Pole assignment using matrix generalized inverses

The problem of pole assignment in a completely controllable linear time-invariant system dx/t = Ax + Bu, y = Cx is considered. A method using matrix generalized inverses is developed for the computation of a matrix K such that the matrix A + BK has prescribed eigenvalues which need satisfy only the condition that a certain number of them are distinct and real; then a feedback law of the form u = r + Kx can be used to achieve the desired pole-placement. The method does not require solution of sets of non-linear equations or manipulation of polynomial matrices, and no knowledge of eigenvalues and/or eigenvectors of A is necessary. If the computed matrix K and the given matrix C satisfy a consistency condition, a matrix Kν such that KνC = K can be directly obtained from K and the desired pole-placement can be realized by an output feedback law u = r + Kνy.     

An algorithm for pole assignment in high order multivariablesystems

An efficient computational method is presented for solving the problem of pole assignment by state feedback in linear multivariable systems with large dimensions. A given multiinput system is first transformed to an upper block Hessenberg form by means of orthogonal state coordinate transformations. The state feedback problem is then reformulated in terms of the Sylvester equation. The transformed system matrices, along with certain assumed block forms for unknown matrices, enable the Sylvester equation to be decomposed and solved effectively. A distinct point of the proposed algorithm is that the solution procedure can be tailored to parallel implementation and is therefore fast. Computational aspects of the algorithm are discussed and numerical examples are provided     

A pole-placement design approach for systems with multipleoperating conditions

The author proposes design procedures based on state-space pole-placement techniques for systems with multiple operating conditions. This is the so-called simultaneous pole-placement problem. First, the full state feedback problem is studied, in which a nonlinear local pole-placement solution is proposed. The design condition is formulated in terms of the rank condition of a multimode controllability matrix. Then, the output feedback problem is approached using a multimodel controller design, which is an extension of the observer design to multimode systems. The design is decomposed into separated global pole-placement subproblems and a local pole-placement subproblem. For a system with some operating conditions having modes on the j ω-axis, but no modes at the origin in the open right-half of the complex plane, stabilizability and detectability conditions for the design of an asymptotically stabilizing control are established, without any restriction on the number of inputs or outputs. Relations of this approach to other simultaneous control design approaches are pointed out     

Optimization of multiple-input systems with assigned poles

In pole assignment for multi-input systems, the constant gain feedback control to achieve desired closed-loop poles is not unique. This design freedom is used to minimize a given performance index, thus combining the pole-placement and optimal control design techniques.     

Asymptotic stabilization of a class of uncertain composite systems

A class of stabilizing, generalized feedback controls is presented for a class of uncertain dynamical systems. The uncertain systems are based on a composite prototype system consisting of two nonlinearly coupled subsystems, with a non-asymptotically-stabilizable linearization. To encompass all possible realizations of uncertainty, a problem formulation based on differential inclusions is adopted. The generalized feedback controls, described in terms of set-valued maps, have practical analogues in the form of discontinuous feedback controls. An intrinsic element of the control strategy is designed to render a prescribed nonlinear manifold, containing the state origin, invariant and globally finite-time attractive. For each uncertain system, the generalized feedback controls guarantee global asymptotic stability of the zero state.     

On constraints in pole assignment by static feedback

A design method based on pole assignment considerations for an nth order linear time-invariant single-input p-output system by means of static (constant gain) feedback is presented for the case in which the rank p of an output matrix C is less than the number of system states n. The constraints in pole assignment are described in the form of linear relationships between the coefficients of the closed-loop characteristic polynomial and some open-loop system parameters. It has been shown that the problem of pole assignment is equivalent to the problem of solving a set of n linear equations with p unknowns in such a way that n – p equations are made linearly dependent by proper choice of coefficients of the closed-loop characteristic polynomial. As a result, all n system poles can be shifted to selected locations which satisfy the constraints. Of particular interest are the design methods for the cases when the number of arbitrarily assigned poles are equal to p and p – I.     

Stabilization of Single‐Input LTI Systems by Proportional‐Derivative Feedback

This article presents an efficient solution to the stabilization pole placement problem for single‐input linear time‐invariant (LTI) systems by proportional‐derivative (PD) feedback. For a controllable system, any arbitrary closed‐loop poles can be placed in order to achieve the desired closed‐loop system performance. Its derivation is based on the transformation of linear system into Hessenberg form by a special coordinate transformation before solving the pole placement problem. The available degrees of freedom offered by PD feedback are utilized to obtain closed‐loop systems with small gains. So, the minimization problem for a suitably chosen cost function is formulated. Simulation results are included to show the effectiveness of the proposed approach.     

On p-normal forms of nonlinear systems

Using the differential-geometric control theory, we present in this note a necessary and sufficient condition under which an affine system is locally feedback equivalent to, via a change of coordinates and restricted smooth state feedback, a generalized normal form called p-normal form, which includes Brunovsky canonical form and feedback linearizable systems in a lower-triangular form as its special cases. We also give an algorithm for computing the appropriate coordinate transformations and feedback control laws.     

Singular PDEs and the single-step formulation of feedback linearization with pole placement

The present work proposes a new formulation and approach to the problem of feedback linearization with pole placement. The problem under consideration is not treated within the context of geometric exact feedback linearization, where restrictive conditions arise from a two-step design method (transformation of the original nonlinear system into a linear one in controllable canonical form with an external reference input, and the subsequent employment of linear pole-placement techniques). In the present work, the problem is formulated in a single step, using tools from singular PDE theory. In particular, the mathematical formulation of the problem is realized via a system of first-order quasi-linear singular PDEs and a rather general set of necessary and sufficient conditions for solvability is derived, by using Lyapunov's auxiliary theorem. The solution to the system of singular PDEs is locally analytic and this enables the development of a series solution method, that is easily programmable with the aid of a symbolic software package. Under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear state feedback law computed through the solution of the system of singular PDEs, both feedback linearization and pole-placement design objectives may be accomplished in a single step, effectively overcoming the restrictions of the other approaches by bypassing the intermediate step of transforming the original system into a linear controllable one with an external reference input.     

Active vibration control for nonlinear vehicle suspension with actuator delay via I/O feedback linearization

The problem of nonlinear vibration control for active vehicle suspension systems with actuator delay is considered. Through feedback linearization, the open-loop nonlinearity is eliminated by the feedback nonlinear term. Based on the finite spectrum assignment, the quarter-car suspension system with actuator delay is converted into an equivalent delay-free one. The nonlinear control includes a linear feedback term, a feedforward compensator, and a control memory term, which can be derived from a Riccati equation and a Sylvester equation, so that the effects produced by the road disturbances and the actuator delay are compensated, respectively. A predictor is designed to implement the predictive state in the designed control. Moreover, a reduced-order observer is constructed to solve its physical unrealisability problem. The stability proofs for the zero dynamics and the closed-loop system are provided. Numerical simulations illustrate the effectiveness and the simplicity of the designed control.     

An approach for pole assignment in singular systems

Pole assignment in a singular system Edx/dt=Ax+Bu is discussed. It is shown that the problem of assigning the roots of det(sE-(A +BF)) by applying a proportional feedback u=Fx+r in a given singular system is equivalent to the problem of pole assignment of an appropriate regular system. An immediate application of this result is that procedures and computational algorithms that were originally developed for assigning eigenvalues in regular systems become useful tools for pole assignment in singular systems. The approach provides a useful tool for the combined problem of eliminating impulsive behavior and stabilizing a singular system     

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.     

On the decentralized control of interconnected systems

Recently, it was shown [1] how to reduce the order of the decentralized control problem for interconnected systems by using local dynamic feedback. The analysis was carried out with polynomial matrices. In this letter, these results are reproduced in state space using the Generalized Hessenberg Representation (GHR). This analysis shows how the interaction between the interconnection structure and the local system structure generates the result above. In addition, this analysis leads directly to numerical algorithms.     

A gradient flow approach to on-line robust pole assignment for synthesizing output feedback control systems

Pole assignment is a basic design method for synthesis of feedback control systems. In this paper, a gradient flow approach is presented for robust pole assignment in synthesizing output feedback control systems. The proposed approach is shown to be capable of synthesizing linear output feedback control systems via on-line robust pole assignment. Convergence of the gradient flow can be guaranteed. Moreover, with appropriate design parameters the gradient flow converges exponentially to an optimal solution to the robust pole assignment problem and the closed-loop control system based on the gradient flow is globally exponentially stable. These desired properties make it possible to apply the proposed approach to slowly time-varying linear control systems. Simulation results are shown to demonstrate the effectiveness and advantages of the proposed approach.     

State feedback design using reduced-order nonsquare descriptions

A pole-placement technique is proposed for computing state feedback in multi-input linear systems. The algorithm has good numerical behavior and utilizes nonsquare pencils and the notion of decomposability. The fact that the method deals with reduced-order pencils and exploits the Schur-Hessenberg form improves the conditioning of the problem     

仿射非线性奇异系统的反馈控制与稳定化

研究非线性奇异系统的反馈稳定化问题，首先给出仿射非线性奇异系统反馈稳定化的概念；然后利用零动态算法构造的局部坐标变换给出仿射非线性奇异系统的一种标准型，并将其用于研究仿射非线性奇异系统的反馈控制和系统稳定化问题；最后证明了对于正则仿射非线性奇异系统，当其零动态渐近稳定时，该系统可通过反馈控制实现系统的稳定化。     

Hessenberg and Hessenberg/triangular forms in linear system thcory†

In this paper we distinguish between free Hessenberg forms, where only the system matrix is transformed into a special structure, and system Hessenberg forms, where either an input or output matrix is simultaneously simplified. We show how the freedom in the choice of the transformation matrix for the free Hessenberg form can be used to obtain a unique system Hessenberg form. We discuss numerical aspects and outline some basic applications for both types of Hessenberg forms in control and system theory. We show that the system Hessenberg form can be extremely sensitive to perturbations, a property not necessarily shared by the free Hessenberg form. Several examples are explored, and a first-order perturbation analysis is given. Analogous aspects are also discussed for the Hessenberg/triangular forms resulting from reduction of a matrix pencil. The main emphasis of this paper is on systems with one input.