Control Systems Design in Frequency Domain: Mechanical Engineering at Georgia Tech, Study notes of Mechanical Engineering

Download Control Systems Design in Frequency Domain: Mechanical Engineering at Georgia Tech and more Study notes Mechanical Engineering in PDF only on Docsity! 1 Mechanical Engineering Georgia Institute of Technology Design of control systems in frequency domain Systems dynamics and control ME-3015 2 Classic control theory G(s) + - G(s) )(1 )( sG sG + (1)Linear, time-invariant, single-input single output systems (2)Controller design, gain tuning, stability analysis based on open-loop characteristics Root-locus : plot of closed-loop poles based on open-loop poles and zeros. Rooth-Hurwitz stability criterion: Stability of a closed-loop system from its characteristic equation )(1 sG+ (1+open-loop system’s TF) Performance and stability analysis in Frequency domain Analysis & design Close the loop 5 2nd order system 22 2 2 )( nn n ss sG ωςω ω ++ = )/2(])/(1[ 1 )2()()(2)( )( 2 22 2 22 2 nn nn n nn n j jjj jG ωςωωω ωςωωω ω ωωςωω ωω +− = +− = ++ = nωω << 1)(log20 =ωjG nωω >> ]dB[log40)/( 1log20)(log20 2 nn jG ω ω ωω ω −== -40dB/decade GAIN: PHASE: 2 1 2 )/(1 /2tan )/2(])/(1[ 1 n n nn j ωω ωςω ωςωωω − −= +− ∠ − #19 6 2nd order system (2) -40 -30 -20 -10 0 10 M ag ni tu de (d B ) 10-1 100 101 -180 -135 -90 -45 0 Ph as e (d eg ) Frequency nω ω -40dB/decade 2.0=ς 0dB/decade Intersection at nωω = deg00 →=ω deg180−→∞=ω deg90−→= nωω 7 Recall: System types G(s)R(s) C(s) + - Unity feedback (special case where K(s)=1) )())(( )())(()( 21 21 n N m pspspss zszszsKsG −−− −−− = L L (1) Plant only (2) Controller + Plant Type 0 system if N=0. Type 1 system if N=1. Type 2 system if N=2…. Number of poles at the origin. 10 Type 1 system: open-loop and closed-loop )1( 1)( + = ss sGOpen-loop TF: Type 1 system Closed-loop TF: 1 1 )1( 11 1)( 2 ++ = + + = ss ss sH Gain (dB) -40dB/decade1=ω Gain (dB) -40dB/decade 0dB/decade Bandwidth -20dB/decade Phase 0deg -180d 11 Type 2 system: open-loop and closed-loop )1( 1)( 2 + = ss sGOpen-loop TF: Type 2 system Closed-loop TF: 1 1 )1( 11 1)( 23 2 ++ = + + = ss ss sH Gain (dB) -60dB/decade 1=ω Gain (dB) -60dB/decade 0dB/decade Bandwidth -40dB/decade Open-loop gain plot: Check system type at low-frequency range Closed-loop gain plot does not tell much In terms of system type Phase 0deg -270deg 12 Closed loop TF )2(3 )2)(1( 11 )2)(1( 1 )( 2 +++ = ++ + ++= kss k ss k ss k sH Closed-loop TF 22 2 22 2)2(3 )( nn n n ss k kss ksH ωςω ω ω ++ = +++ = 2 21 1, ςωςω −±−= nn jpp )2)(1( 1)( ++ = ss sG 15 Recall: Sinusoidal transfer function • Bode plot (cont.) and Nyquist Plot • Sinusoidal Transfer Function System Input Output ttu ωsin)( = )(ty)(sG )()( ωjGsG → )( )()( ω ωω jU jYjG = )(/)()( ωωω jUjYjG ∠=∠ GAIN: PHASE: 16 )( )()( ω ωω jU jYjG = )(/)()( ωωω jUjYjG ∠=∠ GAIN: PHASE: M ag ni tu de [d B ] P ha se [d eg ] ω0 0 G ω Bode plot Im Re O )( ωjG )( ωjG )( ωjG∠∞→ω Nyquist plot Nyquist Plot 17 Nyquist stability criterion (general) G(s) + - )(sGNyquist plot of open-loop TF: Im Re O when a representative point s traces the nyquist path: ∞−= js ∞= js P: # of poles of in the right-half s-plane (unstable poles))(sG )(1 sG+ The closed-loop system is stable if N+P=0 N: # of clockwise encirclement of the (-1,0) Z(=N+P): # of zeros of in the right-half s-plane or unstable poles of closed-loop system You can check the stability of closed-loop systems from corresponding open-loop poles and Nyquist plots The above condition holds = No unstable poles in the closed-loop TF (include loop-gain) 20 Example 2: Nyquist stability criterion general case )31)(31)(2( )4)(2( )102)(2( )4)(2()( 2 jsjss ssk sss ssksG +−−−+ +− = +−+ +− = P: 2 j31± G(s) + - -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 -4 -3 -2 -1 0 1 2 3 4 Real Axis Im ag in ar y A xi s k increases Unstable Stable Unstable 21 Example 2: Nyquist stability criterion general case (cont.) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Nyquist Diagram Real Axis Im ag in ar y A xi s -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Nyquist Diagram Real Axis Im ag in ar y A xi s -2.5 -2 -1.5 -1 -0.5 0 0.5 -1.5 -1 -0.5 0 0.5 1 1.5 Nyquist Diagram Real Axis Im ag in ar y A xi s k=1 N=? Stable or unstable? k=2 N=? Stable or unstable? k=3 N=? Stable or unstable? For stability N+P=0. Now P=2. Then N= is needed. 22 Example 2_sol: Nyquist stability criterion general case (cont.) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Nyquist Diagram Real Axis Im ag in ar y A xi s -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Nyquist Diagram Real Axis Im ag in ar y A xi s -2.5 -2 -1.5 -1 -0.5 0 0.5 -1.5 -1 -0.5 0 0.5 1 1.5 Nyquist Diagram Real Axis Im ag in ar y A xi s k=1 N=0: unstable k=2 N=-2: stable k=3 N=-1: unstable For stability N+P=0. Now P=2. So N=-2 is needed (counter clock-wise encirclements, twice) 25 Example 4 and 5: Simplified Nyquist stability criterion + - )22)(1( 2 2 +++ ssss12 +s PlantPD controller -1 0 -6 -4 -2 0 2 4 6 Real Axis Im ag in ar y A xi s -3 -2 -1 0 1 0 1 Real Axis Im ag in ar y A xi s -1 + - )22)(1( 2 2 +++ ssss15 +s PlantPD controller Increase of Derivative Gain Stability of Open-loop system? Simplified Nyquist Stability Criterion? 26 Phase margin and Gain margin Im Re O-1 ∞→ω 0=ω )( ωjG How much the vector locus is far from (-1,0) deg180)( −=∠ ωjG cωω = Example 3: Trade-off between stability and performance Increasing k for better performance decreases gain and phase margins. Phase margin [deg] (measured from -180 line) Cross-over frequency Larger is better. Gain margin [dB] -20log10(OC) C Larger is better 27 Recall closed-loop characteristics kG(s) + - M ag ni tu de [d B ] P ha se [d eg ] ω0 0 x ω Low-frequency range 0dB (r=x), i.e., output x tracks the input High-frequency range: noise-cut Bandwidth (Magnitude>-3dB) Almost zero phase delay bω Cutoff frequency steep roll-off after bω bω r 30 Increasing Loop-gain kP(s) + - ω Phase [deg] -180 cωω = Phase margin [deg] Gain margin [dB] (1) Cross-over frequency (control bandwidth) increases (2) Gain and phase margins decrease Gain [dB] 0 [dB] Trade-off!! 31 Use Open-loop bode plot [Stability] Keep Phase margin and gain margin – Phase margin 40-60 deg – Gain margin 10-20 dB [Performance] Wider control bandwidth (higher cross-over frequency) Higher gain at lower frequency (steady-state error compensation) Sufficient damping (overshoot compensation) Lower gain at higher (than cross-over) frequency (noise avoidance) Controller tuning in frequency domain In general, trade-off 32 Controller-design using open-loop bode plot (frequency domain) • Bode plots can be obtained experimentally (complex, or unknown dynamic systems) • Closed-loop bode plot does not tell much. • Control bandwidth only. • Hard to know about the controller (no problem to disclose closed-loop plots of products) 35 Phase-lead and Phase-lag compensators For fine tuning. (Finer tuning than PID) • Lead compensator (PD controller with filter) • Lag compensator (PI controller with filter) • Lag-lead compensator (PID controller with filter) P(s) + - s kskk ivp ++ PID: 3 parameters P: Control bandwidth D: Damping I : Steady-state error 36 Phase-Lead compensator )10(1 1 1 )1( 1 1 1 1 << + + = + + = α α α α T s T s k sT sTkK plead -20 -10 0 10 20 30 40 M ag ni tu de (d B ) 10-3 10-2 10-1 100 101 102 0 45 90 Ph as e (d eg ) Bode Diagram Frequency (rad/sec) )1( 1 )1( 1 1 += + = sTksTkKPD α α c.f. PD controller pleadK PDK pleadK PDK PD + Low-pass filter Phase-lead compensator avoids unnecessary high-gain & cuts off high frequency noise 1 1 T Both PD and phase-lead improve phase-margin 37 Phase-Lag compensator )1(1 1 1 )1( 2 2 2 2 > + + = + + = β β β β T s T s k sT sTkK plag )11()1( 22 2 sT k sT sTkKPI += + = β β 2 1 T c.f. PI controller 0 10 20 30 40 50 60 70 80 M ag ni tu de (d B ) 10-4 10-3 10-2 10-1 100 101 -90 -45 0 P ha se (d eg ) Bode Diagram Frequency (rad/sec) plogK PIK plogK PIK PI + High-pass filter Phase-lag compensator avoids unnecessary phase-delay Both PI and phase-lag improve stead-state error Note: Theoretically, phase-lag compensators cannot perfectly eliminate the steady-state errors. 40 Example: Open-loop controller design )2)(1( 2)( ++ = sss sP P(s) + - K(s) 1 )1( 1 )1()( 2 2 1 1 + + + + == sT sT sT sTkKsK pleadlag β β α α α=0.2 T1=2 β=5 T2=3 K=2.7 1)( =sK (Simple P controller)(1) (2) 41 Example: Open-loop controller design 0 50 100 150 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Step Response Time (sec) A m pl itu de -10 -8 -6 -4 -2 0 M ag ni tu de (d B ) 10-3 10-2 10-1 100 101 102 -60 -30 0 30 60 Ph as e (d eg ) Bode Diagram Frequency (rad/sec) -150 -100 -50 0 50 100 M ag ni tu de (d B ) 10-3 10-2 10-1 100 101 102 -270 -225 -180 -135 -90 Ph as e (d eg ) Gm = 14.2 dB (at 2.42 rad/sec) , Pm = 53 deg (at 0.761 rad/sec) Frequency (rad/sec) -150 -100 -50 0 50 100 M ag ni tu de (d B ) 10-2 10-1 100 101 102 -270 -225 -180 -135 -90 Ph as e (d eg ) Gm = 9.54 dB (at 1.41 rad/sec) , Pm = 32.6 deg (at 0.749 rad/sec) Frequency (rad/sec) (1) Simple P controller (2) Phase-lead-lag compensatorPhase-lead-lag compensator bode plot (1) (2)