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 by pruning. The reason for by pruning. The reason for gained this is, however, that the network network this is, however, that the has been trained using has been trained using regularization. egularization.” ” r � 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 • . - - > One must run multiple pruning > One must run multiple pruning great deal . great deal sessions, each one started with a different set sessions, each one started with a different set of network weights. of 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- - called called made between this type of result and so “ genuine predictions genuine predictions” ” , where test sets are , where test sets are “ strictly used for validation. - - > gives more > gives more strictly used for validation. 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 � oscillatory oscillatory 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 – – training error training error x o – – test error test error o � 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 that the training that the training absolutely vital 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 – – training error training error solid dashed – – test error test error dashed dot - - dashed dashed – – simulation on test simulation on test dot � 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- - loop when conducting the loop when conducting the closed 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 - - point. point. set
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- - state error. state error. steady � 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 � open. The water output flow and open. 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- - frequency frequency able to produce reliable high outputs of small magnitude. outputs of small magnitude.
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