A Nonlinear Programming Approach for Control Allocation

Abstract—In this paper, a novel nonlinear programming based control allocation scheme is developed. The performance of this nonlinear control allocation algorithm is compared with that of other control allocation approaches, including a mixed optimization scheme, a redistributed pseudo-inverse approach, and a direct allocation (geometric) method. The control allocation methods are compared using open-loop measures such as the ability to attain commanded moments for a prescribed maneuver. The methods are then compared in closed-loop with a dynamic inversion-based control law. Next, the performance of the different algorithms is compared for different reference trajectories under a variety of failure conditions. Finally, we perform some preliminary studies employing “split actuators” that increase available control authority under failure conditions. All studies are conducted on a re-entry vehicle simulation.
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  A Nonlinear Programming Approach for Control Allocation † Vishnu L. PoonamalleeStephen Yurkovich  Andrea SerraniDepartment of Electrical EngineeringThe Ohio State University, Columbus, OHDavid B. DomanMichael W. OppenheimerAir Force Research LaboratoryWPAFB, OH  Abstract —In this paper, a novel nonlinear programmingbased control allocation scheme is developed. The performanceof this nonlinear control allocation algorithm is comparedwith that of other control allocation approaches, includinga mixed optimization scheme, a redistributed pseudo-inverseapproach, and a direct allocation (geometric) method. Thecontrol allocation methods are first compared using open-loopmeasures such as the ability to attain commanded momentsfor a prescribed maneuver. The methods are then comparedin closed-loop with a dynamic inversion-based control law.Next, the performance of the different algorithms is comparedfor different reference trajectories under a variety of failureconditions. Finally, we perform some preliminary studiesemploying “split actuators” that increase available controlauthority under failure conditions. All studies are conductedon a re-entry vehicle simulation. I. I NTRODUCTION One of the primary limitations of traditional controlallocation algorithms is the dependence on assumptions of linearity. Most approaches are based on the assumption thatthe control variables and rates are linear functions of theeffector deflections. However, the forces and moments pro-duced by aerodynamic control effectors are often nonlinearfunctions of the effector deflection.The concept of control allocation has therefore be-come increasingly more important as flight control systemsemploy multiple actuators, with numerous constraints, toachieve multiple control objectives. There are several ob- jectives that must be satisfied by these algorithms. Thealgorithms must be computationally fast since they must runin real time and they must provide a guarantee that they willcompute a solution in the time allotted by the flight controlsystem. To be considered flight-worthy, the algorithm mustalso reliably produce smoothly varying actuator commandsthat do not chatter from one time step to the next. Anotherimportant goal is to make reconfiguration possible when oneor more control surfaces fail. Ideally, the algorithm shouldalso be able to deliver a set of effector inputs that minimizethe difference between actual and desired commands in theevent that the latter cannot be produced.Several control allocation and control mixing algorithmshave been developed, and a few survey papers exist [2],[5] that point out the advantages and disadvantages of the control allocation schemes. Existing control allocation † This work is supported by the AFRL/AFOSR Collaborative Center of Control Sciences at the Ohio State University.  Corresponding author. Email: algorithms are capable of dealing with systems wheremoments are linearly related to control effector positions,and have the ability to account for position constraints.Therefore, most existing algorithms assume that a linearrelationship exists between the controlled variables (CVs)and the effector variables. In cases where this assumptionfails, errors in the control allocation schemes must bemitigated by the robustness resulting from feedback controllaws. A reasonable goal, therefore, is to free some of the burden on the feedback portion of the control law byincreasing the accuracy of the control allocator. Recently,a piecewise linear control allocation approach has beenintroduced that effectively accommodates separable nonlin-earities using piecewise linear programming methods [8].The effects of the nonlinear relationships between forces,moments and control surfaces are directly considered in thispaper, and a straightforward nonlinear control allocation lawis proposed. In this work, actuator dynamics are assumedto have a negligible effect; the case of non-negligibleactuator dynamics is addressed in a companion paper [7].Nonlinear programming techniques are used to find theeffector positions given the nonlinear relationship betweenthe moments and the effector positions. While nonlinearprogramming does not offer guarantees of convergence in afinite period of time, it does provide a performance metricagainst which other methods can be compared and is areasonable first step that can be used to assess the potentialbenefits of nonlinear control allocation approaches.The results of this approach are compared to severallinear techniques, including a mixed optimization controllaw, a redistributed pseudo inverse method, and a directcontrol allocation law. The algorithms are tested in bothclosed and open loop architectures. Because of the penaltyfor adding additional weight, re-entry vehicles have littlehardware redundancy. Due to the availability of a limitednumber of control surfaces, a control effector failure canseverely affect the vehicle’s performance and safety. Hence,failures are explicitly addressed in this paper, and theireffect on control allocation are observed in a comparativestudy. Preliminary studies employing “split effectors” toincrease control authority in failure situations are alsoinvestigated.The re-entry vehicle considered in this study has fouraerodynamic control surfaces (right/left tails and right/leftflaps), and is capable of flying through different flightregimes, spanning a wide envelope of speeds and altitude. Proceeding of the 2004 American Control Conference Boston, Massachusetts June 30 - July 2, 2004 0-7803-8335-4/04/$17.00 ©2004 AACCWeP12.11689  Such a vehicle would be carried to orbit, reenter the atmo-sphere at hypersonic speeds, and finally land horizontally.II. D YNAMIC  I NVERSION To set this work in proper context, we first consider thefeedback loop within which the control allocation algorithmfunctions. An inner-loop control law is designed usingdynamic inversion [1] so that the vehicle tracks body-ratecommands generated by an outer-loop guidance and controlsystem. A nonlinear control law is fashioned that globallyreduces the dynamics of selected controlled variables (CVs)to a set of integrators. A closed-loop implicit model follow-ing system is then constructed to make the CVs exhibit adesired set of dynamics that replace the existing dynamics.Such a control scheme has several advantages, includinggreater generality for re-use across different flight regimesand greater flexibility for handling different flight models. Fig. 1. Dynamic Inversion The overall control scheme is depicted in Figure 1.The body-axis angular acceleration commands  ˙ ω des  aregenerated as the output of an inner-loop prefilter block,which is used to shape the dynamic response. An outer-loopguidance and control system generates body-axis angularacceleration commands, which drive the inner-loop. Theinner-loop dynamic inversion control law is designed so thatthe vehicle tracks these body-rate commands. The prefilteris selected so that the closed-loop system has the propertiesof a desired model when the dynamic inversion is perfect.As such, the prefilters and dynamic inversion combinationforms the implicit model following system [9]. The reentryvehicle rotational dynamics can be written as ˙ ω  =  f  ( ω,P  ) +  g ( P,δ  )  (1)where  ω  = [  p q r ] T  is the angular velocity vector,  δ   isthe vector of actuator deflections, and  P   denotes measur-able or estimable quantities that influence the body ratestates. The parameter vector  P   includes variables such asMach number, angle of attack, sideslip angle and vehiclemass properties such as moments of inertia. Equation (1)expresses the body-axis rotational accelerations as a sumthat includes control dependent accelerations  g ( P,δ  )  andaccelerations due to the wing-body,  f  ( ω,P  ) . It is assumedthat the mass properties of the vehicle are time-invariant,so that the inertia matrix  I   satisfies  ˙ I   = 0  and equation (1)can be written as ˙ ω  =  I  − 1 ( G B − ω × Iω ) ,G B  =  G WB ( ω,P  )+ G δ ( P,δ  ) =  LM N   WB +  LM N   δ ,G WB ( ω,P  )  is the moment generated by the wing-bodyaerodynamics, and  G δ  is the moment produced by thecontrol effectors. In this representation,  L ,  M  , and  N   arethe moment vector components in a body-axis coordinatesystem. Thus, f  ( ω,P  ) =  I  − 1 ( G WB ( ω,P  ) − ω × Iω ) ,g ( P,δ  ) =  I  − 1 G δ ( P,δ  ) . In this work we employ a simulation of the re-entry vehicle,based on a large array of flight conditions generated byMissile Datcom. The aerodynamic database provides forceand moment coefficient data taken at a moment referencepoint (MRP), located at the center of gravity of the emptyvehicle. Control derivative information is extracted from thetables in the database for the values of Mach number, angleof attack and sideslip angle encountered along the trajectory.As stated previously, most conventional control allocationapproaches require that the control dependent portion of themodel is affine in the control. Thus, in order to compare thenonlinear approach to conventional linear approaches, wedevelop a linear approximation of the control dependentportion as G δ ( P,δ  ) ≈ G δ ( P  ) δ. The model used for the design of the dynamic inversioncontrol law, for use with linear control allocators, has theform ˙ ω  =  f  ( ω,P  ) +  I  − 1 G δ ( P  ) δ  and our objective is to find a control law that provides directcontrol over  ˙ ω  so that  ˙ ω  = ˙ ω des . Therefore, the inversecontrol law must satisfy ˙ ω des − f  ( ω,P  ) =  I  − 1 G δ ( P  ) δ.  (2)Similarly, for the nonlinear control allocator, the control lawmust satisfy ˙ ω des − f  ( ω,P  ) =  I  − 1 G δ ( P,δ  ) .  (3)III. L INEAR  C ONTROL  A LLOCATION  A. Overview Since there are three controlled variables and four controleffectors, a control allocation law can be used to ensure thatEquation (2) is satisfied. Equation (2) can be represented as, Bδ   =  d des  ,  (4)where  B  =  I  − 1 G δ ( P  )  and  d des  = ˙ ω des  −  f  ( ω,P  )  aregiven, and  δ   is to be determined such that the followingconstraints are satisfied δ  min  ≤ δ   ≤ δ  max ˙ δ  min  ≤  ˙ δ   ≤  ˙ δ  max  .  (5)The above inequalities express the position and rate limits of the actuators. Given these constraints, an exact solution may 1690  not exist, despite the redundancy. Furthermore, a solution(either exact or approximate) cannot be assumed to beunique. If a failure occurs, a new control input must befound that accounts for the modified entries of the matrix B , or changes in rate or position limits.  B. Mixed Optimization with Intercept Correction Problem The mixed optimization with intercept correction (MOIC)problem considers both error minimization and the controlminimization problem [2]. The error minimization problemcan be formulated as follows. Given a matrix  B , find avector  δ   such that  J   =  Bδ  − d des   1  is minimized subjectto the constraint (5). The control minimization probleminvolves minimization of   J   =   δ   − δ   p   1 , where  δ   p  is apreferred control vector position. The mixed optimizationapproach essentially combines the error and control mini-mization problems into a single one, and is formulated asfinding a control vector  δ   such that, J   =  Bδ  − d des   1  + λ  δ  − δ   p   1 is minimized, subject to the constraints (5). If the parameter λ >  0  is small, priority will be given to error minimizationover control minimization, as desired. The LP problemwhich results from this type of performance index can besolved using the simplex algorithm [2].Included in this control allocation scheme is the interceptcorrection term [4]. This term takes into account some of the nonlinearities in the aerodynamic data and manifestsitself by simply modifying  d des . Instead of using a linearrelationship to represent  G δ ( P,δ  ) , an affine relationship isutilized such that G δ ( P,δ  ) =  G δ ( P  ) δ   +   ( P,δ  )  (6)so that equation 2 becomes ˙ ω des − f  ( ω,P  ) =  I  − 1 G δ ( P  ) δ   +  I  − 1  ( P,δ  ) .  (7)Then, it is easily seen that  d des  = ˙ ω des  −  f  ( ω,P  )  − I  − 1  ( P,δ  )  and  B  =  I  − 1 G δ ( P  ) .In the results to follow, the intercept correction term isincluded only in the mixed optimization intercept correctionscheme and is not used in the pseudo-inverse or directallocation approaches. C. Redistributed Pseudo-Inverse The redistributed pseudo-inverse (RPI) method [5] in-volves finding the control vector  δ   that minimizes J   =  W  δ   δ  − δ   p   22  (8)subject to  Bδ   =  d des . For the unconstrained case, thesolution is given by a biased weighted pseudo-inverse, δ   =  δ   p  +  W  − 1 δ  B T  ( BW  − 1 δ  B T  ) − 1 ( d des − Bδ   p ) .  (9)In presence of constraints of the kind (5), this approachmay produce solutions that violate the effector limits. Thefollowing method is used to handle unattainable solutions.First, a control vector that solves the unconstrained problem(8) is found using (9). If the solution violates the constraints,the individual commands that saturate are clipped at theirrespective limit, their contributions are subtracted from d des , and the inverse is computed again. The procedure isrepeated until a feasible solution is found or until all thecomponents have saturated. The major problem with thismethod is that it is not able to make use of the full controlauthority. Otherwise, it is simple, and often effective.  D. Direct Control Allocation In the direct allocation (DA) proposed by Durham [6],the objective is to find a control vector  δ   that results inthe best approximation to the commanded moment  d des  ina given direction. The Attainable Moment Set (AMS) isdefined as the set of all moments that can be produced bya set of control effectors constrained within a set of knownlimits. If a desired moment lies outside the boundary of theAMS, then it is not attainable; in such a case, the solutionwhich lies on the boundary of the AMS and preserves thedirection of   d des  is used. If the desired moment lies withinthe AMS, the control effector solution is scaled down suchthat equation (4) is satisfied.IV. N ONLINEAR  C ONTROL  A LLOCATION The assumption that aerodynamic control effectors pro-duce moments that are linear is often violated in practice.The impact of this assumption becomes especially importantin the event of failure of one or more control effectors.The effectors can then be forced to operate in highlynonlinear regions of the moment-deflection curves. To relaxthe linearity assumption, we propose a nonlinear control al-location (NCA) method that minimizes the sum of weightedsquare distances between the commanded moments and thecorresponding moment functions: min L [ w l ( L d − L ( δ  )) 2 +  w m ( M  d − M  ( δ  )) 2 + w n ( N  d − N  ( δ  )) 2 ]  (10)subject to δ  min  ≤ δ   ≤ δ  max ˙ δ  min  ≤  ˙ δ   ≤  ˙ δ  max where  w l ,  w m , and  w n  are weighting factors that can beused to weight the relative importance of achieving theindividual moments.The functions  L ( δ  ) ,  M  ( δ  )  and  N  ( δ  )  are defined viaa nonlinear curve fit based on information obtained fromthe aerodynamic database. Therefore, the approach uses atwo-stage search. From the specified desired moments, thevolume of AMS is searched for the points closest to thedesired moment. The minimum and the maximum range of the deflections are calculated to produce a resulting momentclosest to the desired one. This point is used to initializean optimization algorithm that uses a Sequential QuadraticProgramming (SQP) approach to solve for the control de-flections that yield the desired moment to within some pre-specified tolerance. The algorithm has been implementedusing the gradient-based optimization tools available in 1691  Matlab. The following set of basis functions are used tocurve fit the aerodynamic data: Moment  = 4  i =1 a i δ  i  + 4  i =1 b i δ  2 i  + 4  i =1 c i δ  3 i  +  k i  (11)where  a i , b i , c i  and  k i  represent the coefficients of eachpolynomial approximation.The basis function structure was obtained after a carefulinvestigation of the moment versus deflection plots overa wide range of operating conditions, as provided by theaerodynamic database. To illustrate the nonlinearity inherentin the relationship between moment and effector deflection,consider the pitching moment for a tail effector depicted inFigure 2. It is evident from the plot that for a portion of this moment-versus-deflection curve, a linear approximationis adequate to represent the mapping (approximately inthe region of   − 10 ◦ to  10 ◦ of deflection). However, failureconditions can drive the effectors to a region where thelinear approximation is no longer valid. −30 −20 −10 0 10 20 30−0.4−0.3−0.2− Deflection o    P   i   t   c   h   i   n   g   M   o   m   e   n   t   C   o   e   f   f   i   c   i   e   n   t Curve−fitting: Linear Vs Cubic ApproximationMomentCubicLinearIC Fig. 2. Curve-fitting result for moment-versus-deflection curve. The curvemarked with “*” represents the moment obtained from the aerodynamicdatabase, the solid curve shows the curve fit using (11), the dashed lineshows an approximation for this mapping that is used in the MOIC, andthe cross-hatched line shows the linear approximation that is used for theRPI and DA. Levenberg-Marquardt or Gauss-Newton methods can beused to obtain the curve fit. It is important to note thatthe coefficients are obtained off-line for different operat-ing regimes specified by Mach number, angle of attack,sideslip and altitude. They are stored in databases for futureretrieval. Cubic interpolation is then employed to extractthe appropriate coefficients corresponding to the operatingpoint. Position and rate constraints are included in thecontrol allocation law by applying the most restrictive at thecurrent operating condition. Specifically, the performanceindex is minimized subject to  δ   ≤ δ   ≤ δ  , where  δ   and  δ   arethe most restrictive lower and upper bounds on the controleffector deflection, given by δ   =  min ( δ  max , ∆ T   ˙ δ  max  +  δ  ) δ   =  max ( δ  min , − ∆ T   ˙ δ  max  +  δ  ) where  δ  max ,δ  min  are the lower and upper position limitsvector,  ˙ δ  max  is a vector of effector rate limits, and  ∆ T   isthe inner-loop flight control system sampling rate.The proposed two-stage nonlinear algorithm (curve fitand nonlinear programming optimization) successfully ac-counts for the nonlinearities inherent in the re-entry vehiclemodel. Superior performance is possible, as would beexpected, over the linear control allocation schemes, partic-ularly in the event of failures. The main disadvantage of thismethod lies in its computational complexity: its feasibilityfor utilization in real-time control allocation schemes is asubject of current investigation.  A. Results and Comparison A test trajectory is used here to compare the different con-trol allocation schemes. The trajectory is a rather stringenttest for the control allocation schemes, since it drives thesystem into the nonlinear regions in the pitch, roll and yawmoment characteristics. The performance of the differentcontrol allocation schemes is shown in Figures 3 and 4. Forthese results, the control allocator was isolated (open-loop).Hence, there were no feedback loops being closed on theinner-loop. In Figure 3 the dotted trace displays the desiredmoment or acceleration to be produced by the controleffectors (corresponds to  d des ). Each control allocationscheme computes a set of control effector deflections at eachtime instant. These deflections are then used to interrogatethe aerodynamic database to find the moments producedby the control effectors for each allocation algorithm. Forthe sake of clarity, the results for the redistributed pseudoinverse technique (which are the poorest of the methodscompared) are omitted. The mixed optimization with in-tercept correction scheme implemented in this comparativestudy follows the work of [2], but implements a localslope of the control moment curve with an added interceptterm to account more accurately for the nonlinear behaviorof aerodynamic control effectors (see [4]). From Figure3, it is clear that once the nonlinear region is reachedthe nonlinear technique, as expected, tracks the desiredmoments quite well. Table I summarizes the comparativeperformance for the nominal test trajectory (no failures)in Figure 3. The mean squared error corresponds to theerror in the commanded moment and the moment obtainedby substituting the allocated controls into the nonlinearaerodynamic database and summing the contributions of the individual moment effectors. The average number of control effector saturations occurred is also given. Figure 4illustrates the error performance of the three methods.  B. Actuator Failures As mentioned previously, when an actuator failure occurs,control allocation schemes must be able to accommodate theincreased demands on the remaining effectors in order toproduce desired and attainable moments. Among the variouspossible failures, those addressed in this study are: 1692
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