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Control Systems 2 Lecture 13: Digital controller design Roy Smith 2014-5-25 13.1 Digital control system design Sampled-data closed-loop ZOH ZOH equivalence ZOH ZOH 2014-5-25 13.2

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Zero-order hold equivalence — transfer function ZOH "# ZOH Input: )= =0 =0 ,u )=step( step( Output: )= Ts We now sample this, and take the -transform, ZOH )= Ts =(1 2014-5-25 13.3 Zero-order hold equivalence — state space ZOH "# ZOH Integrating over a single sample period ( kT to kT ): kT )= AT kT )+ kT kT kT Bu ZOH -equivalence AT Bd 2014-5-25 13.4

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Zero-order hold equivalence — frequency domain Example: )= (2 (2 +1)( +2) ,T =0 Magnitude T/ ZOH 01 log (rad/sec) 10 /T Phase (deg.) T/ ZOH 180 90 log (rad/sec) 10 /T 2014-5-25 13.5 Digital control system design Sampled-data closed-loop ZOH approximation ZOH 2014-5-25 13.6

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Design approaches ZOH Continuous-time design ZOH -equivalence of and sample/hold Discrete-time design Approximation of with 2014-5-25 13.7 Design approaches ZOH Continuous-time design ZOH -equivalence of and sample/hold Discrete-time design Approximation of with Sampled-data design 2014-5-25 13.8

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Design by approximation 1. Design a continuous-time controller, Verify stability, performance and bandwidth Verify margins and robustness 2. Select a sample-rate, 3. Find approximating 4. Calculate the ZOH -equivalent ZOH 5. Check the stability of the ZOH loop 6. Simulate with (including sample/hold). Verify simulated performance Examine intersample behaviour 2014-5-25 13.9 Controller approximation Approach: approximating the integrators /s If /s ,then, )= AB CD sx AB CD zx 2014-5-25 13.10

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Integration /s )= (0) + kT kT+T x(t) x(kT+T) x(kT) y(kT+T) - y(kT) The signal, ,overasingle second sample period is, kT )= kT )+ kT kT 2014-5-25 13.11 Trapezoidal approximation kT )= kT )+ Tx kT )+( kT kT )) T/ Taking -transforms, )= )+ Tx )+ 1) Approximation: )= +1 The substitution is therefore, +1 kT kT+T x(t) x(kT+T) x(kT) bl (kT+T) bl (kT) This is known as a bilinear (or Tustin) transform. 2014-5-25 13.12

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Frequency mapping Pole locations under bilinear transform: real( bilinear || stable stable. real imag /T /T 1) +1) real imag 2014-5-25 13.13 Bilinear frequency distortion discrete-frequencies: e Frequency mapping: Continuous frequencies, to discrete frequencies, Substitute and into +1 (1 + sin( T/ 2) cos( T/ 2) tan( T/ 2) Frequency distortion: tan T/ 2) 2014-5-25 13.14

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Bilinear frequency distortion tan T/ 2) Continuous frequency ( ): [rad/sec] Discrete frequency ( ): [rad/sec] = Tustin/bilinear transform The line is the sampling mapping. 2014-5-25 13.15 Prewarping 1) +1) (0 /T ,maps real to {| Modifying the frequency distortion Select a frequency Solve for such that )= The “prewarped” transform makes )= at =0 and 1) +1) tan( T/ 2) which implies that: tan( T/ 2) 2014-5-25 13.16

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Prewarping Frequency distortion (bilinear:) tan T/ 2) Frequency distortion (with prewarping): tan -150 -100 -50 50 100 150 -50 50 Continuous frequency: [rad/sec] Discrete frequency: [rad/sec] Bilinear distortion Prewarping distortion Prewarping frequency = 2014-5-25 13.17 Example Compare bilinear and prewarped: =50 rad/sec. 10 10 10 10 Frequency [rad/sec] 10 10 -100 -50 50 100 Frequency [rad/sec] Phase (degrees) Magnitude Prewarping frequency Prewarping frequency C(j C(j Bilinear with prewarping Bilinear with prewarping Bilinear Bilinear C( e C( e C( e C( e 2014-5-25 13.18

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Choosing a prewarping frequency The prewarping frequency must be in the range: /T =2 /T (standard bilinear) corresponds to =0 Possible options for depend on the problem: The cross-over frequency (helps preserve the phase margin); The frequency of a critical notch; The frequency of a critical oscillatory mode. The best choice depends on the most important features in your control design. Remember: stable implies stable. But you must check that (1 + ZOH )) is stable! 2014-5-25 13.19 Example Plant model: )= 5(1 s/z rhp (1 + +2 +2 where =0 ,z rhp =70 =20 =0 05 and =1 IMC desgn ideal )= 3rd order Butterworth ﬁlter with bandwidth: 25 [rad./sec.] )= ideal mp )=( )) 2014-5-25 13.20

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Loopshapes Magnitude 001 01 10 100 log (rad/sec) 10 100 Phase (deg.) 270 180 90 90 log (rad/sec) 10 100 2014-5-25 13.21 Sensitivity and complementary sensitivity Magnitude 10 log (rad/sec) 10 100 2014-5-25 13.22

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Step response Output Amplitude time (sec) 2014-5-25 13.23 Step response Actuation Amplitude time (sec) 2014-5-25 13.24

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Bilinear/Trapezoidal/Tustin transform Bilinear transform Nyquist frequency: 100 radians/second 100 )= +1 Discrete-time analysis ZOH ZOH 2014-5-25 13.25 Loopshapes: bilinear transformed controller Magnitude ZOH 001 01 10 100 log (rad/sec) 10 100 Phase (deg.) ZOH 270 180 90 90 log (rad/sec) 10 100 2014-5-25 13.26

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Sensitivity and complementary sensitivity: bilinear transformed controller Magnitude 10 log (rad/sec) 10 100 2014-5-25 13.27 Step response: bilinear transformed controller Output Amplitude time (sec) 2014-5-25 13.28

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Step response: bilinear transformed controller Actuation Amplitude time (sec) 2014-5-25 13.29 Prewarped Tustin transform Prewarped Tustin transform Nyquist frequency: 100 radians/second 100 Select the prewarping frequency at (20 radians/sec.). )= +1 where, tan( T/ 2) 2014-5-25 13.30

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Loopshapes: prewarped Tustin controller Magnitude ZOH 001 01 10 100 log (rad/sec) 10 100 Phase (deg.) ZOH 270 180 90 90 log (rad/sec) 10 100 2014-5-25 13.31 Sensitivity and complementary sensitivity: prewarped Tustin controller Magnitude 10 log (rad/sec) 10 100 2014-5-25 13.32

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Step response: prewarped Tustin controller Output Amplitude time (sec) 2014-5-25 13.33 Step response: prewarped Tustin controller Actuation Amplitude time (sec) 2014-5-25 13.34

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Sample rate selection Sample rate selection is critical to digital control design. Main considerations Sampling/ZOH will (approximately) introduce a delay of T/ seconds. Anti-aliasing ﬁlters will need to be designed and these will also introduce phase lag. The system runs “open-loop” between samples. Very fast sampling can introduce additional noise. Very fast sampling makes all of the poles appear close to 1. The controller design can become numerically sensitive. 2014-5-25 13.35 Designing for digital implementation Sampled-data implementation ZOH Continuous-time design sT/ 2014-5-25 13.36

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Sensitivity function We want a similar discrete sensitivity function up to the frequency where returns to 1. Magnitude 01 log (rad/sec) 10 )=( sT/ )) 2014-5-25 13.37 Sensitivity function In this example, for 20 rad./sec., 1= /T =20 is about the minimum. Magnitude 01 (rad/sec) 10 20 )=( sT/ )) 2014-5-25 13.38

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Loop-shaping interpretation For where 10 we want ZOH ZOH is the ZOH-equivalent of real imag 1+ 2014-5-25 13.39 Fast sampling Fast sampling period: Control appropriate (slower) sampling period: (typically MT for integer M> ). af as Digital downsampling Digital ﬁlter Fast sampler Anti-aliasing ﬁlter 2014-5-25 13.40

1 Digital control system design Sampleddata closedloop ZOH ZOH equivalence ZOH ZOH 2014525 132 brPage 2br Zeroorder hold equivalence transfer function ZOH ZOH Input 0 0 u step step Output Ts We now sample this and take the transform ZOH Ts 1 201 ID: 23774

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Page 1

Control Systems 2 Lecture 13: Digital controller design Roy Smith 2014-5-25 13.1 Digital control system design Sampled-data closed-loop ZOH ZOH equivalence ZOH ZOH 2014-5-25 13.2

Page 2

Zero-order hold equivalence — transfer function ZOH "# ZOH Input: )= =0 =0 ,u )=step( step( Output: )= Ts We now sample this, and take the -transform, ZOH )= Ts =(1 2014-5-25 13.3 Zero-order hold equivalence — state space ZOH "# ZOH Integrating over a single sample period ( kT to kT ): kT )= AT kT )+ kT kT kT Bu ZOH -equivalence AT Bd 2014-5-25 13.4

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Zero-order hold equivalence — frequency domain Example: )= (2 (2 +1)( +2) ,T =0 Magnitude T/ ZOH 01 log (rad/sec) 10 /T Phase (deg.) T/ ZOH 180 90 log (rad/sec) 10 /T 2014-5-25 13.5 Digital control system design Sampled-data closed-loop ZOH approximation ZOH 2014-5-25 13.6

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Design approaches ZOH Continuous-time design ZOH -equivalence of and sample/hold Discrete-time design Approximation of with 2014-5-25 13.7 Design approaches ZOH Continuous-time design ZOH -equivalence of and sample/hold Discrete-time design Approximation of with Sampled-data design 2014-5-25 13.8

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Design by approximation 1. Design a continuous-time controller, Verify stability, performance and bandwidth Verify margins and robustness 2. Select a sample-rate, 3. Find approximating 4. Calculate the ZOH -equivalent ZOH 5. Check the stability of the ZOH loop 6. Simulate with (including sample/hold). Verify simulated performance Examine intersample behaviour 2014-5-25 13.9 Controller approximation Approach: approximating the integrators /s If /s ,then, )= AB CD sx AB CD zx 2014-5-25 13.10

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Integration /s )= (0) + kT kT+T x(t) x(kT+T) x(kT) y(kT+T) - y(kT) The signal, ,overasingle second sample period is, kT )= kT )+ kT kT 2014-5-25 13.11 Trapezoidal approximation kT )= kT )+ Tx kT )+( kT kT )) T/ Taking -transforms, )= )+ Tx )+ 1) Approximation: )= +1 The substitution is therefore, +1 kT kT+T x(t) x(kT+T) x(kT) bl (kT+T) bl (kT) This is known as a bilinear (or Tustin) transform. 2014-5-25 13.12

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Frequency mapping Pole locations under bilinear transform: real( bilinear || stable stable. real imag /T /T 1) +1) real imag 2014-5-25 13.13 Bilinear frequency distortion discrete-frequencies: e Frequency mapping: Continuous frequencies, to discrete frequencies, Substitute and into +1 (1 + sin( T/ 2) cos( T/ 2) tan( T/ 2) Frequency distortion: tan T/ 2) 2014-5-25 13.14

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Bilinear frequency distortion tan T/ 2) Continuous frequency ( ): [rad/sec] Discrete frequency ( ): [rad/sec] = Tustin/bilinear transform The line is the sampling mapping. 2014-5-25 13.15 Prewarping 1) +1) (0 /T ,maps real to {| Modifying the frequency distortion Select a frequency Solve for such that )= The “prewarped” transform makes )= at =0 and 1) +1) tan( T/ 2) which implies that: tan( T/ 2) 2014-5-25 13.16

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Prewarping Frequency distortion (bilinear:) tan T/ 2) Frequency distortion (with prewarping): tan -150 -100 -50 50 100 150 -50 50 Continuous frequency: [rad/sec] Discrete frequency: [rad/sec] Bilinear distortion Prewarping distortion Prewarping frequency = 2014-5-25 13.17 Example Compare bilinear and prewarped: =50 rad/sec. 10 10 10 10 Frequency [rad/sec] 10 10 -100 -50 50 100 Frequency [rad/sec] Phase (degrees) Magnitude Prewarping frequency Prewarping frequency C(j C(j Bilinear with prewarping Bilinear with prewarping Bilinear Bilinear C( e C( e C( e C( e 2014-5-25 13.18

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Choosing a prewarping frequency The prewarping frequency must be in the range: /T =2 /T (standard bilinear) corresponds to =0 Possible options for depend on the problem: The cross-over frequency (helps preserve the phase margin); The frequency of a critical notch; The frequency of a critical oscillatory mode. The best choice depends on the most important features in your control design. Remember: stable implies stable. But you must check that (1 + ZOH )) is stable! 2014-5-25 13.19 Example Plant model: )= 5(1 s/z rhp (1 + +2 +2 where =0 ,z rhp =70 =20 =0 05 and =1 IMC desgn ideal )= 3rd order Butterworth ﬁlter with bandwidth: 25 [rad./sec.] )= ideal mp )=( )) 2014-5-25 13.20

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Loopshapes Magnitude 001 01 10 100 log (rad/sec) 10 100 Phase (deg.) 270 180 90 90 log (rad/sec) 10 100 2014-5-25 13.21 Sensitivity and complementary sensitivity Magnitude 10 log (rad/sec) 10 100 2014-5-25 13.22

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Step response Output Amplitude time (sec) 2014-5-25 13.23 Step response Actuation Amplitude time (sec) 2014-5-25 13.24

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Bilinear/Trapezoidal/Tustin transform Bilinear transform Nyquist frequency: 100 radians/second 100 )= +1 Discrete-time analysis ZOH ZOH 2014-5-25 13.25 Loopshapes: bilinear transformed controller Magnitude ZOH 001 01 10 100 log (rad/sec) 10 100 Phase (deg.) ZOH 270 180 90 90 log (rad/sec) 10 100 2014-5-25 13.26

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Sensitivity and complementary sensitivity: bilinear transformed controller Magnitude 10 log (rad/sec) 10 100 2014-5-25 13.27 Step response: bilinear transformed controller Output Amplitude time (sec) 2014-5-25 13.28

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Step response: bilinear transformed controller Actuation Amplitude time (sec) 2014-5-25 13.29 Prewarped Tustin transform Prewarped Tustin transform Nyquist frequency: 100 radians/second 100 Select the prewarping frequency at (20 radians/sec.). )= +1 where, tan( T/ 2) 2014-5-25 13.30

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Loopshapes: prewarped Tustin controller Magnitude ZOH 001 01 10 100 log (rad/sec) 10 100 Phase (deg.) ZOH 270 180 90 90 log (rad/sec) 10 100 2014-5-25 13.31 Sensitivity and complementary sensitivity: prewarped Tustin controller Magnitude 10 log (rad/sec) 10 100 2014-5-25 13.32

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Step response: prewarped Tustin controller Output Amplitude time (sec) 2014-5-25 13.33 Step response: prewarped Tustin controller Actuation Amplitude time (sec) 2014-5-25 13.34

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Sample rate selection Sample rate selection is critical to digital control design. Main considerations Sampling/ZOH will (approximately) introduce a delay of T/ seconds. Anti-aliasing ﬁlters will need to be designed and these will also introduce phase lag. The system runs “open-loop” between samples. Very fast sampling can introduce additional noise. Very fast sampling makes all of the poles appear close to 1. The controller design can become numerically sensitive. 2014-5-25 13.35 Designing for digital implementation Sampled-data implementation ZOH Continuous-time design sT/ 2014-5-25 13.36

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Sensitivity function We want a similar discrete sensitivity function up to the frequency where returns to 1. Magnitude 01 log (rad/sec) 10 )=( sT/ )) 2014-5-25 13.37 Sensitivity function In this example, for 20 rad./sec., 1= /T =20 is about the minimum. Magnitude 01 (rad/sec) 10 20 )=( sT/ )) 2014-5-25 13.38

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Loop-shaping interpretation For where 10 we want ZOH ZOH is the ZOH-equivalent of real imag 1+ 2014-5-25 13.39 Fast sampling Fast sampling period: Control appropriate (slower) sampling period: (typically MT for integer M> ). af as Digital downsampling Digital ﬁlter Fast sampler Anti-aliasing ﬁlter 2014-5-25 13.40

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