Fm_ag-12_tp-120_11 Open.pdf

 

GROUP FOR AERONAUTICAL RESEARCH AND TECHNOLOGY IN EUROPE

ORIGINAL: ENGLISH

GARTEUR FM(AG12)/TP-120-11

Sep. 19, 2002 GARTEUR Limited A state-space approach for stability analysis of a combat aircraft in presence of rate limits involving pilot-induced-oscillation by Isabelle Queinnec and Sophie Tarbouriech and Germain Garcia

This report has been published under auspices of the Flight Mechanics Group of Responsables of the Group of Aeronautical Research and Technology in EURope (GARTEUR)

GoR Report Resp. Project Man. Monitoring Resp.

: : : :

Flight Mechanics Sophie Tarbouriech Binh Dang Vu H.T. Huynh

Action Group : FM(AG12) Version : X Draft: Y Completed : Sep. 19, 2002 c GARTEUR 2002

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List of authors Isabelle Queinnec Sophie Tarbouriech Germain Garcia1 LAAS-CNRS 7 avenue du colonel Roche 31077 Toulouse cedex 4 France

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1

This author is also with INSA, Toulouse.

Date: Sep. 19, 2002 GARTEUR TP-120-11

Date: Sep. 19, 2002 GARTEUR TP-120-11

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Summary

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This report is devoted to the stability analysis of the Admire model in presence of actuator limitations. The solution is given through the computation of admissible domains of state and reference input, denoted 0 and D0 , respectively. These domains are obtained by solving an optimization problem subject to matrix inequalities constraints given in Proposition 1.

3

It must be noted that the approach cannot be directly applied to the linearized state-space representation of the Admire model since, when no saturations occur, the closed-loop system has to be asymptotically stable. This is not the case of the basic model. Three steps of model reduction have then been achieved. The rst one only consisted in removing the states and dynamics which are not activated by the pilot action (input related to disturbances not used in this project). This step did not modify the whole system structure. The second step of model reduction consisted to separate the longitudinal and lateral dynamics which are not preponderant at the same ight conditions. The nal state-space model used for stability analysis is then given by some standard model reduction techniques which removes the uncontrollable and unobservable modes of the closed-loop system which input and output are the pitch angle reference and signal, respectively.

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The stability analysis approach proposed in this report is demonstrated on the pitch-axis control of the concept aircraft ADMIRE in presence of standard rate-limits.

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Date: Sep. 19, 2002 GARTEUR TP-120-11

Contents

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List of authors

i

Summary

ii

List of gures

v

List of tables

vii

List of symbols and abbreviations

viii

Distribution List

x

1 Introduction

1

2 Description of the ADMIRE model

2

2.1 2.2 2.2.1 2.2.2 2.2.3 2.2.4

Introduction and primarily model . . . First reduction of the ADMIRE model Control system . . . . . . . . . . . . . Aircraft system . . . . . . . . . . . . . Closed-loop system with saturations . Validation of the modications . . . .

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3 Reduction of the ADMIRE model 3.1 3.1.1 3.1.2 3.2 3.2.1 3.2.2

Reduction for ADMIRE longitudinal control system The longitudinal model . . . . . . . . . . . . . . . . The state space model used for the analysis . . . . . Reduction for ADMIRE lateral control system . . . . The lateral model . . . . . . . . . . . . . . . . . . . . The state space model used for the analysis . . . . .

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4 Theoretical aspects in stability analysis of saturated systems

14

5 Stability analysis of the reduced ADMIRE longitudinal model

18

4.1 4.2 4.3

5.1 5.2 5.3 5.4 5.5

Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rate saturation modelling . . . . . . . . . . . . . . . . . . . . . . . . Polytopic representation of the saturation function . . . . . . . . . .

Preliminaries . . . . . . . . . . . Error coordinates representation Polytopic modelling . . . . . . . Main proposition . . . . . . . . . Results . . . . . . . . . . . . . . .

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Date: Sep. 19, 2002 GARTEUR TP-120-11 5.5.1 5.5.2 5.5.3 5.5.4 5.5.5

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Numerical data . . . . . . . . . . . . . . . . . . . . . . . . . . . Inuence of the pilot stick saturation . . . . . . . . . . . . . . . Inuence of the actuators (canard angle and elevon) saturation Domains of safe behavior . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1

6 Main conclusion

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7

References

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List of gures 2.1 2.2 2.3 2.4 2.5 3.1 3.2 3.3 3.4 3.5 3.6 4.1 4.2 5.1 5.2 5.3

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5.4 5.5 5.6

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5.7 5.8 5.9

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Date: Sep. 19, 2002 GARTEUR TP-120-11

5.10 5.11 5.12

Original linear ADMIRE model . . . . . . . . . . . . . . . . . . . . . . . Saturation block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear ADMIRE model with pilot . . . . . . . . . . . . . . . . . . . . . Reduced linear ADMIRE model with pilot . . . . . . . . . . . . . . . . . Bode diagrams ( and ) . . . . . . . . . . . . . . . . . . . . . . . . . . Reduced longitudinal model . . . . . . . . . . . . . . . . . . . . . . . . . Bode diagrams - Flight condition FC1 . . . . . . . . . . . . . . . . . . . Bode diagrams - Flight condition FC2 . . . . . . . . . . . . . . . . . . . State-space representation of the longitudinal system at the boundaries of the saturation elements . . . . . . . . . . . . . . . . . . . . . . . . . . . Reduced lateral model . . . . . . . . . . . . . . . . . . . . . . . . . . . . State-space representation of the lateral system at the boundaries of the saturation elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surrogate structure for the rate saturation . . . . . . . . . . . . . . . . . Response to a sinusoidal input with A=0.7 and f=0.3 . . . . . . . . . . Longitudinal model subject to the saturations . . . . . . . . . . . . . . . Bode diagram - Flight condition FC1 - reduction of the uncontrollable and unobservable modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-response - Flight condition FC1 - reduction of the uncontrollable and unobservable modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of the maximal pitch angle setpoint with respect to the saturation allowance of the pilot stick position . . . . . . . . . . . . . . . . . . . . . Evolution of the maximal pitch angle setpoint with respect to the saturation allowance of the canard angle position . . . . . . . . . . . . . . . . . . . Evolution of the maximal pitch angle setpoint with respect to the saturation allowance of the elevon position . . . . . . . . . . . . . . . . . . . . . . . Evolution of the maximal pitch angle setpoint with respect to the saturation allowance of the canard angle rate . . . . . . . . . . . . . . . . . . . . . Evolution of the maximal pitch angle setpoint with respect to the saturation allowance of the elevon rate . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of the maximal pitch angle setpoint with respect to the saturation allowance of the canard angle position . . . . . . . . . . . . . . . . . . . Evolution of the maximal pitch angle setpoint with respect to the saturation allowance of the elevon position . . . . . . . . . . . . . . . . . . . . . . . Evolution of the maximal pitch angle setpoint with respect to the saturation allowance of the canard angle rate . . . . . . . . . . . . . . . . . . . . . Evolution of the maximal pitch angle setpoint with respect to the saturation allowance of the elevon rate . . . . . . . . . . . . . . . . . . . . . . . . .

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Date: Sep. 19, 2002 GARTEUR TP-120-11

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5.13 Projection in two-dimensionnal subspaces of the ellipsoid of admissible state 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14 Ellipsoid of admissible reference input D0 . . . . . . . . . . . . . . . . .

1

32 33

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Date: Sep. 19, 2002 GARTEUR TP-120-11

List of tables 5.1 Position and rate actuator limitations . . . . . . . . . . . . . . . . . . . 5.2 Inuence of the pilot stick limitations on the admissible reference . . . . 5.3 Inuence of the canard angles and elevons limitations on the admissible reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Date: Sep. 19, 2002 GARTEUR TP-120-11

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List of symbols and abbreviations Symbols 



upitch u0 u1 A B C Dbare A B C Dfcs A B C Dlon A B C Dlat 0

D

0

1 2 3 4

pitch angle roll angle amplitude limitation on the stick force pitch amplitude limitation vector of the actuators rate limitation vector of the actuators state-space representation of the Admire aircraft bare model state-space representation of the Admire control system state-space representation of the reduced Admire longitudinal model at the inputs and outputs of the saturation block, with pilot loop state-space representation of the reduced Admire lateral model at the inputs and outputs of the saturation block, with pilot loop state-space domain of safe behavior set of admissible pitch angle reference and derivative

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Notations

<

23 24

n

n-dimensional real Euclidean space A 2 B(A  B) Matrix A ; B is posivite denite (positive semi-denite) uv u(i)  v(i) trace(A) trace of matrix A

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Abbreviations ADMIRE AG ARE BMI DASA FC1

Aero-Data Model In Research Environment Action Group Algebraic Riccati Equation Bilinear Matrix Inequality Daimler Chrysler Aerospace Flight condition 1 : Mach 0.25, altitude 500m, pitch-loop gain 1278

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FC2 FFA FMAG GARTEUR LAAS LMI OLOP PIO SAAB

Date: Sep. 19, 2002 GARTEUR TP-120-11

Flight condition 2 : Mach 0.8, altitude 5000m, pitch-loop gain 328 Flygtekniska Forsoksanstalten (Aeronautical research institute of Sweden Flight Mechanics Action Group Group for Aeronautical Research and Technology in EURope Laboratoire d'Analyse et d'Architecture des Systemes Linear Matrix Inequality Open-loop onset point Pilot-induced (Pilot-In-the-loop) oscillations SAAB AB

Date: Sep. 19, 2002 GARTEUR TP-120-11

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Distribution List

1 2

(distribution is via e-mail and Wide Area Network if not otherwise specied) GARTEUR Executive Committee (hardcopy) F. Abbink A. Amendola P. Garcia Samitier D. Nouailhas - Chairman XC W. Riha O.K. Sodha B. Uggla GARTEUR Secretary (hardcopy) E. Maire

4 5

(NL) (IT) (ES) (FR) (DE) (UK) (SE)

NLR CIRA INTA ONERA DLR DERA FMV

6 7 8 9 10 11 12 13 14 15

(FR)

GARTEUR Flight Mechanics Group of Responsables (hardcopy) W. de Boer (NL) B. Brannstrom (SE) J. Hall (UK) H.T. Huynh - Monitoring Responsable FM(AG12) (FR) A. Kroger (DE) F. Muoz - Chairman FM-GoR (ES) R. Rodlo (DE) L. Verde (IT)

ONERA

16 17 18 19

NLR FMV DERA ONERA EADS INTA DLR CIRA

GARTEUR Flight Mechanics Industrial Points of Contact (hardcopy) J. Enhagen (SE) SAAB L. Goerig (FR) DAv GARTEUR FM(AG12) Participants F. Amato S. Bennani J.M. Biannic M. Crouzet B. Dang-Vu - Chairman FM(AG12) E. Filippone G. Garcia L. Goerig G. Hovmark A. Hyden

3

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

(IT) (NL) (FR) (FR) (FR) (IT) (FR) (FR) (SE) (SE)

UNAP DUT ONERA CEV ONERA CIRA LAAS DAv FOI FOI

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Date: Sep. 19, 2002 GARTEUR TP-120-11

R. Iervolino A. Knoll A.M. Kraeger J.A. Mulder H.C. Oelker I. Postlethwaite I. Queinnec L. Rundqwist H. Smali S. Tarbouriech M. Turner P. Vicente T. Wilmes

(IT) (DE) (NL) (NL) (DE) (UK) (FR) (SE) (NL) (FR) (UK) (ES) (DE)

UNAP EADS DUT DUT DASA UL LAAS SAAB NLR LAAS UL INTA DLR

Others M. Bauschat L. Forssell G. Honger F. Karlsson U. Korte E. Kullberg R. Luckner J.F. Magni A. Martinez D. Moormann M. Selier J. Schuring A. Varga

(DE) (SE) (DE) (SE) (DE) (SE) (DE) (FR) (ES) (DE) (NL) (NL) (DE)

DLR FOI EADS SAAB EADS SAAB (retired) EADS ONERA INTA DLR NLR NLR DLR

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Date: Sep. 19, 2002 GARTEUR TP-120-11

1 Introduction Due to technological and safety reasons, the actuators cannot deliver an unlimited energy to the controlled plant. This fact can be translated into bounds on control and state variables. Actually the presence of input bounds can be source of parasitic equilibrium points and limit cycles, or can even lead the closed-loop system to an unstable behavior. These facts have motivated the renewed interest in the study of linear systems subject to input saturations. Hence in the last years, the problem of stabilizing linear systems with saturating actuators has been widely studied: see 2], 18] for some overviews. Signicant results have lately emerged in the scope of global 17] and semi-global stabilization 9]. They inherently require stability assumptions on the open-loop system. Relaxing these open-loop stability assumptions, results concerning the local stabilization have also been attained (see 19] and references therein). On the other hand, the rate saturation problem has rst received a special interest in the aeronautic eld, where the tradeo between hight performance requirements and the use of hydraulic servos presenting rate limitations is always present (see for example, 11], 14] and references therein). A natural feature following the stabilization problem is the problem of output tracking of references. These references may be generated by an external system, so-called the exosystem. See, in particular, the book 15] proposing an interesting panel in the semiglobal stabilization context of this problem. Moreover, it is well-known that the presence of any constant disturbance in a linear system can be eliminated by using the so-called integral error feedback. When a linear system is subject to input saturations, it becomes nonlinear when the control applies and therefore some limitations may arise. The most important of these limitations is related to the stability and the problem generated by the windup phenomenon. Addressing the problem of integrator windup due to saturation, several approaches are proposed in the literature 7], 8], 5], 24]. In these references, the anti-windup compensator is obtained by adding dead-zone nonlinearities. In 8] the parameters of the controller are computed through extensive simulations and no domain of local stability is analytically determined. The method developed in 24] is closely related to 8] with the advantage of providing a guaranteed and local domain of attraction. In 3], the authors propose a method for tracking a reference signal that converges to a constant admissible value. A tracking domain of attraction is dened as a set of initial states from which the reference signal can be asymptotically tracked without constraints violation. An interesting result is proved. Any domain of attraction is also a tracking domain of attraction. From a stabilizing controller associated with a domain of attraction, the authors build a tracking controller for which the tracking convergence is connected to the speed convergence of the stabilization controller when the reference is constant. In the present report, we consider the problem of output tracking for a small generic ghter aircraft with one engine, which is an exponentially unstable plant with actuator saturation and for which the output must track a certain time-varying reference. Our objective is double in the sense that we want to caracterise a domain of admissible references and a

Date: Sep. 19, 2002 GARTEUR TP-120-11

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safety region over which the stability of the resulting closed-loop saturated system is ensured. Moreover, let us underline that the considered aircraft is a system subject to both amplitude and rate saturation, corresponding to the position limitation on the pitch stick (upitch ), to the position and rate limitations of the canard angles and inboard/outboard elevons (u0 , u1). The solutions are based on the resolution of some matrix inequalities, which can be written as Linear Matrix Inequalities (LMIs), provided the use of some relaxation schemes. Constructive conditions based on the use of quadratic Lyapunov functions, S -procedure and a polytopic saturation nonlinearity description dierential inclusions for modeling the behavior of the closed-loop nonlinear system are proposed. The estimates of the domain of attraction and the admissible reference set are formulated via the resolution of convex optimization problems subject to LMI constraints. Furthermore, we focus more especially our attention on a local stability analysis approach since the open-loop system is unstable unlike the semi-global or global context developed in 10] and 1]. This report may be viewed as an extension to the continuous-time case and to the amplitude and rate actuator limitations case of some results developed in 23] for the discrete-time case (with only amplitude actuator limitations). The outline of the paper is the following. Section 2 describes the ADMIRE model we consider. Section 3 is dedicated to provide a reduced ADMIRE model. In section 4, the theoretical aspects of saturated system are addressed. Section 5 deals with the stability analysis of the reduced ADMIRE longitudinal model. Finally, section 6 ends the report through some concluding remarks.

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Date: Sep. 19, 2002 GARTEUR TP-120-11

2 Description of the ADMIRE model 2.1 Introduction and primarily model All the work presented in this report is based on the existing "ADMIRE release 3.4c, AG-12 version" from a small generic ghter aircraft with one engine. For a general survey see the gure 2.1.

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[rt_time,rt_Fes/4]

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0

drie

[rt_time,rt_Fas/4]

24

Mach No and rate limiter data are taken from Workspace

dlie dFrp

dloe

dFrp dr

0

All inital states empty

dle_in

dle 0

27

drie dlie dloe dr

dle

dle

tss

ldg

y0_lin

dty_in

ldg

tss

dty

dty

dtz

dtz

1

Selector Select 32 to 59

dty 0

Selector Select 1 to 31

dtz_in

dtz dist

u_dist

u_dist

v_dist

v_dist

w_dist

w_dist

y0bare(1:31)

Disturbance

DisturbParam (Sub block) Feedback

p_dist

p_dist

ADMIRE_fcs_Linear

ADMIRE_bare_Linear

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droe

ldg_in

19

23

dFas

Stick Force Roll

0

22

dlc

Saturators, Ratelimiters and Actuators

droe

dVt

ldg

21

drc

dVt

18

20

drc dlc

0

15

dFes

Stick Force Pitch

Memory

Fig.2.1 Original linear ADMIRE model

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Behind 'ADMIRE-fcs-Linear' stands a state space model for the control system. The control system is dened through the matrices Afcs = dim(4  4), Bfcs = dim(4  40), Cfcs = dim(16  4), Dfcs = dim(16  40). Consequently we have 40 inputs, 16 outputs and 4 states, describing the following generic system:

(

x_ = Afcsx + Bfcsu y = Cfcsx + Dfcsu

(2.1)

The state space model for the aircraft response can be found behind 'ADMIRE-bareLinear' with the matrices Abare = dim(28  28), Bbare = dim(28  16), Cbare = dim(59  28), Dbare = dim(59  16). The generic model involving 16 inputs, 59 outputs and 28 states is then described by: ( x_ = Abarex + Bbare u (2.2) y = Cbarex + Dbare u The position and rate limiters and actuators block is plotted in Figure 2.2. The input signal is rst saturated in position, then passes through the rate saturation block and nally through the transfer function which represents the dynamics of the acturators.

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Date: Sep. 19, 2002 GARTEUR TP-120-11

1 FCS_ae_rl_drc_out

2

2

3

0.05s+1

out

FCS_ae_rl_drc

1

FCS_drc

1

1

RL2

drc_actuator (with initial state)

2

FCS_dlc

1 0.05s+1

FCS_ae_rl_dlc

RL3

3

dlc_actuator (with initial state)

4

3

5

FCS_droe

1

FCS_ae_rl_droe

FCS_ae_rl_dlc_out

0.05s+1

RL4

FCS_ae_rl_droe_out

droe_actuator (with initial state)

4

4

0.05s+1

RL5

5

1

6

RL6

5

dlie_actuator (with initial state)

FCS_ae_rl_dloe

FCS_ae_rl_dlie_out

9

6

10

FCS_dloe

1 0.05s+1

7

RL7

FCS_ae_rl_dr

dloe_actuator (with initial state)

FCS_ae_rl_dloe_out

11

7

12

FCS_dr

1 0.05s+1

8

RL8

FCS_ae_rl_dle

FCS_ae_rl_dr_out

dr_actuator (with initial state) 1

RL9

FCS_ae_rl_tss

8

dle_actuator (with initial state) 2s+1 Engine Dynamics (with initial state)

S−Function act_pos_lim

FCS_ae_rl_dle_out

14

9

15

FCS_tss

1 [Mach]

13

FCS_dle

0.05s+1

9

8

FCS_dlie

0.05s+1

in

7

FCS_ae_rl_drie_out

drie_actuator (with initial state)

FCS_ae_rl_dlie

6

FCS_drie

1 FCS_ae_rl_drie

FCS_ae_rl_tss_out

16 17

Fig.2.2 Saturation block

18

The whole model represented in Figure 2.1 then has 12 inputs, 31 outputs and 4+28=32 states. Till now there is no pilot taken in account. That's our starting point.

20

19

We are interested in the inuence of the saturations in the system with pilot, whose objective is to control the lateral dynamics, the longitudinal dynamics or both. The block diagram of the ADMIRE model including the pilot loop is shown on Figure 2.3. The outer pilot loops feed back the pitch (longitudinal) and roll (lateral) angles,  and , respectively, through the pilot gain and a saturation block representing the stick limits.

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em

phi, theta

32

Selector

33

pilot gain fes K1

dFes

drc

drc

1 Stick limit 1

theta_c 0

dVt droe

dVt

drie

K2 2 phi_c

pilot gain fas 0

dFas dlie

Stick limit 2 dFrp

dloe

dFrp dr

0

dle_in

dle 0

34

dlc

dlc

Saturators, Ratelimiters and Actuators Mach No and rate limiter data are taken from Workspace All inital states empty

droe

35

drie

36

dlie dloe

37

dr

dle

dle

tss

ldg

Selector

y0_lin ldg_in

ldg 0

dty_in

Select 1 to 31 ldg

tss

dty

dty

dtz

dtz

Selector

dty 0

Select 32 to 59

dtz_in

dtz dist

u_dist

u_dist

v_dist

v_dist

w_dist

w_dist

Disturbance

DisturbParam (Sub block) Feedback

p_dist

p_dist

ADMIRE_fcs_Linear

ADMIRE_bare_Linear

Memory

Fig.2.3 Linear ADMIRE model with pilot

y0bare(1:31)

1

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The main problem related to this linear ADMIRE model comes from some zero modes of the control system which lead to some instabilities of the closed-loop aircraft response. The rst step before system analysis then concerns model reduction to only consider the relevant part of the system.

2.2 First reduction of the ADMIRE model 2.2.1 Control system The columns 19, 22-30 and 32-40 of Bfcs and Dfcs are null. We then reduce the input vector from 12+28=40 to 12+9=21. So the new matrices are:

Afcs2 = Afcs Cfcs2 = Cfcs

h

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Furthermore the inputs 2, 4, 5-12 are always zero in this project (they correspond to parameter uncertainties, disturbances or unused inputs relative to other GARTEUR projects) thus we can cross out the corresponding columns in the Bfcs2 and Dfcs2 matrices. That means that we reduce the inputs to 2+9=11. So we achieve the following matrices:

Afcs3 = Afcs2 Cfcs3 = Cfcs2

h

Afcs1 = Afcs3 Bfcs1 = Bfcs3

(

38

41 42 43 44 45 46 47 48

"

Cfcs1 = Cfcs3(1 : 7 :) Cfcs3(9 :)

#

"

Dfcs1 = Dfcs3 (1 : 7 :) Dfcs3 (9 :)

#

Finally, the state space representation of the control system follows, with 11 inputs, 8 outputs and 4 states:

37

40

i

Bfcs3 = Bfcs2(: 1) Bfcs2 (: 3) Bfcs2(: 13 : 21) h i Dfcs3 = Dfcs2 (: 1) Dfcs2(: 3) Dfcs2 (: 13 : 21)

The lines 8 and 10-16 from Cfcs3 and Dfcs3 are equal to zero, then eliminated, such that one only considers 8 outputs. Due to the fact that the line 8 is zero, one has to take care that the eighth saturation disappears as well. One gets:

36

39

i

Bfcs2 = Bfcs (: 1 : 18) Bfcs (: 20 : 21) Bfcs (: 31) h i Dfcs2 = Dfcs (: 1 : 18) Dfcs(: 20 : 21) Dfcs(: 31)

30 31

Date: Sep. 19, 2002 GARTEUR TP-120-11

x_ fcs = Afcs1xfcs + Bfcs1 u y = Cfcs1xfcs + Dfcs1u

(2.3)

2.2.2 Aircraft system As a result of the reduction of the control system outputs, one has to decrease the number of inputs of the aircraft response. The corresponding inputs which are nonrelevant are 8, 9, 11-16. Note that the output 9 of the control system is associated to the input 10 of the aircraft response. So Bbare and Dbare are reduced as: The notations are derived from Matlab notations.

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Date: Sep. 19, 2002 GARTEUR TP-120-11

h

1

i

2

Bbare2 = Bbare (: 1 : 7) Bbare (: 10) h i Dbare2 = Dbare (: 1 : 7) Dbare (: 10)

3 4 5

Concerning the ouptut of the aircraft response, the block is in fact subdivided into a plane part and a sensor part, the last one allowing the feedback to the control system. In practice, according to the reduction of Bfcs and Dfcs , one only needs the lines 1-6, 8, 9 and 19 from the output of the sensor part, which are the outputs 32-37, 39, 40 and 50 in the aircraft response system. One can easily verify that these outputs only use states 1, 2, 4, 5, 13-21, 23, 25, 27 and 28. They in turn need the states 3, 8, 9, 12, 22 and 24. By this way we found out that the states 7, 10, 11 and 26 are useless as the corresponding lines in the Bbare2 are as well zero. For this reason we cancel them:

h i Abare2 = Abare(: 1 : 6) Abare(: 8 : 9) Abare (: 12 : 25) Abare(: 27 : 28) h i Cbare2 = Cbare(: 1 : 6) Cbare(: 8 : 9) Cbare(: 12 : 25) Cbare(: 27 : 28) h i Abare1 = Abare2(1 : 6 :) Abare2(8 : 9 :) Abare2 (12 : 25 :) Abare2 (27 : 28 :) h i

9 10 11 12 13 14 15

18 19 20 21 22

3 2 3 2 Cbare2 (1 : 6 :) Dbare2 (1 : 6 :) 66 C (8 : 9 :) 77 66 D (8 : 9 :) 77 66 bare2 77 66 bare2 77 C (12  :) D (12  :) 77 Dbare1 = 66 bare2 77 Cbare1 = 666 bare2 77 7 6 C (32 : 37  :) D (32 : 37  :) 77 66 bare2 66 bare2 4 Cbare2(39 : 40 :) 5 4 Dbare2(39 : 40 :) 75

23 24 25 26 27 28 29 30 31 32 33 34

Dbare2 (50 :)

35

For the state space model of the aircraft response we achieve now 8 inputs, 18 outputs and 24 states with the following representation:

x_ bare = Abare1 xbare + Bbare1 u y = Cbare1xbare + Dbare1 u

8

17

In addition we simplify the system by eliminating the outputs 13-31, which can be reproduced by the rst twelve outputs and we already said that we need the outputs 32-37, 39, 40 and 50 for the state feedback. Hence it results in:

(

7

16

Bbare1 = Bbare2(1 : 6 :) Bbare2 (8 : 9 :) Bbare2(12 : 25 :) Bbare2 (27 : 28 :)

Cbare2 (50 :)

6

(2.4)

2.2.3 Closed-loop system with saturations At last, some comments concern the saturation block of Admire model. In fact, examination of Cfcs1, Dfcs1 and the saturation block reveals that some inputs of the aircraft model may be duplicated from one single output of the fcs control block. This represents

36 37 38 39 40 41 42 43 44 45 46 47 48

; 7; 1 2 3 4

some common eect on the right and left canard angles (drc and dlc), right inboard and outboard elevons (drie and droe) and left inboard and outboard elevons (dlie and dloe). The number of outputs of fcs1 block is then reduced to 5 as follows:

2  :) 66 CCfcs1(1 Cfcs1 = 66 fcs1(3 :) 4 Cfcs1(5 :)

5 6 7 8

Cfcs1(7 : 8 :)

9 10 11 12 13

Date: Sep. 19, 2002 GARTEUR TP-120-11

3 2 fcs1 (1 :) 77 66 D 77 Dfcs1 = 66 Dfcs1(3 :) 5 4 Dfcs1(5 :)

Dfcs1 (7 : 8 :)

3 77 77 5

Finally, the reduced ADMIRE model involving the aircraft model and sensors bare1, the control feedback fcs1, the position and rate saturation blocks and the pilot loop is shown on Figure 2.4.

14 15

phi

16

theta

em

17

phi, theta Selector 8, 7

18

1 0.05s+1

19

drc et dlc

20 21 22

pilot gain fes

0.05s+1

K2 2 phi_c

droe et drie

Stick limit 1

theta_c

23 24

1

K1 1

pilot gain fas

Mux Stick limit 2

x’ = Ax+Bu y = Cx+Du

em

Mux 1

fcs1

0.05s+1

25

x’ = Ax+Bu y = Cx+Du

Selector

Avion+capteurs

Select 1 to 9

1

bare1

dloe et dlie

26 27

1

28 29

:6);y0bare(8:9);y

0.05s+1 dr 1 2s+1

Selector Select 10 to 18

tss

30 31 32 33

Fig.2.4 Reduced linear ADMIRE model with pilot

34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

2.2.4 Validation of the modications When we appraise the Bode diagram (Figure 2.5) of the original model and the reduced model rst between the input c and the output  and afterwards between the input c and the output , we can pinpoint that there are no big dierences. A zoom of the Bode diagrams around the cutting frequency would show that only the Bode diagrams relative to the lateral mode () presents some dierences. Furthermore the eigenvalues of the reduced linear system and those of the original linear one completely agree. This validates the reduced model.

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Date: Sep. 19, 2002 GARTEUR TP-120-11

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Bode Diagrams

16 From: U(1)

17

50

18

0

19 20

Phase (deg); Magnitude (dB)

−50

21

−100

To: Y(1)

22 −150

23

200

24

100

25

0

26

−100

27

−200

28

−300 −4 10

29 −3

10

−2

10

−1

10

0

10

1

10

Frequency (rad/sec)

Fig.2.5 Bode diagrams ( and )

2

10

3

10

30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

; 9; 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Date: Sep. 19, 2002 GARTEUR TP-120-11

3 Reduction of the ADMIRE model 3.1 Reduction for ADMIRE longitudinal control system 3.1.1 The longitudinal model The longitudinal model concerns the control of the pitch angle , with pilot action on the stick force pitch (fes). That means that  doesn't play a role for the longitudinal model, and there is no pilot loop on it. All the parts of the system which are related to the control , i.e., activated by the stick force roll (fas) are then always zero and may be removed. Furthermore we found out that the inputs dr and tss are almost zero and may be removed. According to the symmetry of the aeroplane in the longitudinal mode, the actions on the elevons droe, drie, dlie and dloe are identical, and only one saturation may be used to transfer the action on the elevon on the aircraft. The longitudinal structure of the system is then achieved as follows:

18 19

theta

20 21 22 23 24 25

1

pilot gain fes

0.05s+1

−K−

drc and dlc

1 theta_c

Stick limit fes Mux

x’ = Ax+Bu y = Cx+Du

em

Mux

fcs longitudinal

26

x’ = Ax+Bu y = Cx+Du

Selector

bare+sensors

Select 1 (theta)

1

1 0.05s+1

27

droe and drie Selector Select 2 to 10

28

y0bare(8)

29 30 31 32

Fig.3.1 Reduced longitudinal model

33 34 35 36

The new control and aircraft systems, (Aflo, Bflo , Cflo, Dflo) and (Ablo , Bblo , Cblo, Dblo ), respectively, are directly derived from the above discussion as :

37 38 39 40 41 42 43 44 45 46 47 48

Aflo = Afcs1 h i Cflo = Cfcs1(1 : 2 :)

h

i

Bflo = Bfcs1(: 1) Bfcs1 (: 3 : 11) h i Dflo = Dfcs1(1 : 2 1) Dfcs1 (1 : 2 3 : 11)

Ablo = Abare1 " # Bblo = B" bare1(: 1 : 6) # C D bare 1(7 :) bare 1 (7 1 : 6) Cblo = Dblo = Cbare1(10 : 18 :) Dbare1(10 : 18 1 : 6) The reduced longitudinal model has been validated with respect to the original ADMIRE system (Afcs , Bfcs , Cfcs, Dfcs , Abare , Bbare , Cbare, Dbare ) through the Bode diagrams

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Date: Sep. 19, 2002 GARTEUR TP-120-11

obtained for various inputs, pilot gains, Mach numbers and Altitudes.

1

In the following, two main ight conditions are considered:

3

2

4

 FC1:

Mach 0.25, altitude 500m, pitch-loop gain 1378. This ight condition was believed to be PIO prone in the pitch axis.

 FC2:

Mach 0.8, altitude 5000m, pitch-loop gain 328. This ight condition was believed to be PIO prone in the roll axis.

5 6 7 8 9 10 11 12

The pilot gains relative to the pitch loop (and roll loop in the next section) were determined in the report by Postlethwaite, Turner and Prempain 13] through an OLOP evaluation.

13 14 15 16

Bode Diagrams

17

From: U(1) 50

18 0

19

Phase (deg); Magnitude (dB)

−50

20

−100

21

−150

22

0

23

−50

24

To: Y(1)

−100 −150

25

−200

26

−250 −300 −3 10

−2

−1

10

0

10

1

10

2

10

3

10

10

27 28

Frequency (rad/sec)

29

Fig.3.2 Bode diagrams - Flight condition FC1

30 31 32

Bode Diagrams

33

From: U(1) 0

34

−20

35

−40 −60

36

Phase (deg); Magnitude (dB)

−80

37

−100 −120

38

−140

39

0 −50

40

To: Y(1)

−100

41

−150 −200

42

−250 −300 −4 10

−3

10

−2

10

−1

10

0

10

1

10

Frequency (rad/sec)

Fig.3.3 Bode diagrams - Flight condition FC2

2

10

3

10

43 44 45 46 47 48

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Date: Sep. 19, 2002 GARTEUR TP-120-11

3.1.2 The state space model used for the analysis First of all, with respect to the representation of the longitudinal system proposed in Figure 3.1, we rst derive the state-space representation of the system on the inputs and outputs of the saturation elements as it is shown in Figure 3.4.

7 theta

8 9 10 11 pilot gain fes

12 13 14

K 1 In1

2

2

3

3

Out2

In2 fess

Out3 y3

In3 u3

fes

Mux

theta_c

15

x’ = Ax+Bu y = Cx+Du

em

x’ = Ax+Bu y = Cx+Du

Selector

Aircraft+sensors

Select 1

Mux

fcs

16

4

4

Out4

In4 u4

y4

17 18

1 Out1 theta

Selector Select 2 to 10

19 20 21 22 23

Fig.3.4 State-space representation of the longitudinal system at the boundaries of the saturation elements

24 25 26 27 28

The state-space model associated to the system described in Figure 3.4, with fcs and aircraft blocks described by (Aflo, Bflo , Cflo, Dflo) and (Ablo , Bblo , Cblo , Dblo ), respectively, is then stated as:

8 > x_ = Alon x + Blon2 fess + Blon3 u > <  = Clon1x > fes = Clon2x + Kc > :

29 30 31 32

y

33 34 35

wherey

y = y3 y4

37 38

"

39

41

"

42

"

 u = u3 u4

#

# "

 Blon2 = Bflo(: 1)

#

0

Blon3 = Bflo(: 2 : 10)Dblo(2 : 10 1 : 2)12 Bflo (: 2 : 10)Dblo(2 : 10 3 : 6)14 Bblo (: 1 : 2)12 Bblo (: 3 : 6)14

43 44 45 46

48

#

Alon = Aflo Bflo (: 2 : 10)Cblo(2 : 10 :) 0 Ablo

40

47

= Clon3x + Dlon32fess + Dlon33u

"

36

(3.1)

y

This system takes into consideration that Dblo (1 :) is a zero matrix. 1m denotes a m-dimensional vector of ones, i.e., 1m = 1 1 1 ]0 2 <m .

#

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Date: Sep. 19, 2002 GARTEUR TP-120-11

h

Clon1 = 0 Cblo(1 :)

i

h



h

;KCblo(1 :)

Clon2 = 0

Clon3 = Cflo Dflo (: 2 : 10)Cblo(2 : 10 :)

i

i

1 2 3 4

Dlon32 = Dflo(: 1)

5

h

Dlon33 = Dflo(: 2 : 10)Dblo(2 : 10 1 : 2)12 Dflo(: 2 : 10)Dblo(2 : 10 3 : 6)14

i

6 7 8 9

Remark 1 It may be noted that for ight conditions corresponding the longitudinal control

10 11

system, in fact for Mach lower than 0.58, Dlon33 is equal to 0 and may eventually be removed.

12 13 14

3.2 Reduction for ADMIRE lateral control system

15 16 17

3.2.1 The lateral model

18 19

The lateral model concerns the control of the roll angle , with pilot action on the stick force roll (fas). That means that, in this case, the pitch angle  is not involved in the pilot loops. All the part of the control which are related to the pilot roll loop is then removed. Furthemore, we pointed out that, in this case, in all the ight conditions tested, the right and left canard angles ('drc' and 'dlc') never saturated. The lateral model is then shown in Figure 3.5.

20 21 22 23 24 25 26 27

phi

28 29 30 31 1

32

0.05s+1

33

drc et dlc

34

1 0.05s+1

phi_c

pilot gain fas

35

droe et drie

2 1

Mux Stick limit

Mux

x’ = Ax+Bu y = Cx+Du fcs

1

em

0.05s+1

x’ = Ax+Bu y = Cx+Du

Selector

Avion+capteurs

Select 1 (phi)

1

36 37 38

dloe et dlie

39

1 0.05s+1

y0bare(9)

dr

40 41

1 2s+1 tss

Selector Select 2 to 10

42 43 44 45

Fig.3.5 Reduced lateral model

46 47 48

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Date: Sep. 19, 2002 GARTEUR TP-120-11

For this system, the new matrices for the control and aircraft systems are:

3 4 5 6 7

Afla = Afcs1 Bfla = Bfcs1 (: 2 : 11) Cfla = Cfcs1 Dfla = Dfcs1 (: 2 : 11)

"

Cbla = Cbare1 (8 :) Cbare1 (10 : 18 :)

Abla = Abare1 Bbla = Bbare1

8

#

"

Dbla = Dbare1(8 :) Dbare1(10 : 18 :)

9 10 11 12 13 14 15 16

3.2.2 The state space model used for the analysis As it has been done in the previous section for the longitudinal model, the state-space representation of the system at the boundaries of the saturation elements is rst set on the basis of the structure given in Figure 3.6.

17

phi

18 19

1

20

0.05s+1

21 22 23 24

K

1 In1 u1=phi_c

2 pilot gain fas Out2 y2=fass

3 Out3 2 In2 u2=fas Mux

x’ = Ax+Bu y = Cx+Du control system

25 26 27 28 29

em

y3

3 In3 u3

4 Out4 y4

4 In4 u4

5 Out5

5 In5

y5

u5

6 Out6

6 In6

y6

u6

Mux

x’ = Ax+Bu y = Cx+Du Avion+capteurs

Selector phi 8

1 Out1 y1=phi

y0bare(9) 1 2s+1

Selector Select 10 to 18

30 31 32 33 34 35 36 37 38 39

Fig.3.6 State-space representation of the lateral system at the boundaries of the saturation elements

2 3 2 3 y3 u3 6 7 6 By denoting y35 = 4 y4 5  u35 = 4 u4 75, it happens: y5

u5

40 41 42 43 44 45 46 47 48

8 x_ = Alatx + Blat2fass + Blat35u35 + Blat6 u6 > > > <  = Clat1x fas = Clat2x + Kc > > y = C x + D fass + D u > : y356 = Clatlat63x + Dlatlat6232fass + Dlatlat635335u3535

(3.2)

#

Date: Sep. 19, 2002 GARTEUR TP-120-11

;14;

wherez

1

2 Afla Bfla (: 2 : 10)Cbla (2 : 10 :) Bfla (: 2 : 10)Dbla (2 : 10 1 : 2)12 Bfla (: 2 : 10)Dbla (2 : 10 8) 6 Bbla (: 1 : 2)12 Bbla (: 8) blo Alat = 664 20C 0 (1 :) 20D (1 2 : A10) Cbla (2 : 10 :) 20(Dfla (1 2 : 10)Dbla (2 : 10 1 : 2)12 ) ; 1) 20Dfla (1 2 : 10)Dbla (2 : 10 8) fla fla 0 0 ;0:5 2 20 3 Bfla (: 1) 3 0:2cm]0 6 7 6 7 0 7 Blat2 = 664 20D 0 (1 1) 775  Blat6 = 664 7 5 0 fla 0 0 : 5 3 2 Bfla (: 2 : 10)Dbla (2 : 10 3 : 4)12 Bfla (: 2 : 10)Dbla (2 : 10 5 : 6)12 Bfla (: 2 : 10)Dbla (2 : 10 7) 7 6 B (: 5 : 6)12 B (: 7) B =4 B (: 3 : 4)12 5 lat35

h

fla

20Dfla (1 2 : 10)Dbla (2 i : 10 3 : 4)12 0 Cbla (1 :) 0 0

fla

20Dfla (1 2 : 10)Dbla (2 : 10 5 : 6)12

fla

20Dfla (1 2 : 10)Dbla (2 : 10 7)

Clat1 = h i Clat2 = 0 ;kCbla (1 :) 0 0 i h Clat3 = Cfla (3 : 5 :) Dfla (3 : 5 2 : 10)Cbla (2 : 10 :) Dfla (3 : 5 2 : 10)Dbla (2 : 10 1 : 2)12 Dfla (3 : 5 2 : 10)Dbla (2 : 10 8) h i Clat6 = Cfla (6 :) Dfla (6 2 : 10)Cbla (2 : 10 :) Dfla (6 2 : 10)Dbla (2 : 10 1 : 2)12 Dfla (6 2 : 10)Dbla (2 : 10 8) Dlat32 = Dfla (3 : 5 1) h i Dlat335 = Dfla (3 : 5 2 : 10)Dbla (2 : 10 3 : 4)12 Dfla (3 : 5 2 : 10)Dbla (2 : 10 5 : 6)12 Dfla (3 : 5 2 : 10)Dbla (2 : 10 7) Dlat62 = Dfla (6 1) h i Dlat635 = Dfla (6 2 : 10)Dbla (6 3 : 4)12 Dfla (6 2 : 10)Dbla (2 : 10 5 : 6)12 Dfla (6 2 : 10)Dbla (2 : 10 7)

3 77 75

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

This system takes into consideration the null terms of (Afla , Bfla, Cfla , Dfla ) and (Abla, Bbla , Cbla , Dbla). z

46 47 48

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Date: Sep. 19, 2002 GARTEUR TP-120-11

4 Theoretical aspects in stability analysis of saturated systems 4.1 Problem statement The main objective of our task is to study the asymptotic stability of the ADMIRE closedloop system. Note that this objective is not concerned with time-response properties of the closed-loop system. In fact, the feedback is given in the ADMIRE scheme we are concerned with, and we do not have to modify it. Our problem is then only an analysis problem which can be stated as follows:

Problem 1 Determine the sets of admissible states x 2
e ! 0 when t ! 1 8 admissible x(0) r(t) = r0 = constant where e is the tracking error of dimension l.

In such a work, we are not interested in modifying the rate limiter block, since we consider it as the given facts of our problem. However, there is some real interest in studying the inuence on the size of domains of admissible states and tracking references of the bounds upitch, corresponding to the position limitiation on the pitch stick, u0, u1 , corresponding to the position and rate limitations of the canard angles and inboard and outboard elevons, respectively, and/or of the pilot gain.

Remark 2 Ths synthesis problem consisting in modifying the rate-limiter block of the

simulink representation (such as de ned in the AG12 GARTEUR) is dicult to understand. Relatively to the approach we develop, the rate limiter of the aircraft is considered as a given fact and therefore it is not modi ed. Of courses, it can be intersting to bring some modi cation to the given controller a posteriori (as for exemple in the anti-windup approach) implying in some cases to maybe change the structure of the rate-limiter. Moreover, it seems obvious that if we try to increase the size of safe domains, it is to the detriment of the time-response performance (closed-loop poles, delays...). Hence, what we want to do is to give some sets of state and reference guaranteeing a safe behaviour of the aircraft a priori.

Remark 3 From the synthesis point of view, the problem should be to build a new feedback

block where both time-response, frequency performance and constraints would be taken into account. Since this problem is not in the AG12 agenda, some suboptimal solution would be to add another loop between the state of the aircraft and the input of the rate limiter to anticipate the eects of the saturation (cf what we have done in previous reports). But it must be clear that since we do not have the speci cation requirements, we can only focus on the size of safety domains, i.e., to the detriment of the actual feedback.

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Date: Sep. 19, 2002 GARTEUR TP-120-11

1

We are concerned with the analysis problem of determination of state space and reference regions guaranteeing a safe behaviour of the aircraft. Two problems can be studied:

 Saturations not allowed.

In such a case, we are only concerned with the linear behaviour of the aircraft, and we seek domains included in the domain of linearity of the system.

 Saturations allowed. described above.

We are then concerned with the nonlinear polytopic model

2 3 4 5 6 7 8 9 10 11 12

For both cases, two subcases can also be studied:

13

 determination of maximal amplitude of the constant reference#  determination of both the maximal amplitude of the time-varying reference and the maximal rate of variation.

14 15 16 17 18

In the rst case, since nothing is said about the rate of variation of the tracking reference, it understands that steps reference can be applied. The second case is then expected to lead to larger domains of admissible reference but with constraints on the rate of variations. Note that this is the case of reference proles that have been furnished to us by FFA.

4.2 Rate saturation modelling

19 20 21 22 23 24 25 26

At this stage, the rate limiter has to be replaced by a certain "equivalent" saturation model in order to obtain a mathematical representation suitable for system analysis. For that reason we composed a structure, which has the same behavior as the rate limiter. Note however that other structures for the description of the rate limiter could be considered 20]. Both structure may be compared through the Simulink le proposed in Figure 4.1.

27 28 29 30 31 32 33 34 35

1

36

Tr.s+1 v1

37

drc or dlc

38 Signal Generator

1

1/Tr drc or dlc1

Gain

rlimit_c

v2

Scope

s Integrator

39 40 41 42

Fig.4.1 Surrogate structure for the rate saturation With a sinusoidal input with amplitude 0.7 and frequency 0.3 Hz, position and rate saturated outputs for the two structures are identical.

43 44 45 46 47 48

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Date: Sep. 19, 2002 GARTEUR TP-120-11

1 0.6

2 0.4

3 0.2

4 0

5 −0.2

6 −0.4

7 −0.6

8 0

2

4

6

8

10

12

14

16

18

20

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Fig.4.2 Response to a sinusoidal input with A=0.7 and f=0.3 We then consider the following position and rate saturation model, between the input v1 and the output v2 , as follows : 1  v_ = sat (sat (v )) ; 1 v (4.1) 2

rate

Tr

position

1

Tr

2

4.3 Polytopic representation of the saturation function Let us consider a generic state-space system of dimension n, with input u = Fx of dimension m. The saturation function satu0 (Fx(t)) is generically described by its components:

8 > < u0(i) if F(i)x(t) > u0(i) satu0 (F(i)x(t)) = > F(i)x(t) if ;u0(i) F(i)x(t) u0(i)  u0(i) > 0 8i = 1 : : :  m : ;u0(i) if F(i)x(t) < ;u0(i)

(4.2)

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

where F(i) denotes the ith row vector of matrix F . Our results are based on a polytopic model representation of the saturated system: see, for example, 12], 21], 6], 16]. Thus, the saturation term satu0 (Fx) dened in equation (4.2) can be written as:

satu0 (Fx(t)) = D((x(t)))Fx(t)

(4.3)

where D((x(t))) 2 <m m is a diagonal matrix whose diagonal elements (i) (x(t)), i = 1 ::: m are dened by:

8 u0(i) > < F(i)x(t) if F(i)x(t) > u0(i) (i)(x(t)) = > 1 if ;u0(i) F(i) x(t) u0(i) : ; Fu0(xi()t) if F(i)x(t) < ;u0(i) (i )

(4.4)

Note that the coe$cient (i) (x(t)) can be viewed as an indicator of the saturation degree of the ith entry of u(t). Thus, the smaller (i) (x(k)), the farther the state vector x is from

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Date: Sep. 19, 2002 GARTEUR TP-120-11

the region of linearity S (F u0) = fx 2
h

1

i0

Consider now lower bounds ` = `(1)  `(m) for (x(t)) dened above. By considering all the possible m-order vectors such that the ith entry takes the value 1 or `(i) , we can conclude that there exist 2m dierent vectors associated to ` . By denoting each of these vectors j (` ), j = 1   2m , respectively, we can dene the following matrices:

Dj (` ) = diag ( j (` ))

(4.5)

S (u0 `) = x 2 <

n#

 u0(i) jF (i)xj `(i)  i = 1 ::: m

(4.6)

satu0 (Fx(t)) = where

X j =1

j (x(t))FDj (` )x(t)

5 6 7 8 9 10

13 14 15 16 17

(4.7)

Hence it follows that at instant t, if x(t) 2 S (u0 ` ), there exist nonnegative scalars

j (x(t)) j = 1   m, such that satu0 (Fx(t)) can be computed as follows: 2m

4

12

Notice that for all x(t) 2 S (u0 ` ), one has:

0 < `(i) (i) (z (t)) 1  i = 1   m

3

11

Furthermore, associated to ` , we can dene the following region in


2

(4.8)

18 19 20 21 22 23 24 25

2m X

26

j =1

28

j (x(t)) = 1.

27

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

;19; 1 2 3 4 5 6 7 8 9 10 11 12

Date: Sep. 19, 2002 GARTEUR TP-120-11

5 Stability analysis of the reduced ADMIRE longitudinal model 5.1 Preliminaries Let us go back to the state-space description of the ADMIRE longitudinal model given by system (3.1). We can now add the saturation terms to close the loops between u3 and y3 , u4 and y4 , fess and fes. The nal Longitudinal structure of the ADMIRE reduced model is then shown on Figure 5.1.

13

+/−u1

14

1

15

s Integrator

+/−u0 1/Tr

16

rate saturation

Gain

position saturation

17 +/−upitch

18 19 20

1

21 22 23 24

theta 1

Stick limit fes theta_c

Mux fess u = [u3;u4]

x’ = Alon x + Blon u y = Clon x + Dlon u

fes

em Mux

longitudinal system

y = [y3;y4]

25

Fig.5.1 Longitudinal model subject to the saturations

26 27 28 29 30 31 32 33 34 35 36

According to the description of the rate saturation given in equation (4.1), we then state the longitudinal saturated model as follows:

8 > x_ = Alon x + Blon2 satupitch (Clon2x + Kc) + Blon3u > <  1 1 ;C x + D sat (C x + K ) + D u u + sat u _ = sat ; u lon3 lon32 upitch lon2 c lon33 u 0 1 Tr Tr > > :  = Clon1 x

(5.1)

37 38 39 40 41 42 43 44 45 46 47 48

where satu0 and satu1 are the saturation vectors corresponding to the position and rate limitations of the canard angles and inboard and outboard elevons, respectively. To solve the stability analysis problem, we rst dene an augmented system related to an augmented state vector involving both the state of the longitudinal model and of the actuator. This augmented vector is dened by:

"

z~(t) = x(t) u(t)

#

2
+m

;20;

Date: Sep. 19, 2002 GARTEUR TP-120-11

System (5.1) may then be written as: 8 z~_ (t) = A z~(t) + B sat (C z~(t) + K (t)) > o 1o upitch 1o c < ; ;  +B 2o satu1 K1o z~(t) + K 2o satu0 C 2o z~(t) + D 2o satupitch (C 1o z~(t) + Kc (t)) (5.2) > : (t) = C z~(t) 3o where " # " # " # A B B 0nm lon lon 3 lon 2 Ao =  B 1o =  B 2o = h 0nm 0mm i h 0ml i h Im i C 1o = Clon2 0lm  C 2o = Clon3 Dlon33  C 3o = Clon1 0lm

h

i

= 0mm ; Tr1 Im  K2o = Tr1 Im  D 2o = Dlon32 In absence of control constraints, i.e., upitch , u0 and u1 ! + 1, on gets for system (5.2): 8_ > > < z~(t) = (A o + B 1o C 1o + B 2o K1o + B 2o K2o C 2o + B 2o K2o D 2o C 1o )~z(t) (5.3) +(B 1o K + B 2o K2o D 2o K )c (t) > > : (t) = C 3o z~(t) It may be quickly veried by checking the controllability and observability matrices that c only acts on 4 modes of the system and that only 3 modes are observable through . It is then strongly recommended to operate some model reduction. This may be done thanks to the function minreal of MATLAB, which also returns an orthogonal matrix U relative to a Kalman decomposition. The state transformation is nally given as: z~ = Tzr  zr 2
T = U0

"

Inr

#

h

i

and T # T = Inr  T # = Inr 0nr n+m;nr U

0n+m;nr nr and the model reduction is applied to the Longitudinal augmented system: 8 > < z_r (t) = A r zr (t) +; B 1r satupitch (C 1r zr (t);+ Kc(t))  + B 2r satu1 K1r zr (t) + K2r satu0 C 2r zr (t) + D 2r satupitch (C 1r zr (t) + Kc (t)) > : (t) = C z (t) 3r r (5.4) where A r = T # A o T  B 1r = T # B 1o  B 2r = T # B 2o  D 2r = D 2o C 1r = C 1o T  C 2r = C 2o T  C 3r = C 3o T  K 1r = K1o T  K 2r = K2o System (5.4) is the state space description of the longitudinal Admire model that is used in the following sections to solve Problem 1. The dimension of the reduced state zr is denoted nr . This model reduction is validated through the Bode diagram (gure 5.2) and the time-response of the linear closed-loop system submitted to a step input (gure5.3) for the ight condition FC1.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

;21; 1

Bode Diagrams

2

From: U(1) 50

0

4

−50

7 8 9

−100

−150

−200 0 −50 −100

To: Y(1)

6

Phase (deg); Magnitude (dB)

3

5

10 11

−150 −200 −250

12

−300

−2

−1

10

13

0

10

16 17

1

10

2

10

3

10

4

10

5

10

10

Frequency (rad/sec)

14 15

Date: Sep. 19, 2002 GARTEUR TP-120-11

Fig.5.2 Bode diagram - Flight condition FC1 - reduction of the uncontrollable and unobservable modes Linear Simulation Results

1.4

18 19

1.2

20

1

21

23

0.8

To: Y(1)

Amplitude

22

0.6

24

0.4

25 0.2

26 27

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (sec.)

28 29 30 31 32 33 34 35 36 37

Fig.5.3 Time-response - Flight condition FC1 - reduction of the uncontrollable and unobservable modes

5.2 Error coordinates representation First we introduce an error coordinates representation which objective is to remove the reference input c from the saturation terms 22]. c then becomes an exogeneous input. The error coordinate transformation is obtained using the new state vector

38

z = M2zr ; Ec  E =

39 40 41 42 43 44 45 46 47 48

M2 2
nr

"

Il

0nr ;ll

#

 z 2
being a nonsingular matrix solution to the following conditions: C 3r M2;1 E

= Il ; 1 C 1r M2 E = K C 2r M2;1 E = 0m:l K1r M ;1 E = 0ml 2

;

;22;

Date: Sep. 19, 2002 GARTEUR TP-120-11

Remark 4 There is some degrees of freedom in the choice of M2 induced by the structure of the conditions. However, by an appropriate choice of M2 , matrix inequalities above are satis ed to within 10;4.

1 2 3 4 5

Remark 5 In the classical error transformation, the rst component of z corresponds to the error e =  ; c 22]. This is not the case in the transformation proposed here, but the objective of the transformation is rather to remove the reference input c from the saturation terms.

Then, this transformation, zr = M2;1 z + M2;1 Ec , the exogeneous input h 0 with i 0 c _c0 the closed-loop saturated longitudinal system writes:

8 > > < z_(t) = A z (t) + B 1 satupitch (C 1 z(t)) +B 2 satu1 (K1 z (t) + K2 satu0 (C 2 z (t) + D 2 satupitch (C 1 z (t)))) + B 3 d(t) > > : (t) = C z(t) + 1 0]d(t)

d =

7 8 9 10 11 12 13 14 15 16 17

(5.5)

18 19 20

3

21

where A

6

22

= M2A r M ;1 2

= C 1r M2;1  K 1 = K1r M2;1 

C1



= M2 B 1r  C 2 = C 2r M2;1  K 2 = K 2r 

B1

= M2 B 2r  C 3 = C 3r M2;1 D 2 = D 2r

B2

B3

= M2A r M ;1 E 2

; E]

23 24 25 26 27 28

5.3 Polytopic modelling

29 30

Let us dene the diagonal matrices D(p), D(0 ) and D(1) whose diagonal elements are respectively dened by:

31

u p(i) (z (t)) = min 1  pitch(i)   i = 1   l (5.6) C 1(i) z (t) !  u 0(i)   i = 1   m (5.7) 0(i) (z (t)) = min 1  C 2(i) z (t) + D 2(i) D(p )C 1 z (t)  ! u 1(i)   i = 1   m(5.8) 1(i) (z (t)) = min 1  K1(i) z (t) + K2(i) D(0 )(C 2 + D 2 D(p)C 1 z (t))

34



!

33

35 36 37 38 39 40 41 42

From the above denition, system (5.5) is equivalent to the following one:

z_ (t) = A z (t) + B 1 D(p )C 1 z (t) +B 2 D(1)(K1 z (t) + K2 D(0)(C 2 z (t) + D 2 D(p )C 1 z (t))) + B 3 d(t)

32

43 44

(5.9)

45 46 47 48

;23; 1 2 3 4 5 6

Furthermore, let us dene the following regions of


S (upitch p` ) = z 2 < S (u0 0`) =

7 8 9 10 11 12 13 14 15

S (u1 1`) =

2l  \

j =1

20

25 26 27 28 29

32

0`(i)

z_ (t) =

33

+

34

36

38 39 40 41 42 43 44 45 46 47 48

(5.11)



z 2
2l X 2m X 2m X

(5.13)

f

j k q A z (t) + B 1 Dj (p`)C 1 z(t) + B 2 Dk (1`)K1 z(t) j =1 k=1 q=1 B 2 Dk (1` )K2 Dq (0` )C 2 z (t) + B 2 Dk (1`)K2 Dq (0` )D 2 Dj (p` )C 1 z (t) + B 3 d(t)

where

2l X

j =1

g

(5.14)

35

37

(5.10)

Moreover, matrices Dj (p` ), Dk (0` ) and Dq (1` ) dene the 2l, 2m and 2m vertices of polytopes of diagonal matrices whose elements take values 1 or p`(i) , 1 or 0`(i) and 1`(i), respectively. D(p), D(0) and D(1) belong to the convex sets formed by their respective vertex matrices. Hence, it follows that at instant t, if z (t) 2 S (upitch  p`) \ S (u0 0`) \ S (u1 1`), there exist nonnegative scalars j (z (t)) j = 1   2l, k (z (t)) k = 1   2m and q (z (t)) q = 1   2m such that z_ (t) can be computed as follows:

30 31

  u0(i) 2(i) z + D 2(i) Dj (p` )C 1 z   i = 1 ::: m

0 < p`(i) p(i) (z (t)) 1  i = 1   l 0 < 0`(i) 0(i) (z (t)) 1  i = 1   m 0 < 1`(i) 1(i) (z (t)) 1  i = 1   m

19

24

z2<

p`(i)

 u1(i) (5.12) D 2 Dj (p` )C 1 )z j 1`(i)  i = 1 ::: m For all z (t) 2 S (upitch p` ) \ S (u0 0` ) \ S (u1 1` ), p` , 0` and 1` dene lower bounds on p , 0 and 1 , respectively, i.e.:

18

23

  upitch(i)  i = 1 ::: l 1(i) z

nr # C

j =1k=1

17

22

nr # C

2l \ 2m \ 

16

21

Date: Sep. 19, 2002 GARTEUR TP-120-11

j (z (t)) =

2m X

2m X

k=1

q=1

k (z(t)) =

q (z (t)) = 1.

5.4 Main proposition Let us dene, for j = 1   2l, k = 1   2m, q = 1   2m:

M(j k q) = A 0 P + P A + C 0 Dj (p`)B 0 P + P B

Dj (p`)C 1 + K01 Dk (1`)B 02 P (5.15) +P B 2 Dk (1` )K1 + C 02 Dq (0` )K02 Dk (1` )B 02 P + P B 2 Dk (1`)K 2 Dq (0` )C 2 0 0 0 0 +C 1 Dj (p` )D 2 Dq (0` )K2 Dk (1` )B 2 P + P B 2 Dk (1` )K2 Dq (0` )D 2 Dj (p` )C 1 1

1

1

;24;

Date: Sep. 19, 2002 GARTEUR TP-120-11 Let us also dene Fe

1

= ; (A + B 1 C 1 + B 2 K 1 + B 2 K2 C 2 + B 2 K 2 D 2 C 1

);1 B

2

3

(5.16)

Aiming to solve our problem, we can state the following proposition.

2
"

PB3 < 0 ;!R

B 03 P

"

P

p`(i)C 1(i)

"

p`(i)C 01(i) u2pitch(i)=

#

0

P

#

R

?

C 1(i) Fe

"

R

u2pitch(i)= ?

R

9 10 11 12

(5.18)

15 16 17

0 #

i = 1   m

(5.19)

19 20 21 22

0

i = 1   m (5.20)

23 24 25 26

0 #

(C 2 + D 2 C 1 )(i)Fe u20(i)=

"

8

18

?

#

7

13

i = 1   l

1`(i)(K1 + K2 Dk (0`)(C 2 + D 2 Dj (p` )C 1 ))(i) u21(i) =

"

(5.17)

6

14

P ? 2 0`(i)(C 2 + D 2 Dj (p`)C 1 )(i) u0(i)=

"

nr ,R

#

M(j k q) + P

4 5

= R0 > 0, R 2 <2l 2l , positive scalars ! , ,  ,  ,  and vectors p` , 0` and 1` satisfying the following matrix inequalities:

Proposition 1 If there exist P = P 0 > 0, P

3

i = 1   l

27

(5.21)

28 29 30

0

?

(K 1 + K2 (C 2 + D 2 C 1 ))(i)Fe u21(i) =

#

i = 1   m

31

(5.22)

32 33 34

0

i = 1   m

35

(5.23)

36 37 38

; + ! 0 0 < p`(i) 1  i = 1   l 0 < 0`(i) 1  i = 1   m ? is the substitute for blocks ensuring matrix symmetry.

(5.24)

39 40 41

(5.25)

42 43 44

(5.26)

45 46 47 48

;25; 1 2 3 4 5 6 7 8 9

Date: Sep. 19, 2002 GARTEUR TP-120-11

0 < 1`(i) 1  i = 1   m (5.27) 8 j = 1   2l, k = 1   2m, q = 1   2m, then the asymptotic stability of the longitudinal ADMIRE model is preserved for all z (t) 2 0 de ned by:   0 = z 2
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Proof. The satisfaction of (5.18), (5.19) and (5.20) means that the ellipsoid 0 dened

in (5.28) is included in the set S (upitch  p`) \ S (u0 0`) \ S (u1 1` ). The satisfaction of relation (5.17) means that the ellipsoid 0 is a domain of invariance for the trajectories of system (5.14) which represents the saturated system (5.5). Moreover, let us consider the quadratic function V (z ) = z 0 Pz and compute V_ (z ) along the trajectories of system (5.14). Hence, in order to prove that 0 is a domain of invariance, we have to prove that V_ (z ) 0 for any z (t) 2 0 , and any admissible reference, i.e. d 2 D0 . By using the S -procedure 4] it is possible to show that it is equivalent to seek  > 0 and ! > 0 such that V_ (z(t)) +  ;z(t)0Pz(t) ;  + ! ; ; d(t)0Rd(t) 0 (5.30) Thanks to convexity properties, if relations (5.17) and (5.24) are veried then inequality (5.30) holds. Finally, relations (5.21), (5.22), (5.23) ensure that for any constant reference input d belonging to D0, the equilibrium point ze belongs to the domain of linearity of the system, i.e., when no saturation occurs. It is obtained, when z_ (t) = 0, as:

ze = Fe d

29 30 31 32

with Fe dened in equation (5.16) and has to satisfy:

34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

;upitch C ze upitch ;u (C + D C )ze u ;u (K + K C + K D C 1

33

2

0

2

2 1

0

1

1

2 2

2 2 1

)ze u1

(5.31)

Proposition 1 provides a su$cient condition in order to determine a domain in which the system can be initiated and a domain of admissible reference input dened by c and its time derivative. It is then interesting to derive a solution in order to obtain ellipsoids 0 and D0 the largest as possible. In this sense, we suggest the following optimization problem: minf0 trace (R) + 1 trace (P ) ; 2 ; 3  g (5.32) subject to Relations (5:17) ; (5:27) where i > 0, i = 0   3, are tuning parameters to aect some weights to each element.

;26;

Date: Sep. 19, 2002 GARTEUR TP-120-11

Remark 6 Note that relations (5.17)-(5.23) are not LMIs since it appears many products

1

in the variables. However, the problem may be solved with some of the variables a priori xed, or by using some relaxation techniques to recursively solve dierent subproblems.

2

5.5 Results

5

3 4

6 7

5.5.1 Numerical data

8

This section is devoted to the stability analysis of the pitch-axis control described through the longitudinal Admire submodel (5.5). The ight condition used for numerical evaluation are: Mach 0.25, altitude 500m. Actuator limitations are given in Table 5.1. Canard angles position limits Elevon position limits Canard angles rate limits Elevon rate limits Pilot stick limits

 25 deg  30 deg  50 deg/s  150 deg/s  80 Newtons

Table 5.1 Position and rate actuator limitations

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

For this ight condition, the pitch-loop pilot gain K was chosen as 1378 13]. The reduced longitudinal Admire model (5.4) is of dimension nr = 7, which corresponds to remove 23 states from the augmented model (5.2) formed of the orginal state vector x of dimension n = 28 and the input vector u of dimension m = 2. According to Remark 6, some parameters have been frozen for all the numerical evaluation:

 = 0:8 

! = 0:1

Note however that the results are related to this particular choice.

24 25 26 27 28 29 30 31 32 33 34 35 36 37

5.5.2 Inuence of the pilot stick saturation Table 5.2 shows the inuence of the saturation allowance on the pilot stick limit during pitch-axis control. The saturation allowance is caracterised by the value of p` belonging to the interval ]0  1]. p` = 1 corresponds to the linear case when no saturation on the pilot stick is allowed. The admissible pitch angle setpoints are caracterised through the ellipsoidal set D0 in terms of amplitude c and rate of variation _c . No saturation of the canard angles and

38 39 40 41 42 43 44 45 46 47 48

;27; 1 2 3 4 5 6 7 8

Date: Sep. 19, 2002 GARTEUR TP-120-11

Volume D0 Volume 0 c max _c max (deg) (deg/s) ; 8 1 8.74 2.22 10 2.96 2.96 ; 6 0.335 26.72 8.16 10 5.17 5.17

p`

Table 5.2 Inuence of the pilot stick limitations on the admissible reference

9 10 11 12 13

elevons actuators are allowed (0 and 1 equal to 1). The maximum of c is obtained for p` = 0:335. Figure 5.4 plots the optimal values for the pitch angle reference obtained for various values of p` .

14 15 16

5.2

17

5.1

18

5

19 4.9

21

4.8

22

θc

20

4.7

23 24

4.6

25

4.5

26

4.4

27 28 29 30 31 32 33

4.3 4.2 0.25

0.3

0.35 0.4 saturation degree on the pilot stick position (αp)

0.45

0.5

Fig.5.4 Evolution of the maximal pitch angle setpoint with respect to the saturation allowance of the pilot stick position

34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

5.5.3 Inuence of the actuators (canard angle and elevon) saturation Next, the inuence of the saturation degrees allowed for the position and rate limitations of the canard angles and elevons is evaluated. The smaller the component is, the larger the saturation allowance on the actuator is allowed. Figures 5.5, 5.6, 5.7 and 5.8 show the inuence of the saturation degrees of the actuators on the maximal admissible pitch angle reference. These plots have to be related to the corresponding ellipsoids 0 of safe behavior, shown in Figures 5.9, 5.10, 5.11 and 5.12, respectively. The saturation allowance on the pilot stick is set to p` = 0:335.

;28;

Date: Sep. 19, 2002 GARTEUR TP-120-11

1

5.2

2 5.15

3 4

5.1

5 5.05 θc

6 7

5

8 4.95

9 10

4.9

11 4.85 0.8

0.82

0.84

0.86 0.88 0.9 0.92 0.94 saturation degree on the canard angle position (α

0.96

0.98

1

)

12

0(1)

13

Fig.5.5 Evolution of the maximal pitch angle setpoint with respect to the saturation allowance of the canard angle position 5.5

14 15 16 17

5

18 19

4.5

20

4

θc

21 3.5

22 23

3

24

2.5

25 2

1.5 0.8

26 0.82

0.84

0.86 0.88 0.9 0.92 0.94 saturation degree on the elevon position (α )

0.96

0.98

1

0(2)

Fig.5.6 Evolution of the maximal pitch angle setpoint with respect to the saturation allowance of the elevon position From these gures, it can be concluded that saturation allowance on the actuators position and rate does not increase the size of admissible pitch angle references for which the stability of the aircraft Admire longitudinal model is guaranteed a priori. However, saturation allowance of the canard angle rate seems to have a positive eect of the size of the domain of safe behavior of the aircraft Admire longitudinal model expressed through the ellipsoidal set 0 (cf Figure 5.11 and to a certain degree Figure 5.12).

27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

In Table 5.3, the inuence of the saturation degrees allowed for the position and rate limitations of the canard angles and elevons is evaluated. The smaller the component is, the larger the saturation allowance on the actuator is allowed. The saturation allowance on the pilot stick is set to p` = 0:335.

43 44 45 46 47 48

;29; 1

Date: Sep. 19, 2002 GARTEUR TP-120-11

5.22

2 5.2

3 4

5.18

5 5.16 θc

6 7

5.14

8 5.12

9 10

5.1

11 5.08 0.8

12

0.82

0.84

0.86 0.88 0.9 0.92 0.94 saturation degree on the canard angle rate (α )

0.96

0.98

1

1(1)

13 14 15 16

Fig.5.7 Evolution of the maximal pitch angle setpoint with respect to the saturation allowance of the canard angle rate 5.3

17 5.2

18 19

5.1

20

5

θc

21 4.9

22 23

4.8

24

4.7

25 4.6

26 27

4.5 0.8

0.82

0.84

29 30 31

0.86 0.88 0.9 0.92 0.94 saturation degree on the elevon rate (α )

0.96

0.98

1

1(2)

28

Fig.5.8 Evolution of the maximal pitch angle setpoint with respect to the saturation allowance of the elevon rate −6

10

x 10

32 9

33 34

36 37 38 39

volume of stability ellipsoid Ξ

35

8

7

6

5

4

40 41 42 43 44 45 46 47 48

3

2 0.8

0.82

0.84

0.86 0.88 0.9 0.92 0.94 saturation degree on the canard angle position (α

0.96

0.98

1

)

0(1)

Fig.5.9 Evolution of the maximal pitch angle setpoint with respect to the saturation allowance of the canard angle position

;30;

Date: Sep. 19, 2002 GARTEUR TP-120-11

1

−6

7

x 10

2 3

6

4

volume of stability ellipsoid Ξ

5

5 4

6 7

3

8 2

9 10

1

11 0 0.8

0.82

0.84

0.86 0.88 0.9 0.92 0.94 saturation degree on the elevon position (α0(2))

0.96

0.98

1

12 13

Fig.5.10 Evolution of the maximal pitch angle setpoint with respect to the saturation allowance of the elevon position −6

12

x 10

14 15 16 17

volume of stability ellipsoid Ξ

11

18

10

19

9

20

8

21

7

22 23

6

24 5

25

4

26

3 0.8

0.82

0.84

0.86 0.88 0.9 0.92 0.94 saturation degree on the canard angle rate (α1(1))

0.96

0.98

1

28

Fig.5.11 Evolution of the maximal pitch angle setpoint with respect to the saturation allowance of the canard angle rate −6

volume of stability ellipsoid Ξ

14

27

x 10

29 30 31 32

13

33

12

34

11

35

10

36

9

37

8

38

7

39

6

40

5

41

4 0.8

0.82

0.84

0.86 0.88 0.9 0.92 0.94 saturation degree on the elevon rate (α )

0.96

0.98

1

1(2)

Fig.5.12 Evolution of the maximal pitch angle setpoint with respect to the saturation allowance of the elevon rate

42 43 44 45 46 47 48

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Date: Sep. 19, 2002 GARTEUR TP-120-11

canard angle elevon Volume D0 Volume 0 c max _c max position rate position rate (deg) (deg/s) ; 6 1 1 1 1 26.72 8.16 10 5.17 5.17 ; 5 0.975 0.98 0.965 0.975 23.02 1.02 10 4.80 4.80 ; 6 0.975 0.98 1 1 23.93 3.59 10 4.89 4.89 ; 6 1 1 0.965 0.975 26.24 5.00 10 5.12 5.12 ; 5 0.975 1 0.965 1 25.78 1.35 10 5.08 5.08 ; 5 1 0.98 1 0.975 23.49 1.12 10 4.85 4.85 ; 6 0.8 1 1 1 23.98 3.12 10 4.90 4.90 ; 7 1 0.8 1 1 2.84 1.87 10 1.69 1.69 ; 6 1 1 0.8 1 26.32 5.67 10 5.13 5.13 ; 6 1 1 1 0.8 21.04 5.67 10 4.59 4.59 ; 7 0.8 0.8 1 1 2.66 1.15 10 1.63 1.63 ; 6 1 1 0.8 0.8 20.62 3.47 10 4.54 4.54 ; 6 0.8 1 0.8 1 23.41 9.98 10 4.84 4.84 1 0.8 1 0.8 0.88 1.69 10;7 0.94 0.94 ; 7 0.8 0.8 0.8 0.8 1.59 1.42 10 1.26 1.26 Table 5.3 Inuence of the canard angles and elevons limitations on the admissible reference

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

5.5.4 Domains of safe behavior Finally, the inuence of the saturation allowance is illustrated on Figures 5.13 and 5.14. The projection of the admissible sets of states and pitch references are plotted on these gures for three cases:

 No saturation allowance, i.e. p` = 1,  ` = 1 1]0 and  ` = 1 1]0 in solid line#  Saturation of the pilot stick allowed. The solution which maximizes the size of 0

1

admissible pitch angle reference is given for p` = 0:335 and plotted in dashdot line#

 Saturation allowed.

Saturation of the canard angles and elevons position and rate do not increase the size of admissble reference. However, it increases the domain of safe behaviour (stability domain) of the aircraft Admire.

5.5.5 Conclusion Main comments relative to the numerical evaluation of the inuence of position and rate saturations of the canard angles and elevons are:

 The saturation of the canard angle and elevon position and rate have rather a decreasing eect on the size of state domain of safe behavior for the aircraft Admire

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Date: Sep. 19, 2002 GARTEUR TP-120-11

1

0 0 x2

2

0

x7

1

0 −5 −2 2

x7

−1 −2 1

0 x1

0 x2

0

3 4 5

0 x1

1

6 7

0

8

−2 −1 2

10

9

0 x1

1

11 12

0 −2 −2 5

2

2

0 −5 −1 2

1

x5

0 x1

0 x1

0 −1 −1 5

x4

x3

−1 −1 1

1

0 −2 −1 1

x6

0 x1

x6

x5

−2 −1 2

0

x4

0

1

5 x4

1 x3

x2

2

13 14

0 x2

2

15 16

0

17 18

1

0

x7

2

0 −1 −1 1

x6

0 x3

0 x2

−5 −1 2

0 x3

1

0

0 x3

1

19 20 21

0 −2 −1 2

x7

2

0 −2 −1 2

x5

0 x2

−2 −2 1 x6

x5

−1 −2 2

22 23

0 x3

1

24 25 26

0

27

−1 −5 2

5

0 −1 −2

0 x5

2

0 x4

0 −2 −2

−2 −5 2

5 x7

0 x4

x7

x6

−2 −5 1

0 x5

2

0 x4

5

29 30

0 −2 −1

28

31 32

0 x6

1

33 34 35

Fig.5.13 Projection in two-dimensionnal subspaces of the ellipsoid of admissible state 0 longitudinal model.

 Saturation of these actuators leads to decreasing of the admissible pitch angle references which guarantee a safe behavior of the aircraft.

 Saturation allowance of the elevon position has the worst eect on the domains of safe behavior and moreover on the size of admissible pitch angle setpoint.

On the other hand, to a certain degree, the saturation of the pilot stick largely increases the size of both admissible pitch angle references and domain of safe behavior of the aircraft

36 37 38 39 40 41 42 43 44 45 46 47 48

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Date: Sep. 19, 2002 GARTEUR TP-120-11

6

2 4

3 4

6 7 8

c

2 time derivative of θ

5

0

−2

9 10

−4

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

−6 −6

−4

−2

0 θc

2

4

6

Fig.5.14 Ellipsoid of admissible reference input D0 Admire longitudinal model.

Date: Sep. 19, 2002 GARTEUR TP-120-11

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6 Main conclusion This report has been devoted to the stability analysis of the Admire model in presence of actuator limitations. Conditions allowing to exhibit a solution are expressed through matrix inequalities given in Proposition 1. Let us underline that this Proposition presents only a su$cient condition for the solution of Problem 1. The main sources of conservatism here reside in the use of quadratic Lyapunov functions and in the modeling on the saturated system by dierential inclusions. The solution is given through the computation of admissible domains of state and reference input, denoted 0 and D0 , respectively. These domains are obtained by solving an optimization problem submitted to matrix inequalities constraints given in Proposition 1. After some steps of model reduction, the approach has been applied to the longitudinal Admire submodel with standard rate limited actuators.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

The prospective studies will concern rst the stability analysis of the lateral submodel. Another main perspective of this work will concern the stability analysis of the Admire model when SAAB or DASA phase-compensation achemes are used to compensate for rate-limiting.

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Date: Sep. 19, 2002 GARTEUR TP-120-11

References 1] D. Angeli, A. Casavola, E. Mosca, Set-point output tracking under positional and rate actuator saturation, Proc. of the 4th European Control Conference, Karlsruhe (Germany), September 1999. 2] D.S. Bernstein and A.N. Michel (Editors), Special issue: Saturating actuators, Int. J. of Robust and Nonlinear Control, vol.5, n.5, pp.375-540, 1995. 3] F. Blanchini and S. Miani, Any domain of attraction for a linear constrained system is a tracking domain of attraction, SIAM J. Control Optim., vol.38, no.3, pp.972-994, 2000. 4] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan: Linear matrix inequalities in system and control theory, SIAM Studies in Applied Mathematics, 15, 1994. 5] A.H. Glattfelder and W. Schaufelberger, Stability analysis of single loop control sytems with saturation and antireset-windup circuits, IEEE Trans. Autom. Control, vol.28, pp.1074-1081, 1983. 6] J.M. Gomes da Silva Jr and S. Tarbouriech Local stabilization of linear discrete-time systems under amplitude and rate saturating actuators, Workshop systems with timedomain constraints, Eindhoven (Holland), august 30 - september 1, 2000. 7] N.J. Krikelis, State feedback integral control with \intelligent integrator", Int. J. of Control, 32, pp.465-473, 1980. 8] N.J. Krikelis and S.K. Barkas, Design of tracking systems subject to actuator saturation and integrator windup, Int. J. Control, 39, pp.667-682, 1984. 9] Z. Lin and A. Saberi, Semi-global exponential stabilization of linear systems subject to input saturation via linear feedback, Systems & Control Letters, vol.21, pp.225-239, 1993. 10] Z. Lin, A.A. Stoorvogel, A. Saberi, Output regulation for linear systems subject to input saturation, Automatica, vol.32, no.1, pp.29-47, 1996. 11] R.B. Miller and M. Pachter, Manoeuvring ight control with actuator constraints, J. of Guidance, Control and Dynamics, vol.20, no.4, pp.22-23, 1997. 12] A.P. Molchanov and E.S. Pyatniskii: Criteria of asymptotic stability of dierential and dierence inclusions encountered in control theory, Systems & Control Letters, 13, pp.59-64, 1989. 13] I. Postlethwaite, M. Turner and E. Prempain: An H1 approach to rate-limit compensation for pilot-induced-oscillation avoidance: concept, demonstration and comparison (preliminary version), GARTEUR report FM(AG12)/TP-120-06, 2001.

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14] L. Rundqwist and K. Stahl-Gunnarson, Phase compensation of rate limiters in unstable aircraft, AIAA, 1996. 15] A. Saberi, A.A. Stoorvogel, P. Sannuti, Control of linear systems with regulation and input constraints, Springer-Verlag, London (UK), 1999. 16] G. Scorletti, J.P. Folcher and L. El Ghaoui: Convex output feedback control design with input saturations: a comparison, Proceedings of the second IFAC Symposium, Budaptest (Hungary), pp.247-252, june 1997. 17] E.D. Sontag and H.J. Sussmann, Nonlinear output feedback design for linear systems with saturating control, Proc. of the 29th IEEE-CDC, Honolulu (USA), pp.3414-3416, December 1990. 18] A.A. Stoorvogel and A. Saberi (Editors), Special issue: Control problems with constraints, Int. J. of Robust and Nonlinear Control, vol.9, n.10, 1999. 19] S. Tarbouriech and G. Garcia (Editors), Control of uncertain systems with bounded inputs, Lecture Notes in Control and Information Sciences vol.227, Springer-Verlag, 1997.

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20] S. Tarbouriech, G. Garcia and I. Queinnec: Stabilization of linear time-delay systems with anti-stable eigenvalues under amplitude and rate saturation, 3rd IFAC Workshop on TIme Delay Systems (TDS'2001), Santa Fe (USA), december 8-10, 2001.

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21] S. Tarbouriech and J.M. Gomes da Silva Jr.: Admissible polyhedra for discrete-time linear systems with saturating controls, Proceedings of the American Control Conference (ACC), Albuquerque (USA), 6, pp.3915-3919, june 4-6, 1997.

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22] S. Tarbouriech, C. Pittet and C. Burgat: Output tracking problem for systems with input saturations via non linear integrating actions, Int. Journal of Robust and Nonlinear Control, 10, pp.489-512, 2000.

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23] S. Tarbouriech, I. Queinnec, C. Pittet, Output-reference tracking problem for discretetime systems with input saturations, IEE Proc. on Control Theory and Applications, vol.147, no.4, pp.447-455, 2000.

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24] F. Tyan and D.S. Bernstein, Anti-windup compensator synthesis for systems with saturating actuators, Int. J. of Robust and Nonlinear Control, vol.5, pp.521-537, 1995.

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