Conical System Research Articles

Solving Conic Systems via Projection and Rescaling

We propose a simple projection and rescaling algorithm to solve the feasibility problem $$\begin{aligned} \text { find } x \in L \cap \Omega , \end{aligned}$$ where L and \(\Omega \) are respectively a linear subspace and the interior of a symmetric cone in a finite-dimensional vector space V. This projection and rescaling algorithm is inspired by previous work on rescaled versions of the perceptron algorithm and by Chubanov’s projection-based method for linear feasibility problems. As in these predecessors, each main iteration of our algorithm contains two steps: a basic procedure and a rescaling step. When \(L \cap \Omega \ne \emptyset \), the projection and rescaling algorithm finds a point \(x \in L \cap \Omega \) in at most \(\mathcal {O}(\log (1/\delta (L \cap \Omega )))\) iterations, where \(\delta (L \cap \Omega ) \in (0,1]\) is a measure of the most interior point in \(L \cap \Omega \). The ideal value \(\delta (L\cap \Omega ) = 1\) is attained when \(L \cap \Omega \) contains the center of the symmetric cone \(\Omega \). We describe several possible implementations for the basic procedure including a perceptron scheme and a smooth perceptron scheme. The perceptron scheme requires \(\mathcal {O}(r^4)\) perceptron updates and the smooth perceptron scheme requires \(\mathcal {O}(r^2)\) smooth perceptron updates, where r stands for the Jordan algebra rank of V.

Exact Conic Programming Relaxations for a Class of Convex Polynomial Cone Programs

In this paper, under a suitable regularity condition, we establish a broad class of conic convex polynomial optimization problems, called conic sum-of-squares convex polynomial programs, exhibiting exact conic programming relaxation, which can be solved by various numerical methods such as interior point methods. By considering a general convex cone program, we give unified results that apply to many classes of important cone programs, such as the second-order cone programs, semidefinite programs, and polyhedral cone programs. When the cones involved in the programs are polyhedral cones, we present a regularity-free exact semidefinite programming relaxation. We do this by establishing a sum-of-squares polynomial representation of positivity of a real sum-of-squares convex polynomial over a conic sum-of-squares convex system. In many cases, the sum-of-squares representation can be numerically checked via solving a conic programming problem. Consequently, we also show that a convex set, described by a conic sum-of-squares convex polynomial, is (lifted) conic linear representable in the sense that it can be expressed as (a projection of) the set of solutions to some conic linear systems.

The Extended Conic Sector Theorem

Input-output stability theory is an indispensable tool in control engineering. However, classic results such as the Conic Sector Theorem, cannot be applied in some important cases. This technical note extends the original Conic Sector Theorem to apply to open-loop unstable systems and those with negative upper conic bounds and incorporates a careful treatment of degenerate conic systems and alterations accounting for nonzero initial conditions. Simulations highlight how the contributions facilitate robust control in the most challenging of circumstances.

Design of Centralized PI Controller for Interacting Conical Tank System

Background/Objectives: The objective of this paper is to design centralised controller for liquid level in the two interacting conical tank system. Methods/Statistical analysis: Two Centralised Proportional and Integral (PI) controller tuning method (Davison method and Tanttu and Liestehto method) are used for simulation and implemented in real time process. Findings: Centralised PI controller reduces the interaction between the two interacting conical tank and improves the system performance. The performances are compared in terms of Integral Square Error (ISE) and Peak Overshoot. It is observed that Tanttu and Liestehto method produces better response than Davison method. Applications/Improvements: Level control in two interacting conical tank is used in chemical industries and water treatment plants.

Comparison of PI Controller, Model Reference Adaptive Controller and Fuzzy Logic Controller for Coupled Tank System

Objective: The Coupled Tank System is modeled and its level should be controlled. Methods: The conventional Proportional Integral (PI) controller, Model Reference Adaptive Controller (MRAC) and Fuzzy Logic Controller are used, to control the level of the second tank. The MRAC can alter the controller parameters in response to changes in plant and the reference model indicates properties of the desired control system. The Fuzzy Logic Controller uses a set of rules to control the plant. Findings: The simulation is done, which demonstrates that the MRAC controller delivers better response compared to the PI controller and Fuzzy Logic Controller. Applications: The Model Reference Adaptive controller can be implemented in various level control applications such as conical tank system, spherical tank system, etc.

Soft Computing Technique and Conventional Controller for Conical Tank Level Control

In many process industries the control of liquid level is mandatory. But the control of nonlinear process is difficult. Many process industries use conical tanks because of its non linear shape contributes better drainage for solid mixtures, slurries and viscous liquids. So, control of conical tank level is a challenging task due to its non-linearity and continually varying cross-section. This is due to relationship between controlled variable level and manipulated variable flow rate, which has a square root relationship. The main objective is to execute the suitable controller for conical tank system to maintain the desired level. System identification of the non-linear process is done using black box modelling and found to be first order plus dead time (FOPDT) model. In this paper it is proposed to obtain the mathematical modelling of a conical tank system and to study the system using block diagram after that soft computing technique like fuzzy and conventional controller is also used for the comparison.

On Kepler’s system of conics in Astronomiae pars optica

This is an attempt to explain Kepler’s invention of the first “non-cone-based” system of conics, and to put it into a historical perspective.

A Graph Separation Stability Condition for Non-Planar Conic Systems

Abstract: Results related to an extension of the notion of nonlinear conic systems to the non-planar case are presented. Specifically, parametrization of the non-planar dynamic conic sectors in terms of their central subspace and radius is developed. Relationship between (Q,S,R)-dissipativity and non-planar conicity is studied; in particular, it is shown that any (Q,S,R)-dissipative system is non-planar conic, and a procedure for calculation of the central subspace and the radius is presented. A graph separation stability condition for interconnections of non-planar conic systems is formulated and proven. The results presented in this work will form a background for the future extensions of the scattering transformation-based stabilization techniques to the case of interconnections of non-planar conic systems.

Performance characteristics of constant flow valve compensated conical multirecess hybrid journal bearing under micropolar lubrication

This paper presents the theoretical study on the effect of micropolar parameters of the fluid and the semi-cone angle on the performance characteristics of a conical multirecess hybrid journal bearing under micropolar lubrication compensated with constant flow valve. The numerical solution of the modified Reynolds equation for micropolar fluid flow on the conical surface has been done by using finite element method using Galerkin's technique along with the necessary boundary conditions and iterative scheme. The performance characteristics have been presented for a wide range of the semi-cone angle and the restrictor design parameter of the conical hybrid journal bearing system operating under Newtonian and micropolar lubrication. The study suggests that the semi-cone angle of the bearing, constant flow valve restrictor and micropolar parameters significantly influence the performance characteristics of conical bearing. With increase in semi-cone angle of bearing the direct stiffness, damping coefficients and the stability margin have been observed to increase significantly.

Pulverized-coal injection at a blast furnace with a conical charging system

The initial experience with pulverized-coal injection in two blast furnaces at OAO NLMK is analyzed. Factors affecting the pulverized-coal combustion in the tuyere are identified. The operational efficiency of the blast furnace with different pulverized-coal flow rates is discussed. Stable pulverized-coal consumption no lower than 135 kg/t has been attained at a blast furnace with a conical charging system; the corresponding furnace productivity is 70–75 t/m2 per day.

Second order geometry of spacelike surfaces in de Sitter 5-space

The de Sitter space is known as a Lorentz space with positive constant curvature in the Minkowski space. A surface with a Riemannian metric is called a spacelike surface. In this work we investigate properties of the second order geometry of spacelike surfaces in de Sitter space S5 1 by using the action of GL(2,R) X SO(1,2) on the system of conics defined by the second fundamental form. The main results are the classification of the second fundamental mapping and the description of the possible configurations of the LMN-ellipse. This ellipse gives information on the lightlike binormal directions and consequently about their associated asymptotic directions.

Control of Two Tank Conical Interacting Level System using Relay Auto Tuning

Objectives: The main objective of this work is to design a relay feedback based PI controller to maintain the level of two tank conical interacting level system (TTCILS) which is highly a non linear process. Methods: This technique is based on the observation that when the output lags behind the input by π radians, the closed loop system can oscillate with a period of Pu. It is a simple way to tune PI controllers that avoids trial and error process. Findings: A continuous cycling of controlled variable is generated from the relay feedback experiment. The important process information, ultimate gain and ultimate period can be extracted directly from the experiment. Model design and simulation is carried out in MATLAB/SIMULINK environment. Conclusion: The advantage of this method is that, it requires a single relay feedback test. Simulation studies prove that this technique has good tracking and disturbance rejection capability.

Design of Deadbeat Algorithm for a Nonlinear Conical Tank System

Abstract The control of nonlinear process is the main problem in process industries which repeatedly show its nonlinear activity. In this paper the control action of nonlinear framework using Digital controller with Deadbeat Algorithm was presented. The Nonlinear process which has been considered for its application is conical tank system. System identification of this nonlinear method is made using mathematical modeling and it is approximated to a first order system. Simulation is carried out by the MATLAB software and its servo and regulatory operation is resolved.

Analysis and Design of Robust Positivity and Stability for Continuous-Time Linear Uncertain Systems

This paper deals with the analysis and design of positivity and stability of linear continuous conic systems. First, two robustness analysis theorems are proposed for the systems with state-feedback. Second, the state-feedback stabilization problem is solved by using linear programming (LP). Numerical examples are given for illustration. Finally, the conclusions are made.

A Generalization of the Scattering Transformation for Conic Systems

A generalized version of the scattering transformation is introduced which allows for rendering of the input-output characteristics of a nonlinear conic system into an arbitrary prescribed conic sector. The transformation is consequently applied to the problem of stabilization of conic systems interconnections. The stability is achieved by rendering the input-output characteristics of the subsystems into complementary cones such that the graph separation stability condition is satisfied. In the presence of communication delays, the scattering-based technique for stabilization of feedback interconnections is generalized to the case of arbitrary conic subsystems.

Solving second-order conic systems with variable precision

We describe and analyze an interior-point method to decide feasibility problems of second-order conic systems. A main feature of our algorithm is that arithmetic operations are performed with finite precision. Bounds for both the number of arithmetic operations and the finest precision required are exhibited.

On relationships among passivity, positive realness, and dissipativity in linear systems

The notions of passivity and positive realness are fundamental concepts in classical control theory, but the use of the terms has varied. For LTI systems, these two concepts capture the same essential property of dynamical systems, that is, a system with this property does not generate its own energy but only stores and dissipates energy supplied by the environment. This paper summarizes the connection between these two concepts for continuous and discrete time LTI systems. Beyond that, relationships are provided between classes of strictly passive systems and classes of positive real systems. The more general framework of dissipativity is introduced to connect passivity and positive realness and also to survey other energy-based results. The frameworks of passivity indices and conic systems are discussed to connect to passivity and dissipativity. After surveying relevant existing results, some clarifying results are presented. These involve connections between classes of passive systems and finite-gain L2 stability as well as asymptotic stability. Additional results are given to clarify some of the more subtle conditions between classes of these systems and stability results. This paper surveys existing connections between classes of passive and positive real systems and provides results that clarify more subtle connections between these concepts.

Comments on: Farkas’ lemma: three decades of generalizations for mathematical optimization

The celebrated Farkas lemma originated in Farkas (1902) provides us an attractively simple and extremely useful characterization for a linear inequality to be a consequence of a linear inequality system on the Euclidean space Rn . This lemma has been extensively studied and extended in many directions, including conic systems (linear, sublinear, convex), infinite/semi-infinite convex inequality systems and some classes of nonconvex systems such as the difference of convex (DC) systems and composite convex systems, and the rich results are spread in the literature.

Modelling and Controlling of Two Conical Tank Interacting Level System Using Regime Based Multi Model Adaptive Concept

The implementation of modelling and control algorithms for multi-input/multi-output (MIMO) systems is often complicated due to variations in process dynamics that occur because of change in operating point and characteristics of nonlinear dynamic coupling. Such difficulties often affect performance of modelling techniques and existing industrial controllers such as decoupled based decentralized fixed gain PID and Gain scheduled PID controllers unsatisfactory. In this paper, authors have represented nonlinear dynamics of process as a family of local linear models and local Dynamic Matrix Controllers (DMC) have been designed on the basis of linear models. Using Regime based Multi Model Adaptive Control (MMAC) strategy, Adaptive Dynamic Matrix Controller (ADMC) has been designed to control the nonlinear process. The applicability of the developed ADMC scheme using Regime based Multi Model Adaptive concept has been demonstrated on Two Conical Tank Interacting Level System (TCTILS) which exhibits dynamic nonlinearity and coupling dynamics. Simulation results show that the designed ADMC scheme overcome coupling effects among each degree of freedom and meet the design specifications for each loop independently in each operating regime.

Round-off estimates for second-order conic feasibility problems

We present the analysis of an interior-point method to decide feasibility problems of second-order conic systems. A main feature of this algorithm is that arithmetic operations are performed with finite precision. Bounds for both the number of arithmetic operations and the finest precision required are exhibited.