Predicting the Effects of Unmodeled Dynamics on an Aircraft Flight Control System Design Using Eigenspace Assignment

Transcription

1 NASA Technical Memorandum 58 Predicting the Effects of Unmodeled Dynamics on an Aircraft Flight Control System Design Using Eigenspace Assignment Eric N. Johnson, John B. Davidson, and Patrick C. Murphy June 99

2 NASA Technical Memorandum 58 Predicting the Effects of Unmodeled Dynamics on an Aircraft Flight Control System Design Using Eigenspace Assignment Eric N. Johnson The George Washington University Washington, D.C. John B. Davidson and Patrick C. Murphy Langley Research Center Hampton, Virginia National Aeronautics and Space Administration Langley Research Center Hampton, Virginia 38- June 99

3 Abstract When using eigenspace assignment to design an aircraft ight control system, one must rst develop a model of the plant. Certain questions arise when creating this model as to which dynamics of the plant need to be included in the model and which dynamics can be left out or approximated. The answers to these questions are important because a poor choice can lead to closed-loop dynamics that are unpredicted by the design model. To alleviate this problem, a method has been developed for predicting the eect of not including certain dynamics in the design model on the nal closed-loop eigenspace. This development provides insight as to which characteristics of unmodeled dynamics will ultimately aect the closed-loop rigid-body dynamics. What results from this insight is a guide for eigenstructure control law designers to aid them in determining which dynamics need or do not need to be included and a new way to include these dynamics in the ight control system design model to achieve a required accuracy in the closed-loop rigid-body dynamics. The method is illustrated for a lateraldirectional ight control system design using eigenspace assignment for the NASA High Alpha Research Vehicle (HARV). Introduction Fidelity of the design model is a chief concern in any control law design process. In this context, delity of the design model corresponds to how well a control law, which is designed using this model, achieves design objectives when applied to the actual system. A delity issue was raised in the design of a research lateral-directional control law (ref. ) for NASA's F/A-8 High Alpha Research Vehicle (HARV). The control law was synthesized with the CRAFT (control power, robustness, agility, and ying-qualities trade-os) design methodology (ref. ), which is a graphical method that uses eigenspace placement methods (refs. 3 and ). Only rigid-body dynamics were considered; other dynamics, such as actuators, were neglected. It was noted, however, that when the other dynamics were included, the closed-loop rigid-pole locations varied from those predicted by the low-order design model. In this report, a method is developed to determine which dynamics need to be included in a control law synthesis procedure that uses eigenspace assignment. Elements of the aircraft rigid-body eigenspace are well understood, and much is known about desirable dynamic characteristics (ref. 5). However, the aircraft has dynamics besides those of the rigid body (e.g., actuators, control system lters, and transport delays). Exactly which dynamics will signicantly aect the design is normally not known at the outset of the control law design process. A controller can be synthesized using a system with dynamics beyond those of the rigid body, but this occurs at a cost. First, the relationship between the desired eigenspace and the dynamics of the closed-loop aircraft becomes less obvious. Second, the speed at which a given set of feedback gains can be generated for an iteration of the desired eigenspace is reduced. This reduction results in a trade-o between the simplicity and speed of a ight control system design iteration and the accuracy of the nal design. The fundamental question is: what error will result if certain dynamics are neglected? The answer to this question will provide insight into the relationship between given unmodeled dynamics and their eect on the rigid-body dynamics of the full-order, closed-loop system. To provide a clear exposition of the key results, the report is organized as follows. The HARV control law design is presented in detail as the motivator of this research. The rigid-body system used to synthesize the controller and the unmodeled higher order dynamics are described. Next, a single-input, singleoutput (SISO) example of the eect of unmodeled dynamics is presented. This example provides the conceptual basis for the multiple-input, multiple-output (MIMO) work presented. Symbols A B stability matrix control matrix

4 E eigenspace transformation matrix angle of attack, deg e number of eectors sideslip angle, rad G feedback gain matrix ight path angle, rad g gravitational acceleration, g units, g =3: ft/sec control eector deection angle, deg damping ratio I K k k FB L v;p;r;() l M m N N v;p;r;() n n y p Q R r s T TF(s) t d u V V T v identity matrix feed-forward gain matrix reduction factor feedback gain roll moment dimensional stability and control derivatives number of measurements measurement matrix number of controls feed-through matrix yaw moment dimensional stability and control derivatives number of states of low-order system lateral acceleration, g units roll angular rate, rad/sec feed-through term eect on augmentation matrix real number set yaw angular rate, rad/sec Laplace transform variable time constant matrix transfer function transport delay time, sec control vector eigenvector matrix airspeed, ft/sec higher order states 3 eigenvalue matrix eigenvalue or pole s eigenvector time constant bank angle, deg! natural frequency, rad/sec Subscripts: ail aileron as asymmetric stabilator b body-axis measurement c commanded comp computed den denominator e associated with lag in eectors F unmodeled lters in parallel f unmodeled lter FB feedback FO full-order, closed-loop system HO unmodeled, higher order dynamics i index l associated with lag in measurement LO low-order, closed-loop system m associated with lag in control num numerator pilot pilot commanded rb rigid-body pole () rb rigid-body dynamics of ( ) RO roll-o lter v s lateral velocity, ft/sec rtv roll thrust vectoring w unmodeled lter output rud rudder x state vector of rigid-body system s state Y v;p;() side force dimensional stability and control derivatives sens slow sensed value increased time constant matrix z measurement vector used for feedback ytv yaw thrust vectoring

5 Superscripts: T transpose. time derivative b predicted * conjugate transpose Abbreviations: CRAFT DEA HARV HATP MIMO SISO Background control power, robustness, agility, and ying-qualities trade-os direct eigenspace assignment High Alpha Research Vehicle High-Angle-of-Attack Technology Program multiple-input/multiple-output single-input/single-output HARV Description The analysis presented is motivated by a research lateral-directional ight control system design for the High Alpha Research Vehicle (HARV), which is shown in gure. The HARV is part of the NASA High-Angle-of-Attack Technology Program (HATP), and it will provide ight validation of HATP research and technology. The HARV is a preproduction F/A-8 that has been modied with a thrust vectoring system, as shown in gure. The thrust vectoring is designed to provide additional control moments for high angle-of-attack ight. The HARV has a research ight control system designed to simplify the installation and the modication of control laws. One intent is to provide ight validation of experimental high angle-of-attack control systems. Low-Order Aircraft Model The research lateral-directional ight control system is designed using linear models of the aircraft at various ight conditions. For these models, the rigidbody dynamics are fourth order. The states include lateral velocity, roll rate, yaw rate, and bank angle. The control eectors are aileron, rudder, asymmetric stabilator, yaw thrust vectoring, and roll thrust vectoring. The measurements are roll rate, yaw rate, lateral acceleration, and computed sideslip rate. The lateral acceleration sensor is located near the pilot station, thus preventing the similarity with sensed sideslip rate that would occur if it were at the aircraft center of gravity. The low-order, open-loop aircraft model can be written as _x = Ax + Bu z = Mx + Nu The state, measurement, and control eector vectors are x T =(v s ;p s ;r s ; s ) z T =(p b ;r b ;n ysens ; _ comp ) u T =( ail ; rud ; as ; ytv ; rtv ) The measurement equation z is dened by M and N matrices to distinguish it from the traditional output equation. These measurements are assumed to have no noise. The elements of stability, control, measurement, and feed-through matrices, at a single ight condition, are dened as A = B = M = N = Y v Y p V T g cos L v L p L r N v N p N r tan 3 5 Y ail Y rud Y as Y ytv Y rtv L ail L rud L as L ytv L rtv N ail N rud N as N ytv N rtv cos sin sin cos n yv n yp n yr _ v p _ r n yail n yrud n yas n yytv n yrtv _ ail _ rud _ as _ ytv _ rtv 3 Flight conditions range from angles of attack of.5 to at a constant altitude of 5 ft in unaccelerated ight. For this HARV control law design, the states chosen above lead to classically dened spiral, roll, and Dutch roll modes (at low angle of attack). The models used for all examples presented in this work are listed in appendix A. Low-Order, Closed-Loop System The components of the ight control system corresponding to the low-order, closed-loop system, shown in gure 3, consist of feed-forward and feedback 5 3

6 - - Figure. High Alpha Research Vehicle (HARV). L Figure. HARV thrust-vectoring system. L-9-3

7 u pilot + + u c Feed-forward gains u FB K u Feedback gains G Low-order plant. x = Ax + Bu z = Mx + Nu z with n states, m controls, l measurements, and e effectors, thus making x R n, u c R m, z R l, and u R e. To derive an expression for the closed-loop system, equations () and (5) can be substituted into equation (3) to get u = KGz + Ku pilot Equation () is then used for z such that u = KGMx + KGNu + Ku pilot Figure 3. Low-order, closed-loop system. gains. The feed-forward gain matrix maps two inputs into the ve aircraft control eectors. These two inputs are commanded roll and yaw angular accelerations such that u T c =(_p c;_r c ) The feed-forward gain matrix is a Jacobian of a control mapping algorithm, which is discussed in reference. For the purposes of the research presented, it is assumed that the feed-forward gain matrix is given. The feedback gains will be used to place closed-loop dynamics. Here, the commanded angular acceleration is referred to as the controls, and the ve inputs to the aircraft are called eectors. The controls then become the sum of pilot input and feedback. The feedback gain matrix maps the four measurements into these two controls. The control system synthesis technique used to generate the lateral-directional ight control system was the CRAFT approach based on the direct eigenspace assignment (DEA) of references, 3, and. The CRAFT process provides a graphical approach to trade agility, robustness, ying qualities, and control power. This approach utilizes DEA to achieve the desired dynamics selected using the CRAFT technique. The DEA generates linear measurement feedback gains as a function of the design model and a desired closed-loop eigenspace. Here, the design model contains only the rigid-body aircraft dynamics. The design model and control law can be expressed as _x = Ax + Bu (System dynamics) () z = Mx + Nu (System measurements) () u = Ku c (Feed-forward control) (3) u c = u FB + u pilot (Feed-forward control) () u FB =Gz (Feedback control) (5) Solving for u, u =(IKGN) KGMx +(IKGN) Ku pilot By using this equation for u in equation (), the closed-loop system can be stated as or in shorthand as _x =[A+B(IKGN) KGM]x + B(I KGN) Ku pilot _x = A LO x + B LO u pilot As previously discussed, the feed-forward gains de- ne the eector blending used, and the feedback gains are synthesized to achieve a desired closed-loop eigenspace. For the system described, the feedback gain matrix will place l eigenvalues. The DEA will also exactly place m elements of the l corresponding eigenvectors, or alternatively, it will achieve a leastsquares t of i elements of an individual eigenvector, where m<in. For the purposes of feedback gain synthesis, there are four states, four measurements, ve eectors, and two controls. This implies exact placement of the poles for the low-order, closed-loop system. Also, two eigenvector elements can be exactly placed for each mode. The feedback gains are designed to be scheduled with angle of attack. The feed-forward gains are scheduled primarily with angle of attack, dynamic pressure, and thrust. Appendix B contains baseline feed-forward and feedback gain matrices. Full-Order, Closed-Loop System The complete aircraft plus control system has dynamics beyond that of the system previously described. This includes other aircraft dynamics, such as actuator dynamics, and additional elements of the control system, such as structural notch lters. The layout of this control system and plant model is shown in gure. 5

8 Rolloff Feed-forward Low-order u pilot u c gains plant u z HO HO LO HO LO + + Second-order Actuator filters models Feedback Second-order gains filters LO HO Figure. Layout of HARV lateral-directional control system and plant model which dierentiates low-order and higher order, unmodeled dynamics. Models of the actuator dynamics are available in reference. These actuator models range from rst order to eighth order for each of the ve actuators. Various ight control system lters are also not included in the design process. These include rstorder, roll-o lters of 5 rad/sec, which are placed on each of the controls as part of the ight control system design. A notch lter is also placed on each of the controls as well as on each of the measurements. An exception is the sensed lateral acceleration channel that has two notch lters in series. Here, the HARV full-order, closed-loop system will have 5 states. In the following development, however, not all unmodeled dynamics will be initially considered. The term full-order, closed-loop system will be used to describe a closed-loop system consisting of the open-loop, rigid-body dynamics, the feed-forward and feedback gains, and the particular unmodeled dynamics under consideration. Spectral Decomposition To evaluate full-order, closed-loop system dynamics, consider the portion of the eigenspace that corresponds to the rigid-body system. When the unmodeled dynamics are of suciently higher frequency than the low-order, closed-loop system, or they are outside the bandwidth of the pilot, the dynamics that are dominant to the pilot will be determined by the rigid-body eigenspace. The rigid-body eigenspace is a combination of the rigid-body eigenvalues and eigenvector elements that are associated with the rigid-body modes. As an example, the rigid-body roll pole of the low-order, closed-loop system and of the 5-state system is shown in gure 5 as a function of angle of attack. Roll mode pole, rad/sec Low-order system roll mode Full-order system roll mode 5 Figure 5. Low-ordersystem and full-order, closed-loop system roll pole. The full-order, closed-loop system has dynamics dened by _x = A FO x + B FO u pilot The rigid-body eigenspace is characterized here by a spectral decomposition of A FO such that A FO V = V3 The eigenvalue and eigenvector matrices are then partitioned to separate the eigenspace of interest so that V V V = V V 3 3 = 3 where 3 is a square matrix containing achieved rigid-body eigenvalues on its diagonal and V is a square matrix containing achieved rigid-body eigenvector elements associated with rigid-body states. Ideally, one would like to compare the eigenspace of the low-order, closed-loop system with the eigenspace of (A FO ) rb V 3 V (i.e., the rigid-body eigenspace of the full-order, closed-loop system). This matrix has the same dimensions as the low-order, closed-loop system matrix A LO.

9 Although this reduced-order system is not a substitute for a look at frequency response and time histories of the full-order system, it does allow the control law designer to compare the dynamics of the full-order, closed-loop system with well-understood aircraft rigid-body dynamics. Also, knowledge of the achieved rigid-body eigenspace is important because its relationship with the desired dynamics fundamentally determines whether or not certain dynamics can be omitted from the design model. u pilot + + Actuator model τ ail s + Feedback gains k FB δ ail Low-order plant L δail s - L p p Eigenspace Transformation Matrix A concept that will appear repeatedly in this work is that of a single matrix transformation in the form A LO = E(A FO ) rb where A LO is the low-order, closed-loop system matrix and (A FO ) rb is composed of the rigid-body portion of the full-order, closed-loop system eigenspace, as described in the section entitled \Spectral Decomposition." The term E will be referred to as an eigenspace transformation matrix. Although A LO and (A FO ) rb are not actually the eigenspaces, their eigenvalues and eigenvectors are the rigid-body eigenspace of the low- and full-order closed-loop systems, respectively. All three of these matrices are square, and they have dimensions equal to the number of states in the rigid-body system. As the unmodeled dynamics become less signicant to the achieved rigid-body eigenspace, the eigenspace transformation matrix should approach the identity matrix. A look at the most signicant contributors to the eigenspace transformation matrix may provide insight into what causes the rigid-body dynamics to dier from those of the low-order, closed-loop system. If the eect on the rigid-body eigenspace can be easily calculated, such a calculation could be useful in determining whether certain dynamics can be left out of the design model. SISO Example To facilitate the discussion of the multiple-input / multiple-output (MIMO) results in this report, a single-input/single-output (SISO) example is now presented. The SISO system (g. ) presented is a rst-order roll mode approximation of the HARV at an angle of attack of 5. Aileron deection is the plant input, and roll rate is the plant output. The rst-order lag aileron actuator model is the unmodeled dynamics in this example. Figure. SISO example with rst-order roll mode approximation at angle of attack of 5. For the purposes of control law design, the rstorder lag aileron actuator is not included in the design model. This SISO low-order, closed-loop design model can be written as p(s) u pilot (s) = L ail s L p L ail k FB () With the actuator model, this full-order, closed-loop system has two states, and it can be written as p(s) u pilot (s) = L ail ( ail s + )(s L p ) L ail k FB () In state space, this full-order, closed-loop system is expressed as _p _ ail = " L p L ail k ail FB ail # p ail + u pilot ail The low-order, closed-loop system has one state, with the eigenvalue LO = L p + L ail k FB (8) which is the desired roll mode pole. The full-order, closed-loop stability matrix is A FO = " L p L ail k ail FB ail Of interest are the eigenvalues and eigenvectors of this closed-loop system corresponding to the rigidbody dynamics, or roll mode. The eigenvalue is FOrb and the eigenvector is when this full-order, closed-loop system matrix is written as the spectral decomposition Lp L ail = FO rb k ail FB HO ail #

10 Note that FOrb and are scalars and that, in the absence of, is arbitrary. In this development, only the upper left and lower left partitions of the above matrix equation are needed such that L p + L ail = FOrb (9) ail k FB ail = FOrb () Now, the goal is to eliminate and solve for LO in terms of FOrb. The method that proves successful when working with the MIMO system is the following. Equations (9) and () are rewritten as L p FOrb = L ail () k FB = + ail FOrb () Equation () is expanded into an innite series such that each term contains the right-hand side of equation (). To start this expansion into an innite series, add and subtract L p ail FOrb and group terms to give L p FOrb = L ail ( + ail FOrb ) + L ail ail FOrb Continuing in a similar manner by adding and subtracting () i L ail i ail i FO rb (i =;3;:::;) One can now relate the low-order (eq. (8)) and fullorder, closed-loop rigid-body pole as LO = FOrb + L ail ail k FB FOrb L ail ail k FB FO rb + L ail 3 ail k FB 3 FO rb ::: This is an innite series that converges as long as j ail FOrb j < In the case where the actuator has a much lower time constant than the rigid-body mode j ail FOrb j, the rst two terms constitute a good approximation to the innite series such that LO +L ail ail k FB FOrb or FOrb b FOrb +L ail ail k FB LO where b FOrb denotes an approximation of FOrb. A plot of this approximation is shown in gure. Note that b FOrb approximates FOrb up to an actuator time constant of approximately. sec, which is three times slower than the actual actuator time constant..5 -λ - LO and grouping terms to get the right-hand side of equation () in each term yields the innite series L p FOrb = L ail ( + ail FOrb ) + L ail ail ( + ail FOrb ) FOrb L ail ail ( + ail FOrb ) FO rb + L ail 3 ail ( + ail FOrb ) 3 FO rb ::: Then eliminate by using equation () so that Time constant, sec λ - FOrb Closed-loop actuator pole Open-loop actuator time constant, sec -λ - FOrb Complex pair formed L p FOrb = L ail k FB + L ail ail k FB FOrb L ail ail k FB FO rb Figure. Eect of rst-order actuator model on SISO system and approximation. + L ail 3 ail k FB 3 FO rb ::: Dividing out the now arbitrary, the above becomes Lp FOrb = L ail k FB +L ail ail k FB FOrb L ail ail k FB FO rb + L ail 3 ail k FB 3 FO rb ::: Because of the divergence of LO and, a FO rb point will exist where the open-loop actuator time constant is high enough that it becomes necessary to include this actuator model in the design process. This point will be determined by the required level of accuracy in the nal design. A point will also exist where the rigid-body roll mode and the actuator 8

11 mode couple together to form a single oscillatory mode. Approximating the Transformation First-Order Actuators In this section, an approximation of the eigenspace transformation is derived and tested for unmodeled dynamics placed at the input of the rigidbody aircraft. The eigenspace transformation relates the rigid-body eigenspace of the low-order, closedloop system used for the design with the full-order, closed-loop system. Here, the SISO results shown in the section entitled \SISO Example" are generalized for rst-order lag actuator dynamics applied to the MIMO design. Without feed-through term. Consider the following system, shown in gure 8, which represents the HARV rigid-body system and the rst-order lag actuator models: _x = Ax + Bv z = Mx T e _v = v + u u = KGz + Ku pilot (System dynamics) (System measurements) (Actuator dynamics) (Control law) Low-order Feed-forward Actuators system gains u pilot u v A, B, z K + τ i s + M + Feedback gains G (3) Figure 8. System block diagram with rst-order lag actuators as unmodeled dynamics. The measurement feed-through term is omitted from this derivation for clarity. The actuator matrix is diagonal and contains the time constant associated with each actuator: T e... e This formulation corresponds to actuator dynamics with the transfer functions v i(s) u i(s) = i s (i=;;:::;e) This full-order, closed-loop system now has n + e states with dynamics expressed as _x _v = z =[M ] A B T e KGM T e x v x + v T e K u pilot The closed-loop stability matrix for the low-order system is A LO = A + BKGM and the closed-loop stability matrix for this full-order system is A FO = A B T e KGM T e Of interest are the eigenvectors and eigenvalues of this closed-loop system corresponding to the rigidbody dynamics. These correspond to V and 3 when this full-order system matrix is rewritten as the spectral decomposition, A B T e KGM T e V V V V 3 = V V V V 3 () Here, we need only the upper left and lower left partitions of the above matrix equation, or or equivalently, AV + BV = V 3 (5) T e KGMV T e V = V 3 () V 3 AV = BV () KGMV = V + T e V 3 (8) As an intermediate step to obtain the eigenspace transformation approximation, equation () is expanded as an innite series. To generate this series, add and subtract BT e V 3 and group terms corresponding to the right-hand side of equation (8). This becomes V 3 AV = B(V + T e V 3 ) BT e V 3 Then add and subtract BT ev 3 before such that and group as V 3 AV = B(V + T e V 3 ) BT e (V + T e V 3 )3 + BT ev 3 9

12 where A AA, A 3 AAA;::: Continuing to add and subtract BT i ev 3 i (i =3;;:::;) and following the same grouping strategy yields the following innite series: V 3 AV = B(V + T e V 3 ) BT e (V + T e V 3 )3 + BT e (V + T e V 3 )3 BT 3 e (V + T e V 3 )3 3 + ::: Equation (8) can now be substituted into each term on the right to get V 3 AV = BKGMV BT e KGMV 3 + BT ekgmv 3 ::: (9) Postmultiply equation (9) by V. The innite series that results relates the rigid-body eigenspace of the low-order, closed-loop system with that of the full-order, closed-loop system in which where A LO =(A FO ) rb + BT e KGM(A FO ) rb BT ekgm(a FO ) rb + BT 3 ekgm(a FO ) 3 rb ::: () (A FO ) rb = V 3 V as dened in the section entitled \Spectral Decomposition." As the amount of augmentation increases (K and G), the eect of the unmodeled dynamics on the rigid-body dynamics is increased. As the actuators get faster (i.e., as the elements of T e approach ) the eect on the closed-loop eigenspace is decreased. The eigenspace of the low-order, closed-loop system and the rigid-body eigenspace of the full-order, closedloop system become equal. The convergence of equation (9) is equivalent to the convergence of X i= () i T i ekgmv 3 i This series can be bounded using the matrix -norm and the Schwarz inequality (ref. 8) such that kt i ekgmv 3 i kkt ek i kkgmv kk3 k i (i =;;:::;) The matrix -norm of A equals max (AA 3 ) =. By virtue of being eigenvectors, V can be multiplied by an arbitrary scalar. This factor is chosen such that the matrix norm of KGMV is unity, so that kt i ekgmv 3 i k(kt e kk3 k) i Convergence of equation (9) is therefore guaranteed when kt e kk3 k < Feed-through term. For many systems, the loworder model of interest may have a feed-through or N matrix term. This is the case for the HARV control law design. Here, the previous analysis and the development are extended to obtain the eigenspace transformation matrix for this type system. To begin, consider the system that represents an aircraft with rst-order lag actuator models in which _x = Ax + Bv z = Mx + Nv T e _v = v + u u = KGz + Ku pilot (System dynamics) (System measurements) (Actuator dynamics) (Control law) The system has n + e states, with the closed-loop dynamics _x _v = A B T e KGM T e KGN T e + Te K z =[M N] x v u pilot A B Te KGM Te KGN Te = 3 x v V V The spectral decomposition of the system matrix is V V V V 3 V V where 3 and V correspond to the rigid-body eigenspace. As before, only the upper left and lower left partitions of this equation are used, or where V 3 AV = BV () KGMV = Q e V + T e V 3 () Q e I KGN

13 This is a useful denition for the development to follow, since from the section entitled \Low-Order, Closed-Loop System" A LO = A + BQ e KGM As in the previous section, equation () is expanded as an innite series with terms grouped to match the right-hand side of equation (). The series is generated as before: add and subtract BQ e T e V 3 to equation (), so that -norm of terms in infinite series Second term Third term Fourth term V 3 AV = BQ (Q e ev + T e V 3 ) BQ e T e V 3 Then continue in a similar manner as before and apply equation () as follows: V 3 AV = BKGMV BQ e T e Q e KGMV 3 + BQ e T e Q e T e Q e KGMV 3 ::: Convergence is guaranteed when kq e T e kk3 k <. To obtain the nal form, postmultiply by V, so that A LO =(A FO ) rb + BQ e T e Q e KGM(A FO ) rb BQ e T e Q e T e Q e KGM(A FO ) rb + ::: (3) An important simplication of equation (3) is suggested by multiplying through by (A FO ), so rb that A LO (A FO ) rb = I + BQ e T e Q e KGM BQ e T e Q e T e Q e KGM(A FO ) rb + ::: The -norm of each successive term on the right-hand side, except the identity term, is shown in gure 9. Clearly, the rst two terms constitute a reasonable approximation to the innite series for this example. An advantage of using the rst two terms as an approximation to the series is the following prediction of the rigid-body eigenspace of the full-order, closed-loop system expressed in terms of the loworder, closed-loop system: (A FO ) rb (I + BQ e T e Q e KGM) A LO 3 5 Figure 9. -norm of terms in innite series. This expression leads to the concept of a single matrix (an eigenspace transformation matrix) which transforms the low-order, closed-loop stability matrix to approximate the rigid-body eigenspace of the fullorder, closed-loop system such that (A FO ) rb E A LO () For this full-order, closed-loop system, the eigenspace transformation matrix is simply Application E = I + BQ e T e Q e KGM The results of the previous development are applied to the HARV lateral-directional ight control system design. The validity of the previous analysis in predicting the eect of rst-order actuator models on the rigid-body dynamics is shown. The low-order system was used to generate the feedback gains. Because there are four measurements, the rigid-body poles are exactly placed. Moreover, the low-order, closed-loop eigenspace is the desired eigenspace. In this example, the actuators are modeled as the following transfer functions: TF ail (s)= TF as (s)= TF rtv (s) = s + TF rud (s) = 8 s + TF ytv(s) = 3 s + s s +

14 The approximate eigenspace transformation matrix is then E = I + BQ e Q e KGM (5) The expression E A LO is postulated to be an approximation to the rigid-body eigenspace of the fullorder, closed-loop system (i.e., V 3 V ). The rst check on the validity of the approximation will be on the convergence of the innite series used to derive the eigenspace transformation matrix. The test kq e T e kk3 < is applied using the -norm. At an angle of attack of, the left side of the equation is equal to.3. This value indicates convergence of the series. To examine the range in which the eigenspace transformation matrix is valid, the poles of the actuators are gradually slowed down. This procedure is done by multiplying the actuator poles by a reduction factor k which is varied from to such that where T slow = kt e T slow _v = v + u The error associated with using this eigenspace transformation matrix in predicting the roll mode pole and Dutch roll natural frequency for a ight condition with an angle of attack of is shown in gure. The actuators have been slowed by a factor of 5 when the aileron actuator and the roll mode poles form an oscillatory pair. The unmodeled dynamics will need to be much closer in frequency to the loworder, closed-loop system for there to be a problem with the assumptions made. Figure shows how well E A LO predicts the rigid-body roll mode pole. The low-order characteristics are those of the design model, and they represent placed dynamics. The full-order characteristics are those of the low-order plant plus the actuator dynamics. The predicted characteristics are obtained from equation (), and they are expected to predict the full-order characteristics. The Dutch roll mode pole is shown in gure. Note that the variation of the roll mode and Dutch roll frequencies are captured along with the variation in Dutch roll damping. The Roll mode pole, rad/sec Dutch roll natural frequency, rad/sec Low order Full-order rigid body Predicted rigid body k k Figure. Plots that verify accuracy of eigenspace transformation matrix approximation with angle of attack of. Roll mode pole, rad/sec Low order With actuators Predicted 5 Figure. Roll mode pole prediction with actuator models.

15 Dutch roll damping ratio Dutch roll natural frequency, rad/sec Low order With actuators Predicted 5 φ-to-β ratio in Dutch roll eigenvector β-to-p ratio in roll mode eigenvector, sec Low order With actuators Predicted Figure. Dutch roll mode pole prediction with actuator models. Figure 3. Eigenvector prediction with actuator models. negligible variation of the spiral mode pole, which is not shown, is also captured by the approximation. For all cases, the approximate eigenspace transforma - tion matrix accurately predicts the rigid-body poles of the full-order, closed-loop system. The predicted eect of actuators on the desired eigenvector elements is shown in gure 3. The rst of these elements is the -to- ratio in the Dutch roll mode. The second element is the -to-p ratio in the roll mode. The magnitude of the ratio of each of these two eigenvector elements is shown. These ratios give an indication of how much the roll and Dutch roll modes are coupled. The low-order -to-p ratio in the roll mode is for all cases. The low-order -to- ratio in the Dutch roll mode coincides with that of the full-order, closed-loop system. The -to-p ratio is in seconds, and the -to- ratio is nondimensional. The eigenvectors appear to be aected considerably less than the eigenvalues by unmodeled dynamics. This result will show up repeatedly in the examples that are presented. Again, the approximate eigenspace transformation matrix accurately predicts the fullorder, closed-loop rigid-body eigenspace. Dynamics in Other Locations General Form of Eigenspace Transformation Matrix One would like a general expression for the eigenspace transformation matrix for dynamics at other points, including the input to the plant, in the control system structure. It is possible to construct the system with a rst-order lag at three locations within this system. The eigenspace transformation matrix for a rst-order lag in any or all of these locations will then be readily available from the general form. 3

16 u pilot + + u c Feed-forward gains v m u T m K T e u FB Feedback gains G v l T l v e Low-order system A, B, M, N Figure. System block diagram with rst-order lags in three locations. Consider the system shown in gure, in which z The eigenvector matrix has been partitioned such that the rst-column partition contains n columns, and the second contains e+m+l columns. The rst-, second-, third-, and fourth-row partitions contain n, e, m, and l rows, respectively. Equations in the rstcolumn partition of the matrix equation () are V 3 AV = BV () KV 3 = V + T p V 3 (8) GV = V 3 + T m V 3 3 (9) MV = NV + V + T l V 3 (3) To get the feed-through term in a useful form, a second version of equations (8), (9), and (3) is needed, so that _x = Ax + Bv e T e _v e = v e + u KV 3 = Q e V + T e V 3 + KGNV (3) u = Kv m u FB = Gv l z = Mx + Nv e T m _v m = v m + u c T l _v l = v l + z u c = u FB +u pilot where x R n, v e R e, v m R m, and v l R l. When the equations governing this system are combined, the full-order, closed-loop system becomes 8 >< >: _x _v e _v m _v l 9 >= >; = + A B T e T e K T m T mg T l M T l N T l 8 >< >: T m 9 >= >; u pilot 38 >< 5>: x v e v m v l 9 >= with n+e+m+l states. The spectral decomposition of the system matrix becomes A B T e Te K Tm Tm G T l M T l N T l = V V V V V 3 V 3 V V >; V V V V V 3 V 3 V V 3 5 () where V and 3 are the eigenvector elements and eigenvalues associated with the rigid-body eigenspace of the full-order, closed-loop system, respectively. GV = Q m V 3 + T m V GNKV 3 (3) MV = Q l V + T l V 3 + NKGV NV (33) where Q e I KGN Q m I GNK Q l I NKG As in the previous development, expansion obtained via adding and subtracting needed terms will turn equations () through (3) into an innite series. It is again assumed that the speed of the unmodeled dynamics is faster than that of the loworder, closed-loop system. Hence, each new term of the series will be smaller than its predecessor. To form an analogous approximation to the series, all terms containing two or more time constant matrices and two or more eigenvalue matrices will be dropped. Expand equation () so that equation (3) can be used; thus, V 3 AV = BQ e (Q ev + T e V 3 ) BQ e T e Q e (Q ev + T e V 3 )3 + BQ e T e Q e T e Q e (Q ev + T e V 3 )3 ::: Then apply equation (3) and drop the terms with higher powers of T e and 3, so that V 3 AV BQ e (KV 3 KGNV ) BQ e T e Q e (KV 3 KGNV )3 For the remainder of the development, this equation will be stated as an approximation; all terms with

17 more than a single appearance of T and 3 will be dropped as they occur. Expand the above equation again so that the right-hand side of equation (9) appears as V 3 AV BQ e K(V 3 + T m V 3 3 GNV ) BQ e KT m V 3 3 Then by equation (9), BQ e T e Q e K(V 3 + T m V 3 3 GNV )3 V 3 AV BQ e K(GV GNV ) BQ e KT m V 3 3 BQ e T e Q e K(GV GNV )3 Equation (3) will now be used to eliminate the last V3 term, so that V 3 AV BQ e KG(V NV ) BQ e KT m Q m (Q m V 3 + T m V 3 3 )3 BQ e T e Q e KG(V NV )3 and then V 3 AV BQ e KG(V NV ) BQ e KT m Q m G(V NKV 3 )3 BQ e T e Q e KG(V NV )3 Equation (8) is then used; thus, V 3 AV BQ e KG(V NV ) BQ e KT m Q m G(V NV )3 BQ e T e Q e KG(V NV )3 Expand the above equation again so that the right-hand side of equation (3) will allow V to be eliminated: V 3 AV BQ e KG(V + T l V 3 NV ) BQ e KGT l V 3 BQ e KT m Q m G(V + T l V 3 NV )3 BQ e T e Q e KG(V + T l V 3 NV )3 Then apply equation (3), so that V3 AV BQ e KGMV BQ e KGT l V3 BQ e KT m Q m GMV3 BQ e T e Q e KGMV3 Next expand the above expression so that equation (33) is used and V 3 AV BQ e KGMV BQ e KGT l Q l (V + T l V 3 )3 BQ e KT m Q m GMV 3 BQ e T e Q e KGMV 3 or V 3 AV BQ e KGMV BQ e KGT l Q l (MV NKGV + NV )3 BQ e KT m Q m GMV 3 BQ e T e Q e KGMV 3 Equations (9) and (8) are subsequently applied to eliminate the last V term; thus, V 3 AV BQ e KGMV BQ e KGT l Q l (MV NV + NV )3 BQ e KT m Q m GMV 3 BQ e T e Q e KGMV 3 All unwanted terms are eliminated, so that V 3 AV BQ e KGMV Finally, note that BQ e KGT l Q l MV 3 BQ e KT m Q m GMV 3 BQ e T e Q e KGMV 3 Q e KG =(IKGN) KG= KG(I NKG) = KGQ l Q e K =(IKGN) K = K(I GNK) = KQ m 5

18 by the matrix inversion lemma. The approximation is then converted to the familiar form: V 3 AV BQ e KGMV BKGQ l T l Q l MV 3 BKQm T m Qm GMV 3 BQe T e Qe KGMV 3 Postmultiplying the result by V yields A LO (A FO ) rb + BKGQ l T l Q l M(A FO ) rb where + BKQ m T m Q m GM(A FO ) rb + BQe T e Qe KGM(A FO ) rb A LO = A + BQe KGM (A FO ) rb = V 3 V The eigenspace transformation matrix is now E = I + BKGQ l T l Q l M + BKQ m T m Q m GM + BQ e T e Q e KGM (3) Note that the previous derivation for rst-order lag actuator models is a subset of this result. If the rst-order lag at any one of these points is not present, then the associated time constant is. In addition, within each matrix of time constants, any of the elements can be when that particular signal does not have a lag to be modeled. Two or more rst-order lags in series at a particular location in the system are handled in a simple way. The time constants associated with each channel only need to be summed before applying equation (3). For example, the approximate eigenspace transformation matrix of two rst-order lags in series with time constants of. sec and. sec :s + :s + = :s +:3s + :3s + will be the same as that of a single rst-order lag with a time constant of.3 sec. Note that this substitution will be good over the frequency range of the rigid-body dynamics. Application By using the subset of equation (3) that corresponds to a system with rst-order lag in controls, one obtains where E = I + BKQ m T m Q m GM (35) Q m I GNK A LO = A + BQe KGM These results will be used to predict the eect of the roll-o lters shown in gure. The roll-o lter transfer function is TF RO(s) = 5 s + in each channel. Therefore, for this example, the eigenspace transformation matrix is E = I + BKQ m " # 5 5 Q m GM (3) The purpose of the roll-o lter is to attenuate high-frequency commands. This attenuation should reduce potential problems caused by structural dynamics or noise. Results similar to those of the actuators are expected, since both the actuators and the roll-o lters are rst-order lags. Figure 5 shows the predicted eect of the lters on the roll mode pole. The predicted eect of the lters on the Dutch roll mode pole is shown in gure. Figure shows the predicted eect of the lters on modal coupling. The eect of these unmodeled dynamics is more pronounced because the roll-o lters are slower than the actuators. Again, the approximate eigenspace transformation matrix accurately predicts the rigid-body eigenspace of this full-order, closedloop system. Higher-Order Filters Equivalent First-Order Lag Now that the eigenspace transformation is available to predict the eects that rst-order lags have on the rigid-body eigenspace of the full-order, closedloop system, it is useful to generali ze the transformation to higher order lters. More specically, one would like to handle dynamics such as lead or lag compensators, transport delays, and notch lters. Handling these dynamics is essential when faced with the second-order lters found in the HARV ight control system. A system with unmodeled higher order lters on the controls is shown in gure 8. The feedthrough term has been removed for this analysis.

19 Roll mode pole, rad/sec Low order With roll-off Predicted φ-to-β ratio in Dutch roll eigenvector 5 3 Low order With roll-off Predicted Dutch roll natural frequency, rad/sec Figure 5. Roll mode pole prediction with roll-o lters Low order With roll-off Predicted β-to-p ratio in roll mode eigenvector, sec Figure. Eigenvector prediction with roll-o lters Dutch roll damping ratio u pilot + + Command second-order filters u c A F, B F, M F, N F u FB Feed-forward w gains u K Feedback gains G Low-order system A, B, M z Figure. Dutch roll mode pole prediction with roll-o lters. Figure 8. System block diagram with second-ordercommand lters as unmodeled dynamics.

20 The system can be written as follows: _x = Ax + BKw z = Mx u FB = Gz u c = u FB +u pilot _v = A F v + B F u c w = M F v + N F u c (System dynamics/ feed-forward gains) (System measurements) (Feedback control) (Feed-forward control) (Filter system dynamics) (Filter measurements) where the lters in parallel channels are realized as A F = B F = M F = N F = A f... A f m B f... B f m M f M f m N f... N f m thus giving each higher order lter a transfer function of the form M f i si Af B + N i f (i =;;:::;m) i fi The closed-loop system matrix spectral decomposition for this system is A + BKNF GM = B F GM V V BKM F A F V V V V V V where again 3 and V correspond to the rigidbody eigenspace. When multiplied out, this equation yields four matrix equations. Two of these equations, which correspond to the upper left and lower left partitions, can be written as V 3 (A+BKN F GM)V = BKM F V (3) A B F FGMV = V A V F 3 (38) where it is assumed that the higher order lters have no poles at the origin. Equation (3) is expanded by methods used in previous sections to yield the innite series V 3 AV BKN F GMV = BKM F (V A F V 3 )+BKM F A F (V A F V 3 )3 + BKM F A F (V A F V 3 )3 + ::: Substituting equation (38) yields V 3 AV BKN F GMV = BKM F A F B FGMV BKM F A F B FGMV 3 BKM F A 3 F GMV 3 ::: Postmultiplying by V,weget (A FO ) rb = A + BKN F GM BKM F A F B FGM BKM F A F B FGM(A FO ) rb BKM F A 3 F B FGM(A FO ) rb ::: (39) The type of higher order lters considered is restricted to those with a steady-state gain of through all channels. The steady-state gain is the limit of the lter transfer function as s goes to. This gain of yields the relations M f i A fi B fi + N fi = or M F A B F F + N F = I Using this result to combine the rst three terms on the right-hand side of equation (39), we get A LO =(A FO ) rb + BKM F A B F FGM(A FO ) rb + BKM F A 3B F FGM(A FO ) + ::: () rb The approximate eigenspace transformation matrix of any higher order lter with a steady-state gain of is the same as that of a rst-order lag with an equivalent time constant matrix equal to M F A B F F. Second-order lter. A second-order lter modeled as the transfer function w i (s) u i (s) =! den i s! num i + numi! numi s +! num i s + deni! deni s +! den i! 8

21 has the following system matrices A fi = B fi = "! den i deni! deni! den i! num i M fi =[ ] N fi =! den i! num i " # (i =;;:::;m)! num i! den i numi! numi deni! deni The equivalent time constant is M fi A f i B fi or i = den i num i! deni! numi! # () As an example, consider a notch lter with a denominator damping ratio of., a numerator damping ratio of., and equal natural frequencies in the numerator and the denominator. From equation (), we see that the eect of the lter on the rigid-body eigenspace will be virtually the same as that of a rst-order lag with a break frequency of 83 percent of the natural frequency of the notch lter. Time delay. The rst-order Pade approximation to a transport delay is written as w i (s) u i (s) = t di = s + t di = s The system matrices can be written as A fi = t di B fi = t di M fi = N fi = The appropriate rst-order lag for this transfer function is found to have a time constant equal to the transport delay magnitude such that i = t di Lead/lag compensator. Another type of lter that has been considered is a lead/lag compensator of the form w i (s) u i (s) = s + num i deni s + or in state-space form A fi = deni B fi = den i numi den i M fi = N fi = num i deni The rigid-body eigenspace of the full-order, closedloop system that includes this lter can be accurately predicted using the eigenspace transformation matrix corresponding to a rst-order lag with a time constant of i = deni numi This result implies that such a compensator could be designed to cancel much of the eect of unmodeled dynamics on the rigid-body eigenspace. Application The HARV lateral-directional second-order lters have the transfer function TF(s)=! den! num s + num! num s +! num s + den! den s +! den with characteristics shown in table. The rst two characteristics listed are the command lters ; the remaining ve are measurement notch lters. The two lateral acceleration lters are congured in series. Each of these two lter sets will now be considered. Table. HARV Second-Order Filters Filtered! den,!num, signals rad/sec den rad/sec num _p c.. _r c.. p b r b n y sens _sens For the command lters, equations (3) and () imply that the eigenspace transformation matrix should be E = I + BKQ m :9 :9! Q m GM () Although a feed-through term was not included when deriving the equivalent time constants for higher order lters, it has been postulated that the feedthrough term should be handled in the same way as suggested in previous sections. 9

22 Dutch roll natural frequency, rad/sec Dutch roll damping ratio Roll mode pole, rad/sec Low order With command filter Predicted 5 Figure 9. Roll mode pole prediction with command secondorder lters Low order With command filter Predicted 5 5 Figure. Dutch roll mode pole prediction with command second-order lters. β-to-p ratio in roll mode eigenvector, sec φ-to-β ratio in Dutch roll eigenvector Low order With command filter Predicted 5 5 Figure. Eigenvector prediction with command secondorder lters. From the previous discussion, these second-order command lters, having zeros with frequencies of rad/sec and poles with frequencies of rad/sec, will aect the rigid-body eigenspace of the full-order, closed-loop system in the same manner as a rollo lter with a time constant of.9 sec. The eigenspace results are shown in gures 9,, and. Again, the approximate eigenspace transformation matrix accurately predicts the resulting rigid-body eigenspace. The second-order lters in the measurement loop are notch lters designed to cancel resonant peaks in the structural model. There are two secondorder lters in series for the n ysens channel, so the approximate time constant associated with each will be summed to approximate the eect of both. The eigenspace transformation matrix associated with the

23 Roll mode pole, rad/sec Low order With measurement notch Predicted Figure. Roll mode pole prediction with measurement notch lters. Dutch roll natural frequency, rad/sec Dutch roll damping ratio Low order With measurement notch Predicted 5 5 Figure 3. Dutch roll mode pole prediction with measurement notch lters. β-to-p ratio in roll mode eigenvector, sec φ-to-β ratio in Dutch roll eigenvector Low order With measurement notch Predicted Figure. Eigenvector prediction with measurement notch lters. measurement notch lters is, from equations (3) and (), E = I+BKGQ l 3 :55 :83 :39 :5 5 Q l M (3) The eects of these notch lters are shown in gures, 3, and. The eigenvalues and eigenvectors of E ALO predict very well the resulting rigid-body eigenspace. Only a slight degradation in the prediction of the small -to-p ratio in the roll mode eect is observed. Hence, the approximate rst-order lags are a suitable substitute for the higher order lters in this example. This section has shown how the eects of higher order lters on the closed-loop rigid-body eigenspace can be evaluated using rst-order lags placed at the same loop location. This result is expected as long as

24 the steady-state gains of the unmodeled higher order lters are unity. HARV Full-Order, Closed-Loop System An approach is now proposed to predict the simultaneous eect of all unmodeled dynamics (which is introduced in the section entitled \Full-Order, Closed-Loop System") on the HARV control law design. The strategy is straightforward: approximate the HARV full-order, closed-loop system of gure using the closed-loop system in gure by replacing all unmodeled dynamics with equivalent rstorder lag components. Equation (3), which denes the eigenspace transformation matrix in this simpler case, is then directly applied. The full-order, closed-loop system is placed in the simpler form as follows. Note that the actuators are already modeled by rst-order lags; therefore, nothing needs to be done. Turning to the other unmodeled dynamics, the second-order command and measurement notch lters are replaced by approximate rst-order lags derived in the section entitled \Higher Order Filters." Here, it is postulated that the results of that section, which were only considered lters at a single-loop location, extend to lters at multiple-loop locations. Next, the lters in series (i.e., the rst-order roll-o and approximate command lters) are replaced by single rst-order lags. The time constant of the approximate lag equals the sum of the time constants of the replaced lags; this leads to a system of the form shown in gure with rst-order lag elements at three separate loop locations. Equation (3) can then be used to obtain the eigenspace transformation matrix, where (from eq. (5)), T e = :8 :5 :333 :8 :8 From equations (3) and (), : :9 Tm = + : :9 From equation (3), T l = 3 5 :59 = :59 :55 :83 :39 :5 Figure 5 shows the prediction of the roll mode pole. Figure shows the Dutch roll mode pole. 3 5 Roll mode pole, rad/sec Dutch roll natural frequency, rad/sec Dutch roll damping ratio Low order Full order Predicted 5 Figure 5. Roll mode pole prediction with full-order model Low order Full order Predicted 5 5 Figure. Dutch roll mode pole prediction with full-order model.