Abstract
The robustness of asymptotic stability with respect to measurement noise for discrete-time feedback control systems is discussed. It is observed that, when attempting to achieve obstacle avoidance or regulation to a disconnected set of points for a continuous-time system using sample and hold state feedback, the noise robustness margin necessarily vanishes with the sampling period. With this in mind, we propose two modifications to standard model predictive control (MPC) to enhance robustness to measurement noise. The modifications involve the addition of dynamical states that make large jumps. Thus, they have a hybrid flavor. The proposed algorithms are well suited for the situation where one wants to use a control algorithm that responds quickly to large changes in operating conditions and is not easily confused by moderately large measurement noise and similar disturbances.
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References
A. Bemporad, F. Borrelli, and M. Morari. Optimal controllers for hybrid systems: Stability and piecewise linear explicit form. Proc. IEEE Conf. on Decision and Control, Vol. 2, pp. 1810–1815, (2000).
F.H. Clarke, S. Yu, S. Ledyaev, L. Rifford, and R.J. Stern. Feedback stabilization and Lyapunov functions. SIAM Journal on Control and Optimization, Vol. 39, pp. 25–48, (2000).
P. Collins. A trajectory-space approach to hybrid systems. Proc. of the International Symposium on Mathematical Theory of Networks and Systems, (2004).
J-M. Coron and L. Rosier. A relation between continuous time-varying and discontinuous feedback stabilization. Journal of Mathematical Systems, Estimation, and Control, Vol. 4, pp. 67–84, (1994).
R. Goebel, J.P. Hespanha, A.R. Teel, C. Cai, and R.G. Sanfelice. Hybrid systems: Generalized solutions and robust stability. Proc. NOLCOS, Vol. 1, pp. 1–12, (2004).
R. Goebel and A.R. Teel. Results on solution sets to hybrid systems with applications to stability theory. Proc. American Control Gonf., Vol. 1, pp. 557–562, (2005).
R. Goebel and A.R. Teel. Solutions to hybrid inclusions via set and graphical convergence. Automatica, Vol. 42, pp. 573–587, (2006).
G. Grimm, M.J. Messina, S.E. Tuna, and A.R. Teel. Examples when nonlinear model predictive control is nonrobust. Automatica, Vol. 40, pp. 1729–1738, (2004).
G. Grimm, M.J. Messina, S.E. Tuna, and A.R. Teel. Model predictive control: for want of a local control Lyapunov function, all is not lost. IEEE Trans. Auto. Control, Vol. 50, pp. 546–558, (2005).
W.P.M.H Heemels, B. De Schutter, and A. Bemporad. Equivalence of hybrid dynamical models. Automatica, Vol. 37, pp.1085–1091, (2001).
Y. Kuwata and J. How. Receding horizon implementation of MILP for vehicle guidance. In Proc. American Control Gonf., pp. 2684–2685, (2005).
T. Lapp and L. Singh. Model predictive control based trajectory optimization for nap-of-the-earth (NOE) flight including obstacle avoidance. Proc. American Control Conf., Vol. 1, pp. 891–896, (2004).
J. Lygeros, K.H. Johansson, S.N. Simić, J. Zhang, and S.S. Sastry. Dynamical properties of hybrid automata. IEEE Trans. Auto. Control, Vol. 48, pp. 2–17, (2003).
D.Q. Mayne, J.B. Rawlings, C.V. Rao, and P.O.M. Scokaert. Constrained model predictive control: Stability and optimality. Automatica, Vol. 36, pp. 789–814, (2000).
M.J. Messina, S.E. Tuna, and A.R. Teel. Discrete-time certainty equivalence out-put feedback: allowing discontinuous control laws including those from model predictive control. Automatica, Vol. 41, pp. 617–628, (2005).
P. Mhaskar, N.H. El-Farrah, and P.D. Christofides. Robust hybrid predictive con-trol of nonlinear systems. Automatica, Vol. 41, pp. 209–217, (2005).
H. Michalska and D.Q. Mayne. Robust receding horizon control of constrained nonlinear systems. IEEE Trans. Auto. Control, Vol. 38, pp. 1623–1633, (1993).
P. Ogren and N.E. Leonard. A convergent dynamic window approach to obstacle avoidance. IEEE Trans. on Robotics, Vol. 21, pp. 188–195, (2005).
C. Prieur. Perturbed hybrid systems, applications in control theory. Nonlinear and Adaptive Control NCN4 2001 (Lecture Notes in Control and Information Sciences Vol. 281), pp. 285–294, (2003).
R. G. Sanfelice, R. Goebel, and A.R. Teel. Invariance principles for hybrid systems with connections to detectability and asymptotic stability. To appear in IEEE Trans. Auto. Control, (February 2008).
E.D. Sontag. Clocks and insensitivity to small measurement errors. Control, Op-timisation and Calculus of Variations, Vol. 4, pp. 537–557, (1999).
A. van der Schaft and H. Schumacher. An Introduction to Hybrid Dynamical Systems (Lecture Notes in Control and Information Sciences Vol. 251), (2000).
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Tuna, S.E., Sanfelice, R.G., Messina, M.J., Teel, A.R. (2007). Hybrid MPC: Open-Minded but Not Easily Swayed. In: Findeisen, R., Allgöwer, F., Biegler, L.T. (eds) Assessment and Future Directions of Nonlinear Model Predictive Control. Lecture Notes in Control and Information Sciences, vol 358. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72699-9_2
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DOI: https://doi.org/10.1007/978-3-540-72699-9_2
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