Automation Systems Lecture 6 - Place and role of controller in control system Jakub Mozaryn

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Automation Systems

Lecture 6 - Place and role of controller in control system

Jakub Mozaryn

Institute of Automatic Control and Robotics, Department of Mechatronics, WUT

Warszawa, 2017

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Controller

Figure 1 : Place of the controller in control system

controlled variable: y (t) process variable: ym(t) (PV ) set point: w (t) (SP)

control error: e(t)

control variable u(t) - (CV )

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Controller

Role of the controller

Controller generates the control signal u(t) (CV, Control Variable) based on the comparison of output signal y_m(t)(PV, Process Variable) generated by the sensor that represents the y (t) Controlled Variable with the reference signal y_r (SP, Set Point).

The result of this comparison - called Deviation of the Process Variable e(t) - is defined as:

e = ym− w ; ⇒ e = PV − SP (1)

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Structures of the control systems single loop

Figure 2 : Structure of the control system - version 1

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Role of the controller - Steady State

In the steady state, when Deviation of the Process Variable e(t) = 0, the controller should generate a control signal which causes activation of actuator ensuring achievement of predetermined value of the Controlled Variable - operation point.

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Role of the controller - Set Point

Set Point following: 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)).

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Role of the controller - Disturbances

Figure 6 : Structure of the control system - version 2 Disturbance minimization: Occurencee 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.

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Structures of the control systems - normal and reverse modes

Normal mode

In the case of process, where increase of the control signal u(t) is connected with decrease of the output signal (transfer function Gr(s) is negative), the negative feedback can be obtained as follows:

e_normal(t) = y_m(t) − w (t). (2)

Reverse mode

In the case of process, in which increase of the control signal u(t) causes an increase in output (transfer function G_ob(s) is positive), the following action the controller shall be used to obtain negative feedback:

ereverse(t) = w (t) − ym(t). (3)

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Structures of the control systems - normal and reverse modes

The increase in signal from the controller (u(t)) closes the valve - normal mode.

The increase in signal from the controller (u(t)) opens the valve - reverse mode.

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Technical realization of controllers

Figure 7 : Technical realization of controllers

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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

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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

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PID controllers

Control Algorithm

The dynamic properties of controllers are realized by control algorithm.

The most commonly used control algorithm (95 %) is called PID algorithm (Proportional - Integral - Derivative). Its possible to realize simpler algorithms: P, PI, PD, by setting gains of PID controller (kP, Ti, Td).

P controller

G_r(s) = ∆u(s)

e(s) = k_p (4)

I controller

Gr(s) = ∆u(s) e(s) = 1

Tis (5)

PI controller

Gr(s) = ∆u(s) e(s) = kp

1 + 1

Tis

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Transfer functions of PID controllers

PD controller - ideal

Gr(s) = ∆u(s)

e(s) = kp(1 + Tds) (7) PD controller - real

Gr(s) = ∆u(s) e(s) = kp







1 + T_ds T_d kd

s + 1







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Transfer functions of PID controllers

PID controller - ideal

Gr(s) = ∆u(s)

e(s) = kp(1 + 1

Tis + Tds) (9)

PID controller - real

Gr(s) = ∆u(s) e(s) = kp





 1 + 1

Tis + T_ds T_d

kd

s + 1







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Block diagram of PID controller

PID controller - real

Gr(s) = ∆u(s) e(s) = kp





 1 + 1

Tis + T_ds T_d

kd

s + 1







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Figure 8 : Block diagram of PID controller - paralell realization

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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

k_p100% (15)

The proportional range descibes percentage, in relation to the full range of the signal, the change in deviation e(t) that is required to induce changes of the control signal u(t) of the full range.

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I controller

Transfer function

G_r(s) = ∆u(s) e(s) = 1

Tis (16)

T_id ∆u(t)

dt = e(t) (17)

where

∆u(t) = u(t) − u(0) (18)

∆u(t) = u(0) + 1 Ti

t

Z

0

e(τ )d τ (19)

Step response (2 components)

u(t)|_e(t)=e₀_1(t)= u(0) + 1 T_i

t

Z

0

e(τ )d τ = u(0) + e₀ t

T_i (20) Static characteristic

e = 0 (21)

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I controller

Figure 9 : Step response of I controllerFigure 10 : Static characteristic of I controller - astatic algorithm

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PI controller

Transfer function

G_r(s) = ∆u(s)

e(s) = k_p(1 + 1

Tis) (22)

∆u(t) = u(0) + kpe(t) + 1 Ti

t

Z

0

e(τ )d τ (23)

Step response (2 components)

∆u(t)|_e(t)=e₀_1(t)= e0kp1(t) + e0kp

t

T_i (24)

u(t)|_e(t)=e₀_1(t)= ∆u(t) + u(0) = e₀k_p1(t) + e₀k_p t Ti

+ u₀ (25) Static characteristics

e = 0 (26)

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PI controller

Figure 11 : Step response of PI controller

Integral time constant Ti

Component that describes integral action increases with time from an initial value, reaching in time t = Ti a value of a proportional

component, which means doubling the gain in the output signal relative to the proportional component.

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PD controller - ideal

Transfer function Gr(s) = ∆u(s)

e(s) = kp(1 + Tds) (27) Step response (2 components)

∆u(t)|_e(t)=e₀_1(t)= kpe0[1 + δ(t)]

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REMARKS:

PD algorithm doesn’t have technical realisation because kd= 1

Td

→ ∞.

There is no use of Pd controller 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)

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PD controller - real

Transfer function

G_r(s) = k_p







1 + T_ds Td

k_ds + 1





 (29) Step response (2 components)

∆u(t)|_e(t)=e₀_1(t)= k_pe₀[1+k_de^−kd^Td ]

(30) Figure 13 : Step response of PD controller (real)

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PD controller

Figure 14 : Ramp response of PD controller - (a) ideal and (b) real

Derivative time constant - Td

Ramp response of 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.

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PID controller - ideal

Transfer function Gr(s) = ∆u(s)

e(s) = kp

1 + 1

Tis + Tds

(31) Step response (3 components)

∆u(t)|_e(t)=e₀_1(t)= k_pe₀[1+ t Ti

+δ(t)]

(32) Figure 15 : Step response of PID controller (ideal)

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PID controller - real

Transfer function

Gr(s) = kp





 1 + t

T_i + Tds Td

kd

s + 1





 (33) Step response (3 components)

∆u(t)|_e(t)=e₀_1(t)= kpe0[1+ t

T_i+kde^−kd^Td ] (34)

Figure 16 : Step response of PID controller (real)

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PID controller - real

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Technical realization of PID controllers

Figure 17 : A diagram illustrating the functional characteristics of the industrial PID controller

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