Pole Assignment Problem Research Articles

Controllability and control for discrete periodic systems based on fully-actuated system approach

The problem of controllability and control of linear discrete-periodic systems is investigated in this paper. The high-order fully-actuated models for linear discrete periodic time-varying systems are constructed, and a controllability criterion based on fully-actuated system models is proposed. On this basis, stabilization as the fundamental issue is studied and periodic state-feedback control laws are designed via fully-actuated systems approach and parametric design, which converted the original problem into the pole assignment problem for linear constant systems. Finally, a numerical example is given to verify the validity and feasibility of the proposed method.

A parametric poles assignment algorithm for high-order linear discrete periodic systems

The periodic state feedback pole assignment problem of high-order periodic discrete systems is investigated, and the pole assignment problem for such systems is transformed into a class of problems for resolving periodic Sylvester matrix equations with constraints. Using the technique of cyclic lifting, such equations can be transformed into high-order time-invariant Sylvester matrix equations and solved using a parametric method. Lastly, a numerical example is provided to demonstrate the validity of the procedure.

Pole Assignment by Proportional-plus-derivative State Feedback for Multivariable Linear Time-invariant Systems

In this paper the pole assignment problem by proportional-plus-derivative state feedback for multivariable linear-time invariant systems is studied. In particular, explicit necessary and sufficient conditions are established for a given polynomial with real coefficients to be characteristic polynomial of closed-loop system obtained by proportional-plus-derivative state feedback from the given multivariable linear time-invariant system. A procedure is given for the calculation of proportional-plus-derivative state feedback which places the poles of closed-loop system at desired locations. Our approach is based on properties of polynomial matrices.

A method for robust quadratic pole assignment

A new method is proposed for robust quadratic pole assignment problem. We first give the results on the solution of quadratic pole assignment problem. Then the Schur form corresponding to closed-loop system is established and normality departure is used to measure the robustness of closed-loop system. On these grounds, a method for robust quadratic pole assignment problem is proposed by minimizing the normality departure of closed-loop system. The proposed method avoids to transform the second-order control system to the first-order control system. Moreover, it is implemented in real operations. Numerical experiments show that our method has good performance on the robustness of closed-loop system.

On pole assignment of high-order discrete-time linear systems with multiple state and input delays

<p style='text-indent:20px;'>This paper studies the problem of pole assignment for high-order discrete-time linear systems with multiple state and input delays. When the number of state delays is larger than or equal to that of input delays, an effective predictor feedback controller is proposed based on the augmented technique, and the design process for the feedback gain is also presented. In addition, it is proved that the pole assignment problem is solvable if and only if the solutions to a linear matrix equation are such that a matrix is nonsingular. When the number of state delays is smaller than that of input delays, the original system is first transformed into a delay-free system with keeping the system controllability invariant, and then, the corresponding controller with designable feedback gain is established. To characterize all of the feedback gains, a factorization approach is introduced which can provide full degree of freedom. Numerical examples are employed to illustrate the effectiveness of the proposed approaches.</p>

A receptance method for partial quadratic pole assignment of asymmetric systems

A new receptance method is proposed for partial quadratic pole assignment problem (PQPAP) of asymmetric systems with multi-input. We use receptance matrices of symmetric parts of open-loop systems to express the solution of PQPAP. Then we derive the sensitivities of changed eigenvalues of close-loop system and give the measure of robustness. On these grounds, an algorithm for solving the robust and minimum norm PQPAP is put forward. Our method utilizes receptance matrices of symmetric parts of asymmetric systems and it is applicable to the systems with multi-input. Finally, numerical results are given to show the efficiency of our method.

Revisiting IRKA: Connections with Pole Placement and Backward Stability

The iterative rational Krylov algorithm (IRKA) is a popular approach for producing locally optimal reduced-order ${\mathscr{H}}_{2}$ -approximations to linear time-invariant (LTI) dynamical systems. Overall, IRKA has seen significant practical success in computing high fidelity (locally) optimal reduced models and has been successfully applied in a variety of large-scale settings. Moreover, IRKA has provided a foundation for recent extensions to the systematic model reduction of bilinear and nonlinear dynamical systems. Convergence of the basic IRKA iteration is generally observed to be rapid—but not always; and despite the simplicity of the iteration, its convergence behavior is remarkably complex and not well understood aside from a few special cases. The overall effectiveness and computational robustness of the basic IRKA iteration is surprising since its algorithmic goals are very similar to a pole assignment problem, which can be notoriously ill-conditioned. We investigate this connection here and discuss a variety of nice properties of the IRKA iteration that are revealed when the iteration is framed with respect to a primitive basis. We find that the connection with pole assignment suggests refinements to the basic algorithm that can improve convergence behavior, leading also to new choices for termination criteria that assure backward stability.

Parametric solutions to generalized periodic Sylvester bimatrix equations

The problem considered in this paper is to solve generalized periodic Sylvester bimatrix equations. By utilizing bimatrix mapping tool and some algebraic technology, a parametric algorithm is presented to derive the complete and explicit solutions to this type of equations. The exact solutions derived by the proposed algorithm possess countless freely parameters, which can provide sufficient degree of freedom. The method is also applied to the pole assignment problem of complex-valued linear periodic system. Finally, two practical examples are employed to verify the correctness and validity of the proposed approach.

Pole assignment of high-order linear systems with high-order time-derivatives in the input

This paper studies pole assignment of a class of high-order (HO) linear systems, which contains HO time-derivatives of both the state vector and the input vector, and can be used to model some practical control systems directly. Based on an equivalent first-order linear system, whose state vector contains HO time-derivatives of both the state vector and the input vector of the original HO linear system, it is shown that the pole assignment problem is solvable if and only if solutions to a linear matrix equation, which takes a similar form as the original HO linear system, are such that a matrix is nonsingular. Meanwhile, all the feedback gains are characterized by solutions to the linear matrix equation. Explicit solutions to the linear matrix equations are then proposed by using right-coprime factorization of the original HO linear system. Explicit feedback gains are also established for some special cases. A numerical example is worked out to illustrate the effectiveness of the proposed approach.

Stabilization of Linear Control Systems and Pole Assignment Problem: A Survey

Stabilization of Linear Control Systems and Pole Assignment Problem: A Survey

Pole assignment for continuous-time fractional order systems

The problem of pole assignment for linear continuous-time commensurate fractional order systems is addressed. The closed-loop poles are assigned in the adequate stability region according to the fractional derivative order α in or in . First, sufficient stability conditions are established for both cases. Then, link is made to a pole assignment procedure enabling one to ensure the closed-loop asymptotic stability. Furthermore, robustness of the pole assignment against parametric uncertainties for the fractional order differential system has been investigated for both cases. Finally, ease of application and effectiveness of the proposed approach are shown through a practical example.Abbreviations: FOS: Fractional order system; LHP: left half plan; RHP: right half plan; LMI: linear matrix inequality

Disturbance rejection for interconnected positive systems and its application to formation control of quadrotors

This study deals with disturbance rejection of interconnected positive (IP) systems by output feedback laws that are installed as an outer loop of the IP system. Since the IP system without disturbances has a first integral, which is a linear function of the state and takes a constant value along the trajectories, this study utilises this function as an index of detecting the disturbances. Then, dynamic and static disturbance rejection output feedback laws with the first integral are proposed, respectively. Although the pole assignment problem of the IP system with the static output feedback law is not solvable in general, this study reveals that the system matrix of the closed-loop system has a structure called rank-one updated matrices. This special structure implies that the simple eigenvalue zero of the original IP system moves to a stable one by the feedback law while the other stable eigenvalues never move. This study applies these feedback laws to formation control of quadrotors based on IP systems where the dynamics of a quadrotor are linearised to be positive and stable second-order subsystems by dual quaternion techniques. Numerical examples illustrate the quadrotors achieve a designed shape of formation against the disturbances.

Stabilization of linear control systems. Pole assignment problem. A survey

This paper reviews the problems of stabilization and pole assignment in the control of linear time-invariant systems using the static state and output feedback. The main results are presented, along with a literature review. Stationary and nonstationary stabilizations with the static feedback are considered. The algorithms of low- and high-frequency stabilization of linear systems for solving Brockett’s stabilization problem are provided. The effective necessary and sufficient conditions for stabilization of two- and three-dimensional controllable linear systems are given in terms of the system parameters. The pole assignment problem and the related issues for linear systems are considered.

Proportional Local Assignability of Lyapunov Spectrum of Linear Discrete Time-Varying Systems

We consider a local version of the pole assignment problem for linear discrete time-varying systems. Our aim is to obtain sufficient conditions to place the Lyapunov spectrum of the closed-loop system in an arbitrary position within some neighborhood of the Lyapunov spectrum of the free system using an appropriate time-varying linear feedback. Moreover, we assume that the norm of the matrix of linear feedback should be bounded from above by the distance between these two spectra with some constant multiplier. We prove that diagonalizability, Lyapunov regularity, and stability of the Lyapunov spectrum each separately are the required sufficient conditions provided that the open-loop system is uniformly completely controllable.

Necessary and Sufficient Conditions for Assignability of the Lyapunov Spectrum of Discrete Linear Time-Varying Systems

We consider a version of the pole assignment problem for linear discrete time-varying systems with linear state feedback. Our aim is to prove that all the systems from the closure (in the topology of pointwise convergence) of all shifts of the original system have an assignable Lyapunov spectrum if and only if the original system is uniformly completely controllable. We also provide an example to show that uniform complete controllability is not a necessary condition for assignability of the Lyapunov spectrum of the original system.

Pole Assignment in Hybrid Differential-Difference Systems

We present a general approach to the pole assignment problem for linear stationary hybrid differential-difference systems as a coefficient control problem for their characteristic equations. Various scales (classes) of linear feedback controllers are considered. Special attention is paid to the solvability of the pole assignment problem for such systems in the scale of general differential-difference controllers and in a general scale that, along with differential-difference controllers, contains integral controllers whose kernels are compactly supported functions. A general scheme for constructing such controllers is proposed based on the algebraic properties of the shift operator, the Paley–Wiener theorem on compactly supported functions, and the methods of interpolation theory in the class of entire functions of exponential type. Examples and counterexamples illustrating the results are given.

Solutions to linear bimatrix equations with applications to pole assignment of complex-valued linear systems

We study in this paper solutions to several kinds of linear bimatrix equations arising from pole assignment and stability analysis of complex-valued linear systems, which have several potential applications in control theory, particularly, can be used to model second-order linear systems in a very dense manner. These linear bimatrix equations include generalized Sylvester bimatrix equations, Sylvester bimatrix equations, Stein bimatrix equations, and Lyapunov bimatrix equations. Complete and explicit solutions are provided in terms of the bimatrices that are coefficients of the equations/systems. The obtained solutions are then used to solve the full state feedback pole assignment problem for complex-valued linear system. For a special case of complex-valued linear systems, the so-called antilinear system, the solutions are also used to solve the so-called anti-preserving (the closed-loop system is still an antilinear system) and normalization (the closed-loop system is a normal linear system) problems. Second-order linear systems, particularly, the spacecraft rendezvous control system, are used to demonstrate the obtained theoretical results.

Schur method for robust pole assignment of descriptor systems via proportional plus derivative state feedback

ABSTRACTThe pole assignment problem for descriptor systems is a classical inverse algebraic eigenvalue problem, which has attracted attention for decades. In this paper, we propose a direct method to solve the problem with the application of the proportional plus derivative state feedback, intending to obtain a robust closed-loop system. Theorems on the feasibility of our method will be presented. Numerical examples show that our method yields poles of high relative accuracy.

Causal polynomial pole assignment and stabilisation of SISO strictly causal linear discrete processes

ABSTRACTThe paper considers the pole assignment and the stabilisation problem under the practically essential constraint, that the designed controllers have to be causal, because only in that case they will be realisable in real time. It is shown that in case of a strictly proper process, when the set of solutions of the pole assignment or stabilisation problem is not empty, then it always contains a non-empty subset of non-causal controllers. Moreover, according to the set of causal controllers, we have one of the three situations: (1) the set is empty, (2) the set has actually one element and (3) there exists a subset of causal solutions. The paper provides complete solutions for both problems in the form of instructions for building the corresponding sets of causal controllers. Drawing on examples of a double integrator with delay, the procedures are illustrated.

Modal control of hybrid differential-difference systems and associated delay systems of neutral type in scales of differential-difference controllers

We study the statements and solvability of the modal control problem (the pole assignment problem) for linear time-invariant hybrid difference-differential systems in symmetric form and for the associated delay systems of neutral type. We obtain constructive necessary and sufficient parametric conditions for the modal controllability of the systems in question in various scales of difference-differential controllers. Methods for the construction of such controllers solving the corresponding modal controllability problem are indicated. The results are illustrated by examples and counterexamples.