Control Theory Bode Stability Criterion

Other view on stability of CL Where the PHASE of the open loop TF equals -180°(+/-n.360°), we have positive feedback. If the AMPLITUDE RATIO at these frequencies > 0db: unstable closed loop.

Two important measures 1.GAIN MARGIN = How much dB of amplitude ratio we can still add in the open loop before the amplitude ratio goes above 0dB at a frequency where the phase crosses -180° 2. PHASE MARGIN = ?

Phase Margin = A.How much the phase can still be increased before it reaches 0° at a frequency where the amplitude ratio is 0dB. B.How much the phase can still be decreased before it reaches -180° at a frequency where the amplitude ratio is 0dB. C.None of the above makes sense.

Phase Margin = A.How much the phase can still be increased before it reaches 0° at a frequency where the amplitude ratio is 0dB. B.How much the phase can still be decreased before it reaches -180° at a frequency where the amplitude ratio is 0dB. C.None of the above makes sense.

rad/s A.GM = 50 dB, PM = 40° B.GM = 50 dB, PM = 90° C.GM = 30 dB, PM = 40° D.GM = 30 dB, PM = 90° E.None of the above

Given the previous Bode plot of the OPEN LOOP, A.GM = 50 dB, PM = 40° B.GM = 50 dB, PM = 90° C.GM = 30 dB, PM = 40° D.GM = 30 dB, PM = 90° E.None of the above

1) What is a time delay? Typical example: measurement comes too late, e.g.: TRTR T R,m TRTR tdtd Influence of a time delay

Relationship between T R,m (t) and T R (t)? A.T R,m (t) = T R (t) B.T R,m (t) = T R (t-t d ) C.T R,m (t) = T R (t+t d ) D.None of the above TRTR T R,m tdtd

Relationship between T R,m (t) and T R (t)? A.T R,m (t) = T R (t) B.T R,m (t) = T R (t-t d ) C.T R,m (t) = T R (t+t d ) D.None of the above TRTR T R,m tdtd

Which is true? A.A time delay = a pure phase shift, B is false B.A time delay = a non-linear subsystem, A is false C.Both are true D.Both are wrong

Which is true? A.A time delay = a pure phase shift, B is false B.A time delay = a non-linear subsystem, A is false C.Both are true D.Both are wrong

In other words: Dead time t d f(t) f(t-t d ) This is of course a pure phase shift. In Laplace (see table): Dead time t d F(s) e -t d s F(s) A dead time is given as e -t d s in the s domain: it’s non-linear! On the Bodeplot? AR = |e -t d jω |=1 φ =  (e -t d jω ) = -t d ω (in rad) Pure phase shift Influence of a time delay

A- Bode plot of a time delay: 2) What is the influence on a feedback system? Influence of dead time CHAPTER 3. PID CONTROL

φ =  (e -t d jω ) = -t d ω * 180/π (in degrees) This can have severe impact on the stability: The information comes too late. How can we see this in our analysis? - Based on TF: difficult: time delay = non-linear thing - … but it is a pure phase shift… on Bode plot? Influence of a time delay 2) What is the influence on a feedback system?

On the phase margin The bigger the phase margin, the less overshoot in the closed loop. First approximation: The “damping ratio” of the closed loop = PM in degrees / 100 Example: How big do you think the overshoot will be if the open loop TF is

The estimated overshoot is A.ca. 15% B.ca. 30% C.ca. 45% D.ca. 60%

A.ca. 15% B.ca. 30% C.ca. 45% D.ca. 60%

The estimated overshoot is A.ca. 15% B.ca. 30% C.ca. 45% D.ca. 60%

A.ca. 15% B.ca. 30% C.ca. 45% D.ca. 60%

What about PI

We can now state that the “disadvantage” of the I action is A.that it increases the OL gain at low frequencies B.that it increases the OL gain at high frequencies C.that it decreases the OL phase at low frequencies D.that it both decreases the OL phase and increases the OL gain at low frequencies

We can now state that the “disadvantage” of the I action is A.that it increases the OL gain at low frequencies B.that it increases the OL gain at high frequencies C.that it decreases the OL phase at low frequencies D.that it both decreases the OL phase and increases the OL gain at low frequencies

We can use the stability criterion to design controllers as well GROUP TASK 1: A second order process with gain 2, damping ratio 0.5 and natural eigenfrequency 20 rad/s is controlled with a P controller. The time delay in the loop is 0.01s. What is the maximally allowed control gain - in order for the CL to be stable - in order for the overshoot to be smaller than 50%?

Group Task II Drug-induced anasthesia Reaction of the patient’s arterial blood pressure to a drug may vary. Therefore a closed loop system is used. However: 2(s+5) 2e -sT /s 2/(s+2) Remark: What kind of control? Why? a)What is the maximum time delay of the body’s response before the system will become unstable? b)Determine the PM and the GM when T=0.05s? When T=0.1s? c)What is the influence of T on the step response? Desired pressure Blood pressure Amount of drug supplied to the patient BodyController Sensor

Open loop Bode plot

Exercise: Drug-induced anasthesia a) Maximum T? ω PM = 8.94 rad/s: PM = 73.4° Zonder de dode tijd! T max = s Use of the Bode plot in control

Exercise: Drug-induced anasthesia: b) PM and GM when T=0.05s? When T=0.1s? PM = 73.4° ω PM = 8.94 rad/s Zonder de dode tijd! A- Without time delay: B- Influence of T: T = 0.05s: -25.6° T = 0.1s: -51.2° Use of the Bode plot in control

c) Influence on the step response? Exercise: Drug-induced anasthesia: Use of the Bode plot in control