sliding mode output tracking with application to a multivariable high temperature furnace problem

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, VOL. 7, 337—351 (1997)

SLIDING MODE OUTPUT TRACKING WITH APPLICATIONTO A MULTIVARIABLE HIGH TEMPERATURE FURNACE

PROBLEM

CHRISTOPHER EDWARDS* AND SARAH K. SPURGEON

Control Systems Research, Department of Engineering, University of Leicester, Leicester LE1 7RH, U.K.

SUMMARY

The paper describes a theoretical framework for the design of a robust multivariable output trackingcontroller using sliding mode concepts. The approach assumes that only measured outputs are available anduses a sliding mode observer to reconstruct estimates of the internal system states for use in a fullinformation sliding mode control law. This scheme is applied to a control problem associated with hightemperature furnaces. The paper describes the synthesis of the proposed control scheme from design throughto implementation on an industrial test facility. ( 1997 by John Wiley & Sons, Ltd.

Int. J. Robust Nonlinear Control, Vol. 7, 337—351 (1997)(No. of Figures: 4 No. of Tables: 0 No. of Refs: 16)

Key words: sliding mode control; sliding mode observers; output tracking; temperature control

1. INTRODUCTION

This paper describes a nonlinear controller/observer pair, based on sliding mode concepts whichprovides robust output tracking of a reference signal using only measured output information.Usually the theory associated with sliding modes is developed within a state-space frameworkand assumes that all of the internal states are available for the control law. In practice this is oftennot the case and this limits the practical applicability of such schemes. To circumvent thisdifficulty workers have suggested the use of Luenberger (linear) observers to generate estimates ofthe unavailable internal states for use in the control law.1—3 More recently Edwards andSpurgeon4 proposed the use of a sliding mode observer and proved that under certain conditionsthe well-known insensitivity to matched uncertainty, provided by sliding mode controllers whenfull state information is available, is retained by the controller/observer pair. This paper considersa robust output tracking control scheme which utilizes the full state information controllersuggested by Davies and Spurgeon5 in conjunction with a sliding mode observer which is similarin structure to that proposed by Walcott and Zak6 although the explicit solution is an extensionto the work of Edwards and Spurgeon.7

The controller/observer scheme is applied to a multivariable high temperature heating systemwhere control of both temperature and excess oxygen is considered. Until recently control oftemperature was the primary concern. However, in view of future environmental legislation withregard to harmful emissions resulting from combustion, in addition to temperature control,

* Correspondence to: C. Edwards.

CCC 1049-8923/97/040337—15$17.50( 1997 by John Wiley & Sons, Ltd.

attention is now being focused on the problem of combustion efficiency and its effect on theformation of noxious by-products.8 One way of monitoring combustion efficiency is to measurethe concentration of excess oxygen present in the flue gases. Oxygen analysers, adapted fromautomotive technology, situated in the exhaust flue are able to provide on-line measurements ofoxygen concentration for use by the control system. Currently, control of both temperature andexcess oxygen is achieved by independent single-input—single-output control loops. This is clearlynot necessarily the optimal solution.This short-coming is addressed in this paper which treats theproblem in a multivariable context. The paper describes the development of a control schemefrom design through to implementation for a test facility at the Gas Research Centre atLoughborough.

The paper is organized as follows: in Section 2 the theory associated with the sliding modecontroller/observer pair is described; Section 3 introduces the control problems associated withhigh temperature furnaces and describes the synthesis of the proposed control scheme; Section4 presents some results obtained from implementation of the control scheme on a test furnace atthe Gas Research Centre; and finally Section 5 makes some concluding remarks.

The following notation will be used throughout: E · E will denote the Euclidian norm for vectorsand the spectral norm for matrices; j ( · ) will denote the spectrum of a matrix, j

.!9( · ) the largest

eigenvalue and i ( · ) the condition number (i.e., the ratio of the maximum and minimum singularvalues).

2. A ROBUST OUTPUT TRACKING CONTROL SCHEME

Consider an uncertain dynamical square system of the form

xR (t)"Ax(t)#Bu(t)#f (t,x, u)

y(t)"Cx(t) (1)

where x3Rn, u3Rm and y3Rp with m"p(n. Assume that the nominal linear system (A,B,C)is known and that the input and output matrices B and C are both of full rank. The unknownfunction f :R

`]Rn]RmPRn represents any nonlinearities plus any model uncertainties in the

system which are assumed to be matched and bounded, i.e.,

f (t,x, u)"B(g1(t,x, u)u#g

2(t,x)) (2)

where

Eg1(t,x, u)E)k

g1and Eg

2(t,x)E)a(t, y) (3)

for some known scalar kg1(1 and known function a :R

`]RpPR

`. In addition, it is assumed

that the nominal layer system (A,B,C) satisfies

(A1) the pair (A,B) is controllable(A2) det(CB)O0(A3) the invariant zeros of (A,B,C) are in C

~

The assumption that the system is square is required since the sliding mode observer formula-tion of Walcott and Zak6 requires there to be at least as many outputs as inputs. Conversely, thecontrol scheme of Davies and Spurgeon5 requires at least as many inputs as outputs—a squaresystem is therefore required. Assumptions A2 and A3 are synonymous with the system beingminimum phase and relative degree one.

338 C. EDWARDS AND S. K. SPURGEON

( 1997 by John Wiley & Sons, Ltd. INT. J. ROBUST NONLINEAR CONTROL, VOL. 7, 337—351 (1997)

It can be assumed without loss of generality that the system is already in the so-called ‘regularform’ usually used in sliding mode design, so that

A"CA

11A

12A

21A

22D B"C

0

B2D C"[C

1C

2] (4)

where A11

3R(n~m)](n~m), B23Rm]m and C

23Rp]p. The square matrix B

2is non-singular

because the input distribution matrix is assumed to be of full rank. Therefore since CB"C2B2

isnon-singular by assumption it follows that C

2is non-singular.

2.1. Controller formulation

Consider initially the development of a tracking control law for the nominal linear system

xR (t)"Ax(t)#Bu(t) (5)

where the matrix pair (A,B) is assumed to be in regular form as in (4). The control law describedhere is based on that described by Davies and Spurgeon5 and utilizes an integral actionmethodology. Consider the introduction of additional states x

r3Rp satisfying

xRr(r)"r(t)!y (t) (6)

where the differentiable signal r (t) satisfies

rR (t)"! (r(t)!R) (7)

with !3Rp]p a stable design matrix and R a constant demand vector. The matrix ! can, in somesense, be thought of as defining an ideal transient response to the step input R. It also serves toprovide a differentiable ‘reference signal’ for use in the controller. Augment the states with theintegral action states and define

xJ s"Cxr

xD (8)

The (augmented) nominal system can then be conveniently written in the form

xJ Q1(t)"AI

11xJ1(t)#AI

12xJ2(t)#B

rr(t) (9a)

xJQ2(t)"AI

21xJ1(t)#AI

22xJ2(t)#B

2u (t) (9b)

where xJ is partitioned as xJ13Rn and xJ

23Rm and

AI11

AI12

AI21

AI22

s"

0 !C1

!C2

0 A11

A12

0 A21

A22

(10)

The input distribution matrix for the ‘demand’ signal r(t) in the null space dynamics is given by

Br"C

Ip0D (11)

SLIDING MODE OUTPUT TRACKING 339


The proposed controller seeks to induce a sliding motion on the surface

S"M(xJ1, xJ

2)3Rn`p : s(xJ , r)"0N (12)

where the switching function is defined by

s (xJ , r) s"S1xJ1#S

2xJ2!S

rr (13)

and the matrices S13Rm]n, S

23Rm]m and S

r3Rp]p are design parameters which govern the

reduced order motion. Let S2""B~1

2where " is a non-singular diagonal design matrix which

satisfies

kg1

i (")(1 (14)

If a controller exists which induces an ideal sliding motion onS, then the ideal sliding motion isgiven by

xJQ1(t)"(AI

11!AI

12M )xJ

1(t)#(AI

12S~12

Sr#B

r)r (t) (15)

where M s"S~12

S1. The sliding surface design problem may then be viewed as one of choosing

a state-feedback matrix M to stabilize the pair (AI11

,AI12

). It can be shown4 that because thenominal linear system (A,B,C ) is assumed to have no invariant zeros at the origin, the pair(AI

11, AI

12) is completely controllable. Many methods have been described in the literature to solve

this problem (for an introduction see, for example, Zinober and Dorling9). Suppose the matrixM has been chosen by any of these methods so that the dynamics of the sliding motion

AM11"AI

11!AI

12M (16)

satisfies the performance specifications where stability is clearly a minimum requirement. Thecontrol action used to induce a sliding motion is similar to that of Ryan and Corless10 andSpurgeon and Davies.11 Let '3Rm]m be any stable design matrix and let PM

2be the unique

symmetric positive definite solution to the Riccati equation

PM2'#'TPM

2#PM

2QM

2PM2"0 (17)

for some positive definite design matrix QM2. Define a linear operator

uL(xJ , r)"¸xJ #¸

rr#¸

rRrR (18)

where the gains are defined as

¸"!B~12

[(MAI11#AI

21!'1 M) D (MAI

12#AI

22!'1 )] (19a)

¸r"!B~1

2('1 S~1

2Sr#MB

r) (19b)

¸rR"B~1

2S~12

Sr

(19c)

with '1 s"S~12

'S2. The control law is then given by

u"uL(xJ , r)#l

c(20)



where lcis the discontinuous vector given by

lc"G

!oc(u

L, y)"~1

PM2s(xJ , r)

EPM2s(xJ , r)E

if s (xJ , r)O0

0 otherwise

(21)

and the scalar function

oc(u

L, y)"E"E

(kg1

EuLE#a (t, y)#k

g1ccE"~1E#c

o)

(1!kg1

i ("))#c

c(22)

where co

and ccare two positive design scalars. It can be verified that applying the nonlinear

control law (18)—(22) to the uncertain system (1) implies

sR (t)"'1 s(t)!oc(u

L, y)

PM2s (t)

EPM2s (t)E

#"g1(t,x, u)u#"g

2(t, x), s(t)O0 (23)

It is straightforward to verify that »(s)"sTPM2s is a Lyapunov function for s and in particular

»Q )!sTPM2QM

2PM2s!2(c

c#DD"DDc

o) EPM

2sE (24)

A sliding motion therefore takes place on S. It can also be seen that the stable design matrix'1 governs the linear free-motion decay of the states of the switching function s to the origin.

Remarks

The seemingly unnecessarily complicated notation PM2

is used because a Lyapunov function ofthe form x8

1TPM

1xJ1#sTPM

2s will be constructed for the closed-loop system resulting from applying

the control law (18)— (22) to the uncertain system (1).The reason for the use of the Riccati equation (17) rather than the Lyapunov equation used in

the original formulation of Davies and Spurgeon5 will be described in Section 2.3.This control law relies on all the internal states being available. Under this circumstance, the

nonlinear component guarantees quadratic stability. In this paper it is assumed that only theoutput information y (t) is available to the control law. An observer will thus be incorporated toprovide estimates of the internal states.

2.2. Nonlinear observer formulation in regular form

Consider the observer structure given by

zR (t)"Az(t)#Bu(t)!GCe (t)#Blo

(25)

where z represents an estimate of the true states x, and e s"z!x is the state estimation error. Theoutput error feedback gain matrix G is chosen so that the closed-loop matrix A

0s"A!GC is

stable and has a Lyapunov matrix P satisfying both

PA0#AT

0P"!Q (26)

for some positive definite design matrix Q and the structural constraint

PB"CTFT (27)



for some non-singular matrix F3Rm]m. The discontinuous vector lois given by

lo"G!o

o(u

L, y)

FCe

EFCeEif CeO0

0 otherwise(28)

where oo(u

L, y) is the scalar function

oo(u

L, y)"(o

c(u

L, y)!c

c)/E"E (29)

The formulation given in (25)—(29) is essentially that of Walcott and Zak6 although theexposition above follows that of Edwards and Spurgeon.7 Edwards12 shows that assumptions A2and A3 are necessary and sufficient conditions for the existence of such an observer which isinsensitive to matched uncertainty and induces a sliding motion on

So"Me3Rn : FCe"0N (30)

Since the system is square and the matrix F is non-singular, sliding on the surfaceSoimplies the

output of the observer is identical to that of the plant. The original formulation of Walcott andZak6 required the use of symbolic manipulation to synthesize the matrices G and P whichcompletely define the observer. More recently Edwards and Spurgeon4 proposed an analyticsolution which is described below. Let As

22be a stable design matrix and let

G"CA

12C~1

2A

22C~1

2!C~1

2As

22D (31)

which implies

A0"C

A11!A

12C~1

2C

10

A21!A

22C~1

2C

1!C~1

2As

22C

1C~1

2As

22C

2D

It can be verified13 that the invariant zeros of (A,B,C ) are given by j(A11!A

12C~1

2C

1) and

therefore since C~12

As22

C2

is stable, by construction A0

is stable. If P2

is a symmetric positivedefinite matrix satisfying the Lyapunov equation

P2As

22#(As

22)TP

2"!Q

2(32)

for some symmetric positive definite design matrix Q2

then define

F s"(P2C

2B2)T (33)

If P1

is a symmetric positive definite matrix satisfying the Lyapunov equation

P1(A

11!A

12C~1

2C

1)#(A

11!A

12C~1

2C

1)TP

1#AT

21P2Q~1

2P

2A

21#Q

1"0 (34)

where A21

s"C1(A

11!A

12C~1

2C

1)#C

2(A

21!A

22C~1

2C

1) and Q

1is a symmetric positive

definite matrix then the symmetric positive definite matrix

P"CP1#CT

1P1C

1CT

1P2C

2CT

2P2C

1CT

2P2C

2D (35)

is a Lyapunov matrix for A0"A!GC which satisfies PB"CTFT. For further details see

Reference 4.



Remark

This provides a method of designing a Walcott and Z0 ak observer, without the use of symboliccomputation. From the numerical computation viewpoint the Lyapunov equation given in (34)need not be solved and the matrix P in (35) need not be evaluated—they are only required foranalysis purposes.

2.3. Closed-loop considerations

Although both the controller and observer strategies provide robustness against matcheduncertainty it is by no means clear that the overall system, i.e., the uncertain plant with theobserver/controller forming a closed-loop system, retain these properties. However, Edwards andSpurgeon14 prove that quadratic stability of the closed loop is guaranteed. An outline of theconstruction of the quadratic Lyapunov functions is given below.

Let Ac"AI !BI ¸ represent the closed-loop system matrix where the pair (AI ,BI ) represents the

original system (A,B) augmented with the integral action states and ¸ is defined in equation (19a).Also define

GM "CIn

0

S1

S2D C

!Ip

GD (36)

where G is the observer gain matrix in (25) and S1

and S2

are components of the switchingfunction (13). It should be noted that the partitions in equation (36) are not compatible althoughthe overall expression is correct. Edwards and Spurgeon4 demonstrate that for an appropriateclass of symmetric positive definite matrices QM the Riccati equation

PM Ac#AT

cPM #PM GM CQ~1CTGM TPM #PM QM PM "0 (37)

where Q is defined in equation (26), has a symmetric positive definite solution

PM "CPM1

0

0 PM2D

where PM13Rn]n and PM

2satisfies the Riccati equation (17). Finally let P

rbe the unique symmetric

positive definite solution to the Lyapunov equation

Pr!#!TP

r"!GM

rQM ~1GM T

r!Q

r(38)

where ! is the stable design matrix from (7), the matrix

GMr

s"[(Br#AI

12S~12

Sr)T 0

p]p]T (39)

and Qris any symmetric positive definite design matrix. Then it can shown4 that the function

» (xJ1, s, e, r)"(xJ

1#AM ~1

11(B

r#AI

12S~12

Sr)R)TPM

1(xJ

1#AM ~1

11(B

r#AI

12S~12

Sr)R)

#sTPM2s#eTPe#(r!R)TP

r(r!R) (40)

is a Lyapunov function for the uncertain system and controller/observer pair combined and thatquadratic stability is guaranteed. Furthermore it can shown that sliding motions take place onthe controller and observer sliding surfaces S and S

0respectively.



Remark

In establishing quadratic stability it is sufficient to ensure that equation (29) holds. Thisequation relates the scalar functions o

c( · ) and o

c( · ) which premultiply the unit vector compo-

nents of the controller and observer respectively. There is no other dependence between theobserver and the controller and a form of separation principle is seen to hold.

3. CONTROLLER DESIGN AND IMPLEMENTATION ISSUES

The remainder of this paper considers the application of the theoretical results presented inSection 2 to a high temperature furnace control problem. The Gas Research Centre at Lough-borough has an experimental furnace which is representative of a kiln for the firing of pottery.*The furnace can be thought of as a gas filled enclosure bounded by insulating surfaces. Forconvenience, the effect of a load is simulated by a network of pipes covering the interior surfacesof the furnace through which water is circulated. Heat input is achieved by a burner located in oneof the end walls, and the combustion products are evacuated via a flue in the roof. The burner isfed by a fuel and air supply and the flow rate of these gases can be modulated independently bymeans of motorized ‘butterfly’ valves present in the respective flow lines.

From a control systems perspective the outputs of the ‘system’ are the furnace temperature (asmeasured by the thermocouple) and the percentage of oxygen present in the combustion products(as measured by the oxygen analyser in the flue). In an industrial situation, the internal furnacetemperature would be required to exhibit a specific time/temperature profile comprising, say,a period of low fire, ramp to a higher temperature, a period of soak and finally a return to ambienttemperature. During normal furnace operation, efficient fuel combustion is desirable. For a givenmass of fuel, a theoretical mass of oxygen is required to completely oxidize the hydrocarbons (socalled stoichiometric combustion). An inadequate air supply will result in incomplete combustionwith a corresponding loss in thermal energy release. Conversely, excess air, whilst guaranteeingcomplete combustion, will give rise to unnecessary enthalpy losses through the flue due to theincreased flue flow rate. Therefore, efficient combustion is ensured by minimizing the amount ofexcess oxygen present in the combustion products in the flue. In addition to the potential energysavings, it is argued by Disdell, Burnham and James14 that accurate control of both temperatureand excess oxygen is of paramount importance from the point of view of reducing pollutantemissions.

For safety reasons, such furnaces usually operate at a fixed fuel/air ratio as a result of anElectronic Ratio Controller (ERC).15 The fuel and air flows to the burner are measured andfeedback to the ERC which makes appropriate adjustments to the air valve position. The devicealso provides an additional safety feature in the form of a ‘shut down’ alarm which isolates the fuelsupply when the furnace persistently operates ‘off ratio’, i.e., away from the required fuel/air ratioset-point. This framework is usually described as ‘gas led’ since the air flow is modulated asa result of changes in the fuel flow to maintain the appropriate fuel/air ratio necessary for efficientcombustion. An additional input to the ERC exists referred to as the ‘trim signal’. This allows thefuel/air ratio set-point to be dynamically adjusted. In this way, as far as this paper is concerned,the control inputs to the system are the fuel valve positioner signal and the ERC trim signal.These inputs will be manipulated to ensure the temperature and excess oxygen levels followspecified reference profiles. This multivariable approach is distinct from current practices which

* Although the control problems described in this paper pertain to a single burner furnace, the problems are notunrepresentative of those which occur in more general multi-zone furnace designs.



Figure 1. Schematic of proposed multivariable control scheme

control the excess oxygen and temperature via independent single-input—single-output controlloops. The proposed scheme is illustrated in Figure 1.

Physical modelling of such a system is clearly non-trivial and does not lead to a system ofequations conducive to controller synthesis; instead an identification approach has been adopted.The MATLAB† system identification toolbox was used to fit linear models of different orders andstructures to available furnace input—output data. A low order model, which provides goodagreement with input—output data, is given by

A"

!0·4738 0·0279 0·0644 0·0573

0·1319 !0·1965 !0·0482 0·0692

!0·0473 0·0217 !0·0783 !0·0670

!0·0144 !0·0001 0·0076 !0·0260

B"

!0·4726 !0·0201

0·1528 0·0789

!0·0144 !0·0677

0·0038 !0·0074

C"C0 0 !0·1197 0·6694

0 0 !0·2811 !0·2850D (41)

The system is relative degree one and has invariant zeros at M!0·1633, !1·2059N and so thetheory from Section 2 is applicable. The first input channel represents the fuel valve positionsignal and the second, the oxygen trim signal. The first output is the furnace temperature and thesecond output is the percentage of excess oxygen in the flue. Using an orthogonal change ofco-ordinates obtained from ‘QR’ reduction of the input distribution matrix, the system given in

† MATLAB is a registered trademark of the Math¼orks, Inc.



(41) can be written in regular form as

A"

!0·0186 !0·0065 0·0190 0·0129

0·0026 !0·1354 0·0310 0·0040

!0·0972 0·0695 !0·1273 0·0530

!0·0193 !0·0155 !0·1121 !0·4934

B"

0 0

0 0

0 !0·0960

0·4969 0·0453

C"C0·6707 !0·1085 !0·0286 0·0086

!0·2750 !0·1933 !0·2175 0·0060D (42)

The following section describes the design procedures adopted to synthesize a controller/observerpair. The design of an observer is discussed first.

3.1. Observer design

Since the system is square, the reduced order dynamics of the estimation error are completelydetermined by the invariant zeros of the system. By choosing a priori the stable matrix As

22to be

diagonal, any diagonal positive definite matrix P2

will always be a Lyapunov matrix for As22

. Thediagonal matrix P

2can then be considered independently as a design matrix. Here it has been

chosen in an effort to ensure the diagonal entries of the matrix F"(P2C

2B

2)T are of compatible

order. It was reasoned that the diagonal elements of F act as weighting parameters which governthe distribution of the nonlinear action between the individual input channels (for diagonalmatrices this is certainly the case). By choosing

P2"C

10 0

0 2Dit follows that

F"C0·0426 0·0060

0·0313 0·0423DThe diagonal elements of As

22were chosen to make the condition number of the observer

closed-loop matrix A0"A!GC small. By inspection, it was found that choosing

As22"C

!0·5 0

0 !0·5Dfulfils this requirement, giving a matrix condition number for A

0of approximately 38·26. From

equation (31) the linear gain matrix is given by

G"

1·7201 !0·3132

0·6269 !0·2251

8·1321 !2·7820

0·4512 0·4562



3.2. Design of the controller

Forming the augmented plant from (10) it follows that

KCAI

11AI

12AI

21AI

22D"

0 0 !0·6707 0·1085 0·0286 !0·0086

0 0 0·2750 0·1933 0·2175 !0·0060

0 0 !0·0186 !0·0065 0·0190 0·0129

0 0 0·0026 !0·1354 0·0310 0·0040

0 0 !0·0972 0·0695 !0·1273 0·0530

0 0 !0·0193 !0·0155 !0·1121 !0·4934

Early attempts to select the poles of the reduced order sliding motion by inspection resulted incontrollers which exhibited high levels of control activity. To circumvent this, a preliminary linearLQR design was made based on the augmented system, with the cost functional biased towardspenalizing the use of control effort. The four slowest poles of the resulting closed-loop matrix wereused as initial values for the poles of the reduced-order motion. These were subsequentlymanually adjusted in an effort to improve the condition number of the eigenvectors of theclosed-loop matrix and hence maximize the robustness properties. Ultimately the sliding modeeigenvalues were taken to be M!0·0849, !0·0118, !0·0357$0·0220iN. The matrix M fromequation (15) which acts as a feedback matrix for the pair (AI

11,AI

12), was obtained using the

MATLAB command, ‘place’, which uses the robust pole placement algorithm of Kautsky,Nichols and Van Dooren.16 The resulting matrix which defines the hyperplane is given by

M"C0·0376 0·0379 !0·1258 !2·7493

!0·2317 !0·0930 5·7274 4·0709DThe design parameter " was chosen as

""C0·1 0

0 0·1Dwhich completes the hyperplane design. If the stable design matrix '1 , which assigns the poles ofthe range space dynamics, is block diagonal, a diagonal solution for PM

2can be attained from the

Riccati equation (17) by a suitable choice of QM2. A diagonal structure for PM

2is preferable since

this ‘decouples’ the nonlinear components of the control action. The eigenvalues for the rangespace dynamics were take to be the unused poles from the initial LQR design, namelyM!0·1800!0·5009N. The diagonal elements of '1 were arranged so that the slowest pole wasassociated with the oxygen trim channel to circumvent possible violation of the fuel/air ratio shutdown alarm. The linear components of the control law from equations (19a)—(19c) can be shownto be

¸"C0·1080 0·0379 !2·7802 !0·5288 !0·2363 0·2641

0·0706 0·0711 !1·4509 !0·4258 !0·2679 0·4139D¸r"C

5·6968 !0·3288

2·4412 3·3682D¸rR"C

20·9894 !1·9125

11·3840 16·5172D



In this particular case study

[S1

S2]"C

!0·0431 !0·0151 1·1407 0·5580 0·0950 0·2012

!0·0392 !0·0395 0·1310 2·8634 !1·0415 0·0000DThe hyperplane defined in Section 2 is given by

S"MxJ 3Rn`p : s (xJ , r)"0N (43)

where s(xJ , r)"S1xJ1#S

2xJ2!S

rr which is now completely defined except for the design para-

meter Sr3Rm]m. One possible choice is of course to let S

r"0; alternatively S

rmay be viewed as

affecting the values of the integral action components since from equation (15) at steady state

(AI11!AI

12M)xJ

1#(AI

12S~12

Sr#B

r)r"0 (44)

where Bris defined in equation (11). One possibility is therefore to choose a value of S

rso that, for

the nominal system at steady state, the integral action states are zero. If Ks

s"!BTrAM ~1

11AI

12then

provided it is non-singular, choosing

Sr"S

2K~1

sBT

rAM ~1

11Br

(45)

implies xr"0. It is shown in Edwards and Spurgeon9 that because (A,B,C) does not have any

zeros at the origin Ksis non-singular. Substituting the appropriate values into equation (45) gives

Sr"C

2·0989 !0·1912

1·1384 1·6517DThe design matrix ! from equation (7) has been chosen to tailor the step response of theclosed-loop system in the nominal case. The stable design matrix has been chosen to be diagonal,since it makes no sense to introduce coupling between the reference signals. Therefore themultivariable equation (7) can be represented as the pair of scalar equations

rRi(t)"!

ii(ri(t)!R

i(t! 1

!ii)) for i"1, 2

where !11

and !22

are the diagonal elements of !. If R®i(t)"0 on some interval then it can be

shown that ri(t)PR

i(t) asymptotically. In this particular design !

11"!0·02 and !

22"!0·05.

This reflects the different speeds of response required in the temperature and oxygen channelsrespectively.

Because of the identification process adopted, the scalars from equation (22) were chosenexperimentally during the trials at the Gas Research Centre.

4. IMPLEMENTATION RESULTS

The control scheme was implemented on a portable PC containing appropriate interface cards toperform the required analogue% digital conversions. The plant output signals were sampled andthe control outputs updated ten times a second which represents a high sampling rate comparedwith the dominant time constant of the process. However, this is usual with high temperature gasfed plant for reasons of monitoring and safety.

The temperature demand profile shown in Figure 2 was used which is representative of thatrequired by typical industrial processes. It can be seen that the condition R®

i(t)"0 is satisfied



Figure 2. Typical temperature reference signal

Figure 3. Furnace temperature compared with the reference

almost everywhere since the reference profile is piecewise linear. It was observed by Goodhart8that the excess oxygen reference signal cannot be chosen without regard to the temperature setpoints. Care has therefore been taken to provide an excess oxygen profile that is both realistic andattainable.

The scalars comprising the nonlinear gain functions were chosen initially very conservativelyunder the restriction that the gain o

c( · ) be bounded by unity. The rationale for this conservatism

was that, generally speaking, increasing the nonlinear control component increases the ‘aggres-siveness’ of the control signal, which was considered to be undesirable in the oxygen trim channelbecause of the safety shut-down mechanism which exists in this input channel.

Figures 3 and 4 present results obtained from using the scalar gains

oc(u

L, y)"0·02 Eu

L( · )E#0·02 EyE#0·075 (46)

and

oo(u

L, y)"0·2 Eu

L( · )E#0·2 EyE#0·25 (47)

From Figure 3 it can be seen that the temperature tracking performance is very good. This isconfirmed by the mean absolute error of 1·627° which represents a level of accuracy comparable



Figure 4. Excess oxygen in the flue compared with the reference

with that of the measurement device. Visually the tracking performance in the excess oxygenchannel is also good (Figure 4); unfortunately no details of the performance levels of other controlschemes are available for comparison. Overall it can be seen that the multivariable approach hasresulted in good tracking performance in each channel with minimal interaction.

5. SUMMARY

A theoretical framework for designing a robust multivariable output tracking controller usingsliding mode concepts has been outlined. This scheme has been applied to the problem ofcontrolling both the temperature and oxygen content in the combustion products in an industrialfurnace. The oxygen trim signal to the ERC, which alters the fuel/air ratio set-point, has been usedas an additional dynamic input which results in a two-input—two-output multivariable system.Because a model was not available to test the controller prior to implementation, the designscalars that define the nonlinear gain functions were selected in situ during the trials. Overall,good tracking performance was obtained for both reference signals.

ACKNOWLEDGEMENTS

Financial support from the provision of a Research Scholarship by British Gas plc is gratefullyacknowledged. The invaluable assistance and technical expertise provided by Dr. Sean Goodhart,Ruth Davies and Patrick Holmes during the plant trials at the Gas Research Centre is alsoacknowledged.

REFERENCES

1. Breinl, W. and G. Leitmann, ‘State feedback for uncertain dynamical systems’, Journal of Applied Mathematics andComputation, 22, 65—87 (1987).

2. Utkin, V. I., Sliding Modes in Control Optimization, Springer-Verlag, Berlin, 1992.3. Young, K.-K. D. and H. G. Kwatny, ‘Variable structure servomechanism design and applications to overspeed

protection control’, Automatica, 18, 385—400 (1982).4. Edwards, C. and S. K. Spurgeon, ‘Robust output tracking using a sliding mode controller/observer scheme’,

International Journal of Control, 64, 967—983 (1996).5. Davies, R. and S. K. Spurgeon, ‘Robust tracking of uncertain systems via a quadratic stability condition’, Proceedings

of the 12th IFAC ¼orld Congress, Sydney, Australia, 1993, pp. 43—46.6. Walcott, B. L. and S. H. Z0 ak, ‘Observation of dynamical systems in the presence of bounded nonlinearities/

uncertainties’, Proceedings of the 25th Conference on Decision and Control, Athens, Greece, 1986, pp. 961—966.



7. Edwards, C. and S. K. Spurgeon, ‘On the development of discontinuous observers’, International Journal of Control,59, 1211—1229 (1994).

8. Goodhart, S. G., ‘An industrial perspective on temperature control’, Proceedings of the 10th International Conferenceon Systems Engineering, Coventry University, 1994, pp. 399—404.

9. Zinober, A. S. I. and C. M. Dorling, ‘Hyperplane design and CAD of variable structure control systems’, inDeterministic Control of ºncertain Systems, A.S.I. Zinober (Ed), Peter Peregrinus, Stevenage, U.K., 1990, pp. 52—79.

10. Ryan, E. P. and M. Corless, ‘Ultimate boundedness and asymptotic stability of a class of uncertain dynamical systemsvia continuous and discontinuous control’, IMA Journal of Mathematical Control and Information, 1, 223—242 (1984).

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14. Disdell, K. J., K. J. Burnham and D. J. G. James, ‘Developments in furnace control for improved efficiency andreduced emissions’, in IEE Colloquim on ‘Improvements in Furnace Control: Current Developments and New ¹echnolo-gies’ Digest No: 1994/018, 1994, IEE, Stevenage, UK, London, U.K., 211—216.

15. Hammond, P. S., ‘Advanced kiln and furnace control’, Proceedings of the International Gas Research Conference,Orlando, Florida, 1992, pp. 2217—2226.

16. Kautsky, J., N. K. Nichols and P. Van Dooren, ‘Robust pole assignment in linear state feedback’, InternationalJournal of Control, 41, 1129—1155 (1985)



sliding mode output tracking with application to a multivariable high temperature furnace problem

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