ABSTRACT. This paper presents an overview of recent developments in the theory of singular perturbations and time scales (SPaTS) in discrete control systems.
FP4 = 2:30
proceedings oi the 26th Conlerance on Decision and Control Lor Angeles, CA December 1987
SINGULAR PERTURBATIONS AND TIME SCALES (SPaTS)
IN
DISCRETE CONTROL
SYSTEMS-AN OVERVIEW
Dr. D. S. Naidu D.Dr. B. Dr. Price J. L. Hibey Associate Professor (Research) Assistant Associate HeadProfessor & Comp. Engu., Dept. of Elect. & Comp.Engg.SpacecraftControlBranchDept.ofElect. OldDominionUniversity NASA Langley ResearchCenterOldDominionUniversity Hampton, VA, 23665 Norfolk, VA, 23508 Norfolk, VA, 23508 ABSTRACT This paper presents an overview of recent developments in the in theory of singularperturbationsandtimescales(SPaTS) discretecontrol systems. The focus is inthreedirections: modeling, analysis, and control. First, sources of discrete models and the effect of the discretizing interval on the modeling are reviewed. Thenthe analysis of two-timescale systems is presented to bring out typical characteristic features of SPaTS. systems, the Finally, in thecontrol of thetwo-timescale important issue of multirate sampling is addressed. The bibliography containing over100 titles is included. I. INTRODUCTION The dynamics of many control systems are described by highorder differential equatlons. However, the behavior is governed by a few dominant parameters,with a relatively minor role being played by the remaining parameters such as small time constants, masses, moments of inertia, inductances, and capacitances. The presence of these "parasitic" parameters is often the source for the increased order and the "stiffness" of the system. The "curse" of the dimensionality coupled with the stiffness posesformidablecomputationalcomplexitiesfor the analysis and control of such large systems. Singularly perturbed systems are thosewhoseorder is reducedwhentheparasitic parameter is neglected. The methodology of singular perturbationsandtimescales(SPaTS) is a "gift" to control engineers in tackling these large scalesystems. As such it is very desirable toformulate many controlproblems to fit intothe framework of the mathematical theory of SPaTS which has a rich literature [l-141. The theory of SPaTS in continuous control systems has attained a reasonable level of maturity and is well documented [15-241. The methodology of SPaTShas an impressiverecord of applications in a wide spectrum of fields such as circuits [25], networks [26], electrostatics [27], electromagnetics [28], electrical machines [29], power systems[30], semiconductors [31], fluid mechanics [l], structural mechanics [32], soil mechanics [33],flight mechanics [34], celestial mechanics [35], geophysics[36], chemistry [37], thermodynamics [38], nuclear reactor dynamics [39], acoustics [40], oceanography [41], biology [42], biochemistry [43],ecology[44], lasers [45], androbotics [461. Discretesystems are verymuch prevalentinscienceand engineering. There are three sources of discrete models described by difference equations containing several parameters [47]. The first source is digital simulation,whereordinary differential equationsare approximated by thecorresponding difference equations [48-511. The study of sampled-data control systems and computer-based adaptive control systems leads in a natural way toanothersource of discrete-timemodels [52]. Finally, many economic, biological and sociological systems are represented by discrete models [53]. In spite of the fact that the digital control of systems with widely separated eigenvalues was first considered by Stineman [54], the field of singular perturbationsandtimescales in differenceequationsand its applications to discrete control systems is only of recent origin [55-651. This paper will present an overview of these recent developments in the theory of SPaTS in difference equations and discrete control systems. The focus is in the three areas of modeling, analysis, and control. CH2505-6/87/0000-2096$1.00 0 1987 IEEE 20
11. MODELING IN SPaTS SYSTEMS Source 1: Pure Difference Equations Consider a general linear, time-invariant difference equation with small parameters occurring at the right end, left end, or both ends. Then thestate variable model becomes[66-681, xl(k+ 1) = A11xl(k) + h1-jA12x2(k) + B1U(k) h2ix2(k+1) = biAzlxl(k) + hA22~2(k)+ biBp(k)
(1)
0 6 i \< 1; 0 6 j ,< 1 where, xl(k) and x2(k) are "slow" and "fast" state vectors of "1 and n2 dimensions, respectively, u(k) is an r dimensional control vector, h is small a positive scalar parameter responsible for singular perturbation, andA's and B s are matrices of appropriate dimensionality. The three limiting cases of (1) result in (i) C-model (i=O; j =O),
where the small parameter h appears in the TOW of the system matrix, and (iii) D-model (i = 1; J = l),
1) = A21XlW + '422q(k> + B2U(k)
(4)
wherethesmallparameterh is positioned in an identical fashion to that of the continuous systems described by differential equations. Note: The replacementof x2(k) by hxl(k) in the R-model(3)will result in the C-model(2). Source 2: Discrete Modeline of Continuous Svstems Hereeither numericalsolution or sampling of singularly perturbedcontinuous systemswill result in dlscretemodels. Consider the singularly perturbed continuoussystem as Xl(t) = A11Xl(t) + A12X2(t) + B l W
(5) hx2(t) = A2lXl(t) + A22X2(t)
+
B24t)
Applying the block-diagonalizationtransformations [62], the original variables xl(t) and x (t) are expressed in terms of the decoupled slow and fast varia&es xs(t) and xdt) respectively, as xl(t) = Isxs(t) - hMxAt) ~ 2 ( t )= -Lxs(t) + (If+hLM)xf(t) and the decoupled variablesxs(t) and xdt) are obtained in terms of the original variablesxl(t) and x2(t) as
where L and M satisfy A21 t h u l l - A22L - hLA12L = 0 A12 - h(A11-Al2L)M + M(A22 t hLA12) = 0
(8)
Applying (6) and (7) to thecontinuous system ( 5 ) with a samplehold device, we get a discrete model which depends critically on the sampling interval T [69]. If we choose the fast sampling Tf = h or the slow sampling Ts = [l/h]Tf (where [l/h] is the largest integer < l/h), we get a fast or slow sampling model. Inthe particular case, whenTf = h, we get the fast samplin? modelas x l ( n t 1) = (IsthDll)xl(n) t hD12x2(n) t hD1u(n) X2("+ 1) = D21X1(")
+
D22X2(") + D 2 0 )
(9)
where n denotes the fast sampling instant. Similarly, if Ts = 1> we obtain the slow sampling model as
1) = E11Xl(k)
+
hE&k)
+
+ Bzu(O)(k)
x,(O)(kt 1) = A21xl(')(k)
= )xz(0). Here we note that, withxl(')(k=O) = xl(0); ~ 2 ( ~ ) ( k = O in the process of degeneratlon, x (k) has retained its initial condition x1(0), whereas x (k) has fost its initial condition xz(0). The boundary layer is sai2 to exist at k=O. In order to recover this lost initial condition, a correction subsystem is used [72,73]. The transformations between the origirz! and correction variables are (assuming no inputs for simplicity),
) xlc(k) = xl(k)/hkt '; ~ 2 ~ ( =k x2(k)/hk
(15)
Using (15) in (2), we get
E1@)
x2(k+1) = E21Xl(k) + hE22x2(k) + E 2 4 0
formulate the initial value problem and note that similar results can be obtained for boundary value problems also. The outer (degenerate) subsystem, obtained by zeroth order approximation (Le., by making h = 0) of (2), 1s
(10) h l C ( k + 1) = A11Xlc(k)
where k represents the slow sampling point, and n = k[l/h]. Also, the D and E matrices are related to theA, B, L and M matrices. Remarks (i) The fast sampling model (9) can be viewed as the discrete-timeanalog(either by exact calculation using the exponential matrix or by using the Euler approximation) of the continuous system xl(t') = hA11Xl(t') + hA12~2(t')+ hBlU(t') (11) ~ 2 ( t ' )= A21xl(t') + A22~2(t')+ B2u(t') which itself is obtained from the continuous system (5) using the isusually said thatthe stretchingtransformation t' = t/h.It continuous singularly perturbed systems (5) and (11) arethe slow timescale (t)andthe fasttimescale(t') versions, respectively. (ii) The slow samplin model (10) is the same as the state spacemodel (2) obtained horn the singularly perturbed difference equations [55,67,70]. Thus by discretizing the singularly perturbed continuous system (5) withslow and fast sampling rates, we gettwo different discrete-time models. Time Scale Property The slow sampling model (10) possesses the two-time scale property, if the largest eigenvalue of Ef is much smaller than the smallest eigenvalue of E,, that is [62],
+
A12X2c(k) (16)
x2c(k+ 1) = A21Xlc(k)
+
.4229c(k)
The zeroth orderof (16) becomes,
Rewriting (17), xlc(O)(k) = -A11-'[A12~2~(O)(k)] ~ 2 ~ ( O ) (1) k t= A , , ~ 2 ~ ( ~ ) ( k ) where, A,, = A22 - A21All-lA12. The total solution consists of the outer solution andcorrection solution as xl(k) = [xl(')(k) t kul(l)(k) + ..] thk+l[xlc(0)(k)
+ xlc(l)(k) + ...I
x2(k) = [xl(O)(k) t hxz(l'(k) t ...I + hk[x2,(0)(k)
+ x a(l)(k)
(19) t
...I
For the zeroth order, thetotal solution is xl(k) = xl(O)(k) ~2(k= ) ~2(O)(k)t hkx2,(')(k)
where p's are the eigenvalues and the approximatevalues for E, and Ef are
E,
=
E11; Ef = WE22 -E21E11-'E12)
(13)
= x2(O)(k) t ~2~(')(k)
(20)
where, x2 (O)(k) = hkx2,(0)(k). From (14c), we note h t only x2(k) has !os i s initial condition. Hence (20) gives x2J07(k=0) = ~2.0) - xdoj(0). Our current interest is only in zeroth order approximations. Thus, from (14) and (18),
Similarly, we canobtainthe condition for the fast sampling model (9) to exhibit the two-time scale property[65,69]. 111. ANALYSIS OF SPaTS SYSTEMS In this section, we usesingularperturbation and time-scale approaches,and show thatthe two approaches give identical results. We first consider a singularly perturbed discrete control system. Using a singularperturbationapproach,outerand correction subsystems are obtained. Next, by the ap lication of a time scale approach via block diagonalization translbrmations, the original system is decoupled into slow and fast subsystems. To a zeroth order approximation, the singular perturbation and timescaleapproaches yield equivalent results. This result is similar to a corresponding result in continuous control systems[71].
and the correction functionsas, ~ 2 ~ ( O ) ( k1)+ = Aco~2c(o)(k)or x2r(')(k+ 1) = hAc0x2r(0)(k) O ) - x*(O)(O) where, ~ 2 , ( ~ ) ( k = O=) X ~ ~ ( ~ )=( x2(0) 2. Time Scale Approach Let us consider again the singularly perturbed system (2). We now usethetimescaleapproachandobtain slow and fast subsystems to a zeroth order approximation. For decoupling the original system ( 2 ) into slow and fast subsystems, the block diagonalization transformations relating the decoupled variables in terms of the original variables are [62,74],
A. Slow Sampling Model: Initial Value Problems(IW) 1. Sinmlar Pertuybation ADDroach Considerthe singularly -perturbeddiscrete system ( 2 ) . We 2097
(23) andtransformationsrelatingtheoriginalvariablesand decoupled variables xl(O)(k=N) are xl(k) = Isxs(k) - hMxf(k)
models, if the boundaryconditions arexl(N)and x2(0), the boundarylayercan still occur at k=O,andthetotal series solutionremains the same as (18) [67,68,72,73,75].However, the auxiliary conditionsare
the
= xl(N); xzc(k=O) = x2(0) - x2(O)(O)
D-model, theFor boundary if the conditions (24)x2(N), the total series solution is given by
~ 2 ( k )= -Lxs(k) + (If
+ hLM)xf(k)
whereL(n1xnl)and equations,
M(n2xn2)satisfy
+ hLA12L - A21 = 0 hM(A22 + LA12) - (A11 - hA12L)M + A12 = 0
(25)
-'
whose it rative solutionsstartwith initial values of = AA and Mi = A"A12 By using transformation~24)in ($;wellget the decouplki] slow and fast subsystems as,
+ Bsu(k) model than
xf(k+1) = hAfXf(k) + Bp(k)
(26) and
=
= '411; Afo
= A22 - A21All-lA12
Bso = B1; Bfo = B2 -A21All-'Bl
=
1) = A11xso(k); x&+
IP(Ds) min 1) = hAfoXfo(k)
xso(k=O) = xl(0); xfo(k=O) = ~ 2 ( 0 -) A21A11-'~1(0)
(29)
(30)
Comparing the subsystems (21) and (22) and the solution (20) obtained by using the singular perturbation approach with the correspondingsubsystems (29) and the solution (28), we find thatthey satisfy the sameequations with thesame initial conditions. Hence, xl(O)(k) = xso(k); xz(')(k)
= A21All-'xso(k)
x2Jo)(k) = xfo(k); Ac0 = Afo
of an algebraicRiccati
+ L(I+hD11) - hLD12L = 0
(35)
It is notedthateven if the continuous-timesystem possesses the two-time scale property, i.e.,
(28)
Similarly, using(27) in (23), we obtain,
(31)
Thus,forazerothorderapproximation, we haveshownthat bothsingularperturbationandtimescaleapproaches give identical results. This result is akin tothat in the singularly perturbed continuous systems. It has been shown that such an equivalence does exist for first and high order approximations also [69]. In the slow sampling model, the solution can be expressedas a combination of discrete-time slow and fast subsystems. Here, the two-time scale property of the discrete-time system itself, and the lower sampling rate are assumed. However, it is noted that the fast part 1s treated as dead-beat. As a result, for the slowSam ling model (9) obtained from the continuous system ( 9 , and tIere is bound to be performance degradation between the two systems over the initial interval only. 1
= P{D22+hLD12)(34)
I
(5) (36a)
the fast sampling model (9) does not necessarily satisfy its twotime scale property,i.e.,
A21A11-1xso(k) + Xfo(k)
where xso(k) and xfo(k)satisfy x,#+
= p{1+h(D11-D12L));P(Df)
m= I P ( q I < IP(Af) < min
Using(27) in (24) and (26), we get(omittinginputfor simplicity), Xl(k) = x&); x2(k)
(8)
D21- D22L (27)
+
C. Fast Sampling Model Consider the fast-sampling model (9), which is a more exact Euler the a proximation model of Blankenship [65] and Rajagopalan and 8aidu [63]. The ei envalues of the slow fast parts of fast-sampling the model are given by whereL is thedichotomicsolution equation,
Lo = -A21A11-'; MO = All-lA12 4 0
(33)
(l)(k) + ...I and the boundarylayer is said to exist atthefinal point k = N.
=
f
+ ...I
+hN-k[x,c(o)(k)
%=A L b ; h t g t ;. ForAfthezeroth A22 + LA 2; Bs - (Is + hML)Bp(Ds) order qioximation, we
where, + hMB2, get,
+ xlc(l)(k) + ...I
+ h N - k + l [Xlc
x2(k) = [xz(O)(k) + hxz(')(k)
hA22L - L A 1 1
and
+ ...I
xl(k) = [xl(O)(k) + hxl(')(k) Riccati type algebraic (O)(k)
xs(k+l) = &xs(k)
xl(0) are
I
> > m= lP(Df) I
(36b)
This is in contrast to the slow-sampling model (2) or (lo), which preserves its two-timescalepropertyinthediscretization process. Using a boundary layer method, the solutions of (10) are expressed as xl(n,h) = Xl(t,h)
+ hxlc(n,h), t
= hn,
+ x2c(n,h) x2(n,h) = XZ(tJ-4(37) u(n,h) + U(t,h) +uc(n,h) whereX (t,h), XL(t,h) and U(t,h) correspond tothereduced system oflthe continuous system (5). Thus the solution (37) of (9) can be expressed as a hybrid combination of the continuous slow part which dominates the system behavior over the whole interval, and the discrete-time fast part which dominates over the initial time only. Thus, the analysis and design are performed essentially in the contmuous-time domain [65,69,76,77]. D. Steadv State Analysis Analternativeapproachtoderivingthe slow and fast subsystems is based on quasi-steady state concepts [78-86). For a stable linear discrete system having the time-scale property, the fast modes,corresponding tothe eigenvaluescentered aroundtheoriginareimportant only during the first few discreteinstants(transientperiod).Afterthateriod,they are ehavior of the negligible and the slow modes dominate the discrete systems. Neglecting the effects of fast modes is expressed formallyby letting x2(k+ 1) = x2(k). Then,we get,
xl(k+ 1) = A11zl(k) + A12~2(k)+ B l d k ) - B. B The anaiysis of B W is similar tothat of IVP, withfew x2(k) = (IfA22)-1[A21xl(k) + B2u_(k)l Rdifferences which are describedbelow.ForbothC-or 2098
E
(38)
control is formed from the sum of the slow and fast feedback optimal controls. The main advantage of the method is a considerable reduction thein overall computational the closed-loop optimalcontrol of digital + B&!,(k) (39) requirementsfor x,(k+l) = systems [58,69,72,107-110,121,1221. where, x_l(k) = xs(k), x (k) and u(k) are the slow components of 1. Optimal Control of Original System $he corresponding variailes in (2), and Ao = A1 t &-&A22)Consider the linear discrete system (2) having two-time scale 2 . kast su system IS 821, and Bo = B1 t 8 1 2 ( 1 ~ 4 2 2 ) - ~ B The obtalned by making the assumptlon that x (k) = xs(k) = constant character. The performance index to be minimized is we get the fast and q ( k + 1) = q ( k ) . From (2b)and138c), subsystem as,
where, _A's have a simple relation with A's. Rearranging (38), we get the slow subsystem as
x f ( k t 1) = A2lxdk) t B 2 ~ f j k )
(40)
where, x&k) = x2(k) - q(k); udk) = u(k) - us(k);
u(k) = -R-'BTPII t BR-'BTP]-'Ay(k)
IV. CONTROL OF SPaTS SYSTEMS The traditional control problems such as state feedback control design, eigenvalue assignment, and observer design, are equally applicable to discrete-time systemswith SPaTS [62,79-90,96104,123,1241. However, wewill concentrate on the optimal control of these systems. A. Open-Loop Optimal Control Consider the slow-sampling model (2) having two-time scale character. The performance index to be minimized is hI
J
=
y(N)Sy(N)
+ OSZ[y(k)Qy(k) + u(k)Ru(k)]
(41)
where y(k) = [xl(k), x2(k)], S, and Q are real, positive semidefinite symmetric matrices of ( n l + n2) dimensions, R is a real, positive definite matrix of order rxr, and N is a fixed integer indicating the terminal or final value of time. Using the results of optimal control theory [105], the state [xl(k), x2(k)] and costate [pl(k), pl(k)]equations are obtained as q ( k + 1) = A11Xl(k) + hA12x2(k) - W11Pl(k+ 1) - hW1p2(k+ 1) x2(kt 1) = A21xl(k)
The closed-loop optimal control is given by [ 1051, (46)
where p, of order ("1 +n2)x(n t n ), is the positive definite symmetric solution of the matrix hgezraic Riccati equation P = ATPII
+ BR-'BTP]-'A
t
Q
(47)
The closed-loop optimal system is given by y(k+ 1) = (A - BF)y(k)
(48)
where, F = R-'BTPII t BR-'BTP]-'A. Instead of tackling the original regulator problem described by (2) and (45) directly, we decompose it appropriately into two regulatorproblemsfor slow and fast subsystems. For this, we first need to separate the original performance index into the sum of two performance indices for slow and fast subsystems. The original performance index (45) has to be represented as the sum of the performance indices of the slow and fast subsystems. Using the transformation (6) between the original state variables, [x (k) and x2(k)], andthe subsystem variables, [xs(k) and xfjk)lf in ( 4 9 , and using us(k) = ukk) = u(k), we get
+ hA22~2(k)- W 1 p l ( k t 1) - h W 2 ~ 2 ( k +1) (42)
= Q11~l(k)+ hQ12~2(k)+ A 1 1 ~ l ( k + 1 )+ hA21~2@+1)
~ 2 ( k )= Q21xl(k) + hQ22~2(k)+ A 1 p l ( k + 1 )
+
hA2p2(k+ 1)
and the optimal control is given by u(k) = -R-'[Blpl(k+ 1) +hB@2(k+1)1
where, Qss, Q f, Qfs, and Qff are related to Q.. via L and M. Since J has to 6erepresented as the sum of Js a n i Jf, we need to neglect Qsf and Qfs. Then
(43)
where, wi. = BiR-lB.; i,' = 1,2. The 2 ( n l + n ) order two-point boundary h u e probleh (TPBVP) represented% (42) which is in the singularly perturbed structure, is to be solved using the boundary conditions, X (o), x2(0), p1(N) and P2(N). The series representations for (4lfare given by
As we have neglected Q f and Qfs, it certainly introduces an error, in that J will not 8e equal to the sum of J and J . To ,[ll($.As compensate for this, we need to re-adjust Qss and this is simply a design or synthesis approach, we can first select the performance indices of the subsystems and then formulate the original performance index. Thus, let
bf
=ys T(k)Qssxs(k>
Js &
+
usT(k)Rsus(k)l
If ="exfT(k)Qfpf(k> + ~ f ~ ( k ) R p # ) l where, x1 (k), x2 (k), p1 (k),and ~ 2 ~ ( correspond k) to the outer solu%on, xlxk), x2i@), pli(k), and ~ 2 i ( k )correspond to the initial boundary layer correction, and xlr(k), x2r(k), PI@), and p2r(k) correspond to the final boundary layer correctlon. The details are found in Kando and Iwazumi [loll, Rajagopalan and Naidu [ 1061, Naidu and Rao [72] and Rao andNaidu [ 1071.
€3. Closed-LooD ODtimal Control
(5 1)
(52)
Using the transformation (7) between the subsystem variables x&k) and xkk) and the original system variables xl(k) and x2(k), we get
where, Q is related to Q and Qff via L and M. Thus, in (51)(53), we first select Q s, &f, Rs and Rf, and then using L, and M, we get 52 and E. $ere, we are able to decouple J into Js and J exactly without any approximation. But the original J is dependent on L and M, the decoupling matrices, which may not be of practical advantage.
In this sectl'on,-a two time scale discretecontrol system is considered. The closed-loop optimal linear quadratic re for the system requires solution of a full-order algebraic equation. Alternatively, the original system is decomposed into reduced-order slow and fast subsystems. The closed-loop 2. Qptimal Control of Subsystems o timal control of the subsystems requires the solution of two UsinB the transformation (6), we decompose the original a gebraic Riccati equationsof order lower than thatrequired for system into slow and fast subsystems as the full-order system. A composite, closed-loop suboptimal 2099
P
xf(k+ 1) = Afxf(k) + Bpf(k)
We now try to optimize these slow and fast subsystems with respect to their correspondingperformanceindices(50a)and (Sob) respectively. The slow regulator problem consists of the slow subsystem (54) and the performance index (50a). The fast regulator problem consists of the fast subsystem (55) and the performanceindex(50b).Forconvenience, we write Q = Qs andQff = Qf The optimalfeedbackcontrol of tfieslow subsystem is given by us(k) = -R,-lBsTPSIIs + BsRs-lBsTPs]-lA&k)
(56)
where Ps is a positive definite symmetric solution of the reduced order algebraic Riccati equation Ps = $Ps[Is
+ BsRs-lBSTPs]-l& + Qs
Similarly, the optimalfeedbackcontrol becomes
of the fast subsystem
uf(k) = -RilBfTPfIIf + BfRf1BfTPf]-lAfxf(k)
(58)
where Pf is a positive definite symmetric solutionof the reduced order algebraic Riccati equation
+ Qf Pf = AfPfIIf + BfRf'BfTPf]-'Af (59)
From an intuitive point of view, the measurement and controlot the slow variablescanbedoneatlowersamplingratesin comparison with thefastvariables [69,117-1201. Considerthe singularly perturbed continuous system (5) and the performance index (45). By the process of decomposition and discretization, the continuous system (5) is transformed to the fast sampling model (9) or the slow sampling model (10) depending u on the discretizing interval. Similarly, the performance index ($5) can be transformed. Using the slow (fast) sampling model, and the corresponding performance index, we arrive at the slow (fast) sampling regulator problems,which are solved independently. The slow sampling regulator problemis solved by decomposing it into slow and fast subproblems, where the fast subproblem exhibitsadead-beatbehavior. Similarly, the fast sampling regulator problem is decomposed into slow and fast subproblems,wherethe slow subproblem, of continuous-time nature, dominates thesystem behavior over the wholeinterval.
In the singularly perturbedcontinuous system, x (th)and x (t,h) possess slow and fast behaviors, respectively. %us, the s?ow sampling rate (T,) can be toleratedfor measurement of the slow variable xl(t,h), i.e., x (t,h) can be measured at the slow rate t = kTs (k = 1,2,..). On h e other hand, x2(t,h) is measured at the fast rate t = nTf = k[l/h]Tf By combining the controls of the slow subproblem of the slow sampling regulator and the fast subproblem of the fast sampling regulator, the multirate controlis expressed as
Rewriting the control laws (56) and (58),
us(k) = -Fsxs(k); uhk)
=
-Ffxf(k) (60)
We note that the control laws (60) are optimal with respect to the slow and fast subsystems (54) and (55) only. But, it is computationally simpler to determine these controls laws than the optimal control law (43) of the original system [loll. 3.
The iomposite control is formulated as the sum of the slow and fast feedback controls given by (56) and (58). That is, uc(k) = us(k)
+ uf(k) = -[Fsxs(k) + Ffxf(k)]
(61)
Using the transformation(7) between the slow and fast variables and the original variablesin (61), we get (62) uc(k) = -[Fs$(k) + Ff$(k)l = -Fy(k) where, Fsc and F are related with Fc via L and M. Using the composite controff62) in theoriginal system, Yc(k + 1) = (A-BFc)Y,(k)
(63)
It is known that minimizing the original performance index (41) with respect tothe composite system (63) results in the suboptimal performanceindex [ 1lo],
Jc
=
0.5yT(0)Pd(0)
where P is thepositivedefinitesymmetricsolution discrete fyapunov equation P, = (A-BF,)TP,(A-BF,)
of the
Here, the states x (kT h) and x2(nTf,h) are measurable. But, since the state x ( h f , h y cannot be measured between k[l/h]Tf < nTf < (k+ l)[l/h]Tf, the above state feedback control cannot be implemented. This difficulty is overcome by using the estimates of x (nTf,h). Finally, the multirate control is obtained as [11712Oi uC(nTf,h) = Gqxl(kT,,h)
+ Gsx2(nTf,h) (67)
VI. CONCLUSIONS In this paper we tried to overview the recent developments in the theory of singular perturbations and time scales (SPaTS) in discrete control systems. The focus has been in three directions of modeling, analysis, andcontrol.Inmodeling, we reviewed sources of singularlyperturbeddifferenceequations in their on the equivalent state space representations. Depending discretizing interval, we arrive at slow-sampling models andfastsampling models.The analysis of two-time scalesystems brought outthecharacteristicfeatures of orderreduction,boundary layerphenomena,stretchingtransformations,andcorrection series. In controlling the two-time scale systems, we addressed open-loop and closed-loop optimal control problems, hlghlightmg the important issueof multirate sampling. ACKNOWLEDGEMENTS This research work was supported by NASA Langley Research Center under Grant number NAG-1-736. REFERENCES [l] VanDyke, M., Perturbation Methods in FluidMechanics, Academic Press, New York, 1964.
+ Q + F,TRF, (65)
In an entirely different approach to the closed-loop optimal control of discrete systems possessing two-time scale character, the Riccati coefficient matrix P(k) is partitioned into asinglarly perturbed structure and the analysis 1s carried out on the iccatl e uation [58, 69, 70, 72, 102, 106, 109-111, 120,l21]. The theory SPaTS in adaptive control, time-optimal control and stochasticoptimalcontrol is considered by Delebequeand Quadrat [112], Ioannou and Kokotovic [113], Rao and Naidu [1141, Mahmoud et. al., [ 1151, and Sen and Naidu[ 1161.
09
V. MULTIRATE REGULATOR PROBLEM Singularly perturbed systems exhibit slow and fast behaviors. 21 00
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