Uncertain Linear Model Research Articles

Uncertain linear programming with cloud set constraints integrating fuzziness and randomness

In reality, there is a significant need for processing methods of uncertain programming problems with fuzziness and randomness. Although there are many extensions of fuzzy sets to handle uncertain information, it is rare to organically integrate randomness and fuzziness, such as the cloud set which is the set extension of cloud model. To address the limitations of uncertain linear programming, which lacks consideration of fuzziness and randomness, a new approach called uncertain linear programming with cloud set constraints is proposed. This approach integrates both fuzziness and randomness properties. Firstly, the definition of uncertain linear programming model with cloud set constraints integrating fuzzy and randomness properties is given. Secondly, the construction method of cloud set membership function of uncertain linear programming model is studied. Thirdly, the solution method for uncertain linear programming model based on Ioperation and P operation in cloud set is argued. Finally, the effectiveness of the method is verified through a specific application example and a comparison case with other linear programs. This method of uncertain linear programming with cloud set constraints utilizes the normal cloud set to represent the objective and conditional constraints, effectively capturing the fuzziness and randomness inherent in these constraints. The contribution of this study is that it proposes the use of random membership degree to replace the subjective fixed single membership in the constraints of linear programming. The membership degree of normal cloud set is expressed by a normal random number with a stable tendency, which integrates fuzziness and randomness with a regular, discrete, sparse thickness instead of a fixed curve. It can more effectively express and adapt to the inherent uncertainty of the objective world because various variables in natural science and social science typically follow a normal distribution. So, it has good objectivity and universality.

A new uncertain linear regression model based on equation deformation

When the observed data are imprecise, the uncertain regression model is more suitable for the linear regression analysis. Least squares estimation can fully consider the given data and minimize the sum of squares of residual error and can effectively solve the linear regression equation of imprecisely observed data. On the basis of uncertainty theory, this paper presents an equation deformation method for solving unknown parameters in uncertain linear regression equations. We first establish the equation deformation method of one-dimensional linear regression model and then extend it to the case of multiple linear regression model. We also combine the equation deformation method with Cramer’s rule and matrix and propose the Cramer’s rule and matrix elementary transformation method to solve the unknown parameters of the uncertain linear regression equation. Numerical example show that the equation deformation method can effectively solve the unknown parameters of the uncertain linear regression equation.

Robust state estimation for uncertain linear discrete systems with d‐step state delay

This paper discusses the state estimation problems of an uncertain linear discrete time-varying state space model with d-step state delay. Based on the principle of minimising the expectation of estimation errors and the method of state augmentation, a robust state estimation algorithm is proposed. Specially, this estimator retains the form of Kalman-like filter and the characteristics of fast recursive calculation. Moreover, the conditions of bounded estimation error covariance and the proof of asymptotic unbiasedness of the filter are given. Finally, numerical examples are used to verify the effectiveness and the wide applicability of this estimator.

Uncertain Linear Stationary Model of Population Density in a Bounded Habitat

The purpose of this article is to identify an uncertain linear time-invariant (LTI) model of the spatial distribution of a specimen over 1-D habitual space. Population density is described by a nonlinear reaction-diffusion partial differential equation (PDE) following the population dynamics theory. The nonlinear PDE is approximated with a linear parameter varying state-space model from the 120th order with two inputs and 100 outputs, which describe the spatial distribution where the manipulated variable in the model is the length of the habitat. A random excitation signal is applied corresponding to expansion or contraction of the habitual space to assess the dynamical response of the population density. The collected data are used to estimate 100 Box and Jenkins (BJ) models, which describe the reaction of population density across the habitat. Then, a single uncertain LTI model is identified from the magnitude frequency response of the BJ models to cover all of them.

Fully Pythagorean fuzzy linear programming problems with equality constraints

There are several daily life problems where we have to deal with the uncertainties and we are forced to solve the uncertain linear programming models. Certain methods have been presented for dealing with linear programming problems based on fuzzy sets and intuitionistic fuzzy sets which are characterized by membership degree, membership and non-membership degrees, respectively. In this study, we first extend the concept of crisp linear programming problem in Pythagorean fuzzy environment based on triangular Pythagorean fuzzy numbers. The profit/cost coefficients in objective function, input/output coefficients and right-hand side coefficients and decision variables of a linear programming problem are considered as triangular Pythagorean fuzzy numbers. Further, we present methods for solving fully Pythagorean fuzzy linear programming problems for non-negative and unrestricted triangular Pythagorean fuzzy numbers with equality constraints. We also apply the proposed technique to solve practical models.

Parametric approximate optimal control of uncertain differential game with application to counter terror

Abstract The linear quadratic differential game plays an important role in many fields. It is well known that the saddle point of linear quadratic differential game is given in a feedback form with a solution of Riccati differential equation. However, the control-related Riccati differential equation cannot be solved analytically in many cases. Then the optimal controls may be difficult to be implemented in practice. In order to simplify the forms of optimal controls, in this paper, we investigate a parametric approximate optimal control problem of linear quadratic differential game under uncertain environment. First, we introduce an uncertain linear quadratic differential game model and its analytic optimal controls. Then, an uncertain linear quadratic differential game model with constrained parametric control domain is formulated. Moreover, a parametric approximate optimization method is presented for solving the optimal control parameters. Finally, a counter terror problem is analyzed to show the efficiency of our presented method.

A new uncertain linear regression model based on slope mean

The least squares estimation can fully consider the given data and minimize the sum of squares of the residuals, and it can solve the linear regression equation of the imprecisely observed data effectively. Based on the least squares estimation and uncertainty theory, we first proposed the slope mean model, which is to calculate the slopes of expected value and each given data, and the average value of these slopes as the slope of the linear regression equation, substituted into the expected value coordinates, and we can get the linear regression equation. Then, we proposed the deviation slope mean model, which is a very good model and the focus of this paper. The idea of the deviation slope mean model is to calculate the slopes of each given data deviating from the regression equation, and take the average value of these slopes as the slope of the regression equation. Substituted into the expected value coordinate, we can get the linear regression equation. The deviation slope mean model can also be extended to multiple linear regression equation, we transform the established equations into matrix equation and use inverse matrix to solve unknown parameters. Finally, we put forward the hybrid model, which is a simplified model based on the combination of the least squares estimation and deviation slope mean model. To illustrate the efficiency of the proposed models, we provide numerical examples and solve the linear regression equations of the imprecisely observed data and the precisely observed data respectively. Through analysis and comparison, the deviation slope mean model has the best fitting effect. Part of the discussion, we are explained and summarized.

Robust adaptive model predictive control: Performance and parameter estimation

SummaryFor systems with uncertain linear models, bounded additive disturbances and state and control constraints, a robust model predictive control (MPC) algorithm incorporating online model adaptation is proposed. Sets of model parameters are identified online and employed in a robust tube MPC strategy with a nominal cost. The algorithm is shown to be recursively feasible and input‐to‐state stable. Computational tractability is ensured by using polytopic sets of fixed complexity to bound parameter sets and predicted states. Convex conditions for persistence of excitation are derived and are related to probabilistic rates of convergence and asymptotic bounds on parameter set estimates. We discuss how to balance conflicting requirements on control signals for achieving good tracking performance and parameter set estimate accuracy. Conditions for convergence of the estimated parameter set are discussed for the case of fixed complexity parameter set estimates, inexact disturbance bounds, and noisy measurements.

Reconstructing continental‐scale variation in soil δ15N: a machine learning approach in South America

AbstractSoil nitrogen isotope composition (δ15N) is an essential tool for investigating ecosystem nitrogen balances, plant–microbe interactions, ecological niches, animal migration, food origins, and forensics. The advancement of these applications is limited by a lack of robust geospatial models that are capable of capturing variation in soil δ15N (i.e., isotopic landscapes or isoscapes). Due to the complexity of the nitrogen cycle and general scarcity of isotopic information, previous approaches have reconstructed regional to global soil δ15N patterns via highly uncertain linear regression models. Here, we develop a new machine learning approach to ascertain a finer‐scale understanding of geographic differences in soil δ15N, using the South American continent as a test case. We use a robust training set spanning 278 geographic locations across the continent, spanning all major biomes. We tested three different machine learning methods: cubist, random forest (RF), and stochastic gradient boosting (GBM). 10‐fold cross‐validation revealed that the RF method outperformed both the cubist and GBM approaches. Variable importance analysis of the RF framework pointed to biome type as the most crucial auxiliary variable, followed by soil organic carbon content, in determining the model performance. We thereby created a biogeographic boundary map, which predicted an expected multiscale spatial pattern of soil δ15N with a high degree of confidence, performing substantially better than all previous approaches for the continent of South America. Therefore, the RF machine learning framework showed to be a great opportunity to explore a broad array of ecological, biogeochemical, and forensic issues through the lens of soil δ15N.

Uncertain Box-Cox Regression Analysis With Rescaled Least Squares Estimation

Under the uncertain statistical framework by Liu [19], there is still a lack of an effective fitting method for uncertain linear models with Box-Cox transformation of response variables. For example, for the transformation parameter Λ , the uncertain least squares estimation will produce a severely low estimation result. In this paper, we propose uncertain Box-Cox regression analysis by utilizing the uncertainty theory to model the imprecise data and applying a generalized Box-Cox transformation indexed by its parameter to validate classic regression assumptions. We use rescaled least squares to estimate unknown parameters and provide an estimate for noises followed by residual analysis for these uncertain Box-Cox regression models. We also give the forecast values and confidence intervals and use a numerical example to demonstrate our methodology. Our work sets a uniformed framework for Box-Cox transformation on the uncertain regression, and extends such regression from linear to nonlinear cases, taking the Johnson-Schumacher growth model as an example.

Study on Identification Method for Parameter Uncertainty Model of Aero Engine

The linear model of an aero engine is effective in a small range of the neighborhood of equilibrium points. According to this problem, the identification method for the parameter uncertain linear model of the aero engine was proposed. The identification problem is solved by calculating nonlinear programming. Considering the parameter uncertainty of the model is the critical point of this research during the optimization process. A parameter uncertain model of an aero engine can be obtained, which has large use range. This method is used for DGEN380 aero engine. The two parameters, VDD and VE, are defined for describing error range. Compared with experimental data, the uncertain model of DGEN 380 can simulate the real state of DGEN380 within 1% error range when ΔPLA<22%. Compared with another conventional method of identification (recursive least squares), the parameter uncertain model, established by the method of this research, has a broad application area through parameter uncertainty of the model.

WITHDRAWN: Study on identification method for parameter uncertain model of aero gas turbine

Abstract The linear model of the aero gas turbine is effective in a small application range. According to this problem, the identification method for parameter uncertain linear model of aero gas turbine proposed. The identification problem is solved by calculating nonlinear programming. Considering the parameter uncertainty of the model is the critical point of this research during the optimization process. A parameter uncertain model of an aero gas turbine can be obtained, which has large use-range. This method is used for DGEN380 aero gas turbine. The two parameters, VDD and VE, are defined for describing error range. Compared with experimental data, the uncertain model of DGEN380 can simulate the real state of DGEN380 within 1% error range when ΔPLA

Fractional order MIMO controllers for robust performance of airplane longitudinal motion

This paper presents fractional order multi-input multi-output (MIMO) controllers for the robust performance of airplane longitudinal motion. A novel necessary and sufficient criterion is offered by using the value set concept to analyze the robust performance of fractional order MIMO uncertain systems based on the location of the characteristic equation roots. The criterion is applicable to all linear time-invariant systems of commensurate and incommensurate orders with complex coefficients. The obtained results are applied to an uncertain linear model of a business airplane to improve the robust performance of its longitudinal motion by decentralized MIMO output feedback and MIMO state feedback controllers. Numerical simulations are conducted to confirm the effectiveness of the presented criterion and controllers.

CИНТЕЗ РОБАСТНОГО ПІД-РЕГУЛЯТОРА ДЛЯ НЕВИЗНАЧЕНОЇ НЕЛІНІЙНОЇ МОДЕЛІ БПЛА З ВИКОРИСТАННЯМ ШТУЧНОЇ МОДЕЛІ ПОСИЛЕННЯ І ТИМЧАСОВОЇ ЗАТРИМКИ

Estimations of the flight vehicles aerodynamic coefficients through the theoretical, numerical, wind tunnel and flight-test methods have always errors and uncertainties. Nonlinear dynamics of the unmanned aerial vehicle due to speed variations, as well as variability and uncertainty of the aerodynamic coefficients can be considered as an uncertain model. A robust proportional-integral-derivative controller is designed based on the nonlinear optimization in the time domain for the uncertain nonlinear dynamical model of the unmanned aerial vehicle. Artificial gain and time-delay models are added to achieve required stability margins as system robustness during the robust proportional-integral-derivative control design. The aerodynamic coefficients are divided into two groups on the basis of the output vector's sensitivity to the aerodynamic coefficients. Nonlinear optimization of the criterion is performed in two steps for the robust proportional-integral-derivative controller design. In the first step of the design, nominal values are used for coefficients with low-sensitivity, and upper and lower limits are used only for high sensitive aerodynamic coefficients. In the second step, the upper and lower limits are applied all of the aerodynamic coefficients to evaluate and re-adjust the robust proportional-integral-derivative controller parameters. The robust controller is designed for the uncertain nonlinear roll and lateral dynamic model of the Skywalker X8 flying wing to show the effectiveness of the proposed method to guarantee the stability margins. In the given example during the design of the robust controller, with adding of the artificial gain and time-delay model in the control loop the controller parameters are changed. This regulator increases the phase stability margin by about 10 degrees and the gain stability margin by about two for the family of the equivalent uncertain linear models.

The piecewise parametric optimal control of uncertain linear quadratic models

ABSTRACTThe optimal control of linear quadratic model is given in a feedback form and determined by the solution of a Riccati equation. However, the control-related Riccati equation usually cannot be solved analytically such that the form of optimal control will become more complex. In this paper, we consider a piecewise parametric optimal control problem of uncertain linear quadratic model for simplifying the form of optimal control. By introducing a piecewise control parameter, a piecewise parametric optimal control model is established. Then we present a parametric optimisation method for solving the optimal piecewise control parameter. Finally, an uncertain inventory-promotion optimal control problem is discussed and a comparison is made to show the effectiveness of proposed piecewise parametric optimal control model.

Optimal linear filtering for networked control systems with time-correlated fading channels

In this paper, the problem of filtering for discrete-time networked control systems with time-correlated fading channels is considered where each time-correlated fading channel is described by a one-dimensional discrete-time uncertain linear system model with random disturbance. Using the innovation analysis approach and optimal estimates for the products of channel gain and system state obtained, an optimal linear recursive filter is proposed, that has time-independent complexity, and does not increase computation and storage load with time. An example and simulation are provided to show the effectiveness and potential of the proposed new design technique.

Metro Traffic Modeling and Regulation in Loop Lines Using a Robust Model Predictive Controller to Improve Passenger Satisfaction

Automatic traffic regulation plays an important role in high-frequency metro lines. This paper introduces a discrete event nonlinear traffic model for metro loop lines based on the deviations from nominal departure times. This model considers the transfer of knock-on delay from one train to another one and improves on previous models. The model also includes uncertainty to accommodate variation in passenger demand over the course of a work day. The nonlinear uncertain traffic model is approximated by the linear uncertain model which facilitates the design of a robust predictive controller. The objective of control is increasing the passenger satisfaction by minimizing the deviation from the nominal time schedule and keeping headways close to the nominal headway. The linear robust model predictive controller (RMPC) is employed to compensate the disturbances and to regulate traffic in the presence of operational constraints. YALMIP toolbox was used to design the RMPC. Model verification and simulations were carried out using real data from Tehran metro line 4. The simulation results are presented to show the trueness of the model and effectiveness of the proposed control methodology.

Uncertain multivariable regression model

Uncertainty theory provides an alternative to model human belief degree without the scope of probability theory. Supposing that a response variable is an uncertain variable, the simple uncertain regression has proposed. In many cases, the number of response variables is more than one. This paper proposes uncertain multivariable linear regression model which has multiple response variables. Firstly, this paper discusses generalized least squares estimate of simple uncertain linear regression model. Moreover, this paper introduces a least squares method to estimate unknown parameters of uncertain multivariable linear regression.

Robust model predictive control for path-following of underactuated surface vessels with roll constraints

Abrupt maneuvering or strong disturbances from sea waves probably lead to large roll motion of underactuated surface vessels, which severely affects stability of surface vessels. A novel path-following control for vessels with roll constraints is proposed by combining Kalman filter, disturbance observer and robust constrained model predictive control method. Firstly, a switched linear uncertain model with coupled surge, sway, roll and yaw is constructed for underactuated surface vessels. Then, an improved Sage-Husa Adaptive Kalman Filter is proposed to estimate vessels' full states by partial states information and eliminate the effects of measurement noises. Next, the disturbance observer, which is used to estimate low-frequency disturbances from wind, waves and oceans, is combined with constrained model predictive control to improve the robustness of the closed-loop control system. In this paper, the effectiveness of the proposed algorithm is verified by theoretical analysis, simulations and experiments.

A parametric optimization approach for uncertain linear quadratic models

As it is well known, the optimal control of linear quadratic model is given in a feedback form, which is determined by the solution of a Riccati equation. However, the corresponding Riccati equation cannot be solved analytically in many cases. Even if an analytic solution can be obtained, it might be a complex time-oriented function. In this paper, we introduce an approximate model with parameter for simplifying the form of optimal control of uncertain linear quadratic model. First, we discuss an optimal control problem of uncertain linear quadratic model and deduce an analytic expression of optimal control. Then we formulate an approximate model with parameter and present a parametric optimization method for solving the optimal parameter. Finally, a production planning problem is given to illustrate the efficiency of the proposed approximate model and parametric optimization approach.