Arbitrary Pole Placement by Constant Output Feedback for Linear Time Invariant Systems

Pole placement by parametric output feedback

A solution is presented for the problem of diagonalization (row-by-row decoupling). The problem is solved using a constant precompensator and a dynamic output feedback compensator of a p*m linear time-invariant system. The solvability condition is compact and concerns the dimension of a single subspace defined via the concepts of essential rows and static kernels associated with the transfer matrix. A characterization of the set of all solutions to the problem is also given. In solving this dynamic feedback problem, a complete solution to its state-feedback counterpart, namely, the restricted state-feedback problem of diagonalization, is also presented. >

This paper proposes a simple necessary condition for the stabilisation of an inherently unstable linear system with constant output feedback. This is a sign criterion obtained directly from the linear constraints on the attainable closed-loop characteristic polynomials when constant output feedback is employed. The condition is first established for single-input systems and then extended to multi-input systems, where the output-feedback matrices are restricted to have unity rank. Stabilisation with unconstrained output-feedback matrices is also discussed. Illustrative examples on stabilisation are given.

In this paper a numerical efficient approach to the problem of eigenvalue assignment by constant output feedback is presented. It improves the well known Kimura’s condition by 2, i. e., it is shown that if m + p ≥ n − 1 m+p\ge n-1 generically a solution to this design problem exists where n , m n,m and p denote the dimensions of the system states, inputs and outputs, respectively. The algorithm is based on a cascaded control scheme with up to three design steps. The first two steps merely require standard methods from linear algebra while the last step only in case of m + p = n − 1 m+p=n-1 demands for the numerical solution of a system of three polynomial equations each of order two. The design procedure explicitly embodies all degrees of freedom beyond eigenvalue assignment. Thus, they can be used to account for other design it is shown goals, e. g., to minimize the spectral condition number of the closed-loop system or a norm of the feedback gain as it is shown by numerical examples from literature.

We consider the classic problem of pole placement by state feedback. We offer an eigenstructure assignment algorithm to obtain a novel parametric form for the pole-placing gain matrix that can deliver any set of desired closed-loop eigenvalues, with any desired multiplicities. This parametric formula is then exploited to introduce an unconstrained nonlinear optimisation algorithm to obtain a gain matrix that delivers the desired pole placement with minimum gain.

This paper presents as its main result a new and efficient method for the design of optimal dynamic compensators for linear multivariable control systems. The discussion starts with a fresh look into the design of optimal constant gain output feedback. In this context, three new ideas are introduced that prove their usefulness particularly in the dynamic compensator case: (1) deterministic measurement disturbances are included in the formulation of the underlying optimization problem, (2) the relationship between a deterministic and a stochastic optimal control problem is shown, (3) by proper modifications of the quadratic cost function, the effort involved in the numerical solution of the optimal output feedback problem is considerably reduced. Furthermore, a canonical form is introduced for the dynamic compensator and an algorithm is developed for the numerical computation of its free parameters. It is based on quasi-newtonian gradient methods and an efficient line search procedure. Some features of the new design method are demonstrated by means of a non-trivial benchmark taken from the literature.

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.

Gradient matrices are formulated for the linear multivariable system with output feedback. The incremental motion of the poles and zeros and the coefficients of the numerator and denominator transfer function polynomials is determined as a function of the elements of an output feedback gain matrix. Also considered is the incremental movement of zeros and the coefficients of the numerator transfer function polynomials with respect to the elements of an input gain matrix. In addition, the characteristic polynomial gradient results are used to study the bounds on the number of independently assignable poles for linear multivariable systems with constant output feedback. The investigation is extended to estimate the minimum order dynamic output feedback compensator necessary to establish full polo placement for the expanded order system. Examples are included to illustrate the methods involved and to make comparisons with previous results.

The classic problem of robust pole placement for linear time invariant systems via state feedback has been studied for several decades, and involves obtaining a gain matrix that will assign a certain desired set of closed-loop poles, while also providing a robust eigenstructure that is insensitive to uncertainties in the system matrices. There are several ways of measuring the robustness of the eigenstructure, and numerous methodologies have appeared in the literature to address the problem. In this paper, results from extensive experiments comparing the performance of a number of methods—including variations on two methods previously proposed by the present authors—against a variety of robustness measures are reported. The size of the matrix gain, runtime and accuracy of the pole placement achieved by each method are also compared. The results show some notable differences between the methods surveyed.

A method using the simplest matrix generalized inverse, the {1} -inverse, is developed to find a constant state feedback matrix K that yields a set of prescribed eigenvalues for time-invariant linear multivariable systems that contain any finite number of derivatives of the input vector u. The method requires neither controllability assumptions nor the knowledge of eigenvalues and eigenvectors of the plant matrix A. The matrix K exists if a consistency condition for an algebraic linear system of equations is satisfied.

The paper deals with the identification of linear time invariant (LTI) systems by a special observer. An observer emitting an frequency modulated continuous wave (FMCW) signal and having a stretch processor as receiver will be considered for system identification. A thorough derivation of the gathered baseband signal for arbitrary LTI systems will be given. It is shown, that the received signal is approximately given by the transfer function of the LTI system over the frequency sweep of the FMCW signal. The proof relies on an infinite large time-bandwidth product of the transmit signal, such that errors remain in practical applications with a finite time-bandwidth product. Monte–Carlo simulations are conducted to verify the approximation and to quantify its accuracy and remaining errors. The findings are important for e.g. calibration or derivation of a device model in FMCW radar applications.

This study investigates characteristic root equivalency relations between commensurate order and integer order Linear Time Invariant (LTI) systems. Author introduces some useful properties of a special class of commensurate order systems, which is called characteristic root equivalency class of LTI systems. These properties present potential to facilitate design and analysis efforts of this class of commensurate order systems. In this sense, straightforward stability checking procedures and design approaches for commensurate order root equivalent systems of the first and second order LTI systems are demonstrated. Findings of the study are validated by illustrative examples.

This paper describes an applications oriented approach to the generation of optimal output feedback gains for linear time-invariant systems which is dependent of the open loop stability.

The paper presents the design of piecewise constant periodic output feedback control for a discrete-time singularly perturbed system resulting from the discretization of a continuous-time standard singularly perturbed system. By a suitable linear transformation of state variables, the given continuous-time singularly perturbed model is converted into a block triangular form in which the fast subsystem is decoupled. The discrete-time model corresponding to the transformed model also exhibits a two time scale property if sampling period is larger than the parameter E. Now an output injection matrix is found that stabilizes the slow subsystem. The periodic output feedback gain is then calculated only for the slow subsystem and the same for the fast subsystem is set equal to zero. Finally the periodic output feedback gain for the composite system is obtained using the periodic output feedback gains computed separately for the slow and fast subsystems. An approach has been suggested whereby the determination of periodic output feedback gain for the slow subsystem can be converted into an optimization problem. By minimization of the suggested performance index the closed loop system behavior is improved. The method has been applied to a large pressurized heavy water reactor (PHWR) for control of xenon-induced spatial oscillations. A particular grouping of state variables has been suggested for obtaining the model in standard singularly perturbed form. The periodic output feedback gain is then calculated. The efficacy of control has been demonstrated by simulation of transient behavior of the nonlinear model of the PHWR.

Chapter 4 - Frequency Analysis: The Fourier Series