Solving Differential Equations

1. Solving Differential Equations BIOE 4200

2. Solving Differential Equations • Ex. Shock absorber with rigid massless tire • Start with no input r(t)=0, assume y(0)=y0, y’(0)=0 • use Linearity and Differentiation • Group terms with and without Y(s)

3. Definitions • The equation q(s) = 0 to find the roots of the denominator polynomial, is called the characteristic equation, determines the time response of y(t) • The roots of q(s)=0 are called the poles of the system • The roots of p(s)=0 are called the zeroes of the system • Poles and zeroes can be real, imaginary or complex

4. How to find y(t) from Y(s) • Can we get Y(s) into a form we recognize and find inverse? YES • What do we recognize? • Therefore, if • then • This is exactly what we do – it’s called partial fraction expansion

5. Partial Fraction Expansion • Find poles from q(s)=0 with factoring or with MATLAB roots() function • Nth order differential equation must have N poles • Express • where a1, a2, ..., an are poles • Then • where

6. Partial Fraction Expansion • Highest derivative order in differential equation determines number of poles hence number of exponentials • Poles ai can be complex numbers • If ai is complex, then there will be complex conjugate aj=ai* • Evaluate exp(-ait) with Euler’s formula:

7. Example: Real Poles let Therefore, zeros = -5 poles = -3, -2

8. Example: Real Poles CHECK THIS Overdamped YES

9. Example: Complex Poles let Use MATLAB roots([1 2 2]) Or quadratic formula Now what? Proceed as normal.

10. Example: Complex Poles • If you have a complex pole, you must also have a complex conjugate of the pole (if x = a + jb, x* = a - jb) • Scaling of complex poles ki must also be complex conjugates. If not, you did it wrong!

11. Example: Complex Poles Now what??? EULER’S FORMULA

12. Example: Complex Poles This worked because of complex conjugates We cannot have complex y(t)!

13. Complex Plane • Real roots: zero at -5, poles at –3, -2 • Complex roots: zero at -2, poles at –1  j

14. Complex Plane • How do pole locations relate to differential equation solution? • Once we have poles = s jw we know solution is of the form • Use ICs to find C1 and C2

15. Things we can learn from poles • Poles are real – y(t) is exponential • Poles are imaginary – y(t) is sinusoid • Poles are complex – y(t) is damped sinusoid • The frequency of the sinusoidal oscillations equals the imaginary part of poles • The rate of exponential decay is equal to the real part of poles

16. Things we can learn from poles • The speed of the system is determined by the slowest exponential • The pole with most positive real part or farthest to the right is called the dominant pole or root • If all poles are not to the left of the imaginary axis, then we have a big problem • We say a system is stable if all poles are to the left of imaginary axis, unstable otherwise