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)⇥
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• 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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