Self-adaptive near-optimal control ofdiffusion equations

A.S.I. Zinober, B.Sc, M.Sc. (Eng.), Ph.D., A.F.I.M.A., C.Eng., M.I.E.E.

Indexing terms: Optimal control, Digital simulation, Self-adaptive control, Time-optimal control, Variable-structure systems, Sliding motion

Abstract: The self-adaptive near-optimal control of diffusion equations is described. The variable structurefeedback design is independent of the initial conditions and the plant parameter values. Rather than ident-ifying the precise parameter values by adaptive correction, the online self-adaptive closed-loop controllerorganises itself by identifying certain dynamic characteristics associated with sliding motion. Digital simu-lation results are presented for both linear and nonlinear systems, and the effect of parameter variations andexternal disturbances are discussed. The system responses are close to the time-optimal paths.

1 Introduction

The time-optimal control of linear diffusion equations hasbeen discussed by numerous authors. Approximate sol-utions may be obtained using techniques such as themethod of moments,1 the finite Laplace transform,2

Galerkin's method3 and the approximation of the distrib-uted parameter system by a high-order lumped parametermodel using differential difference equations.4'5 Allmethods yield open-loop bang-bang control solutions whichare highly sensitive to system parameter variations6'7 andexternal disturbances.

In this paper a self-adaptive control system is described.The adaptive variable structure feedback design is indepen-dent of the system's initial conditions and parameter values.Owing to its closed-loop structure it can handle externaldisturbances and a class of parameter variations. Ratherthan identifying the precise system parameter values byonline adaptive correction, the adaptive closed-loop con-troller organises itself by identifying certain dynamiccharacteristics associated with sliding motion by alteringthe position of the switching hyperplane. The adaptivestrategy identifies the position of a certain surface in thestate space associated with the sliding motion of thesystem. The easily implemented control scheme has pre-viously been successfully applied to the near-time-optimalcontrol of linear and nonlinear systems up to tenth order.

2 Problem statement

Consider the linear one-dimensional diffusion equation

9 0 , ^ » 2 0 , .s

where

>0 0)

with initial distribution 0(x,O) = 0o(x) and boundaryconditions

(2)

where /(r) = u. The control has the magnitude constraint

The boundary control corresponds to heat flux controlat the end (x = 0) of a one-dimensional slab. The function/(r) has the form

= F(D)r(t) (3)

Paper 1001D, first received 8th July and in revised form6th September 1980Dr. Zinober is with the Department of Applied & ComputationalMathematics, University of Sheffield, Sheffield SlO 2TN, England

290

F(D) = I a,D'j=o

(4)

with D = d/dt. The case generally considered has s = 0and r{t)=—Ku{t). This represents the direct applicationof the controlling heat flux on the slab. For s > 1dynamical systems between the slab and the controller areintroduced. We shall consider the case s = 0 in the rest ofthis Section.

The time-optimal problem is to determine u(t) such that,in minimum time T, the final distribution <f>(x, T) = 0 isattained from the given initial conditions 0O. Theexistence8 and controllability9 of the time-optimal dif-fusion equation with boundary control has been discussedby numerous authors using infinite dimensional functionspace theory. Goldwyn et al.2 have shown that the transfercan be achieved in a finite time using bang-bang control.The final state 0(x, T) can be made arbitrarily close to zeroas the number of switchings approaches infinity. Accept-able results may be achieved with a finite number (n — 1)of control switchings, where n>4. The eigenvalues are— c-1 ,• 2 „ 2i2n2,i = 0,l,2,. . .,and it can readily be seen thatthey rapidly take on large negative values. Thus a low-ordersystem approximation retaining the dominant eigenvaluesis sufficient.

Ah «thorder differential difference equation describingthe system can be obtained using the finite differencemethod, giving

y(t) =

•••y~n)T, yi =

(5)

, t) and 8x =with . y = O il/(» + I)-

Using a model reduction technique due to Davison,10

Mahapatra5 obtains an accurate fourth-order model(fox c=K= 1)

+ bMu(t) (6)

from eqn. 5 with n = 19, w = (.y4 y8 yl2 yit,)T.The low-order system's eigenvalues are 0, —10-91, — 43-34,— 96-42 in comparison with the precise dominant eigen-values 0 , - 9 - 8 7 , - 3 9 - 4 8 , - 8 8 - 8 3 . This model may beused to obtain three switching times and the final time.The results compare favourably with results using largern values indicating the validity of this low-order modelapproximation. In a later Section we shall use these resultsto assess the accuracy of the self-adaptive controller.

IEEPROC, Vol. 127, Pt. D, No. 6, NOVEMBER 1980

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3 Self-adaptive controller

3.1 Basic strategy

For linear systems of the form of eqn. 5 the author 1 1 " 1 3 hasdescribed in detail self-adaptive control strategies whichyield state trajectories close to the time-optimal responseswi thout the direct identification of the plant parametersin A and b. The basic self-adaptive controller has thecontrol law

u — —

where v is the linear switching functionv = kT

z = qn~2an-2z2

(7)

(8)

the zt are the chosen state variables, q is the variable controlparameter and the at are constants . Many authors , includingFliigge-Lotz14 and Utkin,1 5 have derived the necessarycondit ion for sliding (or chat ter) mot ion on v — 0 wi th qconstant . Sliding mot ion occurs when, at a point on theswitching surface v = 0 , the directions of mot ion along thetrajectories on each side of the surface are towards theswitching surface. We need lim v< 0 and lim v>0 in

z>-»o+ v-*o~the neighbourhood of v = 0. Alternatively, we require vv <0 in the neighbourhood of v = 0. The state point then slideson the surface v = 0 and its dynamic behaviour satisfiesthe equation

v = 0 (9)

imposed by the high-frequency chattering control (u = ±1).Utkin15 has shown that the 'equivalent control' ueq, asmoothed version of u, yields the same dynamic behaviour.Thus, in a practical implementation, the high-frequencycomponent of the control may be removed by passing uthrough a lowpass filter. During sliding motion the variablestructure system is insensitive to a wide class of parametervariations and disturbances.15'16 Substitution of eqn. 9 intov=0 yields an (n — l)-dimensional system associatedwith the switching hyperplane v = 0. The (n — 1) eigen-values of the sliding system are fixed by the choice of thehyperplane.

The adaptive control strategy is depicted in Fig. 1. Attime t = 0 the control parameter is set to the value 5, asmall positive number. The position of the switchinghyperplane in the state space depends on the parameter q.The value of q is continually adjusted by the following rule:

plant I — z

~ IS

Iplant sliding?

I no

yes— c1—q ( 1 *e)

Fig 1 Basic adaptive control strategy (nth-order system)

aj> 0; e > 0; 6 > 0

If sliding motion occurs on the current switching hyper-plane, the next position of the switching hyperplane ischosen to be just ahead of the state point on its trajectoryin the state space by setting

q(t+) = q(t){l+e} (10)

where e is a small positive number. Otherwise q is keptconstant.

The state point can be shown to move from the initialconditions to a certain sliding boundary surfaceS(z,A,b) = 0, on which it remains until the state origin isreached. The reachability of the initial and subsequentpositions of the switching hyperplane can be checked byverifying the necessary condition15 vv<0. The self-adaptive controller identifies at a given time instant duringthe sliding phase the local position of the nonlinear surfaceS(z, A, b) = 0, which is a function of the chosen statevariables and the plant parameters. The sliding boundarysurface So is the boundary between the sliding and non-sliding regions of the state space. In the sliding region wehave vv<0 in the neighbourhood of the switching hyper-plane v = 0 for a certain positive value of q. In the non-sliding region vv> 0. Thus, on the boundary of these tworegions, vv= 0. Therefore, points on the nonlinear surfaceSo are generated by the equations

v = kT{q)z = 0 and ljjj\ = kT{q)z = 0 (11)

for values of positive q. The system's dynamic behaviour onSQ is given by the nonlinear equations

^(kT(q)z) = 0 and ^(kT(q)z) = 0 (12)

which can be solved for q and z. The technique may also beapplied to certain nonlinear and slowly time-varyingsystems. So may be regarded as the 'equivalent switchingsurface'. The system parameters need not be directlyidentified to determine the local position of So.

The author has analysed the properties of second-ordersystems,11'12 and a third-order system13 using nonlinearswitching surface theory. The analysis of higher-ordersystems is rather complex. To illustrate the techniquethe dynamic behaviour of the double integrator system

= x2,x2 = ujb,(b > 0, \u\ < 1) 03)

is analysed, and the properties associated with the slidingboundary surface are derived. Sliding motion occurs on theswitching line

v = (qXl +x2) = 0 (14)

when q is held constant, if vv< 0, yielding the condition

\x2\<(qbTl (15)

Thus, the sliding region of the xtx2 state space consists ofpoints on the switching line satisfying

l*21 < (fib)- i (16)

which lie between points A and A' in Fig. 2. Suppose thestate co-ordinates satisfy eqn. 14 and

q = (p\x2\Yl (17)

Then the point (xx ,x2), A or A', lies on the boundary of

IEE PROC, Vol. 127, Pt. D, No. 6, NOVEMBER 1980 291

the sliding region. If q is allowed to vary, the point (JCX ,x2)describes the sliding boundary surface, from eqns. 14 andIV,

S(x1,x2,b) = Xx + bx2 1*21 = 0 (18)

which is the boundary between the sliding and nonslidingregions.

The self-adaptive controller initially drives the state withu = sgn(— 8x! —x2) to the point A on the switching line

x +x2 = 0 (19)

in Fig. 3, since vv<0 (satisfying the reachability con-dition). For 5 sufficiently small, sliding begins and theswitching line is rotated to position L2 by altering the valueof q to q = 5 (1 + e), e > 0. The state slides on the switchline Lx from the point A to A'. Assuming that slidingmotion can be detected within an infinitesimal timeinterval,/!' is coincident withyl. The new control law is

u = sgn {—5 (1 + e)xi ~x2}

and the control value at point A is now

uv = sgn(-Xi) = sgn(x2)

(20)

(21)

which drives the state to point B on the current switchingline L2

5(1 t + x2 = 0 (22)

Sliding recommences, and the switching line is rotatedIX.,

Fig 2 Sliding region of double-integrator system

IX.

Fig 3 Self-adaptive control of double-integrator system

292

further to position L3. After repeated rotation of theswitching line whenever sliding occurs, the state reachesthe point D in the neighbourhood of the sliding boundarysurface, S = 0. At this point the current switching line Lt

is rotated to the position Li+1,

+x2 = 0 (23)

The control uv drives the state to the point E on the lineLi+1 in the nonsliding region. Sliding does not occur atpoint E and the state enters the region

+x2} < 0 (24)

The state path returns to the line Li+l at point F in thesliding region, and sliding recommences. As e -* 0 the over-shoot of the surface So becomes negligibly small, and thestate moves effectively on the sliding boundary surface tothe state origin within a finite time. The effective value ofthe control maintaining the state on the surface So is\ueq I = h and q > 0 from eqns. 12. The surface So, whichmay be regarded as the 'equivalent switching surface', isin fact the time-optimal switching curve of the double-integrator plant with the parameter 2b, i.e.

+bx2\x2\ = 0 (25)

The analysis of an wth-order system requires the study ofmotion on a nonlinear sliding boundary surfaceS(z, A, b) = 0 in an ^-dimensional state space, zGRn.

3.2 Modified strategy

A modified strategy allows the controller to be reset when-ever a new set of initial conditions or an error conditionoccurs. For initial conditions in certain regions the settlingtime is decreased.12 The following steps are carried outconsecutively in the regions of the state space where theswitching hyperplane assumes positive q values:

Step (i)The hyperplane is rotated until it passes through thecurrent state point.

Step (H)The controller tests for sliding at this point.

Step (Hi)If sliding occurs the hyperplane is rotated further, to bejust ahead of the state point (q is reset in accordance witheqn. 10).

Step (iv)If sliding does not occur, the hyperplane is kept fixed untilsliding recurs (or until step (a) below causes step (i) torotate the hyperplane).

Step (a)An auxiliary switching hyperplane

v(Q, z) = 0 (26)

is continuously rotated so as to pass through the state pointz(t). If the inequality

q<Q<q(l+e) (27)

IEEPROC, Vol. 127, Pt. D, No. 6, NOVEMBER 1980

does not hold, then we reset the value of the control para-meter q to that of the auxiliary parameter Q

Q = Q (28)

i.e. rotate the switching hyperplane until it passes throughthe current state point (step (i) above). The purpose of theauxiliary hyperplane is to reset the control parameter qwhenever a sudden disturbance gives rise to a new set ofinitial conditions. In those regions where v = 0 yields non-positive q values, the control is identical to the basicstrategy.

3.3 Switching hyperplane

The design choice of the vector k with components q'ai ineqn. 8 differs from that of variable structure systemdesign15 with fixed hyperplane. For n = 2 we havea = (1 1)T. Any other choice simply rescales the valueof q. Stability results for second-order systems are given inReference 12. The stability of a triple-integrator system hasbeen studied,13 and the fastest transient motion is attainedwith a = (1 4 / 2 / 3 1)T. For higher-order systems thevector a, with components at as the binomial coefficients,has been found to yield stable motion for a wide variety ofsystems up to tenth order. The (n — 1) repeated eigenvaluesassociated with the sliding hyperplane have the value — q.

4 Self-adaptive control of diffusion equation

To choose the order of the self-adaptive control strategy weshall consider eqn. 1 to be satisfactorily approximated bythe fourth-order model of eqn. 6, and therefore adopt thefourth-order controller with

v = + q2a2z2 = kTz (29)

A suitable, but not necessarily optimal, choice of the at

constants is at = 1, a2 = 3, o3 = 3, a4 = 1. The necessaryconditions for the state point to reach the first position ofthe hyperplane and for sliding motion to occur can beverified. The hyperplane v=0 is reached, and slidingmotion occurs when vv < 0, i.e.

\kTAz\ kTb (30)

Satisfactory results have been obtained for a variety of zvectors. In this paper zT = (y19 yl3 yn yx). Theinitial distribution is 0O = 1, c = I, 5 = 0 - 3 , e = 0-05 andr(t) = —Ku(t) in eqn. 2. The 19th-order finite-differencemodel (eqn. 5) was used in the simulation of the adaptivecontrol with K = 1.

The minimum time T of the time-optimal controllerincreases with the number of switching times. For threeswitchings T = 1-199, whereas for six switchingsT = l - 2 2 1 3 . Mahapatra5 obtains r = 1-182-for threeswitchings using his model-reduction technique. In Fig. 4the final distributions arising from Mahapatra's near-time-optimal control and the author's adaptive technique arecompared. Defining the settling time Ts so that.\4>(x, O K A for t>Ts, the adaptive controller yieldsT?= 120 for A = 0-025, and Ts=l-2l for A = 0013 .The identical controller yields near-optimal state paths fora range of values of 0O (*), c and K.

If q is held constant (q = 5), i.e. we have a standardvariable structure system with fixed switching hyperplane,the settling time is increased to 1-7. The greatly increasedsettling time (see Fig. 5) is because of the exponential

decay of the system states on the hyperplane, givinglim z (0 = 0. In the adaptive design the real parts of thet-*ooeigenvalues are continually being made more negative,yielding ever faster exponential decay to the state originin a finite time.

If we assume a third-order adaptive controller2 (31)v = +a0z3

with a0 = 1, ax = 2 , a2 = 1 and z = (j>19 y10 yx)T,

the less accurate state trajectory of Fig. 5 results.The adaptive controller is robust in that it operates

satisfactorily when it is implemented imprecisely, and can

0 - 0 4

Fig 4 Linear diffusion equation: <t>0(x) = 1; final distributions of<t>(x, Ts)

o o o o near-time-optimal control (Mahapatra5); Ts = 1-182self-adaptive control; Ts= 1-20self-adaptive control; Ts = 1-21

1

#0-2,0

1-4 t

0(0-6.0

• • • •1 i—•-

0(0-8,0

Fig 5 State paths of linear diffusion equation: <j>0 (x) = 1

self-adaptive control (4th-order)• • • • self-adaptive control (3rd-order)x x x x constant variable structure control (4th-order)

IEEPR0C, Vol. 127, Pt. D, No. 6, NOVEMBER 1980 293

handle large disturbances. A 'coarse' controller withe = 0-25 and qmax = 5 yields large discrete jumps in theposition of the switching hyperplane. The settling time isstill satisfactory.

It should be noted that the adaptive controller is, unlikethe open-loop optimal controller, independent of the initialconditions 0O and a wide class of parameter variations andexternal disturbances. For example, a large impulse disturb-ance at t = 1, yielding

0(x, 1+) = 2 0(x, 1") (32)

results in T8 = 1 -2 (see Fig. 6). The state point leaves thevicinity of the switching hyperplane for a finite time andthen recrosses the hyperplane after a period of nonslidingmotion. The adaptive control strategy is then restarted.

0(0-2,t)

0(0-4,0

0(0-6,t)

0(0-8,t)

Jeq

Fig. 6 State paths of linear diffusion equation: <}>0(x) — 1

Disturbance at t — 1

Nonlinear and (slowly) time-varying systems may alsobe controlled. For example, the nonlinear diffusionequation

30dt

(33)

with

= o(34)

and 0O = 1 yields Ts = 0-79.When s > 1 in eqn. 4 we have the control acting on the

system through further dynamics. The case s = 1 withu = 10 drjdt + r, i.e. a lag network, results in the statepaths in Fig. 7.

5 Conclusions

A self-adaptive controller has been shown to yield timeresponses with a settling time exceedingly close to thetime-optimal solution of the linear diffusion equation,without the need for parameter identification. The con-troller is easily designed and implemented. An analogue/digital scheme for a typical variable structure controller isgiven by Young.17 He demonstrates that microcomputerimplementation is feasible. In addition, nonideal switching,e.g. switching delay, has little detrimental effect on theoverall performance. The closed-loop controller successfullyhandles nonlinear diffusion equations and a variety ofinitial conditions, disturbances and parameter variations.The time-optimal open-loop controller is highly sensitive todisturbances and parameter variation, and accurate offlinecalculation of the switching times is necessary. The authoris not aware of straightforward algorithms to determine theoptimal control of nonlinear and time-varying diffusionequations. The optimal choice of vectors a and z in eqn. 8has yet to be studied. In the case of the linear diffusionproblem, the vectors a and z used in the paper are close totheir optimal values, since the result is close to the time-optimal solution.

6 References

1 BUTKOVSKY, A.G.: 'Some approximate methods for solvingproblems of optimal control of distributed parameter systems',Automat. Remote Control, 1961, 22, pp. 1429-1438

2 GOLDWYN, R.M., SRIRAM, K.P., and GRAHAM, M.: Time-optimal control of a linear diffusion process', SIAM J. Control& Optimiz., 1967, 5, pp. 295-308

3 PRABHU, S.S., and McCAUSLAND, I.: 'Time-optimal controlof linear diffusion processes using Galerkin's method', Proc. IEE,1970,117,(7), pp. 1398-1404

0(0-8, t)

0(0-6, t)|

0(0-4, t)

0(0-2, t)

1-9 t

Fig. 7 Linear diffusion equation: 0O (x) = 1

w = 10 drjdt + r

294 IEE PROC, Vol. 127, Pt. D, No. 6, NOVEMBER 1980

4 McCAUSLAND, I.: Time-optimal control of a linear diffusionprocess', ibid., 1965, 112, (3), pp. 543-548

5 MAHAPATRA, G.B.: Time-optimal control of linear diffusionsystems by a spatial discretization procedure', IEEE Trans.,1977, AC-22, pp. 481-482

6 ZINOBER, A.S.I., and FULLER, A.T.: The sensitivity ofnominally time-optimal control systems to parameter variation',Int. J. Control, 1973,17, pp. 673-703

7 RYAN, E.G.: 'On the sensitivity of a time-optimal switchingfunction', IEEE Trans., 1980, AC-25, pp. 275-277

8 FALB, P.: 'Infinite dimensional control problems I: On theclosure of the set of attainable states for linear systems', /. Math.Anal. Appl, 1964, 9, pp. 12-22

9 FATTORINI, H.O.: 'On complete controllability of linearsystems', /. Differ. Equations, 1967, 3, pp. 391-402

10 DAVISON, E.J.: 'A method of simplifying linear dynamicsystems', IEEE Trans., 1966, AC-11, pp. 93-100

11 ZINOBER, A.S.I.: 'Relay control of plants subject to parameteruncertainty'. Ph.D. thesis, University of Cambridge, 1974

12 ZINOBER, A.S.I.: 'Adaptive relay control of second-ordersystems', Int. J. Control, 1975, 21, pp. 81-98

13 ZINOBER, A.S.I.: 'Analysis of an adaptive third-order relaycontrol system using non-linear switching surface theory', Proc.R. Soc. Edinburgh, 1977, 76A, pp. 239-254

14 FLUGGE-LOTZ, I.: 'Discontinuous automatic control'(Princeton University Press, Princeton, USA, 1953)

15 UTKIN, V.I.: 'Equations of sliding mode in discontinuoussystems I', Automat. Remote Control, 1972, 33, pp. 1897-1907

16 DRAZENOVIC, B.: The invariance conditions invariable struc-ture systems', Automatica, 1969, 5, pp. 287-295

17 YOUNG, K.-K.D.: "Controller design for a manipulator usingtheory of variable structure systems', IEEE Trans., 1978,SMC-8, pp. 101-109

Erratum

KOUSSIOURIS, T.G.: 'On the sensitivity of a systemcompensated by the inverse Nyquist array', IEE Proc. D,Control Theory & Appl, 1980, 127, (4), pp. 177-187

On p.178, column 1, line 10 following eqn. 18 should read:Tu (/Wfe) is equal.. .

and in column 2, Lemma l(Jb) should read:H(joS) is nonsingular,. . .

and eqn. 23 should read:

W A I \QkiU")\dk (co) / « I (23)

On p.179, column 2, the last line of eqn. 31 should read:

= I [#( /«)] u\• \Qlt (/co) + /„ | (1 + Bt) (31)

and eqn. 33 should read:

6 = \QiiU")+fu\ (33)

On p. 182, column 2, the numerator in the first line ofeqn. 76 should read:

ETC69 D

IEE PROC, Vol. 127, Pt. D, No. 6, NOVEMBER 1980 295