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I4EEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, vOtL, SMc.4, NO, 6, NOVEMBER 1974
the kth row of A and 8 can be eliminated without loss ofpossible solutions to subprohlem P2.
Condition 4: If the kth row of A has all elements ai,i=- 1,2, ,nIA, equal to zero, and some brk equals one,then .i-ak1=iak-O, for all feasible u1i and >j1B v1b k > 0for any feasible vj, where v, > 0. Thus the rth column (rthpattern) of B can be eliminated without loss of possiblesolutions to sujbproblem P2.
Condition 5' If the kth row of A has all elements ajk,i = 1,29 ,1nA, equjal to one, and some hrk equals zero,then Z7¶A1 utatk 1, for all feasible ui and 27!B1 .b.k < 1for any feasible vj, where vr > 0. Thus the rth column (rthpattern) of B can be elimiinated without loss of possiblesolutions to subproblem P2.
Detailed justifications for these subproblem reductionconditions can be found in Leonard [8].
REFERFNCES(11 C. Y. Chang, "Dynamic programming as applied to feature subset
selection in a pattern recognition system," IEEE Trans. Syst.,Man, Cybern., vol. SMC-3, pp. 166-171, Mar. 1973.
[21 E. Balas, "An additive algorithm for solving linear programs withzero-one variables," Oper. Res., vol 13, pp. 517-546. 1965.
(31 K. S. Fu and y. Tr. Chien, "Sequential recognition uising a non-parametric ranking procedure," IEFEE Trans. Infor"m 7Theory, vol.IT-13* pp. 484-492, July 1967.
[4] K. S. Fu, Y. T. Chien, and GI P. Cardillo, "A. dynamic program-ming approach to sequential pattern recognition," IEEE Trans.Electron. Comput., vol. FC-16, pp. 790-I803, D)ec. 1967.
[5] K. S. Fu, P. J. Min and T. J. Li, "Feature selection in patternrecognition," IEEE Trans Syst. Sci. rybern-, vol. SSC6, pp.33 39. Jan. 1970.
[6] R. S. Garfinkel and G. L. Nemhauser, "A survey of integerprogramming emphasizing computation and relations amongmodels," Dept. Operations Res., Cornell Univ., Ithaca, N.Y.,Tech. Rep. 156, Aug. 1972.
[7] A. M. Geoffrion, "Integer programming by implicit enumerationand Balas' method," SIAM Rev., vol. 9, no. 2, pp. 178-190, 1967.
[81 M. S. Leonard, "Analytical models for diagnostic classificationand treatment planning for craniofacial pain," Ph,D. dissertation.Dep. Indus, and Syst. Fng. Univ of Floridia, CGainesville, 1973,chapter 3.
[9] W. S. Meisel, Computer-Oriented Approaches to Pattern Recog-nition. New York: Academic, 1972.
[10] G. E. Nelson and D. M. Levy, "Selection of pattern features bymathematical programming algorithms," 1EFF Trans. S'Yst. S'ci.C)yhern.. vol SSC6, pp. ?0-25, Jan. 1970.
[111 N. S. Nilsson, Learninsq Machines. New York: McGraw-Hill,1965, pp. 65-90.
[121 J. B. Rosen, "Pattern separation by convex programming," J.Math Anal. Appl., vol. 10, 1965, pp. 123-134.
Stable Adaptive Schemes for System Identificationand Control---Part I
KUMPATI S. NARENDRA AND PRABHAKAR KUDVA
Abstract-General schemes for the adaptive control and identificationof multivariable systems whose entire state vectors are accessible formeasurement are developed. A model reference approach is used here, andLyapunov's direct method is employed to ensure the convergence of theseschemes. An added feature is the simplicity of the stable adaptive laws,which depend explicitly on the state variables of the plant and a model,and on the plant input. Computer simulation results of several examplesare included to illustrate the effectiveness of the proposed schemes.
1. IN!TRODUCTION
IN SPITE of the proliferation of adaptive techniquesthat have been proposed during the last fifteen years,
control engineers generally agree that relatively few practicalmethods are currently available for the identification andcontrol of multivariable systems [25]. The difficultiesassociated with applying such methods to real problems maybe attributed to a variety of reasons among which are:i) the large number of parameters that may have to be
Manuscript received June 10, 1973; revised April 25, 1974 and June26, 1974. This work was supported by the Office of Naval Researchunder Contract N0014-67-A-0097-0020 NR375-131.The authors are with the Becton ("enter, Yale UJniversity, New
Haven, Conro. 06520.
adjusted simultaneously while assuring the stability of theoverall adaptive system, ii) the lack of exact informationregarding the variation of the plant parameters, and iii) thenoise present in the measurements of the plant outputs.Recent developments in model reference adaptive controlusing Lyapunov's direct method [2]-[27] have enabledcontrol engineers to overcome some of these difficultiesand have resulted in schemes that may be attractive inpractical situations. In particular, they have proved to bevery effective in the control of vertical take-off and landing(VTOL) aircraft systems [21].The salient feature of this two part paper is a unified
presentation, including several key results obtained recently,of a general scheme for the adaptive control and identifica-tion of multivariable systems. In Part I, techniques aredeveloped for the identification and control of an unknownplant when all of its state variables are accessible formeasurement. The stable adaptive laws are found to havea simple structure and to depend explicitly on the statevectors of the plant and a model, and on the plant input.Though these laws have been developed for continuous-time systems, they are also applicable, with slight modifi-cations in most cases, to discrete-time systems as well [223.
542
NARENDRA AND KUDVA: SYSTEM IDENTIFICATION AND CONTROL, PART I
- U ' =A x +Bpu' P
(PLANT)
Fig. 1. Configuration of plant with feedforward and feedback gainmatrices.
The second part of this paper [24] is concerned exclusivelywith the identification and control problems using onlyinput-output data. A unified approach is developed for thesynthesis of stable adaptive observers and it is shown thatthe observer outputs can also be used for the adaptive controlof the plant.
The Model Reference Approach Using Lyapunov's MethodAll the adaptive control schemes that are developed in
this paper employ a model reference approach. The surveypaper by Landau [25] provides an exhaustive list ofreferences on the general model reference adaptive systems(MRAS) problem. In this paper, however, we deal specific-ally with the MRAS problem solution using Lyapunov'sdirect method. In this approach, the behavior desired of theplant is provided by a model. The plant (more specifically,the controller) parameters are dynamically adjusted asfunctions essentially of the error between the plant andmodel outputs. These parameters also constitute the statevariables of the overall adaptive system which, therefore, isrepresented by a nonlinear vector differential equation.
Since Lyapunov's direct method is directly applicable tothe stability analysis of dynamical systems, it is used todetermine the sufficient conditions under which the adaptivesystem is unconditionally stable. In fact, all the adaptiveproblems considered in this paper correspond to thestability problem of special classes of time-varying vectordifferential equations. The crucial point in all these prob-lems is to choose the equations for updating the controllerparametei-s so that the overall nonlinear system is asymp-totically stable.
II. STATEMENT OF THE PROBLEM1) Adaptive Control
Consider a linear multivariable system (the plant) de-scribed by the vector differential equation
x~p = Apxp + Bpu(1where the elements of the (n x n) matrix Ap and those ofthe (n x m) matrix Bp are unknown.The behavior desired of the plant is prescribed by the
model
xm = Amxm + Bmu (2)where Am is an (n x n) stable matrix, and Bm is (n x mn).We employ a feedforward (m x m) matrix Q and a
feedback (m x n) matrix F to control the plant (Fig. 1),so that the plant, together with the controller, is charac-terized by
xp = [Ap + BpQF]xp + [BpQ]u. (3)
The problem, then, consists of specifying a scheme thatcontinuously adjusts the elements of the matrices F(t)and Q(t) such that the norm of the error between the plantand model states is reduced. Under certain conditions, it ispossible to reduce this error to zero and it is this case thatis of primary concern in this paper.
2) IdentificationThe identification problem, on the other hand, consists
in specifying a suitable model and in developing a schemefor dynamically adjusting its parameters so that they con-verge to those of the plant (1).A number of special cases of these problems are sum-
marized in tabular form as follows.
TYPE I TYPE II1. Matchable plant Nonmatchable plant2. Time-invariant plant parameters Time-varying plant parameters3. Perfect measurements Noisy measurements4. All state variables of the plant Only outputs of the plant acces-
accessible sible
The cases listed under Type I are, in general, analyticallytractable and are considered here, while their counterpartsin Type II are problems that are considerably more difficultand represent current areas of research.The different cases are briefly considered as follows.1) The term "matchable" has been used to indicate that
the plant together with the controller has enough structuralfreedom to match the model exactly. Situations where suchfreedom does not exist have to be treated individually andthere are no significant results in this case, at the presenttime. In Section VI, a method for simultaneously adjustingthe plant and the model parameters (to achieve exactmatching) is discussed.
2) In the case of a time-invariant plant, the adaptiveprocedure results in the convergence of the controllerparameters to constant values, and the plant follows themodel exactly. However, when the plant parameters aretime-varying, so are the controller parameters, and boundson the amplitude of the error vector can be established.
3) Little theoretical work has been devoted to analyzingthe stochastic stability of the overall nonlinear adaptivesystem when output noise is present. The presence ofoutput noise is an important practical problem and is,therefore, a potential area for future research.
4) The requirement of the accessibility of all the statevariables of the plant is also an important constraint in manypractical situations. The case where only outputs areaccessible is considered in the second part of this paper[24]. In this case, schemes are proposed to identify theplant parameters while simultaneously generating a set ofstate variables of the plant; this information, in turn, isused to design a suitable adaptive controller for the plant.To provide insight into the basic ideas and the various
steps involved in the choice of the controller structure andthe generation of the adaptive laws, we shall first considerthe case where the plants and the models are described byscalar first-order differential equations.
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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, NOVEMBER 1974
III. ADAPTIVE CONTROL OF PLANT DESCRIBED BY ASCALAR DIFFERENTIAL EQUATION
Let the plant and model be described by the scalardifferential equations:
Xm am-xmxm + bmU(t) model
xp- -ap(t)xp + bp(t)u(t) plant without controller.(4)
In (4), am and b. are known positive constants, ap(t) andbp(t) represent the time-varying parameters of the plant,and ii(t) is the input to both the plant and the model. Toimplement the adaptive control scheme, feedback andfeedforward gains, f(t) and q(t), are introduced as follows:
xp = -[ap(t) + bp(t)f(t)]xp + [bP(t)q(t)]u. (5)
The aim of the adaptive procedure is to adjustf(t) and q(t)in such a manner that [ap(t) + bp(t)f(t)] and [bp(t)q(t)]tend to am and bi,m respectively. It is noted that the quantitiesap(t) and bp(t) cannot be measured directly, since otherwisethe entire problem would be trivial and an adaptive schemewould not be necessary.We go one step further and consider the case of an
adaptive regulator where ap(t) and bp(t) are constant butunknown.
1) Adaptive Regulator (Time-invariant Plant)
Xm = -amxm + bmu(t) model
xp = - [ap + bpf(t)]xp + [bpq(t)]u(t) plant. (6)
The output error e(t) A Xm(t) - xp(t) satisfies the dif-ferential equation
e = -ame - [am - ap - bpf(t)]xp + [bm - bpq(t)]u.(7)
The quantities [am- ap - bpf(t)] and [bm - bpqq(t)] areerrors in the parameters and are denoted by +(t) and t(t),respectively. Hence
where
and
(8)=-ame -xp + u
+(t) A [am - ap - bpf(t)])/(t) A [bm- bpq(t)].
The objective is to adjustf(t) and q(t) so that as t -* o
e(t) -O 0, +(t) -* 0, and /(t) -* 0.The differential equations for the dynamic controller are
assumed to be of the form
f(t) = g1[xm,xp,u,fl4(t) = 92[xm,xp,u,q] (9)
or in terms of +(t) and +(t),
¢(t) = kl[xm,xp,u,f]=(t)= k2[xm,xp9,uq]. (10)
The functions k1( ) and k2( ) [or equivalently g1(-) andg2( )] specify the adaptive laws.
Equation (8) for the output error together with (10) forthe parameter errors completely determine the error modelof the system.
Let g denote the error space where the error vectore e & is defined by
eT (e,0,0).The stability of the overall system must, then, be consideredin &. The functions k1( ) and k2( ) are to be chosen suchthat the equilibrium state (e 0O) of the overall system isasymptotically stable.We consider as a candidate for the Lyapunov function
the quadratic form :1
V(e) = 2 [bp sgn (bp)e2 + 12 -+ I 2 ,2 Al A2
"-1 '2 > 0
(11)
which is a positive definite function in S.Differentiating V with respect to time and using (8) yields:
V = - bpsgn (bp)a, e2 + p - bpsgn (bp)exp]
+ Iv [A + bp sgn (bp)eu] (12)
Since the parameter errors +(t) and +(t) are unknown andcannot be measured, the only manner in which V' can bemade negative semidefinite is to choose
+ = Al[bp sgn (bp)]expi = - ;2[bp sgn (bp)]eu. (13)
Since the control parameters aref(t) and q(t), for practicalimplementation, (13) has to be expressed in terms of f(t)and 4 (t) as
f(t) = -A, sgn (bp)e(t)xp(t)4(t) = ¾2 sgn (bp)e(t)u(t). (14)
Equations (14) are the adaptive updating laws. The sign ofbp is assumed to be known to the designer for the implemen-tation of these control laws--an assumption which isgenerally not restrictive. The multivariable aspects of thisassumption are discussed in Section IV.
Let us now consider trajectories along which V 0 ore(t) is identically zero. From (8), this implies that
-q5xp + tu _ 0. (15)
One sufficient condition for this equation to have the trivialsolutions 4 = 0 and / = 0 is that u be a signal with atleast a single frequency component, [(Section IV)Theorem 2]. With this condition satisfied, the overallsystem is uniformly asymptotically stable.
2) Adaptive Coontrol (Time-varying Plant)When the plant is time-invariant, the controller param-
eters converge to constant values and the plant follows the
1 sgn (x) = x/lxl.
544
545NARENDRA AND KUDVA: SYSTEM IDENTIFICATION AND CONTROL., PART I
model exactly. Since the overall system is uniformlyasymrptotically stable, if the plant parameters vary slowlywith time, so will the controller parameters and the plantwould closely follow the m-odel. Narendra and Tripathi[15] have shown that if the variatioins ldpl and bp/bp arebouinded, bounds on thie errot- vector can be established.
IV. ADAPTIVE CONTROL OF MULTIVARIABLE SYSTEMS
f'he analysis of the adaptive system in Section I:1I hasbeen carried out primarily to illustrate the behavior ofmore comsplex adaptive systems. Most of the ideas docarry over in the generalization of these adaptive schernesto multivariable systems.
In the sequel, we are interested in the asymptotic stabilityof systems of periodic nonlinear time-varying differentialequations
x = f(x,t) f(x, t + T) = f(x,t) (16)
For this purpose, the following theorem of LaSalle [28]is found to be directly applicable.
where "'tr" denotes the trace of a inatrix. tUpon differentiat-ing (19) with respect to timne and using (17) and (18)
V =- -.eQe + e"PDIDz + tr ('FT' 1-)eTQe + tr [(D7'Pez" + 1- $)A)]
- teTQe, ( i)
Thus ri is negative semidefinite and the system (18) istherefore, stable.Using Theorem 1, the set Al in this case corresponds to
the manifold e(t) 0- and all the solutions of (18) approaciM as t -* oo, anid
(21)lim 4) = lim [__FDTPez7 - Orto- a, tx-D
Since V" > 0 and V . 0, ?((t) is bounded and (21) irmpiiesthat limt1,, (4(t) is a constant matrix, because (18) is alinear system with periodic coefficients.
Considering the trajectories along which V = 0 ore(t) = 0, we have from (18) that
D&z 0- (22)Tiheoreml 1 (LaSalle)
Let V(t,x) be a real valued function with continuous firstpartial-derivatives, for all (tx). Ass-ume that
a) V(t,x) V(t q- T, x) . 0, for all (t,x)b) i(t,x) _ aV/at -t VVjf(x,t) . 0, for all (t,x).
Let E be the set of all (to,x0), such that V(tO,x)= 0. LetAl be the union over all (t,,x0) in E of all trajectoriesx(t,t0,,x0) with the property that (t,x(t,to,x0)) is in E for all t.Then all solutions of (16) which are bounded approach Alast --ctu.
I0heorem 2 lies at the heart of the proof of stability of allt:he adaptive control and identification schemes discussedin this paper. Together with Theorem 1, it can also be usedto prove the uniform asymptotic stability of these schemes,as showin in Section V.
Theorem 2
-Let A be a stable (n x n) mnatrix, 1) an (n x r) matrixof full rank, F - FT > 0 an (m x m) matrix, P a positivedefinite matrix satisfyinig
c Tp - PA = Q = QO > 0 (17)and z(t) an (r x 1) vector, ,z > ni > r, of arbitrarybounded functionis of time. Then the system of (n + mr)differential equations
= A --t- D(Fz(t)( =-FDT-PeZT(t) (18)
is stable. Furthermore, if the r-coimiponents of the vectorz(tj are signals each with a distinct frequency component,then the system (18) is uniform/v asymptotically stable.
Proof. Consider the following quadratic function as aLyapunov function candidate for the system (18):
where 4( is a constant matrix.Since D is of full-ratnk, (22) implies that
(Fz - O. (23)
Let {ti} denote the column vectors of (F, aiid {zj} tlkelements of the vector z, where i E t 1,2, ,r }; then (23)may be rewritten as
24)E oizi(t) - 0i =1
If {z1(t)} are signals with distinct frequency comiponentsand hence are linearly inidependent functions of timle,(24) implies that
i= 0, forall is {1,2, , r} (25)that is,
( - 0.
Since z(t) is a periodic vector, i.e., z(t) z(t + T),where T is the least common multiple of the timt1e perio)dsof the elements of z, the system (18) is a particular case of(16). Hence we can invoke LaSalle's Theorem 1. Since
e =_ =>( = 0 (26)and Ml contaiins only the origin, the system (18) is uniformlyasymptotically stable.
Since a random signal over any finite interval can heregarded as a periodic signal with an infinite nurnber ofdistinct frequencies, Theorem 2 also holds for the casewhen z(t) is a vector of (almost all) random signals over thatiniterval [27].
The Adaptive Control Problem
Consider a plant and a model, which are described bythe differeintial equations
xp= [Ap + BpQF]xP + [BpQ]u plant(19) -* = APXm + BmuV = I[eTPe + t r ((D "' f '-- 1 (D)]2 miodel (2 7)
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, NOVEMBER 1974
where Am is a stable matrix. The analysis of the adaptiveprocedure is conducted for systems of the Type I class.The assumptions of exact matching implies the existence ofmatrices Q* and F* such that
B Q* = B
Ap + BmF* = Am
-2 0
-1 5
-1.0O
-0 5(28)
(Bi and Bp are both assumed to be of full rank, so that Q*is nonsingular). These conditions, in turn, imply that thecolumn vectors of the error matrices [Am- Ap] and[Bi - Bp] should be linearly dependent on the columnvectors of the matrix B.. The simplest case where thiscondition is satisfied is where Am and Ap are in phase-variable canonical form and Bm and Bp are vectors havingall but the last elements zero.The state error vector e A [Xm- xp] satisfies the
differential equation
e = Ame + [Am - Ap BpQF]xp + [Bm- BpQ]u= Ame + Bm(DXp + BmTQ[u + Fxp] (29)
where (D A [F* - F(t)] and P A [Q-l(t) - Q*-l].Using Theorem 2, stability in the {e,(D,T} space can be
guaranteed if (F and vP are chosen as
(D = -FiBmTPexpT
v = -F2BM Pe(u + Fxp)TQTwhere i) P is a positive definite matrix satisfying
AmTP+ PAm = -R, R = RT > 0
and ii)
0 2 4 6 8 10 SECONDS--
2
06 f2(t)0.40 2
-0 2 2 4 6 8 10 SECONDS
1.00 8 q(t)0.6 q0 40 2
_
o 2 4 6 8 180 SECONDS-Fig. 2. Convergence of controller parameters in Examnple I.
{e,F,Q} space. As a necessary condition, the initial values(30) of e, F, and Q should satisfy the strict inequality
V(to) - 1[e"(tO)Pe(to)(31a) + tr {(F* -F(to))",f- -' * -lF(to))
+ (Q*-I Q1(to))FTF l(Q*$ I Q'(to))81F1 = F,T > 0, (36)
The resulting adaptive control laws are
F = F IBmTPexpO= QF2B.PfPe(u + Fx )TQTQ. (32)
With these control laws, the stability of the overallnonlinear system given by (29) and (30) is assured in aregion where
V= 1[eTPe + tr ((DTFj- + "F2 "')] (33)is a Lyapunov function. Since v is unbounded whendet IQ(t)I = 0, a necessary condition for this scheme isthat
if the overall system is to be stable.
Example 1The adaptive control scheme (32) has been applied to
control the following second-order plant:
1) Plant (with controller):
xCp - [Ap + bpqf71]x-p + bpqu
sgn [det IQ(to)l] = sgn [det IQ*I]. (34)Moreover, stability is assured only in that region of the{e,F,Q} space where the constant V surfaces given by
V = 1[eTPe + tr {(F* - F)TF1-1(F* - F)
+ (Q* 1 _ Q-')TF2-l(Q*-l - Q-1)}]= constant (35)
is closed, so that the initial values of e, F, and Q should liein this region. Since the last term in (35) contains the inverseof the matrix Q, this region does not include the entire
_6 7] b 47
2) Model:
=m AmXim + bmU
An -=L o ] m 2]3) Adaptive control laws:
f = Fxpe'Pbm== ye'Pb,q1(u + fT'x)
=[I I] r=F4 0]andy=9.
-1--
546
fl
F2 = F2T > 0. (3 lb) 2'*- 1)T - 1 Q * .- 11< i tr RQ I'2
f I t
---I --I-
NARENDRA AND KUDVA: SYSTEM IDENTIFICATION AND CONTROL, PART I
TABLE I
Form of the Plant Togetherwith the Controller Assumptions Adaptive Laws
1) kP = Apxp + Bpu The designer has access to every element of the two matrices A,P = rFPexpTAP and B. B=r=2peuT
(Direct Control)
2) |xp = [Ap + BF]x, + Bu | i) Bp Bma F B= rEBTPexpTii) 3 a matrix F*: Ap + BF* A_.
3) x,, = [A, + BpFlxp + Bpu i) the elements of the matrix Bp can be adjusted directly. F = FBiBmTPexXpTii) 3 a matrix F*: Ap + BmF* = Am,, Bp = r2Pe(u + FxP)T
4) 4 = Axp + BpQu i) Ap = Am = Aii) 3 a matrix Q*: BpQ* BmQB,,QFlBmTpeuTQTQ
iii) Bp,Bm are of full rank
4) Input: The input signal u chosen is a periodic rec-tangular pulse of unlit height and with a fundamentalfrequency of 1/(2X) Hz. The digital computer simulationresults are presented in Fig. 2.
Various special cases of this adaptive control problem arenoted in Table I.
In the adaptive laws, the matrices P, F1, and '2 aredefined as in (31).An adjustment in the plant matrix involves a correlation
of the state error e with the plant state xp, while an adjust-ment in the input matrix involves a correlation of e withthe input u.
V. IDENTIFICATION OF MULTIVARIABLE SYSTEMSThe problem of identification, in the model reference
framework, is essentially that of adaptive control. In thiscase, the roles of the plant and model are interchanged sothat the parameters of the model track those of the plant.While the two problems are mathematically equivalent, itis worth noting that the identification problem is somewhatsimpler since the structure of the model (whose parametersare adjusted) unlike that of the plant, can be chosen freelyby the designer. The identification problem is thus devoidof the intricate structural problems that exist in the controlsituation, especially in the multivariable case.
The Identification SchemeLet the plant be described by the differential equation
xp = Apxp + Bpuand the model used to track the plant parameters by
!M = CXm + [Am(t) - C]xp + Bm(t)u
(37)
(38)
where C is a stable matrix and Am(t) and B.(t) are matriceswith adjustable elements. The objective is to devise a schemethat dynamically adjusts these elements so that
lim Am(t) =Apt- QC)
lim Bm(t)-BPt- Co
lim [Xm(t) xp(t)] = lim e(t) = 0.t- o0 t oo
The state error vector e(t) A Xm- xp satisfies the dif-ferential equation
e(t) = Ce + (Dxp+ Tu
where the parameter error matrices are defined as
41 A [Am AP]
(39)
and P A [Bm- BP].(40)
Stability is assured with the adaptive laws
(b = rIl, Pex T
+ = - I"2Peu'where i) P is a positive definite matrix satisfying
CTP + PC = _Q,and ii)
Q = QT> 0
r =p 7T > 0
(41)
(42a)
I2 r2T > 0. (42b)For practical implementation, therefore, the identification
laws are
A..(t) = -FPex T
Bm(t) = -F2PeuT. (43)In the proof of asymptotic stability in Theorem 2, the
elements of the vector z(t) were assumed to be arbitrarylinearly independent functions of time. However, in theadaptive control and identification schemes, the vector z(t)is not completely arbitrary and has the form
z(t) = ____
where xp(t) is not strictly periodic even if u(t) is periodic.In the following, we indicate the lines along which
uniform asymptotic stability of the identification schemegiveii previously can be proved.Assume the plant to be uniformly asymptotically stable.
Since the plant is linear and time-invariant, if the inputvector u(t) is periodic, the state xp(t) is only limit periodic,i.e.,
xp(t) = xP(t) + 6(t)
547
IEEE TRANSACTIONS ON SYSTEMS, MAN, ANt) CYBERNETICS, NOVEMBER 1974
where xp(t) is periodic and e(t) tends to zero exponentially.Hence
z[(t)A P
is a periodic vector. Let us consider the system (18) ofdifferential equations with the vector z(t) replaced by z(t).For this modified system, Theorem 1 ensures that all thesolutions that are bounded approach asymptotically themanifold e(t) 0O, which corresponds to (as t -+ oo)
@ [x (t)] =OLu(t)
sincelim [xp(t) - xP(t)] = 0.t-ooo
02~~~~~~~0o~~~~~~~~~~~~~~~~~~~~~~' 3
0.2
0. [ / 231C)1 2 3 4 5 tOD
-3V3
2.21
It is noted here that e(t) 0O implies that (F is a constant2 --_iJ
matrix. 1 2 3 4 S SECONDS--
It is shown in the Appendix that if 1) the plant is com- (DOMINANT TIME CONSTANT OF THE PLANT -184secs)pletely controllable, 2) the elements of the vector u(t) are Fig. 3. Convergence of model parameters in Exanple 2a) (linearlinearly independent, and 3) each element of u(t) containsq . (n + 1)/2 distinct frequency components, then
[xP =
implies that (F = 0.
This proves the uniform asymptotic stability of the mod-ified system. Since
r0J18(t)Ij dt < 0o
the uniform asymptotic stability of the system (18) with
z(t) = X(t)
follows immediately from Dini-Hukuhara's Theorem [29].
Comments
The identification scheme described in this section formultivariable systems is in the opinion of the authors one
of the more effective methods currently available. Similarschemes using Lyapunov's direct method have been in-dependently suggested by Pazdera and Pottinger [10] anda method for achieving convergence of the model param-eters when the inputs do not contain enough distinctfrequencies has been reported by Lynch and VandeLinde[19]. Further, this scheme has been used recently, withconsiderable success by Narendra and Tripathi [21], forthe control of a VTOL aircraft.
Example 2
This identification scheme has been applied to identifythree parameters of a fourth-order plant.
i) Plant:
=pt) Apxp(t) + Bpu(t)
1 2 3 4 5SECONDS --
Fig. 4. Convergence of time-varying model paramneter in Example 2b)(linear time-varying plant).
-0.0366
A = 0.0482P- a3 *[()
0.4422B = b2e*p -5.52
L 0
0.0271l-1.010.3680
0.151.06331
0- .19-- 1.9
1.90 I
0.17617.594.490 J
where a31*, a33* and b21 are the unknown parameters.ii) Model:
*m(t) = Cxm(t) + [Am(t) --- C: IXp(t) i- Bm(t)u(t).
The matrices Am(t) and Bm(t) are the same as AP and BPwith a31*, a33*, and b21* replaced by a3,(t), a43(t), andb21(t), respectively. C - -10I where I is the (4 x 4)identity matrix. The initial conditions for a31(t), a33(t),and b2l(t) are chosen as: a3(0) 0.1002, a33(0) - -2.35,and b21(0) = 3.54.
iii) Inputs: The input signals u1 and u2 are chosein to berectangular pulses of unit height with fundaamental fre-quencies f1 = 1/(2X) Hz and f2 1/n Hz, respectively.
iv) P and Q matrices: Choosing the positive definitematrix Q = 20kI, where k > 0 is a constant scalar and
548
NARENDRA AND KUDVA: SYSTEM IDENTIFICATION AND CONTROL, PART I
solving (42), the positive definite matrix P obtained isP = kI.
v) Adaptive scheme: The following updating equationsare employed:
I3 l(t) = -0x1[Pe]3Xp1 i33(t) = -a2[Pe]3XP3
b21(t) = -1[Pe]2u1
where al, X2, and /l1 are the adaptive gains, and [Pe]jdenotes the ith element of the vector Pe.
vi) Simulation results: Digital computer simulations havebeen carried out on the following cases.
a) Linear time-invariant plant:The following values are used
1) a31* = 0.2, a33* =- 1.95, b2l* = 5.
2) k = 30, o1 = 0.27, C2 = 1.0, and P1 = 1.0.
b) Linear time-varying plant:All the parameters of the plant except a31 * are assumedto be known; a31* is the time-varying gain a31* = 0.2[1 +0.5 sin (5t)]. The adaptive gains in this case are chosen tobe: k =30 and a, = 10.The results of all the simulations are presented in Figs.
3 and 4. In the time-invariant case, the dominant timeconstant of the plant is 1.84 s. The identification scheme,which converges in about 2 s is thus seen to be quite rapid.
This scheme, developed for linear systems, can be readilyextended to the identification of nonlinear multivariablesystems when the form of the nonlinearity is known exactly.For this purpose, consider a plant described by
x,p = Apxp + Bpf(xp,u) (44)and a model by
xm = Cxm + [Am(t) - C]xp + Bm(t)f(xp,u). (45)Following the approach developed, stability is assuredwith the identification laws
Am = -lFPexpTBm = - F2Pef(xP,U)T. (46)
VI. AN ADAPTIVE CONTROL SCHEME WITH ATIME-VARYING MODEL
In the previous section on the identification problem,a scheme is developed for adjusting the model parametersso that they converge asymptotically to the correspondingones of the plant in the absence of noise. For the adaptivecontrol problem treated in Section IV, similar control lawsare developed when exact matching of the plant with amodel is possible. The matchability conditions as such,however, are found to be far too restrictive in practice;in such cases, the following modification, which is found toincrease the applicability of the adaptive control schemes,is suggested.The modification is based on the realization that there is
rarely in practice one "satisfactory" model for a system.Rather, the model can generally be selected from a set S,which is obtained by perturbing the insensitive parameters
of a typical model. Exact matching in this case consequentlyimplies matching the plant parameters to those of any onemember of the set S.
Within this framework, the adaptive control probleminvolves the simultaneous adjustment of the model and plantparameters. The sensitive parameters of the model areheld constant while the remaining parameters are allowedto vary within a domain in the parameter space generatingthe model set S. If perfect matching between model andplant is obtained when the model belongs to S, satisfactoryadaptation is achieved.
In many practical systems where the plant parametersvary with time, it is known a priori as to which of theparameters appreciably affect the performance of thesystem. In such cases, the parameters of the system can beclassified as sensitive and insensitive parameters. In thefollowing, matrices with subscripts s and i denote, res-pectively, those with sensitive and insensitive parameters;the remaining elements, in either case, are all zero.
Let the plant, together with a feedback controller bedescribed by
xp = (Ap + BF)xp + Bu.
It is assumed that Ap can be decomposed as
Ap = Aps + A.
(47)
(48)APS is assumed to be constant but unknown, while thematrices Ai and B are assumed to be known.
If the feedback matrix BF is similarly decomposed as
BE = (BF)s + (BF)ithe set S of allowable models is defined by
im = [Am + (BF)i]xm + Bu
(49)
(50)where the nonzero elements of the matrix (BF), may bechosen arbitrarily. For adaptation purposes, the time-vary-ing model
Xm = CXm + [Am + (BF)i - C]xp + Bu (51)obtained by feeding part of the feedback signal of the plantinto the model, is chosen. C is a stable matrix. It is furtherassumed that an F* exists such that
Ap + (BF*)s = Am (52)
a matchability condition, which is considerably less re-strictive than (28), as will be seen from Example 3.With this choice of the model, the error between the
model and plant states satisfies the differential equation
e = Ce + [Am + (BF)i- AP - BF]x,= Ce + [(BF*)s- (BF),]xp
where 4 A [(f1 -f1*), (f2 -f2*)T,-A, (f,- f,*)T] X, {fi}being the column vectors of F; and the elements of thematrix X = X(xp) contain linear combinations of theelements of x,.
549
= Ce + Xo (53)
IEEt TRANS.,ClTIONS ON SYSI'5EMS MAN, ANI) (CYBERNETICS, NOVEMBER 1974
Using Theofem 2, stability is achieved with the followingadaptive law
±1 [f /25 5,fn _- F-fX Pedi
o02k
U
(54)
where P satisfies (42) and I" - I 0.The following example is piesented to illustrate the
advantages of this schemwe.
Example 3
a) For the plant, we cornsider the following liiiearizcdequations of a VTOL aircraft flying at 135 knots/h.
-p [Ap BF]xp 4- Bu
where i)
-- 0.0366004821.2170
0.0271- 1.011
0
0.00150.00020.707
0.0375--0. 3310
0
0.0364 0.01451
BP -,-- 0.2918 -06255.52 4,49
0
aP32 and ap34 being the unknown sensitive parameters of theplant. Hence wheni AP is decomnposed into Ap. and Ai.,the matrix AP. contains the param-leters a,,2 and aP334 only,the rest of its elements being zero. For purposes of simula-tiorn, the values of ap,1 aiid a1, are set at 0.1 and 0,2.respectively.
ii) The state variables ate: A-1-horizontal v-elocity; x2-;vertical velocity; X3 -pitch rate, and ¾V4-pitch angle.
iii) The plant inpLuts are: u1-collective and u2-ongitudinal cyclic.
iv) T7he inatrix
10 0 0
The plarit miatrix A p is unstable and is stabilized by a constantfeedback matrix G. The plant equationi tlherefotie becomes
xp = iAp+tBfJ]x + Bv b-u - Gx1,
where
G- L0 0 0.324 0.60210 0 0 0
b) The model equations are chosen to be
12 (t) f2
af I(t3 I,;-L l J i IX Il
-04
0 6
na0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4
SECONDS -
fig. 5. Corlvefgenlce of controller parameters in Example 3.
satisfied with
0
(BF)s =0
00
--- 5.52f0
0000
00-5.52f20
so that 1f *- 0.629 andj2* --0.223.c) The iniput signals ul and u2 are chosen to be the same
as in Examiple 2.d) The equations forfi andf2 are
Jv -'5.52y1x2e3t2 - 5-5 52y2X4e3
where xi and ei denote the zth elements of the vectors xpand t, respectively, and ^; 72 > 0.The adaptive gains are chlosen as
y-- 10 atin 2.5.
The results of the simulations are presented in Fig. 5.
APPENDIX
On the manifold e(t) _ 0, we have (as t -* )
(I.1)
or
Oxp 4 Tu =0 (1.2)
where s A [0 Y] is a constant matrix. From (37) and (1.2)it follows that
[O(pl - AP) "'B + ]u _ 0 (1.3)
where p is the d/dt operator.Let a(A) be the characteristic polynomial of Ap where
c(xj) _ )t" + a,AX + an 1-n 2 + * + a2 1. + a1. (I 4)
Then it is well known that
xm = Cx, + [Am 4 (Bt),- C]x;-+-B
where C - 101 and Am is the sarne as Ap with aP32 andap34 replaced by 4.472 and 1.429, respectively; the matrixAm,, therefore, contains only these elemnents, the remainingelements being zero.
It is obvious that this is a case when the plant and thenmodel cannot be miatched. However, condition (52) is
I n
(Al -* Ap) 1 Z- p1~A~a(,)ij z=where the auxiliary polynomials are defined as
,P)= I, %0(X) =-(i)and
oti-,(;') = );(cxj(A) ai, ie {1,2,s -,n}. (1.6)
(1.5)
vU 0 S
550
XP(t) = 0(Du(t)
NARENDRA AND KUDVA: SYSTEM IDENTIFICATION AND CONTROL, PART I
Hence (1.3) may be rewritten as
[0 E c(p)A 'B + t(p)T] u 0. (1.7)
Using the relations given by (1.6) in (1.7) and rearrangingterms we have
[TPpn + (anT + OB P)pn-
+ (a,1'-I + OBpan + OApBp)pn 2 +
+ (Tal + OBpa2 + OApBpa3 +
+ OAn-2Bpa, + OAnB-Bp)]u= 0. (1.8)
Let u1,2,'' ',Um (the elements of the input vector u) besinusoidal signals of distinct frequencies so that they are
linearly independent. Using the following notation:
T = {bij}, OBp = {aij1}, OApkBp = {akj+}where k e {1,2, ,n - 1}, (1.8) reduces to
m
[bikpn + (anbik + a1ik )pk= 1
+ (an-lbik + anaikI + aik2)pn-2 +
+ (albik + a2aikI + a3aik2 + *. + aik )]Uk
-O (I.9)where ie {1,2,' -,n}.
Since all the uk (k e {1,2, ,m}) are linearly independent,
[bikpn (anbik aik )p
+ (albik + a2aik + + aik )]Uk = (1.10)
with ie {1,2, *,n }andk e {1,2, -,m}.It can be easily shown [30] that if the uk are sinusoidal
signals with qk distinct frequencies and if qk (n + 1)/2,then
ik ik ik ik for all k and i.
This implies T = 0 and
OBP =0, OA Bp = O,- -,OAn B = O (I.12)or
O[BPIAPBPIAP2BPI IAn-IBp] = 0. (1.13)
If the plant is assumed to be completely controllable, the
controllability matrix is of rank n, and thus (1.13) impliesthat 0 = 0. Hence
(D [xp1 t]u(t)
implies that D = 0.
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