Robust input shaper design using Linear Matrix Inequalities

Abstract

This paper proposes an Linear Matrix Inequality based problem formulation to determine input shaped profiles. The cost function is the residual energy, a quadratic function of the amplitude of the shaped profile, for each sampling interval. The Schur complement permits representing the quadratic function as a Linear Matrix Inequality. Augmenting the state space model with the sensitivity of the states to uncertain parameters, input shaped profiles which are robust to model uncertainties can be derived. Finally, a minimax input shaped profile which minimizes the maximum magnitude of the residual energy over the domain of uncertainties is determined using the LMI problem. The proposed technique is illustrated on the single spring-mass-dashpot example. The solutions derived are shown to coincide with the solutions presented in the literature, without the requirement of solving a nonlinear programming problem.

Quantitative Feedback Input Shaping for Flexible-Joint Robot Manipulator

“…Other robust input shapers use different concepts, including enforcing constraints on the vibration amplitude over a userspecified range of natural frequencies [23], solving an optimization problem to minimize the maximum magnitude of the residual states over a range of uncertain parameter values [24], considering the probability distribution of the natural frequency about its modeled value during the shaper design [25,26], and formulating the design problem as an optimization problem having linear matrix inequality constraints [27].…”

“…Disadvantages of the input shaping techniques that have been used with the flexible-joint robots [15][16][17][18][19][20], the previously proposed robust input shapers [3,[21][22][23][24][25][26][27], as well as the previously proposed input shaping with model matching [28][29][30][31][32][33] are that more robustness must come at the price of having longer move time, the plant must be linear-and time-invariant, and vibration induced by disturbance and noise cannot be suppressed.…”

Intelligent Backstepping System to Increase Input Shaping Performance in Suppressing Residual Vibration of a Flexible-Joint Robot Manipulator

“…, , , , l r r r L Link length, dimensions 1 2 3 ,, r r r in Fig. 4(b), unstretched spring length (45) Upon substituting the parameter values in Table 1 into (44), the mode parameters, which are the natural frequency and damping ratio, are given by 16.206 rad/s, 0.052.…”

“…This modified filter ensures a more uniform output for each discrete-time sample as the system parameters vary with time. Conord and Singh in [45] formulated the design of the robust input shaper as an optimization problem having linear matrix inequalities (LMI) constraints. Vaughan et al in [46] and [47] compared among several types of robust input shapers and also investigated combining the specified negative amplitude (SNA) shaper (Singhose et al in [48]) with the SI shaper.…”