Lecture 25 Outline: LTI Systems: Causality, Stability, Feedback Announcements: Reading: “6: Laplace Transform” pp. 37-49.5, 53-63.5, 73; “7: “7: Feedback” pp. 1-4.5, 8-17. HW 8 posted, due today. Free 1-day extension HW 9 will be posted today, due next Friday (no late HWs) Final exam is June 7, 8:30am-11:30am. More on final exam, section next week, and extra OHs on Friday OceanOne Robot Tour will be after class Friday (11:30-12:20) Lunch provided afterwards. Can arrange separate tour for those w/conflicts Causality and Stability in LTI Systems LTI Systems described by differential equations Example: 2nd order Lowpass Systems Feedback in LTI Systems

Review of Last Lecture Laplace for Circuit Analysis Turns DEs into algebraic equations Example: Laplace Analysis of 1st Order LFP Inversion of Rational Laplace Transforms Main idea: Convert complex equation into a sum of terms where the inverse of each term is known Step 1: Extract strictly proper part of X(s): Invert D(s): Step 2: Partial Fraction Expansion on

Causality and Stability in LTI Systems Causal LTI systems: impulse response h(t)H(s) LTI system is causal if h(t)=0, t<0, so is h(t) right-sided For H(s) rational, a causal system has its ROC to the right of the right-most pole Step response is h(t)*u(t) H(s)/s Stable LTI System LTI system is bounded-input bounded-output (BIBO) stable if all bounded inputs result in bounded outputs A system is stable iff its impulse response is absolutely integrable; implies H(jw) exists, equivalently jwROC A causal system with H(s) rational is stable if & only if all poles of H(s) lie in the left-half of the s-plane Equivalently, all poles have Re(s)<0 ROC defined implicitly for causal stable LTI systems

LTI Systems Described by Differential Equations (DEs) Finite-order constant-coefficient linear DE system Poles are roots of A(s), zeros are roots of B(s) If system is causal: for x(t)=d(t), initial conditions are zero: y(0-)=y(1)(0-)=…=y(N-1)(0-) ROC of H(s) is right-half plane to the right of the right-most pole Can solve DEs with non-zero initial conditions using the unilateral Laplace transform: Not covered in this class as we focus on causal stable systems Extra credit reading: “Laplace” pp. 63.5-67, example pp. 71-72.5

Second Order Lowpass System + - x(t) y(t) natural frequency wn and damping coefficient z Poles in left-half of s-plane h(t) causal H(jw) exists We first factor H(s): Three regimes: Underdamped: 0<z<1, gs distinct, complex conjugates Critically Dampled: z=1, gs equal and real Overdamped: 1<z<, gs distinct and real

Frequency Response: Underdamped: 0<z<1, gs distinct, complex conjugates Critically Dampled: z=1, gs equal and real Overdamped: 1<z<, gs distinct and real

Feedback in LTI Systems Equivalent System Motivation for Feedback Can make an unstable system stable Can make transfer function closer to desired (ideal) one Can make system less sensitive to disturbances Can have negative effect: make a stable system unstable Transfer Function T(s) of Feedback System: Y(s)=G(s)E(s); E(s)=X(s)±Y(s)H(s)

Two Examples Stabilizing an unstable system: 1st order system a>0, pole in right-half of s-plane System is stable if K>a, single pole in left half of s-plane Feedback can make a stable system unstable Unstable if K1K2>1

Main Points A causal system with H(s) rational is stable if & only if all poles of H(s) lie in the left-half of the s-plane (all poles have Re(s)<0) ROC defined implicitly for causal stable LTI systems Systems described by differential equations easily characterized using Laplace analysis If system is causal: for x(t)=d(t), initial conditions are zero ROC of H(s) is right-half plane to the right of the right-most pole Second order systems characterized by 3 regimes: underdampled, critically damped, overdamped Feedback very useful in LTI system design Can make an unstable system stable Can make the system transfer function closer to ideal one Can mitigate the effects of disturbances Can also have undesired effects: make a stable system unstable