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: Bhouse extrance 2527
2 Outline 2 Feedforward compensation Connection between the Bode plot of the loopgain and the stability of the closedloop 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 closedloop system transfer function. The loopgain 1+F 1 (s)g(s) is still crucial for the stability and dynamics of the closedloop 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 PIDcontroller 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 PIDcontroller 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 Dpart 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 loopgain G 1 G 3 F appears in both transfer functions. Despite the advanced controller structure, the loopgain is still very important for stability and dynamics of the closedloop system. Conclusion: The loopgain 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 loopgain G O (s), what can then be said about the stability of the closedloop system.
13 Stability analysis with Bode plot 13 Given a bode plot for some loopgain 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 closedloop 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 closedloop 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 closedloop 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 loopgain without causing instability in the closedloop system. The phase margin tells us how much we can decrease the phase in the loopgain without causing instability in the closedloop system.
19 House heating system 19 Σ F(s) G 3 (s) G 1 (s) 1 Consider the bode plot of the loopgain 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: Icontroller 20 K I =1
21 House heating system: Icontroller 21 K I =55.1
22 House heating system: PIcontroller 22 K P =2, K I =1
23 More connections 23 It turns out that the Bode plot of the closedloop system has connections with the Bode plot of the loopgain too. The bandwidth of the closedloop system will typically be the gain crossover frequency,. A closedloop system responds faster if the gain crossover frequency of the loopgain is higher. The resonant peak M p in the closedloop Bode plot depends on the phase margin of the loopgain: The closedloop system has strong oscillation if the phase margin of the loopgain is small.
24 House heating system 24 Bode plot of the closedloop system Step response of the closedloop system
25 Summary of this lecture 25 Measurable disturbances can be rejected by adding a feedforward controller. The loopgain is a crucial component for stability and dynamics of the closedloop system, no matter how advanced the controller structure is. Note the connections between the Bode plot of the closedloop system and the Bode plot of the loopgain.
26 Summary of this lecture 26 Important concepts Gain margin: The gain margin tells us how much we can increase the gain in the loopgain without causing instability in the closedloop system. Phase margin: The phase margin tells us how much we can decrease the phase in the loopgain without causing instability in the closedloop system.
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