Solving Differential Equations Ex. Shock absorber with rigid massless tire Start with no input r(t)=0, assume y(0)=y 0, y(0)=0 – use Linearity and Differentiation – Group terms with and without Y(s)
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
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 – its called partial fraction expansion
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 a 1, a 2,..., a n are poles Then where
Partial Fraction Expansion Highest derivative order in differential equation determines number of poles hence number of exponentials Poles a i can be complex numbers If a i is complex, then there will be complex conjugate a j =a i * Evaluate exp(-a i t) with Eulers formula:
Example: Real Poles let Therefore, zeros = -5 poles = -3, -2
Example: Complex Poles let Use MATLAB roots([1 2 2]) Or quadratic formula Now what? Proceed as normal.
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 k i must also be complex conjugates. If not, you did it wrong!
Example: Complex Poles Now what??? EULERS FORMULA
Example: Complex Poles This worked because of complex conjugates We cannot have complex y(t)!
Complex Plane Real roots: zero at -5, poles at –3, -2 Complex roots: zero at -2, poles at –1 j
Complex Plane How do pole locations relate to differential equation solution? Once we have poles = j we know solution is of the form Use ICs to find C 1 and C 2
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
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