Automation Systems Lecture 8 - The quality of the control system. Tuning of the PID controller Jakub Mozaryn

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Automation Systems

Lecture 8 - The quality of the control system. Tuning of the PID controller

Jakub Mozaryn

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

Warszawa, 2016

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Quality of the control system

In addition to the requirement of asymptotic stability, there are imposed additional requirements on control systems, concerning the behavior of the transient (dynamic) response and steady states. These requirements are generally referred to as quality requirements of the control system.

Requirements relating to the transition response in the control systems are determined by a number of indicators, generally called dynamic quality criterias of the control system.

The requirements relating to the steady state are formulated by determi- ning the so-called static accuracy of the control system - perimissible values of deviations from the set point in steady states.

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Quality of the control system

The task of the control system is to minimize the deviation from the set point, described as:

e(t) = ez(t) + ew(t), (1)

gdzie

ez(t) - deviation caused by disturbance,

ew(t) - deviation caused by change of the set point.

e(t) = ym(t) − w (t), (2)

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Quality of the control system

When rating the quality of the control LTI system, both components of the control deviation can be analyzed separately.

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Static deviation caused by disturbance

Transfer function

Gz(s) = ∆ym(s)

z(s) =ez(s)

z(s) = ±Gz(s)Gob(s)

1 + G_ob(s)G_r(s) (3) ez(s) = ∆ym(s) = ±Gz(s)Gob(s)

1 + Gob(s)Gr(s)z(s) (4) Static deviation of the system in relation to disturbances

ezst.= lim

t→∞ez(t) = lim

s→0s · ez(s) (5)

e_zst.= lim

s→0s · ±Gz(s)Gob(s)

1 + G_ob(s)G_r(s)z(s) (6)

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Deviation caused by change of the set point

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Deviation caused by change of the set point

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Deviation caused by change of the set point

Static deviation caused by change of the set point

Transfer function

Gew(s) = ew(s)

∆w (s) = −1

1 + G_ob(s)G_r(s) (7) ew(s) = −1

1 + Gob(s)Gr(s)∆w (s) (8) Static deviation of the system in relation to changes of the set point

ewst.= lim

t→∞ew(t) = lim

s→0s · ew(s) (9)

ewst.= lim

s→0s · −1

1 + Gob(s)Gr(s)∆w (s) (10)

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Deviations - example

Determine the static deviations in the control system shown in the figure, caused by step changes of disturbances z(t) = 2 and step changes of setpoint ∆w (t) = 5, when using:

P controller PD controller PI controller

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Deviations - example

Transfer function

Gob(s) = kob

(Ts + 1)⁴ (11)

P controller

Gr(s) = kp (12)

PD controller

Gr(s) = kp(1 + Tds) (13) PI controller

Gr(s) = kp

1 + 1

Tis

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z(t) = 2 → z(s) = 2

s (15)

Change of the set point

∆w (t) = 5 → ∆w (s) =5

s (16)

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Deviation caused by change of the disturbances - example, P controller

Deviation caused by change of the disturbances:

ezst.= lim

s→0s Gob(s)

1 + Gob(s)Gr(s)z(s) (17) P controller

ezst.P = lims→0s G_ob(s) 1 + Gob(s)Gr(s)

2 s =

lims→0

kob

(Ts + 1)⁴· 2 1 + kob

(Ts + 1)⁴kp

= lims→0

kob· 2 (Ts + 1)⁴+ kob· kp

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Deviation caused by change of the disturbances - P controller ezst.P = kob· 2

1 + k_obk_p (19)

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Deviation caused by change of the disturbances - example, PD controller

ezst.= lim

s→0s Gob(s)

1 + Gob(s)Gr(s)z(s) (20) PD controller

ezst.PD = lims→0

kob

(Ts + 1)⁴· 2 1 + k_ob

(Ts + 1)⁴kp(1 + Tds)

=

= lims→0

kob· 2

(Ts + 1)⁴+ k_ob· k_p(1 + T_ds)

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Deviation caused by change of the disturbances - PD controller ezst.PD= kob· 2

1 + k_obk_p (22)

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Deviation caused by change of the disturbances - example, PI controller

e_zst.= lim

t→∞e_z(t) = lim

s→0s G_ob(s)

1 + Gob(s)Gr(s)z(s) (23) PI controller

ezst.PI = lims→0

kob

(Ts + 1)⁴· 2 1 + kob

(Ts + 1)⁴kp(1 + 1 Tis)

=

= lim_s→0 kob· 2

(Ts + 1)⁴+ k_ob· kp(1 + 1 Tis)

= 0

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Deviation caused by change of the disturbances - PI controller

ezst.PI = 0 (25)

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Deviations caused by the disturbances - summary

Summary P controller

e_zst.P = kob· 2

1 + k_obk_p (26)

PD controller

ezst.PD= kob· 2 1 + kobkp

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PI controller

e_zst.PI = 0 (28)

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Deviation caused by change of the set point - example, P controller

Deviation caused by change of the set point:

e_wst.= lim

s→0s −1

1 + Gob(s)Gr(s)∆w (s) (29) P controller

ewst.P = lim

s→0s −1

1 + Gob(s)kp

5 s = lim

s→0

−5 1 + k_ob

(Ts + 1)⁴k_p

= −5

1 + kobkp

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Deviation caused by change of the set point - P controller ewst.P = −5

1 + kobkp

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Deviation caused by change of the set point - example, PD controller

e_wst.= lim

s→0s −1

1 + Gob(s)Gr(s)∆w (s) (32) PD controller

e_wst.PD = lim_s→0s −1

1 + Gob(s)kp(1 + Tds) 5 s

= lims→0

−5 1 + kob

(Ts + 1)⁴kp(1 + Tds)

= −5

1 + kobkp

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Deviation caused by change of the set point - PD controller ewst.PD = −5

1 + k_obk_p (34)

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Deviation caused by change of the set point - example, PI controller

ewst.= lim

s→0s −1

1 + Gob(s)Gr(s)∆w (s) (35) PI controller

ewst.PI = lims→0s −1 1 + Gob(s)kp

1 + 1

Tis

5 s

= lims→0

−5 1 + kob

(Ts + 1)⁴kp

1 + 1

Tis

= 0

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Deviation caused by change of the set point - PI controller

ewst.PI = 0 (37)

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Deviations caused by change of the set point - summary

Summary P controller

ewst.P = −5

1 + k_obk_p (38)

PD controller

ewst.PD = −5

1 + k_obk_p (39)

PI controller

ewst.PI = 0 (40)

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Conclusions about deviations

Conclusions:

In a system with a static object and a control algorithm P and PD there are non-zero deviations in relation to the disturbances or setpoint changes respectively.

Increasing the proportional gain of P or PD controller reduces the value of static deviations. Reducing the static deviation by increasing the gain k_p is usually limited due to the stability of the system. (The system with PD controller reaches the border of stability at higher gain control than in the case of the regulator P.).

Integral action in the controller (PI, PID) provides zero deviation in relation to the disturbances or setpoint changes respectively.

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Dynamical quality of control system

In practice, various indicators of dynamical quality are used:

transient response indicators,

indicators descibing the frequency plots of the control system - magnitude and phase margins,

integral indicators of quality.

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Transient response indicators

To evaluate the transient response following indicators are used:

Static deviation caused by change of the disturbances: ezst.

Static deviation caused by change of the set point: ewst.

Maximal deviation (dynamical): em- the maximum value of deviation after the step change of disturbance or setpoint.

Time of control: t_r - time since the introduction of the step change of disturbance or setpoint to the time from which the control deviation does not go beyond the range of values ±∆e . Overshoot: κ = e₂

e1

· 100% - expressed as a percentage ratio of the amplitude of the second deviation e2 from the set value to the amplitude of the first deviation e1from the set value.

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Oscillatory transient response - disturbances

Rysunek :Oscillatory transient response of the control system to disturbances:

a) with non-zero static deviation, b) with zero static deviation

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Aperiodic transient response - disturbances

Rysunek :Aperiodic transient response of the control system to disturbances:

a) with non-zero static deviationą, b) with zero static deviation

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Oscillatory transient response - setpoint

Rysunek :Oscillatory transient response of the control system to setpoint change: a) with non-zero static deviation, b) with zero static deviation

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Aperiodic transient response - setpoint

Rysunek :Aperiodic transient response of the control system to setpoint change: a) with non-zero static deviation, b) with zero static deviation

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Selection of controllers

The basic premise when choosing the type of controller is dynamic characteristics of the controlled process.

Rysunek :Control system

Basic equations, describing the properties of the controlled processes

Gob(s) = ∆y_m(s)

∆u(s) = k_ob

Tzs + 1e^−T⁰^s, Gob(s) = ∆y_m(s)

∆u(s) = 1 Tzse^−T⁰^s

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Selection of controllers

for T0

Tz

< 0, 1 ÷ 0, 2 → switch controllers (two- three- gain),

for 0, 1 ¬ T0

Tz

< 0, 7 ÷ 1 ÷ 0, 2 → continuous controllers,

dla T0

Tz

> 1 → impulse controllers (impulse output signals)

In the case of industrial processes common ratio of T0

T_z is in the range of 0, 2 ÷ 0, 7. Therefore, in industrial control systems the most common controllers are continuous, implementing typical control algorithms P, PI, PD and PID.

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Selection of controllers

Analysis of controler action with the process leads to the following conclusions concerning the selection of the control algorithm:

PI algorithm provides good control only for the low frequencies of setpoint changes or disturbancs. Integral action is necessary to obtain zero static deviation.

PI algorithm provides wider bandwidth than PID algorithm, but poorer performance for the low frequencies of setpoint changes or disturbancs. Derivative action is recommended for objects with higher order lag (such as thermal processes), because it allows the strong interaction of control even at small deviations. PD controller does not ensure the achievement of zero deviation in steady-state . PID algorithm merges to a certain extent the advantages of PI and PD algrithms.

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Selection of controllers

In practice, industrial controllers with continuous algorithm and performance are universal. Their parameters (settings) can be changed (adju- sted) within a wide range, so they can work properly with processes with different dynamical properties. Depending on the requirements of the stability and quality, the controller settings are selected by the special selection procedures.

There are following settings of PID controller:

proportional gain kp= 0, 1 ÷ 100 integral gain Ti = 0, 1 ÷ 3600s derrivative gain Td = 0 ÷ 3600s

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Selection of controllers

Methods of PID controllers tuning:

Experimental methods - do not allow to achieve certain quality of the control system, eg. Ziegler – Nichols, Pessen, Hassen and Offereissen, Cohen-Coon, ¨Astr¨om – Hagglund .

Tabular methods - determining the set of controller parameters based on the parameters of a mathematical model of the controlled process and the required quality criterion of the control system (like the lowest overshoot).

Autotuning, eg. relay method.

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Tuning of the controllers

Ziegler-Nichols method

Type 1:

controller settings are selected on the basis of the parameters of the closed loop control system, brought to the stability limit (by experimental excitation of the system).

It can be used to controller tuning in the control systems where processes are described by static and astatic, higher order lag elements.

Type 2:

It can be used to controller tuning in the control systems where processes are described by static higher order lag elements, controller settings are selected based on the transient response of the controlled process.

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Ziegler-Nichols method, type 1

Rysunek :Functional scheme of real control system

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Ziegler-Nichols method, type 1 - steps 1-3 / 6

Step 1: In the manual mode (M) by changing control variable (CV), adjust the process variable ym (PV) to a state in which it is equal with the required setpoint

Step 2: Set the controller to the proportional action (switch off integral and derivative actions), set the operation point control value of the controller equal to the setting obtained in the Step 1 and set the initial value of the controller gain k_p> 0.

Step 3: Switch the system to automatic control (A) and if the system maintains equilibrium, by changing SP produce an impulse with some amplitude and pulse duration depending on the expected dynamics of the process; observe or record the change in the controlled variable. It is recommended to use a pulse with amplitude of 10 % of the process value changes ym(PV) and pulse duration of about 10 % of the estimated value of the time constant of the controlled process.

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Ziegler-Nichols method, type 1 - steps 4-5 / 6

Step 4: If the transient response is underdamped, set higher values of the proportional gain (Steps 1-3) until a system be on the border of stability (constant oscillations).

Step 5: From the steady oscillations read ’critical’ proportional gain k_pkryt.and oscillation period T_osc.

Step 6: Set the patameters according to the table of setings developed by Ziegler-Nichols.

Rysunek :Changes of the process variable (PV) obtained during Ziegler – Nichols experiment Jakub Mozaryn Automation Systems

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Ziegler-Nichols method, type 1

PID controller setting according to Ziegler-Nichols

Controller tyoe kp Ti Td

P 0, 50kpkryt. - -

PI 0, 45kpkryt. 0, 8Tosc • PID 0, 60kpkryt. 0, 5Tosc 0, 12Tosc

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