Eigenvalues assignment in uncontrollable linear systems

Algorithms are currently available for the solution of certain types of optimal-eigenvalue-assignment problems in which the eigenvalues of a given system are required to be shifted into preassigned locations or region while also minimising an appropriate quadratic-performance criterion. All the known methods for a solution of the above problem are based on manipulation of the original nth-order system matrices even if only r eigenvalues (r ≥ n) of the original system are to be reassigned. On the contrary, the method proposed in this paper, for a solution of the above problem, employs an rth-order equivalent model, which leads to a solution via manipulation of rth-order matrices only. The method also ensures that the remaining n — r eigenvalues of the original system are not disturbed and are carried over to the resultant feedback system. It has been shown in the paper that the suggested procedure brings about a considerable saving in computation time, and also requires less computer storage. Two numerical examples have been included to illustrate the suggested technique.

The robust eigenvalue assignment of systems subject to parameter perturbations is addressed. Based on some essential properties of induced norms and matrix measures, the authors derive some sufficient conditions which ensure the assignment of the system's eigenvalues in the specified region irrespective of the system perturbations. The robustness bounds for eigenvalue assignment are obtained without the need to solve the Lyapunov equation. An example is given to illustrate the effectiveness and ease of the proposed analysis methods. >

A method for solving the descriptor discrete-time linear system is focused. For easily, it is converted to a standard discrete-time linear system by the definition of a derivative state feedback. Then partial eigenvalue assignment is used for obtaining state feedback and solving the standard system. In partial eigenvalue assignment, just a part of the open loop spectrum of the standard linear systems are reassigned, while leaving the rest of the spectrum invariant and for reassigning, similarity transformation is used. Using partial eigenvalue assignment is easier than using eigenvalue assignment. Because by partial eigenvalue assignment, size of matrices and state and input vectors are decreased and stability is kept, too. Also concluding remarks and an algorithm are proposed to the descriptions will be obvious. At the end, convergence of state and input vectors in the descriptor system to balance point (zero) are showed by figures in a numerical example.

This article considers the design of finite-dimensional compensators for distributed-parameter systems using eigenvalue assignment. The proposed compensator consists of an observer estimating additional outputs and a static feedback of the measurable and the estimated outputs. Since the additional outputs can be asymptotically reconstructed, the compensator can be designed using the separation principle, i.e. the closed-loop eigenvalues are given by the observer eigenvalues and the eigenvalues resulting from the static output feedback control. In order to solve the corresponding eigenvalue assignment problem, the parametric approach for the design of static output feedback controllers in finite-dimensions is extended to distributed-parameter systems. By using a parameter optimisation it is possible to assign all closed-loop eigenvalues within specified regions of the complex plane in order to stabilise the system and to assure a desired control performance. A heat conductor is used to demonstrate the proposed design procedure.

In this article stabilization of fractional-order periodic discrete-time linear system by a complete nonlinear parametric approach for pole assignment with respect to stability and reachability of system via state feedback is proposed. To modify the dynamic response of linear system we should change the poles of state feedback matrices. Nonlinear parametric feedback matrix makes some freedoms in conditions such as minimum norm of feedback matrix. But it makes large monodromy matrix and changing all of eigenvalues makes some problems. Reassigning a part of bad spectrums, leaving the rest of the spectrums invariant, we have lower-ordered matrix to modify the dynamic response of linear system and we can make nonlinear parametric for this new lower-ordered matrix by less expenses and better stability conditions. Numerical examples illustrate the effectiveness of the proposed approaches.

In this paper, it is shown that by exploiting the explicit para- metric state feedback solution, it is feasible to obtain the ultimate solution to minimum sensitivity problem. A numerical algorithm for construction of a robust state feedback in eigenvalue assignment problem for a control- lable linear system is presented. By using a generalized parametric vector companion form, the problem of eigenvalue assignment with minimum sensitivity is re-formulated as an unconstrained minimization problem. The derived explicit expressions of the solutions allow minimization of the sensitivity problem by using a powerful search technique.

For a single-output linear time-invariant system, output controllability is shown to be a necessary and sufficient condition for the existence of linear state variable feedback that freely assigns the eigenvalues of the closed-loop system that are observable from the output. An application of this result to the single-output linear servomechanism shows that the ability of such a system to track with zero steady-state error and an arbitrary transient response is equivalent to output controllability of the plant, and is independent of the class of command signals and disturbances.

Numerical algorithms are described for solving the eigenvalue assignment (EVA) problem for multi-input systems by means of state feedback. The multi-input EVA problem is first reduced to one or more single-input EVA problems where the single-input systems are in "upper Hessenberg form." An algorithm based on the well-known implicitly shifted QR algorithm is then described for solving the single-input EVA problem. Some variations on the use of these algorithms are also proposed to handle certain special cases. Numerical properties of the algorithms are discussed and examples are given to illustrate their numerical performance.

This study proposes less conservative conditions for robust linear quadratic regulator controllers using state-derivative feedback (SDF). The algebraic Ricatti equation was formulated using the SDF, and its solution was obtained by linear matrix inequalities. SDF was chosen owing to the presence of accelerometers as sensors. Since accelerometers are the main sensors in mechanical systems, the proposed technique may be used to control/attenuate their vibrations/oscillations. Moreover, to formulate the less conservative conditions, some methods in the specialised literature were used, such as, for example, slack variables by Finler's Lemma. The paper also offers necessary and sufficient conditions for an arbitrary convex combination of square real matrices to be a nonsingular matrix, and thus an invertible one: must be nonsingular and all the real eigenvalues of must be positive. This result is important in the formulation of the proposed less conservative conditions since it was assumed that a given convex combination is nonsingular. A feasibility analysis demonstrates that the proposed conditions reduce the conservatism. Thereby, it is possible to stabilise a higher number of systems and to reduce the guaranteed cost. Furthermore, a practical implementation illustrated the application of the proposed conditions.

A simple method is presented on how to generate the state feedback which makes the closed-loop transfer matrix the inverse of the interactor matrix. This can be regarded as a generalization of decoupling control using state feedback. An arbitrary pole assignment which preserves the lower left triangular form of the inverse of the interactor matrix is also considered. >

The partial eigenvalue assignment problem concerns reassigning afew of undesired eigenvalues of a linear system to suitably chosenlocations and keeping the other large number of eigenvalues andeigenvectors unchanged (no spill-over). This paper considers thepartial eigenvalue assignment problem with time delay robustness.A time delay robustness measure is presented by analyzing thesensitivity of the assigned eigenvalues with respect to timedelay. The problem is formulated as an unconstrained minimizationproblem with the cost function involving the time delay robustnessmeasure. A numerical algorithm with analytical formulation of thegradient for the cost function is provided. A numerical example isincluded to show the effectiveness of the proposed method.

This study presents an eigenvalue assignment control architecture for a class of linear time-varying (LTV) systems in the presence of time-varying disturbances. Eigenvalue assignment architecture is presented as a way to achieve feedback stabilisation for LTV disturbed systems. With an input compensator design, the output of slowly LTV systems with disturbances can practically track the reference. Moreover, the implementation of the proposed eigenvalue assignment technique in practice has been demonstrated by using adaptive estimation of time-varying disturbances.

Eigenvalue assignment problem of a linear scalar system with a single discrete delay is analytically and exactly solved. The existence condition of the desired eigenvalue is established when the current and delay states are present in the feedback loop. Design of the feedback controller is then followed. Furthermore, eigenvalue assignment for the input-delay system is also obtained as well. Numerical examples illustrate the procedure of assigning the desired eigenvalue.

The problem of robust disturbance rejection for left invertible linear systems, with nonlinear uncertain structure and with measurable and nonmeasurable disturbances, is solved via an independent from the uncertainties static measurement output feedback law. The necessary and sufficient conditions for the problem to have a solution are established. The general explicit expression of the measurement output feedback matrix and the measurement disturbances feedback matrix are derived. The feedback invariant poles are determined. Sufficient conditions for robust disturbance rejection with simultaneous robust stability are also established. The rejection of the influence of the rotary gust disturbing the vertical velocity of the longitudinal motion of an aircraft is achieved.

This article deals with the problem of dissipative control synthesis for a class of descriptor systems with uncertainties in the derivative matrix. Attention is focused on the design of a proportional plus derivative (PD) state feedback, which guarantees that the closed-loop system is robustly stable and strict dissipative. Firstly, a sufficient condition for the closed-loop system is robustly stable and strict dissipative is presented by using a simple idea of changing the problem to the corresponding problem of an augmented uncertain system. Then, a PD controller is constructed by solving LMIs. Finally, a numerical example is given to demonstrate that the proposed method is effective.