Baranyi and Yam proposed the TP model transformation[1][2][3][4][5][6][7] as a new concept in quasi-LPV (qLPV) based control, which plays a central role in the highly desirable bridging between identification and polytopic systems theories.
It is also used as a TS (Takagi-Sugeno) fuzzy model transformation.
It is uniquely effective in manipulating the convex hull of polytopic forms (or TS fuzzy models), and, hence, has revealed and proved the fact that convex hull manipulation is a necessary and crucial step in achieving optimal solutions and decreasing conservativeness[8][9][2] in modern linear matrix inequality based control theory.
Thus, although it is a transformation in a mathematical sense, it has established a conceptually new direction in control theory and has laid the ground for further new approaches towards optimality.
For details please visit: TP model transformation.
A free MATLAB implementation of the TP model transformation can be downloaded at [1] or an old version of the toolbox is available at MATLAB Central [2].
Be careful, in the MATLAB toolbox the assignments of the dimensions of the core tensor is in the opposite way in contrast to the notation used in the related literature.
In some variants of the ToolBox, the first two dimension of the core tensor is assigned to the vertex systems.
-dimensional parameter vector which is an element of closed hypercube
As a matter of fact, further parameter dependent channels can be inserted to
in the above LPV model can also include some elements of the state vector
, and, hence this model belongs to the class of non-linear systems, and is also referred to as a quasi LPV (qLPV) model.
-dimensional parameter vector which is an element of closed hypercube
As a matter of fact, further parameter dependent channels can be inserted to
of the system (within the convex hull defined by the vertexes) for all
Note that the TP type polytopic model can always be given in the form where the vertexes are the same as in the TP type polytopic form and the multi variable weighting functions are the product of the one variable weighting functions according to the TP type polytopic form, and r is the linear index equivalent of the multi-linear indexing
, whose TP polytopic structure may be unknown (e.g. it is given by neural networks).
The TP model transformation determines its TP polytopic structure as namely it generates core tensor
If the given model does not have (finite element) TP polytopic structure, then the TP model transformation determines its approximation: where trade-off is offered by the TP model transformation between complexity (number of vertexes stored in the core tensor or the number of weighting functions) and the approximation accuracy.
[10] The TP model can be generated according to various constrains.
Typical TP models generated by the TP model transformation are: Since the TP type polytopic model is a subset of the polytopic model representations, the analysis and design methodologies developed for polytopic representations are applicable for the TP type polytopic models as well.
One typical way is to search the nonlinear controller in the form: where the vertexes
are substituted into Linear Matrix Inequalities in order to determine
In TP type polytopic form the controller is: where the vertexes
The polytopic representation of a given qLPV model is not invariant.
In order to generate an optimal control of the given model
Thus, if we apply the selected LMIs to the above polytopic model we arrive at: Since the LMIs realize a non-linear mapping between the vertexes in
The TP model transformation let us to manipulate the weighting functions systematically that is equivalent to the manipulation of the vertexes.
that can be given by the same vertexes, but with different weighting functions as: where If one of these systems are very hardly controllable (or even uncontrollable) then we arrive at a very conservative solution (or unfeasible LMIs).
Therefore, we expect that during tightening the convex hull we exclude such problematic systems.