This paper integrates various ideas about the tensor product (TP) model transformation into one conceptual framework and formulates it in terms of the Takagi-Sugeno (T-S) fuzzy model manipulation and control design framework. Several new extensions of the TP model transformation are proposed, such as the quasi and “full,” compact and rank-reduced higher order singular-value-decomposition-based canonical form of T-S fuzzy models, and the bilinear-, multi, pseudo-, convex-, partial TP model transformations. All of these extensions together form the generalized TP model transformation, which provides an effective tool to freely and readily manipulate the antecedent sets and rules of T-S fuzzy models and also provides main fuzzy rule component analysis, as well as a means for complexity and accuracy tradeoffs. It is demonstrated in this paper that the proposed manipulation forms a new, effective, and necessary optimization step of T-S fuzzy or polytopic models and linear-matrix-inequality-based control design, and can also decrease conservativeness. Identification techniques are typically constructed according to the available data and measurement set, as well as the type of system to be identified. As a result, they may not always provide good representations for control design frameworks. This paper demonstrates that the proposed TP model transformation is unique in that it bridges between various soft-computing-based identification techniques and T-S fuzzy model-based approaches. Finally, this paper proposes the multi-TP model transformation, which is a tractable and nonheuristic framework to verify the stability of the result of fuzzy or various soft-computing-based control designs. The multi-TP model transformation could provide an answer to the frequently emerging criticisms regarding the lack of mathematical stability verification techniques in the soft-computing-based control design. Control examples are provided in this paper.

The Generalized TP Model Transformation for T–S Fuzzy Model Manipulation and Generalized Stability Verification

Use of a recently introduced numerical robust-control design method to stabilize aeroelastic systems according to different control specifications is studied. This numerical design is based on the tensor-product model transformation and the parallel-distributed-compensation design framework. An alternative description of aeroelastic models is also proposed as a gateway to various recent linear-matrix-inequality-based control theories. This study is conducted through an example that focuses attention on the state-variable-feedback controller design to the prototypical aeroelastic wing section with structural nonlinearity. This type of model has been traditionally used for the theoretical as well as experimental analysis of two-dimensional aeroelastic behavior and exhibits limit-cycle oscillation without control effort. Numerical simulations to provide empirical validation of the resulting controllers are presented. Comparison to former alternative control solutions is also presented.

Tensor-product model-based control of two-dimensional aeroelastic system

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Polytopic Decomposition of Linear Parameter-Varying Models by Tensor-Product Model Transformation

The Tensor Product (TP) model transformation is a recently proposed technique for transforming given Linear Parameter Varying (LPV) state-space models into polytopic model form, namely, to parameter varying convex combination of Linear Time Invariant (LTI) systems. The main advantage of the TP model transformation is that it is executable in a few minutes and the Linear Matrix Inequality (LMI)-based control design frameworks can immediately be applied to the resulting polytopc models to yield controllers with tractable and guaranteed performance. Various applications of the TP model transformation-based design were studied via academic complex and benchmark problems, but no real experimental environment-based study was published. Thus, the main objective of this paper is to study how the TP model transformation performs in a real world problem and control setup. The laboratory concept for TP model-based controller design, simulation and real time running on an electromechanical system is presented. Development system for TP model-based controller with one hardware/software platform and target system with real-time hardware/ software support are connected in the unique system. Proposed system is based on microprocessor of personal computer (PC) for simulation and software development as well as for real-time control. Control algorithm, designed and simulated in MATLAB/SIMULINK environment, use graphically oriented software interface for real-time code generation. Some specific conflicting industrial tasks in real industrial crane application, such as fast load positioning control and load swing angle minimization, are considered and compared with other controller types.

https://bib.irb.hr/datoteka/260703.TP_Model_Transformation_Based_Controller_Design_for_SPG8211

Tensor Product Model Transformation-based Controller Design for Gantry Crane Control System – An Application Approach

Fast and accurate positioning and swing minimization of heavy loads in crane manipulation are demanding and, in the same time, conflicting tasks. For accurate positioning, the main problem is nonlinear friction effect, especially in the low speed region. In this paper authors propose a control scheme for 3D tower crane, that consists of the tensor product transformation based nonlinear feedback controller, with additional neural network based friction compensator. Tensor product based controller is designed using linear matrix inequalities utilizing a parameter varying Lyapunov function. Neural network parameters adaptation law is derived using Lyapunov stability analysis. The simulation and experimental results on 3D laboratory crane model are given.

Control of 3d Tower Crane Based on Tensor Product Model Transformation With Neural Friction Compensation

The tensor product (TP) based models have been applied widely in approximation theory, and approximation techniques. Recently, a controller design framework working on dynamic systems has also been established based on TP model transformation combined with Linear Matrix Inequalities (LMI) within Parallel Distributed Compensation (PDC ) framework. The effectiveness of the control design framework strongly depends on the approximation property of the TP model used. Therefore, the primary aim of this paper is to investigate the approximation capabilities of dynamic T P model. It is shown that the set of functions that can be approx imated arbitrarily well by TP forms with bounded number of components lies no-where dense, i.e. “almost discretely ” in the set of continuous functions. Consequently, this pape r points out that in a class of control problems this drawback necessitates the application of trade-off techniques betw een accuracy and complexity of TP form. Such requirements are very difficult to consider in the analytical framework, but T P model transformation offers an easy way to deal with them.

Péter Barayni

Papers

Approximation capability of TP model forms