Robust nonlinear control of heating plant

C. Edwards S.K. Spurgeon

Indexing terms: Industrial heating plant, Nonlinear control, PID controllers, Temperature control

Abstract: A tutorial description of some recent results in the area of continuous nonlinear control and observation of uncertain systems is presented. The theoretical results are applied to the practical problem of temperature control of an industrial heating plant. The results obtained during plant trials are presented. A detailed comparison is made between the performance and robustness which may be achieved using a sophisticated com- mercial PID controller and that effected by the proposed methodology.

1 Introduction

This paper explores the application of recent nonlinear control and observer theory to the practical problem of temperature control in a gas fired furnace. The furnace under consideration is schematically represented in Fig. 1. It can be considered as a gas-filled enclosure contain-

flue products

insulated wolls J‘/ fuel valve

air valve flame

~ burner load (variable) thermocouple

Fig. 1

ing a heat sink bounded by insulating walls. Heat input is achieved via a burner located in one of the end walls and the combustion products are evacuated through a flue positioned in the roof. Although this represents the sim- plest design possible, a single burner arrangement, such a plant could legitimately represent an industrial furnace for the firing of ceramics.

The control problem considered here involves the manipulation of the fuel flow rate so that the measured temperature at some point in the furnace follows some specified temperature-time profile. It is assumed that a controller already exists which regulates the air flow so

Schematic of the box furnace considered

0 IEE, 1994 Paper 1096D (CS), received 25th June 1993 The authors are with the Control Systems Research, Department of Engineering, University of Leicester, Leicester LEI 7RH, United Kingdom

IEE F’r0c.-Control Theory Appl, Vol. 141, No. 4, July I994

that for a given fuel flow rate an appropriate air flow rate is maintained to ensure combustion efficiency and an appropriate concentration of unburnt oxygen in the flue products. Currently, temperature control of such plant is achieved through the use of traditional proportional- integral-derivative (PID) controllers.

The established technique for the mathematical model- ling of industrial furnaces is the ‘zone method‘ [q. This involves the breaking up of all enclosure surfaces and volumes into a patchwork of subsurfaces and subvolumes which are termed ‘zones’. Each zone must be small enough to be considered isothermal. Radiation exchange factors are then calculated for every possible pair of zones, and the integro-differential equations governing radiative heat transfer are reduced to algebraic equations, which can be solved numerically. A well validated nonlin- ear dynamic model of this type was available for this study [lo]. Unfortunately the zone method modelling approach does not easily lend itself to the problem of controller design. Instead, in the light of the uncertainties associated with the valve model, disturbances caused by changes in the fuel characteristics and the wide range over which the furnace is required to operate, an uncer- tain system approach has been adopted.

Variable Structure Control Systems* with a sliding mode exhibit certain robustness properties. In particular, they provide complete insensitivity to any uncertainty which is implicit in the input channels of the process to be controlled; this is termed ‘matched uncertainty’. The control is designed to drive the system states onto a region of the state-space known as the ‘sliding surface’ and then ensure that the states remain there for all sub- sequent time. When the sliding surface is reached the system exhibits reduced-order dynamics, which are speci- fied by the designer’s particular choice of sliding surface. Inherent in this traditional approach to design is a dis- continuous control structure, no consideration of unmatched uncertainty, i.e. uncertainty not in the input channel, and the necessity for full-state feedback. Ryan and Corless [ e ] develop a continuous nonlinear control framework which has its roots in sliding mode control, yet incorporates consideration of unmatched uncertainty contributions. In this paper a variant of the Ryan and Corless controller is considered. A subclass of the uncer- tainty originally considered is found to be sufficiently

Financial support from the provision of a Research Scholarship by British Gas PLC is gratefully acknowledged. The assistance of H. Porch and Dr. S.G. Goodhart during the plant trials at MRS is greatly appreciated.

* A review of Variable Structure Control Systems is given in DcCarlo et al. [2].

221

general for the practical application of the control meth- odology (Spurgeon and Davies 191). In addition, a track- ing requirement is included. A robust continuous nonlinear controller results. However, it should be noted that this controller does require complete state informa- tion. For the furnace application under consideration, measurement of internal states is not possible and there- fore an appropriate state estimation policy must be deter- mined.

Preliminary analysis indicates that a traditional linear observer is inadequate for this nonlinear uncertain process. The possibilities of using a sliding mode observer for robust-state estimation are explored. An algorithm is presented [4] which facilitates sliding mode observer design, based upon the preliminary work of Utkin I l l ] and Walcott and eak [12,14].

A detailed case study is presented using a nominal linear model identified from the nonlinear furnace simu- lation described previously. The choice of both controller and observer parameters is discussed. Quantitative meas- ures of performance which are typically used for closed- loop analysis within the industry are described. The performance of the proposed nonlinear controller obser- ver pair during both the nonlinear simulation and on-site plant trials can then be explored. For comparative pur- poses, the typical performance of a sophisticated com- mercial PID controller is also presented.

2 Nonlinear controller

A robust nonlinear controller which incorporates a demand-following requirement is now described. Con- sider an uncertain dynamical system of the form

i ( t ) = Ax(t) + Bu(t) + F(t, x, U)

At) = Cx(t) (1) where x E R", U E R", y E Rp, rn < n, p < n. The matrix pair (A, B) defining the nominal linear system is assumed to be known and the input and output matrices B and C are both assumed to be of full rank. The unknown func- tion F. R + x R" x R" + R" represents the system nonlin- earities plus any model uncertainties in the system.

2.1 Design of switching surface Consider first the design of a sliding surface which will prescribe the required desirable performance. Since by assumption the input distribution matrix B has full rank there exists an orthogonal matrix T such that

TB = [ i] where B, E R" and is nonsingular (2)

Define z = Tx and partition the states as - - z = lz:] where z1 E 88"-" and z2 E R"

In these new coordinates, the system is in the so called regular form and the range of the inputs has been effect- ively partitioned. The transformation T may be readily determined using 'QR reduction' [3]. Let the functions F,: R, x R"+ R"-" and F,: R, x R" x R"'+ R" be defined as

= TF L p m J

By construction Fdt, z ) and F,(t, z, U) represent the system's unmatched and matched uncertainty, respect-

228

ively. Notice that it has been assumed that all the uncer- tainty with respect to the control action acts in the image of B. Many possible structures may be placed upon the functions Fw(.) and F,,,(.); here it is assumed that

FAt, 2) = F,(t, z)z + FAt, z)

F,(t, z , 4 = G,(t, z, u)u + G,(t, z) (3) where the functions F,(.), F2(.) , Cl(.) and C,(.) are bounded. In this configuration F,(.) can be interpreted as unmatched parametric uncertainty in the nominal system matrix, whilst F2( .) includes external disturbance effects.

The nonlinear controller aims to provide robust track- ing of the differentiable reference signal w(t) using an integral action methodology. To this end consider x, E Rp where

i ,(t) = T ; '(W(t) - At)) (4)

and T; E Rp is a nonsingular design matrix. Define an augmented state vector i, E R"-'"+p as

Then the augmented uncertain system can then be written conveniently as

where F": R, x R" + R"+p-" is defined as

Define a linear transformation by

where the role of the matrix M E R m x ( n + p f m ) will be for- mally defined later. Let

With respect to the (E,, 4) coordinates

9 , =a,,?, + A 1 2 4 + f Z w + Fu (7)

(8)

whereA,, = A , , - A , 2 M ; ~ z , = M ~ , , + A , , - p , , M and = MA,, + A,, . If the augmented system IS con- trollable the pair (A,,, Alz) is controllable, and an appro- priate matrix M may be chosen by any robust linear design technique to prescribe desirable performance. Clearly stability of a,, is a minimum requirement. Define a switching surface by

4 = 22121 + A224 + B,u + M f s w + MFu + F,

9 = {(El, 4): s*, w = 0 ) (9)

where S+, = - Sw w and S, E R m X p is a design parameter. Suppose a controller exists which constrains the states to this surface, then substituting 4 = Sw w into eqn. 7 and neglecting the unmatched uncertainty contri- bution results in the equation

(10) & = A,,;, + (A12SW + Tz)w IEE hoc.-Control Theory Appl., Vol. 141, No. 4, July 1994

which describes the 'ideal sliding motion'. Assuming that the demand ultimately takes a constant value, and A,, has stable dynamics, then El -+ 0. Noting that the first p states of zll are by definition x,, it follows that 2, -+ 0, and hence f i t ) tracks w(t). The development of a nonlinear controller which ensures bounded motion about the desirable dynamics in the presence of uncertainty will now be considered.

2.2 Nonlinear control law The objective is to drive the system states to a neighbour- hood of 9 using a continuous control action. Let CP E R'" '" be any stable design matrix and let P2 E R" be the unique symmetric positive definite solution to the Lyapunov equation

(11) P2Q + OTP, = - I Consider the control law

U = B;' -dZ1Z, + (CP - dZ2)4 - (CPS, + M f J w (

where 6, is a small strictly positive scalar which smooths the discontinuous unit vector component. In the special case when G, = 0 in eqn. 3, an appropriate choice for the scalar function p,,, X,, is any function which bounds the uncertainty contribution in eqn. 7, specifically

~ x , , x , r 3 llMzFl(t, z)ll llzll + IIMzFl(t, Z) + FAtr z)II

Vz E R" and t E R,

where Mz is the subblock comprising the last n - m columns of M. Spurgeon and Davies [9] consider the general case when GI f 0 and use a function of the form

Px,.x,r = r1llzll + rzIluL(z, x,, 411 + r3

where uL(z,x,) represents the linear component of the control action and the positive scalars r,, r2 and r3 depend on the magnitude of the bounds on Fl(.), ..., Gz(.). In this case if N represents the ellipsoid N = @I, 4): (4 - s, W ) ~ P & - S, w) < r,} where rm is a positive scalar it can be shown that

(i) The solution to the uncertain system in eqn. 1, given any initial conditions, enters the ellipsoid JV in finite time and remains there for all subsequent time

(ii) For motion constrained to N , providing the unmatched arametric uncertainty contribution does not destabilise ill in eqn. 7, the states il ultimately enter a neighbourhood of the origin and remain there for all sub- sequent timet.

(iii) The deviation of il from the specified ideal sliding motion is also bounded. For formal proofs of the above assertions see Spurgeon and Davies [9]. In terms of implementation, it is more convenient to express the controller in the original co- ordinates. To this end define

1 = [;] and a new transformation

~

t This may be explored more formally in terms of a 'quadratic stability' constraint [7, 91.

I E E Proc.-Control Theory Appl., Vol. 141, No. 4, July 1994

where T is defined in eqn. 2. Then it can readily be seen that

Define the gain matrices by

L = B,-'[-d,,(CP -dzz)]T2Tl

L , = - B - , ( CPS, + MFz)

L, = B- , 'S,

S4, , = [MI,,,] T1.t - S, w S?, ,

(13)

(14)

and by noting that

then the control law can be written as

where

= rlllxll + r211uL(%, w)ll + r3 (16) The practical choice of the design parameters will be dis- cussed at greater length below. The controller outlined above requires knowledge of all the states and in practice this may necessitate the use of an observer to generate estimates of unavailable states. In view of the VSC- inspired methodology used for the control law it is natural to consider the use of a nonlinear observer strat- egy for the state estimation.

3 Nonlinear observer

A framework for the development of a robust observer for an uncertain dynamical system using a variable struc- ture systems approach is presented. Despite fruitful research and development activity in the area of variable structure control theory, relatively few authors have con- sidered the application of the underlying principles to the problem of estimator design. Walcott and 2ak [12, 141 use a Lyapunov-based approach to formulate an obser- ver design which, under appropriate assumptions, exhibits asymptotic state error decay in the presence of bounded nonlinearities/uncertainties. In particular, this method seeks to render the observer error system totally insensitive to matched uncertainty. This methodology, which requires only bounds on the nonlinearities to be known, can be shown to compare favourably against other direct nonlinear approaches to observer design which require exact knowledge of the nonlinearities [13]. This Section outlines a methodology described in Edwards and Spurgeon [4] which builds on the design approaches of Utkin [11] and Walcott and Zak [12, 14, 151. Here the emphasis is placed upon the numerical tractability of the solution method.

3.1 Problem formulation and observer structure This Section considers the uncertain dynamical system described in eqn. 1 subject to three additional constraints

(i) It is assumed that p 2 m, i.e. more outputs than inputs

(ii) The uncertain function can be expressed as

q t , x, U) = Bt(t, x, U) (17) where the function r : R, x R" x R" + R" is unknown, but bounded with

IIT(t, x, u)ll < p vx E R", U E R", t 3 0 229

(iii) There exists a nonsingular matrix such that follow- ing a change of coordinates with respect to this trans- formation the system can be written in the form

i l = d l l X 1 + d l 2 Y

j , = d 2 , X I + d , , Y + B,(u + t) (18) where y E Rp are the outputs, x1 E R("-p), and dll has stable eigenvalues.

Remark: It is demonstrated in Edwards and Spurgeon [4] that the third assumption is equivalent to the Lyapu- nov structural constraint imposed by Walcott and eak.

If (i,, j ) represent the state estimates (in the coordin- ate system described above) then associated state estima- tion errors are defined as e, = P, - x1 and e, = j - y . The intention is therefore to formulate a nonlinear obser- ver design for which (el, e,) -+ 0 asymptotically despite the presence of uncertainty. This is accomplished by inducing a sliding motion on the surface

9, = {(el, e,): e, = 0) Let d;, E RPXP be any stable design matrix and let P E RPxP be the unique symmetric positive definite solu- tion to the Lyapunov equation

Pd;, +(d; , ) 'P= -Q (19) where Q E RPXP is a symmetric positive definite design matrix.

Then consider an observer of the form

f, = dllil + d l z j - d l , e ,

j = dZliI + d,, 9 + B, U

- (d,, - 4ZkY + P-'IIB,TPlla (20) where the discontinuous vector U is defined by

(21)

The error system dynamics can then be shown to be asymptotically stable and therefore j + y and 2, + x1 asymptotically [4]. Given any system, the key step is therefore to identify whether assumption (iii) is valid.

U = { -~ey/ l l~ , l l ) if e, # 0 0 otherwise

32 Numerical algorithm for robust observer design This Section provides an algorithm which determines whether the structure given in eqn. 18 is attainable in the special case of equal numbers of inputs and outputs. A more general algorithm, together with proofs appears in Reference 4.

Step 1 : Permute the columns of C until C = [C, C,] where C, E R p x p with det (C,) # 0. Then use the non- singular transformation

to obtain a coordinate system in which the outputs appear as the last p system states.

Step 2: In the new coordinate system let the input dis- tribution matrix be written in the form [::I where B, E R(n-p)xp and B, E R p x p

If B2 is singular, no robust observer exists, and therefore stop.

230

Step 3: Change coordinates with respect to the non- singular transformation

and write the system matrix as

[::: :::I where d 11 ER("-^)^("-^) and d Z , ~ R p X P . If dll is unstable then no robust observer exists and therefore stop.

Step 4 : The system is now in the canonical form

i , = d , , x , + d l z y where dll is stable

g = d Z l X l + d,, y + B,u + B, t Define

where d;, is any stable matrix of appropriate dimen- sion.

Step 5 : Compute the gain matrices C, and C, in the original coordinate system as

C, = T;'T;'G,

where P i s the solution to the Lyapunov equation (19). Step 6 : The observer can be written as

i = A i + Bu - q C P - y ) + C"U where

-P(edll~,ll) ife, f 0 U = { 0 otherwise

Remark: Although this Section has demonstrated the construction of an observer robust against matched uncertainty it must be noted that at no time was it neces- sary to appeal to the fact that B represented the system input distribution matrix. Consequently the design pro- cedure can be used to synthesise the construction of a robust observer for a system with structured uncertainty which can be expressed as

F(t, x, U) = Bst; where B E Rnxl for I d p

and < is a bounded function, by considering the 'system' (A, I, c).

For implementation, the discontinuous unit vector U is replaced by the continuous approximation

e, U, = - p

Ile,ll + 6 , where 6 , is a small positive contant.

The observer is now no longer completely insensitive to matched uncertainty and only motion within a bounded region of 9, can be guaranteed [l].

4 Design and nonlinear simulations

The controller and observer outlined above require knowledge of the nominal linear system triple (A, B, C). To generate such a linear model for the heating plant an

I E E Proc.-Control Theory Appl., Vol. 141, No. 4, July 1994

identification of the nonlinear model was performed using the Matlab Identification Tool Box. A third-order transfer function relating thermocouple voltage to the valve positioner signal was obtained, which, in state- space form can be written as

-0.1029 -0.0082 -0.ooO1 A = [ 0 1 0 1 01 0 B = [ i ]

C = C0.0053 0.0022 0.0O0l]

The design of an observer-controller pair for this system based on the theory presented above follows.

4.1 Observer design for heating plant It should be noted that because of the structure of the realisation, any variation in the elements in the first row of the system matrix (which are the coefficients of the characteristic polynomial) occurs in the image of B, and so can be considered as matched uncertainty. The obser- ver is therefore insensitive to changes in the poles of the system.

Considering the first step of the algorithm, and noting that the third element of the output distribution matrix is nonzero, the simplest choice for the first observer trans- formation is

T3 = I : O 1 0 O l L0.0053 0.0022 0.oo01J

Changing coordinates with respect to this transform- ation, and examining the output distribution matrix gives the matrix sub-blocks

B, = [:] and B, = 0.0053

Since B, is nonzero transformation T4 exists. Following a change of coordinates with respect to this matrix it can be verified that

-0.4106 -0.0134 -77.5491 [".' d12]=[ 1 0 1 188.84981 0.0017 O.oo00 0.3077

d 2 1 d 2 2

Since u(dll) = { -0.3749, -0.0358) the system is in the canonical form described in eqn. 18. Letting the design matrix d;, = -0.2 the linear gain matrix becomes

- 77.5491 C,, = 188.8498 [ 0.507,]

Defining Q = 2 and solving the Lyapunov equation (19) gives P = 5. The gain matrices in the original coordinates can be verified to be [ 18.33211 [:I G, = 188.8498 and G, = 0

Because of the system identification approach adopted, no direct details of the nonlinearities/uncertainties are available to compute p in eqn. 21. A value of p = 10 was found to give adequate performance on the nonlinear simulation as shown in Fig. 2. Clearly this parameter could subsequently be modified if unsatisfactory per-

IEE Proc.-Control Theory Appl., Vol. 141, No. 4, July 1994

formance occurs in the real system. Here the smoothing coefficient is taken as 6 , = 0.05.

4.2 Controller Design for heating plant Since the problem under consideration is single-input- single-output, the design parameter TF1 from eqn. 4 is a

g 1'; $ 0 E - -0 Zo 20 40 60 80 100 120

time,min Fig. 2 Error between observer output and the measured temperature

constant, which has the effect of scaling the integrator state. It has no effect on the linear component of the control action since any two augmented systems gener- ated from different T;' can be considered as different realisations of the same augmented transfer function. Here for convenience T; = 1.

Because of the simple form of B, an orthogonal trans- formation to achieve the 'regular form' is the permutation matrix that interchanges the first and third columns

T=[! 3 Changing coordinates so that the augmented system is in 'regular form', the system becomes

ro -0.0001 -0.0022 I -0.00531

LO -0.OOO1 -0.0082 I-0.10291

The poles of the ideal sliding motion dynamics are chosen to be { -0.025, -0.03 f O.O25i}, which represent dynamics marginally faster than the dominant pole of the open-loop plant. Since the pair (All , AI,) is controllable, the unique M such that u(All + A l , M ) = { -0.025, -0.03 & 0.025i) is given by

M = [-0.5372 0.0019 0.08223

The second transformation T, is now completely deter- mined, and in the new coordinates the system matrix can be shown to be

The remaining pole, associated with the dynamics in the range space of B was assigned the value -0.1 by select- ing the design matrix @ = -0.1. Solving the associated Lyapunov equation (1 1) gives P, = 5.

The sliding surface design parameter S,,, which appears in the equation defining the sliding surface (9) can be shown to determine the value. of x, at steady state. For a single-input-single-output system it is always possible to choose S, so that for the nominal system at steady state

23 1

x, = 0 assuming the sliding surface has been attained. Recalling eqn. 10 which represents the ideal sliding motion of the nominal system, and assuming steady state conditions results in the expression

0 = dl1Zl + (Tz + A,,S,)w Therefore S, is required to make the first element of (d;tTz + A;:A,,S,) identically zero. For the system under consideration S, = 26.1642 can be shown to be the appropriate value. From eqn. 13 the gains can be calcu- lated to be

L = c0.0537 -0.0821 -0.0030 -0.00023

L, = 3.1536

L* = 26.1642

All that remains to be done is to calculate the scalars rlr r2 and r3 that comprise pf,r in eqn. 16 and to select an appropriate value for the smoothing coefficient 6,. Form- ally the r s can be related to the magnitudes of the uncer- tainty bounds. However, as with the observer switching gain, because of the system identification approach adopted, no values for these bounds are available. An estimate of the appropriate magnitudes can be obtained by considering the range of allowable inputs. For the system under consideration the input is restricted to the interval [0,4.5] volts. Consequently since in the region of interest llxll is of the order lo4 therefore rl must be of the order Fig. 3 represents nonlinear simulation tests,

2

9 L' 1 e tl F O

1 - 1

1 "

-2

-3 0 20 40 60 80 100 120

time.min Fig. 3 Simulations with diffmenf nonlinear components

using the observer designed in the previous subsection, with the nonlinear gain

P ~ , ~ = h r l llxll for rl E (0, 4, 1, 14, 21 This demonstrates the increase in performance obtained as a result of increasing rl and hence the nonlinear com- ponent of the control action. In all the simulations the smoothing coefficient 6, = 5.

4.3 Quantitative measure of controller performance To compare the performance of the nonlinear controller- observer pair with other control schemes, a quantitative measure of performance is proposed which is similar to that of Goodhart et al. [SI. If {y i } , {ui} and {wt} are finite sequences which represent the output, input and demand signals, respectively, which are sampled at a fixed rate, then formally the index comprises

232

(i) Mean absolute error: representing the accuracy of the tracking performance, is defined by

where N , is the number of samples

control action used, is defined in the obvious way as (ii) Mean control action: reflecting the amount of

N,

i = l U = ~ u ; / N ,

(iii) Degree of excitation: which is a measure of the degree to which high frequency components appear in the control signal, is defined as

N,

i = l uex = 1 I ui - (uf)i I / N s where {ufJ =f({ul)

and f is a linear low-pass filter designed to remove the high frequency conponents from the control action, which are unrepresentative in terms of the movement of the valve.

4.4 Furnace simulation Fig. 4 represents the response of the furnace simulation under nonlinear control. The tracking performance is

c

6500 20 40 60 80 100 120 time,min

absolute errorlmean controllexcitation/overalI 0.179 I 1.073 I 0.001 11.253

Fig. 4 Nonlinear simulation under nonlinear control

good, especially since much of the demand profile com- prises periods of transients between steady-state oper- ating points for which asymptotic tracking is not guaranteed theoretically. The corresponding control action is given in Fig. 5.

5 Plant trials

British Gas Midlands Research Station (MRS) has an experimental single burner furnace of a design similar to that in Fig. 1. It is fitted with five thermocouples, one on the end wall opposite the burner and a pair on each side wall. Industrial furnaces of this type are currently con- trolled by PID controllers. To examine the effectiveness of the proposed nonlinear scheme it is sensible to compare its performance with that of a 'well tuned' PID. This Section considers the performance of both control- lers with regard to the end-wall thermocouple; this represents the 'nominal' situation. Fig. 6 represents a typical response of the furnace under PID control.

Fig. 7 gives the corresponding control action. The nonlinear controller-observer pair was imple-

mented for trial purposes using a 286 portable PC. An input-output card provided the interface between the PC and the analogue thermocouple and valve positioner signals. The controller-observer algorithm was coded in Turbo Pascal, compiled and run as an executable file. The temperature signal was sampled, the output voltage updated and transmitted ten times a second. This left

IEE Proc.-Control Theory Appl., Vol. 141, No. 4, July I994

sufficient CPU time to 'sample' the observer at a rate of 100 times a second. The selected sample rate of 10 Hz is comparable with that of PID which had an intrinsic sam- pling rate of 8 Hz.

a

1

80 I s 1

40

0 20 40 60 80 100 120 tirne,rnin

Fig. 5 Nonlineor control action

tirne.rnin

absolute errorlrnean contrdlexcitotionloverol 1333 I LO22 I 0287 I 26L3

Fig. 6 Plant under PID control

40 a 5 20 >

'0 20 40 60 80 100 120 tirne,rnin

Fig. 7 PID control action

Fig. 8 gives the response of the furnace, under nonlin- ear control, to a differential approximation to the refer- ence signal used for the PID trial. Here the average

700

-650 0 20 40 60 80 100 120 tirne,rnin

absoluteerrorjmwn control)excitationloverall 0.893 I 1.085 I 0169 I 2.148

Fig. 8 Response ofend-wall thermocouple under nonlinear control

tracking error is under 1" which represents a level of accuracy probably greater than the measurement preci- sion. The control action excitation (Fig. 9) is significantly higher than that of the simulation but is considerably less than that of the PID.

Table 1 gives the performance indices of PID control- ler trials conducted on the same furnace during early 1993. The parameters used were obtained from a self-

IEE Proc.-Control Theory Appl., Vol. 141, No. 4, July 1994

tuning algorithm inherent in the controller. Data from the nonlinear trials is also presented.

The absolute error measure is consistently lower for the nonlinear controller as is the valve excitation. The

. . 0 20 40 60 80 100 120 I

time.rnin Fig. 9 Nonlinear control action

Table 1 : Comparison of performance indices for PID and nonlinear controllers

Test Performance indices for PID trials

Absolute error Mean control Excitation Overall

1 1.333 1.022 0.287 2.643 2 1.433 1.001 0.313 2.757 3 1.440 0.971 0.301 2.712 4 1.321 1.054 0.293 2.668 5 1.355 1.138 0.367 2.860

Test Performance indices for nonlinear controller

Absolute error Mean control Excitation Overall

6 0.893 1.085 0.169 2.148 7 0.901 1.048 0.205 2.155

PID performs better with regard to the mean control signal index. However this measure is heavily dependent on the operating history of the furnace immediately prior to any trial$. The overall measure indicates that the non- linear controller-observer pair is performing at least as well as a commercial PID controller for the nominal test.

5.1 Robustness: Side- wall thermocouple trials To explore the robustness of the nonlinear controller, trials were undertaken where the temperature signal was supplied from one of the side-wall thermocouples. It is known that under normal operating conditions while the end-wall thermocouple operates in the region of 650- 820", the side-wall temperatures are in the range 450- 650". It is therefore not possible to use the original demand signal since the side walls cannot attain such ele- vated temperatures.

Fig. 10 demonstrates the results of a typical side-wall test using the same nonlinear controller-observer pair

tirne.rnin

absolute errorlrnean controllexcitationl overall 0.703 I 0.985 1 0.105 1 1.793

Fig. 10 Response of sidewall thermocouple under nonlinear control

t Tests 1-3 represent three consecutive trials which demonstrate the decreases in fuel usage bust of the retention of heat by the furnace.

233

but a modified reference signal. Paradoxically, in terms of the performance indices, the results are better than those obtained for the end wall. It is suspected that this rep- resents a less difficult control problem, because for the nominal tests, the burner impacts directly upon the thermocouple which results in a high level of valve exci- tation.

Results of a comparable PID test are presented in Fig. 11 ; as in the case of the nonlinear controller the original parameters have been left unaltered.

time, min

absolute e r r o r b n controllexcitation)overall 2.138 I 1.060 I 0.218 13.414

Fig. 11 Response of side-wall thennocouple under PID control

These test results indicate that the nonlinear controller performs better from the point of view of robustness, In an industrial setting, retuning of the PID controller would be necessary. This is circumvented by the use of a single, robust nonlinear controller.

6 Conclusions

This paper has demonstrated that nonlinear theory based on sliding mode concepts can be used to good effect on an industrial process. The practical choice of parameters for the proposed controller-observer pair has been described. Both nonlinear simulation and trial results have been presented to support the assertions of this paper. It should be noted that although the case study presented is single-input-single-output, the theoretical approach is multivanable. Work is currently in progress

to test the controller-observer pair on a multiburner tunnel furnace within the process industry.

7 References

1 BURTON, J.A., and ZINOBER, A.S.I.: ‘Continuous approximation of variable structure control’, Int. J. Syst. Sci., 1986,17, pp. 875-885

2 DECARLO, R.A., ZAK, S.H., and MAlTHEWS, G.P.: ‘Variable structure control of nonlinear multivariable systems: a tutorial‘, Proc. IEEE, 1988,76, pp. 212-232

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