- Email: [email protected]
Dynamic Compensation for a Rotational Motion Simulator
DYNAMIC COMPENSATION FOR A ROTATIONAL MOTION SIMULATOR
A. S. Funada & W. C. Leite Filho*
Instituto de Aeronautica e Espaf;o (IAE / CTA) Space Systems Division 12228-904 - sao Jose dos Campos - sao Paulo - Brazil * e-mail: [email protected]
Abstract. A spacecraft attitude control simulation must be completed by a excitation source on its angular sensors to close and test its control loop. To provide this excitation, an angular motion simulator is necessary in the loop. However, this device has its own dynamical characteristics, which is unpredicted on the original control project and it can cause important changes in the control loop performance. In order to minimize those effects on the simulation loop, this work presents a proposition of a digital compensator design to cope with such undesirable dynamics, and a method to evaluate the obtained models. Copyright © 20011FAC Keywords: compensation, identification, hardware-in-the-loop (HWIL) simulation
1. INTRODUCTION To make an effective test of an attitude control system, it is necessary to stimulate the control loop with its actual angular movements and compare those results with other simulation schemes (e.g. digital). Once these movements are simulations of maneuvers performed in space, it is possible, through these simulations, verify and detect potential problems prior to its launch. A three degree-of-freedom rotational motion simulator with reasonable accuracy is used to perform these tests. Hardware in the Loop Simulation
The attitude control loop simulated in laboratory has many hardware devices similar to those used on an actual flight. Since these devices are structured in a closed control loop, the simulations performed are called "hardware-in-the-loop" (HWIL) simulations (Malyshev et alii, 1996). To execute those tests, the rotational simulation is used to perform an actual stimulation of the inertial and rotational sensors, as showed in fig. 1.
1.1 Simulated Flights In order to create the simulation environment as described above, it is adapted an original digital modeling program to work on a real-time scheme. The adopted host computer were aggregated to a number of hardware interfaces to interact with all other devices (Leite Filho & Carrijo, 1996). These devices were added one-by-one, to ensure proper electrical and mechanical connectivity and keeping track of the simulation behavior for each step (Leite Filho, Oliva & Carrijo, 1999). Since this procedure was fmished, it is obtained the control loop scheme below (fig. 2).
Fig. 1: HWIL simulation illustrative diagram POSTER SESSION
507
This work presents a proposition of a digital compensator design to cope with such undesirable dynamics, and an alternative simulation scheme to evaluate the model is obtained.
added dynamic In
.simulatlon .loop.. , vehicle digital simulator
2. COMPENSATION PROPOSAL Note:
n actual elements
Fig. 2: test bed control loop scheme Comparing those simulations, it is noticed that as more hardware devices were introduced, more nonlinear effects showed up in the simulation records. It is quite clear that the simple introduction of hardware devices changes the results when compared with its digital counterparts. However, HWIL simulations are necessary in order to a more complete detection of actual effects. As tests are performed including the rotational simulator, it can be seen changes in the simulation dynamical characteristics as a whole, due to its heavy inertia (low frequency band). Performance changes became remarkable during the tests with and without the rotational simulator. The behavior obtained due the inclusion of the rotational simulator is undesirable and it does not exist in the actual attitude control loop. The main effect is the limit cycle characteristics changes.
The compensating strategy proposed is to excite the rotational simulator through its inverse transfer function and make it into the spacecraft model simulation program (fig 4). This scheme, based on the' inner model principle (Chen, 1984), can cancel its effects. 9, physical system dinamics d6Jdt
rotational simulator G(s)
digital simulator (IBM-PC host)
Fig. 4: open-loop compensation scheme It is necessary to point out that the main focus in this
work is mainly model identification techniques and its evaluation schemes. To complete the constraints list, the actual inertial sensors in the control loop are the most sensible, which means it cannot be freely stressed
3. MODEL OBTAINMENT 1.2 Limit-Cycle Changes
The rotational simulator role in the simulation loop is to provide adequate stimulus to the inertial and rotational sensors. So, there is no way to perform the use of these sensors without adding it into the loop. Although the limit-cycle effect was expected due to the actuators non-linearity, the inclusion of the rotational simulator in the loop changed its frequency from 1.018 Hz to 0.935 Hz (fig. 3). Since this effect is undesirable, it is needed a compensator to attenuate this . Command
lim~-cyde
samples
Due to these requirements and among the available operation modes available in the rotational simulator, the conjugated angular position and velocity mode was chosen. This mode presents a good precision in both angular positioning and velocity outputs, generating a motion profile that maintains the compromise between position and velocity .
0.15 0.1 0.05
~.05 ."
~O~;
'"
" f
\
I,
I
,', I
"
..
, ,
"
-0.25
106
106
'I
,~
,'I
I',
, , I,
.,
110
"
','
112
114
time(s)
Fig. 3: limit-cycle output
508
"
, ,
.,
-0.2
"
POSTER SESSION
The simulator provides movements driven by electric DC motors and digitally controlled by a dedicated CPU board. Its controller is very complex and it has several algorithms for different levels of angular and velocity errors (Contraves, 1988). Hence, its behavior is linear only within a small range of command values. Besides, the angular position and angular velocity commanded for the simulator are not only function of its movement, but also are dependent of the frequency they are updated. So, the first characteristic that must be known for the (continuous) model of the dynamics is the influence of the commands update rate (sampling rate).
116
Moreover, the rotational simulator has an embedded switching circuitry that provides faster correction for large state (angular position, velocity and acceleration) error levels inside the rotational controller (Contraves, 1988). The drawback is that switching operations introduces non-linearity in the control structure. To avoid this, it is necessary to
keep the input velocity signals at controlled low levels in order to permit a linear signal analysis. The stochastic identification technique (Graupe, 1972) - using white noise excitement - is not applicable due this non-linear characteristic. Other options; such as step or slope functions, has discontinuities only on the initial instant, but since they do not provide a good richness in their signals, they were not considered. Sinusoidal signals are preferable, because the absence of discontinuities on any of their derivatives. Moreover, it is possible to keep the values of their derivatives in order to avoid the algorithm switching.
3.1 Frequency Response In the previous section, the operation method and the theoretical and practical limitations imposed to the modeling process were defined. Following that, it is obtained a Bode diagram of the rotational simulator. was applied simple sinusoidal inputs, one frequency at a time, and make an acquisition of their steady-state output. After repeating this procedure to a number of frequencies, it is obtained the gain and phase shift variation curves in frequency .
It
function (if it exists) and the gains K; in the dynamic equations can be obtained through polynomial division between its numerator and denominator. Based on the resulted transfer function given in the previous section, it is calculated the compensator coefficients and performed a simple test. To apply this compensator, it is simply generated its input parameters (angular position, velocity an acceleration profiles) and got the compensated command signals. Applying these commands directly to the rotational simulator, in a way similar to section 3.1, it is obtained the gain and phase curves below (figs. 5,6). The analysis of output signals gives that the openloop compensation scheme provided a reasonable change in dynamical characteristics, mainly in the output phase. However, the procedures employed in section 3.1 characterized the signals under linear approach only, with no further search of common non-linearity (e.g. a delay operator). Due to these factors, and the need to certify previous analysis, led to another model identification proposal. Open-loop compensated output - - original curve
These data records, and posterior computational analysis (Matlab, 1994), gave us its transfer function on the frequency domain with a quite reasonable accuracy (eq. 1).
8 -
compensated curve
6
iD 4
G (s) = f
40.847s + 6449.7 S2 + 32.526s + 6449.7
(1)
E .5
~
I
2
~ . . . . .~
o
3.2 Open-Loop Compensation Analysis -2 L.1--~~"""""'-O--~-~----"""""'" 2
A simple open-loop test, were performed to verify any dynamical changes in the frequency-domain. The main goal for this analysis, is to provide a simple compensator, obtained preferably from a 2nd order dynamic (as obtained in the previous section). Once obtained this transfer function, it is possible to generate the compensator, creating its inverse dynamics, as shown in the group of equations below (eq.2).
10
10
10
Fig. 5: rotational simulator output gain under compensated input Open-loop compensated output
---\1
o
e c = ko · e+k\ ·e+k 2 ·x ec=ko·O+k\.e+k2·e+k3 · x
10 frequency (Hz)
-20
(2)
x= -a·x+e The command mode employed on the rotational simulator these equations states, with conjugated angular position (et) and velocity (de/dt). The input variable e (as well as its derivatives) is already available from the rocket's dynamic model. So, it helps to implement the compensator, which represents its continuous domain states. The pole corresponds to the zero of the original transfer
I
·60 -80
1- - original curve -
-100
10
·1
compensated curve
o
10 frequency (Hz)
10
1
10
2
Fig. 6: rotational simulator output phase under compensated input
POSTER SESSION
509
3.3 Time-Domain Response Applying the same operation modes (angular position and velocity), it is changed the employed analysis techniques and the applied signals to make a time-domain analysis. Although the frequencydomain analysis required gain and phase in steadystate outputs only, this other approach uses both input and output data in transitory state and a search of a mathematical relation between them. The input signal u(t) employed in this analysis were the summation of various sine-wave signals (eq. 3).
way, it is possible to verify the model validation by repeating the dynamical behavior when the actual device is present. It is necessary to note that the inertial sensors were modeled in these simulations because there is no way to use the actual ones without the rotational simulator. This procedure is necessary to avoid a direct simulation with the compensator, without previous knowledge of effects introduced by its model in the closed-loop simulations. digital model
n
u(t) =
LAj .sinew t) j
(3)
i= 1
As a note, a previous analysis of the obtained measurements errors could be characterized as "white-noise", with no noticeable deviations of its characteristics. So, a simple least-square approach is applicable. Once determined these signals properties, it is applied a mathematical analysis with the ARX algorithm (Ljung, 1987) This algorithm gives a coefficient series, by evaluating the input-output data with the smallest quadratic error. The obtained result is a transfer function in the discrete domain. Although there is no unicity in discrete-to-continuous transform, the modeling through a zero-order-holder (ZOH) were used here, giving the following transfer function : G (s) t
+ 7136.6 = e -{).OO2 s - ,65.08s ------S2
+ 39.06s + 7136.6
Note: (') actual elements
Fig. 8: proposed complete simulation loop When verified the same dynamical behavior in this test, the model accuracy is assured, without the need to use the rotational simulator. Consequently, the inertial sensors are not used and unnecessarily stressed. Once the model (and the compensator) is fully qualified, the complete control loop with the actual inertial sensors, can be simulated.
4. RESULTS
4. J Obtained Transfer Functions (4)
3.4 Closed-Loop Simulation The rotational simulator introduced some changes in the non-linear behavior of movable nozzles command signals. The limit-cycle was the most important, since it is quite easy to detect and to analyze. digital model
For comparison, it is presented just below, some plots for the gain and phase responses obtained from the two identification methods employed. The plusdotted curve is the original frequency-domain gain/phase evaluation points. The qualitative analysis, given by the Bode plot evaluations, shows a reasonable match between the frequency and time-domain identifications (figs. 9, 10). Identified Models Frequency Response 10 r-------~--------~~------~
Note: (') actual elements
+ freq. samples - freq. domaln eval. - - time domain eval.
Fig. 7: simulation loop with digital identified model To validate the obtained models and test its influence, it is opted to run a simulation without the rotational simulator. Instead of the real rotational simulator, the simulation loop were provided with its computational model (fig. 7). Once structured this
510
POSTER SESSION
-~
10
,____D10
~========~ ~ 0
10
10
frequency (Hz)
Fig. 9: comparative Bode gain plot
2
The original graph (fig. 3) presents a perceptible gain and frequency modification after the introduction of the rotational simulator in the loop. Meanwhile, the new plot (fig. 11), shows a relative near gain and oscillation frequency in their limit-cycle outputs, which means a fairly good approximation even the identified model was an almost linear one.
IdenUfled Models Frequency Response
It is necessary to note that the dashed plot (actual device), is the same data used in fig. 3, showing a good match on their simulation behavior (which gives a frequency of 0.927 Hz). This ensures a good confidence level on the identified models.
+ freq . samples
- freq . domain eva!. - - time domain eval.
-100 -1
10
10
o
10
1
frequency (Hz)
Fig. 10: comparative Bode phase plot
5. CONCLUSIONS
As present on the curves above, and the corresponding transfer functions provided by eqs. 1 and 4, the obtained models were quite similar. Perhaps a higher order transfer function (e.g. 3rd order) could generate a better adjusted curve to the obtained data if compared to the adopted 2nd order models. However, the implementation of that inverse would generate a differential equation with a 3rd derivative (d38jdf), which demands a numeric derivation, or a function that gives the corresponding result. To provide a comparative view, the time-domain analysis procedure gave the following function:
o. (s) = 3
6.0198(s+ 172.89)(s+5196.4)
Model identification procedures, as employed in this work, are performed mostly in off-line basis and do not provide a reasonable insight of its influence in a closed-loop scheme. So, the test procedure showed in section 3.4 helps to verify the effects introduced by the model in closed-loop simulations. It helps to save sensible devices, once they are not
required, and closed-loop simulations behavior can confrrrn (or discard) test models. The strategy for compensation proposed showed good results and it was introduced in real time hardware-in-the-Ioop environment to simulate attitude control systems of Brazilian Space Program.
(5)
(s+ 726.57)(S2 + 42.119s + 7443.3)
6. REFERENCES
which has some high-frequency components that are of no use in the simulation loop.
4.2 Closed-Loop Outputs Comparison The plot below shows the limit-cycle effects obtained in performed simulations with the actual device (dashed line) and its digitalized model (solid line).
Carrijo D. S., Oliva, A. P. and Leite Filho, W. c., Hybrid Simulation Software Development for a Assesment of a Satellite Launcher Control System, Space Technology, vol. 19, Nos. 3-4, pp 213-219,1999. Chen, C. T., Linear System Theory and Design, HoltSaunders Ed, 1984. Contraves Goerz Corp., Inertial Guidance Tests System (A-7544), 1988.
Command limit-cycle samples
0.2 0.15
n,
0.1
f"1".
,' '. [
~
0.05
.,
Graupe, D., Identification of Systems, Van Nostrand Reinhold, 1972.
- - rot. simulator - ~odeled simulator
0;
p,
Ljung, L., System Identification: Theory for the User, (Prentice-Hall, Inc., 1987).
.
0
~
:'''~ , ,
'0
[ij.0.05 E E 8 -0.1
, .' ~'
., "
··:\U '.\
~:
-0.15 -0.2
110
112
114
116
118
time(s)
Fig. 11: comparison of simulation in a closed control loop with the actual device and its digital model
Leite Filho, W. C. and Carrijo D. S., Hardware-inthe-Loop Simulation of Brazilian Launcher VLS, Proceedings Third ESA International Conference on Spacecraft Guidance, Navigation and Control Systems, 1996,355-358. MATLAB, Signal Processing Toolbox User's Guide, The Mathworks Inc., 1994. Malyshev, V. V. et alii, Aerospace Vehicle Control: Modern Theory and Applictions, Instituto de Aerorulutica e Esparyo, 1996.
POSTER SESSION
511
A. S. Funada & W. C. Leite Filho*
Instituto de Aeronautica e Espaf;o (IAE / CTA) Space Systems Division 12228-904 - sao Jose dos Campos - sao Paulo - Brazil * e-mail: [email protected]
Abstract. A spacecraft attitude control simulation must be completed by a excitation source on its angular sensors to close and test its control loop. To provide this excitation, an angular motion simulator is necessary in the loop. However, this device has its own dynamical characteristics, which is unpredicted on the original control project and it can cause important changes in the control loop performance. In order to minimize those effects on the simulation loop, this work presents a proposition of a digital compensator design to cope with such undesirable dynamics, and a method to evaluate the obtained models. Copyright © 20011FAC Keywords: compensation, identification, hardware-in-the-loop (HWIL) simulation
1. INTRODUCTION To make an effective test of an attitude control system, it is necessary to stimulate the control loop with its actual angular movements and compare those results with other simulation schemes (e.g. digital). Once these movements are simulations of maneuvers performed in space, it is possible, through these simulations, verify and detect potential problems prior to its launch. A three degree-of-freedom rotational motion simulator with reasonable accuracy is used to perform these tests. Hardware in the Loop Simulation
The attitude control loop simulated in laboratory has many hardware devices similar to those used on an actual flight. Since these devices are structured in a closed control loop, the simulations performed are called "hardware-in-the-loop" (HWIL) simulations (Malyshev et alii, 1996). To execute those tests, the rotational simulation is used to perform an actual stimulation of the inertial and rotational sensors, as showed in fig. 1.
1.1 Simulated Flights In order to create the simulation environment as described above, it is adapted an original digital modeling program to work on a real-time scheme. The adopted host computer were aggregated to a number of hardware interfaces to interact with all other devices (Leite Filho & Carrijo, 1996). These devices were added one-by-one, to ensure proper electrical and mechanical connectivity and keeping track of the simulation behavior for each step (Leite Filho, Oliva & Carrijo, 1999). Since this procedure was fmished, it is obtained the control loop scheme below (fig. 2).
Fig. 1: HWIL simulation illustrative diagram POSTER SESSION
507
This work presents a proposition of a digital compensator design to cope with such undesirable dynamics, and an alternative simulation scheme to evaluate the model is obtained.
added dynamic In
.simulatlon .loop.. , vehicle digital simulator
2. COMPENSATION PROPOSAL Note:
n actual elements
Fig. 2: test bed control loop scheme Comparing those simulations, it is noticed that as more hardware devices were introduced, more nonlinear effects showed up in the simulation records. It is quite clear that the simple introduction of hardware devices changes the results when compared with its digital counterparts. However, HWIL simulations are necessary in order to a more complete detection of actual effects. As tests are performed including the rotational simulator, it can be seen changes in the simulation dynamical characteristics as a whole, due to its heavy inertia (low frequency band). Performance changes became remarkable during the tests with and without the rotational simulator. The behavior obtained due the inclusion of the rotational simulator is undesirable and it does not exist in the actual attitude control loop. The main effect is the limit cycle characteristics changes.
The compensating strategy proposed is to excite the rotational simulator through its inverse transfer function and make it into the spacecraft model simulation program (fig 4). This scheme, based on the' inner model principle (Chen, 1984), can cancel its effects. 9, physical system dinamics d6Jdt
rotational simulator G(s)
digital simulator (IBM-PC host)
Fig. 4: open-loop compensation scheme It is necessary to point out that the main focus in this
work is mainly model identification techniques and its evaluation schemes. To complete the constraints list, the actual inertial sensors in the control loop are the most sensible, which means it cannot be freely stressed
3. MODEL OBTAINMENT 1.2 Limit-Cycle Changes
The rotational simulator role in the simulation loop is to provide adequate stimulus to the inertial and rotational sensors. So, there is no way to perform the use of these sensors without adding it into the loop. Although the limit-cycle effect was expected due to the actuators non-linearity, the inclusion of the rotational simulator in the loop changed its frequency from 1.018 Hz to 0.935 Hz (fig. 3). Since this effect is undesirable, it is needed a compensator to attenuate this . Command
lim~-cyde
samples
Due to these requirements and among the available operation modes available in the rotational simulator, the conjugated angular position and velocity mode was chosen. This mode presents a good precision in both angular positioning and velocity outputs, generating a motion profile that maintains the compromise between position and velocity .
0.15 0.1 0.05
~.05 ."
~O~;
'"
" f
\
I,
I
,', I
"
..
, ,
"
-0.25
106
106
'I
,~
,'I
I',
, , I,
.,
110
"
','
112
114
time(s)
Fig. 3: limit-cycle output
508
"
, ,
.,
-0.2
"
POSTER SESSION
The simulator provides movements driven by electric DC motors and digitally controlled by a dedicated CPU board. Its controller is very complex and it has several algorithms for different levels of angular and velocity errors (Contraves, 1988). Hence, its behavior is linear only within a small range of command values. Besides, the angular position and angular velocity commanded for the simulator are not only function of its movement, but also are dependent of the frequency they are updated. So, the first characteristic that must be known for the (continuous) model of the dynamics is the influence of the commands update rate (sampling rate).
116
Moreover, the rotational simulator has an embedded switching circuitry that provides faster correction for large state (angular position, velocity and acceleration) error levels inside the rotational controller (Contraves, 1988). The drawback is that switching operations introduces non-linearity in the control structure. To avoid this, it is necessary to
keep the input velocity signals at controlled low levels in order to permit a linear signal analysis. The stochastic identification technique (Graupe, 1972) - using white noise excitement - is not applicable due this non-linear characteristic. Other options; such as step or slope functions, has discontinuities only on the initial instant, but since they do not provide a good richness in their signals, they were not considered. Sinusoidal signals are preferable, because the absence of discontinuities on any of their derivatives. Moreover, it is possible to keep the values of their derivatives in order to avoid the algorithm switching.
3.1 Frequency Response In the previous section, the operation method and the theoretical and practical limitations imposed to the modeling process were defined. Following that, it is obtained a Bode diagram of the rotational simulator. was applied simple sinusoidal inputs, one frequency at a time, and make an acquisition of their steady-state output. After repeating this procedure to a number of frequencies, it is obtained the gain and phase shift variation curves in frequency .
It
function (if it exists) and the gains K; in the dynamic equations can be obtained through polynomial division between its numerator and denominator. Based on the resulted transfer function given in the previous section, it is calculated the compensator coefficients and performed a simple test. To apply this compensator, it is simply generated its input parameters (angular position, velocity an acceleration profiles) and got the compensated command signals. Applying these commands directly to the rotational simulator, in a way similar to section 3.1, it is obtained the gain and phase curves below (figs. 5,6). The analysis of output signals gives that the openloop compensation scheme provided a reasonable change in dynamical characteristics, mainly in the output phase. However, the procedures employed in section 3.1 characterized the signals under linear approach only, with no further search of common non-linearity (e.g. a delay operator). Due to these factors, and the need to certify previous analysis, led to another model identification proposal. Open-loop compensated output - - original curve
These data records, and posterior computational analysis (Matlab, 1994), gave us its transfer function on the frequency domain with a quite reasonable accuracy (eq. 1).
8 -
compensated curve
6
iD 4
G (s) = f
40.847s + 6449.7 S2 + 32.526s + 6449.7
(1)
E .5
~
I
2
~ . . . . .~
o
3.2 Open-Loop Compensation Analysis -2 L.1--~~"""""'-O--~-~----"""""'" 2
A simple open-loop test, were performed to verify any dynamical changes in the frequency-domain. The main goal for this analysis, is to provide a simple compensator, obtained preferably from a 2nd order dynamic (as obtained in the previous section). Once obtained this transfer function, it is possible to generate the compensator, creating its inverse dynamics, as shown in the group of equations below (eq.2).
10
10
10
Fig. 5: rotational simulator output gain under compensated input Open-loop compensated output
---\1
o
e c = ko · e+k\ ·e+k 2 ·x ec=ko·O+k\.e+k2·e+k3 · x
10 frequency (Hz)
-20
(2)
x= -a·x+e The command mode employed on the rotational simulator these equations states, with conjugated angular position (et) and velocity (de/dt). The input variable e (as well as its derivatives) is already available from the rocket's dynamic model. So, it helps to implement the compensator, which represents its continuous domain states. The pole corresponds to the zero of the original transfer
I
·60 -80
1- - original curve -
-100
10
·1
compensated curve
o
10 frequency (Hz)
10
1
10
2
Fig. 6: rotational simulator output phase under compensated input
POSTER SESSION
509
3.3 Time-Domain Response Applying the same operation modes (angular position and velocity), it is changed the employed analysis techniques and the applied signals to make a time-domain analysis. Although the frequencydomain analysis required gain and phase in steadystate outputs only, this other approach uses both input and output data in transitory state and a search of a mathematical relation between them. The input signal u(t) employed in this analysis were the summation of various sine-wave signals (eq. 3).
way, it is possible to verify the model validation by repeating the dynamical behavior when the actual device is present. It is necessary to note that the inertial sensors were modeled in these simulations because there is no way to use the actual ones without the rotational simulator. This procedure is necessary to avoid a direct simulation with the compensator, without previous knowledge of effects introduced by its model in the closed-loop simulations. digital model
n
u(t) =
LAj .sinew t) j
(3)
i= 1
As a note, a previous analysis of the obtained measurements errors could be characterized as "white-noise", with no noticeable deviations of its characteristics. So, a simple least-square approach is applicable. Once determined these signals properties, it is applied a mathematical analysis with the ARX algorithm (Ljung, 1987) This algorithm gives a coefficient series, by evaluating the input-output data with the smallest quadratic error. The obtained result is a transfer function in the discrete domain. Although there is no unicity in discrete-to-continuous transform, the modeling through a zero-order-holder (ZOH) were used here, giving the following transfer function : G (s) t
+ 7136.6 = e -{).OO2 s - ,65.08s ------S2
+ 39.06s + 7136.6
Note: (') actual elements
Fig. 8: proposed complete simulation loop When verified the same dynamical behavior in this test, the model accuracy is assured, without the need to use the rotational simulator. Consequently, the inertial sensors are not used and unnecessarily stressed. Once the model (and the compensator) is fully qualified, the complete control loop with the actual inertial sensors, can be simulated.
4. RESULTS
4. J Obtained Transfer Functions (4)
3.4 Closed-Loop Simulation The rotational simulator introduced some changes in the non-linear behavior of movable nozzles command signals. The limit-cycle was the most important, since it is quite easy to detect and to analyze. digital model
For comparison, it is presented just below, some plots for the gain and phase responses obtained from the two identification methods employed. The plusdotted curve is the original frequency-domain gain/phase evaluation points. The qualitative analysis, given by the Bode plot evaluations, shows a reasonable match between the frequency and time-domain identifications (figs. 9, 10). Identified Models Frequency Response 10 r-------~--------~~------~
Note: (') actual elements
+ freq. samples - freq. domaln eval. - - time domain eval.
Fig. 7: simulation loop with digital identified model To validate the obtained models and test its influence, it is opted to run a simulation without the rotational simulator. Instead of the real rotational simulator, the simulation loop were provided with its computational model (fig. 7). Once structured this
510
POSTER SESSION
-~
10
,____D10
~========~ ~ 0
10
10
frequency (Hz)
Fig. 9: comparative Bode gain plot
2
The original graph (fig. 3) presents a perceptible gain and frequency modification after the introduction of the rotational simulator in the loop. Meanwhile, the new plot (fig. 11), shows a relative near gain and oscillation frequency in their limit-cycle outputs, which means a fairly good approximation even the identified model was an almost linear one.
IdenUfled Models Frequency Response
It is necessary to note that the dashed plot (actual device), is the same data used in fig. 3, showing a good match on their simulation behavior (which gives a frequency of 0.927 Hz). This ensures a good confidence level on the identified models.
+ freq . samples
- freq . domain eva!. - - time domain eval.
-100 -1
10
10
o
10
1
frequency (Hz)
Fig. 10: comparative Bode phase plot
5. CONCLUSIONS
As present on the curves above, and the corresponding transfer functions provided by eqs. 1 and 4, the obtained models were quite similar. Perhaps a higher order transfer function (e.g. 3rd order) could generate a better adjusted curve to the obtained data if compared to the adopted 2nd order models. However, the implementation of that inverse would generate a differential equation with a 3rd derivative (d38jdf), which demands a numeric derivation, or a function that gives the corresponding result. To provide a comparative view, the time-domain analysis procedure gave the following function:
o. (s) = 3
6.0198(s+ 172.89)(s+5196.4)
Model identification procedures, as employed in this work, are performed mostly in off-line basis and do not provide a reasonable insight of its influence in a closed-loop scheme. So, the test procedure showed in section 3.4 helps to verify the effects introduced by the model in closed-loop simulations. It helps to save sensible devices, once they are not
required, and closed-loop simulations behavior can confrrrn (or discard) test models. The strategy for compensation proposed showed good results and it was introduced in real time hardware-in-the-Ioop environment to simulate attitude control systems of Brazilian Space Program.
(5)
(s+ 726.57)(S2 + 42.119s + 7443.3)
6. REFERENCES
which has some high-frequency components that are of no use in the simulation loop.
4.2 Closed-Loop Outputs Comparison The plot below shows the limit-cycle effects obtained in performed simulations with the actual device (dashed line) and its digitalized model (solid line).
Carrijo D. S., Oliva, A. P. and Leite Filho, W. c., Hybrid Simulation Software Development for a Assesment of a Satellite Launcher Control System, Space Technology, vol. 19, Nos. 3-4, pp 213-219,1999. Chen, C. T., Linear System Theory and Design, HoltSaunders Ed, 1984. Contraves Goerz Corp., Inertial Guidance Tests System (A-7544), 1988.
Command limit-cycle samples
0.2 0.15
n,
0.1
f"1".
,' '. [
~
0.05
.,
Graupe, D., Identification of Systems, Van Nostrand Reinhold, 1972.
- - rot. simulator - ~odeled simulator
0;
p,
Ljung, L., System Identification: Theory for the User, (Prentice-Hall, Inc., 1987).
.
0
~
:'''~ , ,
'0
[ij.0.05 E E 8 -0.1
, .' ~'
., "
··:\U '.\
~:
-0.15 -0.2
110
112
114
116
118
time(s)
Fig. 11: comparison of simulation in a closed control loop with the actual device and its digital model
Leite Filho, W. C. and Carrijo D. S., Hardware-inthe-Loop Simulation of Brazilian Launcher VLS, Proceedings Third ESA International Conference on Spacecraft Guidance, Navigation and Control Systems, 1996,355-358. MATLAB, Signal Processing Toolbox User's Guide, The Mathworks Inc., 1994. Malyshev, V. V. et alii, Aerospace Vehicle Control: Modern Theory and Applictions, Instituto de Aerorulutica e Esparyo, 1996.
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