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, 2016

Jakub Mozaryn Automation Systems

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Role of the controller

The controller generates the control signal change u(CV ) (Control Value) based on the comparison of output signal y_m(PV )(Process variable), generated by the sensor that represents the controlled variable, with the reference signal yr(SP) (Set Point). The result of this

comparison - called deviation of the process variable e - in control systems is defined as:

e = ym− w ; e = PV − SP (1)

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Regulatory

controlled value y process variable y_m(PV ) set point w (SP) control error e

control variable u - (CV )

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Controller

In the steady state of the system, when control deviation e is zero, the controller should generate a control signal which causes activation of actuator ensurinh achievement of predetermined value of the controlled variable (CV).

The occurencee of the positive value of the deviation e (by

increasing the setpoint yr or decrease of the controlled variable due to disturbances) causes an increase of the control value u and, consequently, the expected increase in the value of the controlled variable (y ) or increase the value of the controlled variable thet compensates the impact of dusturbances (z) on the process.

Analogously it works, when a negative value of deviation e occurs.

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Structures of the control systems

Rysunek :Controlled object structure

Rysunek :Scheme of the control system with object described by negative transfer fuction

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Structures of the control systems

Rysunek :Transformed scheme of the control system with object described by negative transfer fuction

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Structures of the control systems

In engineering practice, one can find objects, where increase of the control signal u is connected with decrease of the output signal (transfer function Gr(s) is negative).

Rysunek :A block diagram of the control system, where object is described by the negative transfer function and controler in normal mode (NL).

e = ym− w (2)

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Structures of the control systems

In the case of controlled objects, in which increase of the control signal u causes an increase in output (transfer function G_ob(s) is positive), the other action the regulator shall be used to obtain negative feedback.

Rysunek :A block diagram of the control system, where object is described by the positive transfer function and controller is in reverse mode (NL) (reverse deviation).

e = w − ym (3)

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Role of the controller

The increase in signal from the controller closes the valve - normal mode

The increase in signal from the controller opens the valve - reverse

mode Jakub Mozaryn Automation Systems

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Control process

Rysunek :Technical realization of controllers

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Classification of the controllers

continous discrete linear nonlinear direct action indirect action

pneumatic hydraulic electric specialized universal

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Classification of the controllers

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: nonlinear

Destination: specialized

universal 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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Classification of the controllers

Hydraulic controller - the controller with indirect action (requires the energy supply)

Temperature controller - direct action controller (retrieves energy from the process)

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Transfer functions of PID controllers

Algorithm of the controllers

The dynamic properties of controllers are referred to as control algorithm . The most commonly used control algorithm is called PID algorithm (Proportional - Integral - Derivative). By setting it’s parameters, it can realize a simpler algorithm: P, PI, PD.

P controller

G_r(s) = ∆u(s)

e(s) = k_p (4)

I controller

Gr(s) = ∆u(s) e(s) = 1

T_is (5)

PI controller

Gr(s) = ∆u(s) e(s) = kp

1 + 1

T_is

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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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Rysunek :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) = kpe(t) + up (13)

where: kp - proportionl gain, up - operating point.

Proportional range

x_p= 1 kp

100% (14)

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

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Regulator P

Rysunek :Examples of static characteristics of the P controller (normal and reverse modes) - static algorithm

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Regulator P

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I controller

Transfer function

G_r(s) = ∆u(s) e(s) = 1

Tis (15)

Ti

d ∆u(t)

dt = e(t) (16)

where

∆u(t) = u(t) − u(0) (17)

∆u(t) = u(0) + 1 T_i

t

Z

0

e(τ )d τ (18)

Step response

u(t)|e(t)=e₀1(t)= u(0) + 1 Ti

t

Z

0

e(τ )d τ = u(0) + e0

t Ti

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Static characteristic

e = 0 (20)

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I controller

Rysunek :Step response of I controllerRysunek :Static characteristic of I controller - astatic algorithm

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PI controller

Gr(s) = ∆u(s)

e(s) = kp(1 + 1

Tis) (21)

∆u(t) = u(0) + kpe(t) + 1 T_i

t

Z

0

e(τ )d τ (22)

Step response (2 components)

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

T_i (23)

u(t)|_e(t)=e₀_1(t)= ∆u(t) + u(0) = e0kp1(t) + e0kp

t Ti

+ u0 (24) Static characteristics (astatic algorithm)

e = 0 (25)

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PI controller

Rysunek :Step response of PI controller

Integral time constant Ti

Component thet describes integral action of the response increases with time from an initial value of zero, 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) (26) Step response

∆u(t)|_e(t)=e₀_1(t)= k_pe₀[1 + δ(t)]

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REMARKS:

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

Td

→ ∞.

There is no use of it because of the the dynamics of the actual devices that require a specific signal duration to be able to react to change

Rysunek :Step response of PD controller (ideal)

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PD controller - real

G_r(s) = k_p







1 + Tds Td

k_ds + 1





 (28) Step response

∆u(t)|_e(t)=e₀_1(t)= k_pe₀[1+k_de^−kd^Td ] (29) PD algorithm (ideal/real) are static algorithms.

Rysunek :Step response of PD controller (real)

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PD controller

Rysunek :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, textbf value of the controller output as the sum of the components P and D is achieved earlier by the time Td in relation to the component P.

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PID controller - ideal

Transfer function G_r(s) = ∆u(s)

e(s) = k_p

1 + 1

T_is + T_ds

(30) Step response - astatic algorithm

∆u(t)|_e(t)=e₀_1(t)= kpe0[1+ t T_i+δ(t)]

(31) ^{Rysunek :}Step response of PID controller (ideal)

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PID controller - real

G_r(s) = k_p





 1 + t

Ti

+ T_ds Td

k_ds + 1





 (32) Step response

∆u(t)|e(t)=e₀1(t)= kpe0[1+ t Ti

+kde^−kd^Td ] (33)

Rysunek :Step response of PID controller (real)

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PID controller - real

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Technical realization of PID controllers

Rysunek :A diagram illustrating the functional characteristics of the industrial PID controller

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