2016, 6(3): 221-239. doi: 10.3934/naco.2016009

Partial stabilizability and hidden convexity of indefinite LQ problem

1. 

Dept. Systems Engineering, University of Valladolid, 47005 Valladolid, Spain

2. 

Dept. Systems Engineering, Research School Of Information Sciences And Engineering, The Australian National University, Canberra, Act 0200, Australia

Received  August 2015 Revised  July 2016 Published  September 2016

Generalization of linear system stability theory and LQ control theory are presented. It is shown that the partial stabilizability problem is equivalent to a Linear Matrix Inequality (LMI). Also, the set of all initial conditions for which the system is stabilizable by an open-loop control (the stabilizability subspace) is characterized in terms of a semi-definite programming (SDP). Next, we give a complete theory for an infinite-time horizon Linear Quadratic (LQ) problem with possibly indefinite weighting matrices for the state and control. Necessary and sufficient convex conditions are given for well-posedness as well as attainability of the proposed (LQ) problem. There is no prior assumption of complete stabilizability condition as well as no assumption on the quadratic cost. A generalized algebraic Riccati equation is introduced and it is shown that it provides all possible optimal controls. Moreover, we show that the solvability of the proposed indefinite LQ problem is equivalent to the solvability of a specific SDP problem.
Citation: Mustapha Ait Rami, John Moore. Partial stabilizability and hidden convexity of indefinite LQ problem. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 221-239. doi: 10.3934/naco.2016009
References:
[1]

M. Ait Rami and L. El Ghaoui, LMI optimization for stochastic Riccati equation,, IEEE Trans. Aut. Contr., 41 (1996), 1666.  doi: 10.1109/9.544005.  Google Scholar

[2]

M. Ait Rami and X. Y. Zhou, Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic control,, IEEE Trans. Aut. Contr., 45 (2000), 1131.  doi: 10.1109/9.863597.  Google Scholar

[3]

M. Ait Rami, J. B. Moore and X. Y. Zhou, Well-posedness and attainability of indefinite stochastic linear quadratic control in infinite time horizon,, Syst. & Contr. Letters, 41 (2000), 123.  doi: 10.1016/S0167-6911(00)00046-3.  Google Scholar

[4]

A. Albert, Conditions for positive and nonnegative definiteness in terms of pseudo-inverse,, SIAM J. Appl. Math., 17 (1969), 434.   Google Scholar

[5]

B. D. O. Anderson and J. B. Moore, Optimal Filtering,, Prentice-Hall, (1979).   Google Scholar

[6]

B. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods,, Prentice-Hall, (1989).   Google Scholar

[7]

B. D. O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis,, Prentice-Hall, (1973).   Google Scholar

[8]

M. Athans, Special issues on linear-quadratic-Gaussian problem,, IEEE Trans. Auto. Contr., AC-16 (1971), 527.   Google Scholar

[9]

A. Ben-Tal and Marc Teboulle, Hidden convexity in some nonconvex quadratically constrained quadratic programming,, Math. Programming, 72 (1996), 51.  doi: 10.1016/0025-5610(95)00020-8.  Google Scholar

[10]

S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequality in Systems and Control Theory,, SIAM, (1994).  doi: 10.1137/1.9781611970777.  Google Scholar

[11]

D. J. Bell and D. H. Jacobson, Singular Optimal Control Problems,, Academic Press, (1975).   Google Scholar

[12]

M. H. A. Davis, Linear Estimation and Stochastic Control,, Chapman and Hall London, (1977).   Google Scholar

[13]

J. C. Doyle, K. Glover, P. P. Khargonekar and B. A. Francis, State-space solutions to standard H2 and H control problems,, IEEE Trans. Aut. Control, 34 (1989), 831.  doi: 10.1109/9.29425.  Google Scholar

[14]

S. Bittanti, A. J. Laub, and J. C. Willems, The Riccati Equation,, Springer-Verlag, (1991).  doi: 10.1007/978-3-642-58223-3.  Google Scholar

[15]

T. Geerts, A necessary and sufficient condition for the solvability of the linear-quadratic control problem without stability,, Syst. Cont. Letters, 11 (1988), 47.  doi: 10.1016/0167-6911(88)90110-7.  Google Scholar

[16]

M. Green and D. N. J. Limebeer, Linear Robust Control,, Prentice-Hall, (1995).   Google Scholar

[17]

D. H. Jacobson, Totally singular quadratic minimization problems,, IEEE Trans. Aut. Control, 16 (1971), 651.   Google Scholar

[18]

R. E. Kalman, Contribution to the theory of optimal control,, Bol. Soc. Mat. Mex., 5 (1960), 102.   Google Scholar

[19]

B. P. Molinari, The time-invariant linear-quadratic optimal control problem,, Automatica, 13 (1977), 347.   Google Scholar

[20]

J. B. Moore, The singular solution to a singular quadratic minimization problem control,, Automatica, 7 (1974), 591.   Google Scholar

[21]

R. Penrose, A generalized inverse of matrices,, Proc. Cambridge Philos. Soc., 51 (1955), 406.   Google Scholar

[22]

R. Penrose, On the best approximate solutions of linear matrix equations,, Proc. Cambridge Philos. Soc., 52 (1955), 17.   Google Scholar

[23]

B. T. Polyak, Convexity of Quadratic transformations and its use in control and optimization,, JOTA, 99 (1998), 553.  doi: 10.1023/A:1021798932766.  Google Scholar

[24]

R. E. Skelton, Increased roles of linear algebra in control theory,, Proc. American Cont. Conf., (1994), 393.   Google Scholar

[25]

H. L. Trentelman, The regulator free-endpoint linear quadratic problem with indefinite cost,, SIAM J. Contr. Opt., 27 (1989), 27.  doi: 10.1137/0327003.  Google Scholar

[26]

H. L. Trentelman and P. Rapisarda, Pick matrix conditions for sign-definite solutions of the algebraic Riccati equation,, SIAM J. Contr. Opt., 40 (2001), 969.  doi: 10.1137/S036301290036851X.  Google Scholar

[27]

L. Vandenberghe and V. Balakrishnan, Semidefinite programming duality and linear system theory: connections and implications for computation,, IEEE CDC Conf., 1 (1999), 989.   Google Scholar

[28]

L. Vandenberghe and S. Boyd, Semi-definite programming,, SIAM Review, 38 (1996), 49.   Google Scholar

[29]

J. C. Willems, Least squares stationary control and the algebraic Riccati equation,, IEEE Trans. Aut. Control, AC-16 (1971), 621.   Google Scholar

[30]

J. C. Willems, A. Kitapci and L. M. Sylverman, Singular optimal control: a geometric approach,, SIAM J. Contr. Opt., 24 (1986), 323.  doi: 10.1137/0324018.  Google Scholar

show all references

References:
[1]

M. Ait Rami and L. El Ghaoui, LMI optimization for stochastic Riccati equation,, IEEE Trans. Aut. Contr., 41 (1996), 1666.  doi: 10.1109/9.544005.  Google Scholar

[2]

M. Ait Rami and X. Y. Zhou, Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic control,, IEEE Trans. Aut. Contr., 45 (2000), 1131.  doi: 10.1109/9.863597.  Google Scholar

[3]

M. Ait Rami, J. B. Moore and X. Y. Zhou, Well-posedness and attainability of indefinite stochastic linear quadratic control in infinite time horizon,, Syst. & Contr. Letters, 41 (2000), 123.  doi: 10.1016/S0167-6911(00)00046-3.  Google Scholar

[4]

A. Albert, Conditions for positive and nonnegative definiteness in terms of pseudo-inverse,, SIAM J. Appl. Math., 17 (1969), 434.   Google Scholar

[5]

B. D. O. Anderson and J. B. Moore, Optimal Filtering,, Prentice-Hall, (1979).   Google Scholar

[6]

B. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods,, Prentice-Hall, (1989).   Google Scholar

[7]

B. D. O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis,, Prentice-Hall, (1973).   Google Scholar

[8]

M. Athans, Special issues on linear-quadratic-Gaussian problem,, IEEE Trans. Auto. Contr., AC-16 (1971), 527.   Google Scholar

[9]

A. Ben-Tal and Marc Teboulle, Hidden convexity in some nonconvex quadratically constrained quadratic programming,, Math. Programming, 72 (1996), 51.  doi: 10.1016/0025-5610(95)00020-8.  Google Scholar

[10]

S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequality in Systems and Control Theory,, SIAM, (1994).  doi: 10.1137/1.9781611970777.  Google Scholar

[11]

D. J. Bell and D. H. Jacobson, Singular Optimal Control Problems,, Academic Press, (1975).   Google Scholar

[12]

M. H. A. Davis, Linear Estimation and Stochastic Control,, Chapman and Hall London, (1977).   Google Scholar

[13]

J. C. Doyle, K. Glover, P. P. Khargonekar and B. A. Francis, State-space solutions to standard H2 and H control problems,, IEEE Trans. Aut. Control, 34 (1989), 831.  doi: 10.1109/9.29425.  Google Scholar

[14]

S. Bittanti, A. J. Laub, and J. C. Willems, The Riccati Equation,, Springer-Verlag, (1991).  doi: 10.1007/978-3-642-58223-3.  Google Scholar

[15]

T. Geerts, A necessary and sufficient condition for the solvability of the linear-quadratic control problem without stability,, Syst. Cont. Letters, 11 (1988), 47.  doi: 10.1016/0167-6911(88)90110-7.  Google Scholar

[16]

M. Green and D. N. J. Limebeer, Linear Robust Control,, Prentice-Hall, (1995).   Google Scholar

[17]

D. H. Jacobson, Totally singular quadratic minimization problems,, IEEE Trans. Aut. Control, 16 (1971), 651.   Google Scholar

[18]

R. E. Kalman, Contribution to the theory of optimal control,, Bol. Soc. Mat. Mex., 5 (1960), 102.   Google Scholar

[19]

B. P. Molinari, The time-invariant linear-quadratic optimal control problem,, Automatica, 13 (1977), 347.   Google Scholar

[20]

J. B. Moore, The singular solution to a singular quadratic minimization problem control,, Automatica, 7 (1974), 591.   Google Scholar

[21]

R. Penrose, A generalized inverse of matrices,, Proc. Cambridge Philos. Soc., 51 (1955), 406.   Google Scholar

[22]

R. Penrose, On the best approximate solutions of linear matrix equations,, Proc. Cambridge Philos. Soc., 52 (1955), 17.   Google Scholar

[23]

B. T. Polyak, Convexity of Quadratic transformations and its use in control and optimization,, JOTA, 99 (1998), 553.  doi: 10.1023/A:1021798932766.  Google Scholar

[24]

R. E. Skelton, Increased roles of linear algebra in control theory,, Proc. American Cont. Conf., (1994), 393.   Google Scholar

[25]

H. L. Trentelman, The regulator free-endpoint linear quadratic problem with indefinite cost,, SIAM J. Contr. Opt., 27 (1989), 27.  doi: 10.1137/0327003.  Google Scholar

[26]

H. L. Trentelman and P. Rapisarda, Pick matrix conditions for sign-definite solutions of the algebraic Riccati equation,, SIAM J. Contr. Opt., 40 (2001), 969.  doi: 10.1137/S036301290036851X.  Google Scholar

[27]

L. Vandenberghe and V. Balakrishnan, Semidefinite programming duality and linear system theory: connections and implications for computation,, IEEE CDC Conf., 1 (1999), 989.   Google Scholar

[28]

L. Vandenberghe and S. Boyd, Semi-definite programming,, SIAM Review, 38 (1996), 49.   Google Scholar

[29]

J. C. Willems, Least squares stationary control and the algebraic Riccati equation,, IEEE Trans. Aut. Control, AC-16 (1971), 621.   Google Scholar

[30]

J. C. Willems, A. Kitapci and L. M. Sylverman, Singular optimal control: a geometric approach,, SIAM J. Contr. Opt., 24 (1986), 323.  doi: 10.1137/0324018.  Google Scholar

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