1 Adaptive Neural Network Control of Nonlinear Systems S. Sam Ge Department of Electrical & Computer Engineering National University of Singapore Singapore

2 Work Place Inaugural IEEE Multi-conference on Systems & Control 16th IEEE Conference on Control Applications (CCA) 22 nd IEEE International Symposium on Intelligent Control F. L. Lewis, Sponsorship Chair S. Jagannathan, ISIC Program Chair T. Parisini, Conference Editorial Board Chair

3 Work Place

4 Content 1.Introduction 2.System Descriptions 3.Neural Network Approximation 4.State-Feedback Control for SISO 5.Output-Feedback Control for SISO 6.Simulation Study 7.Conclusion

5 1. Introduction Neural network control has gone through the pioneering works, the pains against the skeptics and doubts, and the graceful acceptance, and maturity as a powerful tool for control of nonlinear systems. Thanks to the many distinguished individuals and their families: Narendra, Levin, Lewis, Calise, Polycarpou, Hovakimyan, Jagannathan, Slotine, Ge, …

6 1 Intelligent Control The most intelligent system in nature! Info. Feedback Real-time Control Decision Making

7 1 Adaptive NN Control Cycling or driving ， we never thinking of the so-called mathematical models ！ Plant Info. feedback Adaptation & Learning Control Law

8 1 Adaptive NN Control

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10 1 Adaptive NN Control

11 1 Adaptive NN Control

12 1. Adaptive NN Control System Modeling is usually more difficult than control system design Model based control though rigorous, it depends too much on model building 。 Before 90s ： Off-line NN Training After 90s ： Combining adaptive control, and NN parametrization, on-line adaptive NN control is investigated.

13 Continuous to Discrete Owing to different analytical tools used, results in continuous time are not necessarily hold in discrete time.

14 Discrete-time SISO system where 2 System Descriptions

Assumptions

System Descriptions Discrete-time MIMO system

System Properties The inputs are in triangular form. There are both inputs and states interconnections. The system cannot be expressed as  (k+1)=F(  (k))+G(  (k))U(k) which makes the feedback linearization method not applicable. We have the following observations:

18 3 Neural Network Approximation In control engineering, different types of neural networks, including LPNN (RBF, HONN) and MLNN, have been used as function approximators over a compact set. LPNN:

19 3 Neural Network Approximator For clarity of analysis, consider HONNs where

20 3 Neural Network Approximation The particular choice of NN is used for analysis only, similar results can be obtained for (extended to) other linearly parametrized networks, radial basis function networks, polynomials, splines functions, fuzzy systems, and, the multiple layer neural networks (Nonlinear). Different choices affect performance though.

21 Part I

22 The non-causality is one of the main problems for strict-feedback nonlinear system through backstepping in discrete time. 4 State-Feedback Control 1.Non-causal Problem, 2.System Transformation, 3.Desired Control, 4.Stable Control System Design The following issues are in order

23 Consider the discrete SISO system given 4.1 Non-causal Problem Direct application of backstepping, the following ideal fictitious controls are in order:

n-step Ahead Predictor

25 Re-examining the system 4.2 System Transformation

System Transformation

System Transformation

28 Through the coordinate transformation, we have 4.2 System Transformation

29 The desired (virtual) controls are given by: 4.2 Desired Virtual Controls

30 The desired controls are functions of unknown functions, thus are not feasible. As such NN control is called upon to construct a feasible controller. Let us consider the fictitious controls and the control as: 4.4 Adaptive Neural Control

31 The errors are defined as Neural network weight update laws are 4.4 Adaptive Neural Control

Stability Analysis

Stability Analysis

Stability Analysis

Stability Analysis

Stability Analysis

Stability Analysis

38 Part II Before

39 Part II After

40 For output feedback control, the strict-feedback form is transformed into a cascade form. For equivalent transformation of coordinates, it is important to ensure that the transformation map is diffeomorphism. The following issues will be highlighted: 1.Coordinate Transformation 2.Diffeomorphism 3.Cascade Form 4.Control Design 5 Output-Feedback Control

Coordinate Transformation

Coordinate Transformation

Diffeomorphism

Diffeomorphism

Diffeomorphism

Diffeomorphism

Cascade Form

Cascade Form

Cascade Form

Cascade Form

Control Design

Control Design

Control Design

Control Design

Control Design

Control Design

57 Part III

58 6 Simulation Study Consider a nonlinear discrete-time SISO plant where

59 3: Simulation Studies (cont.) 6.1 State Feedback Control

60 3: Simulation Studies (cont.) 6.2 Direct Output Feedback Control

61 8. Conclusion 1.Adaptive full state feedback NN control has been presented via backstepping for a class of nonlinear unknown discrete- time SISO systems in strict-feedback form. 2.By transforming the system to sequential decrease cascade form, the non-causal problem was solved. 3.High order neural networks are used as the emulators of desired virtual and practical controls, which avoids possible control singularity problem. 4.By transforming the system into cascade form, an adaptive direct output feedback control scheme has also been presented using neural networks.

62 Research “… I seem to have been only like a boy playing on the seashore, diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, while the great ocean of truth lay all undiscovered before me.” Isaac Newton

63 Latest Pebble or Shell With Appreciation

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65 Eye Fusion