NEO-Fuzzy State-Space Predictive Control

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IFAC-PapersOnLine 48-24 (2015) 099–104 NEO-Fuzzy State-Space Predictive Control NEO-Fuzzy State-Space Predictive Control Yancho V. Todorov*. Margarita N. Terziyska*. Michail G. Petrov**.

Yancho V. Todorov*. Margarita N. Terziyska*. Michail G. Petrov**. *Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, *Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria (Tel: 359-2-9792928; e-mail: [email protected], [email protected]). Sofia, Bulgaria (Tel: 359-2-9792928; [email protected], *** Technical e-mail: University-Sofia, branch Plovdiv, [email protected]). *** Technical University-Sofia, branch Plovdiv, Plovdiv, Bulgaria, (e-mail: [email protected]) Plovdiv, Bulgaria, (e-mail: [email protected]) Abstract: This paper describes the development of a novel state-space model predictive controller. The proposed modelling structure used to capture and predict the nonlinear process dynamics lies on the Abstract: This paper describes the development of a novel state-space model predictive controller. The concept for a neo-fuzzy neuron, deployed in state-space. The introduced approach represents a set of proposed modelling structure used to capture and predict the nonlinear process dynamics lies on the simple fuzzy inferences along the temporal in behaviour of each input node, whose dynamics is expressed state-space. The introduced approach represents concept for a neo-fuzzy neuron, deployed a set of as a singleton function. The learning algorithm for the proposed modelling structure is realized as a simple fuzzy inferences along the temporal behaviour of each input node, whose dynamics is expressed gradient descentfunction. procedure. the basis of the obtained state-spacestructure model, a isfuzzy predictor as a singleton TheOnlearning algorithm for the neo-fuzzy proposed modelling realized as a for the purpose of predictive control is developed. The achieved predictions are used to optimize the gradient descent procedure. On the basis of the obtained neo-fuzzy state-space model, a fuzzy predictor future system response by implementing a quadratic programming optimization procedure along the for the purpose of predictive control is developed. The achieved predictions are used to optimize the stated controller horizons. The potentials of the proposed approach are studied by simulation experiments future system response by implementing a quadratic programming optimization procedure along the to modelling and control of a nonlinear drying plant. stated controller horizons. The potentials of the proposed approach are studied by simulation experiments to and control of aFederation nonlinear plant. © modelling 2015, IFAC (International ofdrying Automatic Control)optimization, Hosting by Elsevier Ltd. All Keywords: neo-fuzzy neuron, predictive control, modeling, state-space, QPrights reserved. Keywords: neo-fuzzy neuron, predictive control, modeling, optimization, state-space, QP The fusion of the fuzzy logic with the neural networks has 1. INTRODUCTION become popular in the last few decades. It allows to combine The fusion of the fuzzy logic with the neural networks has 1. INTRODUCTION Predictive control does not designate a specific control the learning and computational ability of neural networks become popular in the last few decades. It allows to combine the human reasoning ofability a fuzzy A major strategy butcontrol a wide does range not of control algorithms whichcontrol make with the learning andlike computational of system. neural networks Predictive designate a specific drawback of the classical neuro-fuzzy systems is the great explicit use of a (predictive) process model in a cost function strategy but a wide range of control algorithms which make with the human like reasoning of a fuzzy system. A major number of parameters under adaptation during their on-line minimization to obtain the control signal (Huang et al.,2008). explicit use of a (predictive) process model in a cost function drawback of the classical neuro-fuzzy systems is the great This makes the MPC an open framework to implement operation. In such cases a consecutive calculation in a minimization to obtain the control signal (Huang et al.,2008). number of parameters under adaptation during their on-line different modeling and optimization strategies taking into number of discrete instants in order to schedule the rules This makes the MPC an open framework to implement operation. In such cases a consecutive calculation in a premise and consequent parameters is performed. This account the specific requirements of the controlled process. different modeling and optimization strategies taking into number of discrete instants in order to schedule the rules problem inspired many researchers to propose different premise and consequent parameters is performed. This account the specific requirements the controlled process. In the MPC, the objective of theofmodeling procedure is to approaches do decrease the number of the trainable problem inspired many researchers to propose different determine a model that can be quickly in order to save time, while In the MPC, the objective of numerically the modelingevaluated procedure is to parameters approaches decrease the computational number of the trainable and that adequately the process dynamics. To preserving thedoaccuracy of the network (Chadli et al., 2012). determine a model thatdescribes can be numerically evaluated quickly parameters in order to save computational time, while effectively develop describes such models, we need to blend and that adequately the process dynamics. To preserving the accuracy of the network (Chadli et al., 2012). information of different nature: experience of experts and It has been proposed many strategies to manipulate the effectively develop such models, we need to blend operators, measurements and first principle knowledge number of the generated fuzzy rules. For instance, in (Wan et information of different nature: experience of experts and It has been proposed many strategies to manipulate the formulated by mathematical equations. Thus, in the al., 2011) are proposed ideas to design an appropriate operators, measurements and first principle knowledge number of the generated fuzzy rules. For instance, in (Wan et knowledge-based construction, the three different kinds of structure and learning algorithm for minimum rules Takagiformulated by mathematical equations. Thus, in the al., 2011) are proposed ideas to design an appropriate models considered are the mental model, verbal model and Sugeno model, as well as an interesting application for knowledge-based construction, the three different kinds of structure and learning algorithm for minimum rules TakagiEvolving Takagi–Sugeno fuzzy model based on switching to the mathematical model. models considered are the mental model, verbal model and Sugeno model, as well as an interesting application for neighboring models is discussed in (Kalhor 2013). to A Takagi–Sugeno fuzzy model based et onal., switching the mathematical model. From experience, intuition and expert knowledge, we build Evolving novel approach for distribution of the input data space into models is discussed in (Kalhor et al., 2013). A mental model in intuition our mind. model we is build then neighboring clouds in order for to distribution decrease theof calculations in space notioninto to From experience, and The expertverbal knowledge, novel approach the input data formulated using “If…then…” which is is a very procedures in a typical fuzzy-neural network is mental model in our mind. Therules, verbal model then fuzzyfication clouds in order to decrease the calculations in notion to common of description in everyday life. The presented in (Angelov, 2012). As well, in our recent research, formulatedmean using “If…then…” rules, which is a verbal very fuzzyfication procedures in a typical fuzzy-neural network is model can also be formulated based on fuzzy or uncertain proposed many developments generation of a set common mean of description in everyday life. The verbal we presented in (Angelov, 2012). As allowing well, in our recent research, descriptions. Hence, fuzzy sets based serve on as afuzzy smooth interface of distributed fuzzy-neural inferences (Todorov et al., 2015) model can also be formulated or uncertain we proposed many developments allowing generation of a set between qualitative and numerical domains of the and a simplified semi-fuzzy-neural network (Terziyska et al., descriptions. Hence, variables fuzzy sets serve as a smooth interface of distributed fuzzy-neural inferences (Todorov et al., 2015) inputs and outputs of the model serve as a smooth interface 2014) as structures operating with small number of fuzzy between qualitative variables and numerical domains of the and a simplified semi-fuzzy-neural network (Terziyska et al., between andserve numerical domains of the rules. inputs andqualitative outputs ofvariables the model as a smooth interface 2014) as structures operating with small number of fuzzy inputs andqualitative outputs ofvariables the modeland (Ismail et al., domains 2009), (Passino between numerical of the rules. et al., 1998). inputs and outputs of the model (Ismail et al., 2009), (Passino To overcome such deficiencies of the classical fuzzy-neural networks, it has been introduced the idea for the Neo-Fuzzy To overcome such deficiencies of the classical fuzzy-neural et al., 1998). networks, it has been introduced the idea for the Neo-Fuzzy Copyright © 2015, IFAC 2015 99 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Peer review under responsibility of International Federation of Automatic 99 Control. Copyright © IFAC 2015 10.1016/j.ifacol.2015.12.064

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Network (NFN), as special tool for modeling of complex dynamical behavior. This modeling structure was first proposed by Yamakawa in the beginning of 90’s but its wide application in practice is still not very wide spread. The concept of the Neo-Fuzzy Neuron relies on the conventional n-inputs artificial network of neurons, where instead of usual synaptic weights it contains the so-called nonlinear synapses (Uchino et al., 1994). When an input of an NFN is fed by a vector signal, its output is defined by both the input membership functions and the tunable synaptic weights (Bodyanskiy et al., 2005). The most important advantages of the NFN are the learning rate, the high approximation properties, the computational simplicity and the possibility of finding the global minimum of the learning criterion in real time (Bodyanskiy et al., 2013).

number of the neo-fuzzy neurons needed to represent the system. Each neuron comprises a simple fuzzy inference which produces reasoning to singleton weighting consequents:

~ R (i ) : if zi is Ali then wli

Each element of z(k) is being fuzzified using the available three Gaussian fuzzy sets relevant to each neo-fuzzy neuron: 2 1 z −c µli ( zi ) = − exp  i li   2  σ li 

(4)

where µ is the membership degree defined by a Gaussian membership function with corresponding c centre (mean) and σ width (standard deviation).

Besides some reported applications the use of the NFN concept in purpose to control was not yet studied. Therefore, in this paper a design approach for a simple NFN using the state-space approach is proposed. The obtained modelling structure is included into model based predictive control scheme in order to be investigated the prospective of the designed intelligent control solution. The expected outcomes are studied by simulation experiments to control a nonlinear drying plant.

The fuzzy inference should match the output of the fuzzifier with fuzzy logic rules performing fuzzy implication and approximation reasoning in the following way:

µli ( zi ) = ∏ l =1 µli ( zi ) h

(5)

where h is the number of the fuzzy rules in each neuron and l is the relevant synaptic link. The output of the network is produced by implementing consequence matching and linear combination as follows:

2. STATE-SPACE NEO-FUZZY NETWORK

xˆ1 (k + 1)

A great challenge when modeling dynamical systems using the state-space approach is to estimate the system of equations: xˆ (k + 1) = Axˆ (k ) + Bu (k ) yˆ (k ) = Cxˆ (k )

(3)

xˆ2 (k + 1)

(∑ = ∑ (∑ = ∑ i =1 N

h l =1

N

h

i =1

i =1



(1)

) )z

µli ( zi ) wli 1 zi

µli ( zi ) wli 2

i

(6)



xˆn (k + 1) = ∑ i =1 N

(∑

h l =1

)

µli ( zi ) wli n zi

which in fact represents a weighted product composition of the i-th input to l-th synaptic weight to the corresponding n-th rule base.

where x(k) is an input vector of the states, u(k) is the control signal and y(k) is the system output. In order to assume the occurring variances of the parameters and the uncertainties into the data space, a natural choice is to design a fuzzy-neural structure. Such an approach enables the possibility to incorporate knowledge, expressed as simple fuzzy rules while estimating nonlinear kernels to model the complex dynamics.

Thus, the neo-fuzzy implementation of a state-space system for a typical second order system can be represented as: xˆ1 (k + 1) = ( µli ( x1 )a11 xˆ1 (k ) + µli ( x2 )a12 xˆ2 (k ) + µli (u )b1u (k ) ) xˆ2 (k + 1) = ( µli ( x1 )a21 xˆ1 (k ) + µli ( x2 )a22 xˆ2 (k ) + µli (u )b2 u (k ) ) (7) y (k ) = c11 xˆ1 (k ) + c12 xˆ2 (k )

The classical Takagi-Sugeno approach is an often used method but due to its properties, the rule explosion problem easily occurs when dealing with multidimensional state-space problems. An available approach to cope with the mentioned problem is to build a network of fuzzy neurons.

where the corresponding outputs of the neo-fuzzy neurons represent the elements of the matrices A and B. The schematic representation of the designed modeling structure is given in Fig. 1. 2.1 Learning algorithm for the proposed State-Space Neo-

Thus, using the methodology proposed by Yamakawa, we can rewrite the last equation as:

xˆ1 (k + 1) xˆ2 (k + 1)

(∑ = (∑ =

N i =1 1

f ( zi ) z i

N i =1

 xˆn (k + 1) =

) f (z )z )

Fuzzy Neural Network.

2

i

i

(2)



(∑

N i =1

f n ( zi ) zi

)

To train the proposed state-space model an unsupervised learning scheme has been used. For that purpose, a defined error cost term is being minimized at each sampling period. Thus, in order to update the weights in the consequent part of the fuzzy rules is defined:

E =ε

where the input space vector is represented by z(k)=[x(k), u(k)] and n is number of system states and N=n+1 is the 100

2

2

and ε ( k ) = xi ( k ) − xˆi ( k )

(8)

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101

where τd =0.7 and τi=1.05 are scaling factors and kw=1.41 is the coefficient of admissible error accumulation (Osowski, 2000). 3. NEO-FUZZY MODEL PREDICTIVE CONTROL Using the designed NEO-Fuzzy state-space model the optimization algorithm computes the future control actions at each sampling period, by minimizing the following cost: J (k ) = ∑ i =2N yˆ (k + i ) − r (k + i ) 2 Q + N

1

+∑

Nu i = N1

∆u (k + i )

2

(13)

R

subject to Ω∆U ≤ γ

which can be expressed in vector form as:

J (k ) = Y (k ) − Τ(k ) 2 Q + ∆U (k )

where x(k) is the reference input (state) measured form the process and xˆ(k) is the state being estimated by the model. As learning approach is used the well known back propagation approach where the synaptic weights are being adjusted using the following notation:

∂E (k )    ∂β (k ) 

 ∆u ( k + N1 )    ∆U ( k ) =   ∆u ( k + N )   u 

(9)

where η is the learning rate and β is a vector of the trained parameters: the synaptic links in the consequent part of the rules. Form (8) and (9) it can be derived using the chain rule notation:

∆β = −η

∂E ∂E ∂xˆi ∂xˆ = −η = −η ( xi − xˆi ) i ∂β ∂xˆi ∂β ∂β

R

(14)

 y ( k + N1 )   r ( k + N1 )       Y (k ) = Τ( k ) =   y (k + N )   r (k + N )    2  2 

Fig. 1. Schematic diagram of the NEO-Fuzzy State-Space model.

β (k + 1) = β (k ) + ∆β = β (k ) + η 

2

(15)

where, Y is the matrix of the estimated plant output, Τ is the reference matrix, ∆U is the matrix of the predicted controls and Q and R are the matrices, penalizing the changes in error and control term of the cost function:

Q =

 Q ( N1 )  0    0

0   0  Q ( N1 + 1)      0  Q( N2 )  

R =

 R ( N1 )  0    0

0   0  R ( N1 + 1)       0  R( Nu ) 

(10)

0

0

(16)

Using the same approach the parameters in the output matrix C are easily adjusted. It should be mentioned that compared to classical neuro-fuzzy systems, the neo-fuzzy network do not impose adjustment of the fuzzy rule premise parameters, only the rule consequents are under adaptation during each discrete time learning instant.

Taking into account the general prediction form of a linear state-space model (Maciejowski, 2002), (Wang. 2009), we can derive:

2.2 Adaptive Learning Rate Scheduling

Y(k ) = ΨX (k ) + Γu (k − 1) + Θ∆U (k )

In order to overcome the deficiencies of the Gradient Descent approach, a simple adaptive solution to define at each iteration step the learning rate η, has been employed. The idea lies on the estimation of the Root Squared Error:



M

 k 1 ( x(k )  xˆ (k ))

2

Ψ =

(11)

Afterwards, the following condition is applied:

if i  i1k w i 1  i  d if i  i1k w i 1  i i

(12)

101

 CA   CA2   CA3     CA N2 

Γ =

 CB  + CAB CB    CA2 B + CAB + CB    N − 1 i 2  C ∑i =0 A B 

(17)

(18)

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Θ=

CB   CAB + CB        C ∑ Nu Ai B i =1     N 2 −1 i  C ∑ i =1 A B

      

  0  CB   CAB + CB       N 2 − Nu i  C ∑ i =1 A B  0

(19)

Then we can define the system error as: Ε(k)=Τ(k)-ΨX(k)Гu(k-1). This expression is assumed as tracking error in sense of that it is the difference between the future target trajectory and the free response of the system, that occurs over the prediction horizon if no input changes were made, if ΔU=0 is set. Using the last notation, we can write: T

T

Φ = −2Θ QΕ(k ), H = Θ QΘ + R T

300

temperature, K

J (k ) = ∆U H ∆U + ∆U Φ + Ε QΕ, T

T

On Fig. 2 and Fig. 5 are presented the obtained results in control of the drying plant for selected values of Qi =0.7.10-4 and Ri=0.19.10-3, considering two cases for additive noise (amplitude ± 2 K and sampling period of 1.5s), added solely to control signal and to control signal and state x2, in order to assess the controller capabilities in presence of disturbances. On both figures are presented the transient responses of the control and output signals, as well as the response of the control error in logarithmic units. The provided simulation experiments are performed on equal initial conditions for the model.

260

240

(20)

220

Differentiating the gradient of J with respect to ΔU, gives the Hessian matrix: ∂2J(k)/∂ΔU2(k)=2H=2(ΘTQΘ+R). If Q(i)≥0 for each i (ensures that ΘTQΘ≥0) and if R≥0 then the Hessian is certainly positive-definite, which is enough to guarantee the reach of minimum.

system output y control signal u

280

0

200

400

600

800

1000 time, s

1200

1400

1800

1600

1

control error

0.8

3.1 Constraints formulation

0.6 0.4 0.2 0

3

2

10

10

Since, U(k) and Y(k) are not explicitly included in the optimization problem, the constraints can be expressed in terms of ΔU signal:

time, s (log)

0.8

state x1 state x2 output y

0.6

RMSE

− F2 u (k − 1) + f  F1     GΘ  ∆U ≤  −G (ΨX (k ) + Υu (k − 1)) + g  (21)     w  W    The first row represents the constraints on the amplitude of the control signal, the second one the constraints on the output changes and the last the constraints on the rate change of the control.

Fig. 2. Simulation results in control and estimated control, output and control error signals (with noise on the control action).

0.4

0.2

4. RESULTS AND DISCUSSION

0

2

3

10

10 time, s (log)

The performed simulation experiments are conducted to control a nonlinear lyophilization plant for drying of 50 vials filled with glycine in water adjusted to pH 3, with hydrochloric acid. A detailed description of the process is given in (Shoen, 1995). The objective is to control the temperature of the product (y) by manipulating the temperature of the heating fluid (u), in terms of estimating the process dynamics through the interface position (x1) and the temperature in the frozen zone (x2). The following initial conditions for simulation experiments are assumed: N1=1, N2=5, Nu=3; System reference r=255K; Initial shelf temperature, before the start of the primary drying Tsin=228K; Initial thickness of the interface front x=0.0023m; Thickness of the product L=0.003m. The following constraints on the optimization problem are imposed: constraints on the amplitude of the control signal- the heating shelves temperature228K
1

state x2 state x1 output y

0.8

RSE

0.6 0.4 0.2 0

3

2

10

10

time, s (log)

Fig. 3. The trnsient responces of the RMSE and RSE of the modeling errors in logarithmic units (with noise on the control action). As can be seen from the transient responces in both cases of controller operation, the control error is sucessfully minimized and all stated constraints are satisfied. Due to fuzzy proparties of the model to generalize the input signals, a significant deterioration of the model/controller operation cannot be observed.

102

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103

state x1 state x2 output y

0.6 1

RMSE

control increment delta u

0.8 2

0

200

0

400

600

800

1000 time, s

1200

1400

0.4

1800

1600

0.2

-8

x 10

0

10 time, s (log)

1

additive noise on te system input u, degrees of K

-1

0

200

400

600

800

1000 time, s

1200

1400

1800

1600

state x2 state x1 output y

0.8 0.6

2

0.4 0 0.2 -2

0

200

400

600

800

1000 time, s

1200

1400

0

1800

1600

10 time, s (log)

Fig. 6. The transient responses of the RMSE and RSE of the modeling errors in logarithmic units (with noise on the control action and the state).

control increment delta u

300

system output y control signal u

280

3

2

10

Fig. 4. Bar plots of the instant valuses of the control increment ∆u and the cost function J (with noise on the control action).

temperature, K

3

2

10 -0.5

RSE

cost function

0

260

2

1

0

200

400

600

800

1000 time, s

1200

1400

1600

1800

0

200

400

600

800

1000 time, s

1200

1400

1600

1800

0

200

400

600

800

1000 time, s

1200

1400

1600

1800

0

240 -8

0

200

400

600

800

1000 time, s

1200

1400

1600

cost function

0 220

1800

1

x 10

-0.5

-1

0.6

additive noise on the sysem input and the state x 2

control error

0.8

0.4 0.2 0

2

3

10

10

2

0

-2

time, s (log)

Fig. 5. Simulation results in control and estimated control, output and control error signals (with noise on the control action and the state).

Fig. 7. Bar plots of the instant valuses of the control increment ∆u and the cost function J (with noise on the control action and the state).

On Fig. 3 and Fig. 6 are presented the transient responces of of the RMSE and RSE of the modeling errors for both cases. As can be seen, they have a smooth nature reaching values closer to zero, which guarntees the proper operation of the model providing correct estimates of the system dynamics. In the second case the modeling errors are slightly increased as expected. Despite of that reason the dynamics of the model is still unchanged.

When designing a predictive controller a careful consideration of the type and the structure of the model should be made in order to address many issues. Usually, it is required to be designed a modeling structures allowing capturing the process dynamics in fast and accurate manner, rejecting the influence of uncertain factors. On the other hand, the computational simplicity is crucial in order to save computational time in respect of system dynamics. A major problem for implementation of fuzzy-neural statespace controllers is the dimensionality of the input space. When it is large, the number of associated training parameters increases sharply and requires sometimes an extensive computational effort.

On Fig. 4 and Fig. 7 are demonstrated the bar plots of the instant values of the control increment ∆u and the cost function J. The calculated values for ∆u are within the admissible bound of the imposed the constraints, which guarantees that we move smoothly on the surface of the defined dynamic optimization problem without violating them. The defined cost function is successfully minimized at each sampling period reaching values closer to zero.

In Table 1, a comparative study between the proposed neofuzzy state-space controller and another previously proposed distributed state-space algorithm on the basis of the classical Takagi-Sugeno approach is shown (Todorov et al., 2015). As 103

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can be observed, the computational time for the proposed algorithm is twice less than the case of using the classical neuro-fuzzy approach. It should be mentioned also that, the chosen model structure affects not only the computational simplicity of the model but the computational abilities of the constructed predictor along the defined controller horizons. Thus, the chosen structure of the model impacts the operation of the controller at a whole.

Huang, B., Kadali, R., (2008). Dynamic Modeling, Predictive Control and Performance Monitoring, LNCIS, Springer, volume 374. Ismail, R., Jussoff, K., Ahmad, T., Ahmad R., (2009). Fuzzy State-Space Model of Multivariable Control Systems, Jornal of Computer and Information Science, volume 2(1), 19-25. Kalhor, A., Araabi B.N., Lucas, C., (2013), Evolving Takagi– Sugeno fuzzy model based on switching to neighboring models, Applied Soft Computing, volume 13, 939-946. Maciejowski, J., (2002). Predictive Control with constraints, Pearson. Osowski, S., (2000). Neural Networks for information processing, Oficyna Wydawnycza Policehniki Warzawsikej. Passino, K., Yourkovic S., (1998). Fuzzy Control, AdissonWesley. Schoen M., (1995). A Simulation model for primary drying phase of Freeze-drying, International Journal of Pharmaceutics, volume 114, 159-170. Terziyska, M., Doukovska L., Petrov, M., (2014). Implicit GPC Based on Semi Fuzzy Neural Network Model, In Proc. of 7th IEEE International Conference Intelligent Systems IS’2014, Poland, Mathematical Foundations Theory, Analyses, volume 1, 695-706. Todorov Y., Terziyska M., (2014). State-Space Fuzzy-Neural Network for Modeling of Nonlinear Dynamics, In Proc, of the Int. IEEE symposium Innovations in Intelligent Systems and Applications, INISTA'2014, Alberobello, Italy, 212-217. Todorov, Y., Terziyska, M., Doukovska, L., (2015), Distributed State-Space Predictive Control, In. Proceedings of the International IEEE Conference “Process Control’15”, Stribske Pleso, 31-36. Uchino, E., Yamakawa, T., (1994). Neo-Fuzzy-Neuron Based New Approach to System Modeling, with Application to Actual System, In Proc. of Sixth Int. Conf. on Tools with Artificial Intelligence, 564 – 570. Wan, F. (2011). Generation of Takagi-Sugeno fuzzy systems with minimum rules in modeling and identification, In Proc. of FUZIEEE, 1910– 1917. Wang, L., (2009). Model Predictive Control System Design and Implementation Using MATLAB, Springer, Verlag.

Table 1. Average processor time for algorithm execution Algorithm Neo-Fuzzy model Dist. TS model

Model [s] 0.0014 0.0026

Optimization [s] 0.0694 0.0885

5. CONCLUSIONS An approach for designing a simple Neo-fuzzy state-space predictive controller is presented. The proposed modeling procedure is realized as a set of fuzzy neurons, capturing the temporal dynamics of their own inputs while their outputs are used to represent a complex nonlinear behavior. The used state-space representation simplifies additionally the structure of the model and the controller, as well as the incorporated fuzzy predictor, while the implemented Quadratic Programming procedure assumes the process constraints. The achieved results show the potential applicability of the proposed controller in purpose to control of nonlinear systems. A major outcome of the proposed approach is the provided computational simplicity and reliable operation when noisy conditions occur. ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support provided by the Ministry of Education and Science of Bulgaria, Research Fund Project FNI I 02/6. As well, the research work reported in the paper is partly supported by the project AComIn "Advanced Computing for Innovation", grant 316087, funded by the FP7 Capacity Programme (Research Potential of Convergence Regions). REFERENCES Angelov, P., (2012). Autonomous Learning Systems: From Data Streams to Knowledge in Real-time, Wiley. Bodyanskiy, Y., Pliss, I., Vynokurova, O., (2013). Flexible Neo-fuzzy Neuron and Neuro-fuzzy Network for Monitoring Time Series Properties, Inf. Technology and Management Science, volume 16, 47-52. Bodyanskiy, Y., Kokshenev, I., Kolodyazhniy, V., (2005). An Adaptive Learning Algorithm for a Neo-Fuzzy Neuron, Proceedings of the 3rd Conference of the European Society for Fuzzy Logic and Technology, 375—379. Chadli, M., Borne, P., (2012), Multiple Models Approach in Automation: Takagi-Sugeno Fuzzy Systems, Wiley.

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