Välkomna till TSRT15 Reglerteknik Föreläsning 8

Transcription

1 Välkomna till TSRT15 Reglerteknik Föreläsning 8 Summary of lecture 7 More Bode plot computations Lead-lag design Unstable zeros - frequency plane interpretation

2 Summary of last lecture 2 W(s) H(s) R(s) Σ F(s) Σ G(s) Y(s) Feedforward compensation: Add a feedforward term H(s) to reduce the impact of the measurable disturbance W(s) on the output Y(s). In other words, let the input signal depend on both the control error and the measurable disturbance -1

3 Summary of last lecture 3 R(s) Σ G O (s) Y(s) -1 Stability: Assume G O (s) stable The feedback system is stable if the gain G O (iω) is smaller than 1 when the phase arg(g O (iω)) is -180º. Alternatively, stable if the phase is larger than -180º when the gain is 1.

4 Summary of last lecture 4

5 5 Σ F(s) -1 Question today: How can we design the controller F(s) to satisfy specifications on the closed-loop system, given a Bode plot of the initial loop-gain

6 6 We will use the methodology presented today to design a controller for the lateral positioning of a crane Input: Force applied to the rolling cart Output: Lateral position of load Tricky control problem due to pendulum dynamics and cable flexibility

7 7 Hard to model mathematically (due to the flexible cables) Instead, experiments have been carried out with sinusoidal input, and generated a Bode plot for the system According to our stability criteria, a P-controller is stabilizing if the gain is less than 2.77 (since the amplitude gain in the loopgain then is 1 when phase is -180º)

8 8 Step-response with P- controller using K=2 Too large overshoot Can we design a better controller in a structured way by using information in the open-loop Bode plot?

9 9 Typical specifications: The closed-loop system is typically specified in terms of bandwidth (relates to speed), resonance peak (relates to overshoot) and stationary control error Bandwidth: Related to crossover frequency Resonance peak: Related to phase margin Stationary control error: determined by loop-gain in ω=0, e.g.

10 10 Synthesis method: Design F(s) to place the crossover frequency (i.e. F(iω)G(iω) =1 at desired bandwidth), obtain a sufficiently large phase margin in this frequency, and obtain a sufficiently large gain in the frequency 0 to achieve specified control error In other words, given the Bode plot G(s), design F(s) to ensure that F(s)G(s) satisfies loop-gain specifications Naive requirement: High phase (>>-180º) everywhere, large gain in all frequencies up until bandwidth, then very low gain

11 11 What does the Bode plot look like for two systems in series, given the Bode plot of the two separate systems G 1 G 2 Additive! (this is the basis for our method to draw Bode plot approximately)

12 12 Example: Series connection with (stable) zero G 1 G 2

13 13 Does not influence gain significantly up to roughly ω=a, then followed by an increase of 20dB per decade The phase is increased in all frequencies and goes towards an increase of 90º

14 14 Example: Series connection with (stable) pole G 1 G 2

15 15 Can be used to give extra gain up to roughly ω=a (and then the gain drops by 20dB per decade) The phase is decreased in all frequencies and goes towards a decrease of 90º

16 16 Example: Series connection with pole-zero, 0<b<a G 1 G 2

17 17 Gives extra gain from ω=b and above, but does increase it significantly after ω=a Phase increases everywhere, largest increase between a and b

18 18 Example: Series connection with pole-zero, 0<a<b G 1 G 2

19 19 Gives extra gain from ω=b and below, but does increase it significantly more before ω=a Phase decreases everywhere, largest decrease between a and b

20 20 Pole-zero with 0<a<b is called phase-advancing Pole-zero with a>b>0 is called phase-retarding Design method: 1. Design a phase-advancing controller to obtain a sufficiently high phase margin in desired crossover frequency 2. Append (in series) a phase-retarding controller in order to obtain desired low-frequency gain (if needed)

21 21 Phase-advance: The phase-advancing part (called the lead-link) is often written in the following form When is the phase-advance the highest? (the phase-advancement should be highest in the desired crossover frequency) At most in the frequency

22 22 Methodology for the phase-advance link (lead-link) 1. Pick a desired crossover frequency (based on the desired closed-loop bandwidth) 2. How much phase is missing for desired phase margin (which comes from specified resonance peak) 3. Compute β to obtain a corresponding maximum phaseadvance 4. Compute K and τ D to actually place the crossover frequency in the desired crossover frequency used when computing β 5. Evaluate closed-loop, adjust and perhaps iterate

23 23 Back to the crane: Desired bandwidth: 1 rad/s (i.e. desired crossover frequency ω C,desired =1) Desired phase margin: 50º (should give reasonably small resonance peak) Current phase in desired crossover frequency: -143º Required phase-advance:13º

24 24 Required phase-advance:13º This maximum phase-advance should occur at the desired crossover frequency ω = 1 The loop-gain amplitude gain should be 1 at this frequency

25 25

26 26 Phase-retardation: The phase-retarding part (lag-link) is often written in the following form (i.e. zero in 1/τ I and pole in γ/τ I ) The purpose of the phase-retarding link is to increase the amplitude gain at the frequency 0 to obtain a smaller stationary control error The gain is increased with 1/γ Unfortunately, the lag-link decreases the phase slightly, which counter-acts the lead-link which was designed to increase the phase in order to improve the phase margin

27 27 Methodology for the phase-retardation link (lag-link) 1. Pick a desired stationary control error (i.e. e 0 ) 2. Compute γ to achieve (e.g.) 1/(1+ F lead (0)G(0) /γ) e 0 (standard final value theorem calculations) 3. Let τ I be 10/ω C,desired (rule of thumb ) The rule of thumb in step 3 gives a phase-loss from the laglink of roughly 6º in the designed crossover frequency. If you know you will add a lag-link when you design the lead-link, it is therefore reasonable to add an extra phase margin of 6º in the lead-design.

28 28 Complete controller E(s) U(s) Note: With γ=0 the lag-link can be interpreted as a PI-controller The lead-link can be interpreted as an approximate PD-controller (pole on the differentiation operator)

29 Non-minimum phase systems 29 We can now understand why it is hard to control systems with unstable zeros Same gain! Lower phase!

30 Summary 30 Summary of todays lecture In frequency based compensation, we try to form the gain and phase of the loop gain F(s)G(s) by designing F(s) based on the Bode plot of G(s) Specifications on bandwidth and resonance peak on the closed-loop system are translated to specifications on crossover frequency and phase margin on the loop-gain. These specifications are translated to parameters in a lead-link Specifications on stationary control error are translated to the static gain of the loop-gain. This requirement is translated to parameters in the laglink Unstable zeros are tricky to control since they decrease the phase, thus reducing the phase margin.

31 Summary 31 Important concepts Compensation: Another name for control, but with an emphasis on the frequency plane approach (compensate for phase loss etc) Link: Another name for a part of a control structure (typically coupled in series) Lead-link: The part of the controller which is used to adjust the crossover frequency and phase margin Lead-link: The part of the controller which is used to adjust the stationary control error Non-minimum phase: System with unstable zero. Gives same amplitude gain but lower phase compared to a similar system with a mirrored stable zero