PI Compensation

2. Chapter 6 The Frequency-Response Design Method PI Compensation • In many problems, it is important to keep the bandwidth low and also to reduce the steady-state error. • For this purpose, a proportional-integral (PI) is useful, with the transfer function • Frequency-Response of PI Compensation

3. Chapter 6 The Frequency-Response Design Method PI Compensation • The desirable aspect of this compensation is the infinite gain at zero frequency, which reduces the steady-state errors. • The cost is a phase decrease at frequencies lower than the break point at ω = 1/TI. • Therefore, TI is usually located at a frequency substantially less than the crossover frequency so that the system’s PM is not affected significantly. • Frequency-Response of PI Compensation

4. Chapter 6 The Frequency-Response Design Method Lag Compensation • The lag compensation approximates PI control. • The convenient way to write the transfer function of the lag compensation in the Bode form is where α is the ratio between the zero/pole break-point frequencies. • The fact that α>1 causes the pole to have a lower break-point frequency than the zero. • This produces low-frequency magnitude increase and phase decrease (lag), clearly seen in the frequency-response plot. • Frequency-Response of Lag Compensation

5. Chapter 6 The Frequency-Response Design Method Lag Compensation • The typical objective of lag-compensation design is to provide additional gain of α in the low-frequency range and to leave the system sufficient PM. • The phase lag is not a useful effect, thus the pole and zero are selected to be at much lower frequencies than the uncompensated system crossover frequency, in order to keep the effect on the PM to a minimum. • In this way, the lag compensator increases the open-loop DC gain, thereby improving the steady-state response characteristics, without changing the transient response characteristics significantly. • Frequency-Response of Lag Compensation

6. Chapter 6 The Frequency-Response Design Method Design Procedure for Lag Compensation Determine the open-loop gain K that will meet the PM requirement without compensation. Draw the Bode plot of the uncompensated system with crossover frequency from Step 1, and evaluate the low-frequency gain. Determine α to meet the low-frequency gain error requirement. Choose the corner frequency ω = 1/T (the zero of the lag compensator) to be one octave to one decade below the new crossover frequency ωc. The other corner frequency (the pole location of the lag compensator) is then ω = 1/αT. Iterate on the design. Adjust compensator parameters (pole, zero, and gain) to meet all the specifications.

7. Chapter 6 The Frequency-Response Design Method First Design Using Lag Compensation Find a compensation for G(s)=1/[s(s+1)], this time using lag compensation. Fix the low-frequency gain in order to meet the error requirement of Kv = 10; then use the lag compensation to meet the PM requirement of 45°. • Kαmust be greater than 10, so we pick K = 10. • The frequency response of KG(s)=10/[s(s+1)] shows PM=20° at ωc=3 rad/s. • The task: select the lag compensation break points so that the crossover frequency is lowered and more favorable PM is obtained. (Gain is not an option, since it must be 10, as required).

8. Chapter 6 The Frequency-Response Design Method First Design Using Lead Compensation • Choose T = 10, α = 10, thus ωz = 0.1 rad/s and ωp = 0.01 rad/s. • Examine the resulting PM, it is now becoming 50° (if checked in Matlab, it is actually 45°), which satisfies the specifications.

9. Chapter 6 The Frequency-Response Design Method PID Compensation • For problems that need PM improvement at ωc and low-frequency gain improvement, it is effective to use both derivative and integral control. • By combining the respective previous equation, we obtain a PID control, with the transfer function • This compensation is roughly equivalent to combining lead and lag compensators in the same design. • Hence, it can provide simultaneous improvement in transient and steady-state responses. • Frequency-Response of PID Compensation

10. Chapter 6 The Frequency-Response Design Method End of the Lecture