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 and 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), 1666-1671. doi: 10.1109/9.544005.

[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), 1131-1143. doi: 10.1109/9.863597.

[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), 123-133. doi: 10.1016/S0167-6911(00)00046-3.

[4]

A. Albert, Conditions for positive and nonnegative definiteness in terms of pseudo-inverse, SIAM J. Appl. Math., 17 (1969), 434-440.

[5]

B. D. O. Anderson and J. B. Moore, Optimal Filtering, Prentice-Hall, Englewood Cliffs, N.J., 1979.

[6]

B. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods, Prentice-Hall, Englewood Cliffs, N.J., 1989.

[7]

B. D. O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis, Prentice-Hall, Englewood Cliffs, N.J., 1973.

[8]

M. Athans, Special issues on linear-quadratic-Gaussian problem, IEEE Trans. Auto. Contr., AC-16 (1971), 527-869.

[9]

A. Ben-Tal and Marc Teboulle, Hidden convexity in some nonconvex quadratically constrained quadratic programming, Math. Programming, 72 (1996), 51-63. doi: 10.1016/0025-5610(95)00020-8.

[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.

[11]

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

[12]

M. H. A. Davis, Linear Estimation and Stochastic Control, Chapman and Hall London, 1977.

[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), 831-847. doi: 10.1109/9.29425.

[14]

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

[15]

T. Geerts, A necessary and sufficient condition for the solvability of the linear-quadratic control problem without stability, Syst. Cont. Letters, 11 (1988), 47-51. doi: 10.1016/0167-6911(88)90110-7.

[16]

M. Green and D. N. J. Limebeer, Linear Robust Control, Prentice-Hall, Englewood Cliffs, 1995.

[17]

D. H. Jacobson, Totally singular quadratic minimization problems, IEEE Trans. Aut. Control, 16 (1971), 651-658.

[18]

R. E. Kalman, Contribution to the theory of optimal control, Bol. Soc. Mat. Mex., 5 (1960), 102-119.

[19]

B. P. Molinari, The time-invariant linear-quadratic optimal control problem, Automatica, 13 (1977), 347-357.

[20]

J. B. Moore, The singular solution to a singular quadratic minimization problem control, Automatica, 7 (1974), 591-598.

[21]

R. Penrose, A generalized inverse of matrices, Proc. Cambridge Philos. Soc., 51 (1955), 406-413.

[22]

R. Penrose, On the best approximate solutions of linear matrix equations, Proc. Cambridge Philos. Soc., 52 (1955), 17-19.

[23]

B. T. Polyak, Convexity of Quadratic transformations and its use in control and optimization, JOTA, 99 (1998), 553-583. doi: 10.1023/A:1021798932766.

[24]

R. E. Skelton, Increased roles of linear algebra in control theory, Proc. American Cont. Conf., (1994), 393-397.

[25]

H. L. Trentelman, The regulator free-endpoint linear quadratic problem with indefinite cost, SIAM J. Contr. Opt., 27 (1989), 27-42. doi: 10.1137/0327003.

[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), 969-991. doi: 10.1137/S036301290036851X.

[27]

L. Vandenberghe and V. Balakrishnan, Semidefinite programming duality and linear system theory: connections and implications for computation, IEEE CDC Conf., 1 (1999), 989-994.

[28]

L. Vandenberghe and S. Boyd, Semi-definite programming, SIAM Review, 38 (1996), 49-95.

[29]

J. C. Willems, Least squares stationary control and the algebraic Riccati equation, IEEE Trans. Aut. Control, AC-16 (1971), 621-234.

[30]

J. C. Willems, A. Kitapci and L. M. Sylverman, Singular optimal control: a geometric approach, SIAM J. Contr. Opt., 24 (1986), 323-337. doi: 10.1137/0324018.

show all references

References:
[1]

M. Ait Rami and L. El Ghaoui, LMI optimization for stochastic Riccati equation, IEEE Trans. Aut. Contr., 41 (1996), 1666-1671. doi: 10.1109/9.544005.

[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), 1131-1143. doi: 10.1109/9.863597.

[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), 123-133. doi: 10.1016/S0167-6911(00)00046-3.

[4]

A. Albert, Conditions for positive and nonnegative definiteness in terms of pseudo-inverse, SIAM J. Appl. Math., 17 (1969), 434-440.

[5]

B. D. O. Anderson and J. B. Moore, Optimal Filtering, Prentice-Hall, Englewood Cliffs, N.J., 1979.

[6]

B. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods, Prentice-Hall, Englewood Cliffs, N.J., 1989.

[7]

B. D. O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis, Prentice-Hall, Englewood Cliffs, N.J., 1973.

[8]

M. Athans, Special issues on linear-quadratic-Gaussian problem, IEEE Trans. Auto. Contr., AC-16 (1971), 527-869.

[9]

A. Ben-Tal and Marc Teboulle, Hidden convexity in some nonconvex quadratically constrained quadratic programming, Math. Programming, 72 (1996), 51-63. doi: 10.1016/0025-5610(95)00020-8.

[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.

[11]

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

[12]

M. H. A. Davis, Linear Estimation and Stochastic Control, Chapman and Hall London, 1977.

[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), 831-847. doi: 10.1109/9.29425.

[14]

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

[15]

T. Geerts, A necessary and sufficient condition for the solvability of the linear-quadratic control problem without stability, Syst. Cont. Letters, 11 (1988), 47-51. doi: 10.1016/0167-6911(88)90110-7.

[16]

M. Green and D. N. J. Limebeer, Linear Robust Control, Prentice-Hall, Englewood Cliffs, 1995.

[17]

D. H. Jacobson, Totally singular quadratic minimization problems, IEEE Trans. Aut. Control, 16 (1971), 651-658.

[18]

R. E. Kalman, Contribution to the theory of optimal control, Bol. Soc. Mat. Mex., 5 (1960), 102-119.

[19]

B. P. Molinari, The time-invariant linear-quadratic optimal control problem, Automatica, 13 (1977), 347-357.

[20]

J. B. Moore, The singular solution to a singular quadratic minimization problem control, Automatica, 7 (1974), 591-598.

[21]

R. Penrose, A generalized inverse of matrices, Proc. Cambridge Philos. Soc., 51 (1955), 406-413.

[22]

R. Penrose, On the best approximate solutions of linear matrix equations, Proc. Cambridge Philos. Soc., 52 (1955), 17-19.

[23]

B. T. Polyak, Convexity of Quadratic transformations and its use in control and optimization, JOTA, 99 (1998), 553-583. doi: 10.1023/A:1021798932766.

[24]

R. E. Skelton, Increased roles of linear algebra in control theory, Proc. American Cont. Conf., (1994), 393-397.

[25]

H. L. Trentelman, The regulator free-endpoint linear quadratic problem with indefinite cost, SIAM J. Contr. Opt., 27 (1989), 27-42. doi: 10.1137/0327003.

[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), 969-991. doi: 10.1137/S036301290036851X.

[27]

L. Vandenberghe and V. Balakrishnan, Semidefinite programming duality and linear system theory: connections and implications for computation, IEEE CDC Conf., 1 (1999), 989-994.

[28]

L. Vandenberghe and S. Boyd, Semi-definite programming, SIAM Review, 38 (1996), 49-95.

[29]

J. C. Willems, Least squares stationary control and the algebraic Riccati equation, IEEE Trans. Aut. Control, AC-16 (1971), 621-234.

[30]

J. C. Willems, A. Kitapci and L. M. Sylverman, Singular optimal control: a geometric approach, SIAM J. Contr. Opt., 24 (1986), 323-337. doi: 10.1137/0324018.

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