Automatic Control (TSRT15): Lecture 7

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1 Automatic Control (TSRT15): Lecture 7 Tianshi Chen Division of Automatic Control Dept. of Electrical Engineering Phone: Office: B-house extrance 25-27

2 Outline 2 Feedforward compensation Connection between the Bode plot of the loop-gain and the stability of the closed-loop system

3 Feedforward compensation 3 F 2 (s) R(s) Σ E(s) F 1 (s) Σ U(s) G(s) Y(s) -1 More than one controller can be used. The feedforward component F 2 (s) can be used to add or cancel the poles and zeros of the closed-loop system transfer function. The loop-gain 1+F 1 (s)g(s) is still crucial for the stability and dynamics of the closed-loop system.

4 House heating system 4 Modeling (get the ODE upon some physical laws) T I TR T O T O : Outdoor temperature T I : Indoor temperature T R : Radiator temperature u : Control signal to radiator Heat balance for the room Heating of radiator

5 House heating system 5 Transfer function and block diagram T O G 2 (s) U(s) G 3 (s) T R Σ G 1 (s) T I

6 House heating system 6 Feedback control system with PID-controller T O PID G 2 (s) R(s) Σ F(s) U(s) G 3 (s) T R Σ G 1 (s) T I -1 We test a PID-controller and simulate to see what would happen when the reference temperature is 22º and the outdoor temperature varies between 0º and 10º.

7 House heating system 7 Indoor temperature Outdoor temperature Question: Can we do better if we can measure the outdoor temperature?

8 Disturbance rejection using feedforward compensation 8 Add a feedforward controller that makes use of the measurable outdoor temperature. T O H(s) G 2 (s) R(s) Σ F(s) Σ T R G 3 (s) Σ G 1 (s) T I -1

9 Disturbance rejection using feedforward compensation 9 To eliminate the effect of the outdoor temperature, we design H(s) so that Ideally, we need In practice, this choice is however not possible. Similar to the D-part in the PID controller, H(s) is not proper, i.e. not physically realizable. As a remedy, we pick an H(s) making the term small in the frequency regions where the disturbance is large (in our case, the disturbance has a period of 24 hours).

10 Disturbance rejection using feedforward compensation 10 Indoor temperature Outside temperature The outdoor temperature is almost completely rejected.

11 Feedforward compensation 11 Besides the new method to compensate for measurable disturbances we note an important fact The loop-gain G 1 G 3 F appears in both transfer functions. Despite the advanced controller structure, the loop-gain is still very important for stability and dynamics of the closed-loop system. Conclusion: The loop-gain is crucial for control design.

12 Stability analysis with Bode plot 12 All feedback control systems can be put into R(s) Σ G O (s) Y(s) -1 G O (s) is the (equivalent) loop gain. Question: If we know the Bode plot of the loop-gain G O (s), what can then be said about the stability of the closed-loop system.

13 Stability analysis with Bode plot 13 Given a bode plot for some loop-gain What happens if we close the loop?

14 Stability analysis with Bode plot 14 Drive the system initially with a sinusoidal. After transient components fades away, we have G O (s) Set the frequency ω=1 rad/s and thus we obtain the output Asin(t) Now disconnect the external signal and quickly close the loop G O (s) Nothing happens! The signal Asin(t) is still on the input. -1

15 Stability analysis with Bode plot 15 The case when the phase is -180º and the gain is 1 puts us precisely on the margin to stability. The signal neither increases or decreases in amplitude when it goes around in the loop A phase is -180º and a gain >1 would lead to an increasing amplitude when the signal goes around in the loop, thus indicating instability of the closed-loop system A phase is -180º and a gain <1 would lead to an decreasing amplitude when the signal goes around in the loop, thus indicating stability of the closed-loop system

16 Stability analysis with Bode plot 16

17 Stability analysis with Bode plot 17 Gain crossover frequency (ω g ): frequency at which the magnitude gain is 1. Phase crossover frequency (ω p ): frequency at which the phase is -180º. Gain margin (A m ): the opposite of the magnitude gain at ω p. Phase margin (φ m ): the phase at ω g plus 180º If G O (s) is stable and minimum phase (all zeros are stable), the closed-loop system is stable if A m >0 and φ m >0.

18 Stability analysis with Bode plot 18 The gain margin tells us how much we can increase the gain in the loop-gain without causing instability in the closed-loop system. The phase margin tells us how much we can decrease the phase in the loop-gain without causing instability in the closed-loop system.

19 House heating system 19 Σ F(s) G 3 (s) G 1 (s) -1 Consider the bode plot of the loop-gain G O (s)=f(s)g 3 (s)g 1 (s) Assume α 1 =0.1, α 2 =0.01 and α 3 =50. Let s first try F(s) = K I /s

20 House heating system: I-controller 20 K I =1

21 House heating system: I-controller 21 K I =55.1

22 House heating system: PI-controller 22 K P =2, K I =1

23 More connections 23 It turns out that the Bode plot of the closed-loop system has connections with the Bode plot of the loop-gain too. The bandwidth of the closed-loop system will typically be the gain crossover frequency,. A closed-loop system responds faster if the gain crossover frequency of the loop-gain is higher. The resonant peak M p in the closed-loop Bode plot depends on the phase margin of the loop-gain: The closed-loop system has strong oscillation if the phase margin of the loop-gain is small.

24 House heating system 24 Bode plot of the closed-loop system Step response of the closedloop system

25 Summary of this lecture 25 Measurable disturbances can be rejected by adding a feedforward controller. The loop-gain is a crucial component for stability and dynamics of the closed-loop system, no matter how advanced the controller structure is. Note the connections between the Bode plot of the closed-loop system and the Bode plot of the loop-gain.

26 Summary of this lecture 26 Important concepts Gain margin: The gain margin tells us how much we can increase the gain in the loop-gain without causing instability in the closed-loop system. Phase margin: The phase margin tells us how much we can decrease the phase in the loop-gain without causing instability in the closedloop system.

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