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High performance PID control of a cascade tanks system as an example for control teaching

Conference Paper · August 2017

DOI: 10.1109/MMAR.2017.8046931

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Robert Bieda

Silesian University of Technology

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Marian J Blachuta

Silesian University of Technology

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Rafal T Grygiel

Silesian University of Technology

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High Performance PID Control of a Cascade Tanks

System as an Example for Control Teaching

Robert Bieda, Marian Blachuta, Rafal Grygiel

Department of Automatic Control Silesian University of Technology

16 Akademicka St., Gliwice, PL 44-101, Poland Email: marian.blachuta@polsl.pl

Abstract—A detailed analysis of PID level control in the

second tank of the cascade of two tanks is performed with

respect to both, load disturbance attenuation and set-point

change Approximate formulas for extrema of time responses and

for certain performance indices are derived, giving guidelines

for controller settings A simple method of choosing controller

parameters is proposed that bases on time scale separation For

a reasonable transfer from one operating point to another, under

control signal limitations, command signal generators with a

feed-forward from the reference are proposed and the effect of

anti-windup controller augmentation is examined

A model of a water tank can be found in almost every

textbook, [6], [7], [8], [12] on control engineering Classical

P, PI and PID control for systems of water tanks like the

one depicted in Fig.1 is the subject of many control teaching

laboratory experiments, both physical[2], [1] and virtual [11]

As a result, every year hundreds if not thousands of students all

Fig 1 Schematic diagram of a two tanks cascade system

over the world get in touch with this problem Unfortunately,

despite its popularity and importance for control teaching, its

theory seems to be poor The most advanced approach to

be found in the literature [2], [15], based on pole placement

which neglects the system zeros, does not allow to determine

important control system properties As a result, no appropriate

tuning rules are available, and the controller tuning is to

be completed experimentally Unfortunately, there is neither

appropriate guidance on how to do it, nor it is known what

is the performance limit In consequence even poorly tuned

the paper is to fill this gap with respect to a two tanks

1 A detailed analysis of PI control for a single tank system is given in [5].

cascade system We also propose certain solutions to cope with large set-point changes that include not only anti-windup, but also model following, which can be accomplished using time-variable reference from a reference signal generator and possibly a feed-forward signal based on reference signal

In this section equations of tanks systems known from the literature will be written using normalized variables

h1(t)

h2(t)

appropriate conditions of equilibrium

A Basic equations The double tank system depicted in Fig.1 is described by the following set of equations

p

p

with

of discharge, h1(t) and h2(t) liquid levels and q(t) the input

levels in tank 1 and 2 respectively that fulfill

from which it follows

a2

Then the normalized equations are

1

p

p

where

A

cda2

s h2N

and di(t) = qi(t)/qN, i = 1, 2 Figure 2 shows water

levels when filling up the tanks and then draining them It

is interesting that when the input flow equals to zero then the

upper tank becomes empty earlier than the lower one

Fig 2 a) Water tank filling up and draining to zero levels, b) zoom

B Linearized model

Linearization of these equations leads to the transfer function

∆Y (s)

where

and

The value of this coefficient varies in extreme cases from 0 for

the first-order system to 1 for the the second-order one with

identical time constants

C Model parametrization and properties

From the control point of view, systems to be controlled

where 0 < γ ≤ 1 for double tank system This allows for

following transfer function

1

2 This can be attained using another time variable t0= t/2T N and related

Laplace transform variable s0= 2T N s, but we retain t and s in order not to

complicate the notation

1

Exemplary characteristics of the upper tank, and of the lower one described in (16), are displayed in Figures 3 and 4 It is interesting to notice that while the high frequency gain of a single tank does not depend on the working point defined by u0, the high frequency gain of the cascade tanks system does depend on that point

Fig 3 Bode plots for the model linearized at different values of u 0 ; a) first tank, b) cascade tanks

Fig 4 Step responses of linearized model for different values of u 0 ; a) single tank, b) cascade tanks

Let us consider a control system depicted in Fig.5 with a PID controller The ideal controller with the transfer function

Fig 5 PID control system



 (17)

is unfortunately nonrealizable, and its realizable form is



Ds

µs + 1

 , (18) where µ is a small number Relationships between zero-pole and classical three term form are as follows:

(19)

804

These relationships are valid for µ = 0, and approximately

valid for 0 < µ  1 Observe that the high frequency gain of

the controller in (18) equals to kc/µ, and for small µ it can

be large It is a common practice to consider control systems

with the ideal controller, and then to use its filtered version

at an implementation stage [2], [15] As long as dynamic of

a filter is negligible compared to the closed loop dynamic,

the influence of µ can be neglected Otherwise, as in our

case, µ should be considered as an important design factor

A characteristic feature of the considered system is that for

slow mode is responsible for the disturbance response, while

the fast mode, depending basically on µ, is responsible for

both, the control signal that attenuates the disturbance and

the reference response Our approach consists of two stages

Firstly we choose the parameters of the ideal controller so as

to obtain assumed dynamical characteristics of slow modes,

and then we choose the value of µ so that the slow mode is

not affected by the fast one, and that the fast mode reaches

assumed dynamical characteristics

A Analysis of systems with an ideal PID controller

Let us consider the control system with an ideal PID

controller Relevant transfer functions are

There is also Gyr(s) = −Gud(s) The characteristic

polyno-mial of the closed loop system is

the characteristic polynomial

Comparison of (23) and (22) leads to

2

with

Depending on the type of roots three cases can be distiguished:

where θ = ω/σ Then

∆

2) Different real roots:

where φ = δ/σ ≤ 1 Then

∆

B Disturbance responses

In this section the response y(t) caused by step-wise change of d1(t) = 1(t) is analyzed depending on the character

of roots

1) Complex roots:

Let us split y(t) into two parts

For α ≥ 5 there is

there is

2dα



q



√



2) Real roots:

y(t) =

σ2[(1 − α)2− φ2]

(47) Taking into account that y(t) = yα(t) + yσ(t) and assuming

α ≥ 5 one gets

dαh2Ω−2φα − Ω−1+φ2φ

2φ

where

3) Triple root:

unchanged

Fig 6 Dependence of σ2y m from α Solid line - complex roots, dotted line

- real roots.

C System with a filtered differential component

Assume now that we use the controller of (18), and we

want to choose µ in such way, that the control quality will

remain the same as for µ = 0 For µ = 0 and values of α

−ασ is at the real axis in the s-plane The degree of the system

with nonzero µ equals to 4, and its relative degree equals to

2 As a result, there will be an asymptote perpendicular to the

real axis crossing it at σa, where

1

A comparison between an ideal PID and a real PID control

systems is presented in Fig.7 It is worth to notice that

introduction of a filter practically does not affect the slow mode

roots s1,2

Fig 7 PID controller for s 1,2 = −2(1±j), s 3 = −10 (α = 5, θ = 1, σ = 2) with parameters: k c = 3.83, c 1,2 = 2.02 ± j1.69 a) ideal controller

µ = 0, s 1,2 = −2 ± 2j, s 3 = −10 b) real controller µ = (2σα)−1= 0.05,

s 1,2 = −2.04 ± j1.97, s 3,4 = −9.20 ± j10.65 Refer to zoom windows for details.

D Examples Exemplary disturbance outputs and control signals are displayed in Fig.8 It is clear that the faster the control signal approaches the value of disturbance with an opposite sign, i.e the faster the disturbance is cancelled, the smaller the control error Therefore, to obtain a good disturbance attenuation it

Fig 8 Disturbance output and control for large values of α and d = 1, σ =

2, θ = 1 Ideal PID controller with µ = 0 Solid line - complex roots, dotted line - real roots.

responses for various values of µ from (53) are displayed in Fig.9 It is interesting to notice that compared to the µ = 0 case of Fig.8 only the control signals have slightly changed

almost the same

Fig 9 Disturbance output and control for large values of α and d = 1, σ =

2, θ = 1 Real PID controller with µ > 0 Solid line - complex roots, dotted line - real roots.

For comparison with classical settings of [2], [15], plots for

806

in Fig.11 for different values of the working point uo

1, σ = 2, θ = 1 Ideal PID controller with µ = 0 Solid line - complex roots,

dotted line - real roots.

a) k c = 40, α = 20

b) k c = 3, α = 2

Fig 11 Influence of the working point PID controller with σ = 2; θ = 1,

µ = 0.0125 a) our design, b) classical design

Assume a new variable p = µs related to the fast time scale

3 Huge progress has been done since early 80’s of the previous century,

when K.J ˚ Astr¨om and A-B ¨ Osterberg [2] used 8-bit ADC’s with sampling

frequency of 10 Hz Contemporary aquisition devices with 12-20 bit effective

resolution and sampling frequency exceeding human hearing range allow using

PID controllers with smaller µ and higher gain, which calls for new tuning

rules for better control performance.

4 The method is motivated by[13], [14], [4]

and

lim

get the approximate transfer function

is

For µ small enough this leads to

s kc(s + c1)(s + c2)

poles

From these considerations the following simple design procedure results: Choose the values of α µ and σ such that µσ = 1/(2α), where α ≥ 5 determines the time scale

result, taking (19) into account there is

2

1

4αµ

GENERATORS

If the value of controller gain is high, and the derivative part detects fast changes of the control error, the control system

is prone to saturation Therefore a controller with the anti-windup augmentation, see e.g Fig.12, is necessary However,

as indicated in Fig.15.a, for a wider range of working points and reference changes, the behavior of the control system is not fully predictable In particular, one can observe overshoots and large changes of water level in the first tank, which can result in overflow As shown in Fig.16, anti-windup can also

be activated by disturbance d2(t) acting in the lower tank

A reasonable solution to prevent saturation is to deliver the reference signal which the control system is able to follow smoothly without exceeding the control signal limitations This can be done by passing the step-wise reference through a second order reference filter

r

with appropriately chosen parameters Finally, a feedforward signal calculated from the setpoint can facilitate the reference following

Fig 12 PID with anti-windup and feedforward signal input

Fig 13 Second order set-point and feedforward generator

Fig 14 Feedback and feedforward control system

Fig 15 a) PID with antiwindup and stepwise reference changes, b) 2nd order

reference model and PID with antiwindup; K ref (s) = 1.69/(s + 1.3) 2

Fig 16 Effect of disturbance acting in the upper tank (a) and the lower tank

(b)

investigated We proposed a very simple method of controller parameter settings that bases on asymptotic properties of the control system for larger values of controller gain and faster filter of the controller differential term than those used in classical approach This results in two time-scale behavior of the controlled system with very fast control signal and slower error signal We showed that the control error caused by the disturbance in the upper tank can be reduced to extremely small values, and that it is invariant to working point changes

As a result, we obtained better performance of control than that based on known literature The method of controller synthesis

is very simple and intuitive, and as such very well suited for introductory control course based either on laboratory experiment or on virtual interactive control course [11]

Con-sulting Inc., Canada, (1999).

[2] Astr¨om, K.J and A-B ¨˚ Osterberg A Teaching Laboratory for Process Control, Control Systems Magazine, 1986, Vol 6, Issue 5 pp 37-42.

Systems as Control Teaching Tools, accepted for the 20th World Congress of the International Federation of Automatic Control, IFAC

2017, 9-14 July 2017, Touluse, France [4] Blachuta M.J and V.D Yurkevich, ”Comparison Between Tracking and Stabilizing PI Controllers Designed via Time-Scale Separation Tech-nique”, Proc of the 12th Int.Conf on Actual Problems of Electronics Instrument Engineering (APEIE-2014), Novosibirsk, Russia, 2-4 Oct.

2014, Vol 1, pp 733-738 [5] Blachuta M, R Bieda and R Grygiel High Performance Single Tank Level Control as an Example for Control Teaching, accepted for 25th Mediterranean Conference on Control and Automation, (MED 2017) 3-6 July 2017, Valletta, Malta.

[6] Dorf R.C and R.H Bishop, Modern Control Systems, Prentice Hall, 2011

[7] Franklin G.F., J.D Powell and A Emami-Naeini, Feedback Control of Dynmic Systems, Prentice Hall, 2014

[8] Golnaraghi F and B.C Kuo Automatic Control Systems, Wiley, 2010

Apparatus as a Control Teaching Tool., 16th Int.Conf on Actual Prob-lems of Electronics Instrument Engineering (APEIE-2016), October 3

- 6, 2016, Novosibirsk, Russia.

[10] Grygiel R., R Bieda, M Blachuta On significance of second-order dynamics for coupled tanks systems, 21st International Conference on Methods and Models in Automation and Robotics, MMAR 2016, 29th August - 1st September 2016, Miedzyzdroje, Poland.

[11] Guzm´an, J.L., R Costa-Castell´o, S Dormido and M Berenguel An Interacivity-Based Methodology to Support Control Education: How to Teach and Learn Using Simple Interactive Tools, IEEE Control Systems Magazine, vol 36, No 1, 2016

[12] Ogata K Modern Control Engineering, Prentice Hall, 2010 [13] Yurkevich V.D Design of Nonlinear Control Systems with the Highest Derivative in Feedback World Scientific, 2004.

[14] Yurkevich V.D and D.S Naidu, ”Educational Issues of PI-PID con-trollers” Proc of the 9th IFAC Symposium Advances in Control Education, June 19-21, 2012, Nizhny Novgorod, Russia, 2012, pp 448-453.

[15] Reglerteknik AK, Laboration 2, Modellbygge och ber¨akning av PID-regulatorn, Assistenthandledning, Lund tekniska h¨ogskola, 2013

5 The paper has been supported by the Department of Automatic Control Grant No BK-204/RAu1/2017

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