Control Theory (2) Jeremy Wyatt School of Computer Science University of Birmingham

Aims Understand first order models Be able to analyse them See their generality Analyse PE control for our DC motor Meet PI and PID control

Modelling a Simple Process  = temperature Q=rate of heat input What is the relationship between the heat supply and the temperature of the room? What is f?

Modelling a Simple Process There are two ways in which heat is “used up” heats the room leaks out through the walls We combine these to give a first order differential equation model C is the heat capacity of the room L is the loss coefficient (how leaky the room is) [1]

First order processes The behaviour is characteristic of what we call a first order lag process We can analyse the equation to tell us: –The final temperature –The transient behaviour  t

Steady State Behaviour If we hold Q constant the temperature will eventually level out At that point so If then must rise If then must fall is the steady state temperature is the temperature at time t

Transient Behaviour If we solve we obtain tells us the final temperature tells us how fast we reach it is the initial temperature is the temperature at time t  t [2]

The time constant tells us how fast we reach the steady state We can express this another way Where  is pronounced “tau” and is called the time constant of the process Any first order lag process reaches ~0.63 of its steady state value when t=   tt

Standard Form Any first order lag process reaches –~0.63 of its steady state value when t=  –~0.95 of its steady state value when t=3  –~0.99 of its steady state value when t=5  Recall A simple rearrangement gives us a standard form:  tt [1] [3] in general

Solution by Inspection Once in standard form we can solve by inspection: GU is the steady state –where U is the process input (or control action) –G is called the steady state gain Here [3]

Our DC Motor example revisited Our DC motor can also be modelled as a first order process A motor generates a back voltage proportional to its speed The net voltage is related to current by Ohm’s law [4] is a motor constant is the motor speed is the resistance of the motor is the current drawn

Our DC Motor example revisited The torque produced is proportional to the current drawn We can combine these to make a first order model [5] is a motor constant is the mass being driven [4]

Questions 1.Combine the equations to form the first order model 2.What is the steady state speed? 3.What is the time constant  ?

Adding Gears We can alter the properties of the process by adding gears  is the gear ratio (  >1 means a step down gear) input speed =  output speed  input torque = output torque [5] [4]

Questions 1.Combine the equations to form the new first order model 2.What is the new steady state speed? 3.What is the new time constant  ?

Analysing our PE Controller Recall our proportional error control law where - +  G

Properties of PE Control The PE controller will not reach the target speed The difference is called the steady state error How big will it be? s t sTsT

Analysing our PE Controller The open loop behaviour was described by We can remodel the new system by substituting [7] back into [6] For now assume that [6] [7]

Analysing our PE Controller The new system is described by So asthe steady state error disappears and it reaches the steady state more quickly But big causes instability [7] Steady state Time constant

Proportional Integral Control (PI) IDEA: Add a proportion of the accumulated error to the control signal GOOD: there is no steady state error BAD: –Hard to tune –More lag –More unstable s t sTsT

PID control IDEA: Add a third term which looks at how fast the error is changing s t sTsT t

PID control GOOD: –fast initial response to increases in error (D component) –No steady state error (I component) –As error decreases P 0 and D < 0, therefore stable BAD:very hard to tune s t sTsT t

Summary Seen generality of first order model Used it to analyse PE control Seen principles of PI and PID control