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Temporal and differential stabilizability and detectability of piecewise constant rank systems. (English) Zbl 1276.93019
Summary: In a past note, we drew attention to the fact that time-varying continuous-time linear systems may be temporarily uncontrollable and unreconstructable and that this is vital knowledge to both control engineers and system scientists. Describing and detecting the temporal loss of controllability and reconstructability require considering Piecewise Constant Rank (PCR) systems and the differential Kalman decomposition. In this note, measures of temporal and differential stabilizability and detectability are developed for conventional as well as PCR systems. These measures indicate to what extent the temporal loss of controllability and reconstructability may lead to temporal instability of the closed-loop system when designing a static state or dynamic output feedback controller. It is indicated how to compute the measures from the system matrices. The importance of our developments for control system design is illustrated through three numerical examples concerning LQ and LQG perturbation feedback control of a nonlinear system about an optimal control and state trajectory.

93B05 Controllability
93B52 Feedback control
93D15 Stabilization of systems by feedback
93C15 Control/observation systems governed by ordinary differential equations
Full Text: DOI
[1] Athans, The role and use of the linear-quadratic-Gaussian problem in control system design, IEEE Transactions on Automatic Control 16 (6) pp 529– (1971) · doi:10.1109/TAC.1971.1099818
[2] Kalman RE Bertram JE Control system design via the ’Second Method’ of Lyapunov, I continuous-time systems 1960 371 393
[3] Amato, Finite-time control of linear systems: a survey, Current Trends in Nonlinear Systems and Control, Systems and Control: Foundations & Applications pp 195– (2006) · doi:10.1007/0-8176-4470-9_11
[4] Amato, Finite-time stability of linear time-varying systems with jumps, Automatica 45 pp 1354– (2009) · Zbl 1162.93375 · doi:10.1016/j.automatica.2008.12.016
[5] Jameson, Cheap control of the time-invariant regulator, Applied Mathematics & Optimization 1 (4) pp 337– (1975) · Zbl 0307.49040 · doi:10.1007/BF01447957
[6] Kokotovic, Singular perturbations and order-reduction in control theory-an overview, Automatica 12 pp 123– (1976) · Zbl 0323.93020 · doi:10.1016/0005-1098(76)90076-5
[7] Van Willigenburg LG De Koning WL A Kalman decomposition to detect temporal linear system structure 1721 1726
[8] Van Willigenburg, Temporal linear system structure, IEEE Transactions on Automatic Control 53 (5) pp 1318– (2008) · Zbl 1367.93083 · doi:10.1109/TAC.2008.921033
[9] Sandberg, Balanced truncation of linear time-varying systems, IEEE Transactions on Automatic Control 49 (2) pp 217– (2004) · Zbl 1365.93062 · doi:10.1109/TAC.2003.822862
[10] Silverman, Equivalent realizations of linear systems, SIAM Journal on Applied Mathematics 17 (2) pp 393– (1969) · Zbl 0206.14601 · doi:10.1137/0117037
[11] Kalman, Contributions to the theory of optimal control, Boletin de la Sociedad Matematica Mexicana 5 pp 102– (1960) · Zbl 0112.06303
[12] Forth, An efficient overloaded implementation of forward mode automatic differentiation in MATLAB, ACM Transactions on Mathematical Software (TOMS) 32 (2) pp 195– (2006) · Zbl 1365.65053 · doi:10.1145/1141885.1141888
[13] Van Willigenburg, Compensability and optimal compensation of systems with white parameters in the delta domain, International Journal of Control 83 (12) pp 2546– (2010) · Zbl 1205.93097 · doi:10.1080/00207179.2010.532236
[14] Van Willigenburg LG De Koning WL Temporal linear system structure: the discrete-time case 225 230
[15] Van Willigenburg, Optimal reduced-order compensators for time-varying discrete-time systems with deterministic and white parameters, Automatica 35 pp 129– (1999) · Zbl 0937.93054 · doi:10.1016/S0005-1098(98)00138-1
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