Outline. Classical Control. Lecture 2


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1 Outline
2 Outline Outline Review of Material from Lecture 2 New Stuff 
3 Outline Review of Lecture System Performance Effect of Poles Review of Material from Lecture System Performance Effect of Poles 2 New Stuff 
4 Review of Lecture System Performance Effect of Poles Analyzing System Performance System types Continuous system Discrete system Analysis domain Timedomain specifications Frequencydomain specifications Different periods Dynamic (transient) responses Steadystate responses
5 Review of Lecture System Performance Effect of Poles Continuous Time  Dynamic Response Impulse Response Step Response Amplitude 0.6 Amplitude Time (sec) num=[]; den=[ 2 ]; num2=[ 2]; den2=[ 2 3]; impulse(tf(num,den), b,tf(num2,den2), r ) step(tf(num,den), b,tf(num2,den2), r ) Time (sec)
6 Review of Lecture System Performance Effect of Poles Continuous Time  Dynamic Response Step Response Amplitude M p % t p t s Overshoot (M p ) Rise time (t r ) Settling time (t s ) Peak time (t p ) 0. t r Time (sec)
7 Review of Lecture Natural Frequency and Damping System Performance Effect of Poles
8 Review of Lecture Poles in the Continuous splane System Performance Effect of Poles
9 Review of Lecture Poles in the Continuous splane System Performance Effect of Poles
10 Outline Review of Lecture Review of Material from Lecture System Performance Effect of Poles 2 New Stuff 
11 Review of Lecture Short description of different reference inputs Position reference Servomotors Position tracking (robots) Velocity reference ACmotors Vehicles Acceleration reference Space shuttle (during launch) Force exerting systems
12 Review of Lecture Conversion from Time Domain to Laplace Domain Position Time Domain Linear, x(t) Rotational, θ(t) Laplace Linear, X(s) Rotational, Θ(s)
13 Review of Lecture Conversion from Time Domain to Laplace Domain Velocity Time Domain Linear, v(t) = ẋ(t) Rotational, ω(t) = θ(t) Laplace Linear, V(s) = sx(s) Rotational, Ω(s) = sθ(s)
14 Review of Lecture Conversion from Time Domain to Laplace Domain Acceleration Time Domain Linear, a(t) = v(t) = ẍ(t) Rotational, α(t) = ω(t) = θ(t) Laplace Linear, A(s) = sv(s) = s 2 X(s) Rotational, A(s) = sω(s) = s 2 Θ(s)
15 Review of Lecture Why look at steadystate conditions? Different reference inputs Position Velocity Acceleration Steadystate corresponds to the state a system settles in under constant conditions Constant reference can be interpreted differently Constant position (no velocity or acceleration) Constant velocity (ramp/integration in position, no acceleration) Constant acceleration (ramp/integration in velocity, parabola/integration 2 in position)
16 Review of Lecture Diagram of Feedback System General System + D(s) H(s) G(s) System + D(s) G(s)
17 Review of Lecture Continuous Time  SteadyState  Laplace transform Final value theorem Steadystate errors F(s) = 0 f (t)e st dt lim f (t) = lim sf(s) t s 0 s n e ss = lim s 0 s n + K n s k
18 Review of Lecture Continuous Time  SteadyState  Type 0  K p = lim s 0 D(s)G(s) Step input Ramp input Parabolic input e ss +K p K v K a Static error K p =constant K v = 0 K a = 0 Error +K p
19 Review of Lecture Continuous Time  SteadyState  Type  K v = lim s 0 sd(s)g(s) Step input Ramp input Parabolic input e ss +K p K v K a Static error K p = K v =constant K a = 0 Error 0 K v
20 Review of Lecture Continuous Time  SteadyState  Type 2  K a = lim s 0 s 2 D(s)G(s) Step input Ramp input Parabolic input e ss +K p K v K a Static error K p = K v = K a =constant Error 0 0 K a
21 Example 4. Review of Lecture System + k p A τs+ Analysis of SteadyState Error System type is?
22 Example 4. Review of Lecture System + k p A τs+ Analysis of SteadyState Error ( ) System type is 0 L(s) = kpa τs+ SteadyState Error for Step:
23 Example 4. Review of Lecture System + k p A τs+ Analysis of SteadyState Error ( ) System type is 0 L(s) = kpa τs+ SteadyState Error for Step: e ss = +lim s 0 kpa τs+ = +k pa
24 Example 4.2 Review of Lecture System + k p + k I s A τs+ Analysis of SteadyState Error System type is?
25 Example 4.2 Review of Lecture System + k p + k I s A τs+ Analysis of SteadyState Error ( System type is L(s) = (kp+ k I SteadyState Error for Ramp: s )A τs+ = A(kps+k I) s(τs+) )
26 Example 4.2 Review of Lecture System + k p + k I s A τs+ Analysis of SteadyState Error ( System type is L(s) = (kp+ k I s )A τs+ = A(kps+k I) s(τs+) SteadyState Error for Ramp: e ss = lim s 0 s A(kps+k I ) s(τs+) ) = Ak I
27 Review of Lecture Continuous Time  SteadyState Laplace transform Final value theorem Steadystate errors F(s) = 0 f (t)e st dt lim f (t) = lim sf(s) t s 0 e ss = lim s 0 T (s) s k
28 Example 4.3 Review of Lecture System + k p s(τs+) + k t s
29 Review of Lecture Example Continued Analysis of SteadyState Error T (s) e ss = lim s 0 ( s k k p = lim s 0 s k s(τs + ) + ( + k t s)k p s(τs + ) + ( + k t s )k p = lim s 0 s k s(τs + ) + ( + k t s)k p s 2 τ + s( + k t k p ) = lim s 0 s k s 2 τ + s( + k t k p ) + k p = 0 k = 0 = + k tk p k p k = )
30 Review of Lecture Book: Feedback Control Problem 4.6 (a+b) Problem 4.8 (a+b) Problem 4.9
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