Proportional State Feedback Research Articles

Regularization of Linear Discrete-Time Periodic Descriptor Systems by Derivative and Proportional State Feedback

In this paper, we consider the regularization problem for the linear time-varying discrete-time periodic descriptor systems by derivative and proportional state feedback controls. Sufficient conditions are given under which derivative and proportional state feedback controls can be constructed so that the periodic closed-loop systems are regular and of index at most one. The construction procedures used to establish the theory are based on orthogonal and elementary matrix transformations and can, therefore, be developed to a numerically efficient algorithm. The problem of finite pole assignment of periodic descriptor systems is also studied.

Integrated approach to eigenstructure assignment in descriptor systems by PD and P state feedback

Parametric and subspace approaches to eigenstructure assignment are integrated to provide conceptually simple algorithms for eigenstructure assignment in controllable linear descriptor systems by proportional plus derivative state feedback and proportional state feedback. In particular, unlike comparable algorithms the new algorithms do not require right coprime matrix polynomials to be determined. The integrated approach enables the eigenstructure assignable with multiple eigenvalues to be readily identified.

Proportional and derivative state-feedback decoupling of linear systems

We consider the row-by-row decoupling of linear time-invariant systems by proportional and derivative state feedback. Our contribution, with respect to previous results, is that our procedure is based on simple operator (say matrices) manipulations, without any need to use a canonical form. The only assumptions for applying such a decoupling strategy are that the system is right invertible (which is a necessary condition to ensure solvability) and minimum phase. An illustrative example is proposed.

Fixed modes of disturbance decoupling for descriptor systems

We study the disturbance decoupling problem via a proportional state feedback for linear time invariant descriptor systems. The fixed modes are characterized with the help of the supremal obsolute (strong) C-observable normal form that can be computed using orthogonal matrix transformation.

Robust pole assignment in descriptor systems via proportional plus partial derivative state feedback

The problem of eigenvalue assignment with minimum sensitivity in multivariable descriptor linear systems via proportional plus partial derivative state feedback is considered. Different from the purely proportional state feedback case, the number of finite closed-loop eigenvalues is required to be equal to the system dimension. Based on a result in the perturbation theory of generalized eigenvalue problem of matrix pairs, the closed-loop eigenvalue sensitivity measures in terms of the closed-loop normalized right and left eigenvectors are established. By combining these measures and a recently proposed general parametric eigenstructure assignment result for descriptor linear systems via proportional plus derivative state feedback, the robust pole assignment problem is converted into an independent minimization problem. A simple algorithm in sequential order is obtained for solution to the problem of robust pole assignment in descriptor linear systems via proportional plus partial derivative state feedback. The closed-loop eigenvalues may be easily taken as a part of the design parameters and optimized within certain desired fields on the complex plane to improve robustness. Using the proposed algorithm, the optimality of the obtained solution to the robust pole assignment problem is totally dependent on the solution to the independent minimization problem. An example of order six is worked out with multiple solutions, both the indices and the numerical robustness test demonstrate the effect of the proposed approach.

P-D Feedback for Decoupling and Pole Assignment of Singular Systems

The problem of reducing a multi input-multi output system to many single input-single output systems, namely the problem of input-output decoupling, is studied for the case of singular systems i.e., for systems described by dynamic and algebraic equations. The problem of input-output decoupling with simultaneous arbitrary pole assignment, via proportional plus derivative (P-D) state feedback, is extensively solved. The general explicit expression of all P-D controllers solving the decoupling problem is determined. The general form of the diagonal elements of the decoupled closed-loop system is proven to be in a form having a fixed numerator polynomial and an arbitrary denominator polynomial. The necessary and sufficient conditions for the solvability of the problem of decoupling with simultaneous asymptotic stabilizability or arbitrary pole assignment are established. Furthermore, the necessary and sufficient conditions for decoupling with simultaneous impulse elimination, as well as the necessary and sufficient conditions for decoupling with arbitrary assignment of the finite and infinite poles of the closed-loop system, are established.

Derivative and Proportional State Feedback for Linear Descriptor Systems With Variable Coefficients

Derivative and Proportional State Feedback for Linear Descriptor Systems With Variable Coefficients

Robust eigenvalue assignment in second-order systems: a gradient flow approach

In this paper the problem of robust eigenvalue assignment in second-order systems by combined derivative and proportional state feedback is examined. It is shown that almost arbitrary assignment can be achieved by solving a linear matrix equation or symmetric linear system. Based on the fact that the robustness of the spectrum is characterized by the condition number of the eigenvectors, an alternative objective function is defined and minimized by utilizing the gradient flow technique. Numerical tests are performed to illustrate the approach. © 1997 John Wiley & Sons, Ltd.

Derivative and proportional state feedback for linear descriptor systems with variable coefficients

We study linear descriptor systems with rectangular variable coefficient matrices. Using local and global equivalence transformations, we introduce normal and condensed forms and get sets of characteristic quantities. These quantities allow us to decide whether a linear descriptor system with variable coefficients is regularizable by derivative and/or proportional state feedback or not. Regularizable by feedback means for us that there exists a feedback which makes the closed loop system uniquely solvable for every consistent initial vector.

Eigenstructure assignment in descriptor systems via proportional plus derivative state feedback

A complete parametric approach for eigenstructure assignment in the controllable linear descriptor system Ex = Ax + Bu via proportional plus derivative state feedback control u = Kx + Lx is proposed. General complete parametric expressions in direct closed forms for the closed-loop eigenvectors are presented, which are linear in a group of parameter vectors { fk} which represent the degree of the design ij freedom. It is shown that the derivative feedback gain L may be taken to be an arbitrary real matrix satisfying a simple constraint, and the general complete expression for the proportional feedback gain K is composed of both the group of vectors { fk} and the derivative feedback gain L . By properly restricting the ij design parameters in the obtained general results, solutions to eigenstructure assignment in descriptor linear systems via constant-ratio proportional plus derivative state feedback and proportional plus partial derivative state feedback are obtained. The approach does not impose any condition on the closed-loop finite eigenvalues, assigns arbitrarily n finite closed-loop eigenvalues with arbitrary given algebraic and geometric multiplicities and guarantees the closed-loop regularity. An example shows the effect of the proposed approach.

Perfect output control for nonsquare generalised state space systems

The problem of perfect output control, via proportional restricted state feedback, is studied for the case of nonsquare generalised state space systems. The necessary and sufficient conditions for the problem to have a solution are established. Furthermore, the necessary and sufficient conditions for the solution of the problem of command matching are derived, using proportional restricted state and output feedback.

Pole assignment at infinity and polynomial matrix equations

A simple approach to synthesizing a proportional and derivative state feedback gain for assigning poles to infinity is developed. We also apply the synthesized gain to determine a doubly coprime matrix fraction description (MFD) of a state-space system and solve related problems. It is interesting to note that the MFD and the solutions to the related problems could be obtained in an explicit formula by using the synthesized gain.

Proportional and proportional-derivative canonical forms for descriptor systems with outputs

We construct a new canonical form for descriptor systems ( E, A, B, C) under proportional state feedback, proportional output injection and changes of bases in the input space, the output space, the state space (domain) and the state equation space (codomain). Geometric characterizations of the invariants are given. Four pencils are introduced in order to point out the relationships with Kronecker's theory of matrix pencils. This finally allows us to define canonical forms under proportional plus derivative action.

Disturbance rejection of left-invertible generalized state space systems

A new approach is presented for the study of the disturbance rejection of left-invertible generalized state space systems via pure proportional state feedback, reducing the problem to that of solving a linear algebraic system of equations. On the basis of this system of equations, the necessary and sufficient conditions for the problem to have a solution are established in a form of simple algebraic criteria (thus reducing the computational effort over known results), and the general analytical expressions of the controller matrices are derived. The structural properties of the closed-loop system (feedback invariant poles, disturbance rejection with simultaneous elimination of the infinite poles) are studied.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

State feedback uniform decoupling problem of linear time-varying analytic systems

A new approach to the input-output uniform decoupling problem of linear time-varying analytic systems via proportional state feedback is presented. A major feature of the proposed approach is that it reduces the solution of the uniform decoupling problem to that of solving a linear algebraic system of equations. This system of equations greatly facilitates the solution of the three major aspects of the decoupling problem: the necessary and sufficient conditions, the general analytical expressions for the controller matrices, and the structure of the uniformly decoupled closed-loop system.

Generalised command matching for a robot gripping an inertial load

The problem of controlling a robot with end effector rigidly gripping an inertial load is studied using the generalised command-matching technique via proportional state feedback. In particular, the necessary and sufficient conditions for the problem to have a solution are established and the general analytical expressions of the controller matrices are derived. Structural properties of the resulting closed-loop system are also investigated.

Exact model matching of generalized state space systems

The problem of exact model matching for generalized state space (GSS) systems via pure proportional state and output feedback is studied. The following two major issues are resolved here for the first time: The necessary and sufficient conditions for the problem to have a solution and the general analytical expressions for the exact modelmatching controller matrices. The important case of left invertible systems is treated separately wherein simple solutions are established for the above two major issues and results on structural properties of the closed-loop system are reported. Known results on model matching of regular systems are derived as a special case of the GSS systems results, thus unifying the solution of the exact model-matching problem of regular and singular systems.

Model matching of descriptor systems by proportional state feedback

Given a regular, linear descriptor system E x ̇ = Fx + Gu , y = Hx with transfer function T( s) we consider the problem of designing a proportional state feedback u = Lx + Mv with M non-singular such that the resulting system is regular and its transfer function equals a prespecified rational matrix W( s). Necessary and sufficient conditions are given for the existence of such a feedback, as well as for the internal properness and/or stability of the resulting system. These conditions are interpreted in terms of the zeros of T( s) and W( s). A simple design procedure is proposed which consists of finding a constant solution to a polynomial matrix equation.

Disturbance rejection of left-invertible systems

A new approach is presented for the study of the disturbance rejection of left-invertible systems via proportional state feedback, reducing the problem to that of solving a linear algebraic system of equations. On the basis of this system of equations, the necessary and sufficient conditions for the existence of a solution to the disturbance rejection problem are established in a form of simple algebraic criteria (thus reducing the computational effort over known results) and the general analytical expressions of all solutions of the controller matrices are derived. These solutions are parameterized only by an arbitrary matrix.

A new approach to the decoupling problem of linear time-invariant systems

A new approach to the input-output decoupling problem of linear time-invariant systems via proportional state and output feedback is presented. A major feature of the proposed approach is that it reduces the solution of the decoupling problem to that of solving a linear algebraic system of equations. This system of equations greatly facilitates the solution of the following four major aspects of the decoupling problem: necessary and sufficient conditions, general analytical expressions for the controller matrices, general analytical expressions for the diagonal elements of the closed-loop system, and the structural properties of the closed-loop system.