Tensor Product Model Research Articles

Control of Diabetes Mellitus by Advanced Robust Control Solution

The current article investigates the possibilities of designing a robust control solution for a control problem related to type 1 diabetes mellitus. The proposed control methodology exploits linear parameter varying, linear matrix inequality, tensor product model transformation and extended Kalman filtering, four advanced control methods in order to guarantee hard safety control constraints for diabetic patients. In this research we have applied an extension of the minimal model to simulate the glucose-insulin dynamics and the glucose and insulin absorption of a diabetic patient. We have validated our results on numerical simulations by using realistic patient data. During the evaluation we have used randomized glucose intakes both in time and amount (as ”unfavorable” disturbance signals). Furthermore, we did a long-term (30 days) assessment to confirm the stability of the proposed framework. The results have shown that the developed controller effectively intervenes into the process and provides appropriate control action by avoiding hypoglycemia thereby satisfying the predefined quantity and quality requirements.

A Nested Tensor Product Model Transformation

The tensor product model transformation (TPMT) is an emerging numerical framework of the Takagi–Sugeno (T–S) fuzzy (or polytopic) system modeling for a linear matrix inequality based system control design. A nested TPMT (NTPMT) is proposed in this paper, which merges the dimensions of the tensors and performs the TPMT iteratively. The resultant fuzzy model is in a multilevel nested tensor product (TP) structure. The vertex tensor obtained by NTPMT has fewer dimension results than the original TPMT so the number of vertices or fuzzy rules, which have been the main bottleneck for further application of the TPMT in higher dimensional systems, is expected to decrease significantly. It is also proven that the NTPMT contains the hierarchical fuzzy logic, meaning that the NTPMT is capable of conducting hierarchical fuzzy modeling and reduction. Furthermore, because the inclusion of multiple TPMTs is prone to augment the conservativeness of the resultant fuzzy model, a suboptimal convex hull rectification algorithm for the TPMT is developed based on a newly defined tightness measure, and then extended to render the NTPMT as less conservative as possible. Finally, numerical simulations on two real physical systems (two- and four-parameter dimension) are verified to demonstrate the performance of the methods.

Bayesian additive adaptive basis tensor product models for modeling high dimensional surfaces: an application to high-throughput toxicity testing.

Many modern datasets are sampled with error from complex high-dimensional surfaces. Methods such as tensor product splines or Gaussian processes are effective and well suited for characterizing a surface in two or three dimensions, but they may suffer from difficulties when representing higher dimensional surfaces. Motivated by high throughput toxicity testing where observed dose-response curves are cross sections of a surface defined by a chemical's structural properties, a model is developed to characterize this surface to predict untested chemicals' dose-responses. This manuscript proposes a novel approach that models the multidimensional surface as a sum of learned basis functions formed as the tensor product of lower dimensional functions, which are themselves representable by a basis expansion learned from the data. The model is described and a Gibbs sampling algorithm is proposed. The approach is investigated in a simulation study and through data taken from the US EPA's ToxCast high throughput toxicity testing platform.

A Tensor Product Model Transformation Approach to the Discretization of Uncertain Linear Systems

A Tensor Product Model Transformation Approach to the Discretization of Uncertain Linear Systems

Tensor Product Model Transformation-based Parallel Distributed Control of Tumor Growth

Tensor Product Model Transformation-based Parallel Distributed Control of Tumor Growth

Affine Tensor Product Model Transformation

This paper introduces the novel concept of Affine Tensor Product (TP) Model and the corresponding model transformation algorithm. Affine TP Model is a unique representation of Linear Parameter Varying systems with advantageous properties that makes it very effective in convex optimization‐based controller synthesis. The proposed model form describes the affine geometric structure of the parameter dependencies by a nearly minimum model size and enables a systematic way of geometric complexity reduction. The proposed method is capable of exact analytical model reconstruction and also supports the sampling‐based numerical approach with arbitrary discretization grid and interpolation methods. The representation conforms with the latest polytopic model generation and manipulation algorithms. Along these advances, the paper reorganizes and extends the mathematical theory of TP Model Transformation. The practical merit of the proposed concept is demonstrated through a numerical example.

Tensor Product model based PID controller optimisation for propofol administration

This study investigates the computer-regulated propofol administration in anesthesia during medical interventions considering output feedback and robust PID control. The paper applies the Affine Tensor Product Model Transformation to derive the appropriate polytopic quasi-LPV representation of the closed-loop dynamics. This model form enables the use of LMI-based optimisation techniques to evaluate the closed loop performance. Despite the highly non-convex nature of this output feedback problem, the PID gains can be locally tuned through simplex optimisation. The proposed method provides a systematic way of tuning PID-controlled propofol administration for individual patients with theoretically established worst-case performance measures.

Extension of the Multi-TP Model Transformation to Functions with Different Numbers of Variables

The tensor product (TP) model transformation defines and numerically reconstructs the Higher-Order Singular Value Decomposition (HOSVD) of functions. It plays the same role with respect to functions as HOSVD does for tensors (and SVD for matrices). The need for certain advantageous features, such as rank/complexity reduction, trade-offs between complexity and accuracy, and a manipulation power representative of the TP form, has motivated novel concepts in TS fuzzy model based modelling and control. The latest extensions of the TP model transformation, called the multi- and generalised TP model transformations, are applicable to a set functions where the dimensionality of the outputs of the functions may differ, but there is a strict limitation on the dimensionality of their inputs, which must be the same. The paper proposes an extended version that is applicable to a set of functions where both the input and output dimensionalities of the functions may differ. This makes it possible to transform complete multicomponent systems to TS fuzzy models along with the above-mentioned advantages.

A TP-LPV-LMI based control for Tumor Growth Inhibition

Abstract The paper investigates the applicability of an advanced modern control method related to control of tumor growth under angiogenic inhibition. In order to describe the physiological process, a simple mathematical model was applied consisting of two states, the volume of the tumor and the inhibitor value. Extended Kalman Filter (EKF) was applied to estimate the unmeasurable state (inhibitor level). Linear Parameter Varying (LPV) models are used both at controller design (difference based control oriented LPV model) and EKF development (LPV model) level as well. We have used the Tensor Product (TP) model transformation accompanied by Linear Matrix Inequality (LMI) based optimization method in order to design a Parallel Distributed Compensator (PDC) kind TP-LPV-LMI controller considering additive disturbances (on both states) and sensor noise as well. Despite the assumed unfavorable effects the TP-LPV-LMI controller performed well achieving low final tumor volume and less totally injected inhibitor level.

An Adaptive LPV Integral Sliding Mode FTC of Dissimilar Redundant Actuation System for Civil Aircraft

This paper addresses three major issues in dissimilar redundant actuation system when operates in active/active mode. The first one is to reduce the effect of force fighting. The second one is to obtain precise tracking performance of the control surface. The third one is to induce fault tolerance in the presence of faults in hydraulic actuator. The proposed strategy is to design LPV integral sliding mode controller for each actuator in a dissimilar actuation system. The nonlinear dynamics of each actuator is first transformed into LPV form and employed tensor product model transformation to convert into polytopic LPV form. The integral sliding mode FTC is designed based on the LPV model of actuators. To induce the fault tolerance, the nonlinear injection term of proposed integral sliding mode control law is made adaptive subject to the severity of the fault. In order to verify the robustness of the proposed scheme, an external disturbance acting as an airload is applied at the control surface input. Finally, the proposed strategy is applied on the nonlinear model of system to validate the dominant performance as compared to existing methods in the literature. In the simulations, three types of faults in hydraulic actuator are considered and the performance of closed loop system is analyzed.

Control of shimmy vibration in aircraft landing gears based on tensor product model transformation and twisting sliding mode algorithm

Shimmy vibration is a common phenomenon in landing gear systems during either the take-off or landing of aircrafts. The shimmy vibration is undesirable since it can damage the landing gear and discomforts the pilots and passengers. In this work, tensor product model transformation (TPMT) and twisting sliding mode algorithm (TSMA) are utilized to design a robust controller for suppression of the shimmy vibration. The design has two steps. First, the TPMT is applied to determine the first part of the controller to suppress the vibration of the undisturbed system. After that, the TSMA is adopted to obtain another part of the controller to eliminate the remaining vibration caused by disturbances. By integrating these two parts, the proposed controller is obtained. Simulation studies are provided to demonstrate the effectiveness of the controller.

Connes’ embedding problem and winning strategies for quantum XOR games

We consider quantum XOR games, defined in the work of Regev and Vidick [ACM Trans. Comput. Theory 7, 43 (2015)], from the perspective of unitary correlations defined in the work of Harris and Paulsen [Integr. Equations Oper. Theory 89, 125 (2017)]. We show that the winning bias of a quantum XOR game in the tensor product model (respectively, the commuting model) is equal to the norm of its associated linear functional on the unitary correlation set from the appropriate model. We show that Connes’ embedding problem has a positive answer if and only if every quantum XOR game has entanglement bias equal to the commuting bias. In particular, the embedding problem is equivalent to determining whether every quantum XOR game G with a winning strategy in the commuting model also has a winning strategy in the approximate finite-dimensional model.

An Efficient Algorithm for Optimally Reshaping the TP Model Transformation

The tensor product (TP) model transformation is an emerging technique for the ongoing system analysis and design works of recent years, where its integration with the linear matrix inequalities (LMIs)-based methods can be powerful solution. However, one of the main issues is encountered that the performance of the LMI conditions depends heavily on the tightness of the TP models. This brief proposes an efficient TP model reshaping algorithm toward tightening TP models. A novel index is introduced to quantize the tightness of TP models such that an optimal reshaping can be realized. Besides, a random search following a recursive reshaping strategy is developed, which intensively enhances the efficiency of the reshaping process. The efficiency and effectiveness of the algorithm are demonstrated via a series of numerical simulations.

Influence of the Tensor Product Model Representation of qLPV Models on the Feasibility of Linear Matrix Inequality Based Stability Analysis

AbstractThe paper investigates and proves the statement, that the convex hull of the polytopic tensor product (TP) model representation influences the feasibility of linear matrix inequality (LMI) based stability analysis methods. The proof is based on a complex stability analysis example of a given quasi linear parameter varying (qLPV) state‐space model. Specifically, the three degree of freedom (3‐DoF) aeroelastic wing section model including Stribeck friction is used as the tool for the example model. The proof is achieved by utilizing TP model transformation and LMI based tools. As a first step, numerous TP model type control solutions holding different convex hulls are systematically derived of the qLPV model via LMI based control design methods. As a second step, each control solution is further equivalently transformed for different TP model representations holding different convex hulls. Finally, the stability of all solutions over all TP model representations are checked via LMI based stability analysis methods. As a result of the two steps, a two dimensional (2D) convex hull space is attained for the 3‐DoF aeroelastic wing section model. The two dimensions are denoted by the LMI based control design and the LMI based stability analysis for different convex hulls. Based on the numerical results, a detailed, comprehensive analysis is provided. The paper as a novelty proves the statement, that the polytopic TP model representation of a given control solution strongly influences the feasibility of LMI based stability analysis methods.

Control Analysis and Synthesis through Polytopic Tensor Product Model: a General Concept

The paper revisits and renew the concept of Polytopic Tensor Product (TP) Model based control analysis and synthesis by generalizing the use of TP-structured variables in the definite conditions derived from the applied control criteria. For TP forms that can depend on multivariate parameter sets (optionally with two times or larger multiplicities), a compact TP formalism is proposed where the multiplicity vector describes the structure. By setting the multiplicities, the parameter dependencies can be neglected or considered with arbitrary high complexity in the variables of controller-candidate, Lyapunov-function candidate and slack variables as well. The paper points out that the definite conditions on the structures of these variables results in definite conditions on Polytopic TP forms enabling a recursive method to reformulate them into Linear Matrix Inequalities (LMIs), Bilinear Matrix Inequalities (BMIs), etc. according to the design method in consideration. The application of the novel concept is shown through H∞ state feedback design for Linear Parameter Varying (LPV) systems.

Generalization of Tensor Product Model Transformation for Control Design

The paper shows that the separated structure of parameter dependencies within the Polytopic Tensor-Product (TP) model can be exploited during the controller design by applying controller candidates that depend only on certain parameter sets. This approach combines the polytopic uncertainty and Parallel Distributed Compensation (PDC) concepts in such a way that they become special cases of the proposed formulation. Motivated by this recognition and the fact that the separation of parameter dependencies increases the complexity and computational cost, the definition of Polytopic TP representation for LPV/qLPV models is relaxed. The new definition allows for the use of arbitrarily chosen parameter sets in the model - according to the control goals - and the separation can be performed only for these sets. Through the derivation, an appropriate transformation algorithm and the unique Affine TP model is introduced discussing the related concepts of tensor algebra and affine geometry as well.

Polytopic linear parameter varying model-based tube model predictive control for hypersonic vehicles

This article investigates a tube model predictive control scheme ensuring robustness and constraints fulfillment for hypersonic vehicles with bounded external disturbances. The polytopic linear parameter varying model is established using tensor-product modeling method, and tube-model predictive control controller is designed to satisfy all vertices of the polytopic linear parameter varying model. Firstly, the nominal model predictive control law is constructed for the nominal system without disturbances. Secondly, tube invariant set strategy is introduced for the unknown bounded disturbances, and the local robust feedback control law is presented to restrain disturbances. The proposed tube-model predictive control method not only steers the nominal state to track a reference trajectory but also keeps the real states within a tube centered with the nominal trajectory. Simulation results show the effectiveness of the proposed method for the hypersonic vehicle with bounded disturbances.

Design of LPV-Based Sliding Mode Controller with Finite Time Convergence for a Morphing Aircraft

This paper proposes a finite time convergence sliding mode control (FSMC) strategy based on linear parameter-varying (LPV) methodology for the stability control of a morphing aircraft subject to parameter uncertainties and external disturbances. Based on the Kane method, a longitudinal dynamic model of the morphing aircraft is built. Furthermore, the linearized LPV model of the aircraft in the wing transition process is obtained, whose scheduling parameters are wing sweep angle and wingspan. The FSMC scheme is developed into LPV systems by applying the previous results for linear time-invariant (LTI) systems. The sufficient condition in form of linear matrix inequality (LMI) constraints is derived for the existence of a reduced-order sliding mode, in which the dynamics can be ensured to keep robust stability and L2 gain performance. The tensor-product (TP) model transformation approach can be directly applied to solve infinite LMIs belonging to the polynomial parameter-dependent LPV system. Then, by the parameter-dependent Lyapunov function stability analysis, the synthesized FSMC is proved to drive the LPV system trajectories toward the predefined switching surface with a finite time arrival. Comparative simulation results in the nonlinear model demonstrate the robustness and effectiveness of this approach.

Perfect commuting-operator strategies for linear system games

Linear system games are a generalization of Mermin’s magic square game introduced by Cleve and Mittal. They show that perfect strategies for linear system games in the tensor-product model of entanglement correspond to finite-dimensional operator solutions of a certain set of non-commutative equations. We investigate linear system games in the commuting-operator model of entanglement, where Alice and Bob’s measurement operators act on a joint Hilbert space, and Alice’s operators must commute with Bob’s operators. We show that perfect strategies in this model correspond to possibly infinite-dimensional operator solutions of the non-commutative equations. The proof is based around a finitely presented group associated with the linear system which arises from the non-commutative equations.

Improved control performance of the 3‐DoF aeroelastic wing section: a TP model based 2D parametric control performance optimization

AbstractBased on the most recent Tensor Product model transformation solutions, the paper presents an improved control performance for the most recent version of the three Degree of Freedom aeroelastic wing section model including Stribeck friction, according to signals pitch, plunge, trailing edge and control value, based on practical engineering criteria such as overshoot, undershoot, signal end values and settling time. This is achieved through proposing a novel two dimensional parametric convex hull manipulation based method for Tensor Product model transformation based Control Design Frameworks. The approach provides two TP model representations for the different requirements of the controller and observer of a given model, opening the possibility to utilize the TP model transformation's convex hull manipulation potential in control performance optimization for a separate optimization of the two TP model representations. Numerical simulation results are provided to illustrate the control performance improvements of the aeroelastic wing section model through the proposed 2D parametric convex hull manipulation based design method.