Shaping State-Dependent Convergence Rates in Nonlinear Control System Design

Abstract

This paper derives for non-linear, time-varying and feedback linearizable systems simple controller designs to achieve specified state-and timedependent complex convergence rates. This approach can be regarded as a general gain-scheduling technique with global exponential stability guarantee. Typical applications include the transonic control of an aircraft with strongly Mach or time-dependent eigenvalues or the state-dependent complex eigenvalue placement of the inverted pendulum.As a generalization of the LTI Luenberger observer a dual observer design technique is derived for a broad set of non-linear and time-varying systems, where so far straightforward observer techniques were not known. The resulting observer design is illustrated for non-linear chemical plants, the Van-der-Pol oscillator, the discrete logarithmic map series prediction and the lighthouse navigation problem.These results [23] allow one to shape globally the state-and time-dependent convergence behaviour ideally suited to the non-linear or time-varying system. The technique can also be used to provide analytic robustness guarantees against modelling uncertainties.The derivations are based on non-linear contraction theory [18], a comparatively recent dynamic system analysis tool whose results will be reviewed and extended.

Contraction analysis of nonlinear Hamiltonian systems

“…Consider the 2D landmark problem [22], [23] of navigating a vehicle using only azimuth measurements to a fixed point in space, called landmark in the following. The dynamic equations of the landmark's relative motion to the vehicle areẋ = −u with 2D position x = (x 1 , x 2 ) T , vehicle velocity u = (u 1 , u 2 ) T , and azimuth measurement θ = arctan( x1 x2 ).…”

“…Implicit nonlinear measurements enable LTV Kalman-filter tools to be exploited, and information from additional landmarks or features can be recursively incorporated. Concluding remarks are offered in section V. Nonlinear contraction theory [16]- [22], [32], [35] is a comparatively recent dynamic analysis and design tool based on an exact differential analysis of convergence of complex systems. Contraction theory converts a nonlinear stability problem into an LTV (linear time-varying) first-order stability problem by considering the convergence behavior of neighboring trajectories.…”

“…Depending on the application, the metric can be found trivially (identity or rescaling of states), or obtained from physics (say, based on the inertia tensor in a mechanical system as e.g. in [23], [24]). The rate of change of a finite length can now be bounded by where max s (λ max )(min s (λ min )) is the largest (smallest) eigenvalue of the Hermitian part of Λ along the path s. The basic theorem of contraction analysis [17], [18] can hence be stated as Theorem 1: Given the system equationsẋ = f (x, t), where f is a differentiable nonlinear complex function of x within C n , any distance s = min s…”

“…[12]) to the general nonlinear and time-varying case. This leads to the practical controller or observer designs in [6], [16]- [20], [22], [32], [35] and serves as a basis for this paper. For an exhaustive summary of all features of contraction theory the reader is referred to [17].…”

Exact Decomposition and Contraction Analysis of Nonlinear Hamiltonian Systems

“…Nonlinear contraction theory 15,16,17,18,19,20,21,22,32,33 is a comparatively recent dynamic analysis and design tool based on an exact differential analysis of convergence of complex systems. Contraction theory converts a nonlinear stability problem into an LTV (linear time-varying) first-order stability problem by considering the convergence behavior of neighboring trajectories.…”