Keywords

1 Introduction

Recent methods for control of complex dynamical objects, which work is connected with uncertainty, are based on the analytical models, which are represented in form of differential and recurrence equations. Analysis of related publications [1–4] shows that the major part of researches uses traditional methods, which has a set of limitations, such as requirement of subordination to normal distribution law, usage of traditional mean-square criteria for estimation of the optimality of model parameters, usage of the simplest linear models for measurer systems, etc. Here, the questions, which deal with integration of the empirical knowledges of human experts into poorly formalized process models characterized by incompleteness, imprecision and contradictory [5] as well as presence of fuzzy and subjective factors affected on parameter estimation, are still practically unexplored [6–8].

Nowadays, intelligent models are more preferable for modeling of poorly formalized objects. These models are based on knowledges, the main class of which is fuzzy dynamical systems (FDS) [2–4]. The basis of FDS is formalization of the empirical experience and knowledges of human experts, which is represented in linguistic form via fuzzy logic tools. To make FDS be practically usable in controlling systems, the development of effective construction methods as well as estimation and state correction algorithms is required. It is also important to adapt FDS for the real conditions.

This paper considers the decision of general problems, which are referred to identification, prediction and estimation of FDS states, which describe behavior of poorly formalized dynamical objects.

2 Model of Representation and State Estimation for FDS

Among of many well-known ways for discrete-time FDS representation [9–11], the most simple one is FDS construction in form of recurrence equation [7]:

$$ x_{k + 1} = F(x_{k} )\quad (k = 0,1, \ldots ,N), $$
(1)

where x is the state from the state space X of FDS, F is the fuzzy mapping x k  → x k+1, which is given by membership function (MF) \( \mu_{F}(x_{k} ,x_{k + 1} ) \). This function is also convenient to be represented in form of conditional MF \( \mu_{F}(x_{k + 1} |x_{k} ) \).

Real applications consider both FDS and the measuring system, which realize the transformation of states from X into the external observations from Z taking into account affecting noises. Thus, practically useful model of FDS may be represented considering measurer errors and fuzzy noises in form of the following system:

$$ \left\{ {\begin{array}{*{20}c} {x_{k + 1} = F_{k} (x_{k} ,\varepsilon_{k} )} \\ {z_{k} = S_{k} (x_{k} ,\delta_{k} )} \\ \end{array} } \right.\quad k = 0,1, \ldots ,N, $$
(2)

where F k is the state equation for FDS; S k is the nonlinear function of measurer work; x k is the internal state of a system; ε k is the fuzzy system noise described by a defined MF \( \mu_{{\varepsilon_{k} }} \); δ k is the fuzzy measurer error described by a defined MF \( \mu_{{\delta_{k} }} \); k is the discrete time index.

State space X for (2) is the set of values characterizing the position of a FDS in an observed time step and playing the role of initial conditions for the future system behavior. However, real system has uncertainties and, thus, dependency between current and future values is represented by a fuzzy variable. Estimation and correction of this dependency are very important for dynamical objects control in the area of fuzzy modeling. The task of estimation and correction is defined as follows.

Let the initial information about FDS be presented in form of MF µ(x 0 ) and the observed states be presented by vector of observations \( Z_{k} = (z_{0} ,z_{1} , \ldots ,z_{k} ) \) from time interval \( [t_{0} ,t_{k} ] \). It is required to predict fuzzy state x k+1, which is defined by MF µ(x k+1). FDS correction is performed by specifying the MF for the determined fuzzy value of x k+1, when z k+1 is observed at time step t k+1.

3 Recurrent Algorithm for FDS State Estimation

Estimation and correction for the FDS states are performed based on determination and matching prior information and posterior one characterized by conditional MFs \( \mu (x_{k + 1} |Z_{k} ) \) и \( \mu (x_{k + 1} |Z_{k + 1} ) \). The conditional MFs are determined by the following recurrence procedure.

Let the FDS be nonstationary system with discrete time presented by (2). Let initial state MF \( \mu (x_{0} ) \) be a priory determined. Measurement errors, noises and states are independent fuzzy variables [12].

Based on assumption that \( Z_{k + 1} = (Z_{k} ,z_{k + 1} ) \), MF \( \mu (x_{k + 1}|Z_{k + 1} ) \) can be presented as

$$ \mu (x_{k + 1} |Z_{k + 1} ) = \mu (x_{k + 1} |Z_{k} ,z_{k + 1} ). $$
(3)

Conditional fuzzy variables \( \mu (x_{k + 1} |Z_{k} \) and \( (x_{k + 1} |Z_{k + 1} ) \), which belong to (3), are independent because fuzzy noises ε k and δ k affecting on FDS (determining \( (x_{k + 1} |Z_{k + 1} ) \)) and measurer (determining \( (x_{k + 1} |Z_{k + 1} ) \)), respectively, are also independent. Thus, Eq. (3) can be defined as follows:

$$ \mu (x_{k + 1} |Z_{k} ,z_{k + 1} ) = \mu (x_{k + 1} |Z_{k} )\,\& \,\mu (x_{k + 1} |z_{k + 1} ). $$
(4)

Conditional MF of fuzzy variable (x k+1|z k+1) may be expressed by using measurer function from (2) via the MF of fuzzy noise:

$$ \mu (x_{k + 1} |z_{k + 1} ) = \mu (\delta_{k + 1} )\quad \left( {\delta_{k + 1} = S_{k + 1}^{ - 1} (x_{k + 1} ,z_{k + 1} )} \right). $$
(5)

Since nonlinear mapping \( S_{k + 1}^{ - 1} \) from (5) is commonly multivalued, the fuzzy estimation for \( \mu_{{\delta_{k + 1} }} (S_{k + 1}^{ - 1} (x_{k + 1} ,z_{k + 1} )) \) takes maximum possible value via Zadeh extension principle [13]:

$$ \mu_{{\delta_{k + 1} }} (S_{k + 1}^{ - 1} (x_{k + 1,} z_{k + 1} )) = \mathop {\sup }\limits_{{\Delta = S_{k + 1}^{ - 1} (x_{k + 1} ,z_{k + 1} )}} \mu_{{\delta_{k + 1} }} (\Delta ). $$
(6)

Merging (6) and (5), the following equation can be got:

$$ \mu (x_{k + 1} |z_{k + 1} ) = \mathop {\sup }\limits_{{\Delta = S_{k + 1}^{ - 1} (x_{k + 1} ,z_{k + 1} )}} \mu_{{\delta_{k + 1} }} (\Delta ). $$
(7)

Conditional MF \( \mu (x_{k + 1} |Z_{k} ) \) from (4) describes fuzzy mapping \( \Phi :Z_{k} \to x_{k + 1} \), which can be represented in form of the composition of fuzzy mappings:

$$ \Phi = (Z_{k} \to x_{k} ) \circ (x_{k} \to x_{k + 1}). $$
(8)

Conditional MF \( \mu (x_{k} |Z_{k} ) \) characterizes fuzzy mapping \( Z_{k} \to x_{k} \) as well as fuzzy conditional MF \( \mu (x_{k + 1} |x_{k} ) \) characterizes fuzzy mapping \( x_{k} \to x_{k + 1} \). As a result of composition, we get MF \( \mu (x_{k + 1} |Z_{k} ) \) for fuzzy mapping \( \varPhi :Z_{k} \to x_{k + 1} \):

$$ \mu (x_{k + 1} |Z_{k} ) = \mathop {\sup }\limits_{{x_{k} }} [\mu (x_{k} |Z_{k} )\,\&\, \mu (x_{k + 1} |x_{k} )], $$
(9)

where \( \mu (x_{k + 1} |x_{k} ) \) is expressed via fuzzy noise based on the state equation (2):

$$ \mu (x_{k + 1} |x_{k} ) = \mathop {\sup }\limits_{{\Delta = F_{k}^{ - 1} (x_{k + 1} ,x_{k} )}} \mu_{{\varepsilon_{k} }} (\Delta ). $$
(10)

If expression (10) is merged with Eq. (9), MF \( \mu (x_{k + 1} |Z_{k} ) \) can be calculated as

$$ \mu (x_{k + 1} |Z_{k} ) = \mathop {\sup }\limits_{{x_{k} }} [\mu (x_{k} ,Z_{k} )\& \mathop {\sup }\limits_{{\Delta = F_{k}^{ - 1} (x_{k + 1} ,x_{k} )}} \mu_{{\varepsilon_{k} }} (\Delta )]. $$
(11)

Considering (9) and (11), the final recurrence relations for finding the posterior MF of the fuzzy state of FDS at certain step (k + 1) can be expressed in following form:

$$ \left\{ {\begin{array}{*{20}c} {\mu (x_{k + 1} |Z_{k + 1} ) = \mu (x_{k + 1} ,Z_{k} )\& \mathop {\sup }\limits_{{\Delta = S_{k}^{ - 1} (x_{k + 1} ,z_{k + 1} )}} \mu_{{\delta_{k + 1} }} (\Delta )} \\ {\mu (x_{k + 1} |Z_{k} ) = \mathop {\sup }\limits_{{x_{k} }} [\mu (x_{k} |Z_{k} )\& \mathop {\sup }\limits_{{\Delta = F_{k}^{ - 1} (x_{k + 1} ,x_{k} )}} \mu_{{\varepsilon_{k} }} (\Delta )]} \\ \end{array} } \right., $$
(12)

Initial information for implementation of (12) is presented by \( \mu (x_{0} |z_{0} ) \), which is taken in the form of initial fuzzy state \( \mu (x_{0} ) \) determined by the problem statement.

4 Optimal FDS Parameters Estimation

The task of parameter estimation refers to the problem of parameter identification, when the structure of FDS is a prior determined.

Let the structure of FDS is given in form of system (2), where nonlinear state function F k and measurer one S k depend on the set of uncertain parameters at each time step. The parameters are presented by vector A k and B k , respectively. In this case, model of non-stationary FDS with uncertain parameters is expressed by the following system:

$$ \left\{ {\begin{array}{*{20}c} {x_{k + 1} = F_{k} (x_{k} ,{\text{A}}_{k} ,\varepsilon_{k} )} \\ {z_{k} = S_{k} (x_{k} ,{\text{B}}_{k} ,\delta_{k} )} \\ \end{array} \quad k = 0,1 \ldots ,N} \right.. $$
(13)

Let the observation set be presented in form of vector Z = [z 0, z 1, …, z k ], which is determined on time interval [t n , t m ]. It is required to calculate parameters of FDS A k and parameters of measurer B k , which make the system behavior be similar to experimental observations as more as possible.

To formalize the term “as more as possible”, the criterion of identification quality is introduced. This criterion characterizes the rate of correspondence between FDS and experiments. It is calculated based on the matching of prior MF \( \mu (x_{k + 1} |Z_{k} ) \), which reflects the fuzzy estimation of state from X at the time step t k+1 considering the observation of z k at time step t k , and posterior MF \( \mu (x_{k + 1} |Z_{k + 1} ) \), which reflects the fuzzy estimation of FDS state at time step t k+1 considering the observation of z k+1. Difference between prior and posterior MF is expressed by fuzzy error \( e_{k} \) of current estimation of FDS. MF of the error has the following form:

$$ \mu (e_{k} ,{\text{A}}_{k} ,{\text{B}}_{k} ,{\text{B}}_{k + 1} ) = \mathop {\sup }\limits_{{x_{k + 1} }} \mu (x_{k + 1} |z_{k} ,{\text{A}}_{k} ,{\text{B}}_{k} )\,\&\, \mu (x_{k + 1} + e_{k} |z_{k + 1} ,{\text{B}}_{k + 1} ). $$
(14)

Optimality criterion J can be presented by any fuzzy criterion in form of nonlinear dependency on both conditional and posterior MFs. Particularly, it is convenient to use minimum of deviation for MF of fuzzy estimation error e k from its model function defined on interval [e min , e max ], i.e.:

$$ J_{k} ({\text{A}}_{k} ,{\text{B}}_{k} ,{\text{B}}_{k + 1} ) = \int\limits_{{e_{\hbox{min} } }}^{{e_{\hbox{max} } }} {\left( {r(e_{k} ) - \mu \left( {\left. {e_{k} } \right|,{\text{A}}_{k},{\text{B}}_{k} ,{\text{B}}_{k + 1} } \right)} \right)^{2} \,de_{k} } , $$
(15)

where e k is the current error of the estimation, \( \mu (e_{k} |{\text{A}}_{k} ,{\text{B}}_{k} ,{\text{B}}_{k + 1} ) \) is the MF of fuzzy estimation error (14), \( r(e_k) \) is the model function, which is chosen according to the specifications of an identification task.

Estimation problem is concluded in calculation of such vectors A k and B k , which minimize criterion J, i.e.

$$ \mathop {\hbox{min} }\limits_{{{\text{A}}_{k} ,{\text{B}}_{k} ,{\text{B}}_{k + 1} }} J = \mathop {\hbox{min} }\limits_{{{\text{A}}_{k} ,{\text{B}}_{k} ,{\text{B}}_{k + 1} }} \int\limits_{{e_{\hbox{min} } }}^{{e_{\hbox{max} } }} {(r(e_{k}) - \mu (e_{k} |{\text{A}}_{k} ,{\text{B}}_{k} ,{\text{B}}_{k + 1} ))^{2} de_{k} } . $$
(16)

5 Adaptive Network Model of FDS

Determination of the conditional MF for fuzzy state and optimization of the FDS parameters are made by using both adaptive network model (ANM) and iterative algorithm described below. The basis of ANM is the process of a prior and a posterior MF calculation for each time step from the interval [t n , t m ] and also their matching based on chosen criterion J.

Let output signal (or observed state of a system) z k is observed at certain step t k ∊ [t n , t m ]. Then, the fuzzy state of FDS at time step t k+1 can be calculated based on composition of fuzzy relations \( S_{k}^{ - 1} :z_{k} \to x_{k} \) and \( F_{k} :X_{k} \to X_{k + 1} \) determined by measurer equation and state one (13), respectively. Conditional MF µ(x k |x k ) is determined for fuzzy relation \( S_{k}^{ - 1} \) based on measurer equation taking into account fuzzy noise δ k :

$$ \mu (x_{k} |z_{k} ,{\text{B}}_{k} ) = \mu_{{\delta_{k} }} (S_{k} (x_{k} ,{\text{B}}_{k} ) - z_{k} ). $$
(17)

Conditional MF µ(x k+1|x k ) is defined for fuzzy mapping F k based on state equation taking into account fuzzy noise ε k :

$$ \mu (x_{k + 1} |x_{k} ,{\text{A}}_{k} ) = \mu_{{\varepsilon_{k} }} (F_{k} (x_{k} ,{\text{A}}_{k} ) - x_{k + 1} ). $$
(18)

According to (17) and (18), MF µ(x k+1|z k ) has the following form for composition \( S_{k}^{ - 1} { \circ }F_{k} \):

$$ \mu (x_{k + 1} |z_{k} ,{\text{A}}_{k} ,{\text{B}}_{k} ) = \mathop {\sup }\limits_{{x_{k} }} \;[(\mu (x_{k} |z_{k} ,{\text{B}}_{k} )\,\& \,\mu_{{\varepsilon_{k} }} (F_{k} (x_{k} ,{\text{A}}_{k} ) - x_{k + 1} )] $$
(19)

Expression (19) is a prior MF characterizing the fuzzy value of state from X considering observed state z k at step t k .

To calculate posterior MF \( \mu (x_{k + 1} |z_{k} ) \), output signal z k+1 should be assumed. The posterior MF for fuzzy state x k+1 is calculated based on measurer equation (13) considering fuzzy noise δ k+1:

$$ \mu (x_{k + 1} |z_{k + 1} ,{\text{B}}_{k + 1} ) = \mu_{{\delta_{k + 1} }} (S_{k + 1}(x_{k + 1} ,{\text{B}}_{k + 1} ) - z_{k + 1} ) $$
(20)

Calculations for the MFs and criterion J can be shown in form of network structure representing feedforward calculation illustrated by Fig. 1.

Fig. 1
figure 1

General structure of ANM

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Figure 1 shows that each element implements separated stage of transformations for input signals z k and z k+1 into conditional MF \( \mu (x_{k + 1} |z_{k} ,{\text{B}}_{k} ,{\text{A}}_{k} ) \) and \( \mu (x_{k + 1} |z_{k + 1} ,{\text{B}}_{k + 1} ) \).

Output block calculates J according to the given input signals. Identification of the FDS parameters is performed by handling parameters of A k of FDS and both B k and B k+1 of measurer using backpropagation method [14] and modified Nelder-Mead simplex method [15].

6 Example

Implementation of above described method can be illustratively shown on the example of identification of the following FDS:

$$ \left\{ {\begin{array}{*{20}c} {x_{k} = f(x_{k - 1} ,a_{k - 1} ) + \varepsilon_{k} = a_{k - 1} \cdot x_{k - 1}^{1.7} + \varepsilon_{k} } \\ {z_{k} = s(x_{k} ) + \delta_{k} = b_{k} \cdot x_{k}^{1.2} + \delta_{k} } \\ \end{array} } \right. $$

where a k−1 is the identified parameter of the system; b k is the parameter of an observer; ε k is the fuzzy noise presented in the system, which is represented by Gauss MF with zero mean and variance σε = 0.05; δ k is the fuzzy noise presented in the measurer, which is represented by Gauss MF with zero mean and variance σδ = 0.22.

Let a k−1 equal 2, c k equal 1.2 for all k. Then, MFs have the following forms:

$$ \mu (\varepsilon_{k} ) = \exp \;( - \frac{1}{2 \cdot 0,5} \cdot \varepsilon_{k}^{2} ). $$
$$ \mu (\delta_{k} ) = \exp \;( - \frac{1}{2 \cdot 0,22}\delta_{k}^{2} ) $$

To describe the optimality criterion, the minimization of the MF deviation for fuzzy estimation error e k is used:

$$ J = \mathop { \hbox{min} }\limits_{{a_{k} }} \int\limits_{{e_{{\text{min}}} }}^{{e_{{\text{max}}} }} {\left( {r\,\left( {e_{k} } \right) - \mu (e_{k} ,a_{k} )} \right)^{2} \,de_{k} } . $$

Model function of error is determined on interval \( \left[ {e_{\hbox{min} } ,e_{\hbox{max} } } \right] = \left[ { - 1,1} \right] \) of its changing as follows:

$$ \left\{ {\begin{array}{*{20}l} {r(e) = e + 1} \hfill & {if\quad - 1 \le e < 0} \hfill \\ {r(e) = - e + 1} \hfill & {if\quad \, 0 \le e \le 1} \hfill \\ \end{array}} \right. $$

Let initial state x 0 equal 0.8 and the calculation of \( \mathop {\sup}\limits_{{x_{k}}} (*) \) be provided with discretization step Δ = 0.05 for x. Let also the integral from (15) be performed numerically utilizing quadratic formulas with step Δ = 0.05 and infinite limits be changed by finite ones satisfying finite requirements for estimation algorithm (x min  = 0, x max  = 4). Here, the Nelder-Mead method [9] together with ANM optimizing a k−1 plays the main role for providing both the satisfactory computational speed and the required accuracy of results. Presented example consider the imitation of noises be generated programmatically using standard package of Mathematica software.

Calculation results are presented on figures below. Figure 2 illustrates curve of dependency between J and identified parameter a k , when k = 37. The curve shows that the minimum of J is placed near to real value of a k−1 = 2.

Fig. 2
figure 2

Dependency between J and a k for k = 37

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Figure 3 presents the dependency between parameter a k, which is required to be determined, and iteration step k of Nelder-Mead algorithm. Curve illustrates that a k come around its real value and it deviate from this no more than by 10 %, when k increases.

Fig. 3
figure 3

Dependency between a k and iteration number

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The set of 400 experiments was performed to experimentally test efficacy of proposed approach. In the experiments, fuzzy Sugeno models with various numbers of unknown parameters (from 3 to 9) represent the FDS. Results show that the calculated estimations of a k differ from their real values no more than by 10 % in the major part of samples (more than 95 %). Moreover, results proved that identified parameters of FDS are approximately converged to their real values after 20−30 iterations of algorithm in the major part of samples. Small number of required iterations shows the practical possibility of using ANM for FDS identification in real time mode.

7 Conclusions

This paper presents the new approach for estimation of states and identification of parameters in fuzzy systems describing the dynamics of poorly formalized processes. The proposed method utilizes adaptive network model and modified simplex algorithm. Experimental test of the method shows that determined parameters deviated no more than by 10 % from their real values in the major part of samples. Proposed approach for parameter identification has also the set of new properties, among which possibility for integration in the system of expert knowledges, higher level of accuracy versus traditional techniques and possibility of identification in real time are presented.