Contribution to Modeling a Robust Controller for a Flexible Aircraft

Contribution to Modeling a Robust Controller for a Flexible Aircraft

Ranarison Solofo Herizo, Randriamitantsoa Paul Auguste, Randriamitantsoa Andry Auguste,

Reziky Zafimarina S.H.Z.T

Abstract– This study uses the Hamilton method to derive the equation of motion of a flexible aircraft. The purpose is to design a robust controller using the and the loo-shaping methods. The method is used to analyze the system and the

gap, to compare the controllers.

Index Terms–Flexible aircraft, algorithm, loop-shaping,

-analysis, gap.

INTRODUCTION

Modeling the motion of a flexible aircraft is more complex than the case of a rigid aircraft. The flexibility of the structure increases the number of the parameters of the equations of motion. The components of the state vector are intensified, and it is complicating the design of an efficient controller for the system. In this article, only the longitudinal flight of a flexible airplane will be developed.

1. EQUATIONS OF MOTION

   The function of Lagrange is defined by:

   = (1)

   And the Hamiltonian function is defined by:

   = (2)

   Fig. 1. Flexible aircraft model.

   For the flexible aircraft showed in the Fig 1, the equations of energy are:

   T 1 mu2 v2 w2

   2

   p

   Where q represents the generalized coordinates, T is the kinetic

   1 1 n 2

   • p q

   2

   rI q 2 M ii

   energy, U is the potential energy and p is given by:

   p L

   (3)

   r

   1 n

   i1

   (5)

   q

   The equations of the motion of the system is given by the

   U e

   2 2 i M

   i

   i

   2 i1

   x

   Hamilton canonic equations:

   1. mg sin sin cos cos cos y

      g

      H

      z

      q p

      p H Q

      q

      (4)

      u v w : Velocities vector;

      p q r : Angular velocities vector;

      Where Q is the generalized force applied on the system.

      m: Vehicle mass;

      g: Gravity acceleration;

      : The generalized displacement coordinates of the ith vibration mode;

      : The undamped natural frequency of the ith vibration mode;

      : The respective generalized mass of the ith vibration mode;

      Ixx , I yy , Izz : The moments of inertia;

      u

      X u X

      • • g X q

        0 0

        u

        • X

          • X E

            Z Z

            Zq Z

            Z

            Z

            Z

   2. , I

   : The xy and yz inertial product;

   0

   u

   0

   E

   xy yz

   U 0 U 0

   U 0 U 0

   U 0 U 0

   U 0

   (9

   , : The roll and the pitch angle;

   0

   0 0 1

   0 0 0

   0

   0

   q

   E )

   Ue: Elastic potential energy;

   M u M

   0 M q

   M M

   M

   M

   E

   Ug: Elastic potential energy.

   0

   0 0 0 0

   1 0 0

   Then the function of Lagrange:

   u

   0 q

   2

   E

   L 1

   mu 2 v 2 w2

   1 p

   p

   q rI q

   1 M 2

   x u q : The state vector;

   n

   2 2 2 i i

   r

   i 1

   x

   (6)

   E : Thrust and elevator commands;

   1 n 2 2

   U : Equilibrium longitudinal velocity;

   2

   i i M i mg sin

   i 1

   • sin cos

   • cos cos y 0

     z

     And the Hamiltonian function:

     : Short notation for the derivative of

     Qi ;

     Mi

     H mux mvy mwz 1 mu2 v2 w2

     2

     p

   X ,Z ,M

   : The derivative notation of X, Z and M

   3. SYNTHESIS

   1 1 n 2

   3.1.

   synthesis

   p q

   2

   rI q 2 Mi i

   r

   i 1

   x

   (7)

   The problem when designing a controller is to find a controller

   K for a system P (Fig 2) that generates a signal u considering the

   • 1 n 2 2i M mg sin sin cos cos cos y

   information from y to mitigate the effect of w on z. In fact, the

   2 i i

   i 1

   z

   controller is synthesized while minimizing the closed loop norm

   w to z.

   With the Hamilton canonic equations, the longitudinal equation of

   motionis given by:

   Fl P,K

   (10)

   mu rv qw g sin X

   mw qu pv g cos cos Z

   qI yy ( p I xy rI yz ) (I xx I zz ) pr

   (8)

   Fl P,K : The lower linear fractional transformation of P and K, transfer function from w to z;

   : A fixed real number.

   yz

   xy

   xz

   ( pI rI

   )q ( p2 r 2 )I M

   2

   Qi

   i

   i i

   i

   M

   Where: X,Y,Z The longitudinal force, the normal force and the pitching moment;

2. SYSTEM DESCRIPTIONS AND MODELING

With some assumption in the parameters and for one vibration mode, the resulting of state-space model for linear control synthesis is given by:

Fig. 2. Standard interconnection for the synthesis.

3.2. Loop shaping synthesis

Loop shaping procedure shaped the nominal plant using a pre-

~

M 2

~ N

2 1

N

M1

compensator W1 and/or a post-compensator W2. The controller

is synthesized with minimizing the norm of the McFarlan and

2

N

if det M

N1

M

G ,G

2 1

(17)

Glover stability margin, such that:

1 2

I I P K

1 ~ 1

1

N2

N1

K

s Ms

(11)

and wnodet M

M 0

Where Ps W2 PW1 and an optimal stability margin;

1 otherwise

2

1

P ~ -1 ~

is the coprime factor of P .

G N M 1

~ 1 ~ : The normalized right (left) coprime

s Ms N1 s

i i i

Mi Ni

The final feedback controller is:

K W1KW2

1. -ANALYSIS

   M is defined as a transfer function form, w to z:

   M F1P,K

   (12)

   (13)

   factorizations of the plants no denoted the winding number.

   6. SIMULATION

   The large high-speed is adopted as the simulation object. The singular value plot of the system is shown in Fig 3. The peak value is 80 dB at 0.356 rad/s, it is the necessary to design a controller to stabilize the system.

   Singular Values

   80

   60

   The structured singular value of a matrix M is defined as: 40

   Singular Values (dB)

   1

   20

   m in : detI M

   (14) 0

   -20

   • denoted the upper singular value and a set of uncertainty.

     The closed-loop system achieves the nominal performance if only if:

     0

     -40

     -60

     -80

     -2 -1

     10 10

     0 1

     10 10

     Frequency (rad/s)

     2 3

     10 10

     sup M22 1

     (15)

     Fig. 3. Diagram of the singular value

     The closed-loop system achieves the robust stability if, only if:

     Controllers are synthesis with the open-loop bock shown in fig

     sup M11 1

     (16)

     1. The system is perturbed by an additive uncertainty. The objectives of the design are to maintain stability and

        The closed-loop system achieves the robust performance if, only

        if:

        performance in presence of a bounded uncertainty.

        The weights are selected to maximize disturbance rejection, and

        sup M 1

        (16)

        minimize wind gust effect and a sensor noise.

     2. GAP

        The gap metric is defined by the quantification of the distance between any two processes in terms of similarity of behavior when connected to a closed loop.

        W1 1,5

        s 2

        3s 10

        , W2

        12,5

        s 10

        1 2

        +

        +

        +

        With:

        dl s3 +1178s2 + 3075s + 2968

        l11 2,346.10-6 s3 +0,02839s2 +0,562s – 0,00102 l12 – 2,541.10-5 s3 – 0,3548s2 – 6,977s – 0,2226

        l21 – 1,099.10-9 s3 – 8,814.10-6 s2 – 0,000182s – 2,992.10-5 l22 1,253.10-8 s3 +0,0001101s2 +0,002254s +0,0004336 l31 – 3,9.10-9 s3 – 0,0004008s2 – 0,007566s – 0,002176

        Fig. 4. Close-loop system with additive uncertainty.

        The two controllers are obtained:

        l32 – 3,934.10-8 s3 +0,004978s2 +0,09351s+0,02911

        l – 2,964.10-9 s3 – 6,189.10-5 s2 – 0,00121s – 0,001066

        • controller:

          h h

          41

          l42 2,307.10-8 s3 +0,0007675s2 +0,01483s+0,01324

          11

          dh

          p1

          p1

          H

          K dh

          dh

          h41

          12

          dh

          p2 dh p2

          dh

          h42

          The gap of the two controller is:

          KH ,KL 0,0024

          It mean that the two controller are close.

          The close-loop matrix for the -analysis (Fig 5) is :

          W KS W KS

          d d M 2 2

          With:

          h

          h

          Where S 1 GK1 .

          W1S

          W1S

          dh s3 +1,41s2 +0,4838s+0,04285

          p1 1,947.10-7 s3 – 2,89.10-5 s2 + 9,614.10-7 s+1,91.10-7

          p2 – 2,345.10-6 s3 +0,0003394s2 – 6,505.10-6 s – 1,925.10-6

          p1 – 1,029.10-10 s3 + 5,663.10-9 s2 – 7,473.10-10 s -7,442.10-11

          p2 4,165.10-10 s3 – 6,775.10-8 s2 +7,231.10-9 s+7,78.10-10

          p1 4,492.10-9 s3 +1,598.10-7 s2 + 4,066.10-8 s+1,987.10-9

          p2 1,33.10-8 s3 – 1,773.10-6 s2 – 4,542.10-7 s – 2,234.10-8

          h41 8,759.10-10 s3 – 1,615.10-8 s2 + 4,456.10-9 s+ 3,662.10-10

          h42 – 4,752.10-9 s3 +1,983.10-7 s2 – 4,543.10-8 s – 3,889.10-9

        • Loop shaping controller:

   Fig. 5. Close-loop matrix for -analysis.

   The frequency range of the analysis is [104; 104] rad/s.

   Robust Stability: S.S.V of M11

   inf[mu(M11H)]

   sup[mu(M11H)]

   inf[mu(M11LP)]

   sup[mu(M11LP)]

   0.03

   0.025

   l l

   11

   dl

   l21

   K dl

   12

   dl

   l22

   dl

   0.02

   mu

   0.015

   0.01

   L l l

   31

   dl

   32

   dl

   0.005

   l l

   0

   -4 -3

   -2 -1 0 1 2 3 4

   41

   42

   10 10

   10 10

   10 10

   Frequence (rad/s)

   10 10 10

   dl

   dl

   Fig. 6.-plot for robust stabilityanalysis

   TABLE 1 RESULTS OF ROBUST STABILITY ANALYSIS

   Close- loop	( /)	max[(11)]	Guaranteed of stability
   	0,1804	0,0165	1 < 0,0165
   	1.5167	0.0260	1 < 0.0260

   The peak value of (11) is less than one for each case of close- loop (Table 1, Fig 6). This implies that for all perturbations,

   TABLE 2 RESULTS OF ROBUST STABILITY ANALYSIS

   Close- loop	( /)	max[()]	Guaranteed of stability
   	0, 1804	0, 0262	1 < 0, 0262
   	1,5167	0,0307	1 < 0,0307

   The peak value of () is less than one for each case of close-

   < 1 the stability is guaranteed. The guaranteed

   (11)

   loop (Table 3, Fig 8). This implies that for all

   stability is large for thealgorithm.

   Nominal performance: S.S.V of M22

   inf[mu(M22H)]

   sup[mu(M22H)]

   inf[mu(M22LP)]

   sup[mu(M22LP)]

   0.01

   0.009

   perturbations,

   < 1

   ()

   the performance is guaranteed. The

   0.008

   0.007

   0.006

   mu

   0.005

   0.004

   0.003

   0.002

   0.001

   0

   -4 -3 -2 -1 0 1 2 3 4

   10 10 10 10 10 10 10 10 10

   Frequence (rad/s)

   Fig. 7.-plot for nominal performance analysis

   TABLE 2 RESULTS OF NOMINAL PERFORMANCE ANALYSIS

   Close-loop	(/)	max[(22)]
   	0,1804	0,0097
   	0,0001	0,0070

   The peak value of (22) is less than one for each case of close- loop (Table 2, Fig 7). This implies that for all perturbations, nominal performance was achieved. However the performance specification is better for the close-loop with loop shaping algorithm.

   Robust performance: S.S.V of M

   inf[mu(MH)] sup[mu(MH)] inf[mu(MLP)] sup[mu(MLP)]

   0.035

   0.03

   0.025

   mu

   0.02

   0.015

   0.01

   0.005

   guaranteed performance is large for the algorithm.

   1. CONCLUSION

      The large high speed is instable. However, the two controllers designs, and loop shaping guarantee a robust stability, and a nominal and robust performance. The two controllers are close in reference of the gap.

   2. REFERENCES

1. L. Meirovitch, Methods of Analytical Dynamics, Hill, 1970.W.-K. Chen, Linear Networks and Systems. Belmont, Calif.: Wadsworth, pp. 123-135, 1993. (Book style)

2. M.R. Waszak, D.K. Schmidt, Flight Dynamics of Aeroelastic Vehicles, Journal of Aircraft, 1988.

3. D.K. Schmidt, Modern Flight Dynamics, Hill, 2012.

4. C. Zhu, "Robustness analysis for power systems based on the structured singular value tools and the nu gap metric", PHD, Iowa State University, 2001.

5. G. Vinnicombe, Uncertainity and Feedback loop-shaping and the

   metric, Imperial College Press, 2001.

6. K. Zhou, J.C. Doyle, K. Glover, Robust and Optimal Control, Prentice Hall, New Jersey, 1996.

7. K. Zhou, J.Doyle,"Essentials of Robust Control", Prentice Hall, 1999.

RANARISON S. H., received his Master Diploma in Automatic from 2015 at Ecole Supérieure Polytechnique dAntananarivo (ESPA), University of Antananarivo. Currently he is a PhD student at University of Antananarivo in the STII,

RANDRIAMITANTSOA P.A., full Professor, Ecole Supérieure Polytechnique dAntananarivo (ESPA), University of Antananarivo.

RANDRIAMITANTSOA A.A., Doctor, Ecole Supérieure Polytechnique dAntananarivo (ESPA), University of Antananarivo.

REZIKY Z.S.H.Z.T, Doctor, Ecole Supérieure Polytechnique dAntananarivo (ESPA), University of Antananarivo.

0

-4 -3

10 10

-2 -1 0 1

10 10 10 10

2 3 4

10 10 10

Frequence (rad/s)

Figure 6 -plot for robust stability analysis