Tensor Product Model Research Articles

Quasi‐Min‐Max Model Predictive Control for Image‐Based Visual Servoing with Tensor Product Model Transformation

AbstractThis paper presents a novel image‐based visual servoing (IBVS) controller based on quasi‐min‐max model predictive control (MPC). By transforming the image Jacobian matrix (i.e. interaction matrix) into a convex combination of linear time‐invariant vertices form with the tensor‐product (TP) model transformation method, the visual servoing system is represented as a polytopic linear parameter‐varying (LPV) system. A robust controller is designed for the robotic visual servoing system subject to input and output constraints such as robot physical limitations and visibility constraints. The control signal is calculated online by carrying out the convex optimization involving linear matrix inequalities (LMIs) in model predictive control. The proposed visual servoing method avoids the inverse of the image Jacobian matrix and hence can solve the intractable problems for the classical IBVS controller, such as large displacements between the initial and the desired position of the camera. The ability of handling constraints can keep the image features in the boundary of the desired field of view (FOV). To verify the effectiveness of the proposed algorithm, the simulation results on a 6 degrees‐of‐freedom (DOF) robot manipulator with eye‐in‐hand configuration are presented and discussed.

Revisiting the TP Model Transformation: Interpolation and Rule Reduction

AbstractThe tensor‐product (TP) model transformation is a numerical technique that finds a convex representation, akin to a Takagi‐Sugeno (TS) fuzzy model, from a given linear parameter varying (LPV) model of a system. It samples the LPV model over a limited domain, which allows the use of the higher order singular value decomposition (HOSVD) and convex transformations that leads to the TS representation of the LPV model. In this paper, we discuss different strategies that could be used on the sampling step of the TP model transformation (which in turn lead to different membership function properties of a TS fuzzy model). Additionally, this paper discusses how the other steps could be used to reduce the number of rules of a given TS fuzzy model. In cases where nonzero singular values were discarded in the rule reduction, we also show how to obtain an uncertain model that covers the original.

Linear parameter varying switching attitude control for a near space hypersonic vehicle with parametric uncertainties

This paper deals with the problem of linear parameter varying (LPV) switching attitude control for a near space hypersonic vehicle (NSHV) with parametric uncertainties. First, due to the enormous complexity of the NSHV nonlinear attitude dynamics, a slow–fast loop polytopic LPV attitude model is developed by using Jacobian linearisation and the tensor product model transformation approach. Second, for the purpose of less conservative attitude controller design, the flight envelope is divided into four subregions. For each parameter subregion, slow-loop and fast-loop LPV controllers are designed. By the defined switching character function, these slow–fast loop LPV controllers are then switched in order to guarantee the closed-loop NSHV system to be asymptotically stable and satisfy a specified tracking performance criterion. The condition of LPV switching attitude controller synthesis is given in terms of linear matrix inequalities, which can be readily solved via standard numerical software, and the robust stability analysis of the closed-loop NSHV system is verified based on multiple Lypapunov functions. Finally, numerical simulations have demonstrated the effectiveness of the proposed approach.

Tensor Product Model Transformation Based Control and Synchronization of a Class of Fractional‐Order Chaotic Systems

AbstractFractional‐order chaotic systems are the complex systems that involve non‐integer order derivatives. In this paper, tensor product (TP) model transformation‐based controller design for control and synchronization of a class of the fractional‐order chaotic systems is investigated. We propose a novel linear matrix inequality (LMI)‐based stabilization condition for fractional‐order TP models with a controller derived via a parallel distributed compensation (PDC) structure. In the controller design, the controlled system first is transformed into a convex state‐space TP model using the TP model transformation. Based on the transformed TP model, the controller is determined by solving the proposed LMI condition. To the best of our knowledge, this is the first investigation of TP model transformation based design in fractional‐order systems. Several illustrative examples are given to demonstrate the convenience of the proposed LMI condition and the effectiveness of the controller design.

Friction with Hysteresis Loop Modeled by Tensor Product

Engineers encounter the problem of friction in any mechanical system. Friction force is strongly nonlinear and varies considerably while the system is working. In the case of high-precision applications friction makes the situation even more complex, as the stick-slip effect occurs near the target position. This paper introduces a pneumatic servo-system for investigation of the behavior of friction near the target position. A new model is proposed which takes the hysteresis loop of the friction also into consideration emphasizing the importance of hysteresis. This paper presents the tensorproduct (TP) based modeling of friction which is suitable for control design. The main advantage of the TP model transformation is that due to its polytopic model form Linear Matrix Inequality (LMI) can be immediately applied to the resulting model to yield controllers with guaranteed performance. The main contribution of this paper is the application of TP model transformation making the identification of friction parameters unnecessary by utilizing directly the measured data itself.

Non-fragile switching tracking control for a flexible air-breathing hypersonic vehicle based on polytopic LPV model

This article proposes a linear parameter varying (LPV) switching tracking control scheme for a flexible air-breathing hypersonic vehicle (FAHV). First, a polytopic LPV model is constructed to represent the complex nonlinear longitudinal model of the FAHV by using Jacobian linearization and tensor-product (T-P) model transformation approach. Second, for less conservative controller design purpose, the flight envelope is divided into four sub-regions and a non-fragile LPV controller is designed for each parameter sub-region. These non-fragile LPV controllers are then switched in order to guarantee the closed-loop FAHV system to be asymptotically stable and satisfy a specified performance criterion. The desired non-fragile LPV switching controller is found by solving a convex constraint problem which can be efficiently solved using available linear matrix inequality (LMI) techniques, and robust stability analysis of the closed-loop FAHV system is verified based on multiple Lypapunov functions (MLFs). Finally, numerical simulations have demonstrated the effectiveness of the proposed approach.

Tensor product model transformation-based control design for force reflecting tele-grasping under time delay

The improvement of direct human–robot physical interaction has recently become one of the strongest motivation factors in robotics research. Impedance/admittance control methods are key technologies in several directions of advanced robotics such as dexterous manipulation, haptics and telemanipulation. In this paper, we propose a control scheme and a design technique for stabilising shared impedance/admittance-based bilateral telemanipulation under varying time delay. The proposed scheme introduces delay-adaptive non-linear damping to stabilise the impedance model. A modified version of the tensor product model transformation is applied to determine the tensor product type polytopic linear parameter varying (LPV) representation of the impedance controlled interaction model, such that the value of the actual time delay becomes an external parameter rather than an inherent property of the system. The main benefit of the proposed approach is that the model form it produces is amenable to the immediate application of modern, linear matrix inequality (LMI)-based multi-objective synthesis methods. The viability of the proposed methodology is demonstrated through a single DoF force reflecting tele-grasping application. The results are also confirmed through laboratory experiments which help to further highlight the perspectives of this novel approach.

Asian Journal of Control Special Issue on

Asian Journal of ControlVolume 15, Issue 2 p. 636-636 Call for Papers Asian Journal of Control Special Issue on “Tensor Product (TP) Model Transformation-based Control System Design” First published: 19 March 2013 https://doi.org/10.1002/asjc.702Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume15, Issue2March 2013Pages 636-636 RelatedInformation

Representing the Model of Impedance Controlled Robot Interaction with Feedback Delay in Polytopic LPV Form: TP Model Transformation-based Approach

The aim of this paper is to transform the model of the impedance controlled robot interaction with feedback delay to a Tensor Product (TP) type polytopic LPV model whereupon Linear Matrix Inequality (LMI) based control design can be immediately executed. The paper proves that the impedance model can be exactly represented by a finite element TP type poly- topic model under certain constrains. The paper also determines various further TP models with different advantages for control design. First, it derives the exact Higher Order Singular Value Decomposition (HOSVD) based canonical form, then it performs complexity trade-off to yield a model with less number of components but rather effective for LMI design. Then the pa- per presents various different types of convex TP model representations based on the non-exact model in order to investigate how convex hull manipulation can be performed on the model. Fi- nally the presented models are analyzed to validate the accuracy of the transformation and the resulting TP type polytopic LPV models. The paper concludes that these prepared models are ready for convex hull manipulation and LMI based control design.

Tensor Product Model Transformation Based Adaptive Integral-Sliding Mode Controller: Equivalent Control Method

This paper proposes new methodologies for the design of adaptive integral-sliding mode control. A tensor product model transformation based adaptive integral-sliding mode control law with respect to uncertainties and perturbations is studied, while upper bounds on the perturbations and uncertainties are assumed to be unknown. The advantage of proposed controllers consists in having a dynamical adaptive control gain to establish a sliding mode right at the beginning of the process. Gain dynamics ensure a reasonable adaptive gain with respect to the uncertainties. Finally, efficacy of the proposed controller is verified by simulations on an uncertain nonlinear system model.

Novel Tensor Product Models for Automatic Transmission System Control

This paper discusses novel tensor product (TP) models for the control of two complex components of the vehicle automatic transmission systems, namely the drive line without clutch and, the valve–clutch. The TP models are obtained by a transformation of the linear parameter-varying models derived from the first principle nonlinear mathematical models of the controlled processes. Experimental results validate the performance of the proposed TP models.

Slow–fast loop gain-scheduled switching attitude tracking control for a near-space hypersonic vehicle

This article develops a slow–fast loop polytopic linear parameter-varying model and proposes a systematic gain-scheduled switching attitude tracking control scheme for a near-space hypersonic vehicle. First, the dynamics of near-space hypersonic vehicle is modeled as a slow–fast loop polytopic linear parameter-varying model using the Jacobian linearization and tensor-product model transformation approach. Open-loop simulation verification illustrates that the developed polytopic linear parameter-varying model captures the local nonlinearities of the original nonlinear system; therefore, it is suitable for model-based control. Second, for less conservative controller design purpose, the flight envelope is divided into smaller subregions, a family of slow loop and fast loop gain-scheduled controllers are designed, and each of them is suitable for a specific parameter subspace; the slow loop and fast loop gain-scheduled controllers are then switched in order to guarantee the closed-loop near-space hypersonic vehicle system to be asymptotically stable and satisfy a specified performance criteria. The resulting slow loop and fast loop gain-scheduled switching controllers are found by solving a convex constraint problem that can be efficiently solved using available linear matrix inequality techniques. Finally, numerical simulations have demonstrated the effectiveness of the proposed method.

On modeling and control of a flexible air-breathing hypersonic vehicle based on LPV method

This article develops a polytopic linear parameter varying (LPV) model and presents a non-fragile H2 gain-scheduled control for a flexible air-breathing hypersonic vehicle (FAHV). First, the polytopic LPV model of the FAHV can be obtained by using Jacobian linearization and tensor-product (TP) model transformation approach, simulation verification illustrates that the polytopic LPV model captures the local nonlinearities of the original nonlinear system. Second, based on the developed polytopic LPV model, a non-fragile gainscheduled control method is proposed in order to reduce the fragility encountered in controller implementation, a convex optimisation problem with linear matrix inequalities (LMIs) constraints is formulated for designing a velocity and altitude tracking controller, which guarantees H2 control performance index. Finally, numerical simulations have demonstrated the effectiveness of the proposed approach.

Approximation and complexity trade‐off by TP model transformation in controller design: A case study of the TORA system

AbstractThe main objective of the paper is to study the approximation and complexity trade‐off capabilities of the recently proposed tensor product distributed compensation (TPDC) based control design framework. The TPDC is the combination of the TP model transformation and the parallel distributed compensation (PDC) framework. The Tensor Product (TP) model transformation includes an Higher Order Singular Value Decomposition (HOSVD)‐based technique to solve the approximation and complexity trade‐off. In this paper we generate TP models with different complexity and approximation properties, and then we derive controllers for them. We analyze how the trade‐off effects the model behavior and control performance. All these properties are studied via the state feedback controller design of the Translational Oscillations with an Eccentric Rotational Proof Mass Actuator (TORA) System.Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

HOSVD-based LPV modeling and mixed robust H2/H∞ control design for air-breathing hypersonic vehicle

This paper focuses on synthesizing a mixed robust H 2 /H ∞ linear parameter varying (LPV) controller for the longitudinal motion of an air-breathing hypersonic vehicle via a high order singular value decomposition (HOSVD) approach. The design of hypersonic flight control systems is highly challenging due to the enormous complexity of the vehicle dynamics and the presence of significant uncertainties. Motivated by recent results on both LPV control and tensor-product (TP) model transformation approach, the velocity and altitude tracking control problems for the air-breathing hypersonic vehicle is reduced to that of a state feedback stabilizing controller design for a polytopic LPV system with guaranteed performances. The controller implementation is converted into a convex optimization problem with parameter-dependent linear matrix inequalities (LMIs) constraints, which is intuitively tractable using LMI control toolbox. Finally, numerical simulation results demonstrate the effectiveness of the proposed approach.

Computational relaxed TP model transformation: restricting the computation to subspaces of the dynamic model

AbstractThe tensor‐product (TP) model transformation is a recently proposed numerical method capable of transforming linear parameter varying state‐space models to the higher order singular value decomposition (HOSVD) based canonical form of polytopic models. It is also capable of generating various types of convex TP models, a type of polytop models, for linear matrix inequality based controller design. The crucial point of the TP model transformation is that its computational load exponentially explodes with the dimensionality of the parameter vector of the parameter‐varying state‐space model. In this paper we propose a modified TP model transformation that leads to considerable reduction of the computation. The key idea of the method is that instead of transforming the whole system matrix at once in the whole parameter space, we decompose the problem and perform the transformation element wise and restrict the computation to the subspace where the given element of the model varies. The modified TP model transformation can readily be executed in higher dimensional cases when the original TP model transformation fails. The effectiveness of the new method is illustrated with numerical examples. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Parallel Distributed Compensation Based Stabilization of A 3-DOF RC Helicopter: A Tensor Product Transformation Based Approach

This paper presents a control solution for the stabilization of the 3-DOF RC helicopter via the combination of the TP (Tensor Product) model transformation and the PDC (Parallel Distributed Compensation) control design framework. First we recall the nonlinear model of the RC helicopter and its simplified version, in order, to facilitate control design. Then we execute the TP model transformation on the simplified model to yield its TP model representation, that is a kind of polytopic model with specific characteristics, whereupon the PDC framework can immediately be applied. The control design considers practical control specifications such as: good speed of response and physical constrain on the control effort to avoid actuator saturations. We guarantee these specifications by LMI (Linear Matrix Inequality) conditions developed under the PDC frameworks. Further, we avoid the discrepancies, introduced via the simplification of the model, by applying LMI conditions specialized for robust control. By simultaneously solving these LMI conditions, we render a stabilizing nonlinear controller that achieves good speed of response with small control effort without actuator saturations. Simulation results are included to validate the control design. It will be pointed out that the resulting controller is equivalent with the controller successfully applied in the real control experiment of the helicopter presented in a recent paper.The main conclusion of this paper is, that the proposed design process is systematic, non-heuristic and straightforward, the stability proof of the resulting controller is tractable via the feasibility test of LMIs and, hence, exact. The whole design procedure is automatically computed via commercialized mathematical tools (MATLAB LMI Toolbox) without analytical interaction. The computational time is about minutes on a regular PC.

GLOBAL ASYMPTOTIC STABILIZATION OF THE PROTOTYPICAL AEROELASTIC WING SECTION VIA TP MODEL TRANSFORMATION

A comprehensive analysis of aeroelastic systems has shown that these systems exhibit a broad class of pathological response regimes when certain types of non-linearities are included. In this paper, we propose a design method of a state-dependent non-linear controller for aeroelastic systems that includes polynomial structural non-linearities. The proposed method is based on recent numerical techniques such as the Tensor Product (TP) model transformation and the Linear Matrix Inequality (LMI) control design methods within the Parallel Distributed Compensation (PDC) frameworks. In order to link the TP model transformation and the LMI's in the proposed design method, we extend the TP model transformation with a further transformation. As an example, a controller is derived that ensures the global asymptotic stability of the prototypical aeroelastic wing section via one control surface, in contrast with previous approaches which have achieved local stability or applied additional control actuator on the purpose of achieving global stability. Numerical simulations are used to provide empirical validation of the control results. The effectiveness of the controller design is compared with a former approach.

Sliding Sector Design for Nonlinear Systems

The main contribution of this paper is a new method for sliding surface sector design to reduce the chattering. A new approach enables a systematic design based on the key idea that the Tensor Product (TP) model transformation is capable of decomposing sectors, furthermore it can define a High Order Singular Value Decomposition (HOSVD)-based canonical sector system. It is partially combination of sector sliding mode control, the classical manifold design for the linear system and HOSVD-based canonical description of a wide class of nonlinear systems. Experimental results of a DSP-controlled single-degree-of-freedom motion-control system are presented.

COMPLEXITY RELAXATION OF THE TENSOR PRODUCT MODEL TRANSFORMATION FOR HIGHER DIMENSIONAL PROBLEMS

ABSTRACTThe Tensor Product (TP) model transformation method was proposed recently as an automated gateway between a class of non‐linear models and linear matrix inequality based control design. The core of the TP model transformation is the higher order singular value decomposition of a large sized tensor, which requires high computational power that is usually outside of a regular computer capacity in cases of higher dimensionality. This disadvantage restricts the utilization of the TP model transformation to models having smaller dimensionality. The aim of this paper is to propose a computationally relaxed version of the TP model transformation. The paper also presents a 6 dimensional example to show the effectiveness of the modified transformation.