Pressure Regulation in Nonlinear Hydraulic Networks by Proportional

8/2/2019 Pressure Regulation in Nonlinear Hydraulic Networks by Proportional

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PRESSUREREGULATIONINNONLINEAR

HYDRAULICNETWORKSBYPROPORTIONALANDQUANTIZEDCONTROLS

Ankit Deshmukh

11EE64R12


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CONTENTS:

1. Introduction

2. Model:

a) hydraulic networks.

b) assumptions.c) model for nonlinear hydraulic networks.

3. Proportional controllers for practical regulation.

4. Pressure regulation by quantized control.

5. Experiments.

6. References.


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1. INTRODUCTION

DISTRICTHEATINGSYSTEMS

system for distributing heat generated in a centralized

location for residential and commercial heating

requirements.

Consists of a large scale hydraulic network.

Our objective : regulating the pressure at the end-users

to a constant value despite the unknown demands of the

users themselves.


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2. MODELA) HYDRAULICNETWORKS

Valves :

Pipe :

Pump :


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B) ASSUMPTIONS

Obtain a graph : F

a= number of nodes

b=number of edges

Assumption 1. : F is a connected graph.

T = tree with a-1 edges and b a +1 loops

G = set of chords.

The flow through each chord in G is independent of

each other.

We exploit the analogy between the electrical and hydraulic

systems


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Assumption 2.: each user valve isin series with a pipe and a pump.

Each chord in G corresponds to apipe in series with the user valve.

Assumption 3.: There exists one and only onecomponent called the heat source. It correspondsto a valve of the network and it lies in allfundamental loops.


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The set of flows and pressures in the network must

fulfil the well-known Kirchhoffs node and loop laws.

Where B is the fundamental loop matrix: n x b .

If the first n components are chords then

Where I is identity matrix: n x n

and F is : n x a-1 with entries as -1,1,0


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Under these assumptions we can select thedirections of the edges such that the values takenby B are 0,1.

Also by kirchoffs current law to this circuit we get


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C) MODELFORNONLINEARHYDRAULICNETWORKS

The vectors of the flows and the pressure drops of eachedge in the graph

Then each component obeys:

For pump :

For pipe :

For valve :


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Inserting the following terms

We get the nonlinear model:


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3. PROPORTIONALCONTROLLERSFORPRACTICALREGULATION

The output of the control system: set of pressuresacross the user valves.

To design a controller , we have the following form:

Ni is gain of the control law. Define :


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Define error coordinates:

such that

for all

We have to prove that there exist gains

for which we define lyapunov function


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And the control law

Now the derivative of the lyapunov function

defining a set

and

we will define regions and prove that the derivative ofthe lyapunov function is negative.


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Region 1:

for

and

then choosing Ni such that

we get


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Region 2 :

Thus we get the controller gains as

system we are dealing with is largely uncertain. Thegains are tuned by a trial-and-error procedure.


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4. PRESSUREREGULATIONBYQUANTIZEDCONTROL

These controllers take values in a finite set andchange their values only when certain boundariesin the state space are crossed, and thereforecontrol values can be transmitted over a finite-

bandwidth communication channel.

Quantized controllers:

whose quantized version is


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Which is an equation with a discontinuous righthand side. Hence the solution is given in aKrasowskii sense.

The quantizer is of the form:


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5. EXPERIMENTS

Test setup:


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RESULTS:

Step input : 0.2 bar to 0.45 barproportional quantized


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6. REFERENCES

1. De persis, C.; Kallesoe, C.S.; Pressure regulation

in nonlinear hydraulic networks by positive andquantized controls, IEEE transaction on controlsystem technology,volume : 19 No.6,pp.1371-1383,Nov.2011

2. Georgia Kaliorah and Alessandro Astolfi,Stabilization with positive and quantized control,Decision and control, 2002, Proceedings of the 41stIEEE Conference, Publication Year: 2002 , Page(s):1892 - 1897 vol.2.

3. B. Bollobas , Modern graph theory. Springer verlag1998.

4. Hassan khalil, Non linear control systems.

5.District heating systems: www.wikipedia.org
http://www.wikipedia.org/http://www.wikipedia.org/

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Pressure Regulation in Nonlinear Hydraulic Networks by Proportional