### ESTIMATES OF REACHABLE SETS OF CONTROL SYSTEMS WITH BILINEAR–QUADRATIC NONLINEARITIES

#### Abstract

The problem of estimating reachable sets of nonlinear impulsive control systems with quadratic nonlinearity and with uncertainty in initial states and in the matrix of system is studied. The problem is studied under uncertainty conditions with set – membership description of uncertain variables, which are taken to be unknown but bounded with given bounds. We study the case when the system nonlinearity is generated by the combination of two types of functions in related differential equations, one of which is bilinear and the other one is quadratic. The problem may be reformulated as the problem of describing the motion of set-valued states in the state space under nonlinear dynamics with state velocities having bilinear-quadratic kind. Basing on the techniques of approximation of the generalized trajectory tubes by the solutions of control systems without measure terms and using the techniques of ellipsoidal calculus we present here a state estimation algorithms for the studied nonlinear impulsive control problem bilinear-quadratic type.

#### Keywords

#### Full Text:

PDF#### References

Bertsekas D.P. and Rhodes I.B. Recursive state estimation for a set-membership description of uncertainty // IEEE Transactions on Automatic Control. 1971. No. 16. P. 117–128.

Boyd S., El Ghaoui L., Feron E. and Balakrishnan V. Linear Matrix Inequalities in System and Control Theory // SIAM Studies in Applied Mathematics, 1994. Vol. 15. SIAM. 193 p.

Chernousko F.L. State Estimation for Dynamic Systems. CRC Press. Boca Raton. 1994. 320 p.

Chernousko F.L. Ellipsoidal approximation of the reachable sets of linear systems with an indeterminate matrix // Applied Mathematics and Mechanics. 1996. Vol. 60, no. 6. P. 940–950.

Chernousko F.L. and Rokityanskii D.Ya. Ellipsoidal bounds on reachable sets of dynamical systems with matrices subjected to uncertain perturbations // J. of Optimiz. Theory and Appl. 2000. Vol. 104, no. 1. P. 1–19.

Chernousko F.L. and Ovseevich A.I. Properties of the optimal ellipsoids approximating the reachable sets of uncertain systems // J. of Optimiz. Theory and Appl. 2004. Vol. 120, no. 2. P. 223–246.

Filippova T.F. Estimates of Trajectory Tubes of Uncertain Nonlinear Control Systems // Lect. Notes in Comput. Sci. 2010. Vol. 5910. P. 272–279.

Filippova T.F. Trajectory tubes of nonlinear differential inclusions and state estimation problems // J. of Concrete and Applicable Mathematics, Eudoxus Press, LLC, 2010. No. 8. P. 454–469.

Filippova T.F. Set-valued dynamics in problems of mathematical theory of control processes // International J. of Modern Physics. Series B (IJMPB). 2012. Vol. 26, no. 25. P. 1–8.

Filippova T.F. and Lisin D.V. On the estimation of trajectory tubes of differential inclusions // Proc. Steklov Inst. Math.: Problems Control Dynam. Systems. 2000. Suppl. 2. P. S28–S37.

Filippova T.F. and Matviychuk O.G. Algorithms to estimate the reachability sets of the pulse controlled systems with ellipsoidal phase constraints // Automat. Remote Control. 2011. Vol. 72, no. 9. P. 1911–1924.

Filippova T.F. and Matviychuk O.G. Algorithms of estimating reachable sets of nonlinear control systems with uncertainty // Proc. of the 7th Chaotic Modeling and Simulation Int. Conf. 2014. Lisbon, Portugal: 7-10 June. P. 115–124.

Goncharova E.V. and Ovseevich A.I. Asymptotic theory of attainable set to linear periodic impulsive control systems // J. of Computer and Systems Sciences International. 2010. Vol. 49, no. 4. P. 515–523.

Kurzhanski A.B. and Filippova T.F. On the theory of trajectory tubes – a mathematical formalism for uncertain dynamics, viability and control // Advances in Nonlinear Dynamics and Control: a Report from Russia, Progress in Systems and Control Theory, A.B. Kurzhanski (Ed.) 1993. Birkhauser, Boston. Vol. 17. P. 22–188.

Kurzhanski A.B. and Valyi I. Ellipsoidal Calculus for Estimation and Control. Birkhauser, Boston. 1997. 321 p.

Kurzhanski A.B. and Varaiya P. Dynamics and Control of Trajectory Tubes. Theory and Computation. Springer-Verlag, New York. 2014. 445 p.

Matviychuk O.G. Estimation Problem for Impulsive Control Systems under Ellipsoidal State Bounds and with Cone Constraint on the Control // AIP Conf. Proc. 2012. Vol. 1497. P. 3–12.

Mazurenko S.S. A differential equation for the gauge function of the star-shaped attainability set of a differential inclusion // Doklady Mathematics. 2012. Vol. 86, no. 1, P. 476–479.

Milanese M., Norton J., Piet-Lahanier H. and Walter E. (Eds.). Bounding Approaches to System Identification. Springer US, New York. 1996. 567 p.

Polyak B.T., Nazin S.A., Durieu C. and Walter E. Ellipsoidal parameter or state estimation under model uncertainty // Automatica. 2004. Vol. 40. P. 1171–1179.

Rishel R. W. An Extended Pontryagin Principle for Control System whose Control Laws Contain Measures. SIAM J. Control, 1965. Series A Control. Vol. 3, no. 2. P. 191–205.

Schweppe F. Uncertain Dynamic Systems: Modelling, Estimation, Hypothesis Testing, Identification and Control. Prentice-Hall, Englewood Cliffs, New Jersey. 1973. 576 p.

Walter E. and Pronzato L. Identification of parametric models from experimental data. Springer-Verlag London. 1997. 413 p.

#### Article Metrics

### Refbacks

- There are currently no refbacks.