Outline
Outline Outline 1 ler Design 2
What have we talked about in MM2? Sampling rate selection Equivalents between continuous & digital Systems
Outline ler Design Emulation Method for 1 ler Design Emulation Method for 2
ler Design ler Design Emulation Method for Digital controller can be obtained using: Emulation, which finds the discrete equivalent of a continuous controller Direct discrete design (next lecture)
ler Design Frequency Issues Emulation Method for Continuous Systems For a minimum-phase transfer function, the phase is uniquely determined by the magnitude curve: G(jω) n 9 where n is the slope of G(jω) in units of decade of amplitude Discrete Systems The amplitude and phase relationship is lost! The prediction of stability from the amplitude curve alone for minimum-phase systems is lost It is typically necessary to determine both magnitude and phase for discrete systems
ler Design Emulation Method Emulation Method for 1 A continuous controller is designed 2 Sample time is selected 3 Discrete equivalent is computed 4 Evaluation of design
Outline ler Design 1 ler Design Emulation Method for 2
ler Design Case Study: Antenna Control General System Model: J θ + B θ = T c + T d Discarding the disturbances T d gives the transfer function: where a = B J Design Specifications: Θ(s) U(s) = 1 s ( s a + 1) =.1 and u(t) = Tc(t) B. Overshoot to a step input less than 16% (PM 55) Settling time to 1% in less than 1s Tracking error to ramp of slope.1 rad sec less than.1rad Sampling time to give at least 1 samples in a rise-time
ler Design Lead Compensator Design for Antenna (FC pp. 375) Step 1 Design the low frequency gain K with respect to the steady-state error specification Antenna system case: K = 1 Step 2 Determine the needed phase lead sys=tf(1,[1 1 ]); margin(sys) PM=18 at ω =.38
ler Design Lead Compensator Design for Antenna (FC pp. 375) Step 3 Using lead contribution of φ max = 45 should result in PM=63 which is 8 more than needed. Step 4 Determine: α = 1 sinφ max 1 + sinφ max = 1 sin45 1 + sin45 =.1716 Step 5 T = 1 = 1 ω max α ω n = 2 α 2.92 α = 5.248 Giving a zero in s = 1 T =.19 and a pole in s = 1 αt = 1.11.
ler Design Lead Compensator Design for Antenna (FC pp. 375) Step 6 Draw the compensated frequency response, check PM Using the formulation: we use: sysd=tf([5.3 1],[.9 1]) sysc=sys*sysd margin(sysc) step(feedback(sysc,1)) D(s) = Ts + 1 αts + 1
ler Design Lead Compensator Design for Antenna (FC pp. 375) 4 Bode Diagram Gm = Inf db (at Inf rad/sec), Pm = 56.3 deg (at.55 rad/sec) 1.4 Step Response 2 1.2 System: untitled1 Time (sec): 5.65 Amplitude: 1.16 Magnitude (db) 2 4 1 System: untitled1 Time (sec): 13.3 Amplitude: 1.1 6 8 9 Amplitude.8.6 Phase (deg) 135.4.2 18 1 2 1 1 1 1 1 1 2 Frequency (rad/sec) 5 1 15 2 25 Time (sec) Figure: Frequency response Figure: Step response
ler Design Lead Compensator Design for Antenna (FC pp. 375) Step 7 Step 7: Iterate on the design until all specifications are met sysd=tf([1 1],[1 1]) sysc=sys*sysd margin(sysc) syscl=feedback(sysc,1) step(syscl)
ler Design Lead Compensator Design for Antenna (FC pp. 375) Bode Diagram Step Response 2 1.4 Magnitude (db) 2 4 6 8 Amplitude 1.2 1.8.6 System: syscl Time (sec): 3.63 Amplitude: 1.16 System: syscl Time (sec): 8.75 Amplitude:.99 45 Phase (deg) 9 135.4.2 18 1 2 1 1 1 1 1 1 2 Frequency (rad/sec) 2 4 6 8 1 12 Time (sec) Figure: Frequency response Figure: Step response
ler Design Digital Lead Compensator for Antenna - Fast Sampling Continuous lead controller Digitization - Fast Sample Rate D(s) = 1s + 1 s + 1 sysc=tf(1,[1 1 ]); lead=tf([1 1],[1 1]); syslead=sysc*lead; Ts=1/2; leadd1=c2d(lead,ts, zoh ); sysd=c2d(sysc,ts, zoh ); syscld=feedback(sysd*leadd1,1); step(syscld)
ler Design Digital Lead Compensator for Antenna - Fast Sampling 1.4 Step Response 1.2 System: syscldfast Time (sec): 3.65 Amplitude: 1.18 Amplitude 1.8.6 System: syscldfast Time (sec): 8.75 Amplitude:.99.4.2 2 4 6 8 1 12 Time (sec)
ler Design Digital Lead Compensator for Antenna - Slow Sampling Continuous lead controller Digitization - Slow Sample Rate D(s) = 1s + 1 s + 1 sysc=tf(1,[1 1 ]); lead=tf([1 1],[1 1]); syslead=sysc*lead; Ts=1/2; leadd1=c2d(lead,ts, zoh ); sysd=c2d(sysc,ts, zoh ); syscld=feedback(sysd*leadd1,1); step(syscld)
ler Design Digital Lead Compensator for Antenna - Slow Sampling 1.4 System: syscldslow Time (sec): 3.35 Amplitude: 1.35 Step Response 1.2 Amplitude 1.8.6 System: syscldslow Time (sec): 13.5 Amplitude:.99.4.2 2 4 6 8 1 12 14 16 18 Time (sec)
ler Design Effect of Sample Time on Step Response Step Response 1.4 1.2 Continuous Fast sampling Slow sampling 1 Amplitude.8.6.4.2 2 4 6 8 1 12 14 16 18 Time (sec)
ler Design Effect of Sample Time on Frequency Response 5 Bode Diagram Magnitude (db) 5 1 Phase (deg) 15 9 18 Continuous Fast sampling Slow sampling 27 1 2 1 1 1 1 1 1 2 Frequency (rad/sec)
ler Design Effect of Sample Time on Pole Locations 1 Pole Zero Map.8.6.4 System: untitled1 Pole :.5 +.866i Damping:.5 Overshoot (%): 16.3 Frequency (rad/sec): 1 System: syscldslow Pole :.737 +.422i Damping:.3 Overshoot (%): 37.3 Frequency (rad/sec): 1.9 Imaginary Axis.2.2 System: syscldfast Pole :.975.433i Damping:.482 Overshoot (%): 17.7 Frequency (rad/sec): 1.1.4.6.8 1 1.8.6.4.2.2.4.6.8 1 Real Axis
ler Design Incorporating Sampling Delay in System Continuous System G(s) = 1 s(1s + 1) Continuous System with Delay G d (s) = 2/T 1 s + 2/T s(1s + 1)
ler Design Lead Compensator for System using Slow Sampling Rate Inserting T = 1/2 G d (s) = 2/T 1 s + 2/T s(1s + 1) 4 = s(s + 4)(1s + 1)
ler Design Lead Compensator Design for Antenna Step 1 Design the low frequency gain K with respect to the steady-state error specification Steady-state unchanged from original system: K = 1 Step 2 Determine the needed phase lead sys=tf(1,[1 41 4 ]); margin(sys) PM=14 at ω =.38
ler Design Lead Compensator Design for Antenna Step 3 Using lead contribution of φ max = 5 should result in PM=64 which is 9 more than needed. Step 4 Determine: α = 1 sinφ max 1 + sinφ max = 1 sin5 1 + sin5 =.1325 Step 5 T = 1 ω max α = 1.4 (.1325) = 6.869 Giving a zero in s = 1 T =.1456 and a pole in s = 1 αt = 1.99.
ler Design Lead Compensator Design for Antenna Step 6 Draw the compensated frequency response, check PM Using the formulation: we use: sysd=tf([6.9 1],[.9 1]) sysc=sys*sysd margin(sysc) step(feedback(sysc,1)) D(s) = Ts + 1 αts + 1
ler Design Lead Compensator Design for Antenna 5 Bode Diagram Gm = 16.9 db (at 2.5 rad/sec), Pm = 48.6 deg (at.67 rad/sec) 1.4 System: untitled1 Time (sec): 4.76 Amplitude: 1.21 Step Response Magnitude (db) 5 1.2 1 System: untitled1 Time (sec): 13.6 Amplitude: 1.1 1 15 9 Amplitude.8.6 Phase (deg) 135 18.4.2 225 27 1 2 1 1 1 1 1 1 2 Frequency (rad/sec) 5 1 15 2 25 Time (sec) Figure: Frequency response Figure: Step response
ler Design Lead Compensator Design for Antenna Step 7 Step 7: Iterate on the design until all specifications are met sysd=tf([7.5 1],[.68 1]) sysc=sys*sysd margin(sysc) syscl=feedback(sysc,1) step(syscl)
ler Design Lead Compensator Design for Antenna Bode Diagram Step Response 5 1.4 Magnitude (db) 5 1 1.2 1 System: syscl Time (sec): 3.9 Amplitude: 1.16 System: syscl Time (sec): 7 Amplitude: 1.1 System: syscl Time (sec): 11.6 Amplitude: 1.1 15 Amplitude.8.6 Phase (deg) 9 18.4.2 27 1 2 1 1 1 1 1 1 2 Frequency (rad/sec) 5 1 15 2 25 3 Time (sec) Figure: Frequency response Figure: Step response
ler Design Digital Lead Compensator for Antenna - Slow Sampling Continuous lead controller Digitization - Slow Sample Rate D(s) = 7.5s + 1.68s + 1 sysc=tf(1,[1 1 ]); lead=tf([7.5 1],[.68 1]); syslead=sysc*lead; Ts=1/2; leadd1=c2d(lead,ts, zoh ); sysd=c2d(sysc,ts, zoh ); syscld=feedback(sysd*leadd1,1); step(syscld)
ler Design Digital Lead Compensator for Antenna - Comparison Bode Diagram Step Response Magnitude (db) 2 2 4 1.4 1.2 1 Continuous Slow sampling Compensated slow sampling Phase (deg) 6 8 9 18 Continuous Slow sampling Compensated slow sampling Amplitude.8.6.4.2 27 1 2 1 1 1 1 1 1 2 Frequency (rad/sec) 2 4 6 8 1 12 14 16 18 Time (sec) Figure: Frequency response Figure: Step response
ler Design Discretization in Matlab Matlab sysd=c2d(sys,ts,method) method: zoh : Zero order hold foh : First order hold (academic) tustin : Bilinear approximation (trapezoidal) prewarp : Tustin with a specific frequency used for prewarp matched : Matching continuous poles with discrete
ler Design Discretization of Lead Compensator - Fast Sample Rate Bode Diagram Step Response 1 1.2 ZOH Tustin Matched Magnitude (db) 1 2 1 Phase (deg) 3 4 45 9 135 ZOH Tustin Matched Amplitude.8.6.4.2 18 1 1 1 1 1 Frequency (rad/sec) 1 2 3 4 5 6 7 8 Time (sec) Figure: Frequency response Figure: Step response
ler Design Discretization of Lead Compensator - Fast Sample Rate Pole Zero Map.2.15.1.8.7.6.5.4.3.2.1 ZOH Tustin Matched Imaginary Axis.5.9.5.1.15.2.8.82.84.86.88.9.92.94.96.98 1 Real Axis
ler Design Discretization of Lead Compensator - Slow Sample Rate Bode Diagram Step Response Magnitude (db) 2 2 4 1.4 1.2 1 ZOH Tustin Matched Phase (deg) 6 8 45 9 135 18 ZOH Tustin Matched Amplitude.8.6.4.2 225 27 1 1 1 1 1 Frequency (rad/sec) 2 4 6 8 1 12 Time (sec) Figure: Frequency response Figure: Step response
ler Design Discretization of Lead Compensator - Slow Sample Rate Pole Zero Map Imaginary Axis 1.8.6.4.2.8π/T.9π/T π/t π/t.7π/t.6π/t.5π/t.4π/t.1.2.3.4.5.6.7.8.9.3π/t.2π/t ZOH Tustin Matched.1π/T.2.9π/T.1π/T.4.6.8π/T.2π/T.8.7π/T.3π/T.6π/T.4π/T 1.5π/T 1.8.6.4.2.2.4.6.8 1 Real Axis
ler Design Some important things to remember Discretization of compensator Use the method suited for implementation in the system Discrete equivalent of plant Use method corresponding to implementation (usually ZOH) Simulink can combine discrete compensator with continuous plant (digitization of plant not necessary)
ler Design Book: Problem 7.4 Problem 7.5 Problem 7.7