Automation Systems
Lecture 6 - Controller and single-loop control system
Jakub Mozaryn
Institute of Automatic Control and Robotics, Department of Mechatronics, WUT
Warszawa, 2019
Controller
Figure 1:Place of the controller in the control system (the flow process with the valve eg. gas pipelines, wastewater plants)
controlled variable: y (t), process variable (PV ) : ym(t), set point (SP): w (t),
control variable (sym. CV ):
u(t) = f (e(t)),
control error: e(t) = SP − PV ,
Controller
Role of the controller
Controller generates the control signal u(t) (CV, Control Variable) based on the comparison of output signal ym(t)(PV, Process Variable) gen- erated by the sensor that represents the output signal y (t) (controlled variable) with the reference signal yr(t) (SP, Set Point).
The result of this comparison is error signal, called Deviation of the Process Variable, defined as:
e(t) = ym(t) − yr(t); ⇒ e = PV − SP (1)
Structures of the control systems single loop
Figure 2:Structure of the control system - version 1 (w (t) = yr(t))
Figure 3:Structure of the control system - version 2 (w (t) = yr(t))
Role of the controller - Steady State
Figure 4:Structure of the control system - version 2 (w (t) = yr(t)) In the steady state, when e(t) = 0, the controller should generate a con- trol signal which causes activation of the actuator ensuring achieve- ment of the predetermined value of the Controlled Variable - the operation point.
Role of the controller - Set Point
Figure 5:Structure of the control system - version 2
Set Point following: The occurence of the positive deviation e(t) > 0 (by increasing the setpoint yr(t)) causes an increase of the control value u and, consequently, the expected increase in the value of the controlled variable (y (t)).
Role of the controller - Disturbances
Figure 6:Structure of the control system - version 2
Disturbance minimization: Occurence of the positive deviation e(t) > 0 (by decreasing the controlled variable y (t) due to disturbances) increase the value of the controlled variable u(t) that compensates the impact of dusturbances d (t) on the process.
Structures of the control systems - normal and reverse modes
Normal mode
In the case of the process, where an increase of the control signal u(t) is connected with a decrease of the output signal (the transfer function Gr(s) is negative), the negative feedback can be obtained as follows:
enormal(t) = ym(t) − yr(t). (2)
Reverse mode
In the case of process, in which an increase of the control signal u(t) causes an increase in the output (the transfer function Gob(s) is positive), the following action of the controller shall be used to obtain the negative feedback:
ereverse(t) = yr(t) − ym(t). (3)
Structures of the control systems - normal and reverse modes
The increase in the signal from the controller (u(t)) closes the valve - normal mode.
The increase in the signal from the controller (u(t)) opens the valve - reverse mode.
Technical realization of controllers
Figure 7:Technical realization of controllers
Classification of the controllers - pt. 1
Ctiteria Controller type
Type of the processed signals: analogous digital The way of influence on the object: continous
non-continous Compliance with the law of
superposition: linear
nonlinear
Destination: specialized
universal
Classification of the controllers - pt. 2
Ctiteria Controller type
Type of implementation:
mechanical pneumatic hydraulic electrical
Algorithm of control action:
PID controllers other (LQR, state-space, predictive)
The energy required for operation: direct action indirect action
PID controllers
Control Algorithm
Dynamic properties of controllers are realized by the control algorithm.
The most commonly used control algorithm (95 %) is called PID algorithm (Proportional - Integral - Derivative). It is possible to realize simpler algorithms: P, PI, PD, by setting gains of PID controller (kP, Ti, Td).
P controller
Gr(s) = ∆u(s)
e(s) = kp (4)
I controller
Gr(s) = ∆u(s) e(s) = 1
Tis (5)
PI controller
Gr(s) = ∆u(s) e(s) = kp
1 + 1
Tis
(6)
Transfer functions of PID controllers
PD controller - ideal
Gr(s) = ∆u(s)
e(s) = kp(1 + Tds) (7) PD controller - real (the ideal derivative part is substituted with the real derivative part)
Gr(s) = ∆u(s) e(s) = kp
1 + Tds Td
kd
s + 1
(8)
Transfer functions of PID controllers
PID controller - ideal
Gr(s) = ∆u(s)
e(s) = kp(1 + 1
Tis + Tds) (9)
PID controller - real (the ideal derivative part is substituted with the real derivative part)
Gr(s) = ∆u(s) e(s) = kp
1 + 1
Tis + Tds Td
kd
s + 1
(10)
Block diagram of PID controller
PID controller - real
Gr(s) = ∆u(s) e(s) = kp
1 + 1
Tis + Tds Td
kd
s + 1
(11)
Figure 8:Block diagram of PID controller - paralell realization
P controller
Dynamics equation of P controller
∆u(t) = kpe(t) (12)
∆u(t) = u(t) − up (13)
u(t) = kpe(t) + up (14)
where: kp - proportionl gain, up - operating point.
Proportional range
xp= 1 kp
100% (15)
The proportional range describes the percentage (in relation to the full range of the signal) of the change in the deviation e(t) that is required to induce changes of the control signal u(t) of the full range.
I controller
Transfer function
Gr(s) = ∆u(s) e(s) = 1
Tis (16)
Ti
d ∆u(t)
dt = e(t) (17)
where
∆u(t) = u(t) − u(0) (18)
Step response (the response to the step function, 2 components)
u(t)|e(t)=e01(t)= ∆u(t)+u(0) = u(0)+1 Ti
t
Z
0
e(τ )d τ = u(0)+e0 t Ti
(19)
Static characteristic
e = 0 (20)
I controller
Figure 9:Step response of I controllerFigure 10:Static characteristic of I controller - astatic algorithm
PI controller
Transfer function
Gr(s) = ∆u(s)
e(s) = kp(1 + 1
Tis) (21)
∆u(t) = u(0) + kpe(t) + 1 Ti
t
Z
0
e(τ )d τ (22)
Step response (the response to the step function, 2 components)
∆u(t)|e(t)=e01(t)= e0kp1(t) + e0kp
t
Ti (23)
u(t)|e(t)=e01(t)= ∆u(t) + u(0) = e0kp1(t) + e0kp t Ti
+ u0 (24) Static characteristics
e = 0 (25)
PI controller
Figure 11:Step response of PI controller
Integral time constant Ti (double time)
The component that describes an integral action increases with time from an initial value, reaching in time t = Ti a value of the proportional com- ponent, which means double the gain in the output signal relative to the proportional component.
PD controller - ideal
Transfer function Gr(s) = ∆u(s)
e(s) = kp(1 + Tds) (26) Step response (the response to the step function, 2 components)
∆u(t)|e(t)=e01(t)= kpe0[1 + δ(t)]
(27)
REMARKS:
PD algorithm doesn’t have technical realisation because kd= 1
Td → ∞.
The ideal PD controller is not used, because the dynamics of the actual devices requires a specific signal duration to be able to react to change (delay).
Figure 12:Step response of PD controller (ideal)
PD controller - real
Transfer function
Gr(s) = kp
1 + Tds Td
kds + 1
(28) Step response (respone to the step function, 2 components)
∆u(t)|e(t)=e01(t)= kpe0[1+kde−kdTd t]
(29) Figure 13:Step response of PD controller (real)
PD controller
Figure 14:Ramp response of PD controller - (a) ideal and (b) real
Derivative time constant - Td (lead time)
The ramp response of the PD controller (ideal / real) explains the name of the lead time of Td - in the case of ramp input, value of the control variable as the sum of the components P and D is achieved earlier by the time Td than value of the component P.
PID controller - ideal
Transfer function Gr(s) = ∆u(s)
e(s) = kp
1 + 1
Tis + Tds
(30) Step response (the response to the step function, 3 components)
∆u(t)|e(t)=e01(t)= kpe0[1+ t Ti
+δ(t)]
(31) Figure 15:Step response of PID controller (ideal)
PID controller - real
Transfer function
Gr(s) = kp
1 + t
Ti + Tds Td
kd
s + 1
(32) Step response (the response to the step function, 3 components)
∆u(t)|e(t)=e01(t)= kpe0[1+ t Ti
+kde−kdTd ] (33)
Figure 16:Step response of PID controller (real)
PID controller - real
Automation Systems
Lecture 6 - Controller and single-loop control system
Jakub Mozaryn
Institute of Automatic Control and Robotics, Department of Mechatronics, WUT
Warszawa, 2019
