EE3CL4: Compensator Design Introduction to Linear Control Systems - PowerPoint PPT Presentation

EE 3CL4, §9 1 / 56 Tim Davidson Frequency Domain Approach to EE3CL4: Compensator Design Introduction to Linear Control Systems Lead Compensators Section 9: Design of Lead and Lag Compensators using Lag Frequency Domain Techniques Compensators Lead-Lag Compensators Tim Davidson McMaster University Winter 2020

EE 3CL4, §9 2 / 56 Outline Tim Davidson Frequency Domain Approach to Compensator 1 Frequency Domain Approach to Compensator Design Design Lead Compensators Lead 2 Lag Compensators Compensators Lead-Lag Compensators 3 Lag Compensators Lead-Lag 4 Compensators

EE 3CL4, §9 4 / 56 Frequency domain design Tim Davidson Frequency Domain Approach to Compensator Design Lead Compensators • Analyze closed loop using open loop transfer function Lag L ( s ) = G c ( s ) G ( s ) H ( s ) . Compensators Lead-Lag • We would like closed loop to be stable: Compensators • Use Nyquist’s stability criterion (on L ( s ) ) • We might like to make sure that the closed loop remains stable even if there is an increase in the gain • Require a particular gain margin (of L ( s ) ) • We might like to make sure that the closed loop remains stable even if there is additional phase lag • Require a particular phase margin (of L ( s ) ) • We might like to make sure that the closed loop remains stable even if there is a combination of increased gain and additional phase lag

EE 3CL4, §9 5 / 56 Robust stability Tim Davidson Frequency Domain Approach to Compensator Design • Let ˘ G ( s ) denote the true plant and let G ( s ) denote our model Lead Compensators • ∆ G ( s ) = ˘ G ( s ) − G ( s ) denotes the uncertainty in our model Lag Compensators • If ˘ G ( s ) has the same number of RHP poles as G ( s ) , we need to Lead-Lag ensure that the Nyquist plot of Compensators ˘ L ( s ) = G c ( s )˘ G ( s ) = L ( s ) + G c ( s )∆ G ( s ) has the same number of encirclements of − 1 as the plot of L ( s ) . • This will give us a sufficient condition for robust stability

EE 3CL4, §9 6 / 56 Robust stability II Tim Davidson Frequency Domain Approach to Compensator Design Lead Compensators Lag Compensators Lead-Lag Compensators

EE 3CL4, §9 7 / 56 Robust stability III Tim Davidson Frequency Domain Approach to Compensator Design Lead Compensators • Our sufficient condition is | 1 + L ( j ω ) | > | G c ( j ω )∆ G ( j ω ) | . Lag Compensators � � � ∆ G ( j ω ) 1 • That is equivalent to | L ( j ω ) + 1 | > � � Lead-Lag G ( j ω ) � Compensators • That is, we need | L ( j ω ) | to be small at the frequencies where the relative error in our model is large; typically at higher frequencies

EE 3CL4, §9 8 / 56 Frequency domain design Tim Davidson Frequency Domain Approach to Compensator Design • We might like to control the damping ratio of the dominant Lead Compensators pole pair Lag Compensators • Use the fact that φ pm = f ( ζ ) ; Lead-Lag Compensators • We might like to control the steady-state error constants • For step, ramp and parabolic inputs, these constants are related to the behaviour of L ( s ) around zero; i.e., behaviour near DC. Recall K posn = L ( 0 ) and K v = lim s → 0 sL ( s ) . • We might like to influence the settling time • Roughly speaking, the settling time decreases with increasing closed-loop bandwidth. How is this related to bandwidth of L ( s ) ?

EE 3CL4, §9 9 / 56 Bandwidth Tim Davidson Frequency Domain Approach to • Let ω c be the (open-loop) cross-over frequency; Compensator Design i.e., | L ( j ω c ) | = 1 Lead Compensators • Let T ( s ) = Y ( s ) L ( s ) R ( s ) = 1 + L ( s ) . Lag Compensators • Consider a low-pass open loop transfer function Lead-Lag Compensators • When ω ≪ ω c , | L ( j ω ) | ≫ 1, = ⇒ T ( j ω ) ≈ 1 • When ω ≫ ω c , | L ( j ω ) | ≪ 1, = ⇒ T ( j ω ) ≈ L ( j ω ) • Can we quantify things a bit more, and perhaps gain some insight, for a standard second-order system

EE 3CL4, §9 10 / 56 Bandwidth, open loop Tim Davidson ω 2 • For a standard second-order system, L ( s ) = n s ( s + 2 ζω n ) Frequency Domain ω n / ( 2 ζ ) • To sketch open loop Bode diagram, L ( j ω ) = Approach to � � Compensator j ω 1 + j ω/ ( 2 ζω n ) Design • Low freq’s: slope of − 20 dB/decade; Corner freq. at 2 ζω n ; Lead High freq’s: slope of − 40dB/decade Compensators 1 + 4 ζ 4 − 2 ζ 2 � 1 / 2 �� • Crossover frequency: ω c = ω n Lag Compensators Lead-Lag Compensators Circles are the corner frequencies; Observe crossover frequencies

EE 3CL4, §9 11 / 56 Bandwith, closed loop Tim Davidson • To sketch closed-loop Bode diagram, T ( j ω ) = 1 1 + j 2 ζω/ω n − ( ω/ω n ) 2 Frequency Domain • Low freq’s: slope of zero; Double corner frequency at ω n ; Approach to High freq’s: slope of − 40dB/decade Compensator √ Design 1 − 2 ζ 2 (Lab 2) 1 � • For ζ < 1 / 2 ζ √ 2, peak of 1 − ζ 2 at ω r = ω n Lead Compensators 2 − 4 ζ 2 + 4 ζ 4 + 1 − 2 ζ 2 � 1 / 2 , • 3dB bandwidth: ω B = ω n �� Lag ≈ ω n ( − 1 . 19 ζ + 1 . 85 ) for 0 . 3 ≤ ζ ≤ 0 . 8. Compensators Lead-Lag Compensators Asterisks are ω B

EE 3CL4, §9 12 / 56 Bandwidth, open and closed Tim Davidson loops Frequency Domain Approach to Compensator Design Lead Compensators Lag Compensators Lead-Lag Compensators 1 + 4 ζ 4 − 2 ζ 2 � 1 / 2 �� • OL crossover freq.: ω c = ω n 2 − 4 ζ 2 + 4 ζ 4 + 1 − 2 ζ 2 � 1 / 2 �� • CL 3dB BW: ω B = ω n • 2% settling time: T s , 2 ≈ 4 ζω n • Rise time (0 % → 100 % ) of step response: π/ 2 +sin − 1 ( ζ ) √ ω n 1 − ζ 2 • Close relationship with ω c and ω B , esp. through ω n . Care needed in dealing with damping effects.

EE 3CL4, §9 13 / 56 Loopshaping, again Tim Davidson Frequency Domain Approach to Compensator Design 1 G ( s ) L ( s ) Lead E ( s ) = 1 + L ( s ) R ( s ) − 1 + L ( s ) T d ( s ) + 1 + L ( s ) N ( s ) Compensators Lag where, with H ( s ) = 1, L ( s ) = G c ( s ) G ( s ) Compensators What design insights are available in the frequency domain? Lead-Lag Compensators • Good tracking: = ⇒ L ( s ) large where R ( s ) large | L ( j ω ) | large in the important frequency bands of r ( t ) • Good dist. rejection: = ⇒ L ( s ) large where T d ( s ) large | L ( j ω ) | large in the important frequency bands of t d ( t ) • Good noise suppr.: = ⇒ L ( s ) small where N ( s ) large | L ( j ω ) | small in the important frequency bands of n ( t ) ⇒ L ( s ) small where ∆ G ( s ) • Robust stability: = G ( s ) large | L ( j ω ) | small in freq. bands where relative error in model large • Phase margin: ∠ L ( j ω ) away from − 180 ◦ when | L ( j ω ) | close to 1 Typically, L ( j ω ) is a low-pass function,

EE 3CL4, §9 14 / 56 How can we visualize these Tim Davidson things? Frequency Domain Approach to Compensator • Interesting properties of L ( s ) : encirclements, gain Design margin, phase margin, general stability margin, gain at Lead Compensators low frequencies, bandwidth ( ω c ), gain at high Lag frequencies, phase around the cross-over frequency Compensators Lead-Lag Compensators • All this information is available from the Nyquist diagram • Not always easily accessible • Once we have a general idea of the shape of the Nyquist diagram, is some of this information available in a more convenient form? at least for relatively simple systems?

EE 3CL4, §9 15 / 56 Bode diagram Tim Davidson Frequency Domain Approach to Compensator Design Lead Compensators Lag Compensators Lead-Lag Compensators Seems to capture most issues, but How fast can we transition from high open-loop gain to low open-loop gain? This is magnitude. What can we say about phase?

EE 3CL4, §9 16 / 56 Phase from magnitude? Tim Davidson • For systems with more poles than zeros and all the poles and zeros Frequency in the left half plane, we can write a formal relationship between Domain Approach to gain and phase. That relationship is a little complicated, but we can Compensator gain insight through a simplification. Design Lead • Assume that ω c is some distance from any of the corner Compensators frequencies of the open-loop transfer function. That means that Lag around ω c , the Bode magnitude diagram is nearly a straight line Compensators Lead-Lag • Let the slope of that line be − 20 n dB/decade Compensators • Then for these frequencies L ( j ω ) ≈ K ( j ω ) n • That means that for these frequencies ∠ L ( j ω ) ≈ − n 90 ◦ • That suggests that at the crossover frequency the Bode magnitude plot should have a slope around − 20dB/decade in order to have a good phase margin • For more complicated systems we need more sophisticated results, but the insight of shallow slope of the magnitude diagram around the crossover frequency applies for large classes of practical systems