ACC 2018

1. Output-Feedback Adaptive
Control of Discrete-Time Systems
with Unmodeled, Unmatched,
Inaccessible Nonlinearities
The 2018 American Control Conference
Syed Aseem Ul Islam and Dennis S. Bernstein
University of Michigan
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2. Breaking Down the Title
Output-Feedback
Not full-state feedback (𝑦 ≠ 𝑥)
Adaptive Control of Discrete-Time Systems
Retrospective Cost Adaptive Control (RCAC), which requires no controller
discretization for digital implementation
with Unmodeled, (Nonlinearities),
The controller uses no modeling information about the nonlinearity
Unmatched, (Nonlinearities),
The control input and the unknown nonlinearity drive the system differently: Direct
cancellation of the unknown nonlinearity is impossible
Inaccessible Nonlinearities
The nonlinearity is a function of unmeasured states and thus it cannot be computed
even if were known; in addition, its output is not measured so it cannot be cancelled
even if it were matched
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3. Adaptive Control Technique: 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 zeros
• Was developed for linear time-invariant systems
This talk: Apply RCAC to nonlinear systems
The goal is to numerically investigate the ability of RCAC to
adapt to unmodeled nonlinearities
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“Retrospective Cost Adaptive Control:
Pole Placement, Frequency Response,
and Connections with LQG Control,”
IEEE Contr. Sys. Mag., Vol. 37, pp. 28--69,
October 2017

4. Adaptive Control Technique: RCAC
RCAC requires limited modeling information
embedded in the intercalated target model 𝐺f
𝐺f captures the relative degree, NMP zeros (if
any), and sign of the leading numerator
coefficient of 𝐺𝑧𝑢
Controller order 𝑛c, adaptation weight 𝑅 𝜃, and
control weight 𝑅 𝑢 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
4
Adaptive Standard Problem
Linear Time-Varying Controller

5. We consider nonlinear plants of the form
The linear dynamics (𝐴, 𝐵, 𝐶) are asymptotically
stable
The feedback nonlinearity may be matched 𝐵 = 𝐵nl
or unmatched 𝐵 ≠ 𝐵nl
The objective is to minimize 𝑧 in the presence of 𝑑
using minimal plant modeling information
Nonlinear Plant
5
𝑓(𝑥) is a function
of states not
measured by 𝑦0
𝑓(𝑥) and 𝑢 are
unmatched

6. As a preliminary we consider the case of the matched cubic nonlinearity
𝑓 𝑥 = 𝛼𝑥1
3
,
Harmonic command following with 𝑟 𝑘 = cos 𝜔 𝑟 𝑘 , where 𝜔 𝑟 = 0.2 rad/sample.
We pick
RCAC vs LQG: Matched Cubic Nonlinearity
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7. If 𝑥1 is a harmonic with frequency 𝜔𝑟 then
Therefore, the nonlinearity generates a harmonic
signal at frequency 3𝜔𝑟.
(𝐴, 𝐵, 𝐶) is chosen to have lightly damped poles at
frequency 3𝜔𝑟 = 0.6 rad/sample.
As these signals occur in a closed loop the signal
produces additional spectral content at frequencies
5𝜔𝑟, 7𝜔𝑟, 9𝜔𝑟 which in turn will produce even higher
frequency content; all unknown to RCAC and LQG
These signals appear as unmodeled disturbances.
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RCAC vs LQG: Matched Cubic Nonlinearity

8. LQG:
Linear plant is augmented with an internal model
of the harmonic command
 The augmented plant has order 5
LQG controller designed with
𝑄 𝑥𝑢 = diag 0,0,1000,0,0,1 , 𝑄 𝑤𝑣 = 𝐼6
5th–order LQG controller is cascaded with the
2nd–order internal model to give a final controller
order of 7.
LQG requires knowledge of the
command frequency and complete
knowledge of (𝑨, 𝑩, 𝑪).
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RCAC:
𝑛c = 14, 𝑅 𝜃 = 0.0002, 𝑅 𝑢 = 0.01, 𝐺f 𝐪 = −𝐪−1
RCAC requires limited information about
(𝑨, 𝑩, 𝑪) only. Namely, relative degree, NMP
zeros (if any), and the sign of the leading
numerator coefficient
RCAC learns and compensates for the unknown
nonlinearity---including the generated harmonics
.
RCAC vs LQG: Matched Cubic Nonlinearity

9. 9
The error 𝑒0 approaches zero for the values of 𝛼 above using RCAC and LQG.
RCAC is able to drive the error to zero for a much larger range of values of 𝛼.
RCAC vs LQG: Matched Cubic Nonlinearity
𝑓 𝑥 = 𝛼𝑥1
3
,

10. Numerical Examples: Various Nonlinearities
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For all examples we use the same RCAC tuning parameters 𝒏 𝐜 = 𝟑, 𝑅 𝜃 =
0.002, 𝑅 𝑢 = 0.01 and target model 𝐺f 𝐪 = −𝐪−1
No attempt is made to refine RCAC tuning for each example
We set
𝐺 𝑦0 𝑢 has poles at {0.8, 0.7 ± 0.1𝑗} and zeros at {0.3,0.4}
Only 𝑥3 is measured

11. Example 1: Harmonic command following with a quadratic-plus-bias matched nonlinearity
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Nonlinearity is matched 𝐵nl = 𝐵
No sensor noise or disturbance
𝑥 0 = 0, 𝑟 𝑘 = cos 0.2𝑘

12. Example 1: Harmonic command following with a quadratic-plus-bias matched nonlinearity
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We define the control input 𝑢lin to be the
control RCAC applies to the linear plant,
that is, with 𝑓 = 0
We compare 𝑢 with 𝑢lin to determine
how RCAC modifies the control input to
account for the presence of the
nonlinearity
RCAC generates 𝑢 such that 𝑢 + 𝑓 ≈ 𝑢lin
RCAC cancels the nonlinearity despite
the fact that it is unknown with an
unknown bias 𝑓 0 ≠ 0

13. Example 2: Harmonic command following with a discontinuous unmatched nonlinearity
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No sensor noise or disturbance
𝑥 0 = 0
Unmatched NL 𝐵nl = −1 0 0 T ≠ 𝐵
𝜔 𝑟 =
0.2
rad
sample
, 𝑘 ≤ 150,
0.4
rad
sample
, 𝑘 > 150.
Command
Frequency
Changes

14. Example 3: Harmonic command following with an unmatched vector nonlinearity
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𝐵nl =
1 0
0 1
0 0
.
No sensor noise or disturbance
𝑥 0 = 0
𝑟 𝑘 = cos 0.2𝑘

15. Example 4: Step disturbance rejection with a non-Lipschitzian nonlinearity
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𝐵nl = −1 0 0 T ≠ 𝐵, 𝑑 𝑘 = 1
𝐷d = 𝐵, 𝑢 and 𝑑 are matched
𝜈~𝑁(0,0.012)
Signal-to-noise ratio (SNR) for 𝑦0 and
𝜈 is 17.99 dB

16. Example 5: Harmonic disturbance rejection with a harmonic nonlinearity
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𝑑 𝑘 = 0.1 sin 0.32𝑘
𝐷d = 0 0 1 T ≠ 𝐵, 𝑢 and 𝑑 are
unmatched
𝜈~𝑁(0,0.082
)
Signal-to-noise ratio (SNR) for 𝑦0 and
𝜈 is 25.84 dB

17. Conclusions and Future Work
A numerical investigation of output-feedback adaptive control of nonlinear plants with unmatched,
unknown, inaccessible nonlinearities was provided.
RCAC was able to follow harmonic commands and reject step, harmonic disturbances for various
nonlinearities.
RCAC relied on extremely limited modeling information with no knowledge of the nonlinearity.
Future Work:
Investigate nonlinear plants with
 NMP linear dynamics
 Unstable linear dynamics
 Unstable nonlinear zero dynamics
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18. RCXX talks at ACC 2018
“Experimental Identification of the Spatial Spillover Operator for Systems with Insuppressible Disturbances,”
S. Sanjeevini, K. F. Aljanaideh, A. Xie, and D. S. Bernstein, 11:00-11:20 WeA08.4 Regular Session, 103D
“Retrospective Cost Adaptive Control Using Composite FIR/IIR Controllers,”
Y. Rahman, K. F. Aljanaideh, and D. S. Bernstein, 13:30-13:50 WeB03.1 Regular Session, 101C
“Adaptive Attitude Control of a Dual-Rigid-Body Spacecraft with Unmodeled Nonminimum-Phase Dynamics,”
Z. Zhao, G. Cruz, T. Lee, and D. S. Bernstein, 10:00-10:20 ThA03.1 Regular Session, 101C
“Estimation of the Eddy Diffusion Coefficient Using Total Electron Content Data,”
A. Goel, A. Ridley, and D. S. Bernstein, 15:10-15:30 ThB03.6 Regular Session, 101C
“Parameter Estimation for Nonlinearly Parameterized Gray-Box Models,”
A. Goel and D. S. Bernstein, 11:40-12:00 FrA15.6 Regular Session, 202D
“System Identification Using Composite FIR/IIR Models,”
K. F. Aljanaideh and D. S. Bernstein, 14:10-14:30 FrB04.3 Regular Session, 101D
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