September Bound Computation for Adaptive Systems V&V Giampiero Campa September 2008 West Virginia University
September Outline Background / Proposed Effort Bounded Neural Networks Library Improved Bounds Formulation Case of study Future Work
September 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 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 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 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 Proposed Effort
September Outline Background / Proposed Effort Bounded Neural Networks Library Improved Bounds Formulation Case of study Conclusion and Future Work
September 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 Neural Networks Library Adaline Multi Layer Perceptron Extended RBF Extended DCS Demo
September Outline Background / Proposed Effort Bounded Neural Networks Library Improved Bounds Formulation Case of study Conclusion and Future Work
September 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 Δ x and Δ y (uncertainties on both equations) enter the equations in the most general way.
September 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 e nn (t) is the error of the plant
September 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, W e 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 A T P+PA+Q=0 Note that the time derivative of V depends on the evolution equation of both system and network.
September 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 Bounds Calculation The expression for the time derivative is: Developing the above expression leads to an overestimation H i which is usually very long and complicated: Typically in the literature H i is a 2D paraboloid in the scalar variables ||z|| and ||W e ||:
September Typical Bounds Calculations Therefore Is the expression of a simple ellipse in ||z|| and ||W e || 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 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 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 or
September New Bounds Expressions (Norm Space) The function H 2 can be expressed in the form: Note that the coefficients are all scalars
September New Bounds (Norm Space) Where:
September Example The new bounds are considerably smaller that those calculated using existing formulas Bounds calculated with the function H 1 (existing formulas) Bounds calculated with the function H 2 (new formulas)
September Analysis on the Parameters Variation
September Analysis on the Parameters Variation
September Analysis on the Parameters Variation
September Analysis on the Parameters Variation
September Analysis on the Parameters Variation
September New Expression (Absolute State Space) The function H 3 can be expressed in the matrix form: Matrix Vector
September 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 Outline Background / Proposed Effort Bounded Neural Networks Library Improved Bounds Formulation Case Study Conclusion and Future Work
September Case Study We are currently working on a simulation example using F18 dynamics and controller from NASA Dryden.
September 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 rateattack angleairspeedroll rateyaw ratesideslip angle aileronsstabilatorsruddersflaps
September Case Study: no adaptation Blue: actual tracking behavior, green: reference behavior
September Case Study: with adaptation Blue: actual tracking behavior, green: reference behavior
September Case Study : bounding ellipse the extreme point of the blue circle in ||z|| axis is equal to 4.06*10 5, the red line is at the value 2.17*10 5
September 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*10 5 and 2.23*10 5. The real evolution of the system is 6 order of magnitude smaller then the bounds
September 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 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.
September Software for bounds calculation
September Outline Background / Proposed Effort Bounded Neural Networks Library Improved Bounds Formulation Simple Example Conclusion and Future Work
September 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 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.