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Issue 2 (192), article 3

DOI:https://doi.org/10.15407/kvt192.02.044

Kibern. vyčisl. teh., 2018, Issue 2 (192), pp.

Zhiteckii L.S.,
PhD (Engineering),
Acting Head of the Department of Intelligent Automatic Systems
e-mail: leonid_zhiteckii@i.ua
Solovchuk K.Yu.,
PhD Student
e-mail: solovchuk_ok@ukr.net

International Research and Training Center for Information Technologies
and Systems of the National Academy of Science of Ukraine
and Ministry of Education and Sciences of Ukraine, Kiev, Ukraine,
Acad. Glushkova av., 40, Kiev, 03187, Ukraine

ADAPTIVE STABILIZATION OF SOME MULTIVARIABLE SYSTEMS WITH NONSQUARE GAIN MATRICES OF FULL RANK

Introduction. The paper states and solves a new problem concerning the adaptive stabilization of a specific class of linear multivariable discrete-time memoryless systems with nonsquare gain matrices at their equilibrium states. This class includes the multivariable systems in which the number of outputs exceeds the number of control inputs. It is assumed that the unknown gain matrices have full rank.
The purpose of this paper is to answer the question of how the pseudoinverse model-based adaptive approach might be utilized to deal with the uncertain multivariable memoryless system if the number of control inputs is less than the number of outputs.
Results. It is shown that the parameter estimates generated by the standard adaptive projection recursive procedure converge always to some finite values for any initial values of system’s parameters. Based on these ultimate features, it is proved that the adaptive pseudoinverse model-based control law makes it possible to achieve the equilibrium state of the nonsquare system to be controlled. The asymptotical properties of the adaptive feedback control system derived theoretically are substantiated by a simulation experiment.
Conclusion. It is established that the ultimate behavior of the closed-loop control system utilizing the adaptive pseudoinverse model-based concept is satisfactory.

Keywords: adaptive control, multivariable system, discrete time, feedback, pseudoinversion, stability, uncertainty.

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REFERENCES

1 Dahleh M.A., Pearson J.B. l1 optimal-feedback controllers for MIMO discrete-time systems. IEEE Trans. Autom. Contr., 1987, vol. 32, no. 4, pp. 314–322. https://doi.org/10.1109/TAC.1987.1104603

2 McDonald J.S., Pearson J.B. l1 optimal control of multivariable systems with output norm constraints. Automatica, 1991, vol. 27, no. 2, pp. 317–329. https://doi.org/10.1016/0005-1098(91)90080-L

3 Maciejowski J. M. Multivariable Feedback Design. Wokinghan: Addison-Wesley, 1989.

4 Skogestad S., Postlethwaite I. Multivariable Feedback Control. UK, Chichester: Wiley, 1996.

5 Albertos P., Sala A. Multivariable Control Systems: an Engineering Approach. London: Springer, 2006.

6 Francis B., Wonham W. The internal model principle of control theory. Automatica, 1976, vol. 12, no. 5, pp. 457–465. https://doi.org/10.1016/0005-1098(76)90006-6

7 Brockett R. W. The invertibility of dynamic systems with application to control. Ph. D. Dissertation, Case Inst. of Technology, Cleveland, Ohio, 1963.

8 Silverman L. M. Inversion of multivariable linear systems. IEEE Trans. Autom. Contr., 1969, vol. AC-14, no. 3, pp. 270–276. https://doi.org/10.1109/TAC.1969.1099169

9 Lovass-Nagy V., Miller J. R., Powers L. D. On the application of matrix generalized inversion to the construction of inverse systems. Int. J. Control, 1976, vol. 24, no. 5, pp. 733–739. https://doi.org/10.1080/00207177608932859

10 Seraji H. Minimal inversion, command tracking and disturbance decoupling in multivariable systems. Int. J. Control, 1989, vol. 49, no. 6, pp. 2093–2191. https://doi.org/10.1080/00207178908559765

11 Marro G., Prattichizzo D., Zattoni E. Convolution profiles for right-inversion of multivariable non-minimum phase discrete-time systems. Automatica, 2002, vol. 38, no. 10, pp. 1695–1703. https://doi.org/10.1016/S0005-1098(02)00088-2

12 Liu C., Peng H. Inverse-dynamics based state and disturbance observers for linear time-invariant systems. ASME J. Dyn Syst., Meas. and Control, 2002, vol. 124, no. 5, pp. 376–381.

13 Lyubchyk L. M. Disturbance rejection in linear discrete multivariable systems: inverse model approach. Prep. 18th IFAC World Congress, Milano, Italy, 2011, pp. 7921–7926.

14 Pushkov S. G. Inversion of linear systems on the basis of state space realization. Journal of Computer and Systems Sciences International, 2018, vol. 57, vo. 1, pp. 7–17.

15 Pukhov G. E., Zhuk K. D. Synthesis of Interconnected Control Systems via Inverse Operator Method. Kiev: Nauk. dumka, 1966 (in Russian).

16 Skurikhin V. I., Gritsenko V. I., Zhiteckii L. S., Solovchuk K. Yu. Generalized inverse operator method in the problem of optimal controlling linear interconnected static plants. Dopovidi NAN Ukrainy, no. 8, pp. 57–66, 2014 (in Russian).

17 Zhiteckii L. S., Azarskov V. N., Solovchuk K. Yu., Sushchenko O. A. Discrete-time robust steady-state control of nonlinear multivariable systems: a unified approach. Proc. 19th IFAC World Congress, Cape Town, South Africa, 2014, pp. 8140–8145.

18 Zhitetskii L. S., Skurikhin V. I., Solovchuk K. Yu. Stabilization of a nonlinear multivariable discrete-time time-invariant plant with uncertainty on a linear pseudoinverse model. Journal of Computer and Systems Sciences International, 2017, vol. 56, no. 5, pp. 759–773. https://doi.org/10.1134/S1064230717040189

19 Zhiteckii L. S., Solovchuk K. Yu. Pseudoinversion in the problems of robust stabilizing multivariable discrete-time control systems of linear and nonlinear static objects under bounded disturbances. Journal of Automation and Information Sciences, 2017, vol. 49, no. 5, pp. 35–48. https://doi.org/10.1615/JAutomatInfScien.v49.i5.30

20 Fomin V. N., Fradkov A. L., Yakubovich V. A. Adaptive Control of Dynamic Plants. Moscow: Nauka, 1981 (in Russian).

21 Goodwin G.C., Sin K.S. Adaptive Filtering, Prediction and Control. Engewood Cliffs, NJ.: Prentice-Hall, 1984.

22 Landau I. D., Lozano R., M’Saad M. Adaptive Control. London: Springer, 1997.

23 Kuntsevich V. M. Control under Uncertainty: Guaranteed Results in Control and Identification Problems. Kiev: Nauk. dumka, 2006 (in Russian).

24 Zhiteckii L. S., Skurikhin V. I. Adaptive Control Systems with Parametric and Nonparametric Uncertainties. Kiev: Nauk. dumka, 2010 (in Russian).

25 Narendra K. S., Annaswamy A. M. Stable Adaptive Systems. NY: Dover Publications, 2012.

26 Ioannou P., Sun J. Robust Adaptive Control. NY: Dover Publications, 2013.

27 Astrom K. J., Wittenmark B. Adaptive Control: 2nd Edition. NY: Dover Publications, 2014.

28 Albert A. Regression and the Moore-Penrose Pseudoinverse. New York: Academic Press, 1972.

29 Kaczmarz S. Approximate solution of systems of linear equations. Internat. J. Control, 1993. vol. 57, no. 6. pp. 1269–1271. https://doi.org/10.1080/00207179308934446

30 Marcus M., Minc H. A Survey of Matrix Theory and Matrix Inequalities. Boston: Aliyn and Bacon, 1964.

31 Desoer C.A., Vidyasagar M. Feedback Systems: Input–Output Properties. New York: Elsevier, 1975.

Received 29.03.2018

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Issue 3 (189), article 2

DOI:https://doi.org/10.15407/kvt189.03.029

Kibern. vyčisl. teh., 2017, Issue 3 (189), pp.

Zhiteckii L.S., PhD (Engineering), Acting Head of the Department of Intelligent Automatic Systems
e-mail: leonid_zhiteckii@i.ua
Solovchuk K.Yu., Postgraduate Student
e-mail: solovchuk_ok@mail.ru
International Research and Training Center for Information Technologies and Systems of the NAS of Ukraine and Ministry of Education and Science of Ukraine,
Acad. Glushkova av., 40, Kiev, 03680, Ukraine

DISCRETE-TIME STEADY-STATE CONTROL OF INTERCONNECTED SYSTEMS BASED ON PSEUDOINVERSION CONCEPT

Introduction. The problem of controlling interconnected systems subjected to arbitrary unmeasurable disturbances remains actual up to now. It is important problem from both theoretical and practical points of view. During the last decades, the internal model control principle becomes popular among other methods dealing with an improvement of the control system. A perspective modification of the internal model control principle is the so-called model inverse approach. Unfortunately, the inverse model approach is quite unacceptable if the systems to be controlled are square but singular or if they are nonsquare. It turned out that the so-called pseudoinverse (generalized inverse) model approach can be exploited to cope with the noninevitability of singular square and also nonsquare system.
The purpose of the paper is to generalize the results obtained by the authors in their last works which are related to the asymptotic properties of the pseudoinverse model-based method for designing an efficient steady-state control of interconnected systems with uncertainties and arbitrary bounded disturbances and also to present some new results.
Results. In this paper, the main effort is focused on analyzing the asymptotic properties of the closed-loop systems containing the pseudoinverse model-based controllers. In the framework of the pseudoinversion concept, new theoretical results related to the asymptotic behavior of these systems are obtained. Namely, in the case of nonsingular gain matrices with known elements, the upper bounds on the ultimate norms of output and control input vectors are found. Next, in the case of nonsquare gain matrices whose elements are also known, the asymptotic behavior of the feedback control systems designed on the basis of pseudoinverse approach are studied. Further, the sufficient conditions guaranteeing the boundedness of the output and control input signals for the linear and certain class of nonlinear interconnected systems in the presence of uncertainties are derived.
Conclusion. It has been established that the pseudoinverse model-based concept can be used as a unified concept to deal with the steady-state regulation of the linear interconnected discrete-time systems and of some classes of nonlinear interconnected systems with possible uncertainties in the presence of arbitrary unmeasured but bounded disturbances.
Keywords: discrete time, feedback, pseudoinversion, interconnected systems, optimality, stability, uncertainty.

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REFERENCES

1 Davison E. The output control of linear time-invariant multivariable systems with unmeasurable arbitrary disturbances. IEEE Trans. Autom. Contr., 1972, vol. AC-17, no. 5, pp. 621–631.
https://doi.org/10.1109/TAC.1972.1100084

2 Liu C., Peng H. Inverse-dynamics based state and disturbance observers for linear time-invariant systems. ASME J. Dyn Syst., Meas. and Control, 2002, vol. 124, no. 5, pp. 376–381.

3 Lyubchyk L. M. Disturbance rejection in linear discrete Multivariable systems: inverse model approach. Prep. 18th IFAC World Congress, Milano, Italy, 2011, pp. 7921–7926.
https://doi.org/10.3182/20110828-6-IT-1002.02121

4 Skogestad S., Postlethwaite I. Multivariable Feedback Control. UK, Chichester: Wiley, 1996.

5 Freudenberg J. and Middleton R. Properties of single input, two output feedback systems. Int. J. Control, 1999, vol. 72, no. 16, pp. 1446–1465.
https://doi.org/10.1080/002071799220100

6 Francis B., Wonham W. The internal model principle of control theory. Automatica, 1976, vol. 12, no. 5, pp. 457–465.
https://doi.org/10.1016/0005-1098(76)90006-6

7 Brockett R. W. The invertibility of dynamic systems with application to control. Ph. D. Dissertation, Case Inst. of Technology, Cleveland, Ohio, 1963.

8 Sain M. K., Massey J. L. Invertibility of linear time-invariant dynamical systems. IEEE Trans. Autom. Contr.,1969, vol. AC-14, no. 2, pp. 141–149, Apr. 1969.

9 Silverman L. M. Inversion of multivariable linear systems. IEEE Trans. Autom. Contr., 1969, vol. AC-14, no. 3, pp. 270–276.
https://doi.org/10.1109/TAC.1969.1099169

10 Lovass-Nagy V., Miller J. R., Powers L. D. On the application of matrix generalized inversion to the construction of inverse systems. Int. J. Control, 1976, vol. 24, no. 5, pp. 733–739.
https://doi.org/10.1080/00207177608932859

11 Seraji H. Minimal inversion, command tracking and disturbance decoupling in multivariable systems. Int. J. Control, 1089, vol. 49, no. 6, pp. 2093–2191.

12 Marro G., Prattichizzo D., Zattoni E. Convolution profiles for right-inversion of multivariable non-minimum phase discrete-time systems. Automatica, 2002, vol. 38, no. 10, pp. 1695–1703.
https://doi.org/10.1016/S0005-1098(02)00088-2

13 Pukhov G. E., Zhuk K. D. Synthesis of Interconnected Control Systems via Inverse Operator Method. Kiev: Nauk. dumka, 1966 (in Russian).

14 Lee T., Adams G., Gaines W. Computer Process Control: Modeling and Optimization. New York: Wiley, 1968.

15 Skurikhin V. I., Procenko N. M., Zhiteckii L. S. Multiple-connected systems of technological processes control with table of objects. Proc. IFAC Third Multivariable Tech. Systems Symp., Manchester, U.K., 1974, pp. S 35-1 – S 35-4.
https://doi.org/10.1016/S1474-6670(17)69194-8

16 Katkovnik V. Ya., Pervozvansky A. A. Methods for the search of extremum and the synthesis problems of multivariable control systems. Adaptivnye Avtomaticheskie Sistemy, Moscow: Sov. Radio, pp. 17–42, 1973 (in Russian).

17 Skogestad S., Morari M., Doyle J. Robust control of ill-conditioned plants: high purity distillation. IEEE Trans. Autom. Contr., 1988, vol. 33, no. 12, pp. 1092–1105.
https://doi.org/10.1109/9.14431

18 Skurikhin V. I., Zhiteckii L. S., Solovchuk K. Yu. Control of interconnected plants with singular and ill-conditioned transfer matrices based on pseudo-inverse operator method. Upravlyayushchye sistemy i mashiny, 2013, no. 3, pp. 14-20, 29 (in Russian).

19 Zhiteckii L. S., Azarskov V. N., Solovchuk K. Yu., Sushchenko O. A. Discrete-time robust steady-state control of nonlinear multivariable systems: a unified approach. Proc. 19th IFAC World Congress, Cape Town, South Africa, 2014, pp. 8140–8145.
https://doi.org/10.3182/20140824-6-ZA-1003.01985

20 Skurikhin V. I., Gritsenko V. I., Zhiteckii L. S., Solovchuk K. Yu. Generalized inverse operator method in the problem of optimal controlling linear interconnected static plants. Dopovidi NAN Ukrainy, no. 8, pp. 57–66, 2014 (in Russian).

21 Albert A. Regression and the Moore-Penrose Pseudoinverse. New York: Academic Press, 1972.

22 Zhiteckii L. S., Skurikhin V. I. Adaptive Control Systems with Parametric and Nonparametric Uncertainties. Kiev: Nauk. dumka, 2010 (in Russian).

23 Lancaster P., Tismenetsky M. The Theory of Matrices: 2nd ed. With Applications. N.Y.: Academic Press, 1985.

Received 17.02.2017

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Issue 185, article 3

DOI:https://doi.org/10.15407/kvt185.03.021

KVT, 2016, Issue 185, pp.21-34

UDC 681.5

l1-OPTIMIZATION APPROACH TO DESIGN OF DIGITAL AUTOPILOTS FOR LATERAL MOTION CONTROL OF AN AIRCRAFT

Zhiteckii L.S., Pilchevsky A.Yu., Solovchuk K.Yu.

International Research and Training Center for Information Technologies and Systems of the National Academy of Science of Ukraine and Ministry of Education and Sciences of Ukraine, Kiev, Ukraine

leonid_zhiteckii@i.ua , terosjj@gmail.com , solovchuk_ok@mail.ru

Introduction. The optimal digital autopilot needed to control of the roll for an aircraft in the presence of an arbitrary unmeasured disturbances is addressed in this paper. This autopilot has to achieve a desired lateral motion control via minimizing the upper bound on the absolute value of the difference between the given and true roll angles. It is ensured by means of the two digital controllers. The inner controller is designed as the discrete-time PI controller in order to stabilize a given roll rate. This variable is formed by the external discrete-time P controller. To optimize this control system, the controller parameters are derived utilizing the so-called l1-optimization approach advanced in modern control theory. The motion parameters are assumed to be known.

The purpose of the paper is to synthesize a digital autopilot which is able to maintain a given roll orientation of an aircraft with a desired accuracy and to cope with an arbitrary external disturbance (a gust) whose bounds may be unknown.

Results. The necessary and sufficient conditions guaranteeing the stability of the two-circuit feedback discrete-time control system are established. First, the l1-optimal PI and P controller parameters are calculated simultaneously (in contrast with [14]). Second, the aileron servo dynamics are taken into account to establish the stability condition for optimizing the controller parameters. Third, random search algorithm is used to calculate the three optimal values of the autopilot parameters. To support the theoretical results obtained, in this work, several simulation experiments were conducted. We have established that the simultaneous l1-optimization of both controllers was more efficient than the sequential l1-optimization of inner and external controllers.

Conclusion. It was established that the two-circuit l1-optimal PI and P control laws can cope with the wind gust and ensure the desired roll orientation. This makes it possible to achieve the control objective which was stated. A distinguishing feature of the control algorithms is that they are sufficiently simple. This is important from the practical point of view.

Keywords: aircraft, lateral dynamics, digital control system, discrete time, stability, l1-optimization, random search algorithm.

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References:

1 Stevens B.L., Lewis F.L. Aircraft Control and Simulation, 2nd ed. New York: John Willey & Sons, 2003, 680 p.

2 William D.E., Friedland B, Madiwale A.N. Modern conrtol theory for design of autopilots for bank-to-turn missiles. J. Guidance Control, 1987, vol. 10, pp. 378–386. https://doi.org/10.2514/3.20228

3 Teoh E.K., Mital D.P., Ang K.S. A BTT CLOS autopilot design. The EEE Journal, 1992, vol. 4, pp. 1–7.

4 Ang K.S., Teoh E.K., Mital D.P. Adaptive control of a missile autopilot system. Proc. 12th IFAC World Congress, 1993, vol. 1, pp. 293–296.

5 Malaek S.M.B., Izadi H., Pakmehr M. Intelligent Autolanding Controller Based on Neural Networks. Proc. 1st African Control Conference (AFCON2003), Cape Town, South Africa, 2003, vol. 1, pp. 113–119.

6 Khrosravani M.R. Apllication of Neural Network on Flight Control. Int. Journal of Machine Learning and Computing, 2012, vol. 6, pp. 882–885. https://doi.org/10.7763/IJMLC.2012.V2.258

7 Lavretsky E., Wise K. A. Robust and Adaptive Control with Aerospace Aplication. London: Springer-Verlag. 2013, 454 p. https://doi.org/10.1007/978-1-4471-4396-3

8 Astrom K.J., Wittenmark B. Computer Controlled Systems. Theory and Design, 2nd ed. N.J.: Prentice Hall, Englewood Cliffs, 1990, 555 p.

9 Goodwin G.C., Graebe S.F., Salgado M.E. Control Systems Design. N.J.: Prentice Hall, 2001, 908 p.

10 Yuz J.I., Goodwin G.C. Sampled-Data Models for Linear and Nonlinear System. London: Springer-Verlag, 2014, 289 p. https://doi.org/10.1007/978-1-4471-5562-1

11 Dahleh M.A., Pearson J.B. l1-optimal feedback controllers for discrete-time systems. Proc. American Control Conference, Seattle, WA, 1986, pp. 1964–1968.

12 Vidyasagar M. Optimal rejection of persistent bounded disturbances. IEEE Trans. on Autom. Control, 1986, vol. 31, pp. 527–517.

13 Khammash M.H. A new approach to the solution of the l1 control problem: the scaled-Q method. IEEE Trans. on Autom. Control, 2000, vol. 45, pp. 180–187.

14 Melnyk K.V., Zhiteckii L.S., Bogatyrov A.M., Pilchevsky A.Yu. Digital control of lateral autopilot system applied to an UAV: optimal control strategy. Proc. 2013 2nd IEEE Int. Conf. “Actual Problems of Unmanned Air Vehicles Developments”, Oct., 15-17, Kiev, Ukraine, 2013, pp. 189–192. https://doi.org/10.1109/APUAVD.2013.6705322

15 Blakelock J.H. Automatic Control of Aircraft and Missiles, 2nd ed. New York: John Wiley & Sons, Inc., 1991, 672 p.

16 Tou J.T. Digital and Sampled-Data Control Systems. New-York: McGraw-Hill Book Company, 1959, 631 p.

17 Jury E.I. Sampled-Data Control Systems. New York: John Willey & Sons Inc., 1958, 332 p.

18 Polyak B.T., Shcherbakov P.S. Robust Stability and Control. Moscow: Nauka, 2002, 303 p. (in Russian)

19 Polyak B.T. Introduction to Optimization. New-York: Optimization Software Inc., 1987, 438 p.

Received 01.06.16

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ISSUE 181, article 4

DOI:https://doi.org/10.15407/kvt181.01.043

Kibern. vyčisl. teh., 2015, Issue 181, pp.

Zhiteckii L.S., Nikolaienko S.A., Solovchuk K.Yu.

International Research and Training Center for Information Technologies and Systems of the National Academy of Science of Ukraine and Ministry of Education and Sciences of Ukraine, Kiev, Ukraine

ADAPTATION AND LEARNING IN SOME CLASSES OF IDENTIFICATION AND CONTROL SYSTEMS

Introduction. The paper deals with studying the asymptotical properties of the standard discrete-time gradient online learning algorithm in the two-layer neural network model of the uncertain nonlinear system to be identified. Also, the design of the discrete-time adaptive closed-loop system containing the linear multivariable memoryless plant with possibly singular but unknown matrix gain in the presence of unmeasurable bounded disturbances having the unknown bounds are addressed in this paper. It is assumed that the learning process in the neural network model is implemented in the stochastic environment whereas the adaptation of the plant model in the control system is based on the non-stochastic description of the external environment.
The purpose of the paper is to establish the global convergence conditions of the gradient online learning algorithm in the neural network model by utilizing the probabilistic asymptotic analysis and to derive the convergent adaptive control algorithm guaranteeing the boundedness of the signals in the closed-loop system which contains the multivariable memoryless plant with an arbitrary matrix gain in the presence of unmeasurable disturbances whose bounds are unknown.
Results. The Lyapunov function approach as the suitable tool for analyzing the asymptotic behavior both of the gradient learning algorithm in the neural network identification systems and of the adaptive gradient algorithm in the certain closed-loop control systems is utilized. Within this approach, the two groups of global sufficient conditions guaranteeing the convergence of the online gradient learning algorithm in neural network model with probability 1 are obtained. The first group of these conditions defines the requirements under which this algorithm will be convergent almost sure with a constant learning rate. Such an asymptotic property holds in the ideal case where the nonlinearity to be identified can exactly be described by a neural network model. The second group of convergence conditions shows that this property can also be achieved in non-ideal case. It turns out that adding a penalty term to the current error function is indeed not necessary to guarantee this property. It is established that in a worst case where the matrix gain of multivariable plant is unknown and may be singular, and the bounds on the arbitrary unmeasurable disturbances remain unknown, the convergence of the gradient adaptation algorithm and the boundedness of all signals in the adaptive closed-loop system can be ensured.
Conclusions. In order to guarantee the global convergence of the online learning algorithm in the neural network identification system with probability 1, the certain conditions should be satisfied. Also the boundedness of all signals in the closed-loop adaptive control system containing the multivariable memoryless plant whose matrix gain is unknown and possibly singular can be achieved even if the bounds on the unmeasurable disturbances are unknown.

Keywords: neural network, gradient learning algorithm, convergence, multivariable memoryless plant, adaptive control algorithm, boundedness of the signals.

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References

1 Tsypkin Ya.Z. Adaptation and Learning in Automatic Systems. N.Y.: Academic Press, 1971.

2 Tsypkin Ya.Z. Foundation of the Theory of Learning Systems. N.Y.: Academic Press, 1973.

3 Kuntsevich V.M. Control under Uncertainty Conditions: Guaranteed Results in Control and Identification Problems. Kiev: Nauk. dumka, 2006. (in Russian).

4 Zhiteckii L.S. and Skurikhin V.I. Adaptive Control Systems with Parametric and Nonparametric Uncertainties. Kiev: Nauk. dumka, 2010. (in Russian).

5 Suykens J. and Moor B.D. Nonlinear system identification using multilayer neural networks: some ideas for initial weights, number of hidden neurons and error criteria. In Proc. 12nd IFAC World Congress, 1993, vol. 3, pp. 49–52. https://doi.org/10.1016/S1474-6670(17)48485-0

6 Kosmatopoulos E.S., Polycarpou M.M., Christodoulou M.A. and Ioannou P.A. High-order neural network structures for identification of dynamical systems. IEEE Trans. on Neural Networks, 1995, vol. 6, pp. 422–431. https://doi.org/10.1109/72.363477

7 Levin A.U. and Narendra K.S. Recursive identification using feedforward neural networks. Int. J. Control, 1995, vol. 61, pp. 533–547. https://doi.org/10.1080/00207179508921916

8 Tsypkin Ya.Z., Mason J.D., Avedyan E.D., Warwick K. and Levin I.K. Neural networks for identification of nonlinear systems under random piecewise polynomial disturbances. IEEE Trans. on Neural Networks, 1999, vol. 10, pp. 303–311. https://doi.org/10.1109/72.750559

9 Behera L., Kumar S., and Patnaik A. On adaptive learning rate that guarantees convergence in feedforward networks. IEEE Trans. on Neural Networks, 2006, vol. 17, pp. 1116–1125. https://doi.org/10.1109/TNN.2006.878121

10 White H. Some asymptotic results for learning in single hidden-layer neural network models. J. Amer. Statist. Assoc., 1987, vol. 84, pp. 117–134.

11 Kuan C M. and Hornik K. Convergence of learning algorithms with constant learning rates. IEEE Trans. on Neural Networks, 1991, vol. 2, pp. 484 – 489. https://doi.org/10.1109/72.134285

12 Luo Z. On the convergence of the LMS algorithm with adaptive learning rate for linear feedforward networks. Neural Comput., 1991, vol. 3, pp. 226–245. https://doi.org/10.1162/neco.1991.3.2.226

13 Finnoff W. Diffusion approximations for the constant learning rate backpropagation algorithm and resistance to local minima. Neural Comput., 1994, vol. 6, pp. 285– 295. https://doi.org/10.1162/neco.1994.6.2.285

14 Gaivoronski A.A. Convergence properties of backpropagation for neural nets via theory of stochastic gradient methods. Optim. Methods Software, 1994, vol. 4, pp. 117–134. https://doi.org/10.1080/10556789408805582

15 Fine T.L. and Mukherjee S. Parameter convergence and learning curves for neural networks. Neural Comput., 1999, vol. 11, pp. 749–769. https://doi.org/10.1162/089976699300016647

16 Tadic V. and Stankovic S. Learning in neural networks by normalized stochastic gradient algorithm: Local convergence. In Proc. 5th Seminar Neural Netw. Appl. Electr. Eng., 2000, pp. 11–17. https://doi.org/10.1109/NEUREL.2000.902375

17 Zhang H., Wu W., Liu F. and Yao M. Boundedness and convergence of online gradient method with penalty for feedforward neural networks. IEEE Trans. on Neural Networks, 2009, vol. 20, pp. 1050–1054. https://doi.org/10.1109/TNN.2009.2020848

18 Mangasarian O.L. and Solodov M.V. Serial and parallel backpropagation convergence via nonmonotone perturbed minimization. Optim. Methods Software, 1994, pp. 103–106. https://doi.org/10.1080/10556789408805581

19 Wu W., Feng G. and Li X. Training multilayer perceptrons via minimization of ridge functions. Advances in Comput. Mathematics, vol. 17, pp. 331–347, 2002. https://doi.org/10.1023/A:1016249727555

20 Zhang N., Wu W. and Zheng G. Convergence of gradient method with momentum for two-layer feedforward neural networks. IEEE Trans. on Neural Networks, 2006, vol. 17, pp. 522–525. https://doi.org/10.1109/TNN.2005.863460

21 Wu W., Feng G., Li X and Xu Y. Deterministic convergence of an online gradient method for BP neural networks. IEEE Trans. on Neural Networks, 2005, vol. 16, pp. 1–9. https://doi.org/10.1109/TNN.2005.844903

22 Xu Z.B., Zhang R. and Jing W.F. When does online BP training converge? IEEE Trans. on Neural Networks, 2009, vol. 20, pp. 1529–1539. https://doi.org/10.1109/TNN.2009.2025946

23 Shao H., Wu W. and Liu L. Convergence and monotonicity of an online gradient method with penalty for neural networks. WSEAS Trans. Math., 2007, vol. 6, pp. 469–476.

24 Ellacott S.W. The numerical analysis approach. In Mathematical Approaches to Neural Networks (Taylor J.G. ed; B.V.: Elsevier Science Publisher), 1993, pp. 103–137. https://doi.org/10.1016/S0924-6509(08)70036-9

25 Skantze F.P., Kojic A., Loh A.P. and Annaswamy A.M. Adaptive estimation of discrete time systems with nonlinear parameterization. Automatica, 2000, vol. 36, pp. 1879–1887. https://doi.org/10.5555/S0005-1098(00)00106-0 https://doi.org/10.1016/S0005-1098(00)00106-0

26 Loeve M. Probability Theory. N.Y.: Springer-Verlag, 1963.

27 Zhiteckii L.S., Azarskov V.N. and Nikolaienko S.A. Convergence of learning algorithms in neural networks for adaptive identification of nonlinearly parameterized systems. In Proc. 16th IFAC Symposium on System Identification, 2012, pp. 1593–1598. https://doi.org/10.3182/20120711-3-BE-2027.00150

28 Skurikhin V.I., Gritsenko V.I., Zhiteckii L.S. and Solovchuk K.Yu. Generalized inverse operator method in the problem of optimal controlling linear interconnected static plants. Dopovidi Natsionalnoi Akademii Nauk Ukrainy, 2014, no. 8, pp. 57–66. (in Russian).

29 Fomin V.N., Fradkov A.L. and Yakubovich V.A. Adaptive Control of Dynamic Systems. Moscow: Nauka, 1981. (in Russian).

30 Goodwin G.C. and Sin K.S. Adaptive Filtering, Prediction and Control. Engewood Cliffs. NJ.: Prentice-Hall, 1984.

31 Azarskov V.N., Zhiteckii L.S. and Solovchuk K.Yu. Adaptive robust control of multivariable static plants with possibly singular transfer matrix. Electronics and Control Systems, 2013, no. 4, pp. 47–53.

32 Polyak B.T.Convergence and convergence rate of iterative stochastic algorithms, I: General case. Autom. Remote Control, 1976, vol. 12, pp. 1858–1868.

33 Marcus M. and Minc H. A Survey of Matrix Theory and Matrix Inequalities. Boston: Allyn & Bacon Inc. 1964.

Received 06.07.2015