Robust control systems 2018b mk

Nguyễn Công Phương CONTROL SYSTEM DESIGN Robust Control Systems 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 s i tes.google.com/site/ncpdhbkhn Robust Control Systems Introduction Robust Control Systems & System Sensitivity Analysis of Robustness Systems with Uncertain Parameters The Design of Robust Control Systems The Design of Robust PID-Controlled Systems The Robust Internal Model Control System Robust Control Systems Using Control Design Software s i tes.google.com/site/ncpdhbkhn Introduction Td ( s ) R( s ) E a (s ) (−) Gc ( s ) G(s) Controller Process H (s ) Y ( s) N (s ) Sensor • The process model will always be inaccurate representation of the actual physical system because of: – – – – – – Parameter changes, Unmodeled dynamics, Unmodeled time delays, Changes in equilibrium point (operating point), Sensor noise, Unpredicted disturbance inputs • A system is robust when the system has acceptable changes in performance due to model changes or inaccuracies s i tes.google.com/site/ncpdhbkhn Robust Control Systems Introduction Robust Control Systems & System Sensitivity Analysis of Robustness Systems with Uncertain Parameters The Design of Robust Control Systems The Design of Robust PID-Controlled Systems The Robust Internal Model Control System Robust Control Systems Using Control Design Software s i tes.google.com/site/ncpdhbkhn Robust Control Systems & System Sensitivity (1) • A control system is robust when: – It has low sensitivities, – It is stable over the range of parameter variations, – The performance continues to meet the specifications in the presence of a set of changes in the system parameters • The system sensitivity: • The root sensitivity: ∂T / T Sα = ∂α / α ∂ri ri Sα = ∂α / α n ri S α SαT = − i =1 s + ri T  s i tes.google.com/site/ncpdhbkhn Robust Control Systems & System Sensitivity (2) Ex 1 s + α T= = s + α +1 1+ s+α R ( s) ∂T / T Sα = ∂α / α ∂T α −1 α −1 = = = ∂α T ( s + α + 1) T ( s + α + 1) s +α Y (s) T α s + α +1 ∂ ri Sα = ∂(α + 1) ∂α / α ri 1 → Sα = ∂α α = α T= = s + α + s + r1 −α = s + α +1 ri s i tes.google.com/site/ncpdhbkhn Robust Control Systems & System Sensitivity (3) Ex T= s +α +1 R ( s) s +α Y (s) −α Sα = s + α +1 T Sαri = α n ri S α SαT = − i =1 s + ri −α = s +α +1  s i tes.google.com/site/ncpdhbkhn Robust Control Systems & System Sensitivity (4) Ex T Sα = K K K s ( s + 1) T= = K s +s+K 1+ s( s + 1) R ( s) Y (s) K s ( s + 1) ∂T / T = ∂α / α ∂T K ( s + s + K ) − K K = = 2 ∂K T T (s + s + K ) s2 + s = (s + s + K ) K s2 + s = = S (s ) K s +s+K s2 + s + K s i tes.google.com/site/ncpdhbkhn Robust Control Systems & System Sensitivity (5) Ex K s +s T ( s) = , S (s ) = s +s+K s +s+K s i tes.google.com/site/ncpdhbkhn R ( s) Y (s) K s ( s + 1) 10 The Design of Robust PID – Controlled Systems (5) Ex Td ( s ) Obtain an optimum ITAE R(s) performance for a step input and Gp (s) a settling time of less than 0.5 second T (s) = K D s2 + KP s + KI s + ( K D + 2) s + ( K P + 1) s + K I E a (s ) ( −) U (s ) Gc ( s ) Y (s) ( s + 1) → K P = 362.35, K D = 20.75, K I = 2197 ζ = 0.8, ωn = 13 Step Response 1.4 2.5 1.2 Disturbance 10 -3 1.5 0.8 0.6 0.5 0.4 0.2 -0.5 0.2 0.4 0.6 Time (seconds) 0.8 s i tes.google.com/site/ncpdhbkhn 0.2 0.4 0.6 Time (seconds) 0.8 32 The Design of Robust PID – Controlled Systems (6) Ex Obtain an optimum ITAE R(s) performance for a step input and Gp (s) a settling time of less than 0.5 second T (s) = Td ( s ) E a (s ) ( −) U (s ) Gc ( s ) Y (s) ( s + 1) 20.75 s + 362.35 s + 2197 s + 22.75 s + 363.35 s + 2197 Step Response 1.4 T (s) = G p (s) = 20.75 s + 362.35 s + 2197 Without Gp 1.2 With G p s + 22.75 s + 363.35 s + 2197 2197 0.8 s + 22.75 s + 363.35 s + 2197 105.88 → G p (s) = s + 17.46 s + 105.88 0.6 0.4 0.2 0 0.1 s i tes.google.com/site/ncpdhbkhn 0.2 0.3 0.4 0.5 Time (seconds) 0.6 0.7 0.8 33 0.9 Ex The Design of Robust PID – Controlled Systems (7) Td ( s ) Obtain an optimum ITAE R( s ) performance for a step input and Gp ( s ) a settling time of less than 0.5 second E a (s ) (−) U (s ) Gc ( s ) Y (s) ( s + 1) 26 s + 549.4 s + 4096 157.54 ωn = 16 → Gc ( s ) = , G p (s) = s s + 21.13 s + 157.54 Step Response Disturbance 1.4 0.6 Original PID without pref ilter PID with pref ilter 1.2 0.5 0.4 0.8 0.3 0.6 0.2 0.4 0.1 0.2 0 Original With PID -0.1 Time (seconds) s i tes.google.com/site/ncpdhbkhn Time (seconds) 34 The Design of Robust PID – Controlled Systems (8) Td ( s ) R( s ) Gp ( s ) E a (s ) (−) U (s ) Gc ( s ) Y (s) ( s + 1) The design procedure: Select the ωn of the closed-loop system by specifying the settling time, Determine the three coefficients using the appropriate optimum equation & the ωn of step to obtain Gc(s), Determine a prefilter Gp(s) so that the closed-loop system transfer function, T(s), does not have any zeros s i tes.google.com/site/ncpdhbkhn 35 The Design of Robust PID – Controlled Systems (9) Ex Td ( s ) 0.5 ≤ τ ≤ 1, ≤ K ≤ R( s ) E a (s ) Gp ( s ) (−) U (s ) K Y ( s) (τ s + 1)2 Gc ( s ) 26 s2 + 549.4 s + 4096 157.54 K = 1, τ = 1, ωn = 16 → Gc (s) = , G p (s ) = s s + 21.13 s + 157.54 Step Response 1.2 Disturbance 10 -4 20 K= K= K= K= 15 1, 2, 1, 2, = = = = 1 0.5 0.5 0.8 10 0.6 0.4 K K K K 0.2 = 1, = 2, = 1, = 2, =1 =1 = 0.5 = 0.5 0 -5 0.1 0.2 0.3 0.4 Time (seconds) 0.5 0.6 0.7 s i tes.google.com/site/ncpdhbkhn 0.2 0.4 Time (seconds) 0.6 0.8 36 Robust Control Systems Introduction Robust Control Systems & System Sensitivity Analysis of Robustness Systems with Uncertain Parameters The Design of Robust Control Systems The Design of Robust PID-Controlled Systems The Robust Internal Model Control System Robust Control Systems Using Control Design Software s i tes.google.com/site/ncpdhbkhn 37 The Robust Internal Model Control System (1) R ( s ) E a (s ) ( −) Gc ( s ) ( −) G(s) x Y (s) K The internal model principle: if Gc(s)G(s) contains R(s) then y(t) will track r(t) asymptotically (in the steady state), and the tracking is robust s i tes.google.com/site/ncpdhbkhn 38 The Robust Internal Model Control System (2) Ex G(s) = 1/s, obtain a ramp response with a steady-state error of zero K=0 Gc ( s ) = K P + R ( s ) E a (s ) ( −) Gc ( s ) ( −) G(s) x Y (s) K KI s K 1 K s+ K  Gc ( s )G ( s ) =  K P + I  = P I s s s  T (s) = KPs + KI s2 + K P s + K I The Optimum Coefficients of T(s) Based on the ITAE Criterion for a Ramp Input s + 3.2ωn s + ωn2 → T (s) = 3.2ωn s + ωn2 s + 3.2ω n s + ωn2 s i tes.google.com/site/ncpdhbkhn 39 The Robust Internal Model Control System (3) Ex R ( s ) E a (s ) G(s) = 1/s, obtain a ramp response with a steady-state error of zero T (s) = KPs + KI s2 + K P s + K I Tsettling time = ζωn = 1s = 3.2ωn s + ω Gc ( s ) ( −) n ( −) K s + 3.2ω n s + ω n2 → ζ = 0.8, ωn = Ramp input Output Error G(s) x Y (s) → T (s) = 16 s + 25 s + 16 s + 25 0.8 0.5 0.6 Step input Output Error 0.4 -0.5 0.2 -1 10 Time 15 s i tes.google.com/site/ncpdhbkhn Time 40 Ex The Robust Internal Model Control System (4) G(s) = 1/s, G(s) changes gain so that it becomes 2/s R ( s ) E a (s ) ( −) Gc ( s ) ( −) G(s) x Y (s) K s i tes.google.com/site/ncpdhbkhn 41 The Robust Internal Model Control System (5) R( s ) Ex G p ( s) 1 , G2 (s) = s +1 s+2 1 Gˆ1 ( s ) = , Gˆ ( s ) = s + 0.5 s +1 ( −) G c (s ) G1 ( s ) = Achieve a settling time in less than second & a deadbeat response KD s2 + K P s + KI Gc (s ) = s → Ta (s ) = Gp (s) = Ta = ( −) Ka ( −) G1 ( s ) G2 ( s) Y (s) Kb G1G2Gc + K b G1 + K a G1G2 + G1G2Gc K D s2 + KP s + KI s + ( K D + Kb + 3) s + ( K P + K a + Kb + 2) s + K I KI K D s2 + K P s + K I → T (s ) = G p ( s )Ta ( s ) = KI s + ( K D + Kb + 3) s + ( K P + K a + Kb + 2)s + K I s i tes.google.com/site/ncpdhbkhn 42 The Robust Internal Model Control System (6) R( s ) Ex G p ( s) ( −) Gc ( s ) ( −) Ka Achieve a settling time in less than second & a deadbeat response T ( s) = α β 1.90 2.20 Ts = Kb s + ( K D + Kb + 3) s + ( K P + K a + Kb + 2) s + K I Coefficients → T (s ) = G2 ( s) KI System Order 3rd ( −) G1 ( s ) Y (s) γ δ ε Percent Overshoot (%) Percent Undershoot (%) 90% Rise Time 100% Rise Time Settling Time 1.65 1.36 3.48 4.32 4.04 ω n3 s + 1.9ωn s + 2.2ωn2 s + ωn3 4.04 ωn = 0.5 → ω n = 8.08 K a = 10, K b = → K P = 127.6, K I = 527.5, K D = 10.35 s i tes.google.com/site/ncpdhbkhn 43 Ex The Robust Internal Model Control System (7) R( s ) G p ( s) 1 , G2 (s) = s +1 s+2 1 Gˆ1 ( s ) = , Gˆ ( s ) = s + 0.5 s +1 G1 ( s ) = ( −) G c (s ) ( −) Ka ( −) G1 ( s ) G2 ( s) Y (s) Kb Achieve a settling time in less than second & a deadbeat response s i tes.google.com/site/ncpdhbkhn 44 Robust Control Systems Introduction Robust Control Systems & System Sensitivity Analysis of Robustness Systems with Uncertain Parameters The Design of Robust Control Systems The Design of Robust PID-Controlled Systems The Robust Internal Model Control System Robust Control Systems Using Control Design Software s i tes.google.com/site/ncpdhbkhn 45 Ex Robust Control System Using Control Design Software 116 s + 1887 s + 8260 Gc ( s ) = K s R(s) s i tes.google.com/site/ncpdhbkhn (−) Gc ( s ) Y (s) ( s + 1) 46 ... Analysis of Robustness Systems with Uncertain Parameters The Design of Robust Control Systems The Design of Robust PID-Controlled Systems The Robust Internal Model Control System Robust Control Systems. .. Introduction Robust Control Systems & System Sensitivity Analysis of Robustness Systems with Uncertain Parameters The Design of Robust Control Systems The Design of Robust PID-Controlled Systems The Robust. .. 12 Robust Control Systems Introduction Robust Control Systems & System Sensitivity Analysis of Robustness Systems with Uncertain Parameters The Design of Robust Control Systems The Design of Robust