Andrea Serrani

Department of Electrical and Computer Engineering

The Ohio State University

The Role of Zero Dynamics in Aerospace Systems

A Case Study in Control of Hypersonic Vehicles

Outline

q  Issues in Control of Hypersonic Vehicles (HSVs)

q Trajectory Tracking for a Longitudinal HSV Model q Control by Model Inversion q The Zero Dynamics of HSVs q Pitfalls of Approximate Linearization q Shaping the Zero Dynamics: Output Redefinition q Simulation Results

q Conclusions

2

Air-breathing Hypersonic Vehicles

X-51

q  Oxygen taken from the atmosphere – no need to carry oxidant on board q  Increased payload for civilian and military applications q  Part of Two-Stage-to-Orbit concept q  Rocket booster or combined ramjet-scramjet cycle required.

Artist’s rendering of X-51. Image courtesy of NASA

11October 30 - November 1, 2007 FAP Annual Meeting - Hypersonics Project

Two Stage to Orbit Concept

• Reference Vehicle for technology evaluation

• Focus of Level 4 tool development

• TBCC first stage and rocket powered second stage (current version)

Hypersonic Two-Stage-To-Orbit Vehicle Concept (NASA)

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•  Propulsion system integrated in the airframe Fuselage provides compression at the inlet and serves as expansion nozzle Scramjet engine below the CG generates thrust / pitching moment coupling

•  Thrust produced by the scramjet engine affected by the inflow of air Bow shock and spillover of airflow depend on angle-of-attack and Mach no.

•  Structural modes significantly affect aerodynamic and propulsive forces Flexibility effects produce significant changes in lift and pitching moment

•  Elevator-to-Lift coupling generates loss of lift when climbing Non-minimum phase behavior that complicates control system design

Issues in HSV Dynamics: Aerodynamics

�,M1 T

X-43

4

˙V =

1

m

⇥T (�) cos↵�D(q,↵)�mg sin(✓ � ↵)

⇤

˙h = V sin(✓ � ↵)

↵ = Q� 1

mV

⇥L(q,↵, �e) + T (�) sin↵�mg cos(✓ � ↵) ]

˙✓ = Q

˙Q =

1

Iyy

⇥M(q,↵, �e) + zTT (�)

⇤

Longitudinal Vehicle Model

5

D(q,↵) = qSC ↵D (↵), C ↵

D (↵) = C↵2

D ↵2 + C↵D↵+ C0

D

L(q,↵, �e) = qS⇥C ↵

L (↵) + C �L�e

⇤, C ↵

L (↵) = C↵L↵+ C0

L

M(q,↵, �e) = c qS⇥C ↵

M (↵) + C �M�e

⇤, C ↵

M (↵) = C↵2

M ↵2 + C↵M↵+ C0

M

This term is responsible for the “non-minimum phase” behavior: the input appears “too soon” in the equations �e

Elevator-to-Lift Coupling

Output Trajectory Tracking

6

inverse model

feedback controller

plant yuurefyref

xrefx

tracking controller

•  The control action embeds the inversion of the plant model •  How do we compute the inverse? •  What is the resulting dynamics when ? y = yref

yrefyrefuref

x = xref

inverse model

plant model

x = [V, h,↵, ✓, Q]

u = [�, �e]

y = [V, h]

The Zero Dynamics of Control Systems

7

Z!

x(t)

xref(t)

x(0) ⇠(t) = ⇠ref(t)

⌘(t) = q(⌘(t), ⇠ref(t))

e(t) = 0

The set of all forced trajectories of the system compatible with zero tracking error

•  Non-interacting control and disturbance decoupling with stability •  Linearization of the input-output and input-state map •  Tracking and regulation •  Limit of performance of nonlinear control systems

A fundamental concept for a myriad control problems:

−4 −3 −2 −1 0 1 2 3 4−20

−15

−10

−5

0

5

10

15

20

Real Axis

Imag

inar

y Ax

is

PolesZeroes

Flexibleeffects

Pitch DynamicsZeros, one isnonminimumphase

Non-minimum Phase Behavior of HSVs

The system has unstable zero dynamics when

8

(V = u1

h = u28><

>:

✓ = Q

Q =1

IyyM(q, ✓, u1, u2) ,

@

@✓M(q, ✓, 0, 0) > 0

hyperbolic saddle

Controlling altitude via model inversion (linearization by feedback) results in an unstable closed-loop system (even if the tracking error is regulated)

y = [V, h]

Feedback transformation

•  Approximate Linearization: Feedback linearization with NMP coupling strategically ignored to achieve full relative degree (no zero dynamics)

•  Outer-loop compensator achieves stable tracking for the rigid-body model

•  Results in instability when flexible dynamics are included in the model (closed-loop system not robust to dynamic uncertainty)

Naïve Approach: Ignoring the Coupling

9

Beyond (Approximate) Linearization

Approach:

§  Exploiting Control Input Redundancy §  Tool for Robust and Adaptive Stabilization

§  Exploiting System Structure §  Robust Semi-global Design §  Decentralized Adaptive Nonlinear Control

§  Shaping the Zero-Dynamics §  Dynamic Output Redefinition §  Tracking via Integral Control

Control Objective

Control Objective

Track given velocity and altitude references Vref(t) and href(t)

Impose a desired trim value for the angle of attack ⇥�

Assumptions

- the vehicle parameters (e.g., m, S, c, . . .)- the coe�cients of the curve fits (e.g., CT , CM , CL, . . .)

are subject to a possibly large uncertainty (P ⇥ �P)

the initial conditions are chosen inside an arbitrary compact set �0

Since measurements of the flexible states are not availablethe controller uses feedback from the rigid body only

CONTROL DESIGN � Rigid Body Model

(x, ⇤ = ⇤�)

STABILITY ANALYSIS � Flexible Body Model

(x, ⇤ = ⇤ + ⇤�)

FLEXIBLE STATES

RIGID BODY

CONTROLLER

Nonlinear Control of an Air-breathing Hypersonic Vehicle with Structural Flexibility 6 / 17

10

The key is to achieve regulation indirectly by using another output. This new output must be selected such that:

1.  The resulting zero dynamics is stable 2.  Regulation of the “new output” implies regulation of the original

tracking error. Model uncertainty makes it a daunting task.

Redefinition of the Zero-dynamics

The zero-dynamics with respect to the error have an unstable equilibrium at where (�, Q) = (��

�, 0)

T (⇥��, ��

�) cos ⇥�� �D

�⇥��

⇥= 0 , M(⇥�

�, ���) = 0

Trim condition

Redefining the set-point tracking error as and applying the new decoupling input

yield the new 1-dim zero dynamics (the flight path angle dynamics) which has an asymptotically stable equilibrium at

� =D(⇥) + mg sin ⇤

qSCT,�(⇥) cos ⇥� CT (⇥)

CT,�(⇥)

⌅⌅e = �C⇥

M (⇥)C⇤

M

� zT T (⇥,�)qScC⇤

M

eaux = [V � V ��, � � ��

�]

⇥ =L(⇤��� ⇥,��

�)�mg cos ⇥

mV ��

� = 011

e = [V � V ?1, h� h?

1]

Regulation to an Unknown Setpoint

•  Asymptotic stability of the new zero-dynamics suggests to trade with in the regulated output

•  The problem is that is unknown (any discrepancy will lead to , hence to a diverging altitude)

•  Integral augmentation: (to enforce equilibrium at level flight)

•  Change of coordinates: (to remove inputs from the zero-dyn.)

� � � ���

���

limt�⇥

�(t) �= 0

⇥1 = � � �ref

⇥2 = � + µ1(yref)Q + µ2(yref)V

0 1 2 3 4 5 6 7

x 10−5

0.5

1

1.5

2

2.5

3

3.5

4

4.5

αr

[deg

]

ρ [slugs/ft3]

m = 169, Vr = 7500m = 169, Vr = 9500m = 169, Vr = 11000m = 202, Vr = 7500m = 202, Vr = 9500m = 202, Vr = 11000

�ref(yref)parameterizes the angle-of-attack along the reference

µ1(yref) := � 1Vref

Iyy C�L

c mC�M

µ2(yref) :=1

Vref

�zT C�

L

c C�M

1cos �ref(yref)

� tan�ref(yref)⇥

12

•  Letting

where , yields the new stable zero-dynamics

•  The overall system is stabilized by the selection

�

⇥1

⇥2

V , Q, d

V , Q, d

⇥

�

Design with Redefined Zero-dynamics

⇤1 = k1⇥1 + �r, ⇤2 = ⇥2 + ⇤1

0 < k1 < 1

(V , Q, ⇤)

(yref , yref) d

(⇥, �)V , Q

⇤cmd = �k1⌅1 + ⇥r = �⇧1 + ⇥r + �r

Qcmd = �k�[⇤ � ⇤cmd]� k1[⇥ � ⇥cmd]� ⇥ref

⇥1 = �a1(x, yref)�⇥1 �⇥2 + µ1(yref)Q + µ2(yref)V

⇥

+ d1(x, yref)

⇥2 = �a2(x, yref)⇥1 � a3(x, yref)⇥2 + a4(x, yref)�

+ b2(x, yref)Q + b3(x, yref)V + d2(x, yref)

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0 100 200 300 400 500 600 700 8000.2

0.4

0.6

0.8

1

1.2

Φ

0 100 200 300 400 500 600 700 80011.8

12

12.2

12.4

12.6

12.8

δ e[d

eg]

Time [s]

Simulation Results (High-Fidelity Model)

!"" #"" $"" %"" &"" '"" ("" )""!*$&

!*%

!*%&

!*&

!*&&

η1,

[ft

slug1

/2

]

Time [s]

+

+

!"" #"" $"" %"" &"" '"" ("" )""!"*"$%

!"*"$#

!"*"$

!"*"#)

!"*"#'

η2,

[ft

slug1

/2

]

1st bending mode, η1

2nd bending mode, η2

!"" #"" $"" %"" &"" '"" ("" )""!#"

!!&

!!"

!&

"

&,+!"

!$

η1,

[ft

slug1

/2 /

s]

Time [s]!"" #"" $"" %"" &"" '"" ("" )""

!!*&

!!

!"*&

"

"*&

!

!*&,+!"

!$

η2,

[ft

slug1

/2 /

s]

! "!! #!! $!! %!! &!! '!! (!! )!!(!!!

)!!!

*!!!

"!!!!

""!!!

"#!!!

V,V

r[f

t/s]

+

+

Velocity, V

Reference, Vr

! "!! #!! $!! %!! &!! '!! (!! )!!!!,"

!!,!)

!!,!'

!!,!%

!!,!#

!

!,!#

!,!%

Tra

ckin

ger

ror

V[f

t/s]

Time [s]

! " # $ %!&

!

&

-../

0 100 200 300 400 500 600 700 8000.94

0.96

0.98

1

1.02

1.04

1.06

x 105

h,h

r[f

t]

Altitude, h

Reference, hr

0 100 200 300 400 500 600 700 800

1300

1350

1400

1450

1500

1550

q,q r

[psv

]

Time [s]

Dynamic pressure, q

Reference, qr

u

y � yref

(x, u)RIGID- BODY

x

(⇥, ⇥)

Flight-Path Angle

Pitch Angle

Pitch Rate

Altitude

Velocity

FLEXIBLE STATES

CONTROLLER

�

�e

�cmd

⇥cmd

Qcmd

V, Vref

h, href

�1, �2

�

�e

14

Benefits of Adaptation in the Loop

0 100 200 300 400 500 600 700−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

FPA

Trac

king

[deg

]

flight path anglecommand

0 100 200 300 400 500 600 700−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

FPA

Trac

king

[deg

]

Time [s]

flight path anglecommand

Non-adaptive controller Adaptive controller

�, �ref �, �ref

Sizable error in FPA means large error in altitude tracking

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Conclusions

•  The concept of Zero Dynamics plays a fundamental role in virtually all control problems of interest

•  This is especially true for aerospace systems, where typically not all the degrees of freedom are directly actuated

•  Other noticeable examples include: –  Helicopters and Rotorcrafts –  Vertical Take-Off and Landing (VTOL) Vehicles –  Flapping-Wing Micro Air Vehicles –  Under-actuated Satellites –  Fixed-Wing Unmanned Air Vehicles

•  It is impossible to imagine today the field of aerospace without the pivotal contribution of Alberto Isidori to the theory and the practice of flight control system design.

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