This document discusses adaptive flight control for systems with unknown time-varying unstable zero dynamics. The research goals are to improve interceptor performance through reduced miss distance and reduce dependence on aerodynamic models. The approach uses retrospective cost adaptive control (RCAC) and concurrent online system identification through data-driven RCAC (DDRCAC) to mitigate the effects of unknown nonminimum-phase dynamics and time-dependent dynamics. DDRCAC identifies a rudimentary model of the system online and extracts information needed by RCAC to follow commands, even when the system transitions between minimum-phase and nonminimum-phase behavior with unknown time-varying dynamics.

1. Adaptive Flight Control with
Unknown Time-Varying Unstable Zero Dynamics
Syed Aseem Ul Islam, Adam L. Bruce, Tam W. Nguyen,
Ilya Kolmanovsky, and Dennis S. Bernstein
Aerospace Engineering Department
University of Michigan, Ann Arbor, MI
Research supported by ONR under BRC grant N00014-18-1-2211

2. Adaptive Control of Interceptors
• Research goals
• Improve interceptor performance through reduced miss distance
• Reduce dependence on aerodynamic models
• Achieve guaranteed threat engagement
• Goals of this talk
• Mitigate the effect of unknown nonminimum-phase dynamics
• Including time-dependent dynamics
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3. Motivating Example: HTV-2
• NASA model emulates the suspected
failure mode of the HTV
• RCAC papers on the NASA model:
• Represents an important class of
time-varying systems
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4. Time-Varying NMP Zeros
• Unknown transition from minimum-phase to NMP dynamics
• Emulates HTV failure mode (asymmetric ablation)
Continuous time Discrete time
Plant at 𝑡1 is unknown
It is MP
Time of transition is
unknown
Plant at 𝑡2 is unknown
It is NMP
Transfer function from aileron to roll angle
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5. Approach
• Use retrospective cost adaptive control (RCAC) to follow roll angle
commands
• RCAC is effective for LTI NMP plants with known NMP zeros
• RCAC may cancel unknown NMP zeros
• RCAC is effective on time-varying plants if the time dependence is not too fast
• Solution: Data-driven RCAC (DDRCAC)
• Perform concurrent online identification
• Extract closed-loop target model for use by RCAC
• To be explained
• Investigate performance on time-varying plants
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6. Adaptive Standard Problem for RCAC
Controller is linear with time-varying
coefficients that are adapted
Performance
variable
Command or
disturbance
With full-state feedback (𝑦 = 𝑥),
𝐺𝑧𝑢 can still be NMP
All examples in this talk are
output feedback
(only the performance variable z is
measured and y = z)
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7. RCAC Input-Output Controller Structure
• Discrete-time MIMO Input-Output Controller
𝑢 𝑘 =
𝑖=1
ℓc
𝑃𝑖,𝑘 𝑢 𝑘−𝑖 +
𝑖=1
ℓc
𝑄𝑖,𝑘 𝑦 𝑘−𝑖
• Specialization to SISO controller
𝑢 𝑘 = 𝜙 𝑘 𝜃 𝑘 = 𝑢 𝑘 ⋯ 𝑢 𝑘−ℓc
𝑦 𝑘−1 ⋯ 𝑦 𝑘−ℓc
𝑃1,𝑘
⋮
𝑃ℓc,𝑘
𝑄1,𝑘
⋮
𝑄ℓc,𝑘
𝐺c,𝑘 𝐪 =
𝑄1,𝑘 𝐪ℓc−1 + ⋯ + 𝑄ℓc,𝑘
𝐪ℓc − 𝑃1,𝑘 𝐪ℓc−1 − ⋯ − 𝑃ℓc,𝑘
=
𝑁c,𝑘(𝐪)
𝐷c,𝑘(𝐪)
Regressor
Controller
Coefficients
Time-Domain Representation
Controller coefficients
are optimized at each step
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8. • Optimization drives the retrospective performance variable to zero
𝑧 𝑘 𝜃 ≜ 𝑧 𝑘 − 𝐺f 𝐪 [𝑢 𝑘 − 𝜙 𝑘 𝜃]
𝑧 𝑘 = 𝐺 𝑘 𝐪 𝑣0,𝑘( 𝜃)
𝑧 𝑘 ≈ 𝐺f 𝐪 𝑣0,𝑘( 𝜃)
How Does RCAC Work?
Minimization of 𝑧 𝑘
“drives” 𝐺 𝑘 to 𝐺f
Target model
𝜙 𝑘 𝜃
Controller Denominator
Controller Numerator
𝐺c,𝑘 =
𝑁c,𝑘
𝐷c,𝑘
Intercalated injection
𝑢 𝑘
𝑧 𝑘
𝑣0,𝑘( 𝜃)
𝑤 𝑘
𝑦 𝑘
𝐺 𝑘 𝐪 =
𝑁𝑧𝑢(𝐪)𝐪 𝑛c
𝐷c,𝑘 𝐪 𝐷 𝐪 − 𝑁 𝑦𝑢 𝐪 𝐺c,𝑘 𝐪
Actual transfer function from 𝒗 𝟎,𝒌 to 𝒛 𝒌
𝑮 𝐟 must:
• Include all NMP zeros of 𝑮 𝒛𝒖
• Match the relative degree of 𝑮 𝒛𝒖
• Match the sign of 𝑮 𝒛𝒖
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9. Recursive Least Squares (RLS)
𝑧 𝑘 𝜃 ≜ 𝑧 𝑘 − 𝐺f(𝐪) 𝑢 𝑘 − 𝜙 𝑘 𝜃
𝐽 𝑘 𝜃 ≜
𝑖=1
𝑘
𝜆 𝑘−𝑖 𝑧𝑖
T
𝜃 𝑅 𝑧 𝑧𝑖 𝜃 + 𝜆 𝑘 𝜃 − 𝜃0
T
𝑃0
−1
𝜃 − 𝜃0
𝜃 𝑘+1 ≜ argmin 𝜃 𝐽 𝑘 𝜃
Forgetting Factor
This is not computed
𝑃𝑘+1 =
1
𝜆
𝑃𝑘 −
1
𝜆
𝑃𝑘Φ 𝑘
T
𝑁T
𝜆𝑅 𝑧
−1
+ 𝑁Φ 𝑘 𝑃𝑘Φ 𝑘
T
𝑁T
−1
𝑁Φ 𝑘 𝑃𝑘
𝜃 𝑘+1 = 𝜃 𝑘 − 𝑃𝑘Φ 𝑘
T
𝑁T 𝑅 𝑧 𝑁Φ 𝑘 𝜃 𝑘 + 𝑧 𝑘 − 𝑁 𝑈 𝑘
𝑙 𝑧 × 𝑙 𝑧 inverse𝑙 𝜃 × 𝑙 𝜃 matrix
𝑛f 𝑙 𝑢 × 𝑙 𝜃 data buffer 𝑛f 𝑙 𝑢 × 1 data buffer
𝑙 𝑧 × 𝑛f 𝑙 𝑢 filter matrix
𝑙 𝜃 × 1 controller
coefficient vector
RLS
This is implemented
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10. RCAC with MP Plant
• RCAC follows step command with rudimentary target model 𝐺f
x 𝐺c,𝑘
x o 𝐺 𝑧 converges to 0
𝐺 𝐪 =
(𝐪 − 0.3)(𝐪 − 0.7)
(𝐪 − 1.1)(𝐪2 − 1.4𝐪 + 1.052)
𝐺f 𝐪 =
1
𝐪
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11. RCAC with Unmodeled NMP Zeros
• RCAC cancels unmodeled or poorly modeled NMP zeros under
sufficiently high authority
x 𝐺c,𝑘
x o 𝐺
Unmodeled NMP zero cancelled by RCAC
Distance from NMP zero
to closest controller pole
𝑧 diverges
𝑢 diverges
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𝐺 𝐪 =
𝐪 − 1.3
𝐪2 − 0.7𝐪 + 0.48)
𝐺f 𝐪 =
1
𝐪

12. Motivation for Data-Driven RCAC
• All NMP zeros of the plant (𝐺𝑧𝑢) must be modeled
• But these are unknown and time-varying in the HTV application
• The time of transition of the plant from MP to NMP is unknown
• Ad-hoc fixes to update 𝐺f are not practical
• DDRCAC: Identify rudimentary model online using RLS
• Extract information needed for 𝐺f
𝑮 𝐟 must:
• Include all NMP zeros of 𝑮 𝒛𝒖
• Match the relative degree of 𝑮 𝒛𝒖
• Match the sign of 𝑮 𝒛𝒖
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13. Data-Driven Retrospective Cost Adaptive Control
• Fit an IO model to 𝑦 𝑘, 𝑢 𝑘
• Use RLS to minimize
𝑦 𝑘 +
𝑖
𝜂
𝐹𝑖,𝑘 𝑦 𝑘−𝑖 −
𝑖
𝜂
𝐺𝑖.𝑘 𝑢 𝑘−𝑖
• Yields estimates 𝐹𝑖,𝑘, 𝐺𝑖,𝑘
• Use 𝐺𝑖, discard 𝐹𝑖
• 𝐺f,𝑘 𝐪 = 𝑖
𝜂 1
𝐪 𝑖 𝐺𝑖,𝑘
• 𝐺f,𝑘 is FIR
• 𝑧 𝑘 ≜ 𝑧 𝑘 − 𝐺f,𝑘(𝜙 𝑘 𝜃 − 𝑢 𝑘)
• Minimize
𝑖=0
𝑘
𝑧𝑖
T
𝑧𝑖 + 𝜙 𝑘 𝜃 − 𝑢 𝑘
T
𝑅 𝑢(𝜙 𝑘 𝜃 − 𝑢 𝑘)
• Yields updated controller coefficients 𝜃 𝑘+1
System Identification (RLSID) RCAC
Captures leading sign,
NMP zeros,
relative degree of 𝑮 𝒛𝒖
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14. RLS with Variable-Rate Forgetting
Let 𝜆 𝑘 ∈ (0,1], define 𝜌 𝑘 ≜ Π𝑗=0
𝑘
𝜆𝑗 and
𝐽 𝑘(Θ) ≜
𝑖=0
𝑘
𝜌 𝑘
𝜌𝑖
𝑌𝑖 − Φ𝑖Θ T 𝑌𝑖 − Φ𝑖Θ + 𝜌 𝑘 Θ − Θ0
T 𝑃0
−1
Θ − Θ0
The minimizer Θ 𝑘+1 is given by
𝑃𝑘+1 =
1
𝜆 𝑘
𝑃𝑘 −
1
𝜆 𝑘
𝑃𝑘Φ 𝑘
T
𝜆 𝑘 𝐼𝑙 𝑌
+ Φ 𝑘 𝑃𝑘Φ 𝑘
T −1
Φ 𝑘 𝑃𝑘
Θ 𝑘+1 = Θ 𝑘 + 𝑃𝑘+1Φ 𝑘
T
(𝑌𝑘 − Φ 𝑘Θ 𝑘)
Data-dependent VRF:
𝜆 𝑘 =
1
1 + 𝛾𝑓(𝑧 𝑘)
where 𝑓(𝑧 𝑘) is
𝑓 𝑧 𝑘 =
RMS 𝑧 𝑘−𝜏1
, … , 𝑧 𝑘
RMS 𝑧 𝑘−𝜏2
, … , 𝑧 𝑘
− 1, ratio > 1
0, otherwise
𝜆 𝑘 = 1 if 𝑧 𝑘 below noise floor
Prevents forgetting
due to sensor noise;
promotes forgetting
when the error is
large
Forgetting Factor
Learning requires forgetting!
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15. Data-Driven Retrospective Cost Adaptive Control
𝑧
−
𝑟
𝑢 𝑦
𝑮
Online
Identification
Construct 𝑮 𝐟,𝒌
DDRCAC
𝑦
𝑢
Update 𝑮 𝐜
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16. DDRCAC Applied to a NMP Plant
2. Large transient leads
to re-identification
facilitated by VRF-ID
RLSID
RCAC
Converged coefficients
Zeros
4. Re-adaptation leads to
command following3. Re-identification
induces re-adaptation
facilitated by VRF-AC
Identifies NMP zero!
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1. Initially poor model
induces large transient
x o Identified Model
x o True System

17. DDRCAC Hyperparameters
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18. Basic Servo Loop for Sampled-Data Control
Sampled-data control:
Continuous-time plant, controlled with a discrete-time controller
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19. SISO Example
• Harmonic command following for SISO system
with unknown transition from MP to NMP
Zero move to RHP
Lightly damped
mode changes
frequency
𝑤 𝑘 ∼ 𝑁 0,0.00012 , 𝑣 𝑘 ∼ 𝑁(0, 0.0012)
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20. SISO Example
𝑇𝑠 = 0.1, 𝜂 = 8, 𝑛c = 10, 𝑝0 = 1000, 𝑅 𝑢 = 0, 𝛾p = 𝛾c = 0.1, 𝛼 = 90, 𝜏1 = 40, 𝜏2 = 200
Unknown transition from MP to
NMP happens 40 𝑠 < 𝑡 < 50 s
(time of transition is also unknown)
Unknown change in
command frequency
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21. SISO Example
Variable-rate forgettingVRF for
RLSID
VRF for
RCAC
Online-identification coefficients Adaptive controller coefficients
High forgetting during and
right after the transition21 of 28

22. SIMO Example
• Multi-step command following for SIMO system with unknown transition
from stable MP to unstable NMP
• Conflicting commands
Transmission zeros
move to RHP
Plant becomes
unstable
𝑤 𝑘 ∼ 𝑁 0,0.00012
, 𝑣 𝑘 ∼ 𝑁(0, 0.0012
)22 of 28

23. SIMO Example
𝑇𝑠 = 0.1, 𝜂 = 8, 𝑛c = 10, 𝑝0 = 1000, 𝑅 𝑢 = 0, 𝛾p = 𝛾c = 0.1, 𝛼 = 100, 𝜏1 = 40, 𝜏2 = 200
Unknown transition from stable
MP to unstable NMP happens
75 𝑠 < 𝑡 < 95 s
(time of transition also unknown)
Same hyperparameters as
SISO example
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Nonzero command is followed
Zero command is impossible

24. SIMO Example
Variable-rate forgetting
VRF for
RLSID
VRF for
RCAC
Online-identification coefficients Adaptive controller coefficients
High forgetting during and
right after transition24 of 28

25. Hypersonic Aircraft: Lateral Dynamics
Lightly damped
mode changes
frequency
Complex MP zeros
transition to two real
zeros, one NMP
Actuator Rate Saturation:
300 deg/s
𝑤 𝑘 ∼ 𝑁 0,0.00012 , 𝑣 𝑘 ∼ 𝑁(0, 0.0012)
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𝑥 ≜
𝛽
𝑝
𝑟
𝜙
,

26. Hypersonic Aircraft: Lateral Dynamics
𝑇𝑠 = 0.5, 𝜂 = 8, 𝑛c = 10, 𝑝0 = 1000, 𝑅 𝑢 = 0, 𝛾p = 𝛾c = 0.1, 𝛼 = 30∘, 𝜏1 = 40, 𝜏2 = 200
Unknown transition from MP to
NMP happens 90 𝑠 < 𝑡 < 100 s
(time of transition also unknown)
All hyperparameters same
as SISO and SIMO examples.
Only sampling time and
saturation level different.
Unknown change in
command frequency
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27. Hypersonic Aircraft: Lateral Dynamics
Variable-rate forgetting
VRF for
RLSID
VRF for
RCAC
Online-identification coefficients Adaptive controller coefficients
High forgetting during and
right after transition27 of 28

28. Conclusions and Future Work
• DDRCAC was used for linear time-varying plants that transition from
MP to NMP
• Step and harmonic commands
• Unknown transition was estimated online using concurrent system
identification
• Performance was demonstrated in the presence of stochastic disturbances
and sensor noise
• The method was demonstrated for a NMP SIMO system
• Applied to the lateral dynamics of a hypersonic aircraft
• Ongoing work:
• DDRCAC for plants with unknown nonlinear unstable zero dynamics.
• DDRCAC for interceptor autopilot for guaranteed threat engagement
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