LQG/LTR-Based Control of an Integrated Heating System

A. H. Noureddine, Faisal Mnif, and A. Ben SelemaDepartment of Electrical Technology

Buraydah College of TechnologySaudi Arabia

ABSTRACT

In this paper an optimal controller is designed to optimize the operation of an integrated heating system. The design methodology used here is the linear quadratic gaussian with loop transfer recovery widelyknown by LQG/LTR. A linear quadratic gaussian controller is first developed which with the aid of aKalman filter the system is tuned for high robustness to model uncertainties, disturbance rejection by theloop transfer recovery method. The controller was implemented and the simulation shows promising results.

1. INTRODUCTION

As conventional energy resources are dwindling fastwith a corresponding rise in prices, considerable attentionis being focused on utilizing more efficient systems. Traditionally, space heating and domestic hot water(DHW) demands are met with two independent heatingsystems. Hydronic boilers are widely used in areas wherespace-heating demands are high, whereas DHW demandsare met via electric water heaters. In recent years the focushas been shifted towards the use of integrated heatingspace units capable of supplying both heating demands [1]. It has been reported in the literature that integratedheating systems enjoy a 20-25% economic advantage overconventional independent systems [1].

The recent attention towards the use of integratedheating systems provides a solution to a major problem incold areas such as North America. On the other hand, theuse of such heating approach could be extremely beneficialin normally hot areas, like the Middle and North Regionsof Saudi Arabia, that experience long and dry cold seasonswhere freezing temperatures are quite common. In suchareas houses are more or less prepared to deal with hotseasons during which normal temperatures range from 42to 47 degrees Centigrade. As a result, cold weatherpreparation in terms of the dwelling infrastructure is rareexcept for regular heating devices along with the use ofelectric heaters for DHW demands. Average houses usethree electric heaters that are scattered inbathrooms/kitchens. On the other hand, the more commonlarge houses employ a higher number of electric heatersfor DHW needs. Therefore it could be extremely feasible(economically and healthwise) to equip houses withintegrated heating systems.

In this paper, we are interested in the development of a

control strategy to optimize the operation of an integratedheating system. The model used here is based on themodel presented in [1]. In this system, Figure 1, heatfrom combustion is transferred to the water in the boilerthat is circulated in baseboard radiators for space heatingand DHW demands. In an integrated system, only onecircuit is activated at a given time. This is accomplishedusing a three-way valve, Figure 1, to activate either spaceheating or DHW. Normally, boilers are operated in anON-OFF cycle that induces losses, which affect thethermal efficiency significantly. This in turn affects thetemperature of both space and DHW. Hence, thermalcomfort and system efficiency are compromised.

Historically, such efficiency and comfort problems areaddressed via classical single-input single-outputproportional controllers. However, an integrated system isa multivariable system. In this paper, we develop amultivariable robust controller to operate the system shownin Figure 1.

2.SYSTEM MODELING AND REPRESENTATION

A schematic representation for an integrated heatingsystem is shown in Figure 1, [1]. Figure 1, Schematic diagram of an integrated heating system

The system can be described by the following three energybalance equations [1-5].

fgTfgCfgmzTbTbUArsTbTpCUU

rwTbTpCUUHUUdt

bdTbC

&−−−−−

−−=

)()(max11

)(max33max22 (1)

)()(max33 zTwTwUAwqrsTbTpCUUdt

wdTwC −−−−= &

(2)and

)()(

)()(max11

aT

zT

zUA

zT

wT

wUA

zT

bT

bUA

rsT

bT

pCUU

dtz

dT

zC

−−−−

−+−=(3)

Where the following term definitions are used:Tb is the boiler temperature,Cb is the boiler storage capacity,U2U2maxH is the heat input from combustion,U3U3maxCp(Tb-Trw) is the heat supplied to the hot watertank,U1U1maxCp(Tb-Trs) is the heat supplied to the space,UAb(Tb-Tz) is the loss in the boiler exterior surface,UAw(Tw-Tz) is the loss in the boiler exterior surfaces ofthe hot water tank,

fgTfgCfgm& is the loss in the exhaust gases,

and q& is the heat extraction rate.

The heat loss in the exhaust gases can be approximated bythe following [3]:

bTUfgTfgCfgm 2λ=& (4)

and the heat extraction rate is given by

)( wiwpww TTCmq −= && (5)

To consider system efficiency that is influenced by thereturn water temperature, we use the energy balance on theheat exchanger in the DHW tank that is given by thefollowing.

)()( max33max33 wbbwrwbp TTUUUTTCUU −=−(6)

where Ubw is the heat transfer coefficient (generally knownfor a given configuration). Further, if the effectiveness ofthe baseboards is assumed to be constant, one can write:

)()( max11max11 sbbsrsbp TTUUUTTCUU −=−(7)

Then the following mathematical model gives the overallsystem description.

bzbbrsbp

rwbpb

b

TUTTUATTCUU

TTCUUHUUdt

dTC

2max11

max33max22

)()(

)(

λ−−−−

−−−=, (8)

)()()(max33 zTwTwUAwiTwTpCmrwTbTpCUUdt

wdT

wC −−−−−= & (9)

and

)()(

)()(max11

aT

zT

zUA

zT

wT

wUA

zT

bT

bUA

zT

bT

bsUUU

dtzdT

zC

−−−−

−+−=

(10)

It is obvious that equations (10)-(12) must be linearized inorder to obtain a robust multivariable control design tooptimize system performance. Thus, the system can belinearized about an operating point (see Appendix). Theresulting linear system is given by

XCY

aTDwiTDUBXAX

∆=

∆+∆+∆+∆=∆ 21&

(11)

Where X and U are the state and control vectors,respectively. The matrices A, B, C, D1 and D2 are thesystem matrices and are given in the Appendix.

The above system was analyzed to determine thevarious factors needed to be considered during the designprocess. Figure 2 shows the minimum and maximumsingular values of the closed loop system at an operatingpoint (Table 2, Appendix). The plant has widely separatedsingular values that lead to poor robustness characteristics.

Figure 2. Singular values of the uncompensated system

Due to the system's poor robustness properties, afeedback controller must be implemented to ensurestability and good robustness at all operating conditions. A design technique based on the linear quadratic gaussianregulator with loop transfer recovery (LQG/LTR) thatprovides estimates of all unmeasurable states to thecontroller is set forth.

3. DESIGN PHILOSOPHYThis paper uses a robust multivariable loop shaping

design methodology [6-12] in an attempt to improve thecontrol of an integrated heating system. The proposedtechnique should guarantee good tracking performanceand account for modeling errors and parameter variationsin the plant model.

Consider the multivariable feedback control shown inFigure 3. Define

),()()( sFsGsL = (11)

1))(()( −+= sLIsS , (12)

)())()(()( 1 sSIsLrsLsT −=+= − (13)

Figure 3, Multivariable feedback control system

The plant model is only an approximation of thephysical plant. It is a common practice to lump the effectsof all plant uncertainty into a multiplicative perturbation

)(sM∆ as shown in Figure 4. Suppose that the system of

Figure 4 is stable with M∆ being zero, then the size of the

smallest stable )(sM∆ for which the system becomes

unstable is

))((

1))((

jwTjwM σ

σ =∆ (14)

Using this result one would specify the stability margin via

singular value inequality such as, )(13))(( jwWjwT −≤σ

where )(13 jwW − is the size of the largest multiplicative

plant perturbation.

Figure 4, Multiplicative uncertainties

The sensitivity matrix S(s) is the closed loop transfermatrix from the disturbance d to the output y, Figure 3. Adisturbance attenuation performance specification may be

written as )(11))(( jwWjwS −≤σ , where )(1

1 jwW − is

the desired attenuation factor. A successful design thatmeets the performance and robustness specifications issummarized in Figure 5.

Figure 5, Singular value spec’s on S and T

A controller that uses the loop shaping methodology isthe Linear Quadratic Gaussian (LQG). It was shown in[12] that the LQG has a guaranteed gain margin of

),2

1[ +∞ and a guaranteed phase margin of o60± at node

1, Figure 6.

Figure 6, LQG/LTR controller and plant

In order to recover the robustness properties of theLQat node 2 (or plant input), Figure 6, a loop transferrecovery (LTR) must be designed. This is done byadjusting the of the Kalman Filter in such a way as theloop return ratio at node 1 is approximately equal to theloop return ratio at node 2 )( 21 LL ≅ [10].

4. CONTROLLER DESIGN OF THE INTEGRATEDHEATING SYSTEM

In this section we apply the design procedurepreviously discussed and well documented in the literature[special issue] to the integrated heating system whoseparameters are shown in the Appendix. Table I gives theeigenvalues of the compensated and uncompensatedsystems. It is clear that the system's stability is marginal. In other words, stability of the system is not guaranteed assystem parameters undergo changes during normaloperations.

Table I. Eigenvalues of Original and Comp. Systems

ORIGINAL COMPENSATED SYSTEM

-0.1302 -22.6716 ±22.6252i , -7.1341 ± 6.9557i

-1.1685 -19.4006 ±19.3399i , -6.1529 ± 5.9225i

-2.9564 -11.9379 ±11.9062i , -3.7921 ± 3.6337i

Prior to the design an integrator was added to thesystem to boost the behavior of its singular values at lowfrequencies, Figures 2 & 7. The crossover frequency of thesingular values was chosen at 0.97 to ensure good trackingperformance and system robustness. The gain matrix forthe LQR was found to be:

0.60640.0064-0.1370-10.67110.06621.1640-

0.00200.73330.0041-0.06626.55100.2460-

0.11690.15680.92931.1640-0.2460-12.6811

= K

(15)

And the Kalman filter gain matrix was found to be:

38.73370.0609-2.9767-

0.0609-22.91810.1444-

2.9767-0.1444-42.1133

9.87780.0446-1.5894-

0.0963-9.96600.8778-

1.58710.88199.8389

= F

(16)

Figure 7, Singular values of the loop return ratios L1 at node 1and L2 at node 2

Figure 7 gives the plot of the singular values of thecompensated system which shows that the trackingperformance has been enhanced since the magnitude of theminimum singular value at low frequency has beenincreased when compared with Figure 2. In addition,Figure 7 provides a check on the recovery and balancingthe singular values of the system at nodes 1 and 2. If thesingular values are far from each other then the LTR

design is tuned to achieve the desired robustness recoveryand hence completing the design.

5. SIMULATION RESULTS

The compensator was connected to the original linearsystem and several step responses were generated Figures8-10. It is clear from these figures that the controller isproviding the desired output with minimal overshoot. Thesettling time of the outputs in the figures is under a minutein all cases.

Figure 8, Response to step at Tb

Figure 9, Response to step at Tz

Figure 10, Response to step at Ta

Further the system was checked for disturbance rejectionand the results were promising as seen in Figure 11.

Figure 11, Response to step at Twi

CONCLUSION

Multivariable robust control theory was used to designan LQG/LTR controller to optimize the performance of anintegrated heating system. Simulation results show greatimprovement in the system performance and robustnesswhen compared to the classical PI methods.

References

[1]. M. Zaheer-Uddin, "Digital Control of a Heating System forHot Water and Space Heating," Energy, Vol. 16, N0. 10,pp. 1247-1257, 1991, Great Britain.

[2] R.N. Caron, et. al., " Integrated Appliances for Hot Waterand Space Heating," Symposium on Future Alternativesin Residential/ Commercial Space Conditioning, Instituteof Gas Technology, Chicago, IL, 1980.

[3] G. Claus and W. Stephen, ASHRAE Trans. 91, 47, 1985.[4] J.J. Lebrun, et.al., ASHRAE Trans. 91, Part 1B, 60, 1985.[5] T.G. Malmstrom et.al., ASHRAE Trans. 91, 87, 1985.[6] M. Barlaud, et. al., "Computation of Optimal Functions for

Transients of a Photovoltaic Array Inverter InductionMotor Generator," IEEE Proceedings, Vol. 133, No. 1,pp.16-20, Jan. 1986.

[7] D. Briett and Siva S. Banda, Intro. to Robust MultivariableControl, AFWAL-TR-85-3102 Report, Feb. 1986.

[8] J.M. Maciejowski, Multivariable Feedback Design, AddisonWesley Publishing Company, Reading, MA, 1989.

[9] M.J. Grimbel and M.A. Johnson, Optimal Control andStochastic Estimation, Theory and Applications, Vol. 1,John Wiley, NY, 1988.

[10] J.C. Doyle and G. Stein, "Multivariable Feedback DesignConcepts for a Classical/Modern Synthesis," IEEE Trans.on Automatic Control, pp. 4-16, Feb. 1981.

[11] Special issue on Linear Multivariable Control Systems,IEEE Trans. Auto. Cotr., Vol. AC-26, No. 1, Feb.1981.

[12] N. A. Lehtomaki, N. R. Sandell Jr., and M. Athans,“Robustness Results in LQG Based MultivariableControl Designs,” IEEE Trans. Auto. Contr. Vol. AC-26.Feb. 1981, pp. 4-16.

APPENDIX

The system matrices listed above were obtained at thefollowing operating point:

Table 2. System Parameters

Variable Magnitude, Units

UAz 237.5 W/0C

UAb 13.2 W/0C

UAw 13.2 W/0C

U1max 0.1416 Kg/s

HU2max 17.94 KW

U3max 0.1892 Kg/s

mw 0.0756 Kg/s

Tzset 21 0C

Twset 60 0C

Ubw 0.8 KJ/Kg-0C

Ubw 0.6 KJ/Kg-0C

Cb 1188.91 KJ/0C

Cw 798.94 KJ/0C

Cz 374.48 KJ/0C

2.9336- 0.1267 0.4248

0.0594 0.9792- 0.6818

0.1333 0.2291 0.3423-

=A

88.0185 0 0

0 28.61 0

27.5536- 9.6147- 86.775

=B

1 0 0

0 1 0

0 0 1

=C

[ ] [ ]2.382 0 0 = D 0 0.2379 0 = D T2

T1

The system matrices listed above were obtained at thefollowing operating point:

. C10o = T wio

,C10o = Tao ,0.2 = U1o ,0.5 = U3o

,0.4 = U 2o ,C510o 22. = T zo

,C587o 60. = T wo ,C567

o 81. = Tbo