Adaptive non-integer controller for water tank system

Adaptive non-integer controller for water tank system 1

1

1

Jerzy Baranowski , Waldemar Bauer , Marta Zagórowska , Aleksandra

2

1

Kawala-Janik , Tomasz Dziwi«ski , and Paweª Pi¡tek 1

1

AGH University of Science and Technology, Department of Automatics and Biomedical Engineering, Al. Mickiewicza 30, 30-059 Krak?w Email: [email protected], [email protected],[email protected], [email protected], [email protected] 2 Faculty of Electrical Engineering, Automatic Control and Informatics, Opole University of Technology, ul. Proszkowska 76, 45-756 Opole, Poland Email:[email protected]

In this article, the authors consider a new method of designing adaptive controller for non-integer order systems. The theoretical approach was veried with computer simulation of three-tank system. Further research will include implementation in a real system. Abstract.

1

Introduction

General results concerning theory of non-integer order systems can be found in [14, 19, 31]. Oustaloup method was presented in [28] and is analyzed among the others in [27, 29]. This approximation can be eciently used in simulations [11, 12, 16] and with appropriate care experiments [15]. Its sensitivity and stability problems during discretization were discussed in [30]. Dierent method of approximation is based on Laguerre functions and does not poses this sensitivity [3, 9, 36] however it is much more adequate for lters than for the controllers. Stability of non-linear non integer order systems was investigated in [10, 24, 37]. Applications of non-integer order subsystems was investigated among the others in [7, 9, 15]). Works on tuning of non-integer order controllers for linear systems can be found in [12, 15, 20, 26, 31] - both in simulational and in experimental setups. An interesting approach to non-integer fractional controllers was investigated in [2123], where a concept of robust non-integer controllers is described. Water tank system is well described in [8] where a survey of literature concerning its investigation is also given. Adaptive control is a well celebrated area of control theory. Classical results can be found in [2, 33, 35]. Certain interesting results in the theory of non-integer adaptive control can be found in [32, 34]. Results of successful applications can be found in [1, 13, 18, 25].

This paper is organized as follows. In the rst part, we present the system of tanks along with its mathematical description. Then the Oustaloup method is introduced, we included also time-domain approach. The last part of theoretical analysis consists of description of adaptive controllers of non-integer order. Then we present the results of simulation of this approach. In the last part, some extensions and further work are proposed.

2

Laboratory hydraulic cascaded three-tank system

A laboratory hydraulic cascaded three-tank system considered in the paper is depicted on the photograph in the gure 1. In the same gure its schematic diagram is presented as well. The system is located in the Department of Automatics and Biomedical Engineering of AGH. The installation consists of three

Upper tank (1)

Level sensor

Computer PC with universal input/output analog/digital extension card

Middle tank (2)

Lower tank (3)

Buffer tank

Fig. 1.

230 V 50 Hz Measurement interface Power amplifier Hydraulic pump with DC motor

A photograph and a schematic diagram of the laboratory installation

vertically arranged tanks: upper, middle and bottom. A side wall of each container have a dierent shape: rectangular, trapezoidal and quarter circular respectively. Their dimensions are given in the gure 3. The area of a water surface in the upper tank is constant but in two remaining tanks it varies with the water level. There is a fourth tank as well, located in the lowest position, serving as a water buer. A sliding-vane pump driven by a electric DC permanent magnet brush motor pumps water from the buer to the upper tank. From there the water ows through a constant valve to the middle tank and then to the lower and eventually to buer tank in a similar fashion. These ows are driven by gravity force and governed by the Toricelli's law. The electric motor is driven by a PWM signal from a power amplier. Water levels in the three tanks are measured with three pressure sensors and their signals are conditioned in an

m b)

a) S1

a=31 cm b=39.5 cm d=5 cm

V1

b

n=10.3 cm m=51.75 cm d=5 cm c=39.6 cm

h1 d

h2

V2

d n

a Fig. 2.

c

S2

Shapes and dimensions of three water tanks: a) upper, b) middle,

appropriate electronic interface. There is a PC computer dedicated to the laboratory system, equipped with a universal digital-analog input-output extension card RT DAC 4 PCI which measures analog water level signals and provides digital PWM control signal for the pump DC motor. A MATLAB-Simulink environment with Real-Time Workshop (RTW) and Real-Time Windows Target (RTWT) toolboxes is used to develop, build and test a real-time application.

2.1

Mathematical model of the system

A mathematical model of the laboratory installation can be derived from the law of mass conservation which for the three cascaded tanks takes the form of the three following dierential equations

d (% V1 ) = % q0 − % q1 dt d (% V2 ) = % q1 − % q2 dt where tanks,

% is a water density, V1 , V2 , V3 q0 is a control-dependent pump

(1) (2)

are water volumes in three consecutive voluminal ow and

q1 , q2 , q3

are level-

dependent valves voluminal ows. For a partially lled open tank a derivative of a volume

where

S

V

with respect to time

t

can be expressed as follows

dV (h(t)) dV (h) dh(t) ˙ = = S(h(t)) h(t) · dt dh h=h(t) dt

(3)

h

is a water level

is an area of an open water surface in the tank and

measured from the bottom. With an additional assumption of a constant water density one can obtain from (1)(2) and (3) the following state space equations


 1  h˙ 1 (t) = q0 (u) − q1 h1 (t) S1 

 1
q1 h1 (t) − q2 h2 (t) h˙ 2 (t) = S2 h2 (t)

(4)

(5)

where state variables

h1 (t), h2 (t)

and

h3 (t)

areas in

cm3

cm in upper, S3 (h3 ) are the

are the water levels in

middle and bottom tank respectively. Symbols

S1 , S2 (h2 )

and

of open water surfaces given by following formulae

S1 = a d = const   m−n S2 (h2 ) = d n + h2 c

(6) (7)

c, d, m, n and r are tanks physical dimensions dened on gure 3. Function u ∈ [0, 1] of a PWM 3 signal and a voluminal water ow q0 in cm /s produced by the pump. This

where

q0 (u)

characterizes the relationship between a duty factor

mapping can be approximated with a polynomial of fth degree

q0 (u) = w5 u5 + w4 u4 + w3 u3 + w2 u2 + w1 u + w0 Functions

q1 (h1 ), q2 (h2 )

and

q3 (h3 )

(8)

describe dependencies between water levels

and water ows through valves. According to the slightly modied Toricelli's law these relationships can be approximated as

qi (hi ) = Ci where

Ci

and

Di

p

Di + hi ,

i ∈ {1, 2, 3}

are constants which need to be identied and

(9)

i

is the tank

index (1 for upper, 2 for middle and 3 for bottom one).

Table 1.

Results of the identication of the pump  coecients wj of the polynomial

q0 (u) w5 w4 w3 w2 w1 w0

1.9287 −6.5254 8.9191 −6.2047 2.4298 −0.258

Result of identication are gathered in the table 2.

√Results of the identication of valves  parameters Ci and Di in relationships Di + hi

Table 2.

hi = C i

i 1 2

Ci 29.405 33.260

Di 5.9217 4.0083

3

Oustaloup method

Oustaloup lter approximation to a fractional-order dierentiator

G(s) = sα

is

widely used in applications [27]. An Oustaloup lter can be designed as

N Y s + ωi0

Gt (s) = K where:

(10)

s + ωi

i=1

ωi0 = ωb ωu(2i−1−α)/N ωi = ωb ωu(2i−1+α)/N K = ωhα s ωh ωu = ωb

(11)

Approximation is designed for frequencies

ω ∈ [ωb , ωh ]

and

N

is the order of the

approximation. As it can be seen its representation takes form of a product of a series of stable rst order linear systems. As one can observe choosing a wide band of approximation results in large of poles spacing from close to

−ωh

is logarithmic with a grouping near

ωu

and high order

N result in −ωb . This

to those very close to

−ωb

spacing spacing

and causes problems in discretization.

Wide band of approximation is on other hand desirable, because approximation behaves the best in the interior of the interval and not at its boundary, so certain margins need to be kept.

4

Time domain approximation

The proposed approach is to realize every block of the transfer function (10) in form of a state space system. Those rst order systems will be then collected in a single matrix resulting in full matrix realization. This continuous system of dierential equations will be then discretized.

4.1

Realization

One can easily observe that for zero initial condition

s + ωk0 ⇐⇒ s + ωK

(

x˙ k = Ak xk + Bk uk yk = xk + uk

where

Ak = − ω k Bk = ωk0 − ωk Ci = 1 Di = 1

Full Oustaloup approximation can be then realised as

   KB1 A1 0 0 . . . 0  KB2   B2 A2 0 . . . 0         B 3 B 3 A3 . . . 0  x˙ =   x +  KB3  u    ..  . . . . . . . .. .   ..   . . . . B N B N . . . B N AN KBN   y = 1 1 . . . 1 1 x + Ku 

(12)

or in brief

x˙ = Ax + Bu

(13)

y = Cx + Du What can be immediately observed is that the matrix

A is lower triangular. This

is an extremely important in the case of this problem, as all its eigenvalues (poles of transfer function (10) are on its diagonal, so there is no need for eigenvalue products, which would lead to rounding errors. That is why discretization of (12) has a structure preserving property [5].

5

Adaptive non-integer controller

In this paper a new approach to to adaptive non-integer controllers is considered. In literature most popular approach is to either use the fractional variant of MIT rule (with non-integer derivative for evolution of adapted parameter) or using combining non-integer derivative of deviation with steepest descent as a part of adaptation gain. We propose dierent approach - use the classical MIT rule for adapting parameters of non-integer order controller, however the steepest descent is computed using variational equation. The proposed controller structure is

e(t) = r − h2 (t) Z u(t) = Ke(t) + I

t

e(t)dt + θ0C Dµt e(t)

(14)

0 where

r

is the reference value

µ ∈ (0, 1)

is a xed derivative order and

θ

is

the adapted parameter. This controller structure was chosen because of system specics. Integer order integration is needed for elimination of steady state errors, while inuence of non-integer derivative term can vary depending on the conditions system is in. It was verie, that strong derivative action is not needed, and

µ = 0.1

was completely satisfactory. Gain was chosen using adaptive algorithm.

Adaptation of

θ

was realized by the celebrated MIT rule [2].

∂e(t) θ˙ = −γe(t) ∂θ

(15)

Because the controlled system is nonlinear, standard application of adaptation formulas is not possible. That is why a variational equation (see for example [4]) is used. With variational equation derivative of deviation with respect to parameters is given by:

∂e(t) = −ϕ2 (t) ∂θ

(16)

where

ϕ˙ = J(h1 , h2 , e)ϕ + θCψ − θDϕ2 + Cx + De ψ˙ = Aψ − Bϕ2 where

x, A, B, C, D

are given by (12) -(13).

J(h1 , h2 , e)

(17) (18)

is the Jacobi matrix

of nonlinear system with PI part of the controller.

6

Simulation Result

We implemented the above described method in a simulational analysis of tank system. The adaptive controller was compared with two more controllers - classic, integer order PI and fractional PID controller with non-integer derivative. The reference values was 10 and 11 centimetres and we wanted to verify the performance of each type of control. As we can see in gure 3, the adaptive controller has the smallest maximal value. However, it introduces visible oscillations. It is also quite eective concerning tracing of a given curve - despite oscillations it reaches the reference values faster than non-integer order PID and classical PI.

7

Conclusions

This article summarizes an early stage of research concerning fractional adaptive control. The proposed approach works ne for this simulation, however, there are some issues that need further consideration. First of all, the choice of adaptation parameter is an still an open question. Moreover, in order to implement this solution in a real-time system and then in existing system of tanks, we need to take into account the numerical part of implementation (see e.g. [5, 6, 17, 30]). Also, one of the main drawbacks that need further analysis is the fact, that this adaptive approach requires use of mathematical model and its linearisation.

ACKNOWLEDGEMENT Work partially realized in the scope of project titled Design and application of non-integer order subsystems in control systems. Project was nanced by National Science Centre on the base of decision no. DEC-2013/09/D/ST7/03960 and partially from AGH statutory funds.

14

12

10

8

PI Reference 0.1

PID

6

Adaptive PID0.1

4

2

0

0

Fig. 3.

500

1000

1500

2000

Simulation result - comparison of three types of controllers

2500

References

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