Lecture 10. Lab next week: Agenda: Control design fundamentals. Proportional Control Proportional-Integral Control

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1 264 Lab next week: Lecture 10 Lab 17: Proportional Control Lab 18: Proportional-Integral Control (1/2) Agenda: Control design fundamentals Objectives (Tracking, disturbance/noise rejection, robustness) Feedback control Proportional Control Proportional-Integral Control

2 265 Control Design - Objectives Y(s) = G(s) U(s) Overall goal: choose U(s) so that Y(s) behaves as desired 1. Stability closed loop system poles on left half of complex plane 2. Performance how well does Y(s) follow command 3. Disturbance rejection not affected by disturbance 4. Immunity to measurement noise not affected by sensor inaccuracies 5. Robustness not affected by uncertainty in system model G(s)

3 266 Pros and Cons of Feedback Feedback versus dead reckoning Pros: Compensates disturbances Compensates model uncertainty Performance without needing tight tolerance Cons: Potential for instability Subject Noise / inaccurate sensing Need a sensor Sensor

4 267 Simplest feedback controller Proportional Control Consider 1 st order plant (note not all plants are first order!!!!) Formulate closed loop transfer function Use closed loop transfer function to analyze Stability how design parameters affect stability Performance time domain and frequency domain Disturbance rejection and noise immunity Input disturbance, output disturbance Robustness

5 268 Stability Closed loop poles must be on the left-half of the complex plane In theory, 1 st order system can be stable for very high gains In practice, very high gains can destabilize system Model uncertainty High order dynamics ignored (e.g. system is really 3 rd order instead of 1 st order) Saturation and other nonlinearity Try very high gain in experiments!

6 269 Tracking Find G YR (s) How does P gain affect: Steady state response (use final value theorem or D.C. gain) Transient response to step input Frequency response for what s=jw is G YR (s) close to 1 System bandwidth - what kind of input frequencies can it track?

7 270 Disturbance Rejection Find G YD (s) How does P gain affect: Steady state response (use final value theorem or D.C. gain) to step disturbance? What kind of disturbances (frequency content) can it reject? System bandwidth

8 271 Noise Immunity Find G YN (s) Note relationship between G YN (s) and G YR (s) How does P-gain affect immunity to noise Noise frequencies that it is not sensitive to?

9 272 Response to R(s), D(s) and N(s) Consider a system with a proportional control subject to: Command input, R(s) (Input) disturbance, D(s) Sensor noise, N(s) Work out response: Y(s) = G YR (s) R(s) + G YD D(s) D(s) + G YN (s) N(s) Performance objective: G YR (s) = 1 Disturbance rejection objective: G YD (s) = 0 Noise immunity objective: G YN (s) = 0 Note cannot have G YR (s) = 1 and G YN (s) = 0 simultaneously Fortunately, R(s) typically low frequency, N(s) typically high frequency

10 Question: If system model G(s) is off by say, 10%, how will much will the closed loop transfer function G c (s) be off? G(s)! G(s)+dG(s) Robustness G c (s)! G c (s)+dg c (s) 273 Method: dg c (s) G c (s) G c (s) = Find in terms of G(s)C(s) 1+G(s)C(s) dg(s) G(s) dg c (s) G c (s) = apple dgc (s) dg(s) G(s) G c (s) dg(s) G(s)

11 274 Robustness (2) dg c (s) G c (s) = 1 1+G(s)C(s) dg(s) G(s) S(s) * 10% Sensitivity function S(s) Say 10% If S(s) << 1, e.g. by setting C(s) larger, the closed loop system can be very insensitive to plant uncertainty Attenuation is frequency (s= jw) dependent Do this for system with P-controller How does P gain affect robustness?

12 275 P-Control Summary P control allows tracking of low frequency signals All aspects of response (except for noise) improves as P gain increases Transient response Bandwidth Disturbance rejection Steady state error For integrator plant, steady error to step command is 0 but not so for other 1 st order system Increasing P increases influence of noise

13 276 Proportional-Integral Control Proportional control does not eliminate steady state error due to constant disturbance or ramp input, unless gain is infinite Non-zero control effort ONLY if there is error Proportional control increases gain at all frequencies Needed for low frequency Bad for high frequency Approach: estimate the disturbance and then compensate for it!!! PI controller: C(s) = K p + K I /s Infinite gain at s=0 and decreasing gain at high frequency! Can have non-zero u(t) control effort even when error = 0

14 277 Pole locations: Stability and Response Two poles Left half plane for stability Design pole locations for good transient response Calculate K p, and K I to obtain desired pole locations Instead of tuning gains directly Descriptors: Natural frequency, wn Damped natural frequency, wr Time constant (1/real part of pole) Damping ratio Underdamped, overdamped, critically damped

15 278 Steady state response to Step and ramp input Step disturbance How do gains affect these? Performance Transient behavior how do gains affect Convergence time Oscillation Resonance At certain frequency, output amplitude is maximum for a given input amplitude

16 279 P-I Control Summary P-I can eliminate steady error due to constant disturbance or input (ramp for integrator plant) Interpretation: Increases gain for D.C. and decreases gain for high frequency Estimates the constant disturbance (or required input) Design pole locations for good performance and calculate gains to obtain the pole locations Note difference in response to 1 st order system

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