Case studies Case studies Roland Pihlakas Roland Pihlakas 08. Dec - - PowerPoint PPT Presentation

Control with Neural Networks Control with Neural Networks

Case studies Case studies

Roland Pihlakas Roland Pihlakas

• 08. Dec 2008
• 08. Dec 2008

Proof of concept examples Proof of concept examples

• Sunspot Activity:

Sunspot Activity:

• Classical example

Classical example

• Hydraulic actuator for a crane:

Hydraulic actuator for a crane:

• NPC > APC

NPC > APC

• The issue of fast sampling for validation

The issue of fast sampling for validation

• Pneumatic position servomechanism:

Pneumatic position servomechanism:

• Nonlinear

Nonlinear

• Level in a water tank:

Level in a water tank:

• Direct inverse control

Direct inverse control

The Sunspot Benchmark The Sunspot Benchmark

• Optimal Brain Surgeon (OBS) assists

Optimal Brain Surgeon (OBS) assists in: in:

• Selecting the network architecture

Selecting the network architecture

• Selecting the regressors (inputs)

Selecting the regressors (inputs)

The Sunspot Benchmark The Sunspot Benchmark

• Fully connected network has too

Fully connected network has too many adjustable parameters for the many adjustable parameters for the training set training set

• OBS algorithm:

OBS algorithm:

• Prune input

Prune input -

• to

to-

• hidden weights

hidden weights

• Retrain

Retrain

• Remove the least salient unit from the

Remove the least salient unit from the set of units with single input set of units with single input

The Sunspot Benchmark The Sunspot Benchmark

• The training error gets larger during

The training error gets larger during pruning pruning

• Test errors will decrease due to better

Test errors will decrease due to better generalization / less overfitting ... Until generalization / less overfitting ... Until some point. some point.

• Note that FPE

Note that FPE is not too informative is not too informative in current example... in current example...

The Sunspot Benchmark The Sunspot Benchmark

• Matlab:

Matlab: “ “ it looks as if not much is it looks as if not much is gained gained by pruning. The reason for by pruning. The reason for this is, however, that the this is, however, that the network network has been trained using has been trained using r regularization. egularization.” ”

• The result of

The result of pruning: pruning:

The Sunspot Benchmark The Sunspot Benchmark

• Additional notes

Additional notes

• The result of pruning sessions can vary a

The result of pruning sessions can vary a great deal great deal. . -

• > One must run multiple pruning

> One must run multiple pruning sessions, each one started with a different set sessions, each one started with a different set

• f network weights.
• f network weights.
• The test sets w ere in som e sense

The test sets w ere in som e sense “ “actively actively” ” used for pruning used for pruning. A distinction is . A distinction is made between this type of result and so made between this type of result and so-

• called

called “ “ genuine predictions genuine predictions” ” , where test sets are , where test sets are strictly used for validation. strictly used for validation. -

• > gives more

> gives more reliable estimate of generalization error. reliable estimate of generalization error.

Hydraulic Actuator Hydraulic Actuator

• Problem with fast sampling

Problem with fast sampling

• NPC > APC

NPC > APC

Hydraulic Actuator Hydraulic Actuator

• Measured values:

Measured values:

• Valve opening (input)

Valve opening (input)

• Oil pressure (output)

Oil pressure (output)

• Note the

Note the

• scillatory
• scillatory

response response

Hydraulic Actuator Hydraulic Actuator

• First, linear model will be estimated

First, linear model will be estimated

• This is useful as a reference against

This is useful as a reference against more complicated models more complicated models

• ARX(3, 2, 1)

ARX(3, 2, 1) evaluation: evaluation:

Hydraulic Actuator Hydraulic Actuator

• NNARX(3, 2, 1)

NNARX(3, 2, 1)

• 10 network architectures, with 1

10 network architectures, with 1-

• 10

10 hidden units, 5 networks of each hidden units, 5 networks of each

• Legend:

Legend: x x – – training error training error

• –

– test error test error

• Spread of errors

Spread of errors is caused by is caused by local minima local minima

Hydraulic Actuator Hydraulic Actuator

• For comparing model structures it is

For comparing model structures it is absolutely vital absolutely vital that the training that the training must be continued until the weights must be continued until the weights are extremely near the minimum. are extremely near the minimum. Else overfitting will be less Else overfitting will be less pronounced. pronounced.

• Network with 4 hidden units was

Network with 4 hidden units was

• best. It is recommended to choose
• best. It is recommended to choose

then a slightly larger network. then a slightly larger network.

Hydraulic Actuator Hydraulic Actuator

• Next, regularization is performed.

Next, regularization is performed.

• Legend:

Legend: solid solid – – training error training error dashed dashed – – test error test error dot dot -

• dashed

dashed – – simulation on test simulation on test

• Note that again test set was used for

Note that again test set was used for training... training...

Hydraulic Actuator Hydraulic Actuator

• NNARX simulation is better than of

NNARX simulation is better than of the linear model: the linear model:

Pneumatic Servomechanism Pneumatic Servomechanism

• Nonlinear and has poorly damped

Nonlinear and has poorly damped complex pole pair in the operating complex pole pair in the operating point. point.

Pneumatic Servomechanism Pneumatic Servomechanism

• The system has to be operated in

The system has to be operated in closed closed-

• loop when conducting the

loop when conducting the experiment. experiment.

• Manually tuned PI

Manually tuned PI -

• controller is used

controller is used for stabilization of the system during for stabilization of the system during the experiment. the experiment.

Pneumatic Servomechanism Pneumatic Servomechanism

• To cover entire operating range, a high

To cover entire operating range, a high-

• frequency signal is applied in some

frequency signal is applied in some periods of the experiment. periods of the experiment.

Pneumatic Servomechanism Pneumatic Servomechanism

• Mimimum test error was achieved

Mimimum test error was achieved with 12 hidden units, which with 12 hidden units, which corresponds to 121 weights. corresponds to 121 weights.

• 121 weights is small number

121 weights is small number compared to training set compared to training set = > no need for regularization or = > no need for regularization or pruning. pruning.

Pneumatic Servomechanism Pneumatic Servomechanism

• NPC control. Note how controller

NPC control. Note how controller anticipates future changes in the anticipates future changes in the set set -

• point.

point.

Pneumatic Servomechanism Pneumatic Servomechanism

• APC control. The response is similar

APC control. The response is similar to the one of NPC. to the one of NPC.

• But this time there was no more

But this time there was no more steady steady-

• state error.

state error.

• APC is simpler to implement and

APC is simpler to implement and requires much less computations requires much less computations than NPC. than NPC.

Pneumatic Servomechanism Pneumatic Servomechanism

• The poles of the extracted linear

The poles of the extracted linear models: models:

Control of Water Level Control of Water Level

• The water input inlet is controlled.

The water input inlet is controlled.

• The water outlet is uncontrolled and

The water outlet is uncontrolled and

• pen. The water output flow and
• pen. The water output flow and

thus also the system is nonlinear. thus also the system is nonlinear.

Control of Water Level Control of Water Level

• When linearising such nonlinear

When linearising such nonlinear system, one gets different system, one gets different parameters at different operating parameters at different operating points. points.

• This time direct inverse control was

This time direct inverse control was used instead. Such controllers are used instead. Such controllers are very simple to implement. very simple to implement.

Control of Water Level Control of Water Level

• Conducting the experiment:

Conducting the experiment:

• One of linear models was used for

One of linear models was used for conducting the experiment in closed conducting the experiment in closed-

• loop.

loop.

• There should be both small and big

There should be both small and big changes in the output. changes in the output.

• A random signal is added to the control

A random signal is added to the control inputs to ensure that the model will be inputs to ensure that the model will be able to produce reliable high able to produce reliable high-

• frequency

frequency

• utputs of small magnitude.
• utputs of small magnitude.

Control of Water Level Control of Water Level

• Behavior of the model:

Behavior of the model:

• “

“ Bang Bang-

• bang

bang” ” type control type control -

• maximum or

maximum or minimum control input is applied until the minimum control input is applied until the desired set desired set -

• point is achieved.

point is achieved.

• After that the control

After that the control input attains its input attains its steady steady-

• state.

state.