Class Of Affine Control Systems Research Articles

Consumption minimization for an academic model of a vehicle

AbstractThe present article is a study of an optimal control problem having a non‐differentiable, but Lipschitz, cost function. It is inspired by the minimisation of the energy consumption of a car‐like vehicle or robot along a road which profile is known. This problem is stated by means of a simple model of the longitudinal dynamics and a running cost that comprises both an absolute value function and a function that accounts for the efficiency of the energy conversion process. A regularity result that excludes chattering phenomena from the set of solutions is proven. It is valid for the class of control affine systems, which includes the considered problem. Three case studies are detailed and analysed. The optimal trajectories are shown to be made of bang‐bang, inactivated, singular and backward arcs.

Robust structural feedback linearization based on the nonlinearities rejection

In this paper, we consider a class of affine control systems and propose a new structural feedback linearization technique. This relatively simple approach involves a generic linear-type control scheme and follows the classic failure detection methodology. The robust linearization idea proposed in this contribution makes it possible an effective rejection of nonlinearities that belong to a specific class of functions. The nonlinearities under consideration are interpreted here as specific signals that affect the initially given systems dynamics. The implementability and efficiency of the proposed robust control methodology is illustrated via the attitude control of a Planar Vertical Take Off Landing (PVTOL) system.

State‐input constrained asymptotic null‐controllability by a set‐valued approach

This study presents a unified approach to investigate asymptotic null-controllability under possibly mixed constraints on state and control, for standard ordinary differential equation control systems. Using tools from set-valued analysis and viability theory, initial data which can be steered to the origin at infinity are provided through a new type of Lyapunov functions. The corresponding controls are given via selections of adequately designed multifunctions, which are examined in both cases of convex constraints and the class of affine-control systems. Finally, numerical examples from classical mechanics are given to verify the theoretical results.

Passivity and Passive Feedback Stabilization for a Class of Mixed Potential Systems

This paper studies the input-output properties of a class of control affine systems where the drift dynamics is generated by a metriplectic structure. Those systems, related to generalized (or dissipative) Hamiltonian systems, are generated by a conserved component and a dissipative component and appear, for example, in non-equilibrium thermodynamics. In non-equilibrium thermodynamics, the two potentials generating the dynamics are interpreted as generalized energy and generalized entropy, respectively. In this note, passivity and passive feedback stabilization of this class of systems are studied, with the output function taken as the gradient of the conserved component of the dynamics, and the proposed storage function is computed using the dissipative (metric) component of the dynamics.

GLOBAL CLF STABILIZATION OF SYSTEMS WITH POINT-DISSIPATIVE FREE DYNAMICS VIA ADMISSIBLE CONTROLS: CHAOS CONTROL OF A CHUA'S SYSTEM

The aim of this paper is to address the global asymptotic stabilization (GAS) of the class of affine control systems which are open-loop point-dissipatives (have compact global attractors), that include some chaotic systems (e.g. Lorenz, Chen, smooth Chua), via admissible (bounded and regular) feedback controls. The existence of the global attractor for the free dynamics, whose overestimation is achieved by a Lyapunov function V(x), allows us to use V(x) as a basis to obtain a control Lyapunov function (CLF) in order to apply our control design method (entailed from Artstein and Sontag's CLF-stabilization approach) developed for the GAS of affine control systems via admissible feedback controls. We illustrate the results with the chaos control of a smooth (cubic) Chua's system via admissible feedback controls.

Dynamic homotopy and landscape dynamical set topology in quantum control

We examine the topology of the subset of controls taking a given initial state to a given final state in quantum control, where “state” may mean a pure state |ψ⟩, an ensemble density matrix ρ, or a unitary propagator U(0, T). The analysis consists in showing that the endpoint map acting on control space is a Hurewicz fibration for a large class of affine control systems with vector controls. Exploiting the resulting fibration sequence and the long exact sequence of basepoint-preserving homotopy classes of maps, we show that the indicated subset of controls is homotopy equivalent to the loopspace of the state manifold. This not only allows us to understand the connectedness of “dynamical sets” realized as preimages of subsets of the state space through this endpoint map, but also provides a wealth of additional topological information about such subsets of control space.

Generalized Quadratic Linearization of Machine Models

In the exact linearization of involutive nonlinear system models, the issue of singularity needs to be addressed in practical applications. The approximate linearization technique due to Krener, based on Taylor series expansion, apart from being applicable to noninvolutive systems, allows the singularity issue to be circumvented. But approximate linearization, while removing terms up to certain order, also introduces terms of higher order than those removed into the system. To overcome this problem, in the case of quadratic linearization, a new concept called “generalized quadratic linearization” is introduced in this paper, which seeks to remove quadratic terms without introducing third‐ and higher‐order terms into the system. Also, solution of generalized quadratic linearization of a class of control affine systems is derived. Two machine models are shown to belong to this class and are reduced to only linear terms through coordinate and state feedback. The result is applicable to other machine models as well.

Guaranteed cost control of uncertain nonlinear systems via polynomial lyapunov functions

In this paper, we consider the problem of guaranteed cost control for a class of uncertain nonlinear systems. We derive linear matrix inequality conditions for the regional robust stability and performance problems based on Lyapunov functions which are polynomial functions of the state and uncertain parameters. The performance index is calculated over a set of initial conditions. Also, we discuss the synthesis problem for a class of affine control systems. Numerical examples illustrate our method.

Decomposition of homogeneous vector fields of degree one and representation of the flow

We first give a characterization for the set of real analytic diffeomorphisms which transform homogeneous vector fields of certain degree into homogeneous fields of the same degree with respect to an arbitrary dilation δεr. Such a set is constituted by the invertible analytic maps that are homogeneous of degree one with respect to δεr and can be endowed with the structure of a Lie Group whose Lie algebra is the space H1,r(ℝn) of the homogeneous fields of degree one with respect to δεr. Then we prove a decomposition theorem for the elements of the non semisimple Lie algebra H1,r(ℝn). This result is a non linear analog of the Jordan decomposition of a linear field, i.e. for X ∈ H1,r(ℝn), we can write X = S + N, with S linear semisimple and [S, N] = 0. We also give an explicit representation formula for the flow generated by a field in H1,r(ℝn). Finally we apply this result to obtain a simple representation for the trajectories of a class of affine control systems x˙=X0(x)+Bu, with X0 ∈ H1,r(ℝn) and В a constant field, that constitute a natural extension of the linear control systems.

Dynamic recurrent neural network for system identification and control

A dynamic recurrent neural network (DRNN) that can be viewed as a generalisation of the Hopfield neural network is proposed to identify and control a class of control affine systems. In this approach, the identified network is used in the context of the differential geometric control to synthesise a state feedback that cancels the nonlinear terms of the plant yielding a linear plant which can then be controlled using a standard PID controller.