Abstract:
A method, apparatus, article of manufacture, and a memory structure for designing a robust controller. The method comprises the steps of determining a plant model G of the system dynamics; bounding system dynamics unmodeled by the plant model G of the system dynamics by a weighting function W; applying a transform to an augmented plant model T having the plant model G and the weighting function W; defining a controller {tilde over (F)} from the transformed plant model T; and applying an inverse of the transform to the controller {tilde over (F)} defined from the transformed plant model and the weighting function W to generate the robust controller F.

Description:
BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The present invention relates to systems and methods for designing and operating feedback control systems, and in particular to a system and method for robustly controlling spacecraft. 
   2. Description of the Related Art 
   Controllers are often employed to stabilize or improve the stability of an inherently unstable or marginally stable dynamic system. Such controllers are typically designed using well known state-space or Laplace transform techniques. 
   However, most dynamic systems include one or more non-linearities (e.g. actuator saturation, sensor dynamic range, plant operation envelope) that make the derivation of closed form controller solutions difficult or impossible. 
   In most cases, the system non-linearities do not substantially affect controller design or system performance in measures of interest, and can simply be ignored. However, in many cases, system non-linearities have a substantial effect on system performance. In such cases, the system non-linearities must be accounted for in the controller design. 
   One solution to this problem is to generate a linearized model of the system (including the system non-linearities), and design the controller using linear control system design techniques. However, while appropriate for some non-linearities, this technique is not generally applicable to all non-linearities. 
   Another solution is to generate a detailed simulation (e.g. an N degree of freedom, or N DOF simulation) of the system, including the non-linear elements, and use that simulation to design a controller to meet system requirements. This solution, however, can be time and resource intensive, both in generating the simulation and in performing multiple monte-carlo simulations to account for stochastically-modeled processes. Detailed simulations can also be very sensitive to modeling errors and uncertainties. Further, even when a detailed N DOF simulation is available, it can be difficult to design a controller without a starting point that is reasonably close to a stable solution that meets system requirements. 
   These difficulties are particularly apparent when considering the design of a controller for the attitude control system (ACS) of a spinning spacecraft during a transfer orbit mission. During the transfer orbit mission, the spacecraft can be difficult to control due to a strong interaction between the dynamics of fuel sloshing around in the fuel vessels (fuel slosh) and the dynamics of the spinning spacecraft. Fuel slosh dynamics are not only difficult and complicated to model, they are also highly uncertain. 
   Existing spacecraft ACS transfer orbit controller designs include Wheel Gyro Wobble and Nutation Control (WGWANC) systems. Unfortunately, this design is not robust to fuel slosh dynamics and system uncertainties. This is due to the nature of the WGWANC design methodology, which attempts to decouple 3-axis dynamics by absorbing the inverse of the plant inertial into the controller. This seriously degrades ACS controller robustness when the actual plant dynamics differ from model predictions. 
   WGWANC design procedures are also typically based on a classical 2 nd  order approximation of the plant dynamics. The controllers that result require substantial tuning (in the form of gain-scheduling, for example) throughout the mission to assure stability and adequate performance. Such gain scheduling can result in complicated software and mission operation procedures. 
   What is needed is a simple, effective method for designing system controllers that are robust to system uncertainties and non-linearities, for example, an ACS controller for spinning spacecraft that is robust to fuel sloshing effects. 
   The present invention satisfies that need. 
   SUMMARY OF THE INVENTION 
   To address the requirements described above, the present invention discloses a method, apparatus, and article of manufacture for designing a robust controller. The method comprises the steps of determining a plant model G of the system dynamics; bounding system dynamics unmodeled by the plant model G of the system dynamics by a weighting function W; applying a transform to an augmented plant model T having the plant model G and the weighting function W; defining a controller {tilde over (F)} from the transformed plant model T; and applying an inverse of the transform to the controller {tilde over (F)} defined from the transformed plant model and the weighting function W to generate the robust controller F. The apparatus comprises means, such as a digital computer having a processor and a memory, for performing the foregoing operations. The article of manufacture comprises a program storage device tangibly embodying instructions for performing the foregoing method steps. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Referring now to the drawings in which like reference numbers represent corresponding parts throughout: 
       FIG. 1  is an illustration of a spacecraft; 
       FIG. 2  is a diagram depicting the functional architecture of a representative spacecraft attitude and control system; 
       FIG. 3  is a block diagram presenting a representation of a control system model; 
       FIG. 4  is a flow chart presenting illustrative steps that can be used to design a robust controller for a dynamic system; 
       FIG. 5  is a diagram of a linearized model of a spinning body; 
       FIG. 6  is a diagram showing an example of how unmodeled system dynamics can be bounded by a weighting function W; 
       FIG. 7  is a diagram an augmented plant model; 
       FIG. 8  is a diagram showing the operation of a bilinear pole-shifting transform; and 
       FIG. 9  is a diagram of a modified representation of the system model. 
   

   DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
   In the following description, reference is made to the accompanying drawings which form a part hereof, and which is shown, by way of illustration, several embodiments of the present invention. It is understood that other embodiments may be utilized and structural changes may be made without departing from the scope of the present invention. 
   In the following description, reference is made to the accompanying drawings which form a part hereof, and which is shown, by way of illustration, several embodiments of the present invention. It is understood that other embodiments may be utilized and structural changes may be made without departing from the scope of the present invention. 
     FIG. 1  illustrates a three-axis stabilized satellite or spacecraft  100 . The satellite  100  has a main body or spacecraft bus  102 , a pair of solar panels  104 , a pair of high gain narrow beam antennas  106 , and a telemetry and command omni-directional antenna  108  which is aimed at a control ground station. The satellite  100  may also include one or more sensors  110  to measure the attitude of the satellite  100 . These sensors may include sun sensors, earth sensors, and star sensors. Since the solar panels are often referred to by the designations “North” and “South”, the solar panels in  FIG. 1  are referred to by the numerals  104 N and  104 S for the “North” and “South” solar panels, respectively. 
   The three axes of the spacecraft  100  are shown in  FIG. 1 . The pitch axis P lies along the plane of the solar panels  140 N and  140 S. The roll axis R and yaw axis Y are perpendicular to the pitch axis P and lie in the directions and planes shown. The antenna  108  points to the Earth along the yaw axis Y. 
     FIG. 2  is a diagram depicting the functional architecture of a representative attitude control system (ACS). The spacecraft  100  includes a processor subsystem  274 , which may include a spacecraft control processor (SCP)  202  and a communication processor (CP)  276 . 
   The SCP  202  implements control of the spacecraft  100 . The SCP  202  performs a number of functions which may include post ejection sequencing, transfer orbit processing, acquisition control, stationkeeping control, normal mode control, mechanisms control, fault protection, and spacecraft systems support, among others. The post ejection sequencing could include initializing to assent mode and thruster active nutation control (TANC). The transfer orbit processing could include attitude data processing, thruster pulse firing, perigee assist maneuvers, and liquid apogee motor (LAM) thruster firing. During LAM thruster firing, the satellite is typically spin-stabilized, rather than 3-axis stabilized. 
   The acquisition control could include idle mode sequencing, sun search/acquisition, and Earth search/acquisition. The stationkeeping control could include auto mode sequencing, gyro calibration, stationkeeping attitude control and transition to normal. The normal mode control could include attitude estimation, attitude and solar array steering, momentum bias control, magnetic torquing, and thruster momentum dumping (H-dumping). The mechanisms mode control could include solar panel control and reflector positioning control. The spacecraft control systems support could include tracking and command processing, battery charge management and pressure transducer processing. 
   Input to the spacecraft control processor  202  may come from a any combination of a number of spacecraft components and subsystems, such as a transfer orbit sun sensor  204 , an acquisition sun sensor  206 , an inertial reference unit  208 , a transfer orbit Earth sensor  210 , an operational orbit Earth sensor  212 , a normal mode wide angle sun sensor  214 , a magnetometer  216 , and one or more star sensors  218 . 
   The SCP  202  generates control signal commands  220  which are directed to a command decoder unit  222 . The command decoder unit operates the load shedding and battery charging systems  224 . The command decoder unit also sends signals to the magnetic torque control unit (MTCU)  226  and the torque coil  228 . 
   The SCP  202  also sends control commands  230  to the thruster valve driver unit  232  which in turn controls the liquid apogee motor (LAM) thrusters  234  and the attitude control thrusters  236 . 
   Wheel torque commands  262  are generated by the SCP  202  and are communicated to the wheel speed electronics  238  and  240 . These effect changes in the wheel speeds for wheels in momentum wheel assemblies  242  and  244 , respectively. The speed of the wheels is also measured and fed back to the SCP  202  by feedback control signal  264 . 
   The spacecraft control processor also sends jackscrew drive signals  266  to the momentum wheel assemblies  243  and  244 . These signals control the operation of the jackscrews individually and thus the amount of tilt of the momentum wheels. The position of the jackscrews is then fed back through command signal  268  to the spacecraft control processor. The signals  268  are also sent to the telemetry encoder unit  258  and in turn to the ground station  260 . 
   The spacecraft control processor also sends command signals  254  to the telemetry encoder unit  258  which in turn sends feedback signals  256  to the SCP  202 . This feedback loop, as with the other feedback loops to the SCP  202  described earlier, assist in the overall control of the spacecraft. The SCP  202  communicates with the telemetry encoder unit  258 , which receives the signals from various spacecraft components and subsystems indicating current operating conditions, and then relays them to the ground station  260 . 
   The wheel drive electronics  238 ,  240  receive signals from the SCP  202  and control the rotational speed of the momentum wheels. The jackscrew drive signals  266  adjust the orientation of the angular momentum vector of the momentum wheels. This accommodates varying degrees of attitude steering agility and accommodates movement of the spacecraft as required. 
   The CP  276  and SCP  202  may include or have access to one or more memories  270 , including, for example, a random access memory (RAM). Generally, the CP and SCP  202  operates under control of an operating system  272  stored in the memory  270 , and interfaces with the other system components to accept inputs and generate outputs, including commands. Applications running in the CP  276  and SCP  202  access and manipulate data stored in the memory  270 . The spacecraft  100  may also comprise an external communication device such as a satellite link for communicating with other computers at, for example, a ground station. If necessary, operation instructions for new applications can be uploaded from ground stations. The CP  276  and SCP  202  can also be implemented in a single processor, or with different processors having separate memories. 
   In one embodiment, instructions implementing the operating system  272 , application programs, and other modules are tangibly embodied in a computer-readable medium, e.g., data storage device, which could include a RAM, EEPROM, or other memory device. Further, the operating system  272  and the computer program are comprised of instructions which, when read and executed by the SCP  202 , causes the spacecraft processor  202  to perform the steps necessary to implement and/or use the present invention. Computer program and/or operating instructions may also be tangibly embodied in memory  270  and/or data communications devices (e.g. other devices in the spacecraft  10  or on the ground), thereby making a computer program product or article of manufacture according to the invention. As such, the terms “program storage device,” “article of manufacture” and “computer program product” as used herein are intended to encompass a computer program accessible from any computer readable device or media. 
     FIG. 3  is a block diagram presenting representation of a control system model  300 . The model  300  includes a representation of a command input  310 , which is compared to the output  314  to generate an error  312 . 
   The control system model  300  includes a plant model G  302 . In the illustrative example presented herein, the control system model  300  is a satellite control system and plant model G  302  is a representation of satellite rigid body dynamics. The plant model G  302  typically includes only linear or linearized representations of the plant. The satellite control system model  300  also includes a controller F  304 . The controller F  302  is designed to modify the error signal  314  as required to achieve the desired overall system response before providing the signal the rigid body dynamics G  302 . 
   The system model  300  also includes an model for dynamics that are unmodeled by the plant model G  302 . In one embodiment, these unmodeled dynamics include fuel slosh dynamics Δ A    308 , which will be described further herein. The system model  300  also includes a bounding or weighting function W  306 , which bounds the unmodeled dynamics  308  as described further below. 
     FIG. 4  is a flow chart presenting illustrative steps that can be used to design a robust controller for a dynamic system such as that which is presented in system model  300 . In block  402 , a plant model G  302  of the system dynamics is determined. Typically, the plant model G  302  is a linearized plant model, such as the model of the linearized rigid body dynamics as shown below. 
   Linearized Rigid Body Dynamics 
     FIG. 5  is a diagram of a linearized model of a spinning body such as the satellite  100 , when undergoing orbital transfer maneuvers. This example is used in the following discussion to provide an example of how the present invention can be used to design a robust control system. 
   In the illustrated example, the satellite  100  body is a rigid body with a mass, spinning about an axis such as the z-axis as shown. The linearized dynamic model G  302  about a nominal spacecraft spin rate Ω 0  and constant wheel  508  momentum h 0    504  along unit vector axis e w    506  in the satellite  100  body frame (typically [0−1 0] T ) can be represented as 
             [           Δ   ⁢           ⁢     ω   .                 Δ   ⁢           ⁢     h   .             ]     =       T   ⁡     [           Δ   ⁢           ⁢   ω               Δ   ⁢           ⁢   h           ]       +       [             -     I     -   1         ⁢     e   w               1         ]     ⁢           ⁢   τ             
wherein
 
                   T   =     [           -       I     -   1       ⁡     [         ω   0   x     ⁢   I     -       (       I   ⁢           ⁢     ω   0       +     h   0       )     x       ]                 -     I     -   1         ⁢     ω   0   x     ⁢     e   ω               0       0         ]       ;               ω   =     [                 ω   x               ω   y                     ω   z           ]                 
is a vector describing a rotational rate of the spacecraft about axes x, y, and z respectively;
 
           h   =     [                 h   x               h   y                     h   z           ]           
is a vector describing the rotational momentum of the spacecraft about axes x, y, and z respectively;
 
           I   =     [           I   11           I   12           I   13               I   21           I   22           I   23               I   31           I   32           I   33           ]           
is a matrix describing the moment of inertia of the spacecraft and (diagonal components, I 11  I 22  I 33 , of the I matrix are the principal inertia components; the off-diagonal components are the cross product of inertia of the rigid body);
 
               ω   0     =     [               0           0                   ω   s           ]       ,         
wherein ω s  is the spin rate of the spacecraft about the z axis;
 
               h   0     =     [             -     I   13       ⁢     ω   s                   -     I   23       ⁢     ω   s               0         ]       ;         
τ is a torque applied to the spacecraft  100  by the wheel; and
         wherein the operator x is defined such that follows:       
   The cross-product of vectors {right arrow over (a)} and {right arrow over (b)} (i.e. {right arrow over (a)}×{right arrow over (b)}) is 
                         [           i   →           j   →           k   →               a   1           a   2           a   3               b   1           b   2           b   3           ]     =       ⁢     [                     a     2   ⁢               ⁢     b   3       -       a     3   ⁢               ⁢     b   2                       a     3   ⁢               ⁢     b   1       -       a     1   ⁢               ⁢     b   3                             a     1   ⁢               ⁢     b   2       -       a     2   ⁢               ⁢     b   1               ]                   =       ⁢         [         0         -     a   3             a   2               a   3         0         -     a   1                 -     a   2             a   1         0         ]     ⁡     [                 b   1               b   2                     b   3           ]       ⁢     a   x     ⁢     b   .           ⁢   Therefore         ,                           a   x     =     [         0         -     a   3             a   2               a   3         0         -     a   1                 -     a   2             a   1         0         ]       ,       and   ⁢           ⁢     ω   x       =       [         0         -     ω   3             ω   2               ω   3         0         -     ω   1                 -     ω   2             ω   1         0         ]     .         ⁢                       
Since they are not stabilizable, states ω z  and H z  are eliminated from the above dynamics, resulting in a 4-state, two-input, two-output plant dynamic description with only ω x  and ω y  fed back.
 
   To design a robust control system, anticipated plant variations and other uncertainties must be bounded. Such anticipated plant variations include rigid body plant mass property variations (as fuel is expended, for example), and fuel slosh. 
   Returning to  FIG. 4 , in block  404 , the system dynamics that are unmodeled by the plant model G 0    302  are bounded by a weighting function W. 
     FIG. 6  is a diagram showing an example of how unmodeled system dynamics can be bounded by a weighting function W. In this specific example, a Bode plot  602  of the unmodeled fuel slosh dynamics for varying parameters A k  and LK 0 , where A k  is the modal inertia coupling coefficient and LK 0  is the modal frequency parameter. Both parameters define a particular two-dimensional Finite Element Model (FEM) of the fuel sloshing dynamics. Weighting function  604  is defined to bound the unmodeled system dynamics  602 . Note that the higher the bandwidth of the weighting function W ( FIG. 6  illustrates W 1 ), the more fuel slosh uncertainty in the system can be tolerated. 
   Returning to  FIG. 4 , an augmented plant model T is determined from the weighting function W and the system model G, as shown in block  406 . 
     FIG. 7  is a diagram showing an augmented plant model  700 . In this configuration, the plant model T  702  defined the response of the output y 1  to an input u 1 , and includes system model G  302 , the bounding function W  306 , and the controller {tilde over (F)} ( 704 ). 
   Returning again to  FIG. 4 , a transform is applied to the augmented plant model T  700 , as shown in block  408 . In one embodiment, the transform is a bilinear transform such as the transform 
           s   =         s   ~     +     p   1         1   +       s   ~       p   2                 
wherein s is a Laplace operator s=jω, p 1  is a first pole, p 2  is a second pole, and {tilde over (s)} is the Laplace operator s in the transformed (H ∞ ) domain. The first pole p 1  at least partially characterizes the bandwidth of the closed loop system  300  that results from the application of the design methodology of the present invention, while the second pole p 2  at least partially characterizes the damping of the closed loop system  300 . Accordingly, the poles p 1  and p 2  are selected to result in the desired bandwidth and damping of the closed loop system.
 
     FIG. 8  is a diagram showing the operation of the bilinear pole-shifting transform from a first plane (s-plane)  800  to a second ({tilde over (s)}-plane)  850 . The first plane  800  is defined by a real and imaginary axis, upon which the poles and zeros of the closed loop system model are plotted. Poles  806  (denoted by Xs in  FIG. 8 ) and zeros  804  (denoted by “0”s) appear on the imaginary axis. Poles  806  appearing in the right half plane (RHP)  802  indicate an unstable closed loop system, whereas poles appearing in the left half plane indicate a stable closed loop system. 
   When the bilinear transform is applied to the augmented plant model T  702 , the right half plane  802  gets mapped into a region  852 . In the illustrated example, wherein the bilinear transform is defined in terms of poles p 1  and p 2 , the region  852  is a circle that intersects the real axis at −p 1  and −p 2 , as shown. With this transform, poles  806 A- 806 C are mapped to locations  854 A- 854 C on the periphery of the circle  852 . 
   Returning to  FIG. 4 , a controller F({tilde over (s)})  704  (hereinafter alternatively designated as {tilde over (F)}) can be designed from the transformed augmented plant model  700 . In doing so, the uncertainty model is folded into the controller {tilde over (F)}  704  design. 
     FIG. 9  is a modified representation of the system model  300 . Here, he system model  300  can also be represented by its closed loop equivalent
   M=W*{tilde over (F)}* ( I+G{tilde over (F)} ) −1   
   By the small gain theorem, if |Δ A |*|M|&lt;1, the closed loop is stable. In the foregoing case, W  306  bounds the unmodeled fuel slosh dynamics by an additive uncertainty. W  306  is also a weighting function in the H ∞  domain on the plant G  302  to be optimized. Hence, controller {tilde over (F)}  704  in the H ∞  domain can be found that bounds the pre-specified and unmodeled fuel slosh dynamics  308 . 
   This can be accomplished by minimizing the measure of maximum singular value (  σ ) of the augmented plant transfer function [W{tilde over (F)}(I+G{tilde over (F)}) −1 ] to be less than one (Small Gain Stability Theorem). If the transformed augmented plant model T  700  can be expressed in the following state space realization: 
             T     y1   u1       =     [         A         B   1           B   2               C   1           D   11           D   12               C   2           D   21           D   22           ]           
the controller {tilde over (F)} can be parameterized into two game theoretic Riccati equations:
 
             F   ~     =     [           A   +       B   1     ⁢     B   1   T     ⁢   P     +       B   2     ⁢   R     +     Z   ⁢           ⁢   L   ⁢           ⁢     C   2                 -   Z     ⁢           ⁢   L             R       0         ]           
where R=−B 2   T P, L=−QC 2   T , and Z=(I=PQ) −1 , and wherein
 
   
     
       
         
           
             
               
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   The result of the foregoing is a controller, {tilde over (F)}  704  which propagates the closed loop poles from the edge of region  852  to the right half plane (RHP)  858  of the {tilde over (s)}-plane to locations  856 A- 856 C. 
   Returning to  FIG. 4 , an inverse of the applied transform is then applied to the controller {tilde over (F)} defined above to produce controller F  304 . This is shown in block  412 . Continuing the exemplary embodiment discussed above, the inverse transform is defined as 
   
     
       
         
           
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   Referring again to  FIG. 8 , application of the inverse transform maps the left half plane  858  of the {tilde over (s)}-plane  850  to the region  810  of the s-plane  800 . Accordingly, the inverse transform moves the poles  856 A- 856 C from their locations in the LHP  858  of the {tilde over (s)}-plane  850  to locations  808 A- 808 C within the region  810  of the s-plane  800 . Since the mapping compresses the LHP  858  into a smaller region  810  closer to the real axis and further away from the imaginary axis of the s-plane, the stability of the resulting system is enhanced. 
   Therefore, with respect to the example of the stabilization of the spinning spacecraft  100 , it can be seen that the application of the bilinear transform maps the nutation poles onto the circle with end points −p 1  and −p 2  in the RHP  802 . The plant dynamics have become more “unstable” in the new domain. However, as shown above, the direct Ricatti solution maps the open-loop unstable poles in  FIG. 8  to their mirror image in plane  800  on a circle  810  with endpoints p 1  and p 2  as their final closed loop poles. The controller {tilde over (F)} maps the {tilde over (s)} plane  850  closed loop poles further to the left when transformed back to the s plane  800 . 
   This design synthesis technique not only stabilizes the spinning dynamics, but also de-couples the roll/pitch axis automatically while minimizing the sensitivity function of the system. Fuel slosh dynamics, which have highly uncertain behavior, are treated as unmodeled dynamics, and a weighting function W (which is folded into the design process) is imposed on the overall cost function, thus bounding the fuel slosh dynamic uncertainty. The resulting controller F is therefore stable against all fuel slosh uncertainty bounded by the weighting function W. 
   Conclusion 
   This concludes the description of the preferred embodiments of the present invention. The foregoing description of the preferred embodiment of the invention has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed. Many modifications and variations are possible in light of the above teaching. It is intended that the scope of the invention be limited not by this detailed description, but rather by the claims appended hereto. The above specification, examples and data provide a complete description of the manufacture and use of the composition of the invention. Since many embodiments of the invention can be made without departing from the spirit and scope of the invention, the invention resides in the claims hereinafter appended.