Bound Computation for Adaptive Systems VV Giampiero Campa
Bound Computation for Adaptive Systems V&V Giampiero Campa September 2008 West Virginia University September 2008 1
Outline • Background / Proposed Effort • Bounded Neural Networks Library • Improved Bounds Formulation • Case of study • Future Work September 2008 2
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 ? September 2008 3
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 September 2008 4
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. September 2008 5
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 September 2008 6
Outline • Background / Proposed Effort • Bounded Neural Networks Library • Improved Bounds Formulation • Case of study • Conclusion and Future Work September 2008 7
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) September 2008 8
Neural Networks Library Adaline Multi Layer Perceptron Demo Extended DCS Extended RBF September 2008 9
Outline • Background / Proposed Effort • Bounded Neural Networks Library • Improved Bounds Formulation • Case of study • Conclusion and Future Work September 2008 10
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. September 2008 11
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 September 2008 12
Lyapunov Function • 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. September 2008 13
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 September 2008 14
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 2 D paraboloid in the scalar variables ||z|| and ||We||: September 2008 15
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 September 2008 16
Novel Bounds Expressions • By avoiding a number of approximations that are usually made to limit length and complexity, two different expressions can be obtained: or • The first function H 2 is useful for calculating the bounds in the norm space and can be directly compared with the approximation obtained in the function H 1. • The second function H 3 is useful for the calculation of the bounds for each absolute error state variable of the system September 2008 17
New Bounds Expressions (Norm Space) • The function H 2 can be expressed in the form: • Note that the coefficients are all scalars September 2008 18
New Bounds (Norm Space) Where: September 2008 19
Example Bounds calculated with the function H 1 (existing formulas) Bounds calculated with the function H 2 (new formulas) • The new bounds are considerably smaller that those calculated using existing formulas September 2008 20
Analysis on the Parameters Variation September 2008 21
Analysis on the Parameters Variation September 2008 22
Analysis on the Parameters Variation September 2008 23
Analysis on the Parameters Variation September 2008 24
Analysis on the Parameters Variation September 2008 25
New Expression (Absolute State Space) • The function H 3 can be expressed in the matrix form: Vector Matrix Vector September 2008 26
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). September 2008 27
Outline • Background / Proposed Effort • Bounded Neural Networks Library • Improved Bounds Formulation • Case Study • Conclusion and Future Work September 2008 28
Case Study • We are currently working on a simulation example using F 18 dynamics and controller from NASA Dryden. September 2008 29
Case Study • We are using a Multi Input Multi Output Local Linear Model of the aircraft: • Where the states are: pitch rate attack angle airspeed roll rate yaw rate sideslip angle • And the inputs are: ailerons September 2008 stabilators rudders flaps 30
Case Study: no adaptation • Blue: actual tracking behavior, green: reference behavior September 2008 31
Case Study: with adaptation • Blue: actual tracking behavior, green: reference behavior September 2008 32
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 September 2008 33
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 September 2008 34
Case Study: Bounds computation • The bounds were computed using the function H 3 and the Matlab Ellipsoidal Toolbox. • Note: it is convenient think about the bounds as ellipse in 2 D 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. September 2008 35
Software for bounds calculation September 2008 36
Outline • Background / Proposed Effort • Bounded Neural Networks Library • Improved Bounds Formulation • Simple Example • Conclusion and Future Work September 2008 37
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. September 2008 38
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. September 2008 39
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