Pole Placement Problem Research Articles

SOLUTION OF THE EXPONENTIAL DIOPHANTINE EQUATION 10^X + 400^Y=Z^2

Diophantine equations are so important in solving important real-world problems like network flow problems, pole placement problems, business investment problems, and data privacy problems that researchers are becoming more interested in developing new techniques for analyzing the nature and solutions of the various Diophantine equations. In the present study, authors examined the Diophantine equation〖10〗^X+〖400〗^Y=Z^2, where X,Y,Zare non-negative integers, for the non-negative integer solution of this equation. A result of the present study shows that it has no non-negative integer solution.

Pole placement with minimum gain and sparse static feedback

The work addresses the issue of developing polar positioning algorithms for linear control systems with despotic thinning constraints on a static feedback matrix. The design of sparse-limiting feedback control systems is considered to be one of the more complex issues, but the issue of defining the sparse feedback matrix, which allows the selection of eigenvalues or, in other words, the placement of poles, has not been fully studied. Algorithms for determining the eigenvalues, i.e., solving the problem of pole placement and finding the optimal solution of the technological process control system are considered in the work. The results of numerical analysis confirmed the effectiveness of these algorithms, which allows them to be used in solving practical problems of synthesizing adaptive control systems for technological objects.

Solution of the Diophantine equation 22x+40y=z2

Since Diophantine equations play an important role in solving important real-world problems such as business investment problems, network flow problems, pole placement problems, and data privacy problems, researchers are increasingly interested in developing new techniques for analyzing the nature and solutions of the various Dio phantine equations. In this study we investigate the Diophantine problem 22x+40y = z2, where x, y, z are non-negative integers, and discover that it does not have a non-negative integer solution.

Eigenvalue sensitivity minimisation for robust pole placement by the receptance method

The problem of robust pole placement in active structural vibration control by the method of receptance is considered in this paper. Expressions are derived for the eigenvalue sensitivities to parametric perturbations, which are subsequently minimised to improve performance robustness of the control of a dynamical system. The described approach has application to a vibrating system where variations are present due to manufacturing and material tolerances, damages and environment variabilities. The closed-loop eigenvalue sensitivities are expressed as a linear function of the velocity and displacement feedback gains, allowing their minimisation with carefully calculated feedback gains. The proposed algorithm involves curve fitting perturbed frequency response functions, FRFs, using the rational fraction polynomial method and implementation of a polynomial fit to the individual estimated rational fraction coefficients. This allows the eigenvalue sensitivity to be obtained entirely from structural FRFs, which is consistent with the receptance method. This avoids the need to evaluate the M,C,K matrices which are typically obtained through finite element modelling, that produces modelling uncertainty. It is also demonstrated that the sensitivity minimisation technique can work in conjunction with the pole placement and partial pole placement technique using the receptance method. To illustrate the working of the proposed algorithm, the controller is first implemented numerically and then experimentally.

Sufficient conditions for assignability of nonuniform dichotomy spectrum of discrete time-varying linear systems

We consider a version of the pole placement problem for tempered one-sided linear discrete-time time-varying linear systems. We prove a sufficient condition for assignability of the nonuniform dichotomy spectrum by linear feedback. The main result is that the nonuniform dichotomy spectrum is assignable if the system is completely controllable and certain lower asymptotic bound for the controllability Gramian holds.

Constrained state feedback pole placement of coupled lateral plant dynamics of RLV during the reentry phase

Stability and dynamic behaviour of a linear multivariable system mainly depend on the pole locations of the plant in the complex s plane. For general linear time-invariant multi-input multi-output (MIMO) systems, static (constant gain) output feedback pole-placement is an open nondeterministic polynomial time (NP) hard problem. This problem has attracted much attention from the control community for the last four decades. Even today, the general problem has not been solved analytically because of its highly nonlinear nature. In this paper, we have addressed the issue of finding gains without using the procedure of sequential loop closure. We have proposed an iterative numerical algorithm to solve the pole placement problem with a minimum number of feedback gains. The proposed method is used to compute the state feedback gain matrix for coupled yaw-roll dynamics of a typical reusable launch vehicle (RLV) during the reentry phase.

A PI Controller with a Robust Adaptive Law for a Dielectric Electroactive Polymer Actuator

Dielectric electroactive polymer actuators are new important transducers in control system applications. The design of a high performance controller is a challenging task for these devices. In this work, a PI controller was studied for a dielectric electroactive polymer actuator. The pole placement problem for a closed-loop system with the PI controller was analyzed. The limitations of a PI controller in the pole placement problem are discussed. In this work, the analytic PI controller gain rules were obtained, and therefore extension to adaptive control is possible. To minimize the influence of unmodeled dynamics, the robust adaptive control law is applied. Furthermore, analysis of robust adaptive control was performed in a number of simulations and experiments.

Necessary and Sufficient Conditions for Assignability of Dichotomy Spectrum of One-Sided Discrete Time-Varying Linear Systems

We consider a version of the pole placement problem for one-sided linear discrete time-varying linear systems. Our purpose is to prove that uniform complete controllability is equivalent to possibility of arbitrary assignment of the dichotomy spectrum.

Sign function and ANN based pole placement for computing interval controls

This paper deals with estimation of interval controls for interval Linear Time Invariant (LTI) plants using pole placement technique. Generally, the parameters associated with LTI plant system are assumed to be crisp values, but due to the occurrence of error in experimentation, measurement or due to maintenance induced errors, the parameters are imprecise. As such, the LTI problem reduces to interval LTI. The pole placement problem is then reduced to interval algebraic equation viz. interval Diophantine equation using proper transfer function. Further, the interval Diophantine equation is transformed to Interval System of Linear Equations (ISLE) viz. interval Sylvester matrix equation. Initially, using interval arithmetic an interval algebraic approach has been applied to solve the ISLE for computing non-negative or non-positive controls. Then, another approach using Artificial Neural Network (ANN) is proposed for computing the interval controls. Further, algorithms based on sign function and ANN procedures have been discussed. Finally, the efficiency of the proposed algorithms has been verified using different order interval plants.

Parameter estimation of nonlinearly parameterized regressions without overparameterization: Application to adaptive control

In this paper we propose a solution to the problem of parameter estimation of nonlinearly parameterized regressions – continuous or discrete time – and apply it for adaptive control. We restrict our attention to parameterizations that can be factorized as the product of two functions, a measurable one and a nonlinear function of the parameters to be estimated. Although in this case it is possible to define an extended vector of unknown parameters to get a linear regression, it is well-known that overparameterization suffers from some severe shortcomings. Another feature of the proposed estimator is that parameter convergence is ensured with an excitation assumption that is strictly weaker than persistency of excitation. It is assumed that, after a coordinate change, some of the elements of the transformed function satisfy a monotonicity condition. The proposed estimators are applied to design adaptive controllers for nonlinearly parameterized systems. In continuous-time we consider a general class of nonlinear systems and those described by Euler–Lagrange models, while in discrete-time we apply the method to the challenging problem of indirect adaptive pole-placement. The effectiveness of our approach is illustrated with several classical examples, which are traditionally tackled using below par performance overparameterization and assuming persistency of excitation.

Approximation and control of a class of distributed delay systems

Abstract In this paper, we show that a linear system with distributed state delay can be approximated by a linear delay-free system that has the same state dimension as the original system if a so-called smallness condition holds. The smallness condition is an inequality which the maximum lag and norms of the system matrices have to satisfy. The eigenvalues of the approximate system correspond to the dominant eigenvalues of the original system with distributed delay. We provide a numerically stable, iterative algorithm to compute the state matrix of the approximate system. Furthermore, we show that, based on the proposed approximation, the stabilisation, pole placement and setpoint tracking control problems of the addressed class of distributed delay systems can be performed using methods developed for delay-free systems. Simulation results are provided to show the applicability of the proposed approximation and control design method.

Necessary and sufficient conditions for assignability of dichotomy spectra of continuous time-varying linear systems

We consider a version of the pole placement problem for continuous time-varying linear systems. Our purpose is to prove that uniform complete controllability is equivalent to possibility of arbitrary assignment of the dichotomy spectrum. The main ingredients of the proof are the reduction of system to upper triangular form and the use of the concept of uniform complete stabilization. To illustrate the theoretical result, we consider scalar continuous time-varying control systems. For these systems we provide a simple necessary and sufficient condition for uniform complete controllability and if this condition holds, then we construct an explicit control to assign the dichotomy spectrum.

Два метода решения задачи синтеза для линейной управляемой системы

Два метода решения задачи синтеза для линейной управляемой системы

Pole placement for overdetermined 2D systems

We formulate and solve a pole placement problem by state feedback for overdetermined 2D systems modeled by commutative operator vessels. In this setting, the transfer function of the system is given by a meromorphic bundle map between two holomorphic vector bundles of finite rank over the normalization of a projective plane algebraic curve. The obstruction for a solution is given by an existence of a certain meromorphic bundle map on the input bundle. Reducing to the 1D case, this gives a functional obstruction which is equivalent to the classical pole placement theorem. Our result improves on, and gives a new approach to pole placement even in the classical case, and answers a question of Ball and Vinnikov.

Analytical Tuning Method of MPC Controllers for MIMO First-Order Plus Fractional Dead Time Systems

An analytical model predictive control (MPC) tuning method for multivariable first-order plus fractional dead time systems is presented in this paper. First, the decoupling condition of the closed-loop system is derived, based on which the considered multivariable MPC tuning problem is simplified to a pole placement problem. Given such a simplification, an analytical tuning method guaranteeing the closed-loop stability as well as pre-specified time-domain performance is developed. Finally, simulation examples are provided to show the effectiveness of the proposed method.

Robust force control for brake-by-wire actuators via scenario optimization

Clamping force control in Electro Mechanical Brakes (EMBs) is a challenging task, mainly due to the nonlinear dynamics of the system and the uncertainty affecting its physical parameters. In this paper, a robust tuning of a PID control loop for an EMB is proposed. First, a control-relevant linear model of the system is derived. Then, the optimal parameters of the controller are tuned by solving a convex pole-placement problem and probabilistic robustness guarantees are provided according to the scenario theory. Finally, the performance of the proposed strategy is assessed on a complex nonlinear simulator of the EMB dynamics, and compared with the state of the art approach for robust control of EMBs.

Pole placement in uncertain dynamic systems by variance minimisation

Abstract The problem of pole placement in dynamic systems with uncertainties is addressed using a global optimisation approach. Variability between nominally identical systems, which arises from manufacturing tolerances, wear and environmental variability, is considered. In the proposed method, receptances are measured from one of the open-loop, nominal systems chosen at random. In this case, in addition to the variability about the parameter means, the parameter means are also random. The receptance method is then used to place the poles of the measured system such that their spread, due to parameter uncertainty, is minimised. The measure used to assess the spread is the variance, determined efficiently by a polynomial chaos expansion. Among the advantages of the method are: (i) there is no requirement to model the system since experimental receptances are used, (ii) it is not necessary to measure the mean system, and (iii) uncertainty in poles is quantified across its full range by using variances. Numerical and experimental examples are provided to illustrate the working of the proposed method.

Robust pole placement under structural constraints

A new controller synthesis technique is presented which allows the design of output feedback control systems achieving robust regional pole clustering in the presence of parametric uncertainties as well as satisfying prescribed structural constraints. Such features are rarely jointly present in currently available controller synthesis methods. The central idea in the proposed approach consists in reformulating the original robust pole placement problem into an equivalent robust stabilization problem involving highly structured controller and uncertainty. A numerical application corroborating the applicability of the proposed synthesis technique is also presented.

Further Results on Dominant Pole Placement via Stability Mapping Approach

Dominant roots of the closed loop characteristic equation play a crucial role in terms of the performance of Linear Time Invariant (LTI) systems. Within the scope of this study, a dominant pole placement approach which has two main phases is proposed for PI/PID type controllers. In the first phase, characteristic equation is partitioned into its dominant and non-dominant polynomial pairs and dominant poles are placed to predetermined locations. In the second phase, it is required to determine how far the non-dominant poles can be placed. In the current study, this requirement is transformed into a stability problem and Lyapunov Equation-based stability mapping approach is used. This combined approach creates a more flexible design environment compared to the currently existing approaches in literature. In order to demonstrate this flexibility, two benchmark case studies are included with different definitions of dominant pole placement problem.

The resolution of the equation [formula omitted] and the pole assignment problem: A general approach

In this paper, we generalize an existing technique which is relevant to a pole placement problem and state feedback control design with constraints on control inputs. The existing technique concerns only a restrictive class of systems for which many assumptions should be satisfied. The generalization aims then at avoiding those assumptions. The advantages of generalizing this technique are to determine linear feedback controllers that asymptotically stabilize a wide class of systems which are subject to input constraints without violating these constraints, to derive a simple formula for the state feedback and to assign the whole spectrum when stable poles are not suitably located for performances. Moreover, there will be no need to assume a lack of overlap between open and closed-loop spectra. Illustrative examples are presented to point out these advantages.