Transfer Equivalence and Realization of Nonlinear Input-Output Delta-Differential Equations on Homogeneous Time Scales

Abstract

Fig. 4. Constraint variables z (dotted), v (dashed), and v (solid) for the upper constraint of the uncertain active magnetic bearing system for the initial condition x(0) = [0:5; 0; 0] and the safety control law.while this would clearly happen without the invariance control. In Fig. 4 the time trajectories of the three constraints z 1;1 ; v 1;1 ; v 1;2 are shown. All three constraints are kept below zero at all times, hence the overall constraint is satisfied. The constraint v1;2 (solid line) would be the first to become positive, which is avoided by the invariance controller. VI. CONCLUSIONIn this paper, an algorithm is proposed to construct control laws for uncertain nonlinear systems under hard state constraints. Using a recursive design procedure, a set of safe initial conditions is constructed that can be rendered invariant by a suitable control action. A switching control is designed to guarantee positive invariance of the set and to satisfy a nominal control objective whenever possible. With some additional assumptions, stability of the closed loop is established.The current work focuses primarily on single input systems. However simulations show that a generalization to multiinput systems is possible. An important problem that remains open is with regards to the simultaneous satisfaction of state and input constraints. The backstepping approach seems to offer the flexibility to integrate both challenges into the controller design. Future research will aim to provide a sound theoretical analysis of this problem. ACKNOWLEDGMENTThe authors wish to thank the anonymous reviewers, who helped to improve the paper significantly. REFERENCES[1] M. Bürger and M. Guay, "Robust constraint satisfaction for continuous-time nonlinear systems," in Proc. 47th Conf.[8] P. Tsiotras and M. Arcak, "Low-bias control of amb subject to voltage saturation: State-feedback and observer designs," IEEE Trans.Abstract-Nonlinear control systems on homogeneous time scales are studied. First the concepts of reduction and irreducibility are extended to higher order delta-differential input-output equations. Subsequently, a definition of system equivalence is introduced which generalizes the notion of transfer equivalence in the linear case. Finally, the necessary and sufficient conditions are given for the existence of a state-space realization of a nonlinear input-output delta-differential equation.

Explicit formulas for the state coordinates in nonlinear MIMO realization problem on homogeneous time scales

“…In this paper, we consider one of such problems -the nonlinear state-space realization for control systems defined on homogeneous time scale. The problem has been studied before for single-input single-output (SISO) systems in [6] and [2]. In [6] the necessary and sufficient realizability conditions were given.…”

“…The problem has been studied before for single-input single-output (SISO) systems in [6] and [2]. In [6] the necessary and sufficient realizability conditions were given. Note that the state coordinates in [6] were computed using a recursive algorithm, that requires to solve at each step a set of nonlinear equations, that is known to be a bottleneck problem in Computer Algebra Systems (CAS) even for examples of medium-complexity.…”

“…In [6] the necessary and sufficient realizability conditions were given. Note that the state coordinates in [6] were computed using a recursive algorithm, that requires to solve at each step a set of nonlinear equations, that is known to be a bottleneck problem in Computer Algebra Systems (CAS) even for examples of medium-complexity. Therefore, starting from input-output (i/o) system description that can be transformed into the state-space form, in [2] the goal was to find explicit formulas that allow to construct the state equations in an efficient manner, both theoretically and computationally.…”

Static state feedback linearization of nonlinear control systems on homogeneous time scales

“…Though the literature on time scale calculus is rich (see the references in [3,6,9,16,19,20,32,34,38]), there is not so many papers addressing the control problems for nonlinear systems defined on time scale [7,12,28]. In the earlier papers [4,5] the algebraic formalism has been developed that allows to address various analysis and control problems for nonlinear systems, defined on homogeneous time scales.…”

“…In the earlier papers [4,5] the algebraic formalism has been developed that allows to address various analysis and control problems for nonlinear systems, defined on homogeneous time scales. This approach has been successfully applied to address the irreducibility, reduction [28] and realization of the input-output equations in the state space form [12].…”