On a parameter adaptive self-organizing system in the presence of large outliers in observations

. The aim of the given paper is development of a minimum variance control (MVC) approach for a closed-loopdiscrete-timelinear time-invariant(LTI)system when the parametersof a dynamic system as well as that of a controllerare not known and ought to be estimated by processingobservationsin the case of additive Gaussian noise on the output with contaminating outliers uniformly spread in it. Afterwards, the current value of the control signal is found in each operation, and it is used to generate the output of the system. The resultsof numerical simulation by computer are presented and discussed here, too.


Introduction
To provide self-tuning control of a real plant several ordinary control approaches, such as, MVC, generalized MVC (GMVC), incremental GMVC, and so on, are frequently used that take into consideration random disturbances affecting the process. The MVC and GMVC algorithms, as noted in [5], were the first that were designed specially for self-tuning applications and are now considered 'classical' formulations. The algorithms described there can be implemented as self-tuning controllers that underpin the design and development of a modern model based predictive control approach. On the other hand, it has been emphasized in [3] that in designing a robust control system, one ought to determine the type of uncertainties appearing in the system to be controlled. One of the main ones of them is the uncertainty arising in the output disturbance description of a plant model to be used. It is frequently assumed that system's output is affected by Gaussian disturbance. However, nonnormal noise, and particularly the presence of outliers, degrades the performance of a system acting in a closed-loop as well as the parametric identification of the same system. Besides, before calculating the value of the control signal it is important to find the values of the output that have not been harmed by outliers. To this end, we propose here ways and means how to solve these problems.

Statement of the problem
Assume that a system to be observed is a causal and LTI system with one output {y(k)} and one input {u(k)}, expressed by the equation that consists of two parts: a system model G 0 (q −1 ; θ) = B(q −1 ; b)A −1 (q −1 ; a) and a noise model H 0 (q −1 ; ϕ) = A −1 (q −1 ; a). Here k = 1, 2, . . . is the current number of observations of a respective signal, τ is a known time delay, θ T = (b T , a T ) are unknown parameter vectors to be estimated, q −1 is the backward time-shift operator such that q −1 u(k) = u(k − 1), and a 2 , . . . , a n a ) (2) are unknown parameters of respective polynomials. Given the model (1) and measured data {u(1), . . . , u(N), y(1), . . . , y(N)} and assuming that the white noise {ξ(k)}, k = 1, 2, . . . is really a sequence of independent identically distributed variables with an -contaminated distribution of the form are sequences of independent Gaussian variables with zero means and variances σ 2 µ , σ 2 ς , respectively; besides, σ µ < σ ς ; 0 1 is the unknown fraction of contamination. The aim of the given paper is to design a parameter adaptive self-organizing robust system with the MVC law in the case of additive noise {v(k)}, that contains large outliers and corrupts the output {y(k)} of the LTI system.

Design of a self-organizing system
The MVC controller seeks to design the required control signal by minimizing with respect to {u(k)} the quadratic performance function that refers to the variance of the error between set-point r(k) and the controlled output τ -time steps in the future, y(k + τ ) [5].
To implement the self-tuning MVC controller, it is necessary, firstly, to estimate LTI system's model unknown parameters (2) in such a noisy environment using robust M-techniques [1,2,4] and, secondly, to determine the value of control signal u(k) in each current operation by substituting in (3) the values of abovementioned estimateŝ b T = (b 0 ,b 1 , . . . ,b n b ),â T = (â 1 ,â 2 , . . . ,â n a ). However, in such a case, the transfer of meanings of large outliers proceeds in random noise appearing in output observations. Therefore, in each current operation before calculating the value of the control signal {u(k)} it is important to find the values of the output that have not been harmed by outliers. To this end, we propose here to generate an auxiliary output signalŷ(k + τ ) that will be without outliers. A self-organizing MVC strategy is achieved when estimation and control are carried out every current instant k simultaneously.

Simulation example
A closed-loop system to be analysed is described by a linear difference equation of the form while the MV controller design equation is Here a 1 = −1.5, a 2 = 0.7, b 0 = 1 and the value of coefficient b 1 varries from 0.5 to 0.6 over 400 observations. The output {y(k)}, k = 1, 2, . . . , 400 of the closed-loop system will be observed under the additive noise {v(k)}, k = 1, 2, . . . , 400 in the presence of large outliers. Firstly, the initial values of estimatesâ 1 ,â 2 ,b 0 ,b 1 of the true parameters a 1 , a 2 , b 0 , b 1 of Eq. (5) were calculated by the ordinary LS with Mallow's estimator using 23 pairs of observations of u(k), y(k). Secondly, we recursively calculate the estimatesâ 1 ,â 2 ,b 0 ,b 1 of the same parameters a 1 , a 2 , b 0 , b 1 by processing k = 24, 25, . . . , 400 observations of the control signal {u(k)} and the output {y(k)} in each current iteration, using two S-algorithms with a version of Shweppe's GM-estimator [1]. Thus, the output signals {y(k)} of the same system (5) to be processed by both algorithms were different and generated in two ways (see Figs. 1a, b, e, f): with and with whereŷ  (Figs. 1c, d). In such a case, the true output signal {y * (k)} (7) does not track the reference one, if the control signal {u(k)} is calculated according to (8) (see Fig. 1e). Therefore it is important for calculating current values of the control signal {u(k)} to use formulas (9)-(10) because, in such a case, the output signal {y * (k)} of form (7) tracks the reference one (Fig. 1f).

Conclusions
Despite that the MVC approach has been worked out for a random disturbance generated from the statistically independent and stationary sequence, it appears to be also applicable in the presence of large, but rare outliers in output observations in case the robust recursive parametric identification algorithms are used. One can state that the use of auxiliary signal {ŷ(k)} allowed us to increase the efficiency of an adaptive LTI system with a self-tuning MVC controller significantly.