Lecture Control system design: Stability in the frequency domain - Nguyễn Công Phương

This chapter include all of the following content: Mapping contours in the s – plane, the nyquist criterio, relative stability and the nyquist criterion, time – domain performance criteria in the frequency domain, system bandwidth, the stability of control systems with time delays, pid controllers in the frequency domain, stability in the frequency domain using control design software.

Nguyễn Công Phương CONTROL SYSTEM DESIGN Stability in the Frequency Domain Contents I Introduction II Mathematical Models of Systems III State Variable Models IV Feedback Control System Characteristics V The Performance of Feedback Control Systems VI The Stability of Linear Feedback Systems VII The Root Locus Method VIII.Frequency Response Methods IX Stability in the Frequency Domain X The Design of Feedback Control Systems XI The Design of State Variable Feedback Systems XII Robust Control Systems XIII.Digital Control Systems sites.google.com/site/ncpdhbkhn Stability in the Frequency Domain Mapping Contours in the s – Plane The Nyquist Criterion Relative Stability and the Nyquist Criterion Time – Domain Performance Criteria in the Frequency Domain System Bandwidth The Stability of Control Systems with Time Delays PID Controllers in the Frequency Domain Stability in the Frequency Domain Using Control Design Software sites.google.com/site/ncpdhbkhn Mapping Contours in the s – Plane (1) Ex jω s – plane jv F (s) D j2 j2 A A D j1 j1 −2 −1 F(s) – plane − j1 σ −2 −1 − j1 u B C − j2 − j2 B C u = 2σ + F ( s) = s + = 2(σ + jω ) + = (2σ + 1) + j2ω = u + jv →  v = 2ω As = + j1 = σ + jω → AF = u + jv = (2σ + 1) + j (2ω ) = (2 × + 1) + j (2 × 1) = + j Bs = − j1 → BF = (2 × + 1) + j[2 × ( −1)] = − j Cs = −1 − j1 → CF = [2 × ( −1) + 1] + j[2 × ( −1)] = −1 − j Ds = −1 + j1 → DF = [2 × ( −1) + 1] + j (2 × 1) = −1 + j sites.google.com/site/ncpdhbkhn Mapping Contours in the s – Plane (2) Ex F ( s) = s s+2 s-plane F(s)-plane 1.5 1.5 A D D 0.5 0.5 0 jv jω A B -0.5 -0.5 -1 -1.5 -2.5 -1 C -2 -1.5 -1 C B -0.5 σ 0.5 1.5 -1.5 -2.5 -2 sites.google.com/site/ncpdhbkhn -1.5 -1 -0.5 u 0.5 1.5 Mapping Contours in the s – Plane (3) 2 F ( s ) = s + 11 0 jv jω s-plane -1 -1 -2 -2 F(s)-plane 1 0.5 0.5 0 jv jω Cauchy’s -2theorem: Γ3s in the s-plane encircles and P2 poles of F(s) -1 0If a contour -2 -1 Z zeros u σ and does not pass through any poles or zeros of F(s) and the traversal is in the clockwise directions-plane along the contour, the corresponding contour ΓF in the F(s)-plane F(s)-plane 1.5 encircles the origin of the F(s)-plane N = Z – P1.5times in the clockwise direction -0.5 F ( s) = -1 -1.5 -2 -1 σ s-0.5 s + -12 -1.5 -2 sites.google.com/site/ncpdhbkhn -1 u Mapping Contours in the s – Plane (4) Ex s s + 0.5 F(s)-plane s-plane 0.15 0.1 0.05 0 jv jω F ( s) = -2 -0.05 -4 -0.1 -6 -6 -4 -2 σ 0.9 0.95 1.05 u 1.1 1.15 If a contour Γs in the s-plane encircles Z zeros and P poles of F(s) and does not pass through any poles or zeros of F(s) and the traversal is in the clockwise direction along the contour, the corresponding contour ΓF in the F(s)-plane encircles the origin of the F(s)-plane N = Z – P times in the clockwise direction sites.google.com/site/ncpdhbkhn Mapping Contours in the s – Plane (5) Ex F ( s) = s s+z = = F ( s) ∠F ( s) = F ( s) ∠(φz − φ p ) s+2 s+ p If a contour Γs in the s-plane encircles Z zeros and P poles of F(s) and does not pass through any poles or zeros of F(s) and the traversal is in the clockwise direction along the contour, the corresponding contour ΓF in the F(s)-plane encircles the origin of the F(s)-plane N = Z – P times in the clockwise direction F(s)-plane s-plane 1.5 1.5 1 φz φp 0 -0.5 -0.5 -1 -1 -1.5 -2.5 -2 -1.5 -1 -0.5 σ φz − φ p 0.5 jv jω 0.5 0.5 1.5 -1.5 -2.5 sites.google.com/site/ncpdhbkhn -2 -1.5 -1 -0.5 u 0.5 1.5 Stability in the Frequency Domain Mapping Contours in the s – Plane The Nyquist Criterion Relative Stability and the Nyquist Criterion Time – Domain Performance Criteria in the Frequency Domain System Bandwidth The Stability of Control Systems with Time Delays PID Controllers in the Frequency Domain Stability in the Frequency Domain Using Control Design Software sites.google.com/site/ncpdhbkhn The Nyquist Criterion (1) • F(s) = + L(s) = • A feedback system is stable if and only if the contour ΓL in the L(s) – plane does not encircle the (–1, 0) point when the number of poles of L(s) in the right – hand s – plane is zero (P = 0) • (when the number of poles of L(s) in the right – hand s – plane is other than zero) A feedback system is stable if and only if, for the contour ΓL , the number of counterclockwise encirclements of the (–1, 0) point is equal to the number of poles of L(s) with positive real parts sites.google.com/site/ncpdhbkhn jω s – plane r→∞ σ Γs Nyquist contour 10 Time – Domain Performance Criteria in the Frequency Domain (1) R(s) Y ( jω ) Gc ( jω )G ( jω ) T ( jω ) = = R ( jω ) + Gc ( jω )G ( jω ) G (s ) Gc ( s) ( −) = M (ω )e jφ (ω ) Y ( s) H (s ) Gc ( jω )G ( jω ) = u + jv Gc ( jω )G ( jω ) → M (ω ) = + Gc ( jω )G ( jω ) 1.5 M = 1.5 u + jv u + v2 = = + u + jv (1 + u ) + v M = 0.7 M =1 0.5 M =2 M = 0.5 -0.5  M   M  → u − + v =    1− M   1− M   2 -1 -1.5 -3 -2.5 sites.google.com/site/ncpdhbkhn -2 -1.5 -1 -0.5 0.5 1.5 32 2.5 Time – Domain Performance Criteria in the Frequency Domain (2) 4.5 K1 M1 M2 K2 0.5 3.5 M1 Magnitude jv -0.5 2.5 M2 -1 1.5 K1 > K -1.5 0.5 K1 -2 -2.5 -2 K2 -1.5 -1 -0.5 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 ω u Polar plot of Frequency response of Gc ( jω )G( jω ) = L ( jω ) = KP( jω ) sites.google.com/site/ncpdhbkhn T ( jω ) = Gc ( jω )G ( jω ) + Gc ( jω )G ( jω ) 33 Time – Domain Performance Criteria in the Frequency Domain (3) φ = 10 R(s) o ( −) Y ( s) G (s ) Gc ( s) φ = 20o H (s ) T ( jω ) = φ = 30o 1.5 φ = − 30o Y ( jω ) Gc ( jω )G ( jω ) = = M (ω )e jφ (ω ) R( jω ) + Gc ( jω )G ( jω ) M = 1.5 0.5 -2 M =2 φ = −20 -4 -3 -2 -1 M = 0.5 o -0.5 φ = −10o -6 -4 M = 0.7 M =1 -1.5 1   1   u +  +v −  = 1 +  2  2φ  4 φ   -1 -3 -2.5 -2 -1.5 -1 -0.5 0.5 1.5  M2   M  + v = u −    1− M   1− M   sites.google.com/site/ncpdhbkhn 2.5 34 Ex Time – Domain Performance Criteria in the Frequency Domain (4) R(s) 0.47 Gc ( s)G( s ) = , H (s ) = s( s + s + 1) ( −) G (s ) Gc ( s) Y ( s) H (s ) Nichols Chart 20 dB 15 dB Open-Loop Gain (dB) 10 dB -5 -10 -15 -20 -270 -225 -180 -135 -90 Open-Loop Phase (deg) sites.google.com/site/ncpdhbkhn 35 Stability in the Frequency Domain Mapping Contours in the s – Plane The Nyquist Criterion Relative Stability and the Nyquist Criterion Time – Domain Performance Criteria in the Frequency Domain System Bandwidth The Stability of Control Systems with Time Delays PID Controllers in the Frequency Domain Stability in the Frequency Domain Using Control Design Software sites.google.com/site/ncpdhbkhn 36 System Bandwidth (1) Ex 1 T1 ( s) = , T2 ( s) = s +1 5s + T1 T2 20log|T|, dB -3 -6 -8 -10 -18 0.2 ω Step Response Amplitude 0.8 0.6 0.4 T1 0.2 T2 10 Time (seconds) sites.google.com/site/ncpdhbkhn 37 System Bandwidth (2) Ex T1 ( s) = 100 900 , T ( s ) = s + 10 s + 100 s + 30 s + 900 10 Step Response T1 1.2 T1 T2 -3 T2 -10 0.8 Amplitude 20log|T|, dB -20 -30 0.6 -40 0.4 -50 0.2 -60 -70 10 10 10 ω 0 sites.google.com/site/ncpdhbkhn 0.2 0.4 0.6 0.8 Time (seconds) 1.2 1.4 38 1.6 Stability in the Frequency Domain Mapping Contours in the s – Plane The Nyquist Criterion Relative Stability and the Nyquist Criterion Time – Domain Performance Criteria in the Frequency Domain System Bandwidth The Stability of Control Systems with Time Delays PID Controllers in the Frequency Domain Stability in the Frequency Domain Using Control Design Software sites.google.com/site/ncpdhbkhn 39 The Stability of Control Systems with Time Delays (1) • Time delay: the time interval between the start of an even at one point in a system and its resulting action at another point in the system • A pure time delay, without attenuation: Gd(s) = e–sT • A pure time delay does not change the magnitude of the transfer function • The Nyquist criterion remains valid for a system with a time delay sites.google.com/site/ncpdhbkhn 40 Ex T (s ) = The Stability of Control Systems T with Time Delays (2) − s +1 e− sT ≈ 31.5 e−s ( s + 1)(30s + 1)[( s / 3) + s / + 1] T s +1 20log10|T|, dB -2 -4 -6 -0.3 10 -0.2 10 10 -0.1 10 10 0.1 ω -100 With time delay Without time delay o φ( ) -150 Φ without time delay -180 -200 -250 -0.3 10 Φ with time delay -0.2 10 10 -0.1 10 10 0.1 ω sites.google.com/site/ncpdhbkhn 41 Stability in the Frequency Domain Mapping Contours in the s – Plane The Nyquist Criterion Relative Stability and the Nyquist Criterion Time – Domain Performance Criteria in the Frequency Domain System Bandwidth The Stability of Control Systems with Time Delays PID Controllers in the Frequency Domain Stability in the Frequency Domain Using Control Design Software sites.google.com/site/ncpdhbkhn 42 PID Controllers in the Frequency Domain (1) Ex Td (s ) R(s) ( −) KI KP + + KD s s K ωn2g Y (s) (τ s + 1)( s + 2ζωng s + ωn2g ) H (s ) K = –7000, τ = 5s Design the PID controller so that GM ≥ 6dB, 30o ≤ ΦM ≤ 60o, rise time Tr < 4s, time to peak TP < 10s s2 + (K P / K D )s + ( K I / K D ) L( s ) = K ω K D s (τ s + 1)( s + 2ζωng s + ωn2g ) ng ζ = 0.01Φ M = 0.01× 30o = 0.3 2.16ζ + 0.6 2.16 × 0.3 + 0.6 Tr = = < → ωn > 0.31 ωn ωn 2.16 × 0.3 + 0.6 π π ωn = 0.4 → Tr = = 3s, TP = = = 8s 2 0.4 0.4 − 0.3 ωn − ζ sites.google.com/site/ncpdhbkhn 43 PID Controllers in the Frequency Domain (2) Ex K = –7000, τ = 5s Design the PID controller so that GM ≥ 6dB, 30o ≤ ΦM ≤ 60o, rise time Tr < 4s, time to peak TP < 10s s + ( K P / K D )s + ( K I / K D ) L( s ) = K ω K D , ζ = 0.3, ωn = 0.4 2 s (τ s + 1)( s + 2ζωng s + ωng ) ng Bode Diagram 20 Magnitude (dB) 10 GM -10 -20 -30 -40 -45 Phase (deg) -90 -135 ΦM -180 -225 -1 10 10 10 Frequency (rad/s) sites.google.com/site/ncpdhbkhn 44 Stability in the Frequency Domain Mapping Contours in the s – Plane The Nyquist Criterion Relative Stability and the Nyquist Criterion Time – Domain Performance Criteria in the Frequency Domain System Bandwidth The Stability of Control Systems with Time Delays PID Controllers in the Frequency Domain Stability in the Frequency Domain Using Control Design Software sites.google.com/site/ncpdhbkhn 45 Stability in the Frequency Domain Using Control Design Software • • • • • nyquist nichols margin pade ngrid sites.google.com/site/ncpdhbkhn 46 ... Domain Performance Criteria in the Frequency Domain System Bandwidth The Stability of Control Systems with Time Delays PID Controllers in the Frequency Domain Stability in the Frequency Domain. .. F(s)-plane s-plane 1.5 1.5 1 φz φp 0 -0 .5 -0 .5 -1 -1 -1 .5 -2 .5 -2 -1 .5 -1 -0 .5 σ φz − φ p 0.5 jv jω 0.5 0.5 1.5 -1 .5 -2 .5 sites.google.com/site/ncpdhbkhn -2 -1 .5 -1 -0 .5 u 0.5 1.5 Stability in the. .. Criteria in the Frequency Domain System Bandwidth The Stability of Control Systems with Time Delays PID Controllers in the Frequency Domain Stability in the Frequency Domain Using Control Design Software