Singular-perturbation method for discrete modelsof continuous systems in optimal controlP.K. Rajagopalan, B.E., M.S., Ph.D., and D.S. Naidu, B.E., M.Tech., Ph.D., Mem. I.E.E.E.

Indexing terms: Optimal control, Perturbation theory, Closed-loop systems

Abstract: The closed-loop and open-loop optimal controls of a singularly perturbed continuous systemare considered by means of their discrete models. A singular-perturbation method is developed to obtainseries solutions in terms of the outer, inner and intermediate series analogous to that in a continuous system.It is shown that the resulting matrix Riccati difference equation for closed-loop optimal control is notamenable, to singular-perturbation analysis in its original form and has to be properly recast to fit into theframework of singular-perturbation theory. The discrete-model representation has the twin advantages ofthe reduction in order associated with singular perturbation and the reduction in the computation due tothe recursive nature of the solutions in discrete systems. Series solutions are then possible for higher-orderapproximations with considerable reduction in computation. The method is illustrated by an example.

1 Introduction

The study of singularly perturbed continuous optimal-controlsystems has been a flourishing field of research [1], Thetechnique is essentially an adaptation of the basic resultsfor initial- and boundary-value problems in singularlyperturbed differential equations [2, 3 ] . In general, it isexpected that any method developed for continuous controlsystems described by differential equations should be capableof being extended to discrete control systems characterisedby difference equations. But, no serious effort has so farbeen made in this direction, although discrete control theoryis of great importance because of its applications in computercontrol [4]. Recently, Rajagopalan and Naidu [5] developeda singular-perturbation method for initial-value problemsarising in discrete control systems, whereas Phillips [6]considered discrete systems satisfying a two-time-scaleproperty and obtained reduced-order models without con-sidering the initial conditions lost in the process of reduction(or degeneration). Blankeship [7] considered a differentclass of singularly perturbed difference equations arising inoptimal control and used the method of matched asympoticexpansions and a multitime method [8].

In this paper, the closed-loop and open-loop optimalcontrols of a class of discrete models for continuous systemsare considered. The resulting matrix Riccati differenceequation arising in closed-loop optimal control is recastin a form amenable to singular-perturbation analysis. Theopen-loop optimal control gives rise to a discrete two-pointboundary-value problem described by singularly perturbeddifference equations. A method is developed to obtain seriessolutions in terms of the outer, inner and intermediate seriesanalogous to that in continuous systems [9, 10]. An exampleis given to illustrate the method, the results being given up tothe fourth-order approximation and compared with the exactsolution of the continuous system.

The distinguishing features of the present method are(a) the reformulation of the matrix Riccati difference

equation to fit into the framework of singular-perturbationtheory

(b) the development of a singular-perturbation methodfor a class of discrete models in closed-loop and open-loopoptimal-control problems

(c) the full utilisation of the twin advantages of thereduction in order embedded in singular perturbations andthe reduction iri the computation due to the recursive natureof solutions in discrete systems, thereby enabling us to obtain

Paper 1399D, first received 5th January and in revised form 14thMay 1981The authors are with the Department of Electrical Engineering, IndianInstitute of Technology, Kharagpur, 721302, India

series solutions for higher-order approximations with lesscomputational complexity.

2 Problem formulation

Consider a linear, singularly perturbed continuous systemdescribed by

dx— = Axx + A2zdt

dz . , .h — = A3x + A^z

dt

x(t = 0) = x(0) (la)

+ B2u z(t = 0) = z(0) (16)

where x and z are nx- and n2-dimensional state vectors, uis an /--dimensional control vector, and h is a small positivescalar parameter. The A$ and Bts are matrices of appropriatedimensions. The performance index to be minimised is

j = i y\tf)Fy(tf) +1- \ '(y'Qy + u'Ru)dt (2)2 2 Jo

where / = (*', z'), F and Q are {(/ij + n2)x (n1 + n2)\dimensional, positive-semideiinite symmetric matrices, and Ris, an (r x r) positive-definite symmetric matrix. Of the severalways [11-13] available for obtaining discrete models ofsingularly perturbed continuous systems, the simplest onefor digital simulation is the model obtained by using thefirst forward difference [4]. In this method, the function iscomputed at discrete intervals of time T. Furthermore, att = kT

dx

dt

x(k(3)

= kT

where x(k) = x(t)\t=kT. Based on this, the differenceequations describing the discrete models of the continuoussystem given by eqns. 1 and 2 are

x(k + 1)

hzQc + 1)

and

J =l-y

+ TAi TA2

TA3 hI+TAA\[z(k)\ [TB2

2 fe=0

(4)

If the results of discrete optimal-control theory [14] are

•The substitutions Q = TQ and R = TR are made for analyticalconvenience

142 0143-7054/81/040142 + 07 $01.50/0 IEEPROC, Vol. 128, Pt. D, No. 4, JULY 1981

applied to the discrete system given by eqns. 4 and 5, theoptimal control can be obtained in the following cases:

(a) closed-loop form resulting in the matrix Riccatidifference equation

(b) open-loop form resulting in the discrete two-pointboundary-value problem.The two cases are considered separately.

2.1 Case (a): closed-loop controlThe closed-loop optimal control is given by

u(k) = -R-1B'(A'r1(P(k)-Q)x(k)

where P(k) satisfies the matrix Riccati difference equation

with the end conditions specified as

P{kf) = F

where

(6)

(7)

(8)

A =TA2

T

h A

B =TBX

T

It is to be noted that the matrix Riccati difference equation(eqn. 7) is solved backwards from k = kf to k = 0 using theend condition given by eqn. 8.

To cast eqn. 7 into singularly perturbed form, P(k), F andQ are assumed as

P(k) =

Q =

Pi(k) hP2(k)

hP'2{k) hP3(k)F =

Fx hF2

hF2 hF3

(9)

and the standard form for singularly perturbed matrix Riccatiequations is [15]

Px(k) = Gx(Px(k + l),P2(k+ l),P3(k + \),k,h)

hPi{k) = Gi(Pl(k+ l),P2(k + l),P3(k + \),k,h) (106)

i = 2,3

But the presence of an inverse in eqn. 7 prevents us puttingit in the required form given by eqns. 10. This difficulty isovercome by using the binomial expansion [16]

[I+BR~1B'P(k+ I)]"1 = I-BR~lB'P{k+\) (11)

If eqns. 7—11 are used, the singularly perturbed matrix Riccatidifference equation becomes

Pt(k) = Px{k+ l)+77{/>1(A:+ \)AX +A[Px(k+ 1)

+ P2(k+ \)A3 +A'3P2(k+ 1)

-Px(k+

(12a)

-P1(k+l)E3P2(k+

-P2(k + \)E2P'2(k +

hP2(k) = hP2(k+ 1)+ T[Px(k+ \)A2

+ A'^k+V-PtQc+VE^ik+l)

-P2(k+l)E2P3(k+l) + Q2

-h{-A[P2(k+ 1) + Pi(k+ l)ElP2(k+

+ P2(k + \)E'3P2(k + 1)}]

hP3(k) = hP3(k + 1) + T[P3(k + l)A4 +A\P3(k

-P3(k+l)E2P3(k+l) + Q3

-h{-P2(k + l)A2 -A'2P2(k + 1)

+ P2(k+l)E3P3(k+l)

(12c)

where

Ex = BXR Bx, E2 — B2R B2, E3

The end conditions are reformulated as

p.Qc = kf) = F. i = 1 , 2 , 3

2.2 Case (b): open-loop controlThe open-loop optimal control is given by

B

u(k)-R~1(B[v(k+l) 1))

(13)

(14)

where v and w are the costates associated with x and z,respectively. The discrete two-point boundary-value problem(TPBVP) is described by the state and costate differenceequations as [14]

x(k

v{k)

hz{k

hw(k)

= x(k) + T{Axx(k) + A2z(k) -Exv(k

-E3w(k+ 1)}

= v(k+ l)+T{A[v(k+ l) + A'3w(k+

- E3v(k= hz(k) + T{A3x(k)

-E2w(k+1)}

= hw(k) + T{A'2v(k

(15a)

(15b)

(15c)

with boundary conditions

x(k = 0) = x(0) z(k = 0) = z(0) v(kf) = 0

w(kf) = 0 (16)

where the terminal cost is assumed to be zero to simplifythe complexity of the resulting equations.

The main aim of this paper is to develop a method ofobtaining the approximate solutions for case (a), the singularlyperturbed matrix Riccati system given by eqns. 12 and 13,and for case (b), the singularly pertubed discrete two-pointboundary-value problem given by eqns. 15 and 16.

3 Singular-perturbation method

The singular-perturbation method is developed and is analogousto the results for closed-loop continuous optimal-controlproblems [10]. The method consists of seeking the seriessolution in terms of the outer, inner and intermediate series.

IEEPROC, Vol. 128, Pt. D, No. 4, JUL Y 1981 143

The outer series corresponds to the reduced-order (ordegenerate) system with some of the boundary conditionsbeing lost in the process of reduction. Therefore the outer-series solution by itself cannot satisfy all the given boundaryconditions, although it is closer to the exact solution every-where except at the boundaries. The lost boundary conditionsare recovered by considering the stretched form of the givensystem. The inner and intermediate series are constructedfrom the stretched system in such a way that their mainpurpose is to salvage the destroyed boundary conditionsand to contribute negligible effect outside the boundarylayer.

3.1 Case (a): closed-loop controlConsider the singularly perturbed matrix Riccati system ofeqns. 12 and 13.

3.1.1 Outer series: The outer series is assumed in the formof a power series in h as

Pt(k) = Pt°\k) + hP[i}(k) + . . . i = 1,2,3 (17)

If eqn. 17 is used in eqn. 12 and coefficients of like powersof h are compared, a set of recursive equations are obtained.For zeroth-order approximation

P[°\k) = P[°\k T{P[°\k

l)E1P[°\k+

-P\°\k + l)E3P{0)'(k + 1)

-P[0)(k+l)E3Pi°\k+l)

= T{P{°\k+l)A2

-P\°\k+l)E3P3o)(k +

l)E2Pi°\k+

Q3}

with the condition

P\°\kf) = Py(kf) =

(18a)

(18b)

(18c)

(19)

It is to be noted that, owing to the nature of eqns. 186 and18c, the end conditions Pfo)(kf) will not, in general, beequal to Pi(kf), i=2, 3. These end conditions are sacrificedin the process of making h = 0. Once P[°\k) is known,Pi°\k) and Z^0)(A:) are automatically fixed from eqns. 18band 18c.

For first-order approximation,

= P\l\k + 1) + T{P[l\k + lMi

1)A3

+ A'3Pil)'(k + 1)-P[l\k

= P{2°\k

+ 1) E2Pi°y (k + 1)

+ l)E2Piiy(k + 1)}

1) + T{P[l\k + 1)A

4 +A'3Pil)(k

(20a)

-Pl2°\k (20b)

Pio)(k) = \k + 1)A4

-P\l\k + l)E2Pi0)(k

-Pi°\k + 1)E2Pix)(k (20c)

where G\0)(k) and G^0)(k) represent all the terms containingzeroth-order coefficients. Similar equations are obtainedfor higher-order approximations.

The solution of eqns. 20 can be obtained only if the endcondition P[l)(kf) is known explicitly. The determination ofthis end condition is an important step in singular-perturbationtheory and is discussed later. Again, once P\1)(k) is known,P2

l)(k) and P3l)(k) are automatically known from eqns. 206

and 20c.

3.1.2 Inner series: To recover the end conditions P2(kf)and P3(kf) lost in the process of degeneration, it is necessaryto form a stretched discrete model for the matrix Riccatidifference eqns. 12. The stretched discrete model can bedirectly obtained from the discrete model by using thetransformation

Ts = T/h (21)

where Ts is the stretched discretising interval, and T isassumed to be smaller than h.

If this transformation is used in eqn. 12, the stretchedversion of the matrix Riccati equation becomes

Mx(k) = My(k+ l) + hTs{Ml(k

+ M2(k+1)A3

l + A\M1(k+

-Mx(k+ l)E3M'2(k+ 1)

-M2(k + l)E2M2(k + 1) + (22a)

M2(k) = M2(k + l) + Ts[M1(k

+ A3M3(k + 1)-Mi(k + l)E3M3(k + 1)

-M2(k + l)E2M3(k + 1) + Q2

-h{-A\M2(k + l)+Ml(k + l)£iM2(A:

+ M2(k + l)E3M2(k + 1)}]

M3(k) = M3(k + 1) + Ts[M3(k + l)/44 + A'4M3(k

-M3(k+l)E2M3(k + l) + Q3

-h{-M'2(k + 1)A2 -A'2M2(k + 1)

k + l)EiM3(k+ 1)

(22b)

144 IEEPROC, Vol. 128, Pt. D, No. 4, JULY 1981

+ M3(k + 1) E3M2(k + 1)}

-h2{M2(k + \)ExM2(k + 1)}] (22c)

The solution of this stretched discrete model is assumed inthe form of a series, called the inner series, as

Mt = hM}1\k) + ... i = 1, 2, 3 (23)

Substitution of eqn. 23 in eqns. 22 and collection of co-efficients of like powers of h results in a set of equations.For zeroth-order approximation,

M[x\k) = M[0)(k + l) (24*)

M2°\k) = M20)(k + 1) + Ts{M\0)(k + l)A2

+ M20)(k + 1>44 + A'3M3

o)(k+l)

-M\0)(k+l)E3M3°\k + l)

-M\0){k + l)E2M30)(k + 1) + Q2} (24b)

M3°\k) = M3°\k + 1) + Ts{M3°\k + \)A«

+ A\M3°\k + 1) -M3°\k + \)E2Mi°\k+\)

+ <23} (24c)

For first-order approximation,

M[°\k) = M\l\k + 1) + Ts(G~[0)(k) (25a)

M2l)(k) = M2

x)(k + 1) + Ts{M\n(k + \)A2 +Ml2

1)(k + 1>44

+ A'3M3x)(k+ l)-M\1)(k+l)E3M3

0)(k + l)

-M21)(k+l)E2Mi°\k+l)

-M\0)(k + l)E3M31)(k + 1)

-M20)(k + 1) E2M3

l\k + 1) + G20)(k)} (25b)

Mix\k) = M3x\k + 1) + T^M^k + 1)^4

+ A\M3l\k + 1) -M3

x\k + l)E2M3o)(k + l)

-M3°\k + \)E2M3x\k + 1) + G(

30)(A:)} (25c)

Similar equations are obtained for higher-order approxi-mations. The above eqns. 24 and 25 have their end conditionsas

M\0)(kf) = Pt(kf) M\x\kf) = 0 i = 1,2,3

(26)

3.1.3 Intermediate series: The task of evaluation of the endcondition P\x)(kf) is still not finished. To complete theevaluation, a series called the intermediate series is definedas

Mi(k) = M\°\k) + hM\x\k) + ... i = 1, 2,3 (27)

If eqn. 27 is used in eqns. 22 and coefficients of like powersof h are collected, the following equations are obtained.For zeroth-order approximation,

M\°\k) = M\0)(k + 1) (28a)

M2°\k) = M2°\k + 1) + Ts{M\°\k + \)A2

+ M2°\k + l)A4 +A'3M30)(k + 1)

-M\°\k + 1) E3M3°\k + 1)

- ^ ( k + 1) E2M°\k + 1) + Q2) (28b)

IEEPROC, Vol. 128, Pt. D, No. 4, JULY 1981

M3°\k) =

+ A'4M3o)(k + l) •

-M3°\k + l)E2M3°\k + 1) + Q3} (28c)

For first-order approximation,

M\x)(k) = M\x \k + 1) + TsG\0)(k) (29a)

M2x\k) = M2

x)(k + 1) + Ts{M\x\k + \)A2

+ M[x)(k + l)/44 +A3M{x)(k + 1)

-M\x)(k + \)E3M3°\k + 1)

-M[x\k + l)E2M30)(k + 1)

-M\0)(k+l)E3M3x)(k+l)

-M20)(k + \)E2M3

x\k + 1) + G20)(k)}

(29b)

M3x\k) = M3

x)(k + 1) + Ts{M3x\k + 1)^4

+ A\M3x)(k + 1)

-M3x\k+l)E2M3

0)(k+l)

-M3°\k + \)E2M30)(k + 1) + G3

0)(k)}

(29c)

Higher-order equations are obtained in a similar manner.The above intermediate series eqns. 28 and 29 have the

same end conditions as those of the outer series, i.e.

M\j\kf) = Pl]\kf) i= 1,2,3 0 (30)

The end condition P\x)(kf) is determined using the propertythat the value (M\x\k) — M\x\k)) tends to zero as k tendsto — °°, which is similar to that used in the case of continuousoptimal-control systems [10]. This requirement becomes

P[x\kf) = M\x)(kf) = I {(M\x\k + 1) -M\x\k))

-(M[x\k+l)-M[x)(k))} (31)

This result is viewed as the discrete equivalent for findingthe end conditions.

Combining the outer series, inner series, and intermediateseries given by eqns. 17, 23 and 27, respectively, the totalseries solution is given by

Piik) = I (Pji)(k) + Mji)(k)-M^(k))hi (32)

where q is the order of the approximation.

3.2 Case (b): open-loop controlThe open-loop optimal control gives rise to the singularlyperturbed discrete two-point boundary-value problem(TPBVP) given by eqns. 15 and 16. As the boundary con-ditions are specified at both initial and final points, the seriessolution is sought in terms of the outer series for the originalsystem and in terms of the inner and intermediate series forboth initial and final stretched systems, i.e.

.*(*) = Z (xU)(k)+x\i\k)+x\i\k))hi

j=o

j=6

(33a)

(33b)

145

Ij=o

(vU)(k) + v\j)(k) + vfJ)(k))hi (33c)

w(k)= X ( w % ) + wP(^) + wW()t))^' (33J)

where x^\k) corresponds to the outer series, x\J)(k) cor-responds to initial stretching correction series and x^\k)corresponds to final stretching correction series and similarlyfor other functions. The correction series itself consists of

xV\k) = x\J)(k)-x^(k) (34)

where xfJ)(k) and x\J)(k) correspond to inner and inter-mediate series, respectively.

Following along lines similar to the closed-loop controlof case (a), the initial stretched system is described by

xt(k +1) = *,(*) + hTs{A ,*,(*) + A2zflc)

-Elvi(k)-E3wi(k)} (35a)

vt(k +1) = v,(k) ~hTs{QlXi(k) + Q2zt(k)

+ A'1vi(k) + A'3wi(k)} (35b)

Zi(k + 1) = zt(k)+ Ts{A3Xi(k) + A^zt(k)

-E3vt(k)-E^Q2(k)} (35c)

Wi(k + 1) = Wi(k) - Ts{Q'2Xi(k) + Q^k)

+ A 2 vt(k) + A\ w((k)} (3 5d)

The final stretched system is described by

xf(k+l) = xf(k)-hT8{Alxf(k) + A2zf{k)

-Exvf{k)-Ezvfr{k)} (36a)

vf(k + 1) = vf(k) + hTs{QlXf(k) + Q2zf(k)

+ A'lvf(k) + A3wf(k)} (36b)

zf(k + 1) = zf(k) - T8{A3xf(k) + AAzf(k)

-E'3vf(k)-E'2Wf(k)} (36c)

wf(k + 1) = wf(k) + Ts{Q2xf(k) + Q3zf(k)

(36d)

If the series expansions given by eqns. 33 are used in thecorresponding system of equations for the outer series, innerand intermediate series of the initial and final corrections,a set of recursive equations are obtained for zeroth-, first-and higher-order approximations by the normal process ofsubstitution and collection of coefficients. The elaborateequations given elsewhere [17] are omitted here to save spaceand avoid repetition. However, the various boundary con-ditions to be used along with the recursive equations aregiven below.

For zeroth-order approximation,

*,(0)(A; = 0) = x(0)

x$°\k = 0) = JC(O)

v\°\k = kf) = v(kf)

vf\k = kf)= v(kf)

z J0)(fc = 0) = z(0) (37a)

z\0)(k = 0) = z(0)(0) (375)

wf°\k = kf) = w(kf) (37c)

(31d)

For first-order approximation,

xf\h = 0) = 0 z\l\k = 0) = 0 (38a)

x\l\k = 0) = JC(1)(O) z\l)(k = 0) = z(1)(0) (38Z?)

v\l)(k = kf) = 0 wfl)(k = kf)= 0 (38c)

vl\k =vfl\k = kf) = v{ wl)(k = kf) = w(l)

wfl)(k = kf) = w(l)(kf)

(38d)

The initial value xil)(0) is determined using the condition

^ ^ 1 ^ -> 0 as k -> « (39)

whereas the final value v(-1\kf) is determined using the con-dition

0 as k -* —' (40)

4 Illustrative example

To demonstrate the method, consider a second-order con-tinuous system described by [10]

x(t = 0) = x(0) (41a)dx— = zdt

h — = -x-z + u z(t = 0) = z(0) (4lb)dt

The performance index to be minimised is

J=\ jf(x2+u2)dt (42)

The final states are free.The discrete model corresponding to eqns. 4 and 5 becomes

= x(k)+Tz(k) x(k = 0) = x(0)(43a)

hz(k + 1) = hz(k) + T(-x(k) -z(k) + u(k))

z(k = 0) = z(0) (43b)

andkf-X

fe=o(44)

146

4.1 Case (a): closed-loop controlThe singularly perturbed matrix Riccati difference systemcorresponding to eqns. 12 and 13 is given by

Pi(k) = Pi(k+ l)+T{-pl(k+ l)-2p2(k+ 1)+ 1}(45a)

hp2(k) = hp2(k + 1) + T{-p2(k +l)-p3(k+ 1)

-P2(k + I)p3(k +l) + Pt(k+ 1)} (45b)

hp3(k) = hp3(k+l)+T{-p23(k+l)-2p3(k+l).

+ 2hp3(k+l)} (45c)

with the end conditions

Pi(kf) = 0 / = 1,2,3 (46)

If the method developed in Section 3 is used, the series

IEEPROC, Vol. 128, Pt. D, No. 4, JULY 1981

Table 1 : Comparison of various order solutions with the exact solution of the continuous system

kT 0.0 0.1 0.3 0.5 0.7 0.9 1.0

Px(kT)

pAkT)

P3(kT)

zeroth orderfirst ordersecond orderthird orderfourth orderexact solution(continuous case)

zeroth orderfirst ordersecond orderthird orderfourth orderexact solution(continuous case)

zeroth orderfirst ordersecond orderthird orderfourth orderexact solution(continuous case)

0.38830.51150.50890.50720.50650.5039

0.38830.38650.38450.38320.38300.3822

0.00000.38830.07380.07220.07150.0707

0.37960.50280.50040.49850.49630.4931

0.37960.37840.37140.37020.36940.3680

0.00000.37960.07160.07010.06950.0670

0.34810.46800.46530.46120.45980.4560

0.34810.34300.34130.33090.32650.3215

0.00000.34810.06480.06010.05850.0553

0.30560.41640.41170.40920.39140.3863

0.30560.29140.28480.27150.26230.2621

0.00000.30560.05320.04450.03620.0372

0.22510.31150.30180.29130.28430.2687

0.22510.19680.18240.17210.16110.1693

0.00000.22510.03580.02250.01870.0155

0.09250.12940.11890.10050.09940.0985

0.09250.06520.05180.04020.03120.0210

0.00000.09250.01390.01020.00850.0015

0.00000.00000.00000.00000.00000.0000

0.00000.00000.00000.00000.00000.0000

0.00000.00000.00000.00000.00000.0000

r=0.05,/? = 0.

kT

Table 2: Comparison of the various order solutions with the exact solution of the continuous system

0.0 0.1 0.3 0.5 0.7 0.9 1.0

u(kT)

x(kT)

z(kT)

zeroth ordersecond orderfourth orderexact solution(continuous case)

zeroth ordersecond orderfourth orderexact solution(continuous case)

zeroth ordersecond orderfourth orderexact solution(continuous case)

0.78270.39840.39950.4107

2.00002.00002.00002.0000

5.00005.00005.00005.0000

0.59570.30220.31030.3147

1.56641.55761.55891.5592

-4.0013-3.8640•3.8655-3.8662

0.32830.16720.17870.1784

0.92640.95680.95420.9540

-2.48762.3181-2.31912.3193

•0.16770.0867•0.0902-0.0910

0.52950.59360.59120.5904

-1.4912-1.3925-1.3942-1.3944

0.06890.03860.03520.0351

0.30160.37620.37900.3799

0.8677-0.8316-0.8371-0.8375

0.01730.00700.00350.0045

0.17040.23040.24100.2408

0.46220.49780.50240.5021

0.00000.00000.00000.0000

0.12840.18760.19610.1965

-0.3421-0.3912- 0.3904- 0.3902

x(a) = 2.0, z{0) = — 5.0, T = 0.05, h = 0.2

Table 3: Comparison of various order solutions with the exact solution of continuous system

kT 0.0 0.1 0.3 0.5 0.7 0.9 1.0

u{kT)

x(kT)

z(kT)

zeroth ordersecond orderfourth orderexact solution(continuous case)

zeroth ordersecond orderfourth orderexact solution(continuous case)

zeroth ordersecond orderfourth orderexact solution(continuous case)

0.78210.38990.38450.3840

2.00002.00002.00002.0000

5.00005.00005.00005.0000

0.65920.31210.30950.2978

1.54931.55821.55961.5597

4.01803.85893.857935576

0.47310.18010.17620.1734

0.90020.95530.95620.9564

2.55022.28822.30952.3109

0.31250.09970.09120.0907

0.49120.59150.59330.5939

1.59011.39521.39241.3921

0.17950.03980.03510.0355

0.23820.37420.37520.3754

0.95110.83650.83870.8390

- 0.0601-0.0121-0.0047- 0.0046

0.10030.24280.24360.2439

- 0.5200-0.5018- 0.5038- 0.5048

0.00000.00000.00000.0000

0.05850.19850.19910.1992

-0.3612-0.3920-0.3928-0.3930

x(0) = 2.0, z(0) = - 5.0, T = 0.05,h = 0.25

solutions are obtained for Pi(k), p2(k) and p$(k) and theseare shown in Table 1. The control u(k) and the states x(k)and z(k) are evaluated using these Riccati functions andare shown in Table 2.

4.2 Case (b): open-loop controlConsider the same second-order system described in case

(a). The discrete TPBVP corresponding to eqns. 15 and 16is given by

x(k +1) = x(k) + Tz(k) (41a)

v(k) = v(k + 1) + T(x(k) - w(k + 1)) (41b)

hz(k +1) = hz(k) - T(x(k) + z(k) + w(k + 1)) (47c) •

IEEPROC, Vol. 128, Pt. D, No. 4, JULY 1981 147

hw(k) = hw(k + 1) +

with the boundary conditions

T(v(k + 1) - w(k + 1)) (47d)

x(k = 0) = JC(O) z(k = 0) = z(0)

= w(*,) = 0 (48)

The control u(k) and the states x(k) and z(k) are evaluatedusing the method briefly described in Section 3 [17]and are shown in Table 3.

From the results shown in Tables 1—3, it should be notedthat the series solutions are obtained up to fourth-orderapproximation and are compared with the exact solutionof the corresponding continuous system. In Tables 2 and 3,the first- and third-order approximations are omitted forconvenience. It is seen from the results that the series solutionsare quite close to the exact solutions, even for a sufficientlylarge value of h = 0.2.

5 Conclusions

In this paper, it has been shown that the closed-loop andopen-loop optimal-control problems associated with singularlyperturbed continuous systems can be more convenientlysolved by means of their discrete models. The proposedmethod is essentially concerned with obtaining the seriessolution in terms of the outer, inner and intermediate series.The recursive nature of solutions associated with discretemodels has the advantage that it is easy to obtain higher-order approximate solutions. Furthermore, for a given orderapproximation, it is readily seen that, in case of discretemodels, the solutions are obtained with less computationaleffort compared with their continuous counterparts [10].An example has been given to demonstrate the method,and the solutions have been obtained up to fourth-orderapproximation.

6 Acknowledgments

The authors are grateful to Prof. N. Kesavamurthy for helpin the preparation of the original manuscript. Thanks are

due to the Indian Institute of Technology, Kharagpur, forproviding the facilities for research work.

7 References

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2 TIKHNOV, A.N.: 'Systems of differential equations containingsmall parameters multiplying some of the derivatives', Mat. Sb.,1952, 31, pp. 575-586

3 VASILEVA, A.B.: 'Asymptotic behaviour of solutions to certainproblems involving nonlinear differential equations containing asmall parameter multiplying the highest derivatives', Russ. Math.Surveys, 1963, 18, pp. 13-84

4 CADZOW, J.A., and MARTENS, H.R.: 'Discrete-time and computercontrol systems' (Prentice Hall, New Jersey, 1970)

5 RAJAGOPALAN, P.K. and NAIDU, D.S.: 'A singular perturbationmethod for discrete control systems', Int. J. Control, 1980, 32,pp. 925-936

6 PHILLIPS, R.G.: 'Reduced order modelling and control of two-time-scale discrete systems', ibid., 1980, 31, pp. 765-780

7 BLANKENSHIP, G.: 'Singularly perturbed difference equationsin optimal control problems', IEEE Trans., 1981, AC-26 (to bepublished)

8 HOPPENSTEAD, F.C. and MIRANKER, W.L.: 'Multitime methodsfor systems of difference equations', Stud. Appl. Math., 1977,56, pp. 273-289

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10 NAIDU, D.S., and RAJAGOPALAN, P.K.: 'Singular perturbationmethod for a closed-loop optimal control problem', IEE Proc D,Control Theory & Appl., 1980, 127, (1), pp. 1-6

11 BUDAK, B.M.: 'Difference approximations in optimal controlproblems', SIAMJ. Control, 1969, 7, pp. 18-31

12 CULLUM, J.: 'Discrete approximations to continuous optimalcontrol problems', SIAMJ. Control, 1969, 7, pp. 32-49

13 ABRAHAMSSON, L.R., KELLER, H.B., and KREISS, H.O.:'Difference approximations for singular perturbation of systemsof ordinary differential equations', Numer. Math., 197'4, 22, pp.367-391

14 SAGE, A.P., and WHITE, C.C. Ill: 'Optimum systems control'(Prentice Hall, New Jersey, 1977)

15 YACKEL, R.A., and KOKOTOVIC, P.V.: 'A boundary layermethod for the matrix Riccati equation', IEEE Trans., 1973,AC-18,pp. 17-24

16 BEN-ISRAEL, A., and GREVILLE, T.N.E.: 'Generalized inverses:theory and applications' (John Wiley, New York, 1974)

17 NAIDU, D.S.: 'Applications of singular perturbation techniqueto problems in control systems'. Ph.D. thesis, Indian Instituteof Technology, Kharagpur, India, 1977

148 IEE PROC, Vol. 128, Pt. D, No. 4, JULY 1981