1. Modeling Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo

2. Outline of Today’s Lecture Review Feedback Open Loop Systems Closed Loop System Positive Feedback Negative Feedback Basic Control actions Models Dynamics States Phase Plots Example: Predator Prey Model

3. Open Loop Control Usually “set point” systems Advantages Simple Insensitive to environment Set and forget Disadvantages Non correcting Sensitive to disturbances Insensitive to environment Examples Irrigation systems Washing machines SensingComputeActuate

4. Closed Loop Control Adds a feedback loop to the control system For computational purposes, it is shown as Sense Compute Actuate ControllerPlant Sensor Input Output Disturbance + or -

5. Positive Feedback Positive Feedback Clip ControllerPlant Sensor + + + + Vibrating Guitar String Magnetic Pickup Amplifier Speaker Plucked String String Vibrations ControllerPlant Sensor + + + + Sound Guitar String w/ pickup Amplifier Speaker Ambient Sound 2 possible models Background sound Previous Vibrations

6. Positive Feedback Positive feedback is used to increase the actuation in the loop. Advantages Increased results Faster results Finds extremes (maxima and minima) Disadvantages Consumes energy Subject to local extremes (introns) May become unstable May destroy system Examples: M etal finders Searches Stock market programs Genetic Algorithm build population create mutations test performance measure Results Culled from Population BestWorst

7. Negative Feedback Input ControllerPlant Sensor Output Disturbance + - + + Error Signal homeostasis Desired Heart Beat ControllerPlant Sensor Heart Beat Salt + - + + Heart Nerves Parasympathetic/ Sympathetic System

8. Negative Feedback Negative Feedback is used to reduce error Advantages Controls to a set point Robustness to disturbances (uncertainty) Rejection of distortion Disadvantages Prone to oscillation Instability Complexity Coupling Examples Set point control Tracking Chang the system dynamics

9. Basic Control Actions Bang-Bang (Off-On) Fixed two state or multistate control actions Control question: how to chose? Proportional Control in proportion to error Integral Control based on size and duration of error Derivative Control based on size and change of error Combined (PID) All three: Proportional, Integral and Derivative Most used

10. Models A model is a representation of something The something can be an idea, a concrete object or an abstract object It is NOT the real thing: they are simplifications it is a fiction of our imagination Models can take many forms Solid Blocks Equations Computer programs Word descriptions Symbols

11. Models It is NOT the real thing: they are simplifications it is a fiction of our imagination Models are used for visualization understanding explaining to others analysis predict improve The value of a model is how well it serves the purpose used for

12. Models Different models answer different questions As a model developer, you need to chose the right model for your problem How will costs change with the strength of the landing gear? How much vertical force is the landing gear putting on the nose? How high should the tire pressure be? At what point can the pilot rotate the aircraft to take weight off the nose wheel? How much heat is built up in the tire on a takeoff roll?

13. Dynamics defn (TheFreeDictionary): The branch of mechanics that is concerned with the effects of forces on the motion of a body or system of bodies, especially of forces that do not originate within the system itself. Also called kinetics. Things move! If we are to control movement we need to know how they move

14. Dynamics Models of dynamics used in this course: Based on functions of time Differential equations time is considered continuous Difference equations time is considered discrete

15. State A state is a set of variables whose values when known completely define the dynamics (motion) State variables for nose wheel example: The parameters of the example are So, what about ?

16. State So, what about ? These are completely determined by the state variables! We can rewrite the equations as

17. Phase Plot A phase plot is a plot of a state variable vs. another state variable Useful in understanding how the dynamics change with changes in state To see how the dynamics are represented by the phase plot, consider the predator prey problem we will first use differential equations then difference equations

18. Predator Prey Model Volterra – Lotke model Vito Volterra and Alfred J. Lotke independently developed this useful model Explains the growth of a thing that depends on the growth of another thing Lynx and hares or whales and krill are typically used to demonstrate the model but it could be two infantry units fighting each other or two stock firms trying to acquire the same limited commodity Let x(t) represent the prey (hares, krill, xth Inf, etc.) and y(t) represent the predator (lynx, whales, nth Inf, etc.) Prey population growth rate without predators is assumed proportional to population size But….

19. Predator Prey Model But there are predators! The predators [eat, destroy, acquire…] the prey in proportion to the number of prey and predators. The predator population without sufficient prey dies out at a rate of But there are predators that are [eaten, destroyed, acquired …] that sustain the predators in proportion to both populations sizes:

20. Predator Prey Model The model is subject to x(t) = prey population size y(t) = predator population size a = growth rate of prey b = rate of prey predation m = death rate of predators n = rate of predator sustenance Model Solution State Variables: x, y Note the controls in this nonlinear, coupled, model

21. Predator Prey Model Lynx – Hare ( data from Leigh, 1968) Continuous Model

22. Discrete Predator Prey Model To discretize, choose time step, h replace The difference equations are If h = 1 year, the model is Evaluation: with x(0)=80, y(0)=30 a = 0.7, b = 0.03, m=0.99, n = 0.03 Initially h = 1 year

23. Discrete Predator Prey Model Unstable Model! Problem: the time step is too big! with h = 0.1 years Choice of time step is critical to discrete modeling The smaller the time step, the more accurate the model but more computations

24. Summary We discussed modeling Models are simplifications Dynamics: things move! differential equations difference equations States a set of variables whose values when known completely define the dynamics (motion) Phase Plots a plot of a state variable vs. another state variable Example: Predator Prey Model Next: State Space Models