CDC 2016

1. Adaptive Decentralized Control
with
Nonminimum-Phase Closed-Loop
Channel Zeros
55th IEEE Conference on Decision and Control
Syed Aseem Ul Islam, Yousaf Rahman, and Dennis S. Bernstein
University of Michigan
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2. RCAC: Retrospective Cost Adaptive Control
RCAC is a direct, digital, adaptive control technique that
• Applies to stabilization, command following, and disturbance
rejection
• Uses minimal modeling information
• Works on plants with nonminimum-phase zeros
Tutorial session on RCAC at ACC 2016
Forthcoming article in IEEE Control Systems Magazine
Developed for centralized control
This talk: Extensions to decentralized control
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3. Standard problem in state space
As transfer functions (time domain!)
The objective is to minimize 𝑧 in the presence of 𝑤 using minimal modeling information
Standard Problem Framework
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4. Controller Structure
We use the dynamic compensator of order 𝑛c
where
The transfer function of the controller is given by
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5. We define the retrospective performance variable
where 𝐺f is defined by
The rational behind is to replace with , where is obtained through
optimization
It can be seen that minimizing , yields which best matches to
Retrospective Performance Variable
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6. We define the cumulative retrospective cost function
where 𝑅 𝑧, 𝑅 𝑢, and 𝑅 𝜃 are weighting matrices
We use recursive least squares (RLS) minimization of the cumulative retrospective cost to obtain
the unique global minimizer given by
where
Cumulative Retrospective Cost Function
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7. Consider the SISO case
The modeling information needed is
• The relative degree of
• The sign of
• The NMP zeros of
Modeling Information Needed by RCAC
𝑧 𝑘, 𝜃∗
= 𝐺𝑧𝑤
∗
𝐪 𝑤 𝑘 + 𝐺 𝑧 𝑢
∗
𝐪 − 𝐺f 𝐪 𝑢(𝑘)
RCAC Cost Decomposition
Closed-loop performance Target model matching
Alternatively, the
retrospective
performance can
be interpreted as
the sum of the
actual performance
and an adaptation
term.
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8. NMP zeros
RCAC will attempt to cancel NMP zeros that are not zeros of 𝐺f
Cancelling NMP zeros leads to unstable pole-zero cancellation
Therefore, we use the NMP zeros of to construct 𝐺f
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9. RCXX talks at CDC 2016
“Retrospective Cost Adaptive Control with Concurrent Closed-Loop Identification of
Time-Varying Nonminimum-Phase Zeros,” F. Sobolic and D. S. Bernstein
10:00-10:20 MoA11 Regular Session, Starvine 11
“Adaptive Control of Plants That Are Practically Impossible to Control by Fixed-Gain
Control Laws,” Y. Rahman and D. S. Bernstein
10:40-11:00 MoA11 Regular Session, Starvine 11
“Adaptive Input Estimation for Nonminimum-Phase Discrete-Time Systems,” A. Ansari
and D. S. Bernstein
13:30-13:50 MoB09 Regular Session, Starvine 9
“Combined State and Parameter Estimation and Identifiability of State Space
Realizations,” M.-J. Yu and D. S. Bernstein
11:00-11:20 TuA13 Regular Session, Starvine 13
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10. Effective Plants in Decentralized Control
Each subcontroller “sees” a different effective
plant
The NMP zeros of this effective plant are the
NMP channel zeros
We define
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11. Effective Plants in Decentralized Control
Gc1 sees
If
If
For Gc1 the zeros of effective plant are a
combination of poles of the other subcontroller
and the zeros of *
*similarly for effective plant seen by Gc2
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12. Adaptation Schemes for Decentralized RCAC
1CAT or 1 controller-at-a-time
Subcontroller 1 adapts and Controller 2 is frozen; several steps later Subcontroller 2
adapts and Subcontroller 1 is frozen; this is repeated
NMP zeros of effective plant need to be calculated only at switch
Concurrent adaptation
Subcontroller 1 and Subcontroller 2 are adapted at the same time
NMP zeros of effective plant need to be calculated at each step (ideally)
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13. RCAC is a Discrete-Time Method
The following examples are given in terms of continuous-time plants
with a sampled-data interface
RCAC treats each system as a discrete-time plant
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14. Example 1: Disturbance Rejection for a Two-Mode Oscillator
Consider the continuous-time plant
where , and the discretization is at 1 Hz
The objective is to reject the disturbance using a
decentralized controller structure
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15. Gf
For each subcontroller, we construct Gf at each time step by putting any and all NMP
zeros of the effective plant in and choosing d so that the relative degree of Gf
matches the relative degree of
For Gc1, Nf is the polynomial containing the NMP zeros of
For Gc2 , Nf is the polynomial containing the NMP zeros of
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16. We apply RCAC with 1CAT Adaptation
Gc1 first, then Gc2
Gc1 Gc2
Rθ
10,00
0 10
Ru 0 0
nc 4 4
Example 1: Disturbance Rejection for a Two-Mode Oscillator
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17. Gc1 Gc2
Rθ 10 0.1
Ru 0 0
nc 4 4
We apply RCAC with 1CAT Adaptation
Gc2 first, then Gc1
Example 1: Disturbance Rejection for a Two-Mode Oscillator
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18. Example 2: Position and Shape Control of 2 DoF Flexible Body
The continuous-time dynamics are
which are sampled at 100 Hz,
where , and
This plant is unstable due to rigid body mode
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19. The objective is to control the average
velocity and separation of the two
constituent masses in the presence of
disturbance
Achieved by appropriate ramp commands
at r1 and r2 with concurrent RCAC
We set
Gc1 Gc2
Rθ 10 10
Ru 200 200
nc 8 8
Example 2: Position and Shape Control of 2 DoF Flexible Body
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20. We can express a discrete-time transfer function as a Laurent expansions
𝐻𝑖 ≜ 𝐸0 𝐴𝑖−1
𝐵 is the 𝑖th
Markov parameter of
A finite but sufficiently large number of Markov parameters yields an FIR target model with
• Correct relative degree
• Correct first Markov parameter
• Zeros that approximate the NMP zeros of
•We construct Gf for each subcontroller this way using the first 10 Markov parameters at time
0 and use this Gf throughout the simulation
Markov Parameter-Based Gf
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21. Example 2: Position and Shape Control of 2 DoF Flexible Body
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22. Summary
For the two-mode oscillator we show two RCAC subcontrollers with 1CAT
adaptation are able to reject broadband disturbance better than either
subcontroller operating alone
For the 2 DoF flexible body we show that two RCAC subcontrollers are
able to achieve shape and average velocity objectives, with each
subcontroller having access to only one mass each, for observation and
control
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23. Future Work
Decentralized control with MIMO subcontrollers
Further investigation of adaptation schemes 1CAT vs Concurrent
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