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June  2022, 12(2): 353-371. doi: 10.3934/naco.2021010

Second order discrete time-varying and time-invariant linear continuous systems and Kalman type conditions

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 Published  June 2022 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.

Citation: Elimhan N. Mahmudov. Second order discrete time-varying and time-invariant linear continuous systems and Kalman type conditions. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 353-371. doi: 10.3934/naco.2021010
References:
[1]

S. AvdoninJ. 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.

[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.

[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.

[4]

F. GaoW. LiuV. 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.

[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.

[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.

[7]

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

[8]

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

[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.

[10]

C. LiF. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[19]

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

[20]

M. PaskotaV. SreeramK. 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.

[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.

[22]

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

[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.

[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.

[25]

X. ZhaoL. 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.

[26]

X. L. ZhuH. YangY. 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.

show all references

References:
[1]

S. AvdoninJ. 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.

[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.

[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.

[4]

F. GaoW. LiuV. 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.

[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.

[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.

[7]

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

[8]

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

[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.

[10]

C. LiF. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[19]

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

[20]

M. PaskotaV. SreeramK. 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.

[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.

[22]

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

[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.

[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.

[25]

X. ZhaoL. 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.

[26]

X. L. ZhuH. YangY. 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.

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