Pole assignment for three-dimensional systems using two-dimensional dynamic compensators

This short paper deals with the problem of pole assignment with incomplete state observation. It is shown that if the system is controllable and observable, and if n \leq r + m - 1 , an almost arbitrary set of distinct closed-loop poles is assignable by gain output feedback, where n, r , and m are the numbers of state variables, inputs and outputs, respectively. This result improves considerably the ones obtained so far about this problem. Different from the conventional approach using the characteristic equation, an approach based on the properties of the eigenspaces of the closed-loop dynamics is used in this short paper, which gives a new light on the various problems in the linear system theory. It is also shown, as a direct consequence of this result, that the minimum order of the dynamic compensator required for almost arbitrary pole assignment of overall closed-loop system is not greater than n - m - r + 1 .

A simple approach is proposed for disturbance attenuation in multivariable linear systems via dynamical output compensators based on complete parametric eigenstructure assignment. The basic idea is to minimise the H2 norm of the disturbance–output transfer function using the design freedom provided by eigenstructure assignment. For robustness, the closed-loop system is restricted to be nondefective. Besides the design parameters, the closed-loop eigenvalues are also optimised within desired regions on the left-half complex plane to ensure both closed-loop stability and dynamical performance. With the proposed approach, additional closed-loop specifications can be easily achieved. As a demonstration, robust pole assignment, in the sense that the closed-loop eigenvalues are as insensitive as possible to open-loop system parameter perturbations, is treated. Application of the proposed approach to robust control of a magnetic bearing with a pair of opposing electromagnets and a rigid rotor is discussed.

This article investigates region stabilization issue of switched neural networks (SNNs) with multiple modes (MMs) and multiple equilibria (ME) via a pole assignment method. In such an SNN, every neuron is observed with more than one mode and unstable equilibrium point. First, SNNs with MMs and ME are modeled in terms of switched systems with unstable subsystems and ME. Second, a necessary and sufficient condition and a sufficient condition are, respectively, proposed for arbitrary switching paths pole assignment and arbitrary periodic/quasi-periodic switching paths (PSPs/QSPs) asymptotically region stabilizing pole assignment of switched linear time-invariant (LTI) systems with ME. It is shown that to stabilize a switched LTI system, some/all poles of all/some linear subsystems can be assigned to suitable locations of the right-half side of the complex plane. Third, based on the obtained pole assignment results, an asymptotical-region-stabilizing-control law observed as distributed state feedback controllers of MMs, asymptotical-region-stabilizing PSPs/QSPs, and a corresponding algorithm are all designed for asymptotical region stabilization of switched linear/nonlinear neural networks with MMs and ME. Finally, a numeral example is given to illustrate the effectiveness and practicality of the new results.

Pole assignment is a well-established controller synthesis technique used in control engineering. An important drawback of the pole assignment technique is poor numerical accuracy of the resulting compensators especially for high order systems, when fixed precision arithmetic is used in calculations. Obviously there are several parameters that affect the performance of the pole assignment algorithms. It has been, for example, already shown that increasing the order of the system causes an exponential increase in the numerical errors made in pole assignment. In this paper, the effect of the ratio of real poles to all poles of the open-loop (and closed-loop) system (real pole ratio - RPR) to the performance of the pole assignment algorithms is investigated for the first time.

The non-redundant parametrization of the pole assignment problem for a n-th order system with m inputs allows to express the solution of the problem in term of n(m − 1) free parameters. These parameters can be used to fulfill additional requirements on the closed-loop system as for instance minimum norm feedback gain matrix, well conditioned eigenvector set, maximum stability radius. One of reliable numerical methods for pole assignment is the so-called Schur method. An extension of this method is proposed which computes the solution of the pole assignment problem corresponding to a non-redundant parameter set. Several possibilities are further investigated to compute minimum norm feedback matrices. An improved approach to compute minimum Frobenius-norm feedback relying on a redundant parametrization is also discussed.

This paper considers control of an elastic robotic arm of two links based on nonlinear inversion and pole assignment for stabilization. The design is done in two steps. First, based on nonlinear inversion, a nonlinear controller is designed for the trajectory control of the joint angles using joint torquers. Although, the inverse controller accomplishes trajectory control of the joint angles, this excites the elastic modes of the arm. In order to damp the elastic oscillations, a stabilizer is designed for a linearized system about the terminal state using pole assignment technique. In the closed-loop system, first the inverse controller acts when a command is given, and the stabilizer automatically switches when the joint angle trajectory enters a specified neighborhood of the final commanded position.

Transfer function estimation of turboalternator from practical operating data by spectral analysis

This paper considers dynamical compensators design for purpose of pole assignment for discrete-time linear periodic systems. Similar to linear time-invariant systems, it is pointed out that the design of a periodic dynamical compensator can be converted into the design of a periodic output feedback controller for an augmented system. Utilizing the recent result on output feedback pole assignment, parametric solutions for this problem are obtained. The design approach can be used as a basis for the robust dynamical compensator design for this type of systems. Combined with a robustness index presented in this paper, robust dynamical compensator design problem is converted into a constrainted optimization problem. A numerical example is employed to illustrate the validity and feasibility of the methods.

A pole assignment algorithm was recently developed by considering the sensitivity of closed loop eigenvalues to each gain element. The theory is extended here to incorporate the Linear Quadratic Regulator formulation so that the closed loop system satisfies optimality as well as eigenvalue constraints. The approach is to compute the sensitivity of the Hamiltonian matrix to changes in the state weighting matrix, in contrast to the sensitivity of the closed loop system in the pole placement problem. Examples from a six degree-of-freedom lumped-mass model and an Euler-Bernoulli pinned beam system are investigated to demonstrate the procedure. It is shown that the resulting closed loop system behavior sufficiently approximates the desired response.

A rational function whose denominator polynomial has no zeros inside the unit circle, is called a generalized polynomial, and has properties similar to a polynomial. The ring of generalized polynomials is denoted by R∑(σ). A generalized polynomial matrix has the ∑-Smith form similar to the Smith form of a polynomial matrix. Linear neutral time-delay systems can be represented as systems over the field of rational functions whose argument is σ, where σ is a fixed delay operator. R∑(σ)-controllability which is similar to R[σ]-controllability for a time-delay system of the retarded type, is defined. Theorem 1 gives a necessary and sufficient condition for R∑(σ)-controllability. This condition is represented by the ∑-Smith form for the controllability matrix.∑-pole assignability which is the extension of pole assignability for a retarded system and guarantees asymptotic stability of the closed loop system by phiscally realizable state feedback, is defined. Theorem 2 states that R∑(σ)-controllability is equivalent to ∑-pole assignability. Theorem 3 gives a sufficient condition for decoupling by physically realizable state feedback. Finally, it is shown that under the condition of Theorem 3 and R∑(σ)-controllability, there exists a differential-difference compensator of the neutral type to achieve decoupling and arbitrary ∑-pole assignment.

This paper considers pole assignment and robust pole assignment problems for discrete-time linear periodic systems by using linear periodic state feedback. The monodromy matrix of the closed-loop system is represented in a special form. By combining this special form with our recent result on solutions to a class of generalized Sylvester matrix equations, a complete parametric approach for pole assignment via periodic state feedback is proposed. The free parameters existing in the solutions to pole assignment are further used to achieve robustness performances. The robust pole assignment problem is converted into a nonconvex optimization problem. Numerical examples illustrate the effectiveness of the proposed approaches.

The pole assignment problem for distributed parameter systems governed by analytic semigroups is discussed. Introducing unbounded feedback operators, we can solve the problem under a weaker restriction on the separation of the open-loop and the closed-loop poles. These types of feedback systems inherit the spectrum determined growth assumption of the open-loop systems. We first present a characteristic equation whose roots are the poles of the closed-loop system. Next we give a feedback element which realizes the desired pole assignment of the closed-loop system. We show an interesting result that an infinite number of the open-loop poles can be uniformly moved by means of a suitable linear feedback. We apply the results -to a heat system and a flexible structure for the cases of distributed controls and boundary controls.

On the dynamic system identification of the response behaviour of linear viscoelastic materials

This paper considers a parametric approach for quasi-linear systems by using dynamic compensator and multi-objective optimization. Based on the solutions of generalized Sylvester equations, we establish the more general parametric forms of dynamic compensator and the left and right closed-loop eigenvector matrices, and give two groups of arbitrary parameters. By using the parametric approach, the closed-loop system is converted into a linear constant one with a desired eigenstructure. Meanwhile, it also proposes a novel method to realize multi-objective design and optimization. Multiple performance objectives, containing overall eigenvalue sensitivity, $H_2$ norm, $H_\infty$ norm and low compensation gain, are formulated by arbitrary parameters, then robustness and low compensation gain criteria are expressed by a comprehensive objective function which contains each performance index weighted. By utilizing degrees of freedom (DOFs) in arbitrary parameters, we can optimize the comprehensive objective function such that an optimized dynamic compensator is found to satisfy the robustness and low compensation gain criteria. Finally, an example of attitude control of combined spacecrafts is presented which proves the effectiveness and feasibility of the parametric approach.

This paper introduces a mathematical formula for calculating the eigenvalue differential sensitivities of the closed-loop state matrix with respect to a disturbance matrix when the closed-loop system is subjected to an additive structured perturbation. The robust pole assignment problem is then formulated as an optimization task in which the differential sensitivities of the closed-loop poles are minimized. The case of “multi-perturbations” can be considered naturally in our problem formulation. The gradient of the objective function to be minimized is then derived which permits the minimization to be proceeded effectively. Numerical examples are then employed to illustrate the design technique.