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doi: 10.3934/naco.2021010
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Second order discrete time-varying and time-invariant linear continuous systems and Kalman type conditions

Elimhan N. Mahmudov 1,2,

1. 

Department of Engineering Mathematics, Istanbul Technical University, Istanbul, Turkey
2. 

Azerbaijan National Academy of Sciences Institute of Control Systems, Baku, Azerbaijan

Received  November 2020 Revised  February 2021 Early access March 2021

Fund Project: This paper is dedicated to the memory of Professor Lotfi A. Zadeh (1921-2017), Founder of Fuzzy logic and Fuzzy Mathematics

The paper deals with the controllability and observability of second order discrete linear time varying and linear time-invariant continuous systems in matrix form. To this case, we generalize the classical conditions for linear systems of the first order, without reducing them to systems of the first order. Within the framework of Kalman-type criteria, we investigate these concepts for second-order linear systems with discrete / continuous time; we define the initial values and input functions uniquely if and only if the observability and controllability matrices have full rank, respectively. Also a conceptual partner of controllability, that is, reachability of second order discrete time-varying systems is formulated and a necessary and sufficient condition for complete reachability is derived. Also the transfer function of the second order continuous-time linear state-space system is constructed. We have given numerical examples to illustrate the feasibility and effectiveness of the theoretical results obtained.

Keywords: Observability, controllability, second order, Kalman criterion, reachability.
Mathematics Subject Classification: Primary: 11D04, 39A70; Secondary: 93B05, 93B07.
Citation: Elimhan N. Mahmudov. Second order discrete time-varying and time-invariant linear continuous systems and Kalman type conditions. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021010
References:
[1]

S. Avdonin, J. Park and L. de Teresa, The Kalman condition for the boundary controllability of coupled 1-d wave equations, Evol. Equat. Contr. Theory, 9 (2020), 255-273.  doi: 10.3934/eect.2020005.  Google Scholar

[2]

Z. Benzaid, On the constrained controllability of linear time-varying discrete systems, IEEE Trans. Autom. Contr., 44 (1999), 608-612.  doi: 10.1109/9.751361.  Google Scholar

[3]

M. G. Frost, Controllability, observability and the transfer function matrix for a delay-differential system, Inter. Journ. Contr., 35 (1982), 175-182.  doi: 10.1080/00207178208922610.  Google Scholar

[4]

F. Gao, W. Liu, V. Sreeram and K. L. Teo, Characterization and selection of global optimal output feedback gains for linear timeinvariant systems, Optim. Contr. Appl. Methods, 21 (2000), 195-209.  doi: 10.1002/1099-1514(200009/10)21:5<195::AID-OCA673>3.0.CO;2-D.  Google Scholar

[5]

V. Y. Glizer, Novel conditions of Euclidean space controllability for singularly perturbed systems with input delay, Numer. Algebra, Contr. Optim., 11 (2020), 307-320.  doi: 10.3934/naco.2020027.  Google Scholar

[6]

A. Hamidolu and E. N. Mahmudov, On construction of sampling patterns for preserving observability/controllability of linear sampled-data systems, Inter J. Control., 2020. doi: 10.1080/00207179.2020.1787523.  Google Scholar

[7]

R. E. Kalman, On the general theory of control systems, IRE Trans. Automat Contr., 4 (1959), 110-110.   Google Scholar

[8]

H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, Wiley Interscience, 1972.  Google Scholar

[9]

F. L. Lewis and K.M. Przyluski, Hautus-type conditions for controllability of implicit linear discrete-time systems revisited, IEEE Trans. Automat Contr., 38 (1993), 502-505.  doi: 10.1109/9.210157.  Google Scholar

[10]

C. Li, F. Ma and T. Huang, 2-D Analysis based iterative learning control for linear discrete-time systems with time delay, J. Indust. Manag. Optim., 7 (2011), 175-181.  doi: 10.3934/jimo.2011.7.175.  Google Scholar

[11]

C. Liu and C. Li, Reachability and observability of switched linear systems with continuous-time and discrete-time subsystems, Int. Journ. Contr, Automat. Syst., 11 (2013), 200-205.  doi: 10.1049/iet-cta.2011.0317.  Google Scholar

[12]

E. N. Mahmudov, Approximation and Optimization of Discrete and Differential Inclusions, Boston, USA: Elsevier, 2011. doi: 10.1016/B978-0-12-388428-2.00001-1.  Google Scholar

[13]

E. N. Mahmudov, Optimization of Mayer problem with Sturm-Liouville-Type differential inclusions, J. Optim. Theory Appl., 177 (2018), 345-375.  doi: 10.1007/s10957-018-1260-2.  Google Scholar

[14]

E. N. Mahmudov, Approximation and Optimization of higher order discrete and differential inclusions, Nonlin. Diff. Equat. Appl., 21 (2014), 1-26.  doi: 10.1007/s00030-013-0234-1.  Google Scholar

[15]

E. N. Mahmudov, Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints, J. Indust. Manag. Optim., 16 (2020), 169-187.  doi: 10.3934/jimo.2018145.  Google Scholar

[16]

E. N. Mahmudov, Optimal control of second order delay-discrete and delay differential inclusions with state constraints, Evol. Equat. Contr. Theory, 7 (2018), 501-529.  doi: 10.3934/eect.2018024.  Google Scholar

[17]

E. N. Mahmudov, Mathematical programming and polyhedral optimization of second order discrete and differential inclusions, Pacific J. Optim., 11 (2015), 511-525.   Google Scholar

[18]

E. N. Mahmudov, Optimal control of higher order differential inclusions with functional constraints, ESAIM: Control, Optim Calculus Variat., 26 (2020), 1-23.  doi: 10.1051/cocv/2019018.  Google Scholar

[19]

E. N. Mahmudov, Single Variable Differential and Integral Calculus, Mathematical Analysis. Paris, France, Springer, 2013. doi: 10.2991/978-94-91216-86-2.  Google Scholar

[20]

M. Paskota, V. Sreeram, K. L. Teo and A. I. Mees, Optimal simultaneous stabilization of linear single-input systems via linear state feedback control, Int. Journ. Contr., 60 (1994), 483-498.  doi: 10.1080/00207179408921477.  Google Scholar

[21]

K. Ravikumar, M. T. Mohan and A. Anguraj, Approximate controllability of a non-autonomous evolution equation in Banach spaces, Numer. Algebra, Contr. Optim., 2020. doi: 10.3934/naco.2020038.  Google Scholar

[22]

L. M. Silverman and H. E. Meadows, Controllability and Observability in time-variable linear systems, SIAM J. Contr. Optim., 5 (1967), 64-73.   Google Scholar

[23]

W. S. W. Wang, D. E. Davison and E. J. Davison, Controller design for multivariable linear time-invariant unknown systems, IEEE Trans. Automat Contr., 58 (2013) 2292–2306. doi: 10.1109/TAC.2013.2258812.  Google Scholar

[24]

L. A. Zadeh and C. A. Desoer, Linear System Theory: The State Space Approach, McGraw-Hill Series in System Science, New York: McGraw-Hill, 1963.  Google Scholar

[25]

X. Zhao, L. Zhang and P. Shi, Stability of a class of switched positive linear time-delay systems, Inter. J. Robust Nonlin. Contr., 23 (2013), 578-589.  doi: 10.1002/rnc.2777.  Google Scholar

[26]

X. L. Zhu, H. Yang, Y. Y. Wang and Y. L. Wang, New stability criterion for linear switched systems with time-varying delay, Inter. J. Robust. Nonlin. Contr., 24 (2014), 214-227.  doi: 10.1002/rnc.2882.  Google Scholar

show all references

References:
[1]

S. Avdonin, J. Park and L. de Teresa, The Kalman condition for the boundary controllability of coupled 1-d wave equations, Evol. Equat. Contr. Theory, 9 (2020), 255-273.  doi: 10.3934/eect.2020005.  Google Scholar

[2]

Z. Benzaid, On the constrained controllability of linear time-varying discrete systems, IEEE Trans. Autom. Contr., 44 (1999), 608-612.  doi: 10.1109/9.751361.  Google Scholar

[3]

M. G. Frost, Controllability, observability and the transfer function matrix for a delay-differential system, Inter. Journ. Contr., 35 (1982), 175-182.  doi: 10.1080/00207178208922610.  Google Scholar

[4]

F. Gao, W. Liu, V. Sreeram and K. L. Teo, Characterization and selection of global optimal output feedback gains for linear timeinvariant systems, Optim. Contr. Appl. Methods, 21 (2000), 195-209.  doi: 10.1002/1099-1514(200009/10)21:5<195::AID-OCA673>3.0.CO;2-D.  Google Scholar

[5]

V. Y. Glizer, Novel conditions of Euclidean space controllability for singularly perturbed systems with input delay, Numer. Algebra, Contr. Optim., 11 (2020), 307-320.  doi: 10.3934/naco.2020027.  Google Scholar

[6]

A. Hamidolu and E. N. Mahmudov, On construction of sampling patterns for preserving observability/controllability of linear sampled-data systems, Inter J. Control., 2020. doi: 10.1080/00207179.2020.1787523.  Google Scholar

[7]

R. E. Kalman, On the general theory of control systems, IRE Trans. Automat Contr., 4 (1959), 110-110.   Google Scholar

[8]

H. Kwakernaak and R. Sivan, Linear Optimal Control Systems, Wiley Interscience, 1972.  Google Scholar

[9]

F. L. Lewis and K.M. Przyluski, Hautus-type conditions for controllability of implicit linear discrete-time systems revisited, IEEE Trans. Automat Contr., 38 (1993), 502-505.  doi: 10.1109/9.210157.  Google Scholar

[10]

C. Li, F. Ma and T. Huang, 2-D Analysis based iterative learning control for linear discrete-time systems with time delay, J. Indust. Manag. Optim., 7 (2011), 175-181.  doi: 10.3934/jimo.2011.7.175.  Google Scholar

[11]

C. Liu and C. Li, Reachability and observability of switched linear systems with continuous-time and discrete-time subsystems, Int. Journ. Contr, Automat. Syst., 11 (2013), 200-205.  doi: 10.1049/iet-cta.2011.0317.  Google Scholar

[12]

E. N. Mahmudov, Approximation and Optimization of Discrete and Differential Inclusions, Boston, USA: Elsevier, 2011. doi: 10.1016/B978-0-12-388428-2.00001-1.  Google Scholar

[13]

E. N. Mahmudov, Optimization of Mayer problem with Sturm-Liouville-Type differential inclusions, J. Optim. Theory Appl., 177 (2018), 345-375.  doi: 10.1007/s10957-018-1260-2.  Google Scholar

[14]

E. N. Mahmudov, Approximation and Optimization of higher order discrete and differential inclusions, Nonlin. Diff. Equat. Appl., 21 (2014), 1-26.  doi: 10.1007/s00030-013-0234-1.  Google Scholar

[15]

E. N. Mahmudov, Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints, J. Indust. Manag. Optim., 16 (2020), 169-187.  doi: 10.3934/jimo.2018145.  Google Scholar

[16]

E. N. Mahmudov, Optimal control of second order delay-discrete and delay differential inclusions with state constraints, Evol. Equat. Contr. Theory, 7 (2018), 501-529.  doi: 10.3934/eect.2018024.  Google Scholar

[17]

E. N. Mahmudov, Mathematical programming and polyhedral optimization of second order discrete and differential inclusions, Pacific J. Optim., 11 (2015), 511-525.   Google Scholar

[18]

E. N. Mahmudov, Optimal control of higher order differential inclusions with functional constraints, ESAIM: Control, Optim Calculus Variat., 26 (2020), 1-23.  doi: 10.1051/cocv/2019018.  Google Scholar

[19]

E. N. Mahmudov, Single Variable Differential and Integral Calculus, Mathematical Analysis. Paris, France, Springer, 2013. doi: 10.2991/978-94-91216-86-2.  Google Scholar

[20]

M. Paskota, V. Sreeram, K. L. Teo and A. I. Mees, Optimal simultaneous stabilization of linear single-input systems via linear state feedback control, Int. Journ. Contr., 60 (1994), 483-498.  doi: 10.1080/00207179408921477.  Google Scholar

[21]

K. Ravikumar, M. T. Mohan and A. Anguraj, Approximate controllability of a non-autonomous evolution equation in Banach spaces, Numer. Algebra, Contr. Optim., 2020. doi: 10.3934/naco.2020038.  Google Scholar

[22]

L. M. Silverman and H. E. Meadows, Controllability and Observability in time-variable linear systems, SIAM J. Contr. Optim., 5 (1967), 64-73.   Google Scholar

[23]

W. S. W. Wang, D. E. Davison and E. J. Davison, Controller design for multivariable linear time-invariant unknown systems, IEEE Trans. Automat Contr., 58 (2013) 2292–2306. doi: 10.1109/TAC.2013.2258812.  Google Scholar

[24]

L. A. Zadeh and C. A. Desoer, Linear System Theory: The State Space Approach, McGraw-Hill Series in System Science, New York: McGraw-Hill, 1963.  Google Scholar

[25]

X. Zhao, L. Zhang and P. Shi, Stability of a class of switched positive linear time-delay systems, Inter. J. Robust Nonlin. Contr., 23 (2013), 578-589.  doi: 10.1002/rnc.2777.  Google Scholar

[26]

X. L. Zhu, H. Yang, Y. Y. Wang and Y. L. Wang, New stability criterion for linear switched systems with time-varying delay, Inter. J. Robust. Nonlin. Contr., 24 (2014), 214-227.  doi: 10.1002/rnc.2882.  Google Scholar

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