Formulation Method of Gain Calculation at Marginal Stability of a Linear Invariant Control Systems

The stability analysis of a linear invariant control system is based mainly on its characteristic equation. There are various methods to examine, to do analysis, and to design a control system. Those methods become less effective and more complicated for use when those methods are used for a high order of an open loop transfer function that has several poles and several zeros. In this research work a new method was developed to find the gains at marginal stability and the intersection points with the imaginary axis of a single-input and single-output of a linear invariant control system by using two new formulas. First formula is used to construct a new polynomial where its roots are the intersection points with the imaginary axis of the s-plane, and a second formula is used to calculate the gains at the marginal stability of the system. The coefficients of the characteristic equation’s polynomial of the control system are substituted in the first formula to obtain a new polynomial. The roots of the obtained polynomial are substituted in the second formula to obtain the gains at marginal stability. In this research work the derivation of the polynomial’s construction formula, its mathematical proof, and the derivation of the gains formula at marginal stability are presented. The proposed Formulization method is compared with another three common methods in the solution of three examples. The used methods are the proposed method, Routh-Hurwitz criterion, Root Locus technique, and the complex variable s on the imaginary axis. The chosen examples are three control systems where their transfer functions are different in order and in complexity, going from low to high. The comparison shows that the Formulization method is accurate and needs less mathematical operations by the user. It is applicable for any order of a single-input and single-output of invariant control systems. It is an effective method especially for a higher order and for more complicated transfer functions of the control systems

Software for Exact Integration of Polynomials over Polyhedra

We are interested in the fast computation of the exact value of integrals of polynomial functions over convex polyhedra. We present speed ups and extensions of the algorithms presented in previous work. We present the new software implementation and provide benchmark computations. The computation of integrals of polynomials over polyhedral regions has many applications; here we demonstrate our algorithmic tools solving a challenge from combinatorial voting theory.Comment: Major updat

Deep Learning Topological Invariants of Band Insulators

In this work we design and train deep neural networks to predict topological invariants for one-dimensional four-band insulators in AIII class whose topological invariant is the winding number, and two-dimensional two-band insulators in A class whose topological invariant is the Chern number. Given Hamiltonians in the momentum space as the input, neural networks can predict topological invariants for both classes with accuracy close to or higher than 90%, even for Hamiltonians whose invariants are beyond the training data set. Despite the complexity of the neural network, we find that the output of certain intermediate hidden layers resembles either the winding angle for models in AIII class or the solid angle (Berry curvature) for models in A class, indicating that neural networks essentially capture the mathematical formula of topological invariants. Our work demonstrates the ability of neural networks to predict topological invariants for complicated models with local Hamiltonians as the only input, and offers an example that even a deep neural network is understandable.Comment: 8 pages, 5 figure

Modal Identification of Dynamic Properties of the Cylindrical Grinder.

In the paper the method of model identification of the cylindrical grinder dynamic properties by means of experimental modal test was described. The method application, hardware solution as well as the procedure of carrying out the identification modal test of the cylindrical grinder was presented. The experiment was performed in order to acquire the frequency response function (FRF) of the cylindrical grinder. Having obtained the experimental FRF, the mathematical model of the response function was created. That mathematical model of the machine tool dynamic behavior can be applied in grinder and grinding holistic model. The conclusions regarding the application aspects of experimental modal analysis in order to identify dynamic properties of the machine tool were drawn

Rollover Preventive Force Synthesis at Active Suspensions in a Vehicle Performing a Severe Maneuver with Wheels Lifted off

Among the intelligent safety technologies for road vehicles, active suspensions controlled by embedded computing elements for preventing rollover have received a lot of attention. The existing models for synthesizing and allocating forces in such suspensions are conservatively based on the constraint that no wheels lift off the ground. However, in practice, smart/active suspensions are more necessary in the situation where the wheels have just lifted off the ground. The difficulty in computing control in the last situation is that the problem requires satisfying disjunctive constraints on the dynamics. To the authors',knowledge, no efficient solution method is available for the simulation of dynamics with disjunctive constraints and thus hardware realizable and accurate force allocation in an active suspension tends to be a difficulty. In this work we give an algorithm for and simulate numerical solutions of the force allocation problem as an optimal control problem constrained by dynamics with disjunctive constraints. In particular we study the allocation and synthesis of time-dependent active suspension forces in terms of sensor output data in order to stabilize the roll motion of the road vehicle. An equivalent constraint in the form of a convex combination (hull) is proposed to satisfy the disjunctive constraints. The validated numerical simulations show that it is possible to allocate and synthesize control forces at the active suspensions from sensor output data such that the forces stabilize the roll moment of the vehicle with its wheels just lifted off the ground during arbitrary fish-hook maneuvers