Patent Publication Number: US-2005131592-A1

Title: In-flight control system stability margin assessment

Description:
BACKGROUND OF THE INVENTION  
      The present invention generally relates to attitude control systems and, more particularly, to a method of assessing control system stability margins.  
      A current approach to assessing control system stability margins is to provide a dynamic model of the control system as it applies, for example, to a spacecraft, or other vehicle whose physical motion, or attitude, is to be controlled and assume that dynamic model can vary, say, +/−25%, then check the stability margins accordingly in simulation—such as a computer simulation. While being comforting by providing some information where there is a complete lack of data for prediction of stability margins, this approach lacks a rigorous theoretical underpinning and, consequently, can lead to one or another of the following in-flight situations: either an overly conservative prediction of stability margins or a poor prediction of insufficient stability margins. In either case, the design cost and man power to develop the control system have been unnecessarily wasted, the attitude control system is bound to be sensitive to physical uncertainty, and control system stability margin and performance will most likely be poor.  
      In general, stability margins of spacecraft attitude control systems have not been assessed directly in flight due to the possibility of pushing the spacecraft into its instability regions with the attendant risk of driving the spacecraft into instability and not being able to recover control. Actual missions of prior art spacecraft have experienced in-flight “surprises” or anomalies from time to time in terms of lacking control system stability. When such an incident occurs, it can be a very disappointing and costly situation. When the design and analysis work fail to predict control system stability due to lack of in-flight spacecraft dynamics knowledge, entry of the spacecraft into service is typically delayed and additional engineering resources are often spent solving the problem.  
      As can be seen, there is a need for in-flight stability margin assessment that can prevent the kind of anomaly described above. There is also a need for a stability test that identifies the critical stability margins of a closed-loop attitude control system using in-flight data without driving the attitude control system into its instability regions. Moreover, there is a need for verifying the spacecraft stability margins in-flight and obtaining a realistic assessment of control system stability at any particular phase of a mission.  
     SUMMARY OF THE INVENTION  
      In one aspect of the present invention, a method for stability margin assessment includes determining a stability margin from in-flight data.  
      In another aspect of the present invention, a for in-flight stability margin assessment includes steps of: determining a stability gain margin from in-flight data; and determining a stability phase margin from the in-flight data.  
      In still another aspect of the present invention, a method for attitude control system stability margin assessment includes steps of: exciting a control system with a wide band spectrum excitation signal to produce input and output data; using the input and output data to estimate a system sensitivity function of the control system; and determining a stability margin of the control system from the system sensitivity function.  
      In yet another aspect of the present invention, a method for spacecraft attitude control system design includes steps of: exciting a control system with a white noise excitation signal to produce input and output data; storing the input and output data in an on-board computer during operation of a spacecraft mission; downloading the input and output data via telemetry during operation of the spacecraft mission; taking the discrete Fourier transform of the input autocorrelation function of the input data to create an input power spectrum of the input data; taking the discrete Fourier transform of the output autocorrelation function of the output data to create an output power spectrum of the output data; estimating a system sensitivity function by taking the ratio of the output power spectrum to the input power spectrum; determining a first stability margin of the attitude control system from the system sensitivity function by determining a gain margin GM from the formula:  
         1     1   -     a   min         &lt;   GM   &lt;     1     1   +     a   min             
 
 where “a min ” is the reciprocal of the peak of the system sensitivity function; and determining a second stability margin of the attitude control system from the system sensitivity function by determining a phase margin PM from the formula:  
       PM   &gt;     ±       sin     -   1       ⁡     (       a   min     2     )             
 
 where “a min ” is the reciprocal of the peak of the system sensitivity function. 
 
      In a further aspect of the present invention, a system for in-flight stability margin assessment includes: a physical plant; a controller that feeds control signals to the physical plant and receives feedback signals from the physical plant; a signal generator that excites the physical plant with white noise to provide input and output data; and an analysis subsystem. The analysis subsystem uses the input and output data to estimate a system sensitivity function of an attitude control system that includes the physical plant and the controller; and the analysis subsystem determines a stability margin of the attitude control system from the system sensitivity function.  
      In a still further aspect of the present invention, a spacecraft includes an attitude control system. The attitude control system includes a physical plant; a controller that feeds control signals to the physical plant; and a comparator, wherein the comparator receives a reference signal, the comparator receives a feedback signal from the physical plant, and the comparator provides a comparison signal to the controller. The spacecraft further includes a signal generator that excites the physical plant with white noise to provide input and output data from the attitude control system. The attitude control system is connected via telemetry to an analysis subsystem. The analysis subsystem uses the input and output data to estimate a system sensitivity function of an attitude control system that includes the physical plant and the controller; and the analysis subsystem determines a stability margin of the attitude control system from the system sensitivity function.  
      These and other features, aspects and advantages of the present invention will become better understood with reference to the following drawings, description and claims. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       FIG. 1  is a system block diagram showing a summary of an approach for in-flight stability margin assessment according to one embodiment of the present invention;  
       FIG. 2  is a block diagram showing processing of data according to one embodiment of the present invention;  
       FIG. 3  is a time domain and frequency domain graph of a band limited white noise excitation signal in accordance with one embodiment of the present invention;  
       FIG. 4A  is block diagram for defining gain and phase control system stability margins according to an embodiment of the present invention;  
       FIG. 4B  is a graph in the complex plane for determining the system sensitivity function in accordance with an embodiment of the present invention;  
       FIG. 5  is a graph in the complex plane showing an example of determining stability margins as a function of system sensitivity function peak in accordance with an embodiment of the present invention;  
       FIG. 6  is a block diagram for a system simulation using a commercially available system simulation program for in-flight stability margin assessment according to one embodiment of the present invention; and  
       FIG. 7  is a frequency domain graph of a system sensitivity function according to an analytical model compared to a sensitivity function obtained in accordance with in-flight stability margin assessment according to an embodiment of the present invention. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION  
      The following detailed description is of the best currently contemplated modes of carrying out the invention. The description is not to be taken in a limiting sense, but is made merely for the purpose of illustrating the general principles of the invention, since the scope of the invention is best defined by the appended claims.  
      Broadly, one embodiment of the present invention provides a method for assessment of control system stability margins that can be used during the flight of aerospace vehicles and spacecraft such as satellites. Using a special closed-loop stability test, one embodiment solves in a robust fashion the problem that prior art systems are unable to directly assess stability margins of a spacecraft control system in flight due to the difficulty and possibility of pushing the spacecraft into its instability regions. One embodiment includes an innovative method that identifies the critical stability margins of a closed-loop attitude control system using in-flight data without driving the system even anywhere near its instability regions. As a result, one embodiment can be used to verify spacecraft stability margins in-flight and obtain a much more realistic assessment of system stability at any particular phase of a spacecraft&#39;s mission.  
      One embodiment includes algorithms, software implementing the algorithms, and hardware executing the software for a set of in-flight stability measurement tools and procedures to access spacecraft stability margins. In one embodiment, the procedures can be turned on by an on-board computer—on board a satellite, for example—through a series of ground commands, and the on-board computer will telemeter down the time signal for ground processing. In-flight stability margins may then be calculated using the algorithm and formulae that are part of the set of in-flight stability measurement tools and procedures.  
      One embodiment of the present invention provides an opportunity, not present in prior art systems, for robust re-design of an aerospace vehicle attitude control system using the inventive in-flight stability assessment procedure. For example, the closed-loop control system may, first, be excited by the on-board signal generator, then a carefully selected set of closed-loop data may be downloaded via telemetry. The in-flight spacecraft control system stability margins then may be identified via the algorithms presented here. Finally, a sharpened attitude controller may be re-designed, if shown to be necessary, and uploaded to the on-board computer.  
      Referring now to the figures,  FIG. 1  illustrates an exemplary system  100  for in-flight stability margin assessment (IFSMA) according to an embodiment of the present invention. IFSMA system  100  may include any type of entity or physical plant  102  for which an attitude control system is to be designed and provided. Physical plant  102  may include, for example, the physical plant for an aerospace vehicle or spacecraft, such as a satellite. The entire vehicle or body—including, for example, its physical plant, attitude control system, and processors—may be referred to as control system  101 . For purposes of brevity and illustration of one embodiment, system  101  may also be referred to as “spacecraft  101 ”, however, the description is applicable to any type of vehicle or body considered as a system having a physical plant  102  for which it is appropriate to implement an attitude control system.  
      Systems  100  and  101  may include a controller  104 , which may implement the attitude control system used to control physical plant  102 , for example, via control signals  106 . Controller  104  may, for example, embody a control law specifically designed for the particular physical plant  102 —such as for a spacecraft  101 . The control law and operation of the controller  104  may be characterized by a transfer function F, as indicated in  FIG. 1  by the label “F” on controller  104 . Systems  100  and  101  may receive a reference signal  108 . Reference signal  108  may be provided, for example, from an on-board computer or via telemetry from a ground control station. By way of example for illustration purposes, reference signal  108  may be a command to turn spacecraft  101  by ten degrees about some axis. Physical plant  102  may provide a feedback signal  110  for comparison to reference signal  108  and input to controller  104 . Feedback signal  110  may be generated by a sensing device and transducer including, for example, a gyro, star tracker, resolver, or position sensor (not shown). Systems  100  and  101  may include a comparator  112  for comparing feedback signal  110  to reference signal  108  and providing a comparison signal  114  to controller  104 . Continuing the same illustrative example, controller  104  may continue feeding control signals  106  to physical plant  102  until the feedback signal  110  from a position sensor (for example) indicates that a rotation of spacecraft  101  of ten degrees has been achieved so that feedback signal  110  “matches” reference signal  108  producing a null comparison signal  114 , which in turn may be used by controller  104  to provide a control signal  106  to stop further position adjustment of spacecraft  101 .  
      System  100  may include means to provide the input and output signals from physical plant  102  as data to an analysis subsystem  120 . For example, physical plant signal input data  116  may be sampled from control signals  106  and provided via telemetry to analysis subsystem  120 . Also, for example, physical plant signal output data  118  may be sampled from feedback signals  110  and downloaded via telemetry to analysis subsystem  120 . Stability calculation  122  may be performed and attitude control system analysis  124  may be used to update or re-design the control law. For example, updated control parameters  126  may be uploaded via telemetry to controller  104  in order to effect changes to transfer function F that will modify operation of controller  104  and adjust the stability margins of the system (spacecraft)  101 .  
      Still referring to  FIG. 1 , IFSMA in accordance with one embodiment may proceed as follows. First, the spacecraft  101  may be excited by an on-board signal generator for a few minutes with spacecraft inertially held still without any maneuver interruption, for example, reference signal  108  is maintained as a null signal. The signal generator, for example, may be incorporated into controller  104  and the excitation signals generated may result in a small perturbation of control signals  106  with a resultant output of feedback signals  110  from physical plant  102 . The magnitude of the perturbations may be kept small to avoid any potential loss of control involving physical plant  102 . Because a null reference signal  108  is maintained, the spacecraft  101  has a tendency to return to its initial attitude once the transients caused by the perturbations die out. The in-flight data provided by control signals  106  and feedback signals  110  may be stored, for example, by an on-board computer, as input data  116  and output data  118 .  
      At the next communication window in the orbit of spacecraft  101 , the input/output data  116 ,  118  may be transmitted via telemetry down to the ground, for example, to analysis subsystem  120 . The stability margin may be calculated (stability calculation  122 ) based on the in-flight spectrum estimates of the stability function or sensitivity function, using the input/output data  116 ,  118 . If the stability margin is similar to what was predicted, no more control system re-design is needed. Otherwise, IFSMA may include re-designing the controller law and uploading to the spacecraft—for example, by uploading control parameters  126  to controller  104  on spacecraft  101 —for use in service with proper stability margins.  
      The IFSMA procedure may be divided into the following steps, which are described in more detail below: 
          (1) Excite the spacecraft with stability signal generated on-board.     (2) Store the input/output data in an on-board computer.     (3) Telemeter down the I/O data and compute its spectrum estimate.     (4) Plug in pre-determined stability margin formulae and compute for IFSMA.     (5) Re-design control law if necessary.     (6) Telemeter up the new control law, if re-designed, to the controller.        

      Referring now to  FIG. 2 , an outline is diagrammed for the mathematical computation of the stability sensitivity function for a controller and physical plant such as spacecraft  101 . Spacecraft  101  may be considered to be a black box  202  characterized by a transfer function H(z) as represented in  FIG. 2 . For example, H(z) may be the composition of transfer functions of the controller  104  and physical plant  102  so that if transfer function F characterizes the controller  104  and transfer function G characterizes the physical plant  102 , then H may be represented by H=GF. In general, H may be estimated or computed by comparing inputs  204  to black box  202  with outputs  206  from black box  202 . For example, inputs  204  may have the form of a time sequence  208  denoted by x k  in  FIG. 2  and outputs  206  may have the form of a time sequence  210  denoted by y k  in  FIG. 2 . Inputs  204  and corresponding outputs  206  may be generated as in steps (1) through (3) above, for example, by exciting the spacecraft  102  with stability signal generated by an on-board signal generator to provide control signals  106  and feedback signals  110  and recording and transmitting the data as input/output data  116 ,  118 , as described above.  
      Box  212  of  FIG. 2  shows the autocorrelation function φ of input sequence  208  and the discrete Fourier transform Φ of the autocorrelation function φ for inputs  204 . Likewise, box  214  of  FIG. 2  shows the autocorrelation function φ of output sequence  210  and the discrete Fourier transform Φ of the autocorrelation function φ for outputs  206 . The stability function estimate for H may be computed mathematically by taking the discrete Fourier transform of the input and output autocorrelation functions to create the so-called “power spectrum” of the input and output data—such as input/output data  116 ,  118  which may have the form of time sequences  208 ,  210 —in the frequency domain. Then by taking the ratio of the output power spectrum to the input power spectrum, a transfer function magnitude Bode plot of the stability function can be calculated and plotted, such as stability sensitivity function  720  shown in  FIG. 7 .  
      Referring now to  FIG. 3 , a time domain graph  302  and frequency domain graph  304  of a band limited white noise excitation signal  300  are shown in accordance with one embodiment of the present invention. The signal  300  may be supplied by an on-board signal generator as in step (1) above. For example, signal  300  may be fed as control signals  106  to physical plant  102  as shown in  FIG. 1 . To get a good frequency domain approximation of the in-flight stability function—such as stability sensitivity function  720  shown in  FIG. 7 —it is preferred to use a wide band spectrum excitation signal to move the spacecraft. Thus, a Uniformly Distributed white noise may be used for the on-board excitation signal  300  as shown in  FIG. 3 . The white noise, whether it is Uniformly Distributed or Gaussian Distributed, generally has a “flat” spectrum as shown by graph  304  in  FIG. 3 . As the time domain signal (graph  302 ) lasts longer, the spectrum (graph  304 ) turns flatter. This special characteristic ensures that the system  101  can be excited in every frequency range of interest with an equal amount of energy such that the resulting output spectrum—such as a frequency domain graph of output  206 —can be evaluated at all relevant frequencies without missing any significant response of the system  101 .  
      Then, the excitation signal at plant input, for example, input data  116 , and the response at plant output, for example, output data  118 , may be post-processed by the following refinement procedure, which may use Fast Fourier Transform (FFT) techniques: 
          1. Divide the time domain signal data into equal size (FFT N-point) and overlapped pieces called segments.     2. Apply windowing techniques—such as rectangular, tapered rectangular, triangular, Hanning, Hamming, and Blackman—to each segment of the data.     3. FFT the time domain segments into periodograms.     4. Average the periodograms to get final power spectrum estimates.        

      In any practical application of IFSMA, the noise embedded in the physical system—such as system  101 —can be the major obstacle of getting an accurate plant model—such as an mathematical model of physical plant  102 . The IFSMA procedure, according to one embodiment, may window and average out the noise effect, hence producing much more accurate plant models in the frequency ranges of interest.  
      Referring now to  FIGS. 4A, 4B , and  5 , an illustration is given of the principles underlying assessment of stability margins using the stability sensitivity function determined from the collection of in-flight data according to an embodiment of the present invention. System  401  shown in  FIG. 4A  corresponds to system  101  shown in  FIG. 1  and may be used to mathematically represent system  101  and to show how stability margins may be defined. A gain stability margin and a phase stability margin may both be defined with the aid of  FIG. 4A . System  401  may include a transfer function  402  representing the combined operation of controller  104  and physical plant  102  and characterized by transfer function GF, where, as described above, GF may be the composition of transfer function F of the controller  104  and transfer function G of the physical plant  102  so that system  101  may be characterized (in system  401 ) by the transfer function GF, transfer function  402 .  
      Thus, for example, “F” may represent the control law of the control system and “G” may represent the spacecraft dynamics for spacecraft  101 . System  401  may further include a comparator  412  representing system  101  comparator  112 , reference signal  408  representing reference signal  108 , feedback signal  410  representing feedback signal  110 , and comparison signal  414  representing comparison signal  114 . System  401  may include stability margin tester  430  characterized by the complex function exp(jK). Stability margin tester  430  exists only in simulation and does not represent an actual part of system  101 . The value of K, which is a complex number, may be varied to affect the behavior of system  401 . For example, when K=0, exp(jK)=1, so comparison signal  414  is multiplied by 1 in stability margin tester  430  so that test signal  432  is the same as comparison signal  414  and there is no effect on the behavior of system  401 . When, for example, the value of K is varied from zero only in its imaginary part, exp(jK) becomes a purely real number so that the test signal  432  is a real multiple of comparison signal  414 , i.e., only the gain is affected. When, for example, the value of K is varied from zero only in its real part, exp(jK) becomes a value on the unit circle in the complex plane so that the test signal  432  has the same magnitude as comparison signal  414  but the angle is changed according to the angle of exp(jK) on the unit circle, i.e., only the phase is affected.  
      Thus, when K varies on the imaginary axis until system  401  goes unstable, the stability “gain margin” (GM) may be defined. Similarly, when K varies on the real axis until system  401  goes unstable, the “phase margin” (PM) of the system may be defined. Stability margins may be defined mathematically in this manner, however, in real life, no one can afford driving the system—such as the actual spacecraft  101 —to the vicinity of the instability region and claim the measurement of stability margins. It is not done, for example, because one could simply lose an entire billion dollar spacecraft to an out-of-control situation from which no recovery is possible. Thus, in-flight stability margin assessment, as in one embodiment of the present invention, has not been accomplished in the prior art.  
       FIG. 4B  provides a novel approach to the problem of in-flight stability margin assessment via the so called “system sensitivity function” S=1/(1+GF). If one can compute the closed-loop system sensitivity function S with nominal control laws and plant dynamics, the peak of the stability function—such as peak  725  of stability sensitivity function  710  in  FIG. 7 —determines the gain margin and phase margin equivalently and more accurately. In  FIG. 4B , the transfer function GF of the system, for example, transfer function  402  of system  401 , is represented by a curve  442  in the Nyquist plane of complex numbers. At each point X of the curve  442 , a vector  444 , an example of which is shown in  FIG. 4B , may be calculated as X−(−1)=X+1. Thus, the vectors  444  for the transfer function GF of curve  442  may be represented as 1+GF. By the definition of the system sensitivity function S, 1+GF=1/S=S −1 , as indicated in  FIG. 4B . It may be noted that for values of GF close to −1, the system sensitivity function “blows up”, indicating instability of the system.  
       FIG. 5  continues the illustration of  FIG. 4B  using a different example curve  542  for the purpose of providing a clearer illustration. Curve  542 , like curve  442 , should, however, represent the transfer function GF of the system, for example, transfer function  402  of system  401 . Each vector  544 , like vectors  444 , represents a value of S −1= 1+GF, and is (generically) denoted by “a”. The vector “a”, or vector  544 , of minimum length, vector  546  denoted “a min ”, corresponds to the peak of the system sensitivity function. For a system sensitivity function corresponding to stability sensitivity function  720  shown in  FIG. 7 , for example, the minimum length vector  546 , a min , may correspond to peak  725  of sensitivity function  720  when curve  542  corresponds to the transfer function GF of the same sensitivity function  720  and the system sensitivity function S=1/(1+GF).  
      The equations below show the stability margin formulae as determined by the peak of the system sensitivity function, using a min  described above.  
               1     1   -     a   min         &lt;   GM   &lt;     1     1   +     a   min                     PM   &gt;     ±       sin     -   1       ⁡     (       a   min     2     )                   
 
 where “a min ” is the reciprocal of the peak of the system sensitivity function. Therefore, by completing the steps 1 through 4 above—for example, post-processing the data  116 ,  118 —with the above formulae, one may achieve IFSMA with a high degree of accuracy. 
 
      IFSMA can show how much stability margin actually exists during operation in the mission, for example, of a spacecraft. If the gain or phase stability margin is inadequate, for example, smaller than what is expected to be safe, a redesign of the control law may be necessary and may be undertaken. In doing so, the new control law with an increased stability margin may be uploaded to an on-board computer of the spacecraft—such as spacecraft  101 —and used by the controller—such as controller  104 —for the rest of the mission operation. With the updated controller, the overall system should be much more robust and the performance should be superior with accurate IFSMA.  
     EXAMPLE  
      Referring now to  FIGS. 6 and 7 , IFSMA may be illustrated using an example of an analytical physical plant model. The approach of the illustrative example is to identify the sensitivity function S using the white noise excitation signals, compute the spectrum estimate of S and compare the spectrum estimate of S  720  to the system sensitivity function S  710  of the analytical model in the frequency domain.  
      A SIMULINK™ block diagram for system model  601 , shown in  FIG. 6 , models a system with nominal control laws and plant dynamics. Thus, an analytical model can be used to provide the “exact” system sensitivity function S  710  shown in  FIG. 7  of the analytical model of system model  601 . The modeled system may be similar to an actual system such as system  101  shown in  FIG. 1 . Thus, system model  601  includes a controller  604 , model of physical plant  602 , reference signal  608 , feedback signal  610 , comparator  612 , comparison signal  614 , and control signals  606  modeling corresponding parts of system  101 . Block diagram of system model  601  of  FIG. 6  illustrates that we excite the model system  601  from inputs  1  and  2 , i.e. inputs  650 , using white noise signals, and collect the output data at outputs  1  and  2 , i.e. outputs  652 . This process, for example, models the process of collecting input/output data  116 ,  118  after exciting system  101  with white noise—such as white noise excitation signal  300 . Using the power spectrum estimate tools—such as those available in MATLAB™ and SIMULINK™ and described above, for example, at steps 1 through 4—we can compute the spectrum estimate of the sensitivity function  720  shown in  FIG. 7 .  FIG. 7  shows that the spectrum estimate may have very good agreement with the nominal system sensitivity function of the analytical model.  
      It should be understood, of course, that the foregoing relates to preferred embodiments of the invention and that modifications may be made without departing from the spirit and scope of the invention as set forth in the following claims.