PROSPECTS OF GRADIENT METHODS FOR NONLINEAR CONTROL

1. PROSPECTS OF GRADIENT METHODS FOR NONLINEAR CONTROL Ivo BUKOVSKÝ1 Jiří BÍLA1 Noriasu HOMMA2 Ricardo RODRIGUEZ1 1Czech Technical University in Prague 2Tohoku University, Japan

2. PROSPECTS OF GRADIENT METHODS FOR NONLINEAR CONTROL • We consider sample-by-sample adaptation of discrete-time models and controllers by gradient descent weight update system

3. PROSPECTS OF GRADIENT METHODS FOR NONLINEAR CONTROL • Stability monitoring and maintenance of weight update system of adaptively tuned models and controllers significantly contributes to a stable and convergent control loop

4. PROSPECTS OF GRADIENT METHODS FOR NONLINEAR CONTROL • In the paper, we introduce derivation of stability condition for gradient-descent tuned models and controllers • The approach is valid for models and controllers that are nonlinear (incl. linear), but they are linear in parameters • Not suitable for conventional neural networks (MLP, RBF) • Suitable for Higher-Order Neural Units (HONU, also known as polynomial neural networks) (not limited to)

5. PROSPECTS OF GRADIENT METHODS FOR NONLINEAR CONTROL Further in this presentation • Fundamental gradient descent schemes for adaptive identification and control • Static or dynamic Higher Order Neural Units (HONU) • Stability conditions for static and dynamic HONUand its maintenance at every adaptation step • Demonstration of achievements with ONU( NOx prediction – EME I, lung motion prediction, nonlinear control loop of a laboratory system)

6. + - Fundamental gradient descent schemes for adaptive identification and controlPlant Identification by Gradient Descent Plant weight update system • Adaptive model • linear • neural network,

7. + - (Základní schemata adaptivní identifikace a řízení gradientovými metodami)Automatické ladění adaptivního stavového regulátoru Referenční model (požadované chování regulované soustavy) Žádaný průběh chování Žádaná hodnota Regulovanásoustava - + • Adaptivní regulátor • lineární • polynomiální • - klasická neuronová síť Systém adaptovaných vah

9. - - + + Fundamental gradient descent schemes for adaptive identification and control (continue)Tuning of Adaptive Controller in a Feedback Control Loop with Gradient Descent Model of desired behavior adaptive controller - linear PID - neural network, Plant

10. + - - + Fundamental gradient descent schemes for adaptive identification and control (continue)Updating Control Inputs Directly by Gradient Descent Plant • Adaptive model • linear • neural network, eC(k)

11. The question is: • How do we assure stability of nonlinear adaptive control loop? • The ways is to assure stability and convergence of adaptive components in a control loop (plant model + controller) • What nonlinear model to use?

12. Static & Dynamic Higher-Order Neural Units How do we assure stability of the nonlinear adaptive control loop? What model to choose? • MLP or RBF networks as models and controllers • Not linear in parameters • Guaranteeing stability is complicated (not suitable for undergraduate level, difficult for PhD students from non-heavy-math schools) • Guaranteeing stability is complicated and theoretically heavy for practicioners (thus not attractive for practice)

13. Static & Dynamic Higher-Order Neural Units How do we assure stability of the nonlinear adaptive control loop? What model to choose? Example of 2nd-order HONU: Weight-update system:

14. Static & Dynamic Higher-Order Neural Units (continue) Sketch of optimization error surfaces Linear x MLP Networks x HONU convetional NN LNU Approximation strength of neural networks can be improved by adding more neurons or even layers, GA, PSO,… HONU 0 “axis of adapted neural weights”

15. Static & Dynamic Higher-Order Neural Units (continue)Static MLP vs. QNU as MISO models ofhot steam turbine averaged data (“steady states”, batch training by Levenberg-Marquardt) • double hidden layer FFNN • single hidden layer FFNN • static QNU • measured data

16. Static&Dynamic Higher-Order Neural Units (continue) Respiration time series: Training Accuracy for Predicting Exhalation Time -Instances of trained neural architectures trained from different initial conditions by L-M algorithm 2-hidden-layer static MLPs (static feedforward networks) 1-hidden-layer static MLPs (static feedforward networks) static QNUs

17. Trénování predikce nelineárního periodického signálu Trénování predikce polohy plic Static & Dynamic Higher-Order Neural Units (continue) Trénování predikce Mackey-Glass

18. Static & Dynamic Higher-Order Neural Units (continue)

19. Stability of weight-update system • Condition for STATIC HONU • Condition for DYNAMICAL HONU HONU ,

23. Achievements with QNU

24. NOx,CO prediction – EME I

26. Lung Tumor Motion Prediction

27. Lung Tumor Motion Prediction by static QNU sampling 15 Hz, epochs=100, Ntrain=360, 492 neural weights

28. Lung Tumor Motion Prediction by static QNU

29. Nonlinear Control Loop of a Laboratory System [ ] Ladislav Smetana: Nonlinear Neuro-Controller for Automatic Control,Laboratory System, Master’s Thesis, Czech Tech. Univ. in Prague, 2008.

30. Nonlinear Control Loop of a Laboratory System PID Control and Nonlinearity of the Plant Tunned PID controller for 10 cm Tunned PID controller for 20 cm 30

31. Nonlinear Control Loop of a Laboratory System Linear PID QNU as Adaptive Controller (simplest gradient descent) 31

32. Nonlinear Control Loop of a Laboratory System

33. False Neighbor Analysis is a single-scale analysis To train neural networks , input (state) vector must be estimated to minimize uncertainty in training data

34. Děkuji za pozornost

35. False Neighbor Analysis is a single-scale analysis x input data y output data y=f(x) False Neighbors

36. MULTI-SCALE ANALYSISapproach(MSA) number of false neighbours on a main diagonal 150 100 FN 50 0 1 2 3 4 5 6 id...index of a diagonal cell • What is the fundamental idea? • To characterize a system over the range of setups • Power law

37. MULTI-SCALE ANALYSISapproach(MSA) • To characterize a system over the range of intervals • What is the fundamental idea? • The power-law concept r(k)=2,4,8 q … quantity H … characterizing exponent r(k) … discretely growing radius

38. MULTI-SCALE ANALYSISapproach(MSA) • To characterize a system over the range of intervals • What is the fundamental idea? • The power-law concept r(k)=2,4,8 q … quantity H … characterizing exponent r(k) … discretely growing radius

39. MULTI-SCALE ANALYSISapproach(MSA) • What is the fundamental idea?

40. MULTI-SCALE ANALYSISapproach(MSA) (cont.) • How can MSA help to create better neural network models? False Neighbors Matrix: Multiscale False Neighbor Approach

41. MULTI-SCALE ANALYSISapproach(MSA) (cont.) • What are other potentrials for MSA for signal processing? • MSA based signal processing • Variance Fractal Dimension Trajectory (VFDT) • Mutual Information • Multiscale approach to calculate mutual information itself • Mutual information of VFDT processed signals • Everywhere, where a common analysis is subject to a single-parameter setup and changing the setup disqualifies the analysis results.