Variational integrator for constrained mechanical systems with pulsed disturbances and optimal feedback control

Abstract

An approach to minimize the control costs and ensuring a stable deviation control is the Riccati controller and we use it to control constrained dynamical systems (differential algebraic equations of Index 3) with pulsed disturbances. To describe their discrete dynamics, a constrained variational integrator [1] is used. Using a discrete version of the Lagrange-d'Alembert principle yields a forced constrained discrete Euler-Lagrange equation in a position-momentum form that depends on the current and future time steps [2]. The desired optimal trajectory (qopt, popt) and according control input uopt is determined solving the discrete mechanics and optimal control (DMOC) algorithm [3] based on the variational integrator. Then, during time stepping of the perturbed system, the discrete Riccati equation yields the optimal deviation control input uR. Adding uopt and uR to the discrete Euler-Lagrange equation causes a structure preserving trajectory as both DMOC and Riccati equations are based on the same variational integrator. Furthermore, coordinate transformations are implemented (minimal, redundant and nullspace) enabling the choice of different coordinates in the feedback loop and in the optimal control problem. Variational integrator and DMOCToday, a lot of mechanical systems have to operate with an improved performance compared to equal constructions decades ago. Reasons therefore are e.g. higher energy costs and a globalised market with more competitors. To stay competitive, engineers of a mechanical system need to guarantee its optimal operation. This includes both offline optimisation of a desired trajectory as well as online feedback control for eliminating perturbations from the optimal trajectory. In the simulation here, the variational integrator as a variant of a structure-preserving integration scheme is used. A forward rectangular rule is used to approximate the action in one time interval via a discrete LagrangianL(q(s),q(s))ds with configuration sequence q k ≈ q(t k ) for k = 0, . . . , N . The configuration q(t) is approximated by a linear polynomial and the velocity by finite differences. Applying a discrete variational principle δS d ({q k } N k=0 ) = 0, see [1], and an approximation of the virtual work F ± d (q k , q k+1 , u k ) with control sequence {u k } N −1 k=0 , the Lagrange-d'Alembert principle yields a discrete Euler-Lagrange equation in a "position-momentum form that only depends on the current and future time steps" [2]. Part of the virtual work is the discretisation of a pulsed disturbance force F z (t), see [4], acting at time node t ∈ [t k , t k+1 ) and being defined as F z (t) = F t δ (t − t ), outlined in Figure 1. To compute the desired trajectory, initial and final conditions on the configuration and conjugate momentum together with the discrete equations in minimal coordinates serve as non-linear equality constraints for the minimisation of a given objective functional. Applying the DMOC (discrete mechanics and optimal control [3]) algorithm, an optimal trajectory and accordin...