Feedback Control Roundup

1. Feedback Control Roundup

2. Where is this class going? • Why control? • What is feedback? • What are those block diagrams, and how do they relate to the equations? • How does PID work? • How do controllers go out of tune, how can I recognize them, and what do I do?

3. Where is this class going? • Things I want you to learn • “Assume steady state” does not work in real life. • Engineering is a synthesis of what you have learned, not a collection of separate topics • How to analyze and describe a dynamic system • How to understand control systems: • Anything called a “regulator” • Trade-offs in control design • How to save the day when the irresponsible control engineer goes on vacation

4. Where is this class going? • Content remaining • Control applications: feedforward, cascade, decoupling • Ways to analyze the effect of control on closed loop stability • Frequency domain analysis of dynamic systems • State space (very good for mechanical systems)

5. Why control? • We use control to: • Keep a value close to its desired value: • Room temperature, thermostat • Car speed, cruise control • Boiler drum level • Boiler fire-side pressure • Trade off variation between controlled and manipulated variables • Torque and position or speed • Flow and level in a surge tank • Temperature and heat stresses induced by heat

6. Where are we, as mechanical engineers, going to encounter control? • Any time something moves, that motion must be regulated. • Equipment: motors and guides • Fluids: flow rate, pressure and temperature • Control is implemented through mechanical equipment • Motors, valves, dampers • Limitations or faults in this equipment will affect control (and thus system) performance.

7. Disturbances - wind, slope, bad fuel Speed Error in km/h Actual Throttle Position Desired Throttle Position Desired speed in km/h Engine, Transmission, Wheels Cruise Control Throttle Actuator + - Speed in km/h Speed Sensor What is Feedback? • The controlled variable is measured. • It is then compared to a desired value, or setpoint. • The difference between setpoint and measurement is called the error. • The controller calculates a control output depending on the error. • The control output is sent to some physical equipment, a valve or similar, to affect the process. This physical equipment is called the Final Control Element. • The process responds to the change made by the final control element. • or 1. The process response is measured Speed

8. What are those block diagrams? • Block diagrams are a visual representation of the actual dynamic system and its governing equations. • Each line is a signal - something that varies over time, and can be measured or calculated. • Each block is a “functional element” - something that does something, and is governed by differential, integral or algebraic equations, and describable as a transfer function.

9. Disturbances - wind, slope, bad fuel Speed Error in km/h Actual Throttle Position Desired Throttle Position Desired speed in km/h Engine, Transmission, Wheels Speed Cruise Control Throttle Actuator + - Speed Sensor Speed in km/h Block Diagram

10. Standard Nomenclature • PV, or y : controlled variable • SP or ysp : setpoint, desired value • OP or u (text uses p): controller output • e : error, ysp - y • d or l : disturbance or load (wild) variable • ym : measured value of y • G : a transfer function • Gp : process transfer function • Gc : controller transfer function • Gv : final control element (valve) transfer function • Gm : measurement element transfer function • Gd or GL : Disturbance (Load) transfer function • Gcl : closed loop transfer function • K : transfer function gain •  : time constant •  : time delay (dead time) •  : damping ratio (also called damping factor)

11. Block Diagram Disturbances - wind, slope, bad fuel d The purpose of the C.L.T.F. is to describe the entire system with a single transfer function that can be analyzed. Gd or GL Disturbance Process Speed Error in km/h Actual Throttle Position Desired Throttle Position Desired speed in km/h Gp Engine, Transmission, Wheels Speed Gc Cruise Control Gv Throttle Actuator y + ysp u e - ym Gm Speed Sensor Speed in km/h

12. Block Diagram Disturbances - wind, slope, bad fuel d Closed Loop System Gd or GL Disturbance Process Closed Loop Disturbance Transfer Function Speed Error in km/h Actual Throttle Position Desired Throttle Position Desired speed in km/h Closed Loop Setpoint Transfer Function Engine, Transmission, Wheels Gp Speed Cruise Control Gc Throttle Actuator Gv y + ysp u e - ym Gm Speed Sensor Speed in km/h

13. Block Diagram • y = Gd * d + Gp * Gv * u • = Gd * d + Gp * Gv * Gc * e • = Gd * d + Gp * Gv * Gc * (ysp - ym) • = Gd * d + Gp * Gv * Gc * (ysp - Gm * y) • = Gd * d + Gp * Gv * Gc * ysp - Gp * Gv * Gc * Gm * y • y + Gp * Gv * Gc * Gm * y = Gd * d + Gp * Gv * Gc * ysp • y ( 1 + Gp * Gv * Gc * Gm) = Gd * d + Gp * Gv * Gc * ysp • y = (Gd/(1 + Gp Gv Gc Gm)) * d + ( Gp Gv Gv/(1 + Gp Gv Gc Gm)) * ysp • These rules are applied around the block diagram: • y = Gd * d + Gp * Gv * u • u = Gc * e • e = ysp - ym • ym = Gm * y • The block diagram is used to figure out the governing equations directly. • Each transfer function is multiplied by its input to get its output: • u = Gc * e • ym = Gm * y • Summers (circle with x inside) add inputs, or, if the input is marked with a - sign, that input is subtracted • e = ysp - ym Disturbances - wind, slope, bad fuel d Gd or GL Speed Error in km/h Actual Throttle Position Desired Throttle Position Desired speed in km/h Speed Engine, Transmission, Wheels Gp Gc Cruise Control Throttle Actuator Gv y + ysp u e - ym Gm Speed in km/h

14. Block Diagram Disturbances - wind, slope, bad fuel d Closed Loop System Closed Loop Disturbance Transfer Function Gd 1 + Gp Gv Gc Gm Gd or GL Speed Error in km/h Actual Throttle Position Desired Throttle Position Desired speed in km/h Speed Closed Loop Setpoint Transfer Function Gp Gv Gc 1 + Gp Gv Gc Gm Gp Engine, Transmission, Wheels Cruise Control Gc Throttle Actuator Gv y + ysp u e - ym Gm Speed in km/h

15. How does PID work? • PID control acts on the error. • The control action is the sum of three terms: • Proportional: Kc * the error now • Integral: Kc/TauI * the integral of error since the controller was turned on • Derivative: Kc*TauD * the rate of change of the error now

16. P animation

17. PI animation

18. Animation

19. Comparison

20. Comments • P never reaches setpoint. • PID responds before PI does because of the Derivative term.