Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo Solving ODE

Outline of Today’s Lecture Review SIMULINK Two common mathematical problems is controls Numerical Methods Newton Raphson Homogeneous Solution Euler Method Runge Kutta

Simulink

SIMULINK

Two Mathematical Problems Frequently Encountered in Controls Find the roots of an equation Methods Trial and Error (bracketing methods add a bit of science to this) Graphics Closed form solutions (e.g.: quadratic formula) Newton Raphson Find the solution at a given time for given conditions Various differential and difference equations analytic solutions (sometimes reformulated as find the roots problem) Numerical Methods Newton Cotes Methods (trapezoidal rule, Simpson’s rule. etc. for integration) Euler’s Method Runga Kutta/Butcher Methods Many other techniques (Adams-Bashforth, Adams-Milne, Hermite–Obreschkoff, Fehlberg, Conjugate Gradient Methods, etc.)

Numerical Methods Solutions can be approximated using numerical methods Why Numerical Methods? Analytical methods may not exist to solve for the exact roots or the exact solution Use of computers Flexibility of making changes

Numerical Methods Numerical methods follow the procedure Step1: Initialize: Select some initial value Step2: Estimate using (guess, some analytical technique) a new value at increment “i” Step 3: Is the system converging? If not, use something else. We usually know a priori whether a method will converge or not form mathematics. Therefore, this step is often omitted. Step 4: Is the change from the previous value to current value smaller than our acceptable error? If not, make the current value the previous value and return to step 2. If so, stop and accept the new value as the solution.

Newton Raphson Method for finding roots Probably the most common numerical technique simple efficient flexible It can be shown from a truncated Taylor’s Series that Provided that the slope at the test points is consistent, we can iterate to a solution within our error tolerance t f(t) f(t i ) titi t i+1 Problems occur if the slope reverses sign such as in an oscillation or becomes very flat

Example Find the roots of To within 0.01 of the value of s There appears to be a crossing between -2 and 0, but where?

ODE for Linear Control Theory In Linear Control Theory, the equations that are encountered are almost always of the form with constant coefficients subject to initial conditions. Most problems are 2nd order. The electric motor example: Subject to {V a (0), i a (0)=0 and  (0) = 0}

Solution Methods To solve We can use Reduction in order Undetermined coefficients Variation of parameters Laplace Transforms Superposition of particular integrals Cauchy-Euler equation Numerical methods

Reduction of Order The object is to reduce the order of the equation by substitutions until it can be solved Example: A hanging cable with a weight of w per unit length under tension T: W H T  x y s

Homogenous Case When we form the characteristic equation and solve for its roots We have three possible outcomes for the roots that we must consider: 1) Real Distinct Roots : for p distinct real roots, the solution term is of the form 2) Repeated Real Roots: Assume roots r and r+1 are repeated, that is, Then the term in the solution corresponding to the repeated root is and if there were v repeated roots of the same value, the term would be 3) Complex Conjugate Roots: Assume that roots, The corresponding term of the solution is Therefore the complete solution accounting for all possible roots is

Homogenous Example A weight of 15 Kgf is supported (x = 0, at rest,) by a spring with a spring rate of 50 N/m. At t = 0, the weight is extended by 20 centimeters and released. Model and solve.

Euler Method Our goal is to solve equations of the form The theory for the Euler method is the same as that of the Newton Raphson Method: Rather than now solve for an axis crossing, we predict where the next value of the curve will be and then Make successive estimates of y i+1 yiyi xixi Prediction Error } y i+1 x i+1 step h }

Example

Try a smaller time step, say, h = 0.01seconds

Runge Kutta/Butcher Method Has its origins in a 2 variable Taylor Series Expansion The function is called the increment function RK4 is a four factor expansion of the incrementing function For RK4: Butcher’s method uses 5 factors is more accurate than RK4 at a given time step

Example

The result of the RK4 at h = 0.1 is essentially the same as the analytic solution:

Summary Models for Control Systems are either differential equation or difference equations The problems we commonly see will be Finding the roots of the equation Finding the trajectory or path of the equation over time While we have a number of analytical techniques to find exact results, we cannot address all of the equations encountered Numerical methods for finding roots and for trajectories are most commonly used Newton Raphson for finding roots Methods such as the Runge Kutta 4 are used for trajectories Next Class: Analyzing stability