Ackermann's Formula Research Articles

The generalized Ackermann's formula for singular systems

Being an elegant algorithm for state feedback pole placement, Ackermann's (1972) formula had been widely quoted in control texts. In this paper, the formula is extended to solve the root assignment problem for singular systems. Without loss of generality, it is both well known and convenient that any regular generalized system can be transformed into the standard form E x ̇ = Ax(t) + bu(t) , where μE − A = I and μ is a real constant. In the derivation of the generalized Ackermann's formula, the closed-loop characteristic polynomial, det[ sE − A + bk′], is simplified due to the relationship of E and A. If E is nonsingular, the feedback gain k′ can be computed from the generalized Ackermann's formula directly. In this case, only the desired closed-loop characteristic polynomial is required. If E is singular, the feedback algorithm needs both closed-loop and open-loop characteristic polynomials. Two numerical examples are presented to demonstrate our algorithms.

A new approach for computing the state feedback gains of multivariable systems

This note presents some new results in linear systems. First, the relationship between the polynomial matrix description and the state-space representation of multivariable systems is clarified. Then, we show that once such a relationship is determined, the coprime matrix fraction description can be easily computed. And we can further develop a closed-form formula to solve the pole-assignment problem of multivariable systems. Such formula can be thought of as an extension of the Ackermann's formula for MIMO systems. Thus this note potentially gives us a clearer insight into linear systems from the theoretical viewpoint. >

Multivariable generalization of Ackermann's formula

This paper presents the multivariable generalization of Ackermann's formula. For a controllable linear time‐invariant system, hypothetical output is proposed to facilitate the description of a set ...

Extended Ackermann formula for multivariable control systems

The matrix Cayley-Hamilton theorem is first derived to show that Ackermann's formula for the pole-placement problem of SISO systems can be extended to the case of a class of MIMO systems. Moreover, the extended Ackermann formula newly developed by the authors is employed for fast determination of the desired feedback gain matrix for a general MIMO system with an appropriate artificial input. The proposed method will enhance the application of state-space pole-placement concepts to a general class of multivariable control systems.