Close Loop System Identification of Dynamic Parameters of an Airplane. Michal Chvojka

Transcription

1 Close Loop System Identification of Dynamic Parameters of an Airplane Michal Chvojka October 2004

2 Chapter 1 Closed Loop System Identification Identifiability of Closed Loop Systems Sometimes it is important to identify systems with a dynamic feedback It may not be possible to remove the system and identify it as an open loop system because of security reasons etc However, there are several ways how to identify closed loop systems I will mention two of them here Indirect method based on additive test signal is the one giving better results Assuming exact knowledge of the controller, the forward system can be derived from the identified model Transfer function of regulator will be and transfer function of a system G R (z) = Q(z) R(z) = q νz ν + + q 1 z + q 0 r µ z µ + + r 1 z + r 0 (11) G S (z) = B(z) A(z) = b m b z mb + + b 1 z + b 0 a ma z ma + + a 1 z + ax 0 z d (12) The system with dynamic feedback is then Gz = The final transfer function can be expressed as G R(z)G S (z) 1 + G R (z)g S (z) = Q z B z R z A z + Q z B z (13) G(z) = β r z r + + β 1 z a k z k + a k 1 z k a 0 (14) The identification will yield estimatation of coefficients of the closed loop system β i and α i The order of the polynomial in the numerator in (14) is r = ν + m b + d (15) and in the denominator k = max[µ + m a, ν + m b + d] (16)

3 IDENTIFIABILITY OF CLOSED LOOP SYSTEMS 3 To find the coefficients of the open-loop system we just need to compare coefficients of transfer functions (14) and (13) The system of linear equations can be written into matrice q ν q ν 1 q ν q ν q ν 1 q q ν q ν q ν q ν 1 0 r µ r mu 1 r 0 q ν q ν 1 q q ν b 1 b 2 b ν+1 b nu+2 b mb a 1 a 2 a µ+1 a ma = β 1 β 2 β ν+1 β ν+2 β mb β ν+mb α 1 r 1 α 2 r 2 α µ+1 α µ+ma (17) We will get the sought coefficients by solving the linear system of equations (17) The solution can be written as a vector of parameters ˆθ = [ ˆ b mb ˆb 1 ˆ a ma â 1 ] T (18) Direct method with noise source at the output is the other one to identify coefficients of forward system from closed loop system The input value W(z) will be zero and the system will be excited by the disturbance Z(z), see figure (11) Figure 11: The arrangement of experiment for direct identification The order of polynoms of controller must meet requirements max[ µ + m a, ν + m b ] > m a + m b (19)

4 SOLUTION IN MATLAB 4 max[ µ m b, ν m a ] > 0 (110) Keeping these requirements will cancel the correlation between error signal z(t) and signal going into the system u(t) The transfer function of the closed loop system can be expressed as which can be rewritten as Y (z) Z(z) = G p 1 + G R G S = D(z)R(z) A(z)R(z) + B(z)Q(z) (111) A(z)R(z)Y (z) + B(z)Q(z)Y (z) = D(z)R(z)Z(z) (112) For desired value w = 0 is the control deflection e(t) = y(t) The transfer function of the regulator will be then U(z) Y (z) = Q(z) (113) R(z) Inserting (113) into (112) we will get A(z)R(z)Y (z) B(z)R(z)U(z) = D(z)R(z)Z(z) (114) and dividing by R(z) will yield model ARMAX for open-loop system From this we can see that is is possible to estimate parameters of the model directly with disturbance at the output and W (z) = 0 Solution in Matlab Longitudinal Motion For handling qualities prediction, the military standard places requirements on the short period damping ration (ζ sp ), and the product of the short period natural frequency and inverse of the high frequency pitch attitude zero (ω sp T θ2 ), which can be obtained from the pitch rate and normal acceleration LOES model q de = K θ ( ) s + 1 T θ2 s 2 + 2ζ sp ω sp s + ω sp 2 Inserting the parameters of our model will yield Discretization of the model eg c2d(sys sp,005, zoh ) q 1239(s ) = de s s (115) and G S (z) = z z z G R (z) = 01z 005 z 1 The experiment arrangement is on figure (B1) The settings of the generator are Ts = 01, Seed = [23341] Np = 25 The identification is not depended on magnitude of

5 SOLUTION IN MATLAB 5 the input (Np can be varied), and on the dynamic of input Damping ratio in the transfer function of dynamic input can be varied as well The following program creates matrice (17) and computes parameters (18) of the system G s Comparing the results to the model in figure (B1) we will learn that this method provides satisfactory results th = arx([y,u],[3 3 1 ]); q = [ ]; r =[1-1]; [dz nz] = th2poly(th) % dz = [1 a1 a2 a3], dz1 = [a1 a2 a3] for i = 2: length(dz), dz1(i-1) = dz(i); end % nz = [0 b1 b2 b3], nz1 = [b1 b2 b3] for i = 2: length(nz), nz1(i-1) = nz(i); end % α 1 r 1 dz1(1) = dz1(1) - r(2); [res] = mysolve(q,r,dz1,nz1,2,2) end where function mysolve is function [res] = mysolve(q,r,alpha,beta,sa,sb,k) sbeta = length(beta); salpha = length(alpha); a = hank(q,sb,sbeta); c = hank(r,sa,salpha); b = zeros(size(c)); X = [a b k*a c] Y = [beta alpha]; Y = Y ; res = X\Y; Matrice X corresponds to (17) and the submatrices a, c are created by function hank 1 function [X] = hank(q,m1,n); while size(q) < n, q = [q 0]; end for j = 1:n, for i = 1:m1, if (j-i+1) > 0, X(j,i) = q(j-i+1); end 1 Type help hank, help mysolve in Matlab

6 SOLUTION IN MATLAB 6 The estimated coefficients are ˆθ = [ ] T The agreement of the estimated parameters and original ones is obvious Let s consider 4 th order system with phugoid poles The transfer function is G S (z) = z z z z z z z and controller G R (z) = z z z + 01 Using the same procedure we will get a result Gˆ S (z) = 88305e 05z z z z z z z We can see the result is very closed to the original model only zeros differ The reason is that the phugoid poles of the original model are very closed to the right plane and it is difficult to stabilize the system As mentioned in chapter three, the results depend on the input dynamic In the other case we will assume zero input which can imagine zero deflection of elevator and disturbance at the output The experiment arrangement is on figure (B2) The accuracy of identification depends very on the structure of the controller and on the sample time of the error source, which is band limited white noise in our case The settings of the block is Np = 05, Ts = 02, Seed = The procedure is as following: th = armax([y1 u1],[ ]); compare([y1 u1],th) present(th) [dz nz] = th2poly(th); printsys(nz,dz, z ) The result differs significantly from the original model Gˆ 00276z S (z) = z z Comparison of the measured and estimated model for both methods is in appendix (C) Lateral Motion As mentioned above, for the lateral modes, the military standard requires only 3 parameters to be estimated: the roll-mode time constant (T R ), the Dutch roll damping ratio (ζ d ), and the Dutch roll natural frequency (ω d ) These parameters can be obtained eg from β da = A β ( ) ( ) s + 1 T β1 s + 1 T β2 ( s + 1 T S ) ( s + 1 T R ) (s 2 + 2ζ d ω d s + ω 2 d ) (116)

7 SUMMARY 7 Though this would yield all of the parameters required for handling qualities prediction, the model contain a substantial number of parameters which would be difficult to estimate accurately Our main interest is the denominator and this is always easier to identify than zeros Evaluating (116) will yield The controller is G S (z) = 1892e 05z e 05z e 05z 1785e 05 z z z z G R (z) = z 008 z z + 01 The result of the identification does not depend on any settings of the input generator and it is nearly the same as the original model apart from numerator which is not important for predicting the handling qualities Using the same procedure, we will obtain Gˆ S (z) = 31925e 07z e 05z e 05z e 05 z z z z All the results for both longitudinal and lateral motion are in appendix (A) Summary Two methods, indirect and direct were tried Indirect method came out to be very accurate and resistant to any change of the input signal from its dynamic The estimated parameters were nearly the same as those of the original model The best results were obtained when the input signal was brought to the elevator in case of longitudinal motion and angle of attack was recorded In lateral motion, the best result was obtained for relation rudder - roll rate The hankel matrice was used to compute the system from the close loop

8 Bibliography [1] Lennart Ljung - System Identification Toolbox - User s Guide, 2002 [2] Lennart Ljung - System Identification - Theory for the user, 1999 [3] Timothy J Curry - Estimation of the Handling Qualities of an Airplane from Flight Test Data, 2000 [4] Petr Noskievič - Modelování a Identifikace Systémů, 1999 [5] MIL-STD-1797B Military Specifications - Flying Qualities of Piloted Airplanes, January 1990

9 Appendix A Results of Close Loop Identification Input Signals Figure A1: Generator of input signal Settings: Sample time = 01s, Power = 1, Seed = [23341] F ilter1 = s e 005s s s + 21 F ilter2 = s e 005s s s + 21 F ilter3 = s e 005s s s + 21

10 INPUT SIGNALS 10 1st input signal nd input signal rd input signal

11 LONGITUDINAL MOTION 11 Longitudinal Motion Input : elevator - de Output : angle of attack - α Controller: k = G R (z) = 21793e 06 z 2 194z System transfer function: G S (z) = 1326z z z 1301 z z z z Identified system: Gˆ S (z) = 13257z z z z z z z From: elevator To: angle of attack Time (sec) Figure A2: Step Response Table A1: Parameter Values and Flying Qualities Predictions ζ SP ω SP T θ2 Category Level Reason for Not Level B 2 ζ SP < 03

12 LATERAL MOTION 12 Lateral Motion Input : rudder - dr Output : roll rate - p Controller: k = G R (z) = (z2 2222z ) (z z ) System transfer function: G S (z) = z z z z z z z Identified system: Gˆ S (z) = z z z z z z z From: rudder To: roll rate Time (sec) Figure A3: Step Response Table A2: Parameter Values and Flying Qualities Predictions ζ d ζ d ω d T R Category Level Reason for Not Level B 2 T R > 14

13 Appendix B Simulink models Figure B1: Closed loop system - indirect method Figure B2: Closed loop system - direct method

14 Appendix C List of Files for Matlab 60 Matlab Files hank ident fb1 ident fb2 init fb1 init fb2 mysolve setzero th th m th oe thd zero ident mdf1mdl mdf2mdl Compute hankel matrice Identification of longitudinal motion from closed loop Identification of lateral motion from closed loop Initialization of Boeing , longitudinal motion Initialization of Boeing , lateral motion Indirect method of parameter estimation from closed loop Set neglectible values to zero Identify system by ARX method Identify system by ARMAX method Identify system by Output Error method Identify the high frequency pitch attitude zero Display high frequency pitch attitude zeros for all experiments Experiments in closed loop system for longitudinal motion Experiments in closed loop system for lateral motion