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State-of-the-art and prospects of adaptive systems
0005 1098/84$3.00+ 0.00 Pergamon PressLtd. ~'~ 1984InternationalFederationof AutomaticControl
Vol. 20, No. 5, pp. 547 557, 1984 Printed in Great Britain. Automatica,
State-of-the-art and Prospects of Adaptive Systems* A. A. VORONOVt and V. YU. RUTKOVSKYt
Works of Soviet authors in adaptive control describe the structure types of the basic loop and adaptation algorithms as well as possible methods for investigation and design analysis of adaptive systems including optimal adaptive systems. Key Words--Adaptive systems; basic loop; adaptation algorithms; optimal adaptive control.
etc.). This definition is purely technical. There are mathematical definitions of an adaptive system too (Fomin et al., 1981 ; Sragovich, 1981). Application of the principles of adaptation results in higher control accuracy under a considerable change of the plant's dynamic properties and allows optimization of is operation modes with varying coefficients, improves the system's reliability, typifies individual control systems and their units, reduces both technological requirements to manufacturing individual elements of the system and the times of its design and development. The theory of adaptive systems rests upon the general theory of stability, theory of invariance, theory of optimal systems (analytical design) (Letov, 1960; Kalman, 1960; Krasovsky, 1963 and 1969), theory of stochastic control (Pugachev, 1962; A,striSm, 1970) method of stochastic approximation (Tzypkin, 1968) and theory of dual control (Feldbaum, 1965).
Abstract-The theory and applications of adaptive control have developed very intensively. There are two reasons for this: variable parameters of controlled plants change within a broad range, on the one hand, and accuracy requirements to process equipment, aircraft systems etc. increase, on the other. Moreover, the wide use of digital computers permits now even complex adaptation algorithms to be easily realized. The paper attempts to review methods of design analysis and synthesis of adaptive systems (AS). Extreme systems, systems with passive adaptation and identification problems have not been considered. Since contributions of Soviet authors has been largely overlooked by the writers of earlier reviews (A,strfm, 1983; Fujii, 1981; and others) we have decided to focus our attention mainly on the results obtained in the U.S.S.R.
1. INTRODUCTION THE FIRSTadaptive systems had probably appeared by the end of the World War II. The Germans utilized programmed alteration of gain in the FAU2 and WASSERFALL missile control systems. Later gains in aircraft control systems were changed as a function of a dynamic head measured in flight by special sensors. This already has been an openloop adaptation. The term and the concept of adaptive control were introduced in the fifties when the complexity of aircraft led to the need for more efficient control systems for objects whose parameters may vary over a wide range. A system is adaptive if it makes use of the information on external actions, dynamic characteristics of the plant, or its control system obtained in the course of operation, to change the structure or gains of the controller necessary to achieve the required properties of the closed-loop system (for example, the dynamics of a system should be independent of the variable parameters of the plant or should provide an example of some performance criterion under any feasible parameters of the plant,
2. CLASSIFICATION O F ADAPTIVE SYSTEMS
Adaptive systems are classified into two classes (Astr6m, 1983): systems of direct adaptive control and indirect adaptive control. In the first class parameters are readjusted so that the output coordinate of the system coincides with that of the reference model. It is sometimes required that the transfer functions of the closed-loop system and the reference model are identical, too. The parameters are adjusted through the use of the value of mismatch between either the controlled coordinates of the system and the model, or their frequency characteristics, etc. In indirect adaptive control the object is first identified and then the regulator coefficients are adjusted so that the closed-loop system features the given properties. The regulators in such systems are often referred to as self-tuning (AstriSm and Wittenmark, 1973; Peterka, 1970 etc.). Note that identification may be carried out on the basis of
* Received 28 November 1983; revised 14 April 1984. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by guest editor L. Ljung. f Moscow, U.S.S.R. 547
548
A.A. VORONOV and V. Yu. RUTKOVSKY
JfS-'-q
direct adaptive control as well (Margolis and Leondes, 1961 ; Ashimov et al., 1973). Most widely used however are the least squares technique (/~str6m, 1983; Ljung, 1977 and others), statistic approximation technique (Tzypkin, 1968), extended Kalman filter and maximum likelihood methods. Systems with direct adaptive control may be divided into three classes (Petrov et al., 1972): (1) Systems with information on frequency characteristics (Smith, 1962; Krasovsky, 1963; Kukhtenko, 1970; Petrov et al., 1972 and others). (2) Systems with information on time characteristics (Solodovnikov, 1957; Braun, 1959 and others). (3) Model reference adaptive systems (Whitaker et al., 1958; Rutkovsky and Krutova, 1965 and others.
Consider the synthesis of a JTO with the use of the invariance theory (Zemlyakov et al., 1969; Pavlov, 1979 and Petrov et al., 1980). Let the object and model be described by equations
3. SYNTHESIS AND DESIGN OF ADAPTIVE SYSTEMS
~i = A(t)x + D(t)la + C(t)f,
Most widely used in the study of adaptive systems is the principle of decomposition. From physical considerations two subsystems can be isolated: a basic loop and a self-adjusting loop. 3.1. Synthesis o f the basic loop The major task in designing the basic loop is to find a structure of correcting devices, by tuning the parameters of which one may achieve matching of the closed-loop and reference model operators, i.e. a problem of adaptivity arises. The basic loop is fully adaptable by the output of the system with respect to external disturbances if for any vector of the object's variable parameters a single vector of adjusted parameters of correcting devices may be found such that the outputs of the system and the reference model are fully identical under the same initial conditions (Petrov et al., 1976; Yadykin, 1981). The basic loop synthesis may often rest upon the concept of the joint tuned object (JTO). At first a JTO is designed which incorporates the plant proper, sensors, actuators and correcting facilities with tuned coefficients. The law of adjusting these coefficients should be determined so that it promotes the description of the JTO by equations with constant coefficients (equations of the stationary model). Thus the problem of the plant variable coefficients matching is solved. The control for the stationary JTO (Fig. 1) is then synthesized to provide the given dynamic characteristics of the closed-loop system. When optimal control is designed the problem is to be solved by the methods of the optimal system theory. Since JTO is stationary its solution is considerably simple. Isolation of the JTO is a technique, which simplifies the synthesis of adaptive systems (AS). The JTO correcting elements may obviously be structurally combined with the correcting circuits generating the JTO-control.
F. . . . . . . . . . . . . . . . .
l!
If
~Actuotor
f,
JSetfadjusting [ 1-41correcting [-J
I I L .....................
,;
I f°citities °f JTOI
I
[ J
FIG. 1. Block diagram of a self-adjusting system.
~,. = Aox,. + Bou + Cof
(1)
where x is the state vector (n); It is the input coordinates vector of the actuators (m); f is the vector of disturbances (r); x,, is the state vector of the model (n); u is the control vector (m); A(t), D(t) and C(t) are the variable matrices of dimensions (n x n), (n x m), (n x r), respectively; Ao, Bo, Co are stationary matrices (n × n), (n × m), (n × z). To satisfy the condition x = x,,, the correcting JTO loops should be chosen as follows: It = Dd-(Dolu - AKx - ANlt - ARf)
(2)
where Dff is pseudoinverse ( m x n) matrix for Do and Do = D(t) - AD(t) is a constant matrix (n x m) whose coefficients correspond to the nominal operation mode of the plant; D01 is a constant (n x m) matrix and AK, AN and AC are matrices of adjusting coefficients of dimension (n x n), (n × m) and (n × r), respectively. From the condition of parametric invariance and invariance with respect to fwe obtain the following: DoDff A K = AA(t), DoD~ A N = AD(t),
(3)
DoD~ A R =- AC(t) and D o D ~ D o l = Bo
where AA(t) = A(t) - Ao, A C ( t ) = C(t) - Co. The structure obtained (Fig. 2) may be considered as the limit structure and employed as a basis for designing a real system. To realize the system observing devices for x and f estimation should be added. Possible ways of simplifying this structure are considered by Petrov et al. (1976). This structure gives us additive adaptation since the equation of a closed-loop system contains the
State-of-the-art and prospects of adaptive systems
549
Lyapunov method (Petrov et al., 1980), for system (1), (2) and (3) when B o B ~ = E (E is a unity matrix). • Introduce the following notation: AA = IIAa,.jll,
AD = rrAdijrl,
AK = IIAk~jfl,
AN = IIAnJ,
Y=AA-AK, R r = I[rrijl[,
Z=AD-AN, R~ = IIr=J,
dAai~ ryij-
FIG. 2. Basicloop's structure of JTO.
sums o f D o D ~ A K - AA(t) etc. and according to (3) there occurs additive compensation of parametric disturbances. Firsova (1981) suggests the principle of multiplicative adaptation according to which two terms are isolated in the denominator of the system transfer function: a multiplier with a constant coefficient system corresponding to the model transfer function, and a multiplier with adjustible coefficients, which becomes equal to the numerator due to adaptation and therefore cancels. It is evident that the multiplier in the numerator, in any combination of the parameters, should be asymptotically stable. The advantage of this method is the need to measure only one controlled coordinate (its derivatives are not required). In his review paper Fujii (1981) presents a description of another scheme of the basic loop which also rests upon the principle of multiplicative adaptation. Among other methods employed in the basic loop synthesis note the mode control widely used by Elyseev (1977, 1978) and the analytic method for the design of linearized systems described by Sokolov (1966). 3.2 Synthesis o f adaptation algorithms With a selected structure of the basic loop, the main problem in designing AS is synthesis of the adaptation algorithms. These algorithms can be designed by different methods. The most used ones are the Lyapunov method (Zemlyakov, 1969; Zemlyakov and Rutkovsky, 1966 and 1967; Parks, 1966; Prokopov, 1974, and others), the method of hyperstability (Landau, 1969), the method of gradient (Whitaker, 1958; Donalson and Leondes, 1963; Evlanov, 1964; Kazakov, 1965, Fradkov, 1974, 1979 and others), the method of stochastic approximation (Tzypkin, 1968), recursive identification (Ljung, 1977; Peterka, 1970 and others) and other techniques. Let us show the algorithms, obtained by the direct
dt
AC = IIACikll, AR = IIAr~klr, S=AC-AR, R~ = Ilr~k[I,
dAdij
'
i , j = 1, n,
rzij =
dt
dAcik rsik-
'
dt
'
k = 1, r
Adaptation algorithms dAkij dt
- ~a~xj,
-- const > 0,
dAn~# dt dArlk dt
- Ktzi# j, i,j -- 1, n,
- tctrifk, k =
(4)
1,r
where ai are the components of vector Pe, = x - x,,, and P = IlpijJI is the symmetric positive-definite matrix which provides for: (1) Stability of zero solution e=0, Y=O,Z=O,S=O
(5)
and asymptotic stability along coordinate c with Ry = Rz = Rs = 0. (2) Asymptotic stability (8) on the whole, uniformly in to and Y(to), Z(to), S(to) with R r = Rz = R~ = 0 and under the condition that the components of vectors Xm, f and ~t make a linearlyindependent uniformly nondisappearing system of functions (this condition will obviously be satisfied, for instance, when u and f contain a sufficient number of harmonics). (3) Stability (5) under the condition that Xm,f a n d It form the linearly-independent uniformly nondisappearing system of functions with Ry, Rz, R, ~ 0. It should be noted that under asymptotic stability (5) the problem of identification may be solved, i.e. algorithms (4) under these conditions may be used in systems of identification of either the plant or the closed-loop system. Condition Rr, R~, R~ ~ 0 means that the plant coefficients are variable. Consequently (3) guarantees the system operability and with algorithms (4) at least at a low rate of plant parameter variation is possible. The drawback of algorithms (4) obtained by many authors (Zemlyakov and Rutkovsky, 1966;
550
A.A. VORONOV and V. Yu. RUTKOVSKY
Parks, 1966; Aksyonov and Fomin, 1973; and others) is that the adaptation processes depend on control action g(t), disturbance f(t) and plant variable parameters. In a number of papers parametric feed is suggested to decrease the effects influencing the adaptation processes of the plant parameters (Petrov et al., 1972; Yadykin, 1973, and others). Putintsev and Yadykin (1980) suggested nonlinear algorithms which, for the basic loop.
dAki~ -- ~laixj + hc2Akij, h'~, ~c2 = const. dt
= A(t)x + B(t)lt, ~ = Ke, e = u-
y, y = /(x,
(6)
~,. = Aox,. + Bou acquire the following form:
~o 1 dkij kij- i° ~i~j " k 3 dt - 6ij fc2ij
+ ~ ~ ~'o.t,~x~, v=l
respect to model movement cannot be guaranteed but some examples show that these simplified algorithms may ensure both stability and better adaptation as compared with algorithms (4). In some papers (Kudva and Narendra, 1972: Fradkov, 1979) the terms proportional to Ak~,i, An~i and Ar~k are introduced into (4) to improve adaptation processes, i.e. following algorithms are obtained :
(7)
However the introduction of proportional terms leads to Akii(~) = 0 (aA~) = 0) while in the ideal case one aims at getting Ak u = Aa u. Nevertheless, algorithms of the type (9) may give better results as compared to (4) when the plant parameters are rapidly changing. The paper by Rutkovsky and Krutova (1965) provides the grounds for algorithms of the following form:
2=1
Akij = ttl k° 1 dkij_ ~ i~ " k ~ d t
ki~-k ° fl i2 k~
-i
~ e=l
v=l
~ ka-~8o.~*..ej, i , j = l,m. 2=1
Here x, xm, ~t, e, y and u are vectors of dimension n, n, m, m, m and m respectively; A(t), B(t), Ao and Bo = II~'oeill are plant parameter matrices and models of dimension (n x n) (n × m), (n × n) and (n x m); K ( t ) = Ilkull is the matrix of tuned coefficients of the regulator in the direct loop of the (m × m) dimension; /¢(t) = IIk~jII is the matrix of tuned coefficients of the regulator in the (n x m) feedback loop;/(°(t), K°(t) are the matrices of ideal tuning of the regulator coefficients with which the system is described by the equation of the model; = x - Xm; P = IlPq[I is a symmetric positive-definite (n x n) matrix; K + = Ilk~ II is a matrix pseudoinverse with respect to matrix K(t); eu, 6u and flu are constant positive coefficients. Algorithms (7) contain elements ki+ corresponding to parametric feedbacks and/¢o, kO which are computed from the equations
Ao = A(t) - BoI(°(t), Bo = B(t)K°(t).
(8)
To solve these equations matrices A(t) and B(t) should be known, i.e. the plant should be identified, and so using algorithms (7) make no sense. Therefore the self-tuning loops were assumed to be sufficiently fast-responding, and /cu = lc°, kij = k °. In this case the stability of system movement with
[9)
£
aixjdt 4- K20"i.
(10)
These algorithms are not substantiated in a strict mathematical manner but simulation of specific systems shown that the a~-proportional terms improve the adaptation process performance. With values of control action g(t) small in magnitude, i.e. with small ]u(t)] when fit) does not lend itself to measurement (Co = 0 in (1) and AR -= 0), or with large [f(t)l one may observe infinite increase of Ak u and Anij in time. This situation may be avoided via algorithm robustness suggested by some authors (Rutkovsky and Krutova, 1965: Aksyonov and Fomin, 1973 etc.) with the introduction of deadzones into the algorithms of adaptation, for example
dAk° - x~(xmjjaix i dt
(1 I J
where qS(xm) =
{lO with ]'x,,,//~> 61 with IIx.,ll < b b = const > 0.
And finally, with ~c--, oc, algorithms (4) with due regard for the constraints imposed upon the multiplier output coordinates are reduced to the relay type (Petrov et al., 1980)
Akij = Akljsign{olxi), Akii = const > 0.
(121
In this case a sliding mode occurs in the system with respect to the model's motion and AS becomes
State-of-the-art and prospects of adaptive systems a general type system with variable structure (Emelyanov, 1967). However a sliding mode may take place only under limited parametric disturbances Aau ([Aau[ ~ Ak u it breaks loose. On the other hand choosing large Ak u will have a negative effect on the system's operation under noise and therefore it may be useful to use the integral and relay terms simultaneously, i.e. to design an algorithm of the type A k i j = t¢
aixjdt + Ak ij sign (aixj).
(13)
AS operation with the use of algorithms (13) was treated in detail by Petrov et al. (1980). Adaptation algorithms based on Lyapunov's method for discrete systems were suggested by Sebakhy (1976). The structure of adaptation algorithms synthesized with the use of the theory of hyperstability coincides with (4). This is also true of those obtained by the gradient technique in which the regulator coefficients are adjusted on the condition of minimizing some quadratic function J = [F(p)e ]2, d where F(p) is the polynomial of p = ~ - and ~ = X - - X
m.
Minimization of J yields algorithms of the form pk u = - 2~/F(p)~i "F(p)uij, ~x u;j = ~ , 7 = const > 0 ,K u
(14}
Various methods developed within the flamework of the theory of sensitivity are used to compute sensitivity functions u u including, in particular, the auxiliary operator method, the sensitivity points method, etc. Adaptation algorithms close to gradient may be synthesized by means of the recurrent goal-oriented inequalities technique (Yakubovich, 1966; Yakubovich, 1968; Fomin et al., 1981). In contrast to (14) this technique permits determination of the value 7. The recurrent goal-oriented inequalities technique employs finite-converging solution algorithms for infinite recurrent systems of inequalities. With these algorithms a solution is obtained within a finite number of stops. Gradient algorithms (14) include gains 7. The problem of choosing the best, in a certain sense, adaptation algorithm was stated already in Tzypkin (1968) and it requires ~ to be selected in a special manner. In the papers by Polyak and Tzypkin (1980) pseudogradient adaptation methods were suggested as providing optimal asymptotic convergence rate. These algorithms were developed for identification problems by Tzypkin (1982). However they
551
most likely may be applied for adaptation purposes as well. 3.3. The problem of disturbance compensation Note here another problem associated not only with the adaptation algorithms but also with the entire structure of the basic loop. This implies the presence of disturbances f. It was previously assumed that f was measurable, however this is not always the case and it is well known that unmeasurable f results in loss of stability. Most vital therefore is the problem of indirect measuring off in the form of estimates r by means of some observing devices. Two solutions are possible in this case. The basic loop may be designed as two-channel using the B. N. Petrov's principle, in which case r should not go into the model (Co -- 0 in (1)), or the model may recognize f with a given operator. In the first case f is compensated and self-tuning loops should only provide the compensation conditions under variable parameters of the plant. In the second, our goal is to achieve the given response on f. From the viewpoint of estimating the basic loop performance, the first case seems preferable. However in the second case when [' is fed onto the model the self-tuning loops operate with a better accuracy. If for instance g (or u with Co = 0, which is the same) includes a lesser number of harmonics than is required for asymptotic stability (5), the disturbance broadens its spectrum and when fed into the model may compensate lacking harmonics, thus providing us with asymptotic stability. 3.4. Analysis of adaptive systems Adaptive systems designed with the above techniques are nonlinear. To design specific systems both the basic loop and adaptation algorithms should be essentially simplified. Of great importance therefore are linearization techniques, methods of designing systems with the given properties using simplified algorithms and methods of investigating AS under random disturbances. Linearized parametric AS models (Kukhtenko, 1970; Yadykin, 1973; Kosikov et al., 1976) are widely used now for the design of self-adjusting loops for such systems. A parametric model is a stationary linear system whose input is the set of variable parameters of the plant and output, the tuned controller coefficients. The basic loop and nonlinear relationships in the adaptation algorithms are substituted here with an equivalent loop under certain conditions. A structural diagram of a linear parametric model may be presented in a form shown in Fig. 3 where Aa and Ak are the vectors of variable parameters of the plant and re-adustable controller coefficients; f is the disturbance and Wa(s), Wk(s) and Wf(s) are the
552
A.A. VORONOV and V. YU. RUTKOVSKY J f (t)
I w:(s) 1
AO~~ ] s .
.......
AK
I,
A
o
,ve,.rv. FIG. 3. Structural diagram of a linear parametric model. transfer function matrices. The goal of all the above techniques is to find W"(s), Wk(s) and W/(s). The design and study of the self-adjusting loop dynamics with the use of linearized models may be carried out by all methods known for linear control systems. In particular one may apply frequency or analytical methods of synthesizing the correcting devices for linearized systems, find their transfer functions and introduce the appropriate corrections into the simplified adaptation algorithms. Widely used in the basic loop design are the root locus and logarithmic frequency characteristics methods (Krutova, 1972). 4. OPTIMALADAPTIVE SYSTEMS The requirement of optimality was not imposed on the systems considered above. At most, it was required that self-adjusting loops could tune the parameters of the regulator in an optimal way. Optimal systems however are now increasingly implemented. An optimal system with variable parameters of the plant should naturally be adaptive. One should note that the problem of AS design as stated by Solodovnikov (1957) is that of the optimal system synthesis and both Soviet and foreign authors have obtained significant results in this field. 4.1. Optimal AS synthesized by the generalized
perJbrmance criterion In Krasovsky et al. (1977), Bukov and Krasovsky (1974), Bukov (1982), and other papers, some universal optimal adaptive control algorithms were obtained resting upon the use of A. A. Krasovsky's method of analytical design (by the generalized performance criterion). A block-diagram of the designed system is shown in Fig. 4. The control plant is described by the equation = F(x,a,f,t), 6 = u
(15)
where x is the state vector; a the unknown parameters vector; 6 the controller coordinate vector and u the control. The minimized function (the generalized performance function) is
---Iregulat°r
I
]~
FIG. 4. Block diagram of optimal adaptive system by A. A. Krasovsky.
tf
J = Vd(x(t:)) +
ft
Q(x(z)dz
0
I f:o (urK
+ ~
lu + uorpK- lUop)dr"
(16)
Here to and t: are the initial and final time instants; Q and Va, the specified positively defined functions and Uop the optimal control under the observations z = H(x,5,a,t) + ~:
(17)
where ~z is the normal centered white noise with an intensity Sz. It is assumed that the problems of state and parameter estimation (identification) and optimal control design may be treated separately. It was proved that for a high degree of estimation this assumption is true. Estimation of state ~ requires the use of a Kalman filter, identification is carried out with the help of either an adaptive model or a cycle Kalman filter or an algorithm with an 'empirical covariance matrix' or a 'time-saving' algorithm (Krasovsky, 1976; Krasovsky et al., 1977). It is evident that other algorithms of identification may be used as well. When designing algorithms of optimal control, operational algorithms or algorithms with a predicting model may be used. One of the control algorithms with a predicting model is due to Krasovsky et al. (1977). The estimation of the current state of the plant is used to specify the initial conditions for the predicting model d ~ x , . = F(x,.,a,.,6,., z),
6., : 0.
(18)
The predicting model is integrated within the optimization time interval. The results are used to set up initial conditions for the vectors
Px-
~ x ~ ( r : ) ' p~ - (~,,(z:)"
(19)
State-of-the-art and prospects of adaptive systems i = A(t,a)x + B(t,d~)u + Dw,(t)wl,
In reverse accelerated time the following system of equations is integrated: dx,. dr
_
OQT(r) t3xm(r)'
dp~
~FT(r)
OQT(z)
~36,.(r) p'~
06,,(z)
dr
-
(22) z : Cx + Dw2(t)w2
d6m F(xm'am'Sm'Z)' ~-z = O,
dp~ OFt(z) dr - t3x,,(z) p~
where x is the state vector; z is the vector of output coordinates; u the control; A, B, Dw, and Dw2 are the variable matrices of the plant; C is the unknown matrix and wl and w2 the normal centred white noise with unity intensity. To obtain estimates of the BL state use a linear Kalman filter
(20)
x = Ao~ + Bou +/~(z
where all the functions are computed for the predicted motion. The value of the vector po is used to form the optimal control Uop = - Kpa.
(23)
J(x, t) = trM{ [£(t) - x(t)] [£(t) - x(t)]r}. (24) The basic idea of this approach is that the filter equation includes matrices of the BL model Ao and Bo. This is admissible under the assumption that the adaptation unit (AU) provides small values of the errors A(t, a) - Ao and B(t, b) - Bo. Therefore the solution of the Riccati matrix equation for the nonstationary plant equation is reduced to the solution of the corresponding equation of the BL stationary model which significantly simplifies the real time estimation procedure. To adjust the BL parameters one may use a tuned reference model (TRM) of the BL and the Kalman filter which is generally nonstationary which is explained by the variable nature of the factor/( in the Kalman filter. The plus of this approach is in its applicability to optimal adaptive stabilization adaptive programmed control and the possibility to generalize results for the case of nonlinear plant.
4.2. Optimal AS synthesized on the principle of stochastic equivalency Another approach to the synthesis of optimal adaptive systems was suggested in Yadykin (1979) and Danilin and Yadykin (1982). These authors have obtained algorithms of optimal adaptive control of linear stochastic plants using the principle of stochastic equivalency and adaptive control with a reference model. A block diagram of such system is presented in Fig. 5. The basic loop (BL) containing the plant, sensors, drives and correcting devices with tuned coefficients a and b is described by the equations of the form
I x~33 . b uni'l:
AdaptoUon
+k
~,
°i -21 Bosic Loop
Co~),
yielding the estimate £ optimal in terms of minimal variance of the estimation error
The advantages of the above technique are in their applicability to adaptive control of essentially multidimensional plants with a large number of controllers, the possibility to realize multiparameter adaptation when the plant's mathematical model includes a great number of parameters to be identified.
model
-
y = Co~, Co = C
(21)
Tuned reference
553
z
.J Kat.monfiLter [ ~"
~
t 9
FIG. 5. Block diagram of optimal adaptive system by Yadykin.
554
A.A. VORONOV and V. Yu. RUTKOVSKY
4.3. Optimal AS with a stochastic relerence model The design of optimal adaptive systems with a stochastic reference model was suggested by Petrov (1978) and Petrov and Zubov (1981). In the most general scheme the 'cautious' algorithms of adaptive control are synthesized (Wittenmark, 1975), i.e. the re-adjusted parameters are set not only by the plant parameters estimates, but also with due regard for the accuracy of these estimates (with regard for the covariance matrices Px, PII, Px~l, x is the state vector and q is the vector of random parameters of the plant). The estimates ~, 1], A~2 (Ax = x - xm) and the covariance matrices Px, P11, Pxll may be obtained with the help of a nonlinear second-order filter in which one may conventionally isolate the filter x, identifier !1, covariance computer and filter Ax (Fig. 6, w, v are random processes). As a physically realizable unit, the reference model is excluded. It is substituted with the computer AS (filter of Ax). However the model as such is tuned. Random in the model are not only the inputs and initial conditions but also the parameters which may vary with respect to q permitting the efficiency of the system to be increased due to adaptation of ~ts characteristics to those of the plant. The synthesis of the system rests upon the use of the dynamic programming technique and the theory of sensitivity. The minimized function is
J = M{Axrttr)).(ll(ty))Ax(t.r)
~
tf
+
[Axr(t)Q~(ll(t))Ax(t)
(25~
)
+ Aur(t)Q2(q(t))Au(t)]dt where Au = u - urn; 2, Q1, Q2 are the weighting matrices and tf is the final instant of the system operation time. Separation of the control unit into the basic loop regulator and adaptation algorithm was obtained as a result of the optimization problem solution rather than postulated in advance. The block diagram includes parametric adaptation and signal adaptation. The parameters of matrices Lb L2 and L 3 are adjustable. The system above features the advantages of both the system with stochastic models and that with identifiers: the processes of adaptation are of closed nature and the dynamic characteristics of the closed-loop system are optimized in accordance with the plant parameter variation. 4.4. Optimal AS synthesized on the basis o1"
computation oj" Hamiltonian Afanasyev and Danilina (1979, 1983) have suggested an idea of designing the optimal system adaptation algorithms based on computations of Hamiltonian. The plant is described by the equations of the type (22) while matrix B is constant. The functional to be minimized is
o [xr(t)Qx(t) + ur(t)Ru(t)~ dt}.
J= M
(26j The suggested approach may be employed in problems where is presentable in the form J = Jx(g) + Jz(X,U) + J3(x,~;)
(27)
where
1M{ f,? [~'r(t)Q~(t) + ur(t)Ru(t) Jdt}, J3(~,~) = M
f~r(t)Q~(t)dt . to
ALgorithm of adaptation FIG. 6. Block diagram of an optimal adaptive system.
)
It follows from the structure of the functional and the plant's equations that the basic loop should include an observer chosen from the condition of the minimum of 31(~), a regulator to maintain the minimum ,lz(~, u), a plant and a measuring device. It
State-of-the-art and prospects of adaptive systems must be noted that ./3 (x, e) comes to zero under the minimum of Jl(e). The optimal control is u = -R-
1BS(t)~(t)
(28)
where S(t) is the solution of the corresponding Riccati equation. The block diagram of this system is shown in Fig. 7. The adaptation unit (AU) determines the adjustment algorithms for the parameters of plant %, observer c% and regulator ~,. The algorithms for % and %., are found from the modified Wiener-Hopf equation
(29) where the operator L(t, t t)is expressed through the sensitivity functions 0z OC~ ,
= d g: 0.
,t-t1
The adjusted parameters of the regulator 7r (the elements of matrix S(t) in (28) are found from the condition S(t) = 0tr(t)An(f`,u)
(30)
where AH(~,u) = H(~, u) - Hop(f~, u), H(f`, u) is the system's Hamiltonian and Hop(f,, u) is the Hamiltonian at the optimal trajectory. The matrix c~,(t) is chosen so that (28) provide asymptotic system optimization by the regulator parameters: c~(t) --
~AH(f`, u)
(31)
Os(t)
Hence 1
"T
e,(t) = ~M{~(t)~2 (t~) + ~(tl)~2r(t)}
(32)
.21 Plant
I
]
- -
u
I
-[
-I
Sensors
Ada pto~ion unit
~ Regulator l-J-
I-
555
and within the interval d = t - tl the parameters of the plant are assumed constant. Thus the optimal adaptive control theory has been significantly developed and may be applied to the design of highly efficient control systems for most diverse plants. 5. CONCLUSION: TRENDS IN THE DEVELOPMENT AND IMPLEMENTATION OF AS
The overview of literature given above leads one to the conclusion that some interesting and significant results have been obtained in the theory of adaptive control. However this is a new direction of research in control theory which still requires much effort in its further development, design of new adaptation algorithms, new computational and investigation techniques and defining the terminology. Most of the adaptation algorithms have been synthesized under the condition of convergence with t ~ ~ , many algorithms are too complicated and require measurement of the complete state vector. Therefore researchers are facing the problem of designing new algorithms meeting the given requirements of the adaptation process performance, and the problem of meaningful simplification of overcomplicated algorithms. In digital control systems it may be useful to apply information on the post coordinates of the system design. This information may to some extent substitute the actions by the derivatives in control and adaptation algorithms. Most promising is the use of predicting devices which may essentially decrease the number of derivatives from controlled coordinates required for the adaptation algorithm convergence. Very important for adaptive system realization are the observations of disturbances acting upon the plant since the model should be fed with all the actions effecting the system. Application of adaptive systems is at present wide enough. As was noted above many processes in the most diverse industrial areas are characterized with varying parameters and their control by nonadaptive systems is impossible. Besides their basic application--control of nonstationary plants with an accidental change of parameters--adaptive systems must find wide use in computer-aided design, in reconfiguration systems and in systems of automatic tuning for industrial controllers (Novosyolov et al., 1975; Nishikawa et al., 1981).
o8
Observer
FIG. 7. Block diagram of an optimal adaptive system by Afanasyev and Danilina.
REFERENCES Aksyonov, G. S. and V. N. Fomin (1973). On linear adaptive control systems. In Methods of Computation. Leningrad State Univ. Publ., No. 8, pp. 95-116 (in Russian). Afanasyev, V. N. and A. M. Daniline (1979). Motion and tracking of a nonstationary plant along specified trajectory. Aut. Rein. Control, 40, 1778-1783 (in Russian).
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A . A . VORONOV a n d V. YU. RUTKOVSKY
Afanasyev, V. N. and A. N. Daniline (1983). Stabilization problem for linear statistic non-stationary plant on a movable platform, lzv. Tekh. Kibernetika, No. 2, pp. 214 219 (in Russian). Ashimov, A., Dh. Syzdykov and G. M. Tokhtabaev (1973). Searchless self-adjusting identification system. Aut. Rein. Control, 34, 332 336. /~str~Sm, K. J. (1970). Introduction to Stochastic Control Theory. Academic Press, New York. Astr6m, K. J. and Wittenmark {1973). On self-tuning regulators. Automatics, 9. 187 199. AstriSm, K. J. (1983). Theory and applications of adaptive control a survey. Automatics, 19, 471 486. Braun, L. (1959). On adaptive control systems. IRE Trans, 2. Bukov, V. N. and A. A. Krasovsky (1974). Operational optimal control algorithm. Aut. Rein. Control, 35, 1541-1548. Bukov, V. N. (1982). Static properties of dynamic systems that have controllers using a prediction model. Aut. Rein. Control, 43, 1047-1053. Danilin, A. B. and I. B. Yadykin (1982). Suboptimal control of stochastic dynamic plants. Izv., Tekh. Kibernetika, No. 3, pp. 194-204 (in Russian). Donaldson, D. D. and C. T. Leondes (1963). A model referenced parameter tracking technique for adaptive control systems. I, II. IEEE Trans Appl. lnd., No. 68, pp. 241-262. Elyseev, V. D. (1977). Methods of design of multivariable selfadjusting control systems. Aut. Rein. Control, 38, 506-512. Elyseev, V. D. (1978). Modally invariant control systems. Aut. Rein. Control, 39, 1598-1605. Emeylanov, S. V. (1967). Automatic Control Synthesis with l/ariable Structure. Nauka, Moscow (in Russian). Evlanov, L. G. (1964). Self-adjusting system employing gradient search by the auxiliary operator method, lzv., Tekh. Kiberneitka, No. 1, pp. 113-120. (in Russian). Feldbaum, A. A. (1965). Optimal Control Systems, Academic Press, New York. Firsova, Ye. M. ( 198 l). Design of the basic loop structure for nonsearch self-adjustive systems in the absence of derivatives of the plant output variable. Aut. Rein. Control, 43, 481-483. Fomin, V. N., A. L. Fradkov and V. A. Yakubovich (1981). Adaptive Control of Dynamic Plants. Nauka, Moscow (in Russian). Fradkov, A. L. (1974). Synthesis of adaptive system of stabilization of linear dynamic plants. Aut. Rein. Control, 35, 1960 1966. Fradkov, A. L. (1979). Speed-gradient scheme and its application in adaptive control. Aut. Rein. Control, 40, 1333-1342. Fujii, S. (1981). On the Trend of'the researchers in adaptive control. Syst. Control, Japan, 25, 715-726. Kalman, R. (1960). Contributions to the theory of optimal control. Bull. Soc. Mat. Mech., 5. Kazakov, I. E. (1965). Investigation of self-adjustment in systems with gradient search by the auxiliary operator method. In Sell: adjusting Systems. Proceedings of the Ist All- Union Conference on Theory and Practice of Sell:adjusting System Design (Moscow, 10 14 December 1963), pp. 23-33. Nauka, Moscow (in Russian). Kosikov, V. S., I. N. Krutova and B. V. Pavlov (1976). Design of linear models of non-search self-adaptive systems. I, If. Aut. Rein. Control, 37, 49 57; 225-237. Krasovsky, A. A. (1963). Dynamics of Continuous Sell-adjusting Systems. Fizmatgiz, Moscow (in Russian). Krasovsky, A. A. (1969). Analytical Design of Aircraft Control Loops. Mashinostroyenie, Moscow (in Russian). Krasovsky, A. A. (1976). Optimal algorithms in identification problem with an adaptive model. Aut. Rein. Control, 37, 185l 1857. Krasovsky, A, A. V. N. Bukov and V. S. Shendrik (1977). Multipurpose Algorithms for Optimal Control of Continuous Processes. Nauka, Moscow (in Russian). Krutova, I. N. (1972). Synthesis of searchless self-adjusting systems based on the root locus method. I, II. Aut. Rein. Control, 33, 1641 1654; 1873 1981. Kudva, P. and K. S. Narendra (1972). An identification procedure for linear multivariable systems. Tech. Report NCT-48, Yale University, New Haven. Kukhtenko, V. I. (1970). Dynamics qf Sell:adjusting Systems with
Frequency Characteristic Stabilization. Mashin~stroyenic, Moscow tin Russian). Landau, I. D. (1969). Hyperstability criterion for model reference adaptive control systems. IEEE Trans Aut. Contr~d. ,~,C-14. 352 555. Letov, A. M. (1960). Analytical design of control systems, l, 1l, 111. Avtomatika Telemekh., 21, 436 411: 561-568; 661 665 (in Russian). Ljung, L. (1977). On positive real transfer functions on the convergence of some recursive schemes. IEEE Trans Aut. Control, AC-22, 539 551. Margolis, M. and S. T. Leondes (1961). On the theory of selladjusting of control systems: a method of self-learning model. Proceedings o1' the Ist Inter. IFAC Congress. Theory ol Discrete, Optimal and Sell:adjusting Systems (Moscow, 27 June-7 July 1960). AN SSSR Publ., pp. 683 701 (in Russian). Novosyolov, B. V., Yu. S. Gorokhov, A. A, Kobzev and A. J. Schitov (1975). Automata-tuners /or Follow-up System~. Energiya, Moscow (in Russian). Nishikawa, Y., N. Sannomiya, T. Ohta, H. Tanaka and K. Tanaka (1981). A method for auto-tuning of PID-control parameters. Preprints of the 8th Triennial World Congress, Japan, 1981, Vol. VIII, pp. 65 70. Parks, P. C. (1966). Lyapunov redesign of model reference adaptive control systems. IEEE Trans Aut. Control, AC-I1, 362 367. Pavlov, B. V. (1979). Synthesis of structure of basic loop of nonsearching self-adjusting systems. Aut. Rem. Control, 38, 1790-1796. Peterka, V. (1970). Adaptive digital regulation of noisy systems. Proc. of the 2nd IF AC Symposium on Identification and Process Parameter Estimation, Prague. Pergamon Press, Oxford. Petrov, A. I. (1978). Stastical synthesis of terminal control adaptive systems with a reference model. DAN SSSR, 242~ 298-301. Petrov, A. I. and A. G. Zubov (198l). On design of '~cautions" adaptive regulators. DAN SSSR, 256, 306 309 (in Russian). Petrov, B. N., V. Yu. Rutkovsky, I. N. Krutova and S. D. Zemlyakov (1972). Design Principles.lbr Sell:adjusting Control Systems. Mashinostroyenie, Moscow (in Russian). Petrov, B. N., V. Yu. Rutkovsky, S. D. Zemlyakov, 1. N. Krutova and I. B. Yadykin (1976}. Some problems of the non-search self-adjusting system theory. I, I1.1zv., Tekh. Kibernetika, No. 2, pp. 154-163; No. 3, pp. 142-- 154 (in Russian). Petrov, B. N., V. Yu. Rutkovsky and S. D. Zemlyakov (1980). Adaptive Coordinate-Parametric Control of the Nonstationary Plants. Nauka, Moscow (in Russian). Polyak, B. T. and Ya. Z. Tzypkin (1980). Optimal pseudogradient adaptation algorithms. Aut. Rein. Control 41, 1101-1110. Prokopov, B. N. (1974). On the design of model referenced adaptive systems by the direct Lyapunov method, lzv., Tekh. Kibernetiea, No. 2, pp. 167- 172 (in Russian). Pugachev, V. S. (1962). A Theory of Random Functions and its Application to Problems of Automatic Control. Fizmatgiz, Moscow (in Russian). Putintsev, V. A. and I. B. Yadykin (1980). Nonlinear adaptation algorithms for multivariable dynamic plants. A~tomatika Telemekh., No. 6, pp. 85-95 (in Russian). Rutkovsky, V. Yu. and I. N. Krutova (1965). Design principle and some theoretical problems for one class of self-adjusting model referenced systems. In Sell-adjusting Automatic Systems. Proc. of the 1st All-Union Conj'. on Theory and Practice o] Se(ladjusting System Design (10 14 December, Moscow, 1963), Nauka, Moscow, pp. 46 63 (in Russian). Sebakhy, O. A. (1976). A discrete model referenced adaptive system design. Int. J. Control, 23, 799 804. Smith, K. C. (1962). Adaptive control through sinusoidal response. 1RE Trans, 7, 129-139. Sokolov, N. I. (1966), Analytical Design Method jor linearized Automatic Control Systems. Mashinostroyenie, Moscow (in Russian). Solodovnikov, V. V. (1957). Some design principles and theoretical problems of self-adjusting automatic control systems. Proc. of the USSR Academy oJ Sciences Session on Scientific Problems of Automatic Control. U.S.S.R. Academy of Sciences Publ., pp. 143-167 (in Russian).
S t a t e - o f - t h e - a r t a n d prospects o f a d a p t i v e systems Sragovich, V. G. (1981). Adaptive Control. Nauka, Moscow (in Russian). Tzypkin, Ya. Z. (1968). Adaptation and Learning in Automatic Systems. Nauka, Moscow (in Russian). Tzypkin, Ya. Z. (1982). Optimal algorithms of parameter estimates for identification. Avtomatika Telemekh., No. 12, pp 9-23 (in Russian). Whitaker, H. P., A. S. Jamrom and A. Kezer (1958). Design of model reference adaptive control systems for aircraft. Massachussets Teehnol. Instrum. Lab. Rept. Sept., R-164. Wittenmark, B. (1975). Stochastic adaptive control methods: a survey. Int. J. Control, 21, 705-730. Yadykin, I. B. (1973). Frequency methods for the study of nonsearch systems dynamics, lzv., Tekh. Kibernetika, No. 5, pp. 185-194 (in Russian). Yadykin, I. B. (1979). Optimal adaptive control designed on the principle of a non-search self-adjusting system with a learning reference model. Avtomatika Telemekh., No. 2, pp. 65-79 (in Russian). Yadykin, I. B. (1981). On the property of Controller Adaptivity in
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Adaptive Systems. DAN SSSR, 259, 310-313 (in Russian). Yakubovich, V. A. (1968). To the adaptive systems theory. DAN SSSR, 182, 518-521 (in Russian). Yakubovich, V. A. (1966). Recurrent finite-converging algorithms for the solution of inequalities systems. DAN SSSR, 6, 1308-1311 (in Russian). Zemlyakov, S. D. and V. Yu. Rutkovsky (1966). Synthesizingselfadjusting control systems with a standard model. Aut. Rein. Control, 27, 407-414. Zemlyakov, S. D. and V. Yu. Rutkovsky (1967). Generalized adaptation algorithms for a class of searchless self-adjusting systems. Aut. Rein. Control, 28, 935-940. Zemlyakov, S. D., B. V. Pavlov and V. Yu. Rutkovsky (1969). Structural synthesis of adaptive control systems. Aut. Rem. Control, 30, 1233-1242. Zemlyakov, S. D. (1969). Some problems of analysis and design of self-adjusting control systems. In Self-Adjusting Control System Theory. Proceedings of the 2nd International IFAC Symposium on Self-adjusting Systems (U.K., 14-17 September 1965). Nauka, Moscow, pp. 154-158 (in Russian).
Vol. 20, No. 5, pp. 547 557, 1984 Printed in Great Britain. Automatica,
State-of-the-art and Prospects of Adaptive Systems* A. A. VORONOVt and V. YU. RUTKOVSKYt
Works of Soviet authors in adaptive control describe the structure types of the basic loop and adaptation algorithms as well as possible methods for investigation and design analysis of adaptive systems including optimal adaptive systems. Key Words--Adaptive systems; basic loop; adaptation algorithms; optimal adaptive control.
etc.). This definition is purely technical. There are mathematical definitions of an adaptive system too (Fomin et al., 1981 ; Sragovich, 1981). Application of the principles of adaptation results in higher control accuracy under a considerable change of the plant's dynamic properties and allows optimization of is operation modes with varying coefficients, improves the system's reliability, typifies individual control systems and their units, reduces both technological requirements to manufacturing individual elements of the system and the times of its design and development. The theory of adaptive systems rests upon the general theory of stability, theory of invariance, theory of optimal systems (analytical design) (Letov, 1960; Kalman, 1960; Krasovsky, 1963 and 1969), theory of stochastic control (Pugachev, 1962; A,striSm, 1970) method of stochastic approximation (Tzypkin, 1968) and theory of dual control (Feldbaum, 1965).
Abstract-The theory and applications of adaptive control have developed very intensively. There are two reasons for this: variable parameters of controlled plants change within a broad range, on the one hand, and accuracy requirements to process equipment, aircraft systems etc. increase, on the other. Moreover, the wide use of digital computers permits now even complex adaptation algorithms to be easily realized. The paper attempts to review methods of design analysis and synthesis of adaptive systems (AS). Extreme systems, systems with passive adaptation and identification problems have not been considered. Since contributions of Soviet authors has been largely overlooked by the writers of earlier reviews (A,strfm, 1983; Fujii, 1981; and others) we have decided to focus our attention mainly on the results obtained in the U.S.S.R.
1. INTRODUCTION THE FIRSTadaptive systems had probably appeared by the end of the World War II. The Germans utilized programmed alteration of gain in the FAU2 and WASSERFALL missile control systems. Later gains in aircraft control systems were changed as a function of a dynamic head measured in flight by special sensors. This already has been an openloop adaptation. The term and the concept of adaptive control were introduced in the fifties when the complexity of aircraft led to the need for more efficient control systems for objects whose parameters may vary over a wide range. A system is adaptive if it makes use of the information on external actions, dynamic characteristics of the plant, or its control system obtained in the course of operation, to change the structure or gains of the controller necessary to achieve the required properties of the closed-loop system (for example, the dynamics of a system should be independent of the variable parameters of the plant or should provide an example of some performance criterion under any feasible parameters of the plant,
2. CLASSIFICATION O F ADAPTIVE SYSTEMS
Adaptive systems are classified into two classes (Astr6m, 1983): systems of direct adaptive control and indirect adaptive control. In the first class parameters are readjusted so that the output coordinate of the system coincides with that of the reference model. It is sometimes required that the transfer functions of the closed-loop system and the reference model are identical, too. The parameters are adjusted through the use of the value of mismatch between either the controlled coordinates of the system and the model, or their frequency characteristics, etc. In indirect adaptive control the object is first identified and then the regulator coefficients are adjusted so that the closed-loop system features the given properties. The regulators in such systems are often referred to as self-tuning (AstriSm and Wittenmark, 1973; Peterka, 1970 etc.). Note that identification may be carried out on the basis of
* Received 28 November 1983; revised 14 April 1984. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by guest editor L. Ljung. f Moscow, U.S.S.R. 547
548
A.A. VORONOV and V. Yu. RUTKOVSKY
JfS-'-q
direct adaptive control as well (Margolis and Leondes, 1961 ; Ashimov et al., 1973). Most widely used however are the least squares technique (/~str6m, 1983; Ljung, 1977 and others), statistic approximation technique (Tzypkin, 1968), extended Kalman filter and maximum likelihood methods. Systems with direct adaptive control may be divided into three classes (Petrov et al., 1972): (1) Systems with information on frequency characteristics (Smith, 1962; Krasovsky, 1963; Kukhtenko, 1970; Petrov et al., 1972 and others). (2) Systems with information on time characteristics (Solodovnikov, 1957; Braun, 1959 and others). (3) Model reference adaptive systems (Whitaker et al., 1958; Rutkovsky and Krutova, 1965 and others.
Consider the synthesis of a JTO with the use of the invariance theory (Zemlyakov et al., 1969; Pavlov, 1979 and Petrov et al., 1980). Let the object and model be described by equations
3. SYNTHESIS AND DESIGN OF ADAPTIVE SYSTEMS
~i = A(t)x + D(t)la + C(t)f,
Most widely used in the study of adaptive systems is the principle of decomposition. From physical considerations two subsystems can be isolated: a basic loop and a self-adjusting loop. 3.1. Synthesis o f the basic loop The major task in designing the basic loop is to find a structure of correcting devices, by tuning the parameters of which one may achieve matching of the closed-loop and reference model operators, i.e. a problem of adaptivity arises. The basic loop is fully adaptable by the output of the system with respect to external disturbances if for any vector of the object's variable parameters a single vector of adjusted parameters of correcting devices may be found such that the outputs of the system and the reference model are fully identical under the same initial conditions (Petrov et al., 1976; Yadykin, 1981). The basic loop synthesis may often rest upon the concept of the joint tuned object (JTO). At first a JTO is designed which incorporates the plant proper, sensors, actuators and correcting facilities with tuned coefficients. The law of adjusting these coefficients should be determined so that it promotes the description of the JTO by equations with constant coefficients (equations of the stationary model). Thus the problem of the plant variable coefficients matching is solved. The control for the stationary JTO (Fig. 1) is then synthesized to provide the given dynamic characteristics of the closed-loop system. When optimal control is designed the problem is to be solved by the methods of the optimal system theory. Since JTO is stationary its solution is considerably simple. Isolation of the JTO is a technique, which simplifies the synthesis of adaptive systems (AS). The JTO correcting elements may obviously be structurally combined with the correcting circuits generating the JTO-control.
F. . . . . . . . . . . . . . . . .
l!
If
~Actuotor
f,
JSetfadjusting [ 1-41correcting [-J
I I L .....................
,;
I f°citities °f JTOI
I
[ J
FIG. 1. Block diagram of a self-adjusting system.
~,. = Aox,. + Bou + Cof
(1)
where x is the state vector (n); It is the input coordinates vector of the actuators (m); f is the vector of disturbances (r); x,, is the state vector of the model (n); u is the control vector (m); A(t), D(t) and C(t) are the variable matrices of dimensions (n x n), (n x m), (n x r), respectively; Ao, Bo, Co are stationary matrices (n × n), (n × m), (n × z). To satisfy the condition x = x,,, the correcting JTO loops should be chosen as follows: It = Dd-(Dolu - AKx - ANlt - ARf)
(2)
where Dff is pseudoinverse ( m x n) matrix for Do and Do = D(t) - AD(t) is a constant matrix (n x m) whose coefficients correspond to the nominal operation mode of the plant; D01 is a constant (n x m) matrix and AK, AN and AC are matrices of adjusting coefficients of dimension (n x n), (n × m) and (n × r), respectively. From the condition of parametric invariance and invariance with respect to fwe obtain the following: DoDff A K = AA(t), DoD~ A N = AD(t),
(3)
DoD~ A R =- AC(t) and D o D ~ D o l = Bo
where AA(t) = A(t) - Ao, A C ( t ) = C(t) - Co. The structure obtained (Fig. 2) may be considered as the limit structure and employed as a basis for designing a real system. To realize the system observing devices for x and f estimation should be added. Possible ways of simplifying this structure are considered by Petrov et al. (1976). This structure gives us additive adaptation since the equation of a closed-loop system contains the
State-of-the-art and prospects of adaptive systems
549
Lyapunov method (Petrov et al., 1980), for system (1), (2) and (3) when B o B ~ = E (E is a unity matrix). • Introduce the following notation: AA = IIAa,.jll,
AD = rrAdijrl,
AK = IIAk~jfl,
AN = IIAnJ,
Y=AA-AK, R r = I[rrijl[,
Z=AD-AN, R~ = IIr=J,
dAai~ ryij-
FIG. 2. Basicloop's structure of JTO.
sums o f D o D ~ A K - AA(t) etc. and according to (3) there occurs additive compensation of parametric disturbances. Firsova (1981) suggests the principle of multiplicative adaptation according to which two terms are isolated in the denominator of the system transfer function: a multiplier with a constant coefficient system corresponding to the model transfer function, and a multiplier with adjustible coefficients, which becomes equal to the numerator due to adaptation and therefore cancels. It is evident that the multiplier in the numerator, in any combination of the parameters, should be asymptotically stable. The advantage of this method is the need to measure only one controlled coordinate (its derivatives are not required). In his review paper Fujii (1981) presents a description of another scheme of the basic loop which also rests upon the principle of multiplicative adaptation. Among other methods employed in the basic loop synthesis note the mode control widely used by Elyseev (1977, 1978) and the analytic method for the design of linearized systems described by Sokolov (1966). 3.2 Synthesis o f adaptation algorithms With a selected structure of the basic loop, the main problem in designing AS is synthesis of the adaptation algorithms. These algorithms can be designed by different methods. The most used ones are the Lyapunov method (Zemlyakov, 1969; Zemlyakov and Rutkovsky, 1966 and 1967; Parks, 1966; Prokopov, 1974, and others), the method of hyperstability (Landau, 1969), the method of gradient (Whitaker, 1958; Donalson and Leondes, 1963; Evlanov, 1964; Kazakov, 1965, Fradkov, 1974, 1979 and others), the method of stochastic approximation (Tzypkin, 1968), recursive identification (Ljung, 1977; Peterka, 1970 and others) and other techniques. Let us show the algorithms, obtained by the direct
dt
AC = IIACikll, AR = IIAr~klr, S=AC-AR, R~ = Ilr~k[I,
dAdij
'
i , j = 1, n,
rzij =
dt
dAcik rsik-
'
dt
'
k = 1, r
Adaptation algorithms dAkij dt
- ~a~xj,
-- const > 0,
dAn~# dt dArlk dt
- Ktzi# j, i,j -- 1, n,
- tctrifk, k =
(4)
1,r
where ai are the components of vector Pe, = x - x,,, and P = IlpijJI is the symmetric positive-definite matrix which provides for: (1) Stability of zero solution e=0, Y=O,Z=O,S=O
(5)
and asymptotic stability along coordinate c with Ry = Rz = Rs = 0. (2) Asymptotic stability (8) on the whole, uniformly in to and Y(to), Z(to), S(to) with R r = Rz = R~ = 0 and under the condition that the components of vectors Xm, f and ~t make a linearlyindependent uniformly nondisappearing system of functions (this condition will obviously be satisfied, for instance, when u and f contain a sufficient number of harmonics). (3) Stability (5) under the condition that Xm,f a n d It form the linearly-independent uniformly nondisappearing system of functions with Ry, Rz, R, ~ 0. It should be noted that under asymptotic stability (5) the problem of identification may be solved, i.e. algorithms (4) under these conditions may be used in systems of identification of either the plant or the closed-loop system. Condition Rr, R~, R~ ~ 0 means that the plant coefficients are variable. Consequently (3) guarantees the system operability and with algorithms (4) at least at a low rate of plant parameter variation is possible. The drawback of algorithms (4) obtained by many authors (Zemlyakov and Rutkovsky, 1966;
550
A.A. VORONOV and V. Yu. RUTKOVSKY
Parks, 1966; Aksyonov and Fomin, 1973; and others) is that the adaptation processes depend on control action g(t), disturbance f(t) and plant variable parameters. In a number of papers parametric feed is suggested to decrease the effects influencing the adaptation processes of the plant parameters (Petrov et al., 1972; Yadykin, 1973, and others). Putintsev and Yadykin (1980) suggested nonlinear algorithms which, for the basic loop.
dAki~ -- ~laixj + hc2Akij, h'~, ~c2 = const. dt
= A(t)x + B(t)lt, ~ = Ke, e = u-
y, y = /(x,
(6)
~,. = Aox,. + Bou acquire the following form:
~o 1 dkij kij- i° ~i~j " k 3 dt - 6ij fc2ij
+ ~ ~ ~'o.t,~x~, v=l
respect to model movement cannot be guaranteed but some examples show that these simplified algorithms may ensure both stability and better adaptation as compared with algorithms (4). In some papers (Kudva and Narendra, 1972: Fradkov, 1979) the terms proportional to Ak~,i, An~i and Ar~k are introduced into (4) to improve adaptation processes, i.e. following algorithms are obtained :
(7)
However the introduction of proportional terms leads to Akii(~) = 0 (aA~) = 0) while in the ideal case one aims at getting Ak u = Aa u. Nevertheless, algorithms of the type (9) may give better results as compared to (4) when the plant parameters are rapidly changing. The paper by Rutkovsky and Krutova (1965) provides the grounds for algorithms of the following form:
2=1
Akij = ttl k° 1 dkij_ ~ i~ " k ~ d t
ki~-k ° fl i2 k~
-i
~ e=l
v=l
~ ka-~8o.~*..ej, i , j = l,m. 2=1
Here x, xm, ~t, e, y and u are vectors of dimension n, n, m, m, m and m respectively; A(t), B(t), Ao and Bo = II~'oeill are plant parameter matrices and models of dimension (n x n) (n × m), (n × n) and (n x m); K ( t ) = Ilkull is the matrix of tuned coefficients of the regulator in the direct loop of the (m × m) dimension; /¢(t) = IIk~jII is the matrix of tuned coefficients of the regulator in the (n x m) feedback loop;/(°(t), K°(t) are the matrices of ideal tuning of the regulator coefficients with which the system is described by the equation of the model; = x - Xm; P = IlPq[I is a symmetric positive-definite (n x n) matrix; K + = Ilk~ II is a matrix pseudoinverse with respect to matrix K(t); eu, 6u and flu are constant positive coefficients. Algorithms (7) contain elements ki+ corresponding to parametric feedbacks and/¢o, kO which are computed from the equations
Ao = A(t) - BoI(°(t), Bo = B(t)K°(t).
(8)
To solve these equations matrices A(t) and B(t) should be known, i.e. the plant should be identified, and so using algorithms (7) make no sense. Therefore the self-tuning loops were assumed to be sufficiently fast-responding, and /cu = lc°, kij = k °. In this case the stability of system movement with
[9)
£
aixjdt 4- K20"i.
(10)
These algorithms are not substantiated in a strict mathematical manner but simulation of specific systems shown that the a~-proportional terms improve the adaptation process performance. With values of control action g(t) small in magnitude, i.e. with small ]u(t)] when fit) does not lend itself to measurement (Co = 0 in (1) and AR -= 0), or with large [f(t)l one may observe infinite increase of Ak u and Anij in time. This situation may be avoided via algorithm robustness suggested by some authors (Rutkovsky and Krutova, 1965: Aksyonov and Fomin, 1973 etc.) with the introduction of deadzones into the algorithms of adaptation, for example
dAk° - x~(xmjjaix i dt
(1 I J
where qS(xm) =
{lO with ]'x,,,//~> 61 with IIx.,ll < b b = const > 0.
And finally, with ~c--, oc, algorithms (4) with due regard for the constraints imposed upon the multiplier output coordinates are reduced to the relay type (Petrov et al., 1980)
Akij = Akljsign{olxi), Akii = const > 0.
(121
In this case a sliding mode occurs in the system with respect to the model's motion and AS becomes
State-of-the-art and prospects of adaptive systems a general type system with variable structure (Emelyanov, 1967). However a sliding mode may take place only under limited parametric disturbances Aau ([Aau[ ~
aixjdt + Ak ij sign (aixj).
(13)
AS operation with the use of algorithms (13) was treated in detail by Petrov et al. (1980). Adaptation algorithms based on Lyapunov's method for discrete systems were suggested by Sebakhy (1976). The structure of adaptation algorithms synthesized with the use of the theory of hyperstability coincides with (4). This is also true of those obtained by the gradient technique in which the regulator coefficients are adjusted on the condition of minimizing some quadratic function J = [F(p)e ]2, d where F(p) is the polynomial of p = ~ - and ~ = X - - X
m.
Minimization of J yields algorithms of the form pk u = - 2~/F(p)~i "F(p)uij, ~x u;j = ~ , 7 = const > 0 ,K u
(14}
Various methods developed within the flamework of the theory of sensitivity are used to compute sensitivity functions u u including, in particular, the auxiliary operator method, the sensitivity points method, etc. Adaptation algorithms close to gradient may be synthesized by means of the recurrent goal-oriented inequalities technique (Yakubovich, 1966; Yakubovich, 1968; Fomin et al., 1981). In contrast to (14) this technique permits determination of the value 7. The recurrent goal-oriented inequalities technique employs finite-converging solution algorithms for infinite recurrent systems of inequalities. With these algorithms a solution is obtained within a finite number of stops. Gradient algorithms (14) include gains 7. The problem of choosing the best, in a certain sense, adaptation algorithm was stated already in Tzypkin (1968) and it requires ~ to be selected in a special manner. In the papers by Polyak and Tzypkin (1980) pseudogradient adaptation methods were suggested as providing optimal asymptotic convergence rate. These algorithms were developed for identification problems by Tzypkin (1982). However they
551
most likely may be applied for adaptation purposes as well. 3.3. The problem of disturbance compensation Note here another problem associated not only with the adaptation algorithms but also with the entire structure of the basic loop. This implies the presence of disturbances f. It was previously assumed that f was measurable, however this is not always the case and it is well known that unmeasurable f results in loss of stability. Most vital therefore is the problem of indirect measuring off in the form of estimates r by means of some observing devices. Two solutions are possible in this case. The basic loop may be designed as two-channel using the B. N. Petrov's principle, in which case r should not go into the model (Co -- 0 in (1)), or the model may recognize f with a given operator. In the first case f is compensated and self-tuning loops should only provide the compensation conditions under variable parameters of the plant. In the second, our goal is to achieve the given response on f. From the viewpoint of estimating the basic loop performance, the first case seems preferable. However in the second case when [' is fed onto the model the self-tuning loops operate with a better accuracy. If for instance g (or u with Co = 0, which is the same) includes a lesser number of harmonics than is required for asymptotic stability (5), the disturbance broadens its spectrum and when fed into the model may compensate lacking harmonics, thus providing us with asymptotic stability. 3.4. Analysis of adaptive systems Adaptive systems designed with the above techniques are nonlinear. To design specific systems both the basic loop and adaptation algorithms should be essentially simplified. Of great importance therefore are linearization techniques, methods of designing systems with the given properties using simplified algorithms and methods of investigating AS under random disturbances. Linearized parametric AS models (Kukhtenko, 1970; Yadykin, 1973; Kosikov et al., 1976) are widely used now for the design of self-adjusting loops for such systems. A parametric model is a stationary linear system whose input is the set of variable parameters of the plant and output, the tuned controller coefficients. The basic loop and nonlinear relationships in the adaptation algorithms are substituted here with an equivalent loop under certain conditions. A structural diagram of a linear parametric model may be presented in a form shown in Fig. 3 where Aa and Ak are the vectors of variable parameters of the plant and re-adustable controller coefficients; f is the disturbance and Wa(s), Wk(s) and Wf(s) are the
552
A.A. VORONOV and V. YU. RUTKOVSKY J f (t)
I w:(s) 1
AO~~ ] s .
.......
AK
I,
A
o
,ve,.rv. FIG. 3. Structural diagram of a linear parametric model. transfer function matrices. The goal of all the above techniques is to find W"(s), Wk(s) and W/(s). The design and study of the self-adjusting loop dynamics with the use of linearized models may be carried out by all methods known for linear control systems. In particular one may apply frequency or analytical methods of synthesizing the correcting devices for linearized systems, find their transfer functions and introduce the appropriate corrections into the simplified adaptation algorithms. Widely used in the basic loop design are the root locus and logarithmic frequency characteristics methods (Krutova, 1972). 4. OPTIMALADAPTIVE SYSTEMS The requirement of optimality was not imposed on the systems considered above. At most, it was required that self-adjusting loops could tune the parameters of the regulator in an optimal way. Optimal systems however are now increasingly implemented. An optimal system with variable parameters of the plant should naturally be adaptive. One should note that the problem of AS design as stated by Solodovnikov (1957) is that of the optimal system synthesis and both Soviet and foreign authors have obtained significant results in this field. 4.1. Optimal AS synthesized by the generalized
perJbrmance criterion In Krasovsky et al. (1977), Bukov and Krasovsky (1974), Bukov (1982), and other papers, some universal optimal adaptive control algorithms were obtained resting upon the use of A. A. Krasovsky's method of analytical design (by the generalized performance criterion). A block-diagram of the designed system is shown in Fig. 4. The control plant is described by the equation = F(x,a,f,t), 6 = u
(15)
where x is the state vector; a the unknown parameters vector; 6 the controller coordinate vector and u the control. The minimized function (the generalized performance function) is
---Iregulat°r
I
]~
FIG. 4. Block diagram of optimal adaptive system by A. A. Krasovsky.
tf
J = Vd(x(t:)) +
ft
Q(x(z)dz
0
I f:o (urK
+ ~
lu + uorpK- lUop)dr"
(16)
Here to and t: are the initial and final time instants; Q and Va, the specified positively defined functions and Uop the optimal control under the observations z = H(x,5,a,t) + ~:
(17)
where ~z is the normal centered white noise with an intensity Sz. It is assumed that the problems of state and parameter estimation (identification) and optimal control design may be treated separately. It was proved that for a high degree of estimation this assumption is true. Estimation of state ~ requires the use of a Kalman filter, identification is carried out with the help of either an adaptive model or a cycle Kalman filter or an algorithm with an 'empirical covariance matrix' or a 'time-saving' algorithm (Krasovsky, 1976; Krasovsky et al., 1977). It is evident that other algorithms of identification may be used as well. When designing algorithms of optimal control, operational algorithms or algorithms with a predicting model may be used. One of the control algorithms with a predicting model is due to Krasovsky et al. (1977). The estimation of the current state of the plant is used to specify the initial conditions for the predicting model d ~ x , . = F(x,.,a,.,6,., z),
6., : 0.
(18)
The predicting model is integrated within the optimization time interval. The results are used to set up initial conditions for the vectors
Px-
~ x ~ ( r : ) ' p~ - (~,,(z:)"
(19)
State-of-the-art and prospects of adaptive systems i = A(t,a)x + B(t,d~)u + Dw,(t)wl,
In reverse accelerated time the following system of equations is integrated: dx,. dr
_
OQT(r) t3xm(r)'
dp~
~FT(r)
OQT(z)
~36,.(r) p'~
06,,(z)
dr
-
(22) z : Cx + Dw2(t)w2
d6m F(xm'am'Sm'Z)' ~-z = O,
dp~ OFt(z) dr - t3x,,(z) p~
where x is the state vector; z is the vector of output coordinates; u the control; A, B, Dw, and Dw2 are the variable matrices of the plant; C is the unknown matrix and wl and w2 the normal centred white noise with unity intensity. To obtain estimates of the BL state use a linear Kalman filter
(20)
x = Ao~ + Bou +/~(z
where all the functions are computed for the predicted motion. The value of the vector po is used to form the optimal control Uop = - Kpa.
(23)
J(x, t) = trM{ [£(t) - x(t)] [£(t) - x(t)]r}. (24) The basic idea of this approach is that the filter equation includes matrices of the BL model Ao and Bo. This is admissible under the assumption that the adaptation unit (AU) provides small values of the errors A(t, a) - Ao and B(t, b) - Bo. Therefore the solution of the Riccati matrix equation for the nonstationary plant equation is reduced to the solution of the corresponding equation of the BL stationary model which significantly simplifies the real time estimation procedure. To adjust the BL parameters one may use a tuned reference model (TRM) of the BL and the Kalman filter which is generally nonstationary which is explained by the variable nature of the factor/( in the Kalman filter. The plus of this approach is in its applicability to optimal adaptive stabilization adaptive programmed control and the possibility to generalize results for the case of nonlinear plant.
4.2. Optimal AS synthesized on the principle of stochastic equivalency Another approach to the synthesis of optimal adaptive systems was suggested in Yadykin (1979) and Danilin and Yadykin (1982). These authors have obtained algorithms of optimal adaptive control of linear stochastic plants using the principle of stochastic equivalency and adaptive control with a reference model. A block diagram of such system is presented in Fig. 5. The basic loop (BL) containing the plant, sensors, drives and correcting devices with tuned coefficients a and b is described by the equations of the form
I x~33 . b uni'l:
AdaptoUon
+k
~,
°i -21 Bosic Loop
Co~),
yielding the estimate £ optimal in terms of minimal variance of the estimation error
The advantages of the above technique are in their applicability to adaptive control of essentially multidimensional plants with a large number of controllers, the possibility to realize multiparameter adaptation when the plant's mathematical model includes a great number of parameters to be identified.
model
-
y = Co~, Co = C
(21)
Tuned reference
553
z
.J Kat.monfiLter [ ~"
~
t 9
FIG. 5. Block diagram of optimal adaptive system by Yadykin.
554
A.A. VORONOV and V. Yu. RUTKOVSKY
4.3. Optimal AS with a stochastic relerence model The design of optimal adaptive systems with a stochastic reference model was suggested by Petrov (1978) and Petrov and Zubov (1981). In the most general scheme the 'cautious' algorithms of adaptive control are synthesized (Wittenmark, 1975), i.e. the re-adjusted parameters are set not only by the plant parameters estimates, but also with due regard for the accuracy of these estimates (with regard for the covariance matrices Px, PII, Px~l, x is the state vector and q is the vector of random parameters of the plant). The estimates ~, 1], A~2 (Ax = x - xm) and the covariance matrices Px, P11, Pxll may be obtained with the help of a nonlinear second-order filter in which one may conventionally isolate the filter x, identifier !1, covariance computer and filter Ax (Fig. 6, w, v are random processes). As a physically realizable unit, the reference model is excluded. It is substituted with the computer AS (filter of Ax). However the model as such is tuned. Random in the model are not only the inputs and initial conditions but also the parameters which may vary with respect to q permitting the efficiency of the system to be increased due to adaptation of ~ts characteristics to those of the plant. The synthesis of the system rests upon the use of the dynamic programming technique and the theory of sensitivity. The minimized function is
J = M{Axrttr)).(ll(ty))Ax(t.r)
~
tf
+
[Axr(t)Q~(ll(t))Ax(t)
(25~
)
+ Aur(t)Q2(q(t))Au(t)]dt where Au = u - urn; 2, Q1, Q2 are the weighting matrices and tf is the final instant of the system operation time. Separation of the control unit into the basic loop regulator and adaptation algorithm was obtained as a result of the optimization problem solution rather than postulated in advance. The block diagram includes parametric adaptation and signal adaptation. The parameters of matrices Lb L2 and L 3 are adjustable. The system above features the advantages of both the system with stochastic models and that with identifiers: the processes of adaptation are of closed nature and the dynamic characteristics of the closed-loop system are optimized in accordance with the plant parameter variation. 4.4. Optimal AS synthesized on the basis o1"
computation oj" Hamiltonian Afanasyev and Danilina (1979, 1983) have suggested an idea of designing the optimal system adaptation algorithms based on computations of Hamiltonian. The plant is described by the equations of the type (22) while matrix B is constant. The functional to be minimized is
o [xr(t)Qx(t) + ur(t)Ru(t)~ dt}.
J= M
(26j The suggested approach may be employed in problems where is presentable in the form J = Jx(g) + Jz(X,U) + J3(x,~;)
(27)
where
1M{ f,? [~'r(t)Q~(t) + ur(t)Ru(t) Jdt}, J3(~,~) = M
f~r(t)Q~(t)dt . to
ALgorithm of adaptation FIG. 6. Block diagram of an optimal adaptive system.
)
It follows from the structure of the functional and the plant's equations that the basic loop should include an observer chosen from the condition of the minimum of 31(~), a regulator to maintain the minimum ,lz(~, u), a plant and a measuring device. It
State-of-the-art and prospects of adaptive systems must be noted that ./3 (x, e) comes to zero under the minimum of Jl(e). The optimal control is u = -R-
1BS(t)~(t)
(28)
where S(t) is the solution of the corresponding Riccati equation. The block diagram of this system is shown in Fig. 7. The adaptation unit (AU) determines the adjustment algorithms for the parameters of plant %, observer c% and regulator ~,. The algorithms for % and %., are found from the modified Wiener-Hopf equation
(29) where the operator L(t, t t)is expressed through the sensitivity functions 0z OC~ ,
= d g: 0.
,t-t1
The adjusted parameters of the regulator 7r (the elements of matrix S(t) in (28) are found from the condition S(t) = 0tr(t)An(f`,u)
(30)
where AH(~,u) = H(~, u) - Hop(f~, u), H(f`, u) is the system's Hamiltonian and Hop(f,, u) is the Hamiltonian at the optimal trajectory. The matrix c~,(t) is chosen so that (28) provide asymptotic system optimization by the regulator parameters: c~(t) --
~AH(f`, u)
(31)
Os(t)
Hence 1
"T
e,(t) = ~M{~(t)~2 (t~) + ~(tl)~2r(t)}
(32)
.21 Plant
I
]
- -
u
I
-[
-I
Sensors
Ada pto~ion unit
~ Regulator l-J-
I-
555
and within the interval d = t - tl the parameters of the plant are assumed constant. Thus the optimal adaptive control theory has been significantly developed and may be applied to the design of highly efficient control systems for most diverse plants. 5. CONCLUSION: TRENDS IN THE DEVELOPMENT AND IMPLEMENTATION OF AS
The overview of literature given above leads one to the conclusion that some interesting and significant results have been obtained in the theory of adaptive control. However this is a new direction of research in control theory which still requires much effort in its further development, design of new adaptation algorithms, new computational and investigation techniques and defining the terminology. Most of the adaptation algorithms have been synthesized under the condition of convergence with t ~ ~ , many algorithms are too complicated and require measurement of the complete state vector. Therefore researchers are facing the problem of designing new algorithms meeting the given requirements of the adaptation process performance, and the problem of meaningful simplification of overcomplicated algorithms. In digital control systems it may be useful to apply information on the post coordinates of the system design. This information may to some extent substitute the actions by the derivatives in control and adaptation algorithms. Most promising is the use of predicting devices which may essentially decrease the number of derivatives from controlled coordinates required for the adaptation algorithm convergence. Very important for adaptive system realization are the observations of disturbances acting upon the plant since the model should be fed with all the actions effecting the system. Application of adaptive systems is at present wide enough. As was noted above many processes in the most diverse industrial areas are characterized with varying parameters and their control by nonadaptive systems is impossible. Besides their basic application--control of nonstationary plants with an accidental change of parameters--adaptive systems must find wide use in computer-aided design, in reconfiguration systems and in systems of automatic tuning for industrial controllers (Novosyolov et al., 1975; Nishikawa et al., 1981).
o8
Observer
FIG. 7. Block diagram of an optimal adaptive system by Afanasyev and Danilina.
REFERENCES Aksyonov, G. S. and V. N. Fomin (1973). On linear adaptive control systems. In Methods of Computation. Leningrad State Univ. Publ., No. 8, pp. 95-116 (in Russian). Afanasyev, V. N. and A. M. Daniline (1979). Motion and tracking of a nonstationary plant along specified trajectory. Aut. Rein. Control, 40, 1778-1783 (in Russian).
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A . A . VORONOV a n d V. YU. RUTKOVSKY
Afanasyev, V. N. and A. N. Daniline (1983). Stabilization problem for linear statistic non-stationary plant on a movable platform, lzv. Tekh. Kibernetika, No. 2, pp. 214 219 (in Russian). Ashimov, A., Dh. Syzdykov and G. M. Tokhtabaev (1973). Searchless self-adjusting identification system. Aut. Rein. Control, 34, 332 336. /~str~Sm, K. J. (1970). Introduction to Stochastic Control Theory. Academic Press, New York. Astr6m, K. J. and Wittenmark {1973). On self-tuning regulators. Automatics, 9. 187 199. AstriSm, K. J. (1983). Theory and applications of adaptive control a survey. Automatics, 19, 471 486. Braun, L. (1959). On adaptive control systems. IRE Trans, 2. Bukov, V. N. and A. A. Krasovsky (1974). Operational optimal control algorithm. Aut. Rein. Control, 35, 1541-1548. Bukov, V. N. (1982). Static properties of dynamic systems that have controllers using a prediction model. Aut. Rein. Control, 43, 1047-1053. Danilin, A. B. and I. B. Yadykin (1982). Suboptimal control of stochastic dynamic plants. Izv., Tekh. Kibernetika, No. 3, pp. 194-204 (in Russian). Donaldson, D. D. and C. T. Leondes (1963). A model referenced parameter tracking technique for adaptive control systems. I, II. IEEE Trans Appl. lnd., No. 68, pp. 241-262. Elyseev, V. D. (1977). Methods of design of multivariable selfadjusting control systems. Aut. Rein. Control, 38, 506-512. Elyseev, V. D. (1978). Modally invariant control systems. Aut. Rein. Control, 39, 1598-1605. Emeylanov, S. V. (1967). Automatic Control Synthesis with l/ariable Structure. Nauka, Moscow (in Russian). Evlanov, L. G. (1964). Self-adjusting system employing gradient search by the auxiliary operator method, lzv., Tekh. Kiberneitka, No. 1, pp. 113-120. (in Russian). Feldbaum, A. A. (1965). Optimal Control Systems, Academic Press, New York. Firsova, Ye. M. ( 198 l). Design of the basic loop structure for nonsearch self-adjustive systems in the absence of derivatives of the plant output variable. Aut. Rein. Control, 43, 481-483. Fomin, V. N., A. L. Fradkov and V. A. Yakubovich (1981). Adaptive Control of Dynamic Plants. Nauka, Moscow (in Russian). Fradkov, A. L. (1974). Synthesis of adaptive system of stabilization of linear dynamic plants. Aut. Rein. Control, 35, 1960 1966. Fradkov, A. L. (1979). Speed-gradient scheme and its application in adaptive control. Aut. Rein. Control, 40, 1333-1342. Fujii, S. (1981). On the Trend of'the researchers in adaptive control. Syst. Control, Japan, 25, 715-726. Kalman, R. (1960). Contributions to the theory of optimal control. Bull. Soc. Mat. Mech., 5. Kazakov, I. E. (1965). Investigation of self-adjustment in systems with gradient search by the auxiliary operator method. In Sell: adjusting Systems. Proceedings of the Ist All- Union Conference on Theory and Practice of Sell:adjusting System Design (Moscow, 10 14 December 1963), pp. 23-33. Nauka, Moscow (in Russian). Kosikov, V. S., I. N. Krutova and B. V. Pavlov (1976). Design of linear models of non-search self-adaptive systems. I, If. Aut. Rein. Control, 37, 49 57; 225-237. Krasovsky, A. A. (1963). Dynamics of Continuous Sell-adjusting Systems. Fizmatgiz, Moscow (in Russian). Krasovsky, A. A. (1969). Analytical Design of Aircraft Control Loops. Mashinostroyenie, Moscow (in Russian). Krasovsky, A. A. (1976). Optimal algorithms in identification problem with an adaptive model. Aut. Rein. Control, 37, 185l 1857. Krasovsky, A, A. V. N. Bukov and V. S. Shendrik (1977). Multipurpose Algorithms for Optimal Control of Continuous Processes. Nauka, Moscow (in Russian). Krutova, I. N. (1972). Synthesis of searchless self-adjusting systems based on the root locus method. I, II. Aut. Rein. Control, 33, 1641 1654; 1873 1981. Kudva, P. and K. S. Narendra (1972). An identification procedure for linear multivariable systems. Tech. Report NCT-48, Yale University, New Haven. Kukhtenko, V. I. (1970). Dynamics qf Sell:adjusting Systems with
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S t a t e - o f - t h e - a r t a n d prospects o f a d a p t i v e systems Sragovich, V. G. (1981). Adaptive Control. Nauka, Moscow (in Russian). Tzypkin, Ya. Z. (1968). Adaptation and Learning in Automatic Systems. Nauka, Moscow (in Russian). Tzypkin, Ya. Z. (1982). Optimal algorithms of parameter estimates for identification. Avtomatika Telemekh., No. 12, pp 9-23 (in Russian). Whitaker, H. P., A. S. Jamrom and A. Kezer (1958). Design of model reference adaptive control systems for aircraft. Massachussets Teehnol. Instrum. Lab. Rept. Sept., R-164. Wittenmark, B. (1975). Stochastic adaptive control methods: a survey. Int. J. Control, 21, 705-730. Yadykin, I. B. (1973). Frequency methods for the study of nonsearch systems dynamics, lzv., Tekh. Kibernetika, No. 5, pp. 185-194 (in Russian). Yadykin, I. B. (1979). Optimal adaptive control designed on the principle of a non-search self-adjusting system with a learning reference model. Avtomatika Telemekh., No. 2, pp. 65-79 (in Russian). Yadykin, I. B. (1981). On the property of Controller Adaptivity in
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