A vibroacoustic application of modeling and control of linear parameter-varying systems

Abstract

This paper applies recent advances in both modeling and control of Linear ParameterVarying (LPV)

New robust LMI synthesis conditions for mixed gain‐scheduled reduced‐order DOF control of discrete‐time LPV systems

“…To illustrate the aforementioned comments, note that, by choosing a Taylor series expansion of degree ℓ = 1, Theorem is not able to provide a stabilizing solution probably because of the high values of the bounds for the residues ( δ A = 1.2847 and δ B = 0.1018). Considering the same discretized system and ignoring δ A and δ B , theorem 8 in the work of De Caigny et al adapted for stabilization only provides the following stabilizing gain: On the other hand, using a higher degree of Taylor series expansion, for example, ℓ = 3, the bounds for the residues ( δ A = 0.1728 and δ B = 0.0108) are smaller, allowing to find a stabilizing SOF robust controller with Theorem , but only if the decision variables have degree g ≥ 2 of polynomial dependence on the time‐varying parameters. In this case, one obtains the following gain: by Theorem with g = 2, demanding V = 196 scalar variables, and L = 318 LMI rows to solve the optimization problem.…”

“…As expected, the performed simulations show that using the controllers obtained with Theorem , the output trajectories converge to zero (stable closed‐loop system). However, the controller obtained by theorem 8 in the work of De Caigny et al cannot stabilize the closed‐loop system because the bounds δ A and δ B have not been taken into account. Concerning the control of discretized systems obtained from polytopic continuous‐time systems, the result presented in Figure justifies the use of a polynomial representation of the system (coming from the Taylor series expansion of degree ℓ ) and the consideration of the norm‐bounded terms (that represent the residues of the truncated Taylor series).…”

“…Output trajectories of the continuous‐time system in the closed loop with the digital controllers obtained from (A) Theorem with g = 2 and ℓ = 3 considering the bounds for the residues ( δ A = 0.1728 and δ B = 0.0108); (B) Theorem with g = 1 and ℓ = 4 considering the bounds for the residues ( δ A = 0.0817 and δ B = 0.0057); and (C) Theorem 8 in the work of De Caigny et al with ℓ = 1 neglecting the bounds for the residues ( δ A = 1.2847 and δ B = 0.1018) [Colour figure can be viewed at wileyonlinelibrary.com]…”

“…† To illustrate the aforementioned comments, note that, by choosing a Taylor series expansion of degree = 1, Theorem 1 is not able to provide a stabilizing solution probably because of the high values of the bounds for the residues ( A = 1.2847 and B = 0.1018). Considering the same discretized system and ignoring A and B , theorem 8 in the work of De Caigny et al 55 adapted for stabilization only provides the following stabilizing gain:…”

Reduced-order &#x210B;<inf>2</inf>/&#x210B;<inf>&#x221E;</inf> control of discrete-time LPV systems with experimental validation on an overhead crane test setup

“…. , 3 in the LMI (8) results in the standard H ∞ performance characterization for discrete-time LPV systems [16], [17].…”