Systems Analysis and Control


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1 Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 21: Stability Margins and Closing the Loop
2 Overview In this Lecture, you will learn: Closing the Loop Effect on Bode Plot Effect on Stability Stability Effects Gain Margin Phase Margin Bandwidth Estimating ClosedLoop Performance using OpenLoop Data Damping Ratio Settling Time Rise Time M. Peet Lecture 21: Control Systems 2 / 31
3 Review Recall: Frequency Response Input: u(t) = M sin(ωt + φ) Output: Magnitude and Phase Shift Linear Simulation Results y(t) = G(ıω) M sin(ωt + φ + G(ıω)) Amplitude Time (sec) Frequency Response to sin ωt is given by G(ıω) M. Peet Lecture 21: Control Systems 3 / 31
4 Review Recall: Bode Plot Definition 1. The Bode Plot is a pair of loglog and semilog plots: 1. Magnitude Plot: 20 log 10 G(ıω) vs. log 10 ω 2. Phase Plot: G(ıω) vs. log 10 ω BiteSize Chucks: G(ıω) = i G i (ıω) M. Peet Lecture 21: Control Systems 4 / 31
5 Complex Poles and Zeros We left off with Complex Poles: 1 G(s) = ( ( ) ) 2 s ω n + 2ζ s ω n + 1 M. Peet Lecture 21: Control Systems 5 / 31
6 Closing The Loop Now we examine the effect of closing the loop on the Frequency Response. Use simple Unity Feedback (K = 1). ClosedLoop Transfer Function: G cl (ıω) = G(ıω) 1 + G(ıω) We are most concerned with magnitude: G cl (ıω) = G(ıω) 1 + G(ıω)  u(s) + k G(s) Figure : Unity Feedback y(s) M. Peet Lecture 21: Control Systems 6 / 31
7 Closing The Loop On the Bode Plot 20 log G cl (ıω) = 20 log G(ıω) 20 log 1 + G(ıω) Which is the combination of The original bode plot The new factor log 1 + G(ıω) Bode Diagram We are most concerned with the effect of the new term 20 log 1 + G(ıω) Specifically, as 1 + G(ıω) 0 lim 20 log 1 + G(ıω) = 1+G(ıω) 0 An unstable mode! Magnitude (db) Phase (deg) Frequency (rad/sec) Figure : Open Loop: Blue, CL: Green M. Peet Lecture 21: Control Systems 7 / 31
8 Closing The Loop Stability Margin Instability occurs when 1 + G(ıω) = 0 For this to happen, we need: G(ıω) = 1 G(ıω) = 180 Stability Margins measure how far we are from the point ( G = 1, G = 180 ). Definition 2. The Gain Crossover Frequency, ω gc is the frequency at which G(ıω c ) = 1. This is the danger point: If G(ıω c ) = 180, we are unstable M. Peet Lecture 21: Control Systems 8 / 31
9 Closing The Loop Phase Margin Definition 3. The Phase Margin, Φ M is the phase relative to 180 when G = 1. Φ M = 180 G(iω gc ) ω gc is also known as the phasemargin frequency, ω ΦM M. Peet Lecture 21: Control Systems 9 / 31
10 Closing The Loop Gain Margin Definition 4. The Phase Crossover Frequency, ω pc is the frequency (frequencies) at which G(ıω pc ) = 180. Definition 5. The Gain Margin, G M is the gain relative to 0dB when G = 180. G M = 20 log G(ıω pc ) G M is the gain (in db) which will destabilize the system in closed loop. ω pc is also known as the gainmargin frequency, ω GM M. Peet Lecture 21: Control Systems 10 / 31
11 Closing The Loop Stability Margins Gain and Phase Margin tell how stable the system would be in Closed Loop. These quantities can be read from the OpenLoop Data. M. Peet Lecture 21: Control Systems 11 / 31
12 Closing The Loop Stability Margins: Suspension System Bode Diagram Gm = Inf db (at Inf rad/sec), Pm = 42.1 deg (at rad/sec) Magnitude (db) Phase (deg) Frequency (rad/sec) M. Peet Lecture 21: Control Systems 12 / 31
13 Closing The Loop Stability Margins Φ M = 35 G M = 10dB M. Peet Lecture 21: Control Systems 13 / 31
14 Closing The Loop Stability Margins Note that sometimes the margins are undefined When there is no crossover at 0dB When there is no crossover at 180 M. Peet Lecture 21: Control Systems 14 / 31
15 Transient Response Closing the Loop Question: What happens when we Close the Loop? We want Performance Specs! We only have openloop data. Φ M and G M can help us. Unity Feedback: Step Response u(s) + k G(s) y(s) We want: Damping Ratio Settling Time 0 Peak Time Amplitude Time (sec) M. Peet Lecture 21: Control Systems 15 / 31
16 Transient Response Quadratic Approximation Assume the closed loop system is the quadratic Then the openloop system must be ω 2 n G cl = s 2 + 2ζω n s + ωn 2 G ol = Assume that our openloop system is G ol Use Φ M to solve for ζ ω 2 n s 2 + 2ζω n s Set 20 log G cl = 3dB to solve for ω n M. Peet Lecture 21: Control Systems 16 / 31
17 Transient Response Damping Ratio The Quadratic Approximation gives the ClosedLoop Damping Ratio as Φ M = tan 1 2ζ 2ζ 2 + 4ζ Φ M is from the OpenLoop Data! A Handy approximation is ζ = Φ M 100 Only valid out to ζ =.7. Given Φ M, we find closedloop ζ. M. Peet Lecture 21: Control Systems 17 / 31
18 Transient Response Bandwidth and Natural Frequency We find closedloop ζ from Phase margin. We can find closedloop Natural Frequency ω n from the closedloop Bandwidth. Definition 6. The Bandwidth, ω BW is the frequency at gain 20 log G(ıω BW ) = 20 log G(0) 3dB. Closely related to crossover frequency. The Bandwidth measures the range of frequencies in the output. For 2nd order, Bandwidth is related to natural frequency by ω BW = ω n (1 2ζ 2 ) + 4ζ 4 4ζ M. Peet Lecture 21: Control Systems 18 / 31
19 Transient Response Finding ClosedLoop Bandwidth from OpenLoop Data Question: How to find closedloop bandwidth? Finding the closedloop bandwidth from openloop data is tricky. Have to find the frequency when the Bode plot intersects this curve. Heuristic: Check the frequency at 6dB and see if phase is = 180. M. Peet Lecture 21: Control Systems 19 / 31
20 Finding ClosedLoop Bandwidth from OpenLoop Data Example Bode Diagram Magnitude (db) Phase (deg) Frequency (rad/sec) At phase 135, 5dB, we get closed loop ω BW = 1. M. Peet Lecture 21: Control Systems 20 / 31
21 Transient Response Bandwidth and Settling Time We can use the expression T s = 4 ζω n to get ω BW = 4 (1 2ζ T s ζ 2 ) + 4ζ 4 4ζ Given closedloop ζ and ω BW, we can find T s. M. Peet Lecture 21: Control Systems 21 / 31
22 Transient Response Bandwidth and Peak Time We can use the expression T p = π ω n 1 ζ 2 to get π ω BW = (1 2ζ 2 ) + 4ζ 4 4ζ T p 1 ζ 2 Given closedloop ζ and ω BW, we can find T p. M. Peet Lecture 21: Control Systems 22 / 31
23 Transient Response Bandwidth and Rise Time Using an expression for T r, we get a relationship between ω BW T r and ζ. Given closedloop ζ and ω BW, we can find T r. M. Peet Lecture 21: Control Systems 23 / 31
24 Transient Response Example Question: Using Frequency Response Data, find T r, T s, T p after unity feedback. First Step: Find the phase Margin. Frequency at 0dB is ω gc = 2 G(2) = 145 Φ M = = 35 M. Peet Lecture 21: Control Systems 24 / 31
25 Transient Response Example Step 2: ClosedLoop Damping Ratio ζ = Φ M 100 =.35 Step 3: ClosedLoop Bandwidth Intersect at = ( G = 6dB, G = 170 ) Frequency at intersection is ω BW = 3.7 M. Peet Lecture 21: Control Systems 25 / 31
26 Transient Response Example Step 4: Settling Time and Peak Time ω BW = 3.7, ζ =.35 ω BW T s = 20 implies T s = 5.4s ω BW T p = 4.9 implies T p = 1.32 M. Peet Lecture 21: Control Systems 26 / 31
27 Transient Response Example Step 5: Rise Time ω BW = 3.7, ζ =.35 ω BW T r = 1.98 implies T r =.535 M. Peet Lecture 21: Control Systems 27 / 31
28 Transient Response Example Step 6: Experimental Validation. Use the plant G(s) = 50 s(s + 3)(s + 6) We find T p = 1.6s predicted 1.32 T r =.7s predicted.535 T s = 4s Predicted Figure : Step Response M. Peet Lecture 21: Control Systems 28 / 31
29 SteadyState Error Finally, we want steadystate error. SteadyState Step response is lim G(s) = lim G(ıω) s 0 ω 0 Steadystate response is the LowFrequency Gain, G(0). Close The Loop to get steadystate error e ss = G(0) M. Peet Lecture 21: Control Systems 29 / 31
30 SteadyState Error Example A Lag Compensator lim 20 log G(ıω) = 20dB ω 0 So lim ω 0 G(ıω) = 10. SteadyState Error: e ss = G(0) = 1 11 =.091 Phase (deg) Magnitude (db) Bode Diagram Frequency (rad/sec) M. Peet Lecture 21: Control Systems 30 / 31
31 Summary What have we learned today? Closing the Loop Effect on Bode Plot Effect on Stability Stability Effects Gain Margin Phase Margin Bandwidth Estimating ClosedLoop Performance using OpenLoop Data Damping Ratio Settling Time Rise Time Next Lecture: Compensation in the Frequency Domain M. Peet Lecture 21: Control Systems 31 / 31
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