On the stability of receding horizon control with a general terminal cost

We study the stability and region of attraction properties of a family of receding horizon schemes for nonlinear systems. Using Dini's theorem on the uniform convergence of functions, we show that there is always a finite horizon for which the corresponding receding horizon scheme is stabilizing without the use of a terminal cost or terminal constraints. After showing that optimal infinite horizon trajectories possess a uniform convergence property, we show that exponential stability may also be obtained with a sufficient horizon when an upper bound on the infinite horizon cost is used as terminal cost. Combining these important cases together with a sandwiching argument, we are able to conclude that exponential stability is obtained for input-constrained receding horizon schemes with a general nonnegative terminal cost for sufficiently long horizons. Region of attraction estimates are also included in each of the results.

[1]  B. Anderson,et al.  Optimal control: linear quadratic methods , 1990 .

[2]  J. Ball.  OPTIMIZATION—THEORY AND APPLICATIONS Problems with Ordinary Differential Equations (Applications of Mathematics, 17) , 1984 .

[3]  E. Gilbert,et al.  Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: Stability and moving-horizon approximations , 1988 .

[4]  A. Schaft.  L2-Gain and Passivity Techniques in Nonlinear Control. Lecture Notes in Control and Information Sciences 218 , 1996 .

[5]  Andrew R. Teel,et al.  Model predictive control: for want of a local control Lyapunov function, all is not lost , 2005, IEEE Transactions on Automatic Control.

[6]  Ali Jadbabaie,et al.  Control of a thrust‐vectored flying wing: a receding horizon—LPV approach , 2002 .

[7]  Giuseppe Buttazzo,et al.  One-dimensional Variational Problems , 1998 .

[8]  James B. Rawlings,et al.  Constrained linear quadratic regulation , 1998, IEEE Trans. Autom. Control..

[9]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[10]  Arjan van der Schaft.  On a state space approach to nonlinear Hi control , 1990 .

[11]  John R. Hauser,et al.  Control of the Caltech ducted fan in forward flight: a receding horizon-LPV approach , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[12]  Jie Yu,et al.  Unconstrained receding-horizon control of nonlinear systems , 2001, IEEE Trans. Autom. Control..

[13]  A. Jadbabaie,et al.  Unconstrained receding horizon control with no terminal cost , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[14]  A. Schaft.  On a state space approach to nonlinear H ∞ control , 1991 .

[15]  Ali Jadbabaie,et al.  Unconstrained receding horizon control: Stability and region of attraction results , 1999 .

[16]  D. Mayne,et al.  Robust receding horizon control of constrained nonlinear systems , 1993, IEEE Trans. Autom. Control..

[17]  D. Chmielewski,et al.  On constrained infinite-time linear quadratic optimal control , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[18]  G. Nicolao,et al.  Stabilizing receding-horizon control of nonlinear time-varying systems , 1998, IEEE Trans. Autom. Control..

[19]  Lamberto Cesari,et al.  Optimization-Theory And Applications , 1983 .

[20]  B. Dacorogna.  Direct methods in the calculus of variations , 1989 .

[21]  Thomas Parisini,et al.  A receding-horizon regulator for nonlinear systems and a neural approximation , 1995, Autom..

[22]  Frank Allgöwer,et al.  A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability , 1997, 1997 European Control Conference (ECC).

[23]  Andrew R. Teel,et al.  Examples when nonlinear model predictive control is nonrobust , 2004, Autom..

[24]  David Q. Mayne,et al.  Constrained model predictive control: Stability and optimality , 2000, Autom..

[25]  E B Lee,et al.  Foundations of optimal control theory , 1967 .

[26]  J. Hauser,et al.  On the geometry of optimal control: the inverted pendulum example , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[27]  John Doyle,et al.  A receding horizon generalization of pointwise min-norm controllers , 2000, IEEE Trans. Autom. Control..

[28]  F. Allgöwer,et al.  A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability , 1997 .

[29]  L. Magni,et al.  Stability margins of nonlinear receding-horizon control via inverse optimality , 1997 .

[30]  James A. Primbs,et al.  Feasibility and stability of constrained finite receding horizon control , 2000, Autom..