Transfer Function, Stabilizability, and Detectability of Non-autonomous Riesz-spectral Systems

Stability of a state linear system can be identified by controllability, observability, stabilizability, detectability, and transfer function. The approximate controllability and observability of non-autonomous Riesz-spectral systems have been established as well as non-autonomous Sturm-Liouville systems. As a continuation of the establishments, this paper concern on the analysis of the transfer function, stabilizability, and detectability of the non-autonomous Riesz-spectral systems. A strongly continuous quasi semigroup approach is implemented. The results show that the transfer function, stabilizability, and detectability can be established comprehensively in the non-autonomous Riesz-spectral systems. In particular, sufficient and necessary conditions for the stabilizability and detectability can be constructed. These results are parallel with infinite dimensional of autonomous systems

New characterization of controllability via stabilizability and Riccati equation for LTV systems

This paper presents a new characterization of controllability via stabilizability and Riccati equation for linear time-varying systems. An equivalence is given between the global null controllability, complete stabilizability and the existence of the solution of some appropriate Riccati differential equation. © The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved

A Numerical Procedure To Compute Stabilizing State Feedback Gains For Linear Time-varying Periodic Systems

A procedure to synthesize stabilizing controllers for linear time-varying periodic continuous-time systems is proposed in this paper. The controller is a periodic state-feedback gain whose construction is based on the utilization of the transition matrix of the open-loop system, and the stability of the closed-loop system is guaranteed if the system is controllable and if a observability-based condition is satisfied. The periodic state feedback gain is obtained through the numerical integration of two differential matrix equations over two periods, being the resolution of such equations considerably simpler and computationally more viable than the resolution of Ricatti differential equations considered in the standard LQR approach. Some examples illustrates the validity of the technique. © 2012 IFAC.7PART 1678683Danfoss,Grundfos,DONG Energy,VestasAgulhari, C.M., Garcia, G., Tarbouriech, S., Peres, P.L.D., An efficient numerical procedure to compute stabilizing state feedback gains for linear time-varying periodic systems (2012) Technical Report, School of Electrical and Computer Engineering, , www.dt.fee.unicamp.br/~agulhari/Reports/report_ltv_periodic_continuous. pdfAmato, F., Ariola, M., Cosentino, C., Finitetime control of linear time-varying systems via output feedback (2005) Proc. 2005 Amer. Control Conf., pp. 4722-4726. , Portland, OR, USAArtstein, Z., Stability, observability and invariance (1982) J. of Diff. Eqs., 44 (2), pp. 224-248Bittanti, S., Colaneri, P., (2009) Periodic Systems: Filtering and Control, , Springer-Verlag, LondonBittanti, S., Colaneri, P., Guardabassi, G., Periodic solutions of periodic Riccati equations (1984) IEEE Trans. Autom. Control, 29 (7), pp. 665-667Bittanti, S., Guardabassi, G., Maffezzoni, C., Silverman, L., Periodic systems: Controllability and the matrix Riccati equation (1978) SIAM J. Control Optim., 16 (1), pp. 37-40Bittanti, S., Laub, A.J., Willems, J.C., (1991) The Riccati Equation, , Springer-Verlag, New YorkBrunovský, P., Controllability and linear closedloop controls in linear periodic systems (1969) J. of Diff. Eqs., 6 (2), pp. 296-313Calico, R.A., Wiesel, W.E., Control of timeperiodic systems (1984) J. Guidance, Control and Dyna., 7 (6), pp. 671-676Calico, R.A., Wiesel, W.E., Stabilisation of helicopter blade flapping (1986) J. of the American Helicopter Society, 31 (4), pp. 59-64Chen, M.S., Huang, Y.R., Linear time-varying system control based on the inversion transformation (1997) Automatica, 33 (4), pp. 683-688Chen, M.S., Kao, C.Y., Control of linear timevarying systems using forward Riccati equation (1997) J. Dyna. Syst., Measure., Control - Trans. ASME, 119, pp. 536-540Farges, C., Peaucelle, D., Arzelier, D., Daafouz, J., Robust H 2 performance analysis and synthesis of linear polytopic discrete-time periodic systems via LMIs (2007) Syst. Control Letts., 56 (2), pp. 159-166Garcia, G., Peres, P.L.D., Tarbouriech, S., Assessing asymptotic stability of linear continuous time-varying systems by computing the envelope of all trajectories (2010) IEEE Trans. Autom. Control, 55 (4), pp. 998-1003Garcia, G., Tarbouriech, S., Bernussou, J., Finite-time stabilization of linear time-varying continuous systems (2009) IEEE Trans. Autom. Control, 54 (2), pp. 364-369Hewer, G.A., Periodicity, detectability and the matrix Riccati equation (1975) SIAM J. Control, 13 (6), pp. 1235-1251Kabamba, P.T., Monodromy eigenvalue assignment in linear periodic systems (1986) IEEE Trans. Autom. Control, 31 (10), pp. 950-952Kailath, T., (1980) Linear Systems, , Prentice-Hall, Englewood Cliffs, NJ, USAKalman, R.E., Contributions to the theory of optimal control (1960) Boletin Soc. Mat. Mexicana, 5, pp. 102-119Kern, G., To the robust stabilization problem of linear periodic systems (1986) Proc. 25th IEEE Conf. Decision Contr., pp. 1436-1438. , Athens, GreeceKwon, W.H., Pearson, A.E., A modified quadratic cost problem and feedback stabilization of a linear system (1977) IEEE Trans. Autom. Control, 22 (5), pp. 838-842Montagnier, P., Spiteri, R.J., Angeles, J., The control of linear time-periodic systems using Floquet-Lyapunov theory (2004) Int. J. Control, 77 (5), pp. 472-490Phat, V.N., Ha, Q.P., New characterization of controllability via stabilizability and Riccati equation for LTV systems (2008) IMA J. Math. Control Inform., 25 (4), pp. 419-429Poubelle, M.A., Bitmead, R.R., Gevers, M.R., Fake algebraic Riccati techniques and stability (1988) IEEE Trans. Autom. Control, 33 (4), pp. 379-381Sastry, S., Nonlinear systems: Analysis, stability, and control (1999) Interdisciplinary Applied Mathematics, , Springer-Verlag, New YorkSilverman, L.M., Anderson, B.D.O., Controllability, observability and stability of linear systems (1968) SIAM J. Control, 6 (1), pp. 121-130Streit, D.A., Krousgrill, C.M., Bajaj, A.K., Dynamic stability of flexible manipulators performing repetitive tasks (1985) Robot. Man. Automation, 15, pp. 121-136Tornambè, A., Valigi, P., Asymptotic stabilization of a class of continuous-time linear periodic systems (1996) Syst. Control Letts., 28 (4), pp. 189-196Varga, A., On solving periodic Riccati equations (2008) Num. Lin. Alg. Appl., 15 (9), pp. 809-835Willems, J.L., Kučera, V., Brunovský, P., On the assignment of invariant factors by time-varying feedback strategies (1984) Syst. Control Letts., 5 (2), pp. 75-80Zadeh, L.A., Desoer, C.A., Linear system theory - The state space approach (1963) McGraw Hill Series in System Science, , McGraw Hill, New York: McGraw-HillZhou, B., Duan, G.R., Lin, Z., A parametric periodic Lyapunov equation with application in semiglobal stabilization of discrete-time periodic systems subject to actuator saturation (2011) Automatica, 47 (2), pp. 316-325Zhou, B., Zheng, W.X., Duan, G.R., Stability and stabilization of discrete-time periodic linear systems with actuator saturation (2011) Automatica, 47 (8), pp. 1813-182