Leo Lam © Signals and Systems EE235

Leo Lam © Breeding What do you get when you cross an elephant and a zebra? Elephant zebra sin theta

Leo Lam © Today’s menu Friday: LCCDE Zero-Input Response Today: LCCDE Zero-State Response (forced)

Zero input response (example) Leo Lam © steps to solving Differential Equations: Step 1. Find the zero-input response = natural response y n (t) Step 2. Find the Particular Solution y p (t) Step 3. Combine the two Step 4. Determine the unknown constants using initial conditions

From Friday (example) Leo Lam © Solve Guess solution: Substitute: We found: Solution: Characteristic roots = natural frequencies/ eigenvalues Unknown constants: Need initial conditions

Zero-state output of LTI system Leo Lam © Response to our input x(t) LTI system: characterize the zero-state with h(t) Initial conditions are zero (characterizing zero-state output) Zero-state output: Total response(t)=Zero-input response (t)+Zero-state output(t) T  (t) h(t)

Zero-state output of LTI system Leo Lam © Zero-input response: Zero-state output: Total response: Total response(t)=Zero-input response (t)+Zero-state output(t) “Zero-state”:  (t) is an input only at t=0 Also called: Particular Solution (PS) or Forced Solution

Zero-state output of LTI system Leo Lam © Finding zero-state output (Particular Solution) Solve: Or: Guess and check Guess based on x(t)

Trial solutions for Particular Solutions Leo Lam © Guess based on x(t) Input signal for time t> 0 x(t) Guess for the particular function y P

Particular Solution (example) Leo Lam © Find the PS (All initial conditions = 0): Looking at the table: Guess: Its derivatives:

Particular Solution (example) Leo Lam © Substitute with its derivatives: Compare:

Particular Solution (example) Leo Lam © From We get: And so:

Particular Solution (example) Leo Lam © Note this PS does not satisfy the initial conditions! Not 0!

Natural Response (doing it backwards) Leo Lam © Guess: Characteristic equation: Therefore:

Complete solution (example) Leo Lam © We have Complete Sol n : Derivative:

Complete solution (example) Leo Lam © Last step: Find C 1 and C 2 Complete Sol n : Derivative: Subtituting: Two equations Two unknowns

Complete solution (example) Leo Lam © Last step: Find C 1 and C 2 Solving: Subtitute back: Then add u(t): y n ( t ) y p ( t ) y ( t )

Another example Leo Lam © Solve: Homogeneous equation for natural response: Characteristic Equation: Therefore: Input x(t)

Another example Leo Lam © Solve: Particular Solution for Table lookup: Subtituting: Solving: b=-1, =-2 No change in frequency! Input signal for time t> 0 x(t) Guess for the particular function y P

Another example Leo Lam © Solve: Total response: Solve for C with given initial condition y(0)=3 Tada!

Stability for LCCDE Leo Lam © Stable with all Re( j <0 Given: A negative means decaying exponentials Characteristic modes

Stability for LCCDE Leo Lam © Graphically Stable with all Re( j )<0 “Marginally Stable” if Re( j )=0 IAOW: BIBO Stable iff Re(eigenvalues)<0 Im Re Roots over here are stable

Leo Lam © Summary Differential equation as LTI system Stability of LCCDE