Influence of the Tensor Product Model Representation Of QLPV Models on The Feasibility of Linear Matrix Inequality

Abstract

The present paper proves that the vertexes of the tensor product (TP) model type polytopic representation of a given quasi linear parameter varying (qLPV) state‐space model strongly interfere with the feasibility regions of linear matrix inequality (LMI)‐based control design methods. Furthermore this is valid both for the LMI‐based feasibility of the controller and the observer design, but the influence differs for the controller and the observer system components. More specifically, the factors influencing the feasibility regions of the LMI‐based control design include: (i) the manipulation of the vertexes' position; and (ii) the size and complexity of the TP model type polytopic representation, i.e. the number of the vertexes contained in the TP model representation. The proof is based on a complex control design example, where the influence of these factors stated above can be easily and clearly indicated. Furthermore the paper shows via the example that the maximal parameter space of the controller and observer also depends on these factors. The example model consists of the complex Nonlinear Aeroelastic Test Apparatus (NATA) model of the three degree of freedom aeroelastic wing section model including Stribeck friction and the control design method is based on the relaxed TP model transformation‐based control design framework that supports the flexible manipulation of these factors.

Influence of the Manipulation of the Polytopic Tensor Product Model Representation on the Control Performance of LMI Based Design

“…Similar to papers [25]- [27], showing a correlation between the feasibility of the LMI based design and the manipulation of the convex hull of the TP model type polytopic representation, the present paper demonstrates also a tendency of influence extending on the control performance: the results in the paper present that the convex hull manipulation resulted in different results for the different control signals, moreover, a manipulated case even provided better results. This shows that the manipulation leads to further potential optimization possibilities and is a necessary and important step to find the best solution.…”

“…Based on this model, the paper continues a study series investigating the effects of the manipulation of the TP type polytopic representation. The motivation behind this lies in the fact, that previous publications have shown results, where the manipulation of the TP type polytopic representation -namely the convex hull, complexity and parameter space -influences the LMI based design's feasibility [25]- [27]. This indicates that the manipulation of the TP type polytopic representation is important and necessary to achieve the optimal control results for the LMI based design.…”

An Approach to the Design of Nonlinear State-Space Control Systems

“…Since the approach is general and applicable to LTI, LTV, LPV and q-LPV systems, future research will be focused on the stable control of fuzzy systems and systems resulted from Tensor Product model transformation, as nonlinear systems with illustrative results related to general theory [2], type-2 fuzzy control [19], three-tank systems [27], Antilock Braking Systems [28] and q-LPV systems [36]. The optimal design of state-space control systems in terms of classical algorithms applied to drilling processes [10], input shapers [11], large-scale complex systems [13], predictive functional control [34], compartment models [37] and networked control systems [41], and nature-inspired algorithms applied to fuzzy model identification [15] …”

Tensor Product Model Transformation Simplification of Takagi-Sugeno Control and Estimation Laws – An Application to a Thermoelectric Controlled Chamber

“…It is important to note that, even though the choice of the convex hull transformation can be a decisive factor on whether or not an LMI synthesis procedure is successful [14][15][16], they do not affect the feasibility of the LMIs in the strategy proposed in this paper (as they are used simply with the control and estimation laws as opposed to finding a convex representation used in the LMIs). Nevertheless, it is recommended to use the CNO transformation, as it will lead to control and estimation gains that can be more easily interpreted.…”