P control [and first order processes] – part II Control Theory P control [and first order processes] – part II

Last week we ended here… I want to design a control system for the process using a P controller. I choose the bias for a reference r=4 and a disturbance d=3: b=1.5. I choose a control gain Kc=3. What is the steady state error when r->5 and d=3? What is the steady state error when r=4 and d->2? u(t) z(t) P d(t) r(t)

Group Task I want to design a control system for the process using a P controller. I choose the bias for a reference r=4 and a disturbance d=3: b=1.5. I choose a control gain Kc=3. Assume initially the desired equilibrium is reached. z(t),u(t) for servo and control problem on previous slide? u(t) z(t) P d(t) r(t)

http://www.polleverywhere.com/multiple_choice_polls/4TpQYzhU8CRUWck

Result for servo problem (Kc=3) z(t) t

Result for servo problem (Kc=3) u(t) t

Group Task I want to design a control system for the process using a P controller. I choose the bias for a reference r=4 and a disturbance d=3: b=1.5. I choose a control gain Kc=300. Assume initially the desired equilibrium is reached. z(t),u(t) for servo and control problem on slide 2? u(t) z(t) P d(t) r(t)

Result for servo problem (Kc=300) z(t) But… u(t) becomes 301.5 already at t=1s !!! Moreover, no real process is truely first order – see later…

One more time… Using P control, find the closed loop equation LC Disturbance Manipulated Variable Using P control, find the closed loop equation Elaborate on the size of the steady state error for both servo and control problem!

P control if Kce+b < uA,min then uA=uA,min MODEL OF THE P CONTROLLER: if Kce+b < uA,min then uA=uA,min if uA,min< Kce+b < uA,max then uA=Kce+b if Kce+b > uA,max then uA=uA,max uA [% tot. range] 100% CONTROL CHARACTERISTIC => static behavior of the controller. bias 0% r  = r-ym [% tot. range] r-100 prop Definition: PROPORTIONAL BAND = 100/(Kc in %/%) = 100/Kc * (Δactuator)/(Δsensor)

Small exercise Kc = 3 mA/V ; b = 7 mA Sensor range: 0-10V Actuator range: 4-20 mA Proportional Band?

http://www.polleverywhere.com/free_text_polls/gufb6oe4PJMKOws

Some formulations of the definition The proportional band is the error as percentage of range of the measured value to cause a full scale change in the controller output. [20sim] Proportioning Band: is the area , in %, around the setpoint where the controller is actually controlling the process (regardless of the sensor range), i.e. the controller’s output is at some level other than 100% or 0%. [http://iseinc.com/what%20is%20pid.htm]

Group Task 2 Process: We take A=1 m2 and c=0.1 m2/s Assume d=0.04 m3/s Fin,u(t) r(t) Fout(t) Fin,d(t) We take A=1 m2 and c=0.1 m2/s Assume d=0.04 m3/s Actuator range: 4mA -> 0 m3/s 20mA -> 0.1 m3/s Sensor range: 0.2m -> 0V 1.5m -> 5V Draw the static process characteristic (x-axis: z in %, y axis: u in %)

Group Task 3 Process: Actuator: 4mA -> 0 m3/s 20mA -> 0.1 m3/s Fin,u(t) r(t) Fout(t) Fin,d(t) Actuator: 4mA -> 0 m3/s 20mA -> 0.1 m3/s Sensor: 0.2m -> 0V 1.5m -> 5V Assume r = 1m Choose the appropriate bias. Next, draw the control characteristic for a PB of 75%.