Bound Computation for Adaptive Systems V&V

1. Bound Computation for Adaptive Systems V&V Giampiero Campa September 2008 West Virginia University

2. Outline • Background / Proposed Effort • Bounded Neural Networks Library • Improved Bounds Formulation • Case of study • Future Work

3. Background: Adaptive Systems • Adaptive Control Systems are needed to improve: • Autonomy/Self Reliance • Performance in unforeseen conditions • Fault Tolerance • PROBLEM: Can a given adaptive dynamical systems ever exceed certain working limits as a result of its online learning mechanism ? • In other words, can we trust adaptive systems ?

4. The Boundedness Problem • Adaptive Control Systems often rely on some kind of “boundedness proof”, ensuring that the evolution of the system is bounded within certain limits. • Proofs are usually system-specific or not general enough • Bounds are shown to exist but are never calculated • No software tool exist to help the control system designer in the involved tradeoffs. • As a consequence, this kind of proof does not result in a more formal design process

5. Main Goals of the Project • Calculate as-general-as-possible boundedness conditions as well as bounds expressions • Develop software to calculate and visualize such conditions and bounds, given a general adaptive control system. • Perform a detailed analysis of the inherent trade offs among systems parameters and boundedness • Develop software to help the designer within the simulation and implementation phases of the adaptive control system.

6. Proposed Effort • Year 1: • Development of a library for simulation of adaptive control systems containing neural networks as main adaptive element. • Preliminary boundedness study of a simple systems. • Year 2: • Calculation of closed form expressions for bounds of an adaptive system within a general setting • Year 3: • Development of software to calculate and check bounds for a given adaptive system • Case Studies

7. Outline • Background / Proposed Effort • Bounded Neural Networks Library • Improved Bounds Formulation • Case of study • Conclusion and Future Work

8. Neural Networks for Adaptive Control • Neural network software for use within adaptive control needs: • A learning algorithm capable of working on-line, whereas most of the software only allows off-line (batch) learning. • Some sort of “stability modification” of the adaptation laws, other than the usual error-driven “gradient rule”. • The capability of setting limits to the weights. • Seamless integration with both a simulation environment (Simulink) and a Automatic Real Time Code Generation Tool (Real Time Workshop)

9. Adaline Multi Layer Perceptron Demo Extended DCS Extended RBF Neural Networks Library

10. Outline • Background / Proposed Effort • Bounded Neural Networks Library • Improved Bounds Formulation • Case of study • Conclusion and Future Work

11. Plant and Uncertainty Structure • The study assumes: • where u(t) and y(t) are vectors of different dimensions, that is the plant can have several inputs and outputs • The matrix D connecting directly inputs and outputs can be different from zero. • Also note that Δx and Δy (uncertainties on both equations) enter the equations in the most general way.

12. Neural Network • The Neural Network has the following adaptation laws: • Where is the vector containing the weights of the neural network. • ((t)) is the vector of the radial basis functions. • L is the learning rate. •  is the forgetting factor • enn(t) is the error of the plant

13. Boundedness proofs for Adaptive systems are based on Lyapunov Analysis: A positive definite Lyapunov function is defined: Where z is the error in the state of the system, We the error in the state of the Adaptive element, L is the learning rate of the NN, r is a positive constant and P is the solution of ATP+PA+Q=0 Note that the time derivative of V depends on the evolution equation of both system and network. Lyapunov Function

14. Lyapunov Function Lyapunov Function • Derivative of the Lyapunov Function • The Time Derivative of the Lyapunov function depends on the system to be controlled and on the control parameters • We want such derivative to be as negative as possible

15. Bounds Calculation • The expression for the time derivative is: • Developing the above expression leads to an overestimation Hi which is usually very long and complicated: • Typically in the literature Hi is a 2D paraboloid in the scalar variables ||z|| and ||We||:

16. Typical Bounds Calculations • Therefore • Is the expression of a simple ellipse in ||z|| and ||We|| having the center in the origin with following semi-axis (bounds): • This is ok to show that bounds exist for some choice of parameters, but formulas lose significance in many cases and bounds are grossly overestimated

17. Novel Bounds Expressions • By avoiding a number of approximations that are usually made to limit length and complexity, two different expressions can be obtained: • The first function H2 is useful for calculating the boundsin the norm space and can be directly compared with the approximation obtained in the function H1. • The second function H3 is useful for the calculation of the bounds for each absolute error state variable of the system or

18. New Bounds Expressions (Norm Space) • The function H2 can be expressed in the form: • Note that the coefficients are all scalars

19. New Bounds (Norm Space) Where:

20. Example Bounds calculated with the function H2 (new formulas) Bounds calculated with the function H1(existing formulas) • The new bounds are considerably smaller that those calculated using existing formulas

21. Analysis on the Parameters Variation

22. Analysis on the Parameters Variation

23. Analysis on the Parameters Variation

24. Analysis on the Parameters Variation

25. Analysis on the Parameters Variation

26. New Expression (Absolute State Space) • The function H3 can be expressed in the matrix form: Vector Matrix Vector

27. Ellipsoidal Toolbox Formulation • The Matlab Ellipsoidal Toolbox was used to calculate the bounds for each state variable. • In order to use the toolbox the function: has to be transformed to: • The specific form of the function allows to directly calculate the bounds for each of the component of the state vector x(no closed formulas are needed).

28. Outline • Background / Proposed Effort • Bounded Neural Networks Library • Improved Bounds Formulation • Case Study • Conclusion and Future Work

29. Case Study • We are currently working on a simulation example using F18 dynamics and controller from NASA Dryden.

30. Case Study • We are using a Multi Input Multi Output Local Linear Model of the aircraft: • Where the states are: • And the inputs are: pitch rate attack angle airspeed roll rate yaw rate sideslip angle ailerons stabilators rudders flaps

31. Case Study: no adaptation • Blue: actual tracking behavior, green: reference behavior

32. Case Study: with adaptation • Blue: actual tracking behavior, green: reference behavior

33. Case Study : bounding ellipse • the extreme point of the blue circle in ||z|| axis is equal to 4.06*105, the red line is at the value 2.17*105

34. Case Study: projection on p and r axes • the extreme point of the red circle in the error of |p| and |r| axes are equal to 3.57*105 and 2.23*105. The real evolution of the system is 6 order of magnitude smaller then the bounds

35. Case Study: Bounds computation • The bounds were computed using the function H3 and the Matlab Ellipsoidal Toolbox. • Note: it is convenient think about the bounds as ellipse in 2D space or ellipsoid in n+nc+1 dimensional space, but due of the definition of norm and absolute value the bounds have lower limit to zero.

36. Software for bounds calculation

37. Outline • Background / Proposed Effort • Bounded Neural Networks Library • Improved Bounds Formulation • Simple Example • Conclusion and Future Work

38. Conclusion • The boundedness problem for adaptive control systems was studied in deep, using Lyapunov based analysis. • Boundedness conditions and expressions have been calculated using a new, less restrictive method for the norm space case and new approach for the absolute state space error. • A F-18 aircraft model was studied and the calculation of bounds was performed. • Software to calculate and visualize the bounds was developed.

39. Next Steps • Complete the study repeating all the steps and analyze the case study whenever an observer is placed before the adaptive element. • Complete the report with the case of study. • Journal papers submission.