Time Response*, ME451Zeros and poles of a transfer functionTheoremsDC gain or static gain of a stable systemDC Gain of a stable transfer functionPure integratorFirst order systemSlide 8Matlab SimulationFirst order system responseSlide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18First order system – Time specifications.Slide 20First order system – Simple behavior.Second order system (mass-spring-damper system)Polar vs. Cartesian representations.Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Underdamped second order systemSlide 32Impulse response of the second order systemSlide 34Unit step response of undamped systemsUnit step response of undamped systemSlide 37Second order system response.Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Overdamped system responseOverdamped and critically damped system response.Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Second order impulse response – Underdamped and UndampedSlide 59Slide 60Slide 61Slide 62Slide 63Slide 64Slide 65Slide 66Slide 67Slide 68Slide 69Slide 70Slide 71Slide 72Slide 73Slide 74Second order step response – Underdamped and UndampedSlide 76Slide 77Second order step response – Time specifications.Slide 79Slide 80Slide 81Slide 82Slide 83Typical specifications for second order systems.Time Response*, ME451Instructor: Jongeun Choi * This presentation is created by Jongeun Choi and Gabrial GomesZeros and poles of a transfer function•Let G(s)=N(s)/D(s), then–Zeros of G(s) are the roots of N(s)=0–Poles of G(s) are the roots of D(s)=0Re(s)Im(s)Theorems•Initial Value Theorem•Final Value Theorem–If all poles of sX(s) are in the left half plane (LHP), thenDC gain or static gain of a stable system0 0.5 1 1.5 2 2.5 300.20.40.60.811.21.4DC Gain of a stable transfer function •DC gain (static gain) : the ratio of the steady state output of a system to its constant input, i.e., steady state of the unit step response•Use final value theorem to compute the steady state of the unit step responsePure integrator •ODE :•Impulse response :•Step response :•If the initial condition is not zero, then :Physical meaning of the impulse responseFirst order system•ODE :•Impulse response :•Step response :•DC gain: (Use the final value theorem)First order system•If the initial condition was not zero, thenPhysical meaning of the impulse responseMatlab Simulation•G=tf([0 5],[1 2]); •impulse(G)•step(G)•Time constant0 0.5 1 1.5 2 2.5 300.511.522.533.544.55Impulse ResponseTime (sec)Amplitude0 0.5 1 1.5 2 2.5 300.511.522.5Step ResponseTime (sec)AmplitudeFirst order system responseSystem transfer function :First order system responseSystem transfer function : Impulse response :First order system responseSystem transfer function : Impulse response :First order system responseSystem transfer function : Impulse response : Step response : 0 100 200 300 400 500 6000102030405060708090100Step ResponseTime (sec)AmplitudeFirst order system responseRe(s)Im(s)First order system responseUnstableRe(s)Im(s)First order system responseUnstableRe(s)Im(s)-1First order system responseUnstableRe(s)Im(s)-2First order system responseUnstableRe(s)Im(s)faster response slower responseconstantFirst order system – Time specifications.First order system – Time specifications.Time specs:Steady state value : Time constant : Rise time : Settling time : Time to go from toFirst order system – Simple behavior.No overshootNo oscillationsSecond order system (mass-spring-damper system)•ODE :•Transfer function :Polar vs. Cartesian representations.Cartesian representation : … Imaginary part (frequency) … Real part (rate of decay)Polar vs. Cartesian representations.Cartesian representation : … Imaginary part (frequency) … Real part (rate of decay) Polar representation : … damping ratio … natural frequencyPolar vs. Cartesian representations.Cartesian representation : … Imaginary part (frequency) … Real part (rate of decay) Polar representation : … damping ratio … natural frequencyPolar vs. Cartesian representations.Cartesian representation : … Imaginary part (frequency) … Real part (rate of decay) Polar representation : … damping ratio … natural frequency Unless overdampedPolar vs. Cartesian representations.… Overdamped … Critically damped… Underdamped … Undamped Significance of the damping ratio : System transfer function :Polar vs. Cartesian representations.System transfer function : Significance of the damping ratio : … Overdamped … Critically damped… Underdamped … UndampedPolar vs. Cartesian representations.System transfer function : Significance of the damping ratio : … Overdamped … Critically damped… Underdamped … UndampedPolar vs. Cartesian representations.System transfer function : Significance of the damping ratio : … Overdamped … Critically damped… Underdamped … Undamped All 4 casesUnless overdampedUnderdamped second order system•Underdamped•Two complex poles:Underdamped second order systemImpulse response of the second order systemMatlab Simulation•zeta = 0.3; wn=1;•G=tf([wn],[1 2*zeta*wn wn^2]);•impulse(G)0 2 4 6 8 10 12 14 16 18 20-0.3-0.2-0.100.10.20.30.40.50.60.7Impulse ResponseTime (sec)AmplitudeUnit step response of undamped systems•Unit step response :•DC gain :Unit step response of undamped systemMatlab Simulation•zeta = 0.3; wn=1; G=tf([wn],[1 2*zeta*wn wn^2]);•step(G)0 2 4 6 8 10 12 14 16 18 2000.20.40.60.811.21.4Step ResponseTime (sec)AmplitudeSecond order system response.Stable 2nd order system:2 distinct real polesA pair of repeated real polesA pair of complex polesUnstableRe(s)Im(s)Second order system response.2 distinct real polesA pair of repeated real polesA pair of complex polesUnstableRe(s)Im(s)Stable 2nd order system:Second order system response.2 distinct real polesA pair of repeated real polesA pair of complex polesUnstableRe(s)Im(s)Stable 2nd order system:Second order system response.2 distinct real polesA pair of repeated real polesA pair of complex polesUnstableRe(s)Im(s)Stable 2nd order system:negative real partzero real partSecond order system response.2 distinct real polesA pair of repeated real polesA pair of complex polesUnstableRe(s)Im(s)Stable 2nd order system:negative real partzero real partSecond order system response.2 distinct real polesA pair of repeated real polesA pair of complex polesUnstableRe(s)Im(s)Stable 2nd order
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