Outline

Introduction to Control (034040) Course info lecture no. 1

Leonid Mirkin Introduction Faculty of Mechanical Engineering Technion—IIT

Block diagrams

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General info Topics Course web site: http://leo.technion.ac.il/Courses/IC/ 1. Block diagrams Prerequisite (must): Linear Systems M (034032) 2. Modeling, DC motors Musts from LS: 3. Signals and systems in frequency domain − linearization − Laplace transform and signals in s-domain, initial and final value theorems, etc 4. Steady-state and transient requirements − transfer functions and their properties (poles, zeros, static gain, etc) 5. Open-loop control, plant inversion − stability and Routh test 6. Closed-loop control − frequency response and frequency-response plots (Bode and polar) − 1st and 2nd order dynamical systems (definitions, step/frequency responses, etc) 7. Internal stability 8. Root locus method Grading policy: Project with lab parts (provided the final exam grade 55): 30% 9. Nyquist stability criterion − ≥ Final exam: 70–100% 10. Basic loop shaping and stability margins − The use of books / lecture notes is not permitted during exams. 11. PID controllers and their tuning

3/24 4/24 Outline What is “control”? Roughly, control is the discipline studying how to change behavior of systems − Course info (mechanical, electrical, chemical, biological, economical, etc) so that they behave in a desirable manner. −

Introduction Control is relatively young field − (as separate discipline—since ’40s), yet control systems Block diagrams appear nowadays practically everywhere, − in homes, in industry, in communications systems, in vehicles, in scientific instruments, etc. Nature also offers a plenty of examples of sophisticated and successful control systems (biological systems, ecosystems, etc).

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Example 1: steam : centrifugal

Boiler Steam Steam Metal sphere Valve

Governor

Output Output shaft Engine shaft Engine

Problem: maintain rotation speed at a constant predetermined level Solution adapted1 by in 1788 did the trick: if properly tuned, Actuator: valve can regulate the admission of steam into cylinders the governor maintains a near constant speed − Obstacles: unpredictable changes (uncertainties), like whatever the load or fuel supply conditions are. − load − steam pressure − engine / valve aging − ... 1It was not Watt’s invention: centrifugal governors were used to regulate the distance and pressure between millstones in windmills since the 17th century. 7/24 8/24 Centrifugal governor: idea Steam engine: the neverending story

The first automatic control system used in industrial processes − Watt’s governor tamed steam engine and made the Industrial Revolution possible

Tuning centrifugal governors turned out to be not simple at all − Too enthusiastic governor could cause oscillatory motion of steam engine (hunting phenomenon).

This catalyzed the development of rigorous theory (by J. C. Maxwell) − The problem reduces to the question about stability of differential equations

When rotation speed increases, Verifying stability turned out to be not quite simple − balls move outwards, reducing the valve aperture and steam admission − This catalyzed further development of mathematical tools (by Routh, When rotation speed falls, − Hurwitz, et alii) balls move inwards, increasing the valve aperture and steam admission − ... This operation principle called control. −

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Example 2: blood glucose regulation Blood glucose regulation mechanisms Glucose, transported via the bloodstream, is Glucose “stocked” in the form of glycogen, which is produced, stored, and the primary source of energy for body’s cells. cleaved in the liver. − Its When blood glucose levels are in excess, glucose can be stored in the − liver as glycogen (glycogenesis). concentration must be kept within a narrow level around 90 mg/dL − When glucose levels in the blood stream are low, the glycogen can be (normally, 70–120 mg/dL, might be higher after meals). Failure to maintain − catabolised and glucose may re-enter the blood stream. this range consistently (hyperglycemia2 or hypoglycemia) is very dangerous.

Blood glucose level monitored by cells in the pancreas. If blood glucose level falls, ˛-cells of the pancreas release glucagon, a − hormone causing liver cells to release glucose from glycogen. If blood glucose level rises, ˇ-cells of the pancreas release insulin, a − hormone causing liver cells to convert more glucose into glycogen, thus decreasing blood sugar levels.

2Chronic hyperglycemia (even in fasting states) most commonly caused by diabetes. 11/24 12/24 Blood glucose regulation Example 3: breakwater

Problem: maintain blood glucose level constant Actuator: liver (via glycogen) Obstacles: unpredictable changes (uncertainties), like − irregular meals − irregular physical exertions − stress − ... Breakwater is a barrier built out into a body of water to protect a coast or Sensor: pancreas cells harbor from the force of waves. The idea is to Controller: pancreas cells (via glucagon / insulin) dissipate the energy of waves − by properly designed offshore constructions3. This is nothing but a 3 feedback control system Although there is no feedback (even no inputs / outputs) involved, this system can − still be considered control system (yet not of the type we’ll study)—we change behavior of the system to achieve our goals. 13/24 14/24

What this course is about ? Outline This course aims at providing primary insight into the principles of automatic control systems − as well as at introducing some Course info basic theoretical methods of control systems analysis and design. −

Introduction We’ll confine ourselves to continuous-time systems − input / output approach Block diagrams − LTI (linear time-invariant) SISO (single-input / single-output) models − single-loop feedback −

15/24 16/24 Block-diagram of centrifugal governor Block-diagram of blood glucose regulation

dp da dp da

y u Beam r y Blood u Pancreas r Engine Valve Liver linkage glucose cells

Flyball Pancreas y m cells ym

n n

r: required rotation speed dp: plant disturbance r: required glucose level dp: plant disturbance

u: throttle valve opening da: actuator disturbance u: hormones (insulin / glucagon) da: actuator disturbance y: shaft rotation speed n: measurement noise y: blood glucose level n: measurement noise ym: thrust bearing position ym: glucokinase (enzyme)

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Different nature / identical logic Abstract control system

dp da

y u r Plant Actuator Controller

Sensor ym | {z } | {z } Centrifugal governor Blood glucose regulation n

Despite r: reference signal dp: plant disturbance completely different nature of their components, − u: control signal (input) da: actuator disturbance these control systems have y: controlled signal (output) n: measurement noise identical operation logic. − ym: measured signal (output) Hence, from the control viewpoint they can be described by the same block diagram and Control design problem: studied by the same methods. given plant, sensor and actuator − − design controller guaranteeing required behavior of y − 19/24 20/24 Block-diagrams: basic connections Block-diagram manipulation rules: pickoff points

Parallel( y = (G1 + G2)u): Moving pickoff point behind a block:

y1 u y1 u G1 G G y u y u ⇐⇒ G1 + G2 y2 1 G ⇐⇒ 2 G y2

Series / cascade( y = G G u): 2 1 Both diagrams describe the relationships y1 = Gu and y2 = u.

y G G u y G G u 2 1 ⇐⇒ 2 1 Moving pickoff point ahead of a block:

G1 Feedback( y = u): y1 u y1 u 1 G G G G ∓ 1 2 ⇐⇒ y2 G y2 y u G1 ± y G1 u Both diagrams describe the relationship y1 = y2 = Gu. ⇐⇒ 1 G2G1 G2 ∓

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Block-diagram manipulation rules: summing points Example Moving summing point behind a block: y u G G G − G y u y u 5 4 2 1 G 1 G 1 − − ⇐⇒ G3 u2 u2 G

1 m 1

Both diagrams describe the relationship y = G(u1 + u2). G5 G1 y u G5 G4 G2 G1 − − − Moving summing point ahead of a block: G3

y u y u G 1 G 1 m ⇐⇒ 1 G1G5 u2 1 y G4G5 G1G2 u u2 − G 1+ G4G5 1+ G1G2G3

Both diagrams describe the relationship y = Gu1 + u2. m y G1G2G4G5 u (1 + G4G5)(1 + G1G2G3)+ G2G4

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