Nonlinear multivariable flight control Ola Hrkegrd Linkpings - - PDF document

1

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Nonlinear multivariable flight control

Ola Härkegård Linköpings Tekniska Högskola

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Background

Nonlinear

• Different flight cases
• High angle-of-attack
• Rigid body dynamics

Multivariable

• 6 DOF
• Control surface redundancy
• Unconventional control surfaces

2

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

High angle of attack

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Unconventional control surfaces

vs.

3

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Outline

Aircraft Backstepping Control allocation

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Outline

Aircraft Backstepping Control allocation

• Why fly-by-wire?
• Control objectives
• Actuators
• Why fly-by-wire?
• Control objectives
• Actuators

4

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Why fly-by-wire?

control surf. visual info, cockpit displays, etc. Control system sensors stick

• Stabilize aircraft
• Handling qualities
• Advanced control surfaces
• Autopilot functionality

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Control objectives

Longitudinal control V nz

Load factor

α

Angle of attack

q

Pitch rate

5

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Control objectives

Lateral control V

Roll rate

p β

Sideslip angle

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Actuators

Canards Elevons Rudder (TVC)

6

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Actuators Actuators Aircraft dynamics Aircraft dynamics M Control laws Control laws M

System overview

Control allocation Control allocation Flight control sys. Flight control sys. u

Inner control loop: Modular control design:

α, β, p x r u

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Outline

Aircraft Backstepping Control allocation

• What is backstepping?
• Why use it?
• Research at LiTH
• What is backstepping?
• Why use it?
• Research at LiTH

7

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

What is backstepping?

Constructive nonlinear control design method.

( ) ( ) ( )

u , x , , x , x , x f x x , x , x f x x , x f x

n 3 2 1 n n 3 2 1 2 2 2 1 1 1

• =

= = Model structure:

Same requirement as in feedback linearization

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Why use backstepping?

Can benefit from “useful” nonlinearities

May require less control effort modeling information → robustness

Can achieve GAS when feedback linearization fails

8

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Design procedure

( )

2 1 1 1

x , x f x =

• (

) ( )

u , x , , x , x , x f x x , x , x f x

n 3 2 1 n n 3 2 1 2 2

• =

=

2 1 1

x V =

( )

1 des 2 2

x x x =

decreases if

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Step backwards

( ) ( )

3 2 1 2 2 2 1 1 1

x , x , x f x x , x f x = =

• (

)

u , x , , x , x , x f x

n 3 2 1 n n

• =

decreases if

( )

2 des 2 2 1 2

x x V V − + =

( )

2 1 des 3 3

x , x x x =

9

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Step backwards

( ) ( ) ( )

u , x , , x , x , x f x x , x , x f x x , x f x

n 3 2 1 n n 3 2 1 2 2 2 1 1 1

• =

= =

decreases if

( )

2 des n n 1 n n

x x V V − + =

−

( )

n 3 2 1

x , , x , x , x k u

• =

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Research at LiTH

Are there useful nonlinearities in the aircraft dynamics?

Well, at least harmless.

Can backstepping be applied to multivariable flight control?

Yes, applicable to general rigid body dynamics.

10

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

( ) ( ) r

sin F Y mV 1

T

− β − β = β

• r

FT

Sideslip regulation

V β

( )

u , r , N J 1 r

z

β =

• N
• 20
• 10

10 20

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2 x 10

4

Y (N) beta (deg)

Sideforce Y(β)

• Linear
• Independent of Y(β)
• Inverse optimal
• Linear
• Independent of Y(β)
• Inverse optimal

Backstepping N = – k1β – k2r

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Outline

Aircraft Backstepping Control allocation

• What is control allocation?
• Why use it?
• Research at LiTH
• What is control allocation?
• Why use it?
• Research at LiTH

11

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Actuators Actuators System dynamics System dynamics

What is control allocation?

How should the total control effort be distributed among the actuators? How should the total control effort be distributed among the actuators? Control design:

• Determine desired total control effort
• Distribute the control effort among the actuators

Control laws Control laws Control allocation Control allocation

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Applications

12

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Why use control allocation?

Easy to reconfigure Cheap way to handle actuator limits

”Poor man’s MPC”

Necessary for certain control design methods

Feedback linearization (NDI) Backstepping

Modularity

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Example: Hardover

Max deflection after 1 s

13

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Mathematical formulation

• u = true control signal
• v = virtual control signal (total control effort)
• Model: v = g(u)
• Linearization: v = Bu
• Constraints:

max min

u u u ≤ ≤

max min

u u u

• ≤

≤

( ) ( )

t u u t u ≤ ≤    Actuators Actuators v u

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Example

2 1

u u 2 x − − =

• 

  + = − = ⇔

2 1

u u 2 v v x

• 1

u

2

u 2 1 3 v = 4 v = 5 v = Dynamics: Constraints: 2 u 1 u

2 1

≤ ≤ ≤ ≤ Control law: Allocation problem: x v = v u u 2

2 1

= +

14

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Optimization based control allocation

u u u v Bu ≤ ≤ =

• (

) 2

v u u u

v Bu W min arg − = Ω

≤ ≤

• (

) 2

d u u

u u W min arg u − =

Ω ∈

2 1

1

u

2

u 5 . 3 v =

d

u • 2 u 1 u v u u 2

2 1 2 1

≤ ≤ ≤ ≤ = + 1 W I W u

v u d

= = = Minimize cost function. Ω

• 

       = 5 . 1 1 u

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Research at LiTH

Can standard QP methods be used for control allocation in real time?

Yes.

How can filtering be included in the allocation?

Also penalize changes in the control signal.

How is control allocation related to LQ control?

Equivalent without constraints.

15

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Dynamic control allocation

What? Why?

Actuator dynamics ”Practical aspects” G(s) G(s) v u u u u v Bu ≤ ≤ = Constraints:

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Dynamic control allocation

How?

Also penalize changes in the control signal.

( )

( )

v Bu t u W min

2 2 1 t u

=

( ) ( ) ( )

2 2 2

T t u t u W − − +

( ) ( ) ( )

t Gv T t Fu t u + − = ⇒ Stable linear filter

16

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Example: Flight control

10

• 2

10-1 10 10 1 102 10-4 10-3 10-2 10-1 100

Frequency (rad/sec) Control effort distribution —Canards —Elevons —TVC

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Control allocation vs LQ control

u B Ax x

u

+ =

• t

d u R u x Q x min

1 T 1 T u ∫ ∞

+ x L u

1

− = Bu v v B Ax x

v

= + =

• x

L L u

2 3

− = t d v R v x Q x min

2 T 2 T v ∫ ∞

+ x L v

2

− = v Bu då Wu min

2 u

= v L u

3

=

1 1 R

, Q W , R , Q

2 2

LQ LQ u r x LQ LQ v r x CA CA u B B B

v u =

17

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Example

Admire (FOI)

Mach 0.22, 3000 m x = (α β p q r)

Approximations:

Ignore actuator dynamics View control surfaces as moment generators

Model (for control):

c

δ

re

δ

le

δ

r

δ

δ = + = B v v B Ax x

v

• angular acc.

control surfaces

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Simulation results

LQ LQ+CA with constraints

18

Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13

Summary

Control laws Control laws Control allocation Control allocation

Nonlinear dynamics Actuator redundancy Actuator constraints Nonlinear dynamics Actuator redundancy Actuator constraints Configuration can handle