SacMan Control Tuning SacMan Control Tuning

Bert ClemmensBert ClemmensAgricultural Research ServiceAgricultural Research Service

Canal Control ProblemCanal Control Problem• Balance supply with demand.Balance supply with demand.

• Maintain desired delivery rate.Maintain desired delivery rate.

• Above are accomplished byAbove are accomplished by– Control of pool water levelsControl of pool water levels

•which in turn requires control of pool volumeswhich in turn requires control of pool volumes

Demand

Supply

Water-level Control

Gravity Offtakes

Three Aspects Three Aspects of Canal Automation of Canal Automation

• Flow controlFlow control – Ability to control flow rates at key pointsAbility to control flow rates at key points

• Feedforward control of flow ratesFeedforward control of flow rates – Ability to route known major flow changes Ability to route known major flow changes

through the canalthrough the canal

• Feedback control of water levelsFeedback control of water levels – Ability to adjust to disturbances and flow Ability to adjust to disturbances and flow

rate errors with downstream water-level rate errors with downstream water-level feedbackfeedback

Tuning RequirementsTuning Requirements• Gate calibration are important, but not critical when Gate calibration are important, but not critical when

feedback control is used. We use handbook feedback control is used. We use handbook calibration or calibrations provided by operators. calibration or calibrations provided by operators. Nothing special!Nothing special!

• Delay times for routing are important to transient Delay times for routing are important to transient performance. We manually adjust Manning n to performance. We manually adjust Manning n to match predicted and observed delay time for match predicted and observed delay time for feedforward.feedforward.

• We use optimal control methods to obtain water We use optimal control methods to obtain water level feedback control parameters. Canal properties level feedback control parameters. Canal properties are determined from simulation or on-line tests.are determined from simulation or on-line tests.

Typical Check Structure Typical Check Structure HardwareHardware

Automata Hardware

Gate Position Sensor• Two SensorsTwo Sensors

– Digital Digital Output for Output for fine fine resolution of resolution of gate gate movementmovement

– Analog Analog Output for Output for coarse coarse resolution of resolution of gate gate openingopening

Optical Encoder Pulsed Optical Encoder Pulsed OutputOutput

• Pulses count down to zero and motor stopsPulses count down to zero and motor stops

0.95 mm

Calibration of gates at Calibration of gates at CAIDDCAIDD• District has determined from experience, District has determined from experience,

relationship between relative gate position relationship between relative gate position change and flow rate changechange and flow rate change

• This is assumed linear. Then they correct when This is assumed linear. Then they correct when flows do not balance.flows do not balance.

• Sometimes they take into account non-linearity in Sometimes they take into account non-linearity in initial opening.initial opening.

• We use this calibration to determine the amount We use this calibration to determine the amount of gate movement (number of pulses) for a of gate movement (number of pulses) for a desired flow change.desired flow change.

• This is programmed into the SCADA system for This is programmed into the SCADA system for manual controlmanual control

• SacMan also considers upstream water level in SacMan also considers upstream water level in determining gate position changedetermining gate position change

Flow control issuesFlow control issues

• Canal headgates are often not Canal headgates are often not accurate for flow measurementaccurate for flow measurement

• Separate meter downstream can be Separate meter downstream can be used to adjust headgateused to adjust headgate

• Downstream Water-Level Feedback Downstream Water-Level Feedback adjusts for flow errors upstreamadjusts for flow errors upstream

• Incremental flow control allow gradual Incremental flow control allow gradual adjustment to match downstream adjustment to match downstream flowsflows

• Free flow gate downstream can be Free flow gate downstream can be used to adjust head gateused to adjust head gate

If gate is close to head-gate If gate is close to head-gate and is free-flowing, it can be and is free-flowing, it can be alternative measurement alternative measurement devicedevice

SimpleFeedback

ControlCheck Gate tobe Adjusted

Level to beControlled

Canal properties significantly affect the Canal properties significantly affect the performance of any canal automation scheme.performance of any canal automation scheme.

– pool delay times •which limits the responsiveness of the canal and thus the control which limits the responsiveness of the canal and thus the control

possiblepossible

– pool volume changes with flow rate •which influences the routing of flow changes through a canalwhich influences the routing of flow changes through a canal

– downstream water level response to pool volume changes over time•which influences the strength of feedback corrections to water which influences the strength of feedback corrections to water

level errorslevel errors

– Reflection Wave FrequencyReflection Wave Frequency•Which is needed to avoid unstable feedback controlWhich is needed to avoid unstable feedback control

Control Engineering PracticeControl Engineering Practice

• Most industrial controllers use simple Most industrial controllers use simple ““ClassicalClassical”” control, such as PID. control, such as PID.

• So called So called ““ModernModern”” control theory, which control theory, which uses optimization, has never caught on.uses optimization, has never caught on.

• AdaptiveAdaptive-classical control has received -classical control has received more coverage in the literature.more coverage in the literature.

• Several simple Several simple controllers in seriescontrollers in series continues to be a difficult control continues to be a difficult control problem.problem.

Optimization with State-Optimization with State-Feedback Control of Water Feedback Control of Water

LevelsLevels• State-Transition RelationshipState-Transition Relationship

– we use the we use the Integrator-Delay ModelIntegrator-Delay Model

– where, where, y(t)y(t) is the downstream water level at is the downstream water level at time time t t in response to a step change in in response to a step change in upstream flow rate, upstream flow rate, QQ, ,

– is the is the pool time delaypool time delay, and , and – AA is the is the pool backwater surface areapool backwater surface area..

y t y for t

y t tQ

Afor t

( ) ( )

( )

0

Integrator Delay ModelIntegrator Delay Model

• Time delay, Time delay, • Backwater surface area, ABackwater surface area, Ass

Uniform Flow

Backwater

Integrator-Delay ModelIntegrator-Delay Model

Time

Ch

ang

e in

Dep

th

0

0

t

DQ

As

Canals under normal depth follow Canals under normal depth follow this model well this model well (SRP Arizona Canal - (SRP Arizona Canal - Pool 1)Pool 1)

393.5

394.0

394.5

395.0

0:00 1:00 2:00 3:00 4:00 5:00 6:00

Time (hours)

Ele

vati

on

(m

)

Computed response

Linear model

State Transition EquationsState Transition Equations

• Derived from integrator-delay modelDerived from integrator-delay model

x x u

e Cx Du

( ) ( ) ( )

( ) ( ) ( )

k k k

k k k

1

Optimization with State-Optimization with State-Feedback Control of Water Feedback Control of Water

LevelsLevels

• State-Feedback Control LawState-Feedback Control Law

where where uu(k)(k) is the is the control actioncontrol action (change in (change in flow rate) at time step k, flow rate) at time step k,

KK is the is the controller gain matrixcontroller gain matrix, and , and

xx(k)(k) is the is the state vectorstate vector..

u K x( ) ( )k k

Optimization with State-Optimization with State-Feedback Control of Water Feedback Control of Water

LevelsLevels• Linear Quadratic Regulator (LQR)Linear Quadratic Regulator (LQR) with with

Penalty FunctionPenalty Function

where where JJ is the is the costcost, , e(k)e(k) is the is the water level errorwater level error at time step k, at time step k,

and and QQ and and RR are are penaltiespenalties on the water level on the water level

errors and control actions, respectively.errors and control actions, respectively.

J k k k kT

k

T

e Q e u R u( ) ( ) ( ) ( )0

Controller TuningController Tuning

• Centralized PI-controller (with full Centralized PI-controller (with full gain matrix) can be found from gain matrix) can be found from solution of Riccati equationsolution of Riccati equation

• Gradient search procedures are used Gradient search procedures are used to optimize other, more simple to optimize other, more simple controllers, such as a series of local controllers, such as a series of local PI controllersPI controllers

Proportional-Integral Proportional-Integral ControllerController

• We can We can optimally tune a PI controlleroptimally tune a PI controller with the above scheme, with the above scheme, – provided that the state vector, provided that the state vector, xx(k), is (k), is

properly chosen and properly chosen and – when only certain elements are chosen when only certain elements are chosen

within the gain matrix, within the gain matrix, KK..

u k K e k K e kp I( ) ( ) ( ) 1

Three local PI Controllers Three local PI Controllers in seriesin series

u( )

( )

( )

( )

k

u k

u k

u k

1

2

3

K

K K

K K

K K

P I

P I

P I

1 1

2 2

3 3

0 0 0 0

0 0 0 0

0 0 0 0

x( )

( )

( )

( )

( )

( )

( )

k

e k

e k

e k

e k

e k

e k

1

2

3

1

2

3

1

1

1

Expansion of simple PI Expansion of simple PI controllercontroller

• Additional terms are added to state Additional terms are added to state vector to account for vector to account for delaysdelays (as in Smith (as in Smith Predictor used in control theory)Predictor used in control theory)

• Off diagonal elements allow Off diagonal elements allow “decoupling” and “decoupling” and centralized controlcentralized control

Feedback

??

??

• Full gain MatrixFull gain Matrix– Top version highlights PI termsTop version highlights PI terms– Bottom version highlights delay (L) termsBottom version highlights delay (L) terms

e1(k) u1(k-3) u1(k-2) u1(k-1) e2(k) u2(k-2) u2(k-1) e3(k) e4(k) e1(k-1) e2(k-1) e3(k-1) e4(k-1)

Cfs/ft - - - cfs/ft - - cfs/ft cfs/ft cfs/ft cfs/ft cfs/ft cfs/ftu1(k) 59 0.03 0.13 0.13 56 0.05 0.10 51 60 3.6 2.0 1.6 1.0

u2(k) -21 -0.01 -0.05 -0.04 54 0.05 0.10 43 25 -2.6 3.1 1.6 0.9

u3(k) -5 0.00 -0.01 -0.01 -21 -0.02 -0.04 53 27 -0.4 -2.5 3.6 1.2

u4(k) -1 0.00 0.00 0.00 -3 0.00 -0.01 -10 43 -0.1 -0.3 -1.3 3.3

e1(k) u1(k-3) u1(k-2) u1(k-1) e2(k) u2(k-2) u2(k-1) e3(k) e4(k) e1(k-1) e2(k-1) e3(k-1) e4(k-1)

cfs/ft - - - cfs/ft - - cfs/ft cfs/ft cfs/ft cfs/ft cfs/ft cfs/ftu1(k) 59 0.03 0.13 0.13 56 0.05 0.10 51 60 3.6 2.0 1.6 1.0

u2(k) -21 -0.01 -0.05 -0.04 54 0.05 0.10 43 25 -2.6 3.1 1.6 0.9

u3(k) -5 0.00 -0.01 -0.01 -21 -0.02 -0.04 53 27 -0.4 -2.5 3.6 1.2

u4(k) -1 0.00 0.00 0.00 -3 0.00 -0.01 -10 43 -0.1 -0.3 -1.3 3.3

Comparison of Controllers - Test 1-Comparison of Controllers - Test 1-11

40

60

80

100

120

0 100 200 300

Number of Coefficients

Pe

na

lty

Fu

nc

tio

n,

JPI

PI+S

PI+

PI+S+

PI-1+1

PI-1+

PI+S-1+1

PI+S-1+

PI+S-+

PI-+

PI+1

PI+S+1

Conclusions from Conclusions from OptimizationOptimization

• Series of simple PI controllers can be Series of simple PI controllers can be greatly improved upongreatly improved upon

• Adding Smith Predictor should improve Adding Smith Predictor should improve controller performance for this canalcontroller performance for this canal

• Decoupling or sending control signals to Decoupling or sending control signals to other pools should improve controlother pools should improve control

• Sending information to one pool Sending information to one pool downstream and one (or more) pools downstream and one (or more) pools upstream is a good control compromiseupstream is a good control compromise

Simulation TestingSimulation Testing

• Controllers tested with CanalCADControllers tested with CanalCAD

• Tested under tuned and untuned Tested under tuned and untuned conditionsconditions

• 12 different controllers tested for 12 different controllers tested for each test caseeach test case

Test 1-1 with Test 1-1 with NONO gate movement gate movement restrictionsrestrictionsCentralized PI Controller (PILCentralized PI Controller (PIL--

++))

Change at 2 hours had feed-forwardChange at 2 hours had feed-forward

Change at 14 hours was only feed-backChange at 14 hours was only feed-back

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0 4 8 12 16 20 24

Time (hours)

Wat

er le

vel e

rror

(m

) 4 5 83

2

Test 1-1 with gate movement Test 1-1 with gate movement restrictionsrestrictionsCentralized PI Controller (PILCentralized PI Controller (PIL--

++))

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0 4 8 12 16 20 24

Time (hours)

Wat

er le

vel e

rror

(m) 4

1

1

8 5

Test 1-1 untuned (gate move. restr. Test 1-1 untuned (gate move. restr. implied)implied)Centralized PI Controller (PILCentralized PI Controller (PIL--

++))

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0 4 8 12 16 20 24

Time (hours)

Wat

er le

vel e

rror

(m)

4

2

5

5

8

8

5

Test 1-1 untuned Test 1-1 untuned Simple PI ControllerSimple PI Controller

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0 4 8 12 16 20 24

Time (hours)

Wat

er le

vel e

rror

(m)

2/1

2 5

5/2

Test 1-1 untuned Test 1-1 untuned PI PI-1-1

+1+1 Controller Controller

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0 4 8 12 16 20 24

Time (hours)

Wat

er le

vel e

rror

(m) 2

5/2 6 78

2/5

Test 1-1 Comparison: relative to Test 1-1 Comparison: relative to PI+SPI+S--

++

0.0

0.5

1.0

1.5

2.0

PI+S-1+1 PI-1+1 PI

Rel

ati

ve P

erfo

rma

nce

MAE 0-12 Max

MAE 0-12 Ave

MAE 12-24 Max

MAE 12-24 Ave

IAE 0-12 Max

IAE 0-12 Ave

IAE 12-24 Max

IAE 12-24 Ave

StE 0-12 Max

StE 0-12 Ave

StE 12-24 Max

StE 12-24 Ave

IAQ 0-12 Max

IAQ 0-12 Ave

IAQ 12-24 Max

IAQ 12-24 Ave

Test 1-1 Comparison: LQR / Test 1-1 Comparison: LQR / PI+SPI+S--

++

0.00

0.50

1.00

1.50

2.00

MAE0-12Max

MAE0-12Ave

MAE12-24Max

MAE12-24Ave

IAE 0-12Max

IAE 0-12Ave

IAE12-24Max

IAE12-24Ave

StE 0-12Max

StE 0-12Ave

StE12-24Max

StE12-24Ave

IAQ 0-12Max

IAQ 0-12Ave

IAQ12-24Max

IAQ12-24Ave

Rel

ati

ve P

erfo

rma

nce

Test 1-2 untuned Test 1-2 untuned Centralized PI Controller (PILCentralized PI Controller (PIL--

++))

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0 4 8 12 16 20 24

Time (hours)

Wat

er le

vel e

rror

(m)

84

6/7

2

4/24/5

4 6/5/4

5

Test 1-2 untuned Test 1-2 untuned Simple PI Controller (PI)Simple PI Controller (PI)

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0 4 8 12 16 20 24

Time (hours)

Wat

er le

vel e

rror

(m)

25

6/5/22

8 7

5

2

61

Test 1-2 untuned Test 1-2 untuned PI PI-1-1

+1+1 Controller Controller

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0 4 8 12 16 20 24

Time (hours)

Wat

er le

vel e

rror

(m)

6/5

4 2/6/7

8

67

42

5/3

2

Test 1-2 untuned Test 1-2 untuned PIL PIL-1-1

+1+1 Controller Controller

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0 4 8 12 16 20 24

Time (hours)

Wat

er le

vel e

rror

(m)

24

5

87

2

2/4/3

6

4

2/1

32

Test 1-2 Comparison: relative to Test 1-2 Comparison: relative to PI+SPI+S--

++

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

PI+S-1+1 PI-1+1 PI

Rel

ativ

e P

erfo

rman

ce

MAE 0-12 Max

MAE 0-12 Ave

MAE 12-24 Max

MAE 12-24 Ave

IAE 0-12 Max

IAE 0-12 Ave

IAE 12-24 Max

IAE 12-24 Ave

StE 0-12 Max

StE 0-12 Ave

StE 12-24 Max

StE 12-24 Ave

IAQ 0-12 Max

IAQ 0-12 Ave

IAQ 12-24 Max

IAQ 12-24 Ave

Test 1-2 Comparison: LQR / Test 1-2 Comparison: LQR / PI+SPI+S--

++

0.00

1.00

2.00

3.00

4.00

5.00

MAE0-12Max

MAE0-12Ave

MAE12-24Max

MAE12-24Ave

IAE 0-12Max

IAE 0-12Ave

IAE12-24Max

IAE12-24Ave

StE 0-12Max

StE 0-12Ave

StE12-24Max

StE12-24Ave

IAQ 0-12Max

IAQ 0-12Ave

IAQ12-24Max

IAQ12-24Ave

Rel

ati

ve P

erfo

rma

nce

ConclusionsConclusions

• Gate movement restrictions have a big influence on Gate movement restrictions have a big influence on controller performancecontroller performance

• Tuning to actual canal conditions can improve controller Tuning to actual canal conditions can improve controller performanceperformance

• Results suggest passing control actions one pool Results suggest passing control actions one pool upstream and one pool downstream may be good upstream and one pool downstream may be good compromise.compromise.

• While optimization suggests Smith predictor always While optimization suggests Smith predictor always improves performance, simulation results suggest that improves performance, simulation results suggest that it often doesn’tit often doesn’t

• Control with centralized PI controller comparable to Control with centralized PI controller comparable to traditional LQR controllertraditional LQR controller

Simulation results for Upper Arizona Canal when controlling

entire network

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0 50 100 150

Time (hr)

Wat

er L

evel

Err

or (

m)

Granite Reef 1-00.61-01.9 1-03.01-03.4 1-05.01-08.0 1-10.0

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0 50 100 150

Time (hr)

Wat

er L

evel

Err

or (

m)

Granite Reef 1-00.61-01.9 1-03.01-03.4 1-05.01-08.0 1-10.0

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0 50 100 150

Time (hr)

Wat

er L

evel

Err

or (

m)

Granite Reef 1-00.61-01.9 1-03.01-03.4 1-05.01-08.0 1-10.0

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0 50 100 150

Time (hr)

Wat

er L

evel

Err

or (

m)

Granite Reef 1-00.61-01.9 1-03.01-03.4 1-05.01-08.0 1-10.0

Centralized PI w/ feedforward MPC w/ feedforward

Centralized PI w/ feedback only MPC w/ feedback only

Manning n is used to adjust Manning n is used to adjust delay times for volume-based delay times for volume-based

feedforward routingfeedforward routing

0

1

2

3

4

5

6

7

0 20 40 60

Inflow Rate (m^3/s)

Vol

ume

(X 1

00,0

00 m

^3)

0.0140.0180.0220.0260.030

Manningn

Some canal pools do not follow the ID Some canal pools do not follow the ID model. They have “effectively” no model. They have “effectively” no delay, a backwater area, and reflection delay, a backwater area, and reflection waveswaves

463.555

463.56

463.565

463.57

463.575

200 240 280 320 360 400

Time (min)

Ele

vati

on

(m

) .

Influence of reflection wavesInfluence of reflection waves

•Reflection waves can destabilize Reflection waves can destabilize an otherwise stable controlleran otherwise stable controller

•Water level filtering can be used Water level filtering can be used to to

–Minimize the influence of reflection Minimize the influence of reflection waves on controlwaves on control–Remove transducer noiseRemove transducer noise–Provide Anti-aliasingProvide Anti-aliasing

Pseudo-random binary signal can be Pseudo-random binary signal can be used to obtain frequency response of used to obtain frequency response of canal poolcanal pool

Bode (Frequency) Diagram can Bode (Frequency) Diagram can be used to design filtersbe used to design filters

Frequency

Resonance Peak

ActualSignal

FilteredSignal

Filter

ID Model is straight line

Resulting filtered water Resulting filtered water levelslevels

Manual Supervisory Control

• Standard Supervisory Control Standard Supervisory Control Features using iFix Dynamics from Features using iFix Dynamics from Intellution, Inc.Intellution, Inc.

• Added features for canal Added features for canal managementmanagement

Manual Supervisory Control

• iFix allows many types of displays (CAIDD)iFix allows many types of displays (CAIDD)

• Screen allows incremental flow change at gateScreen allows incremental flow change at gate