Adaptive/Self-Tuning PID Control by Frequency Loop-Shaping Elena - - PowerPoint PPT Presentation

CDC 00, Sydney 1

Adaptive/Self-Tuning PID Control by Frequency Loop-Shaping

Elena Grassi, ASU Kostas Tsakalis, ASU Sachi Dash, Honeywell HTC Sujit Gaikwad, Honeywell HTC Gunter Stein, Honeywell HTC

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Outline

• Problem Description: PID Tuning from Input-Output data
• Frequency Loop Shaping

– Off-line tuning – Target loop selection, 1st-2nd order targets

• Direct Adaptation of the PID parameters

– Cost functional – Regressor generation via filter banks – Adaptation – Performance Monitoring Implications

• Simulation Results
• Conclusions

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Problem Description

• Industrial Applications

– Large number of PID loops, often poorly tuned – Reliability and expediency requirements

• A Variety of PID Tuning Strategies

– Complete or partial models. (System identification-based vs. crossover properties) – Control objectives (Time-Frequency domain) – Direct and indirect approaches to adaptation

• Frequency Loop Shaping

– Accounting for uncertainty, several successful applications

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FLS PID Tuning (batch/off-line)

• System ID-modeling from I/O data
• Nominal model & uncertainty bounds
• Control Objective

– Loop-shaping (sensitivity targets) – Disturbance attenuation subject to bandwidth constraints – Guide: “Robust Stability Condition”

• On-line version via indirect adaptation

– Update plant model, re-tune controller – Complete solutions can be computationally demanding – Simple models => off-line construction of look-up table for the PID gains

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Target Loop Selection and FLS PID Tuning

• Typical Targets:

– Target order depends open-loop/closed-loop bandwidth ratio (for input disturbance attenuation) – Uncertainty constraints and RHP pole-zero limitations – More difficult cases via LQ or full-order controller design methods e.g., K=lqr(A,B,Q,R), target: [A,B,K,0]

• FLS Tuning: convex optimization in the frequency domain

– L=loop gain, S=sensitivity, T=complementary sensitivity

! , ) ( ) ( , ) ( ,

2

ε λ λ λ + + + s s a s s a s s

. . . ) ) ( ( min constr t s L GC S

pid L pid

pid

θ θ

θ

∞

− . ) ) ( ( . . ) ) ( ( min

2

constr b L GC S t s L GC S

pid L pid L pid

pid

θ θ θ

θ

≤ − −

∞

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Direct Adaptation with an FLS objective

• Construction of the estimation error (at the plant input)
• Approximate sup by using a filter bank

– Fi: band-pass filters, ||.||2,δ: exponentially weighted 2-norm

2 2

|| || || || sup ) ( ] [ ] [ ] )[ ( u e L CG S u T y SC u L CG S e

e u L e ≠

= − − = − =

∞

δ δ , 2 , 2 2 2

|| ] [ || || ] [ ] [ || max || ] [ || || ] [ ] [ || max ) ( u F u TF y SCF u F u TF y SCF L CG S

i i i i i i i i L

− ≤ − ≅ −

∞

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Direct Adaptation with an FLS objective (cont.)

• Optimization problem
• Recursive computation of Ji,k
• Optimization: min-max of quadratics

k i k i k i k i k i k i k i k T n i n i k n n k k k i k i k i i M

y F SC w u TF z u F m m w z J m J ] [ , ] [ | ] [ | | | ) ( ) ( max min

, , 2 1 , , 2 , , , , , θ θ

λ θ λ θ θ = = + = − =

− = − ∈

∑

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Direct Adaptation details

• Recursive computation of Ji,k
• Each Ji,k+1 is quadratic in the parameters: minimize the

maximum by, e.g., computing a descent direction and performing a line search

T k i k i k i k i k k i k i k i T k i k i k i k i k T k i k i k k i k i k k i T k k T k i k i k i

w z R R P R S w w P P w z J J P S J J

, , , 1 , 1 , 1 , 1 , , , , 1 , 2 1 , 1 , , 1 , 1 , 2 1 1 , 1 , 1 ,

2 , , 2 | | ) ( ˆ ) ( ) ( ) ( ˆ ) ( + = − = + = − + = − − + − − =

+ + + + + + + + + + + +

λ θ λ θ θ λ θ θ θ θ θ θ θ

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Adaptive FLS Properties

• Excitation requirements
• Effects of disturbances and unmodeled dynamics (SNR)
• A dead-zone condition: update when

– Update when the error operator gain drops by at least d0

• Input Saturation does not affect updates
• Linearization offsets (estimation or high-pass filtering)
• The cost functional provides a measure of tuning

confidence

– Feasibility of performance monitoring

2

2 , 1 , ,

> −

− k k i k i T k i

m d S P S

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Example

• Simulation Results for the following plant and target loop:
• Square-wave reference input.

– Excitation injected at the plant input for t<75.

• PID gains converge approximately to the off-line tuning.
• Cost functional has a maximum of 0.32, same as the off-line

fitting error.

s s L s s G 1 ) ( , ) 1 ( 1 ) (

3

= + =

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Simulation Results

• Left: Parameters, output, reference, excitation.
• Right: Square-root of cost functional.

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Conclusions

• Direct adaptation of PID parameters with an FLS objective

– FLS: Operator gain interpretation of fitting error

• Recursive implementation for on-line tuning
• Use of a filter bank to approximate the min-max objective
• Quantitative measures of tuning confidence

– Gain of the error system

Future work:

• On-line monitoring of performance
• On-line adaptation of objective (target loop) based on the cost

functional values