ACC 2019

1. Adaptive Control of Systems
with Unknown Nonminimum-Phase Zeros
Using Cancellation-Based Pseudo-identification
The 2019 American Control Conference
Syed Aseem Ul Islam, Antai Xie, and Dennis S. Bernstein
Aerospace Engineering Department
University of Michigan
Ann Arbor, MI

2. Adaptive Control Using RCAC
• RCAC is a direct, digital, adaptive control
technique that
• Applies to stabilization, command following,
and disturbance rejection
• Uses limited modeling information
• Works on plants with nonminimum-phase
(NMP) zeros when their location is known
within reasonable accuracy
• RCAC cancels unmodeled NMP zeros
• This talk: A technique for applying RCAC to
systems with completely unmodeled NMP
zeros
• Key idea: Take advantage of RCAC's
propensity to cancel NMP zeros by learning
the location of those zeros
“Retrospective Cost Adaptive Control:
Pole Placement, Frequency Response,
and Connections with LQG Control,”
IEEE Contr. Sys. Mag., Vol. 37, pp. 28--69,
October 2017
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3. Why Cancelling a NMP Zero Is Bad
• The control signal 𝑢 grows exponentially, while the output remains
unaffected
• This is a hidden instability
• If 𝑢 is saturated, it gets “stuck” on the saturation
1
𝐪 + 1.5
𝐪 + 1.5
𝐪2 + 0.1𝐪 + 0.2
𝑢 𝑦
Controller Plant
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4. Adaptive Control Using RCAC
 RCAC requires limited modeling information
embedded in the intercalated target model 𝐺f
 𝐺f needs to model the relative degree, NMP
zeros (if any), and sign of the leading
numerator coefficient of 𝐺𝑧𝑢
 Controller order ℓc, adaptation weight 𝑃0 must
also be specified
 The controller coefficients are initialized to be
zero at the start of all numerical examples
 Ensures no additional modeling information is
used
Linear Time-Varying Controller
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5. RCAC Does NOT Cancel NMP Zeros When They Are Modeled
• 𝐺 𝐪 =
𝐪−1.3
𝐪2−0.7𝐪+0.48
• Broadband disturbance rejection
problem
• 𝐺f 𝐪 =
𝐪−1.3
𝐪2
• ℓc = 10, 𝑃0 = 100𝐼𝑙 𝜃
Plant NMP zero
modeled in 𝐺f
Plant NMP zero NOT
cancelled by RCAC
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6. RCAC Cancels Unmodeled NMP Zeros
• 𝐺 𝐪 =
𝐪−1.3
𝐪2−0.7𝐪+0.48
• Broadband disturbance rejection
problem
• 𝐺f 𝐪 =
1
𝐪
• ℓc = 10, 𝑃0 = 100𝐼𝑙 𝜃
Plant NMP zero
not modeled in 𝐺f
NMP zero cancelled by controller
pole leading to instability
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7. RCAC Controller
• MIMO Input-Output Controller
𝑢 𝑘 = 𝜙 𝑘 𝜃 𝑘
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
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Controller
Coefficients
Transfer Function Representation

8. Retrospective Cost Function
• All required modeling information is in the target model 𝑮 𝐟
• We define the retrospective performance variable
𝑧 𝑘 𝜃 ≜ 𝑧 𝑘 − 𝐺f(𝐪) 𝑢 𝑘 − 𝜙 𝑘 𝜃
• Use RLS to minimize the retrospective cost function
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
minimizing 𝐽 𝑘 𝜃 ⟹ 𝑧 𝑘 ≈ 𝐺f(𝐪) 𝑢 𝑘( 𝜃)
• Updated controller coefficients: 𝜃 𝑘+1 = 𝜃opt,𝑘
Target Model
All Modeling Information
Past u data
Controller coefficients
to be re-optimized
Past y and u data
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Tuning Hyperparameter:
Controls adaptation rate
Forgetting Factor

9. Recursive Least Squares (RLS) for RCAC
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
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
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10. How RCAC Works and Why It Cancels Unmodeled NMP Zeros
• RCAC drives the retrospective performance variable to zero
𝑧 𝑘 𝜃 ≜ 𝑧 𝑘 − 𝐺f 𝐪 𝑢 𝑘 where 𝑢 𝑘 ≜ 𝑢 𝑘 − 𝜙 𝑘 𝜃
• Minimization of 𝑧 𝑘 implies 𝑧 𝑘 ≈ 𝐺f 𝐪 𝑢 𝑘( 𝜃)
Minimization of 𝑧 “drives” 𝐺 𝑧 𝑢,𝑘 to 𝐺f
𝐺c,𝑘 =
𝑁c,𝑘
𝐷c,𝑘
𝑁𝑧𝑢 ≠ 𝑁𝑦𝑢
NMP zeros unmodeled by 𝐺f may
be cancelled by closed-loop poles
𝑁𝑧𝑢 = 𝑁𝑦𝑢
NMP zeros unmodeled by 𝐺f may
be cancelled by controller poles
𝐺 𝑧 𝑢,𝑘 𝐪 =
𝑁𝑧𝑢(𝐪)𝐪ℓc
𝐷 𝐪 𝐷c,𝑘 𝐪 − 𝑁𝑦𝑢 𝐪 𝑁c,𝑘(𝐪)
≈
𝑁f
𝐪 𝑛f
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11. RCAC Cancels Unmodeled NMP Zeros (with saturation)
• 𝐺 𝐪 =
𝐪−1.3
𝐪2−0.7𝐪+0.48
• Broadband disturbance rejection problem
• 𝐺f 𝐪 =
1
𝐪
• ℓc = 10, 𝑃0 = 100𝐼𝑙 𝜃
NMP zero cancelled
by controller.
Coefficients
converge
𝑢 saturated at ±20
𝑢 “rides the saturation”
Bad performance!
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12. Cancellation-Based Pseudo-Identification
• Idea: Utilize RCAC’s propensity to cancel unmodeled NMP zeros to LEARN the NMP zeros of a system
• Set 𝐺f =
1
𝐪
(models plant relative degree and sign)
• Select control saturation level 𝛼
• If saturation has been activated at least 𝑛 𝛼,max times, and the controller poles have converged, then
model all unstable controller poles as zeros in 𝐺f, and reset controller update equations
• Formally,
If 𝑘 > 2ℓc AND 𝑛 𝛼,𝑘 > 𝑛 𝛼,max AND
𝜃den,𝑘 −
1
ℓc
𝑖=𝑘−ℓc
𝑘
𝜃den,𝑖 < 𝜀
Then model all unstable poles (SR>1.05) of 𝐺c,𝑘 in 𝐺f, and reset 𝑃𝑘 = 𝑃0, 𝜃 𝑘 = 0𝑙 𝜃
Hyperparameters
to choose:
• ℓc
• 𝑃0
• 𝛼
• 𝑛 𝛼,max
• 𝜀
The plant is assumed to be asymptotically stable with known relative degree, and leading numerator coefficient sign.
The NMP zeros of the plant are unknown.
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13. Example 1: Multi-Step Command Following with 1 Real NMP Zero
• 𝐺 𝐪 =
𝐪−1.4
𝐪2−0.7𝐪+0.61
Controller pole close to NMP zero at 𝑘 = 64.
Subsequently, modeled as zero in 𝐺f.
For ALL examples
• ℓc = 16
• 𝑃0 = 10𝐼𝑙 𝜃
• 𝛼 = 4
• 𝑛 𝛼,max = 10
• 𝜀 = 0.01
• Sensor noise 𝑣 𝑘~𝑁(0,0.012
)
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14. Example 2: Harmonic Command Following with 2 Complex NMP Zeros
• 𝐺 𝐪 =
(𝐪−0.43)(𝐪2−2.1𝐪+1.26)
(𝐪−0.1)(𝐪−0.2)(𝐪2−1.2𝐪+0.61)
Controller poles close to NMP zeros at 𝑘 = 99.
Subsequently, modeled as zeros in 𝐺f.𝑟𝑘 = sin 0.13𝑘
For ALL examples
• ℓc = 16
• 𝑃0 = 10𝐼𝑙 𝜃
• 𝛼 = 4
• 𝑛 𝛼,max = 10
• 𝜀 = 0.01
• Sensor noise 𝑣 𝑘~𝑁(0,0.012
)
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15. Example 3: Harmonic Disturbance Rejection with 3 NMP Zeros: 1
Negative and 2 Complex
• 𝐺 𝐪 =
(𝐪+1.4)(𝐪2−1.6𝐪+1.13)
(𝐪−0.05)(𝐪−0.25)(𝐪2−𝐪+0.61)
Controller poles close to NMP zeros at 𝑘 = 87.
Subsequently, modeled as zeros in 𝐺f.
𝑑 𝑘 = 0.1sin 0.234𝑘
For ALL examples
• ℓc = 16
• 𝑃0 = 10𝐼𝑙 𝜃
• 𝛼 = 4
• 𝑛 𝛼,max = 10
• 𝜀 = 0.01
• Sensor noise 𝑣 𝑘~𝑁(0,0.012
)
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16. Example 4: Broadband Disturbance Rejection with 4 Complex NMP Zeros
• 𝐺 𝐪 =
(𝐪2−2.2𝐪+1.37)(𝐪2−1.1𝐪+1.21)
(𝐪−0.5)(𝐪2−1.6𝐪+0.76)(𝐪2−0.5𝐪+0.93)
For ALL examples
• ℓc = 16
• 𝑃0 = 10𝐼𝑙 𝜃
• 𝛼 = 4
• 𝑛 𝛼,max = 10
• 𝜀 = 0.01
• Sensor noise 𝑣 𝑘~𝑁(0,0.012
)
Controller poles close to NMP zeros at 𝑘 = 631.
Subsequently, modeled as zeros in 𝐺f.
𝑑 𝑘~𝑁(0,0.12
)
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17. Conclusions and Future Work
• Idea: Let the adaptive control attempt to cancel the NMP zeros and use
that weakness to learn
• A heuristic cancellation-based pseudo-identification technique for adaptive
control of NMP plants with UNKNOWN NMP zeros was demonstrated.
• The method exploits RCAC’s propensity to cancel unmodeled NMP zeros.
• This technique was demonstrated for step and harmonic command
following, and harmonic and broadband disturbance rejection.
• Plants with up to four unknown NMP zeros were considered.
Future work:
• Extend this method to unstable systems and MIMO systems
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18. Related Talks at ACC 2019
Wednesday
• “Adaptive Feedback Noise Control for Wide, Square, and Tall Systems,”
A. Xie and D. S. Bernstein, 11:20-11:40, WeA12.5, Room 403
Friday
• “Deadbeat Input Reconstruction for Discrete-Time Linear Time-Varying Systems,”
A. Ansari and D. S. Bernstein, 13:30-13:50, FrB16.1 , Room 407
• “A Modified RLS Algorithm with Forgetting and Bounded Covariance,”
A. Bruce and D. S. Bernstein, 16:00-16:20, FrC04.1, Franklin 4
• “Satellite Drag Estimation Using Retrospective Cost Input Estimation,”
A. Ansari and D. S. Bernstein, 17:20-17:40, FrC16.5, Room 407
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19. RCAC Does NOT need an exact model of the NMP zero
• 𝐺 𝐪 =
𝐪−1.3
𝐪2−0.7𝐪+0.48
• 𝐺f 𝐪 =
𝐪−𝛼 𝑧
𝐪2
Location of plant
NMP zero
• NMP zero location in 𝐺f is varied.
• RCAC works for range of erroneously
modeled NMP zeros.
• RCAC is most robust to 𝑃0 if the NMP zero in
𝐺f exactly models the NMP zero of the
plant.
Largest 𝑃0 for which RCAC does not cancel NMP zero
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