Control Design with Guaranteed Transient Performance via Reachable Set Estimates

Abstract We present a novel way of addressing the problem of control design with guaranteed transient performance for discrete-time linear systems. We establish a theorem that gives necessary and sufficient conditions for the state to evolve from one polyhedral subset of the state-space to another. An algorithm is then presented that, if executed successfully, produces a time-varying state-feedback control law along with a sequence of polyhedral sets within which the control law guarantees the state to evolve. The approach is demonstrated by an example involving the position control of a mass-spring-damper. We then point out various issues that may form the basis of future research.

[1]  Chengyu Cao,et al.  Design and Analysis of a Novel L1 Adaptive Controller, Part II: Guaranteed Transient Performance , 2006, 2006 American Control Conference.

[2]  Robert R. Bitmead,et al.  Transient bounds for adaptive control systems , 1994, IEEE Trans. Autom. Control..

[3]  Stephan Trenn,et al.  Input constrained funnel control with applications to chemical reactor models , 2004, Syst. Control. Lett..

[4]  Yan Gao,et al.  On the Reachability Problem for Uncertain Hybrid Systems , 2007, IEEE Transactions on Automatic Control.

[5]  Charalampos P. Bechlioulis,et al.  A low-complexity global approximation-free control scheme with prescribed performance for unknown pure feedback systems , 2014, Autom..

[6]  Charalampos P. Bechlioulis,et al.  Adaptive control with guaranteed transient and steady state tracking error bounds for strict feedback systems , 2009, Autom..

[7]  Eugene P. Ryan,et al.  Funnel Control With Saturation: Nonlinear SISO Systems $ $ , 2010, IEEE Transactions on Automatic Control.

[8]  B. Krogh,et al.  Computing polyhedral approximations to flow pipes for dynamic systems , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[9]  Chris Sangwin,et al.  Tracking with prescribed transient behaviour , 2002 .

[10]  Alexandre M. Bayen,et al.  A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games , 2005, IEEE Transactions on Automatic Control.

[11]  Daniel E. Miller,et al.  An adaptive controller which provides an arbitrarily good transient and steady-state response , 1991 .

[12]  Miroslav Krstic,et al.  Transient-performance improvement with a new class of adaptive controllers , 1993 .

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

[14]  Lin Yan,et al.  A variable structure MRAC with expected transient and steady-state performance , 2006, Autom..

[15]  Daniel Liberzon,et al.  The Bang-Bang Funnel Controller for Uncertain Nonlinear Systems With Arbitrary Relative Degree , 2013, IEEE Transactions on Automatic Control.

[16]  Charalampos P. Bechlioulis,et al.  Robust Adaptive Control of Feedback Linearizable MIMO Nonlinear Systems With Prescribed Performance , 2008, IEEE Transactions on Automatic Control.

[17]  P. Varaiya,et al.  Dynamic Optimization for Reachability Problems , 2001 .

[18]  G. Bitsoris.  Positively invariant polyhedral sets of discrete-time linear systems , 1988 .