Characteristic Polynomial Assignment Research Articles

Characteristic polynomial assignment for plants with semialgebraic uncertainty: A robust diophantine equation approach

SummaryIn this paper, we address the problem of robust characteristic polynomial assignment for LTI systems whose parameters are assumed to belong to a semialgebraic uncertainty region. The objective is to design a dynamic fixed‐order controller in order to constrain the coefficients of the closed‐loop characteristic polynomial within prescribed intervals. First, necessary conditions on the plant parameters for the existence of a robust controller are reviewed, and it is shown that such conditions are satisfied if and only if a suitable Sylvester matrix is nonsingular for all possible values of the uncertain plant parameters. The problem of checking such a robust nonsingularity condition is formulated in terms of a nonconvex optimization problem. Then, the set of all feasible robust controllers is sought through the solution to a suitable robust diophantine equation. Convex relaxation techniques based on sum‐of‐square decomposition of positive polynomials are used to efficiently solve the formulated optimization problems by means of semidefinite programming. The presented approach provides a generalization of the results previously proposed in the literature on the problem of assigning the characteristic polynomial in the presence of plant parametric uncertainty. Copyright © 2014 John Wiley & Sons, Ltd.

On pole structure assignment in linear systems

The problem of pole structure assignment (PSA) by state feedback in implicit, linear and uncontrollable systems is discussed in the article. It is shown that the problem is solvable if the system is regularisable. Then necessary and sufficient conditions for characteristic polynomial assignment are established. In the case of PSA (invariant polynomials assignment) just necessary conditions have been obtained. But it turns out that these conditions are also sufficient in some special cases. This happens, for example, when the system does not possess any non-proper controllability indexes. A possible application of the achieved results to modelling a constrained movement of a robot arm is outlined, too.

Relatively Optimal Control With Characteristic Polynomial Assignment and Output Feedback

A relatively optimal control is a stabilizing controller such that, if initialized at its zero state, produces the optimal (constrained) behavior for the nominal initial condition of the plant (without feedforwarding and tracking the optimal trajectory). In this paper, we prove that a relatively optimal control can be obtained under quite general constraints and objective function, in particular without imposing 0-terminal constraints as previously done. The main result is that stability of the closed-loop system can be achieved by assigning an arbitrary closed-loop characteristic stable polynomial to the plant. An explicit solution is provided. We also show how to choose the characteristic polynomial in such a way that the constraints (which are enforced on a finite horizon) can be globally or ultimately satisfied (i.e., satisfied from a certain time on). We provide conditions to achieve strong stabilization (stabilization by means of a stable compensator) precisely, we show how to assign both compensator and closed-loop poles. We consider the output feedback problem, and we show that it can be successfully solved by means of a proper observer initialization (based on output measurements only). We discuss several applications of the technique and provide experimental results on a cart-pendulum system.

Robust Controller Design via Neural Network Computing

This paper deals with the problem of robust characteristic polynomial assignment (RCPA) for process systems having parametric uncertainties. A novel computation technique, which is mainly based on the idea of converting the RCPA problem into an equivalent Kennedy and Chua neural network model, is proposed. A measure of uncertain constraint violation is introduced in the solution scheme, which reflects the degree of violation of the uncertain constraints and thus makes the resulting neural model having no uncertain terms and easy to solve. With the Lyapunov-based analysis, the new solution model for the RCPA problem is proven to be stable and without oscillation. To demonstrate the effectiveness and applicability of the presented scheme, two illustrative examples are provided. Extensive simulation results show that the presented approach to the robust controller design is simple, effective and applicable.

Robust characteristic polynomial assignment

In this paper a computational method for designing controllers which attempt to place the characteristic polynomial of an uncertain system inside some prescribed region is presented. An objective function consisting of two terms is proposed, penalizing both the distance to a given controller and the size of the uncertainty region developed to solve the robust assignment problem. The algorithm is based on semi-infinite programming theory, and the special structure of the problem is exploited to get both simpler methods and convergence proof. The way in which the uncertain parameters affect the plant coefficients is realistic, since it includes multinomial dependence. An example of application is given to illustrate the approach.

Mathematical programming with linear uncertain constraints—application to robust control

In this paper we address the problem of optimizing a convex functional subject to linear constraints with uncertainty, which has wide applications in robust control theory. Robust characteristic polynomial assignment is first briefly reviewed to motivate the study. The main result of this work is an algorithm which converges to the solution of the uncertain problem. Moreover, special (but important) cases are considered, where finite convergence is guaranteed.

Algorithm for characteristic polynomial assignment of two-dimensional discrete systems

An algorithm is proposed for characteristic polynomial assignment via state feedback for a wide class of two-dimensional discrete systems. For this class of systems, the required feedback vector is given analytically.

Characteristic polynomial assignment in 2-dimensional discrete systems

The characteristic polynomial assignment for discrete 2-dimensional systems described by the Fornasini-Marchesini model is considered. The similarities and differences of the solutions to this problem, using Roesser's and Fornasini and Marchesini's models, are discussed. Furthermore, the assignment of a part of the characteristic polynomial and the simultaneous calculation of the residual polynomial for the Fornasini-Marchesini model is considered.

On 2-D pole placement

The problem of characteristic polynomial assignment by dynamic output feedback in 2-D systems is solved via 2-D polynomial methods. A simple necessary and sufficient condition is given. The class of all assignable characteristic polynomials is described in a parametric form. Stable and causal solutions are treated.

Characteristic polynomial assignment of 3D discrete systems using a canonical state space model

In this paper the characteristic polynomial assignment (control) problem of single-input single-output (SISO), three dimensional (3D) discrete systems is considered. Two control laws are used for the solution of the above problem, the output feedback and the state feedback control. The output feedback control is a well known technique which unfortunately leads to very restrictive conditions between the coefficients of the closed-loop characteristic polynomial that can be achieved. A canonical state space model representing a given 3D transfer function, recently introduced, makes the state feedback control technique feasible for application. Use of the state feedback control method leads to less stringent conditions between the closed-loop characteristic polynomial coefficients. This advantage of the state versus output feedback control is more important in the case of higher order systems (involving a large number of characteristic polynomial coefficients).

Characteristic polynomial assignment and determination of the residual polynomial in 2-D systems

An approach is presented for simultaneously solving the characteristic polynomial assignment problem and the problem of determining the residual polynomial (i.e., the unassignable part of the open-loop characteristic polynomial) in 2-D systems.