Chapter 12 Case Studies Part B

Control System Design

Two case studies combined - use both neural network and fuzzy logic approaches Single-axis cart positioning system Initial position x(0) and velocity v(0) arbitrary within limits Mass of cart is m, force F is applied (positive from right) Object is to move cart to target location x t in minimum time Objective is met when velocity < v min and

Cart Positioning Problem

System Equations * System equations are known and are given on next slide * Choice of discretized time increment is very important (.02 sec. here) * Maximum number of time steps (perhaps ) * Select mass of cart (2 kg. here)

System Equations Optimal “bang-bang” solution using force F (Bryson 1975); apply F if: otherwise, apply –F Note that system equations often are not available.

Neural Network Approach * Two main uses are made of neural networks: > modeling system to be controlled and overall system with controller > design of controller * First step is to model the nonlinear system without the controller

System Modeling Take data with a sufficient variety of input combinations.

System Model * NLS model predicts system state at next time increment given present state and inputs (including controller output) * Can use system equations or direct measurement * Take data under typical operational conditions * Start out with an appropriate set of initial conditions * Train neural net using data at time t for input and for output

Controller Design (Preview) This is what the controller should look like. x t is the target value.

Controller Design The first step is to model overall system with controller Use overall model to generate (obtain) training pattern pairs Can use system equations, system performance curves, neural net model, or trained operator to operate system while data are taken System operation under typical operating conditions is important

Overall System Model

Develop Controller Combine nonlinear model (weights frozen) with untrained controller network Use patterns from overall system model to train controller Output PE of controller can be input PE of trained system model Replace system model with system; controller is now online

Controller Development

Fuzzy Logic Approach Now use fuzzy logic and evolutionary computation to solve same problem Assume same basic groundwork has been done; same parameters as before First step is to partition domains of input variables (x and v) and the output variable F into fuzzy sets Value of x appearing in fuzzy sets, etc. is really x t – x Objective is to move cart to target position in minimum time

Fuzzy Membership Functions

Fuzzy Rule Evolution Defining fuzzy membership functions can be difficult In example, we define 5 membership functions over domain of each input Therefore, we have a 5 x 5 matrix Don’t make system evolve more than it must–fill in “obvious” matrix locations

Fuzzy Rule Evolution Each matrix position represents a rule consequent.

Representation and Operators Decide on representation; example is {0,1,2,3,4,5}, corresponding to 5 fuzzy membership functions and “no entry” Crossover can be standard 1-point or 2-point Mutation possibilities include: * increment/decrement by one * mutate to any value One GA approach used 31 individuals, crossover rate of.7, mutation rate of.01, and elitist strategy

Fitness Fitness is inversely proportional to number of time steps (other metrics could be added) One form of fitness is (max_steps_allowed – steps_required) Average over wide variety of initial conditions representative of operating environment

Evolution of Membership Functions * Evolve “better” locations for peak points and end points of membership functions * Evolve optimal mix of membership function types (linear, nonlinear) * Probably use same initial conditions and fitness function as previously * Perhaps iterate process

Other Approaches * Genetic programming used by Koza Used Lisp S-expressions Only three functions used: *, ABS, and > Good solution evolved in just 5 generations * Could perhaps use approach of Chapter 8, clustering system state vectors Use equations or simulations to obtain state vectors Cluster vectors using LVQ Develop fuzzy rules using cluster centroids * Other approaches possible

Summary There is usually more than one good way to solve a problem. Computational intelligence tools often provide a good approach when dealing with complex systems in changing environments.