June  2018, 8(2): 211-235. doi: 10.3934/naco.2018013

Asymptotic properties of an infinite horizon partial cheap control problem for linear systems with known disturbances

1. 

Department of Applied Mathematics, ORT Braude College of Engineering, 51 Snunit Str., P.O.B. 78, Karmiel 2161002, Israel

2. 

Department of Mathematics, University of Haifa, Mount Carmel, Haifa 3498838, Israel

* Corresponding author

Received  March 2017 Revised  December 2017 Published  May 2018

An infinite horizon quadratic control of a linear system with known disturbance is considered. The feature of the problem is that the cost of some (but in general not all) control coordinates in the cost functional is much smaller than the costs of the other control coordinates and the state cost. Using the control optimality conditions, the solution of this problem is reduced to solution of a hybrid set of three equations, perturbed by a small parameter. One of these equations is a matrix algebraic Riccati equation, while two others are vector and scalar differential equations subject to terminal conditions at infinity. For this set of the equations, a zero-order asymptotic solution is constructed and justified. Using this asymptotic solution, a relation between solutions of the original problem and the problem, obtained from the original one by replacing the small control cost with zero, is established. Based on this relation, the best achievable performance in the original problem is derived. Illustrative examples are presented.

Citation: Valery Y. Glizer, Oleg Kelis. Asymptotic properties of an infinite horizon partial cheap control problem for linear systems with known disturbances. Numerical Algebra, Control and Optimization, 2018, 8 (2) : 211-235. doi: 10.3934/naco.2018013
References:
[1]

B. D. O. Anderson and J. B. Moore, Linear Optimal Control, Prentice-Hall, Englewood Cliffs, New Jersey, 1971.

[2]

D. J. Bell and D. H. Jacobson, Singular Optimal Control Problems, Academic Press, New York, 1975.

[3]

R. Bellman, Dynamic Programming, Princeton University Press, Princeton, New Jersey, 1957.

[4]

M. U. BikdashA. H. Nayfeh and E. M. Cliff, Singular perturbation of the time-optimal soft-constrained cheap-control problem, IEEE Trans. Automat. Control, 38 (1993), 466-469.  doi: 10.1109/9.210147.

[5]

J. H. BraslavskyM. M. SeronD. Q. Maine and P. V. Kokotovic, Limiting performance of optimal linear filters, Automatica, 35 (1999), 189-199.  doi: 10.1016/S0005-1098(98)00144-7.

[6]

J. ChenS. Hara and G. Chen, Best tracking and regulation performance under control energy constraint, IEEE Trans. Automat. Control, 48 (2003), 1320-1336.  doi: 10.1109/TAC.2003.815012.

[7]

D. J. Clements and B. D. O. Anderson, Singular Optimal Control: The Linear-Quadratic Problem, Lecture Notes in Control and Information Sciences, 5, Springer-Verlag, Berlin, 1978. doi: 10.1007/BFb0004989.

[8]

M. G. Dmitriev and G. A. Kurina, Singular perturbations in control problems, Autom. Remote Control, 67 (2006), 1-43.  doi: 10.1134/S0005117906010012.

[9]

Z. Gajic and M-T. Lim, Optimal Control of Singularly Perturbed Linear Systems and Applications. High Accuracy Techniques, Marsel Dekker Inc., New York, 2001. doi: 10.1201/9780203907900.

[10]

V. Y. Glizer, Asymptotic solution of a cheap control problem with state delay, Dynam. Control, 9 (1999), 339-357.  doi: 10.1023/A:1026484201241.

[11]

V. Y. Glizer, Blockwise estimate of the fundamental matrix of linear singularly perturbed differential systems with small delay and its application to uniform asymptotic solution, J. Math. Anal. Appl., 278 (2003), 409-433.  doi: 10.1016/S0022-247X(02)00715-1.

[12]

V. Y. Glizer, Suboptimal solution of a cheap control problem for linear systems with multiple state delays, J. Dyn. Control Syst., 11 (2005), 527-574.  doi: 10.1007/s10883-005-8818-7.

[13]

V. Y. Glizer, Infinite horizon cheap control problem for a class of systems with state delays, J. Nonlinear Convex Anal., 10 (2009), 199-233. 

[14]

V. Y. Glizer, Solution of a singular optimal control problem with state delays: a cheap control approach, in Optimization Theory and Related Topics, Contemporary Mathematics Series, 568 (eds. S. Reich and A. J. Zaslavski), American Mathematical Society, 2012, 77-107. doi: 10.1090/conm/568/11278.

[15]

V. Y. Glizer, Stochastic singular optimal control problem with state delays: regularization, singular perturbation, and minimizing sequence, SIAM J. Control Optim., 50 (2012), 2862-2888.  doi: 10.1137/110852784.

[16]

V. Y. Glizer, Singular solution of an infinite horizon linear-quadratic optimal control problem with state delays, in Variational and Optimal Control Problems on Unbounded Domains, Contemporary Mathematics Series, 619 (eds. G. Wolansky and A. J. Zaslavski), American Mathematical Society, 2014, 59-98. doi: 10.1090/conm/619/12385.

[17]

V. Y. GlizerL. M. Fridman and V. Turetsky, Cheap suboptimal control of an integral sliding mode for uncertain systems with state delays, IEEE Trans. Automat. Control, 52 (2007), 1892-1898.  doi: 10.1109/TAC.2007.906201.

[18]

V. Y. Glizer and O. Kelis, Solution of a zero-sum linear quadratic differential game with singular control cost of minimizer, J. Control Decis., 2 (2015), 155-184.  doi: 10.1080/23307706.2015.1057545.

[19]

V. Y. Glizer and O. Kelis, Singular infinite horizon zero-sum linear-quadratic differential game: saddle-point equilibrium sequence, Numer. Algebra Control Optim., 7 (2017), 1-20.  doi: 10.3934/naco.2017001.

[20]

R. D. HamptonC. R. Knospe and M. A. Townsend, A practical solution to the deterministic nonhomogeneous LQR problem, J. Dyn. Sys., Meas., Control, 118 (1996), 354-359.  doi: 10.1115/1.2802329.

[21]

A Jameson and R. E. O'Malley, Cheap control of the time-invariant regulator, Appl. Math. Optim., 1 (1974/75), 337-354.  doi: 10.1007/BF01447957.

[22]

P. V. Kokotovic, H. K. Khalil and J. O'Reilly, Singular Perturbation Methods in Control: Analysis and Design, Academic Press, London, 1986.

[23]

G. A. Kurina, A degenerate optimal control problem and singular perturbations, Soviet Mathematics Doklady, 18 (1977), 1452-1456. 

[24]

G. Kurina and Nguyen Thi Hoai, Asymptotic solution of a linear-quadratic problem with discontinuous coefficients and cheap control, Appl. Math. Comput., 232 (2014), 347-364.  doi: 10.1016/j.amc.2013.12.097.

[25]

H. Kwakernaak and R. Sivan, The maximally achievable accuracy of linear optimal regulators and linear optimal filters, IEEE Trans. Automat. Control, 17 (1972), 79-86.  doi: 10.1109/TAC.1972.1099865.

[26]

R. Mahadevan and T. Muthukumar, Homogenization of some cheap control problems, SIAM J. Math. Anal., 43 (2011), 2211-2229.  doi: 10.1137/100811581.

[27]

P. J. Moylan and B. D. O. Anderson, Nonlinear regulator theory and an inverse optimal control problem, IEEE Trans. Automat. Control, 18 (1973), 460-465.  doi: 10.1109/TAC.1973.1100365.

[28]

R. E. O'Malley and A. Jameson, Singular perturbations and singular arcs, Ⅱ, IEEE Trans. Automat. Control, 22 (1977), 328-337.  doi: 10.1109/TAC.1977.1101535.

[29]

J. O'Reilly, Partial cheap control of the time-invariant regulator, Internat. J. Control, 37 (1983), 909-927.  doi: 10.1080/00207178308933019.

[30]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mischenko, The Mathematical Theory of Optimal Processes, Gordon & Breach, New York, 1986.

[31]

L. Qiu and E. J. Davison, Performance limitations of non-minimum phase systems in the servomechanism problem, Automatica, 29 (1993), 337-349.  doi: 10.1016/0005-1098(93)90127-F.

[32]

A. Saberi and P. Sannuti, Cheap and singular controls for linear quadratic regulators, IEEE Trans. Automat. Control, 32 (1987), 208-219.  doi: 10.1109/TAC.1987.1104574.

[33]

M. E. Salukvadze, Analytical design of regulators. Constant disturbances, Autom. Remote Control, 22 (1961), 1147-1155. 

[34]

M. E. Salukvadze, The analytical design of an optimal control in the case of constantly acting disturbances, Autom. Remote Control, 23 (1962), 657-667. 

[35]

M. M. SeronJ. H. BraslavskyP. V. Kokotovic and D. Q. Mayne, Feedback limitations in nonlinear systems: from Bode integrals to cheap control, IEEE Trans. Automat. Control, 44 (1999), 829-833.  doi: 10.1109/9.754828.

[36]

Y. Sibuya, Some global properties of matrices of functions of one variable, Math. Annalen, 161 (1965), 67-77.  doi: 10.1007/BF01363248.

[37]

V. Turetsky and V. Y. Glizer, Robust state-feedback controllability of linear systems to a hyperplane in a class of bounded controls, J. Optim. Theory Appl., 123 (2004), 639-667.  doi: 10.1007/s10957-004-5727-y.

[38]

V. Turetsky and V. Y. Glizer, Robust solution of a time-variable interception problem: a cheap control approach, Int. Game Theory Rev., 9 (2007), 637-655.  doi: 10.1142/S0219198907001631.

[39]

V. TuretskyV. Y. Glizer and J. Shinar, Robust trajectory tracking: differential game/cheap control approach, Internat. J. Systems Sci., 45 (2014), 2260-2274.  doi: 10.1080/00207721.2013.768305.

[40]

K. D. YoungP. V. Kokotovic and V. I. Utkin, A singular perturbation analysis of high-gain feedback systems, IEEE Trans. Automat. Control, 22 (1977), 931-938.  doi: 10.1109/TAC.1977.1101661.

[41]

Y. ZhangD. S. NaiduC. Cai and Y. Zou, Singular perturbations and time scales in control theories and applications: an overview 2002-2012, Int. J. Inf. Syst. Sci., 9 (2014), 1-36. 

show all references

References:
[1]

B. D. O. Anderson and J. B. Moore, Linear Optimal Control, Prentice-Hall, Englewood Cliffs, New Jersey, 1971.

[2]

D. J. Bell and D. H. Jacobson, Singular Optimal Control Problems, Academic Press, New York, 1975.

[3]

R. Bellman, Dynamic Programming, Princeton University Press, Princeton, New Jersey, 1957.

[4]

M. U. BikdashA. H. Nayfeh and E. M. Cliff, Singular perturbation of the time-optimal soft-constrained cheap-control problem, IEEE Trans. Automat. Control, 38 (1993), 466-469.  doi: 10.1109/9.210147.

[5]

J. H. BraslavskyM. M. SeronD. Q. Maine and P. V. Kokotovic, Limiting performance of optimal linear filters, Automatica, 35 (1999), 189-199.  doi: 10.1016/S0005-1098(98)00144-7.

[6]

J. ChenS. Hara and G. Chen, Best tracking and regulation performance under control energy constraint, IEEE Trans. Automat. Control, 48 (2003), 1320-1336.  doi: 10.1109/TAC.2003.815012.

[7]

D. J. Clements and B. D. O. Anderson, Singular Optimal Control: The Linear-Quadratic Problem, Lecture Notes in Control and Information Sciences, 5, Springer-Verlag, Berlin, 1978. doi: 10.1007/BFb0004989.

[8]

M. G. Dmitriev and G. A. Kurina, Singular perturbations in control problems, Autom. Remote Control, 67 (2006), 1-43.  doi: 10.1134/S0005117906010012.

[9]

Z. Gajic and M-T. Lim, Optimal Control of Singularly Perturbed Linear Systems and Applications. High Accuracy Techniques, Marsel Dekker Inc., New York, 2001. doi: 10.1201/9780203907900.

[10]

V. Y. Glizer, Asymptotic solution of a cheap control problem with state delay, Dynam. Control, 9 (1999), 339-357.  doi: 10.1023/A:1026484201241.

[11]

V. Y. Glizer, Blockwise estimate of the fundamental matrix of linear singularly perturbed differential systems with small delay and its application to uniform asymptotic solution, J. Math. Anal. Appl., 278 (2003), 409-433.  doi: 10.1016/S0022-247X(02)00715-1.

[12]

V. Y. Glizer, Suboptimal solution of a cheap control problem for linear systems with multiple state delays, J. Dyn. Control Syst., 11 (2005), 527-574.  doi: 10.1007/s10883-005-8818-7.

[13]

V. Y. Glizer, Infinite horizon cheap control problem for a class of systems with state delays, J. Nonlinear Convex Anal., 10 (2009), 199-233. 

[14]

V. Y. Glizer, Solution of a singular optimal control problem with state delays: a cheap control approach, in Optimization Theory and Related Topics, Contemporary Mathematics Series, 568 (eds. S. Reich and A. J. Zaslavski), American Mathematical Society, 2012, 77-107. doi: 10.1090/conm/568/11278.

[15]

V. Y. Glizer, Stochastic singular optimal control problem with state delays: regularization, singular perturbation, and minimizing sequence, SIAM J. Control Optim., 50 (2012), 2862-2888.  doi: 10.1137/110852784.

[16]

V. Y. Glizer, Singular solution of an infinite horizon linear-quadratic optimal control problem with state delays, in Variational and Optimal Control Problems on Unbounded Domains, Contemporary Mathematics Series, 619 (eds. G. Wolansky and A. J. Zaslavski), American Mathematical Society, 2014, 59-98. doi: 10.1090/conm/619/12385.

[17]

V. Y. GlizerL. M. Fridman and V. Turetsky, Cheap suboptimal control of an integral sliding mode for uncertain systems with state delays, IEEE Trans. Automat. Control, 52 (2007), 1892-1898.  doi: 10.1109/TAC.2007.906201.

[18]

V. Y. Glizer and O. Kelis, Solution of a zero-sum linear quadratic differential game with singular control cost of minimizer, J. Control Decis., 2 (2015), 155-184.  doi: 10.1080/23307706.2015.1057545.

[19]

V. Y. Glizer and O. Kelis, Singular infinite horizon zero-sum linear-quadratic differential game: saddle-point equilibrium sequence, Numer. Algebra Control Optim., 7 (2017), 1-20.  doi: 10.3934/naco.2017001.

[20]

R. D. HamptonC. R. Knospe and M. A. Townsend, A practical solution to the deterministic nonhomogeneous LQR problem, J. Dyn. Sys., Meas., Control, 118 (1996), 354-359.  doi: 10.1115/1.2802329.

[21]

A Jameson and R. E. O'Malley, Cheap control of the time-invariant regulator, Appl. Math. Optim., 1 (1974/75), 337-354.  doi: 10.1007/BF01447957.

[22]

P. V. Kokotovic, H. K. Khalil and J. O'Reilly, Singular Perturbation Methods in Control: Analysis and Design, Academic Press, London, 1986.

[23]

G. A. Kurina, A degenerate optimal control problem and singular perturbations, Soviet Mathematics Doklady, 18 (1977), 1452-1456. 

[24]

G. Kurina and Nguyen Thi Hoai, Asymptotic solution of a linear-quadratic problem with discontinuous coefficients and cheap control, Appl. Math. Comput., 232 (2014), 347-364.  doi: 10.1016/j.amc.2013.12.097.

[25]

H. Kwakernaak and R. Sivan, The maximally achievable accuracy of linear optimal regulators and linear optimal filters, IEEE Trans. Automat. Control, 17 (1972), 79-86.  doi: 10.1109/TAC.1972.1099865.

[26]

R. Mahadevan and T. Muthukumar, Homogenization of some cheap control problems, SIAM J. Math. Anal., 43 (2011), 2211-2229.  doi: 10.1137/100811581.

[27]

P. J. Moylan and B. D. O. Anderson, Nonlinear regulator theory and an inverse optimal control problem, IEEE Trans. Automat. Control, 18 (1973), 460-465.  doi: 10.1109/TAC.1973.1100365.

[28]

R. E. O'Malley and A. Jameson, Singular perturbations and singular arcs, Ⅱ, IEEE Trans. Automat. Control, 22 (1977), 328-337.  doi: 10.1109/TAC.1977.1101535.

[29]

J. O'Reilly, Partial cheap control of the time-invariant regulator, Internat. J. Control, 37 (1983), 909-927.  doi: 10.1080/00207178308933019.

[30]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mischenko, The Mathematical Theory of Optimal Processes, Gordon & Breach, New York, 1986.

[31]

L. Qiu and E. J. Davison, Performance limitations of non-minimum phase systems in the servomechanism problem, Automatica, 29 (1993), 337-349.  doi: 10.1016/0005-1098(93)90127-F.

[32]

A. Saberi and P. Sannuti, Cheap and singular controls for linear quadratic regulators, IEEE Trans. Automat. Control, 32 (1987), 208-219.  doi: 10.1109/TAC.1987.1104574.

[33]

M. E. Salukvadze, Analytical design of regulators. Constant disturbances, Autom. Remote Control, 22 (1961), 1147-1155. 

[34]

M. E. Salukvadze, The analytical design of an optimal control in the case of constantly acting disturbances, Autom. Remote Control, 23 (1962), 657-667. 

[35]

M. M. SeronJ. H. BraslavskyP. V. Kokotovic and D. Q. Mayne, Feedback limitations in nonlinear systems: from Bode integrals to cheap control, IEEE Trans. Automat. Control, 44 (1999), 829-833.  doi: 10.1109/9.754828.

[36]

Y. Sibuya, Some global properties of matrices of functions of one variable, Math. Annalen, 161 (1965), 67-77.  doi: 10.1007/BF01363248.

[37]

V. Turetsky and V. Y. Glizer, Robust state-feedback controllability of linear systems to a hyperplane in a class of bounded controls, J. Optim. Theory Appl., 123 (2004), 639-667.  doi: 10.1007/s10957-004-5727-y.

[38]

V. Turetsky and V. Y. Glizer, Robust solution of a time-variable interception problem: a cheap control approach, Int. Game Theory Rev., 9 (2007), 637-655.  doi: 10.1142/S0219198907001631.

[39]

V. TuretskyV. Y. Glizer and J. Shinar, Robust trajectory tracking: differential game/cheap control approach, Internat. J. Systems Sci., 45 (2014), 2260-2274.  doi: 10.1080/00207721.2013.768305.

[40]

K. D. YoungP. V. Kokotovic and V. I. Utkin, A singular perturbation analysis of high-gain feedback systems, IEEE Trans. Automat. Control, 22 (1977), 931-938.  doi: 10.1109/TAC.1977.1101661.

[41]

Y. ZhangD. S. NaiduC. Cai and Y. Zou, Singular perturbations and time scales in control theories and applications: an overview 2002-2012, Int. J. Inf. Syst. Sci., 9 (2014), 1-36. 

Table 1.  Optimal value of the cost functional (105)
$\varepsilon $ 0.4 0.2 0.1 0.05 0.025 0.0125 0.00625
${\mathcal J}_{\varepsilon}^{*}$ 10.8004 9.2653 8.6776 8.4775 8.4076 8.3809 8.3697
$\varepsilon $ 0.4 0.2 0.1 0.05 0.025 0.0125 0.00625
${\mathcal J}_{\varepsilon}^{*}$ 10.8004 9.2653 8.6776 8.4775 8.4076 8.3809 8.3697
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