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Section 11. Technical sciences
Botirov Tulkin Vafokulovich, Associate Professor, Department of Automation and Control Navoi State Mining Institute, Uzbekistan E-mail: btv1979@mail.ru Latipov Shahriyor Bahtiyorovich, Assistant, Department of Higher Mathematics and Computer Science Navoi State Mining Institute, Uzbekistan E-mail: shahriyorlatipov@mail.ru Boboev Aziz Azimjonovich, Assistant, Department of Automation and Control Navoi State Mining Institute, Uzbekistan E-mail: btv1979@mail.ru
ALGORITHMS FOR THE SYNTHESIS OF PARAMETERS OF REGULATORS BASED ON THE ESTIMATION OF THE STATE VECTOR IN ADAPTIVE CONTROL SYSTEMS AND REFERENCE MODELS
Abstract. The article presents an approach to the development of algorithms for the synthesis of regulators with the current estimation of the intensity of additive noise, allowing to adapt the dynamic filter to varying statistical characteristics of the perturbation at the input of the system. Algorithms for the synthesis of regulators based on estimating the a posteriori probability density of unknown characteristics of the useful signal are proposed, allowing the two-step task of adaptive estimation of controller parameters to be realized based on separation methods and adaptive filtering.
Keywords: control object, controller, adaptive system with reference models, dynamic filtering, regularization, adaptive filtering.
I. Introduction
To estimate the vector of controller parameters, the Kal-man filter algorithm is usually used [1]. A significant disadvantage of the Kalman filter, which has received widespread implementation of control systems in practice, is the phenomenon of divergence. One of the significant reasons for the divergence of the Kalman filter is that, in the course of the operation of a system, its parameters may differ significantly from those of its mathematical model [2]. Therefore, there is a problem of synthesizing adaptive variants of the Kalman filter, in which discrepancies between the parameters of the real system and the parameters of its mathematical model will be taken into account and, thus, the discrepancy of the filter will be eliminated. In other words, this problem lies in the
synthesis of reliable filters that are resistant to previously unpredictable changes in system parameters [3-5].
II. Formulation of the problem
Let us highlight two approaches to solving this problem. The first is based on the synthesis of a filter, in which unknown or changing parameters are estimated, and using this estimate to adjust the parameters of the adaptive Kalman filter [6]. The second is based on the properties of the "renewing" process.
Suppose that the sets [w(k)} and {v(k)} are mutually independent and define stationary processes d(k), x(k).
If there are a priori data about the values a , a you can build a Kalman filter that performs a one-step forecast. It is defined by recurrent formulas.
§(k +1/ k) = §(k / k -1) + K (k) (z (k) - H§(k / k -1)), (1)
x(k /k- 1) = H§1k/k-1), Ow=0, k = 1,2,...,
where x (k / k -1), Q (k / k -1) - estimates of values = on a set
of observations z(I), l = 1,n -1.
The matrix K(k) is also defined by recurrence relations. For further, it is only important that under the assumptions made, the sequence K(k) has a limit lim K (k) = K.
k ^rc
From here, along with the filter (l), one can consider a stationary filter of the form
O(k +1) = O(k) + d(z (k) - Hd\k)), (2)
x* (k) = HO* (k), where d = K. This filter, as follows from the results of [3; 4], should be asymptotically stable. It gives a one-step prediction optimal in the average quadratic sense with an infinite observation time on the class of linear filters. In the case when d(k), v (k) - are Gaussian processes, then estimate (2) is optimal on an arbitrary class of filters and satisfies conditions [2-4]
M[0(k)/ z(k -1),z(k -2),...] = 0\k), (3)
M[x(k)/z(k -1),z(k -2),...] = x (k). Equations (3) can be collapsed into a single scalar equation of the form
x (k) + atx (k -1) +... + a,x (k -1) = (4)
= d(z (k -1) - x* (k -1)) +... + d (z(k -l) - x* (k -l)). The expression (4) will present in equivalent form
x*(k) = VT(k -1)^, (5)
where
V(k -1) = (-x (k -1),...,-x* (k -l),z(k -1),...,z(k - l))T,
* = (p(1),...,p(),d(1),...,d(l))T, p(i) = a(i) + d(i). Observations z(k) can be given the form
z(k) = ¥T (k - 1)M+n(k), (6)
n(k) = x (k) - x* (k) + v(k). Next, we will assume that , a are not set a priori, and we will synthesize an adaptive analogue of the filter (2), which simultaneously performs both state evaluation and parameter setting.
Adaptive assessment x(k) signal x(k) when k > I +1 we find by the formula
x(k) = -p(1,k)x(k -1) -... -p(l, k)x(k -1) + (7)
+d(1,k )z (k -1) + ... + d(l ,k )z (k -1),
where p(i,k), d(i,k), i = 1, l - parameter settings p(i), d(i) values x(1), ... x(() - arbitrary. Based on equality (6) settings ju(k)parameters ¿u (^= (p(1),...,p(l), d(1),...,d(l))T), we find
using vector relations of the form V(k) = n(k -1) + Y(k)0(k - 1)(z(k) -Ot(k -1) v(k -1)),
y-\k) = y-\k-1) + |IO(k-1)||2, y(1 )>0, k = l +1,...,
(8)
where
0(k -1) = (-x(k -1),...,-x(k -1), z(k -1),... z(k -1))T;
), Y(l) are arbitrary. One-step forecast Q (k +1) state vector 9(k) computed by the formula
Q(k +1) = §(k) + d (k )(z (k) - x(k)), (9)
where d(k) is the setting d, which is a simple recount according to n(k), d(k) is set arbitrarily or consistently with x(() [3; 5].
The set of interrelated relations (7), (8), (9) forms an adaptive filter (closed system, determining /j(k), x(k), d(k +1).
Based on the results of [4; 7], we present the conditions indicating that the constructed adaptive filter under certain conditions can be extremely optimal in the sense indicated below.
Let the conditions be fulfilled:
1) M [x (k)/z(k -1), z (k - 2),... ] = x (k),
M [v2(k)/z (k -1), z (k - 2),... ] < L < <» ;
2) filter (4) satisfies the condition of complete controllability with respect to observations z (k).
Then for any finite /j(i),y(l), 0(j),x(1),...,x(l) with probability one we have
lim ¡j.(k) = n,
k ^rc
km 1SU*® - * (l))2=0, (10)
lim1 ZkJ\§(DdI = 0 .
It follows that with probability one, the parameters of the adaptive filter converge to the values of the parameters of the optimal filter (4), and the empirical mean square of the deviation of the adaptive estimates from the optimal ones converges to zero.
The above algorithms of adaptive filtering (2), (7), (8), (9) allow you to implement the adaptation procedure of the Kalman filter, without the limitations inherent in the identification methods and allow you to find the optimal values of the settings of the regulators in the systems of adaptive control of dynamic objects in real noise environment with almost complete a priori uncertainty.
III. Solution of the task
Consider the task of synthesizing a control device in adaptive control systems with reference models [7; 8]. We assume that the equations necessary for estimating the optimal parameters of the controller are given as
0(k +1) = 0(k) + w(k), (11)
z (k) = H (k )d(k) + v(k), (12)
where is 9(k) the n - dimensional vector of controller parameters with discrete time k = 0,N ; z(k) - size measurement vector m; H(k) - size measurement matrix 1 x m; w(k) - vector of perturbations; v(k) is a sequence of random vectors characterizing the errors of observations; 9(0) - some random vector.
To solve the problem of synthesis of the optimal controller, it is necessary to set the joint density of the vector distributions 0(0) and sequences {w(k)}, {v(k)}. We will, however, assume that only the following statistical characteristics are known:
E{w(k)} = 0, E{w(k)wT(j)} = Q(kj), (k, j = 0,N -1),
E{v(k)} = 0, E{v(k)vT(j)} = R(kj), (k, j = ON).
Building optimal filters based on relations (11), (12) requires knowledge of the statistical characteristics of the state vector and the level of interference. For system (11) and (12), knowledge of the statistical characteristics of the noninformation parameters of the state vector is also necessary. In practice, as a rule, there is a significant uncertainty of these characteristics [1; 6]. Often, the uncertainty of the statistical characteristics of the state vector, the output signal and interference is parametric in nature and reduces to the uncertainty of some of their parameters: spectral width, dispersion, spectral density, etc [9; 10]. Under these conditions, in order to take into account the a priori uncertainty of the statistical characteristics of the useful signal and interference, the linear equations (11), (12) of the formation of the state vector and the observed process can be represented as
0(k) = F (X)0(k -1) + T(X)w(k ,1), (13)
z (t) = H (X)d(k) + v (k ,X), (14)
where X is the vector of unknown parameters describing a priori indefinite statistical characteristics of the observed process and interference.
If a priori indefinite characteristics of the state and interference vector are constant during the filter operation, then the description of its working conditions is complemented by the relation
6(k) = 6(k -1). (15)
If these characteristics change in time, then it is convenient to use the description of the vector X such a Markov process as such a model.
e(k) = F,X(k) + wl(k), (16)
where F1 is the matrix that determines the rate of change of parameters X; w1(t) - white noise with the correlation matrix RWl 8(T), which determines together with the matrix F1 the range of parameters X .
If the state and the observed process are described by equations (13) and (14), in which the matrices F, Г ^ Q depend on the vector of unknown parameters X [11], then the optimal filter equations, calculated on the selection of the state vector with known characteristics corresponding to value X = X;, will be:
d0 (k ,Xt) = F (A, )d0 (k - 1,Xt) + +K(k,X) [z(k) - H(X)F(X)d0(k -1,X)],
K (k ,X) = P3 (k ,X)HT (X)[D(k ,X)]-1, D(k ,X) = H (Xt)P3 (k ,X)HT (X) + R, PЭ (k ,X) = F (X)P (k - 1,X)FT (X) + Q(X), P (k ,X) = [I - K (k ,X )H (X )p (k ,X). (17)
We initially assume that the parameter X takes a continuous set of values and consider the equation for the a posteriori probability density p(XI zk0) vector X provided that there is a sequence of discrete values of the input process z(iT) on the interval 0...k. Consider for this conditional distribution density p(X, z (k) I zk-1). Following [10, 88], you can write
p(X I z0) = Hp(X I zk0-l)p(z(k) I z0-l,X), (18)
where p(z(k) I zk-l,X) is the probability density of formation at the time t = kT reference of the input process z(k), found under the condition that a sample of the input process was observed z°-1 and the vector of unknown message parameters is X ; H1 - constant independent of X , which is determined from the condition of density normalization p(XI r0):
H = 1/ jp(XI zk0-1)p(z(k)Izk0-1,X)dX. (19)
Conditional probability density p(z(k)I zk0,X) when observing a vector process (3.12) and measuring (3.13) is Gaussian [96, 98]
p(z(k)I z= 1 exp{-hz(k) - AfB-1 [z(k) - A]}(20) y](2n)m detB l 2 J
where m is the dimension of the input process;
A = M[z(k) I z0-1,X] = H(A)d3(k) = H(X)F(X)d0(k-1) ;
B = M{[z(k) - A] [z(k) - A]T} = R + H(X)P3 (k,X)HT(X) ;
6S)(k) - extrapolated estimate of the state vector at the k-th moment; 0o (k -1) Is the optimal estimate of the state vector at the time instant t = (k - 1)T, equal to the expectation of the posterior distribution of the message at that moment.
Expressions (18) - (20) allow us to sequentially calculate the distribution p(XI z\) at each step of the time sampling [1]. In the transition to a discrete set of possible values of the vector X relations (18), (19) are replaced by the expression for their a posteriori probabilities P(X)
p(X) = p(x IJ) = MI z"(z(k^ z") , (21)
XPX I z0-1)p(z(k)I z0-1,X,) i=1
where probability density p(z(k)I zk0-1 ,X) is determined by expression (20) when X=Xt.
IV. Conclusion
The considered approach to the construction of adaptive filters is based on the separation method. The adaptive filtering procedure in the synthesized filter is divided into two
stages. In this case, at the first stage, state estimates are P(Xi), and at the second stage, these estimates are averaged determined for fixed values \ and posterior probabilities with a weight equal to probabilities P(Xt).
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