0′(s) equivalent to the oscillation model G0(s) of equation (1) is at first assumed:
The parameters of the equivalent baseband model according
to equation (33) are determined in accordance with equation (34) so that the absolute value of G0′(s) at ω=0 coincides with the absolute value of G0(s) at ω=ω0:
According to one embodiment, the oscillator 190 is realized such that ω0>>s0 holds and the relation between the parameters A and A′ is closely approximated by equation (35):
For the discretization of equivalent baseband model G0′(s), equation (36) results, by analogy to equation (21)
From equations (33) and (36) the equivalent discretized baseband model is derived as follows.
The system with the discretized baseband model 190a and the controller model 325a illustrated in
Calculation of the determinant det(z·I−Φ) leads to the characteristic polynomial of this system according to equation (39b) below:
Calculation of the zeros of the characteristic polynomial according to equation (39b) gives the eigenvalues λ1, λ2 of the controlled system, for which the characteristic polynomial can be generally described in the form of equation (40):
(z−λ1)·(z−λ2)=z2−(λ1+λ2)·z+λ1·λ2 (40)
By equating coefficients between equations (39b) and (40), the controller coefficients depending on the eigenvalues λ1 and λ2 (which are to be predetermined) result from equations (41a) and (41b) below.
The equations (41a) and (41b) lead to the equations (42a) and (42b) from which the controller coefficients r1 and r2 of the controller model 325a can be determined from the parameters of the equivalent discrete baseband model and the predetermined eigenvalues:
The amplification factor KP and the integral action coefficient KI of the controller unit 220 according to
KP=r1 (43a)
KI·T=r2 (43b)
According to one embodiment, the eigenvalues λ1, λ2 are predetermined without high dynamics requirements so that the transient oscillation process of the baseband system describes, in close approximation, the envelope of the transient oscillation process of the equivalent bandpass system. In this process, the transferability of the baseband design to the bandpass band only holds approximately, due to the controller dead time of the phase adjustment for the bandpass band system acting as an additional dead time with respect to the baseband system that is not taken into account in the controller design. For this reason, when presetting the eigenvalues with excessively high dynamics requirements, the bandpass band system can be unstable, although the equivalent baseband system is stable. However, by referring to the Nyquist stability criterion, the stability of the bandpass band design can be estimated at any time for the predetermined eigenvalues.
When the method for dimensioning a controller provides presetting of eigenvalues, the position of the two eigenvalues with respect to each other is also predetermined. In contrast, a strong deviation of the two eigenvalues from one other can happen at the dimensioning of the PI-controller for harmonic command variables by pole/zero compensation so that the cancelled system pole remains as eigenvalue in the closed loop and leads to a large time constant at a typically low damping of the oscillator. Indeed the “cancelled” eigenvalue has no influence on the response, but it can be excited by perturbations and can result in long, persistent fading processes. In contrast, the design via eigenvalue presetting allows the presetting of both eigenvalues at approximately the same order of magnitude and thus a positive influence of the perturbation behavior. According to one embodiment, the two eigenvalues are set equal or approximately equal with a maximum deviation of 10% to the larger eigenvalue.
The following exemplary embodiment illustrates the design methods described above for the PI-controller 325 for a controlled system with the following parameters:
As the phase generated by the system dead time at the resonance frequency ω0 is less than 90°, a phase ratio of 180° can be realized by an inverting controller (minus sign in the controller). For the controller dead time βDT the dimensioning rule according to equation (44) then results from equation (24b):
According to a method that provides dimensioning of the discrete PI-controller 325 by pole/zero compensation the amplification factor KP can be, for example, set to KP=− 1/10 in analogy to the example illustrated in
When, in contrast, the discrete PI-controller 325 is dimensioned via eigenvalue presetting, the eigenvalues are chosen for example equally large and according to absolute value, such that the closed loop of the equivalent baseband system has a double real eigenvalue at λ1=λ2=0.98. The controller coefficients r1=0.14004655 and r2=1,41471261·10−3 result from equations (42a) and (42b). Taking into account the minus sign required for the phase adjustment, the values for the amplification factor KP and the integral action coefficient KI of the discrete PI-controller 325 are KP=−0.14004655 and KI·T=−1,41471261·10−3.
The oscillator 190 can have further resonances beside the resonance angular frequency at ω0, such as mechanic structure resonances above or below the resonance angular frequency ω0. The controller extension 328 is formed such that these further resonances are more strongly damped. To this end, a retardation element of first order (PT1-element) with a further pole at the kink frequency beyond the desired bandwidth would be added to a conventional PI-controller in the baseband. This additional controller pole causes the controller to no longer act as a proportional element for high frequencies, but its absolute value frequency drops down with 20 db/decade. The step response y(k) of such an extension in the baseband results from the step function σ(k) as input signal u(k) according to equation (45):
The z transform U(z) of the input signal u(k) corresponds to the z transform of the step signal:
The z transform Y(z) of the output signal y(k) then results:
The transfer function GRE0(z) of such a controller extension in the baseband is derived, thus, in analogy to equation (10):
According to one embodiment, the controller extension 238 in the bandpass band is configured now in analogy to the controller extension in the baseband such that the controller extension 328 responses to an admission with a harmonic oscillation of a resonance angular frequency ω0 modulated by the step function with a harmonic oscillation of the same frequency, wherein the step response of the baseband extension defines the envelope as illustrated on the right side of
u(k)=sin(ω0·T·k)·σ(k) (48)
The controller output signal y(k) is a harmonic oscillation whose envelope corresponds to the step response of the PT1-controller extension in the baseband:
The z-transform U(z) and Y(z) are set forth in equations
(50a) and (50b) below:
The transfer function GRE(z) of the controller extension 328
for the bandpass band follows:
The controller extension 328 with the transfer function GRE(z) acts in series with the discrete PI-controller 325 similarly to a bandpass of first order with the resonance frequency ω0 as midband frequency. The absolute value and phase of the compensated open loop at the resonance angular frequency ω0 in a narrow region around the resonance angular frequency ω0 according to equation (52) remain unchanged.
In this region, the absolute value frequency response of the compensated open loop is barely affected, while, out of this region, a considerable drop of the absolute value happens such that possible undesired resonances can be dropped.
According to one embodiment, the rotation rate sensor 500 includes first force transmission and sensor units 561, 571 (e.g. electrostatic force transmitters and sensors) which excite the system formed from the excitation unit 590 and the detection unit 580 to an oscillation along the direction of excitation 501 and/or are able to capture a corresponding deflection of the excitation unit 590. The rotation rate sensor 500 includes further second force transmission and sensor units 562, 572 (e.g. electrostatic force transmitters and sensors) which act on the detection unit 580 and/or are able to capture its deflection. According to one embodiment at least one of the second force transmission and sensor units 562, 572 is controlled such that it counteracts a deflection of the detection unit 580, caused by a disturbance or, in a closed loop system, caused by a measured variable.
During operation of the rotation rate sensor 500, the first force transmission and sensor units 561, 571 excite, for example, the excitation unit 590 to an oscillation along the direction of excitation 501, wherein the detection unit 580 moves approximately with the same amplitude and phase with the excitation unit 590. When the arrangement is rotated around the axis orthogonal to the substrate plane, a Coriolis force acts on the excitation unit 590 and the detection unit 580, which deflects the detection unit 580 with respect to the excitation unit 590 in the detection direction 502. The second force transmission and sensor units 562, 572 capture the deflection of the deflection unit 580 and, thus, the rotational movement around the axis orthogonal to the substrate plane.
According to one embodiment, at least one of the force transmission and sensor units 561, 572, 562, 572 acts as actuator and either the excitation unit 590 or the detection unit 580 as oscillator within the meaning of one of the devices 200 described above.
According to one embodiment illustrated in
The deflection of the x2-oscillator can be captured via the charge on the common movable electrode, which is formed on the excitation unit 590. The charge can be measured via the attachment structure 551. A charge amplification unit 521 amplifies the measured signal. While typically a demodulation unit modulates the measured signal with a frequency which corresponds for example to the resonance angular frequency ω0 before it is fed into a controller unit, the embodiments of the invention feed the non-demodulated harmonic signal as measurement signal within the meaning described above into a controller unit 520 according to the above.
The damping s0 for the oscillation is considerably smaller than the resonance angular frequency ω0. The signal measured over the excitation frame the excitation unit 590 partly reproduces the movement of the excitation unit 590 along the direction of excitation 501. A disturbance whose source can be outside of the rotation rate sensor 500, or, in a closed loop system, the measurement variable, respectively, superposes the oscillation and modulates its amplitude. The controller unit 520 senses from the modulated harmonic signal a control signal for the second force transmission and sensor units 562, 572 which counteracts the deflection effected by the disturbance or the measurement variable, respectively. An amplification unit 522 transforms the control signal in a suitable reset signal for the electrodes of the second force transmission and sensor units 562, 572. The controller unit 520 is formed and dimensioned according to one of the above described controller units 220. When the amplitude modulation of the harmonic signal reproduces the measurement variable, a demodulation unit can be provided, which generates the rotation rate signal by demodulation of the harmonic control signal with the resonance angular frequency ω0.
The rotation rate sensor 505 illustrated in
According to another embodiment, at least one of the first or second force transmission and sensor units 561, 562, 571, 572 acts as actuator and either the excitation unit 590 or the detection unit 580 or the excitation unit 590 as well as the detection unit 580 act as oscillator according to one of the devices described above, which are operated according to the principle of the bandpass controller. In this process the force transmission and sensor units 561 and 571 act as force transmission and sensor units respectively for the x1-oscillator and the force transmission and sensor units 562 and 572 act as force transmission and sensor units respectively for the x2-oscillator.
A rotation rate sensor according to another embodiment includes two of the arrangements as illustrated in
A further embodiment refers to the controller unit 220 illustrated in
To achieve this, the integral action coefficient of the integrating transfer elements 222, 322 and a amplification factor of the proportional transfer elements 224, 324 is chosen such that the PI-controller 225, 325 for harmonic command variables is suitable for generating a harmonic oscillation of the controller angular frequency □r with rising amplitude at a controller output, with an harmonic input signal of the controller angular frequency ωr modulated by the step function at the controller input.
The PI-controller 225, 325 for harmonic command variables can also be employed for a controller derived from a conventional PI-controller for stationary command variables and differs from it by the position of the poles in the s- or z-plane, respectively.
While the invention has been described with reference to a presently preferred embodiment, it is not limited thereto. Rather it is limited only insofar as it is defined by the following set of patent claims and includes within its scope all equivalents thereof.
Number | Date | Country | Kind |
---|---|---|---|
10 2010 055 631 | Dec 2010 | DE | national |
Filing Document | Filing Date | Country | Kind | 371c Date |
---|---|---|---|---|
PCT/EP2011/006356 | 12/15/2011 | WO | 00 | 11/5/2013 |
Publishing Document | Publishing Date | Country | Kind |
---|---|---|---|
WO2012/084153 | 6/28/2012 | WO | A |
Number | Name | Date | Kind |
---|---|---|---|
7805993 | Spahlinger | Oct 2010 | B2 |
8032324 | Bryant | Oct 2011 | B1 |
20030133495 | Lerner | Jul 2003 | A1 |
20050187650 | Toyama et al. | Aug 2005 | A1 |
Number | Date | Country |
---|---|---|
2004-209176 | Nov 2005 | CN |
2743837 | Nov 2005 | CN |
10 2005 005248 | Aug 2005 | DE |
1227383 | Jul 2002 | EP |
2005-221845 | Aug 2005 | JP |
10-2009-0105065 | Oct 2009 | KR |
WO 2009112526 | Sep 2009 | WO |
Entry |
---|
Wang Jianhui et al., “The Principle of Automatic Control”, Jan. 2005, 4th Edition, Metallurgical Industry Publishing House, pp. 43, 95, 96, 211, 279 through 285,405, 406. |
Number | Date | Country | |
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20140159822 A1 | Jun 2014 | US |