evaluating the performance of control schemes for hybrid

Evaluating the performance of control schemes for hybrid systems

A Practical Approach

JAROSLAV HLAVA* AND BOHUMIL ŠULC**

* Department of Control Engineering

Technical University of Liberec, Faculty of Mechatronics 416 17 Liberec, Hálkova 6 CZECH REPUBLIC

** Institute of Instrumentation and Control Engineering

Czech Technical University in Prague, Faculty of Mechanical Engineering 166 07 Praha 6, Technická 4

CZECH REPUBLIC

Abstract: - Hybrid systems are now one of the most important research topics in the field of system and controltheory. They undoubtedly pose many interesting challenges from the purely theoretical point of view, however,the ultimate motivation for their study is practical. There many industrial plants whose mathematical models arehybrid, i.e. exhibit nontrivial interactions of continuous valued and discrete valued variables and elements.Despite this practical motivation, most papers on the control of hybrid systems are limited to theoretical aspectsof control design and they are concluded with simulational experiments at best. This paper proposes a morepractical alternative. It describes a laboratory scale plant that is designed in such a way that it exhibits most ofthe hybrid phenomena typical of process control applications. The paper includes a detailed description of theplant structure together with its mathematical model. Further, it focuses on hybrid control experiments that canbe performed with this plant in order to test and evaluate various hybrid control schemes.

Keywords: - Hybrid systems, Laboratory scale models, Hybrid control

1 Introduction

Many industrial plants are modelled as systemsthat have neither purely continuous nor purelydiscrete (or in a special case logical) dynamics butinclude both continuous valued and discrete valuedvariables and elements (on/off switches or valves,speed selectors, etc.) Moreover, many plants (e.g.boilers) exhibit considerably different dynamicbehaviours in different operation modes or indifferent working points. Such plants can bemodelled as a combination of several continuousmodels and discrete valued variables that determinewhich of these models is valid in the currentoperation mode or range. In addition, discretefeatures are often introduced by the controller even ifthe system to be controlled is continuous (gainscheduling, controller switching, etc.).

These facts have been the motivation behindgrowing research interest in the theory of hybridsystems (see e.g. [4] for a short survey), whichincludes both continuous and discrete valueddynamics and variables in one general framework.Many important results concerning hybrid systemshave been achieved recently (see e.g. [1] and [10]).

However, this field is still mostly the topic of highlytheoretical papers and PhD level courses. Its highapplication potential remains almost completelyunexploited. The testing of hybrid control schemes isusually done using simulational experiments at best.This paper proposes a more realistic and practicalalternative. It describes a laboratory scale plant thatcan be used to test and evaluate various controlstrategies for hybrid systems in a setting that is muchcloser to real applications. A considerable deal ofattention is devoted to the description of possibleplant configuration variants and control experimentsthat can be performed with the plant. Besides controlexperiments and other research purposes, this plantcan also be used for educational purposes. A detailedtreatment of possible educational applications of thisplant was given by the authors elsewhere [4].

It is quite a sad sign of the preoccupation of thecurrent control literature with purely theoreticalproblems that a laboratory plant similar to the onedescribed in this paper can hardly be found in theliterature. There are several technical reportsdescribing complex laboratory plants designedmainly for the purposes of European research

Proceedings of the 6th WSEAS Int. Conf. on Systems Theory & Scientific Computation, Elounda, Greece, August 21-23, 2006 (pp157-162)

projects (in particular the plants used as case studiesin the Esprit project Verification of Hybrid Systemswww-verimag.imag.fr/VHS/). Other papers justsketch simple model plants that are used to illustratesome ideas described in the paper but were notintended to be built and used for real testing (e.g.,[7]). Still other papers include hybrid control ormodelling experiments with laboratory models.However, although these models exhibit certainhybrid phenomena, they were primarily designed forcontinuous control experiments (see e.g., [6]). Nolaboratory plant designed primarily for experimentswith hybrid control is described anywhere.

2 Hybrid systems

Hybrid phenomena arise as a result of the interactionof continuous and discrete controls, outputs andstates. To avoid misinterpretation, it should perhapsbe emphasised that the words continuous and discreteare used to express the fact that the variables takevalues from a continuous set (usually an interval ofreal numbers) or a discrete set containing a finitenumber of elements. Typically, some of the controlstake only a finite set of values, some of the outputsare quantized and the differential equation

( ))t()t( xfx =& (1)modelling the continuous dynamics depends on somediscrete phenomena. The two most important classesof this dependence are the following. Vector field f(.)can change discontinuously either when the state x(.)hits certain boundaries or in response to a controlcommand change. This behaviour is calledautonomous or controlled switching respectively.State changes or control commands may also resultin state x(.) jumps (or equivalently in vector fieldimpulses). If this is the case, we speak ofautonomous or controlled jumps (impulses).

Many mathematical models of hybrid systemshave been proposed. Most of them are included in ageneral hybrid system model as described e.g. in [2].A hybrid system is given by

( )( ))t(),t(),t()t(

)t(),t(),t()t(

umxgy

umxfx

=

=&(2)

( )( ))t(),t(),t(),t()t(

)t(),t(),t(),t()t(

σumxo

σumxφm

ϕ=

=+

+

(3)

( ))t(),t(),t(),t()t( σumxψx =+ (4)

Equations (2) model the continuous part of thesystem (x(t), u(t) and y(t) are continuous state, inputand output respectively). Unlike state equations of apurely continuous system they include discrete statem(t). This state indexes the vector fields f(.,.,.). Thedevelopment of the discrete state is described by (3)

where σσσσ(t) is discrete input and o(t) is discreteoutput. Equations (3) are formally similar to thestandard state equations of continuous systems.However, the description of the discrete part of thesystem can also be in the form of a finite automatonor a discrete event system, if desirable. Equation (4)models the state jumps (if there are any).

The model described by (2)-(4) is fairly complexand it is not easily tractable in its full generality.Thus, many research results have been obtained forsimpler models, in which state jumps are not present(eq. 4 omitted) and the continuous part has a simplerform. A particularly important special case ispiecewise affine (PWA) systems, where (2) reducesto

))t(()t())t(()t())t(()t(

))t(()t())t(()t())t(()t(

mgumDxmCy

mfumBxmAx

++=

++=&(5)

Despite its simplicity, (5) captures manyimportant phenomena such as switching, relays andsaturations, and it can also be used to approximatenonlinear dynamics with several locally validlinearizations. Moreover, it has recently been proved[3] that PWA systems are equivalent to several otherclasses of hybrid systems (mixed logical dynamicalsystems, linear complementarity systems, etc.). Thus,PWA systems are of special importance becausemost results and approaches in hybrid systems theoryhave been obtained either directly for PWA systemsor for an equivalent class.

3 The laboratory scale plant

The plant is sketched in Fig. 1. Similarly as in mostprocess control applications, the measured and con-trolled variables are water level, temperature andflow. The sensors and actuators were (if possible)chosen from standard industrial ranges. Thus,although the plant does not represent any particularindustrial process exactly, both the plant itself and itsinstrumentation are well representative of manyplants commonly used in the process industries.

The basic components of the plant are three watertanks. Water from the reservoir mounted under theplant can be drawn by pumps 1 and 3 (delivery rate0-5 l/min) to the respective tanks. In both cases, thedelivery rate can be continuously changed and thewater flow rate is measured with a turbine flowmeter(VISION 2000, 0.5-5 l/min).

The flow from pump 3 is fed directly to tank 3.The flow from pump 1 goes through a storage waterheater (2 kW, 5 l) and it is further controlled by anon/off solenoid valve S1. Due to the built-in pressureswitch, the pump is automatically switched off if S1is closed and the internal tank of the heater is full.


The temperature at the heater output is measuredwith a Pt1000 sensor TT4. The power consumptionof the heater can be changed continuously.Depending on the selected control scenario, theinflows to tanks 1 and 3 can be used as manipulatedvariables or as disturbances with specified flow ratesand also with a specified temperature in the case ofinflow 1.

Tanks 1 and 2 are thermally insulated to make theheat losses negligible. Tank 3 and the water reservoirare not insulated, and if the air cooler motor isswitched on, their temperature roughly equals theambient temperature in the laboratory. Tanks 2 and 3have a special shape that introduces a change of thedynamics at a certain level. Their behaviour isdescribed by switched models. Water from tanks 1and 3 is fed to tank 2. The flow from tank 1 iscontrolled by solenoid valves S3, S4 (kv=5 l/mineach), and it can be changed in three steps: no valveopen, one open, both valves open. Closing andopening the valves is instantaneous. Tank 1 can beby-passed by closing S1 and opening S2. The flowfrom tank 3 is controlled by a pump. Its delivery ratecan be changed in four steps (0, 1/3, 2/3, max.).

The other heater (custom-made, 800 W) ismounted at the bottom of tank 2. Water from tank 2is drawn by a pump. The turbine flowmeter and slave

controller FC2 allow us to simulate the changingdemand of the downstream processes. The watergoes back to the reservoir through an air-water heatexchanger where it is cooled. This arrangement isused to keep the water temperature in the reservoirroughly constant during the experiments.

The water levels are measured by pressuresensors. These can be configured either as continuoussensors or as level switches (indicating threedifferent water levels). The plant is controlled from aPC using two data acquisition boards. The realappearance of the plant is shown in Fig. 2.

4 Mathematical model of the plant

A reasonably accurate first principles model of theplant can be derived using mass and energy balanceequations. The liquid (water) can be consideredincompressible. The mass balance equation of asingle tank is then

)t(q)t(q

dt

)t(dh)h(Adx)x(A

dt

d

dt

)t(dV

outin

)t(h

−=

=

= ∫

0 (6)

where h is water level, A(h) is the cross sectionalarea of the tank at level h, and qin and qout are inletand outlet volume flow rates.

Fig. 1 Structure of the plant (FT, LT, TT stands for flow, level and temperature transmitter respectively, FC –flow controller, S–solenoid valve, M – motor, r1= 6 cm, r21= 6 cm, r22= 3 cm, r31= 7 cm, r1= 3 cm), tankheight 80 cm, l1=40 cm, l2=40 cm


Assuming constant liquid heat capacity c,negligible heat losses and well mixed tank, theenergy balance equation of a single tank can bewritten as

( )( ) ( )( ) )t(P)t(c)t(q

)t(c)t(q)t()t(cVdt

d

refout

refininref

+−−

−=−

ϑϑρ

ϑϑρϑϑρ(7)

where P is heater power output, ϑ in inlet watertemperature, ϑ water temperature in the tank. Due tothe assumption of perfect mixing, the outlet watertemperature can be equated with ϑ. Eq. (7) can bemodified to

( ) c)t(P)t()t()t(q)t()h(V inin ρϑϑϑ +−=& (8)The volume flow rate from a tank with water

level h can be expressed by

ghk.q v10= (9)

Combining (6), (8) and (9), the plant model is built

( ))t(gh)t(k.)t(q)A()t(h v 11111 101 σ−=& (10)

( ) )t(hA)t()t()t(q)t( 111411 ϑϑϑ −=& (11)

1 if10

1

0 if10

1

1

202

1122

1

202

1121

2

=

−+

=

−+⟨=

m)t(qq)t(

)t(gh)t(k.)A(

m)t(qq)t(

)t(gh)t(k.)A(

)t(h

v

v

σ

σ

σ

σ

&

(12)

( )

( )

( )

( )1 if

10

0 if

10

11222121

2302

2111

1221

2302

2111

2

=−+

+−+

−

=

+−+

−

⟨=

m)l)t(h(AlA

c

)t(P)t()t(q)t(

)t()t()t(gh)t(k.

m)t(hA

c

)t(P)t()t(q)t(

)t()t()t(gh)t(k.

)t(

v

v

ρϑϑσ

ϑϑσ

ρϑϑσ

ϑϑσ

ϑ&

(13)

1 if

0 if

232

023

222331

023

3

=−

=∆−

−

⟨=m

A

q)t()t(q

m)lr)t(hr(

q)t()t(q

)t(hσ

π

σ

& (14)

12

121 if1

if0

l)t(h

l)t(h)t(m

>

≤⟨=+ (15)

Fig.2 Photo of the actual laboratory scale plant (the plant is composed of two panels connected with a pipe)


23

232 if1

if0

l)t(h

l)t(h)t(m

>

≤⟨=+ (16)

where 2ii rA π= , 3231 rrr −=∆ , m1 and m2 are discrete

state variables, discrete valued input σ1 assumesvalues 0,1,2 (no valve open, S3 open, S3 and S4open), σ2 assumes values 0, 1, 2, 3 (pump 4 deliveryrate is 0, 1/3 max., 2/3 max., maximum), q0 is thedelivery rate of pump 4 running at 1/3 of maximum.

The measurable outputs are identical with thestate variables. The water levels can be used ascontinuous-valued or discrete-valued outputsdepending on the configuration of the level sensors.The equation for ϑ3 is not included in the model. Astank 3 is neither heated nor insulated, ϑ3 is roughlyequal to the ambient temperature, and the mixingdynamics of tank 3 are not important. Valves S1, S3and S4 can be closed and S2 used as a control input.The plant model is then simplified: h1 and ϑ1 areomitted and the square root term in (12), (13) isreplaced by q1(t)σ1(t) (σ1 =0 or 1). The vector fielddefined by (10) to (14) involves both controlled andautonomous switching. Controlled switching due tochanges of discrete valued inputs σ1, σ2 results invector field discontinuity in (10), (12), (13) and (14).Autonomous switching due to changes of h2 and h3results in vector field discontinuity in (12). Thedynamic behaviours of (13) and (14) are alsoswitched, but the vector field remains continuous.

5 Proposed control experiments

Although this plant is fairly simple and easy tounderstand, it allows us to define many controlscenarios that can be used to test and evaluatevarious aspects of hybrid control algorithms. Someinteresting scenarios will be outlined below, howevermany other scenarios are possible.

First, it is possible to define uncomplicatedcontrol tasks such as water level control in tank 2 or3 or temperature control in tank 2. In this case, thecon-trolled system features switched dynamics, andcertain basic classes of hybrid systems can bedemonstrated to the students. For instance, if thecontrol objective is to control h2 using q1 as themanipulated variable (valve S2 constantly open, S1,S3, S4 closed) and q2 being a disturbance (in thisscenario both q1 and q2 can be changed in severalsteps) the controlled system is a simple integratorhybrid sys-tem. Alternatively, the objective can bewater level control in tank 3. The dynamics of (14)are nonlinear for 23 lh ≤ . The nonlinear range of h3can be divided into several sub-ranges and the wholesystem including switching and non-linearity can be

approximated with a piecewise affine (PWA) system.More complicated scenarios use two tanks. One

suggestion (inspired in part by [11]) can beformulated as follows. Tank 1 serves as a buffer thatreceives water from an upstream process. Water flowrate and temperature are disturbances. Since the plantincludes continuously controlled heater H1 and slaveflow controller FC1 these disturbances can follow adefined function of time as required by a particularexperiment. The main control objective is to deliverthe water to a downstream process at a desiredtemperature. The flow demand of the downstreamprocess is another disturbance. Valves S1, S2 and S3are discrete valued manipulated variables and thepower output of heater H2 is a continuousmanipulated variable. The main control objectivenecessarily includes several auxiliary objectives.Tank levels must be kept within specified limits, andoverflow as well as emptying of the tanks must beavoided. It is also necessary to avoid the necessity toclose valve S1 in order to prevent tank 1 overflow. Ina real control situation, closing S1 would mean thatwater from the upstream process cannot flow to thebuffer but must be re-routed to the environment.

The controlled system is again hybrid andnonlinear, allowing approximate modeling, e.g., as aPWA system. However, unlike the previous simplerscenarios it includes nontrivial interactions ofcontinuous and discrete controls. For example, ifvalve S3 is already open and S4 must also be openedto avoid tank 1 overflow, temperature ϑ2 may also besignificantly affected, particularly if temperatures ϑ1and ϑ2 are much different. The plant in thisconfiguration can be used to test various approachesto hybrid system control. It is possible to follow thetraditional approach and to design the water levelcontrol logic and continuous heater controlindependently. Alternatively, a hybrid approach canbe followed and both the discrete and the continuousmanipulated variables can be controlled with a singlecontroller designed using hybrid controller synthesismethods (e.g., those developed in [1]). The controlperformances achieved with both approaches can beevaluated and compared.

Even more complicated interactions can occur inscenarios that use all three tanks. In this case, thereare many possible combinations of manipulated,controlled and disturbance variables. For example,tanks 1 and 3 can be used as buffers that receivewater from two upstream processes. Flow rates q1, q2and q3 as well as temperature ϑ4 (ϑ4>ϑ3) aredisturbances. The control task is to deliver the waterat a desired temperature and flow rate q2 to adownstream process using S3, S4, pump 4 and heaterH2 as control inputs. The control objective must be


achieved while satisfying necessary constraints suchas avoiding emptying and overflow of the tanks.Optimization sub-objectives can be included, e.g. itmay be desirable to make use of the fact that ϑ4>ϑ3in order to minimize the power consumption ofheater H2.

6 Conclusion

The laboratory-scale plant described in this paperexhibits hybrid phenomena typical of process controlapplications. The variety of possible controlscenarios outlined in the previous sectionsdemonstrates the considerable flexibility of the plant.It can be configured just to demonstrate simplehybrid behaviours such as switching between twodynamic models or logic control of continuousplants. However, in its full configuration, it poses agreat challenge to hybrid control synthesis because itincludes several non-linearities and nontrivialinteractions of continuous and logical/discretedynamics. Sufficiently exact modelling using themost important hybrid dynamic models such as PWAor mixed logical dynamical systems is possible.Thus, it can be used to evaluate most of the recentapproaches to hybrid systems control in a setting thatis much closer to real application than meresimulation.

Acknowledgement:

This research has been supported by the CzechScience Foundation within project 101/04/1182.

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