An improved general algorithm for arbitrary pole assignment

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.

Some points of the paper by N.K. Nichols (see ibid., vol.AC-31, p.643-5, 1986), concerning the robust pole assignment of linear multiinput systems, are clarified. It is stressed that the minimization of the condition number of the closed-loop eigenvector matrix does not necessarily lead to robustness of the pole assignment. It is shown why the computational method, which Nichols claims is robust, is in fact numerically unstable with respect to the determination of the gain matrix. In replying, Nichols presents arguments to support the choice of the conditioning of the closed-loop poles as a measure of robustness and to show that the methods of J Kautsky, N. K. Nichols and P. VanDooren (1985) are stable in the sense that they produce accurate solutions to well-conditioned problems.

A short and direct new proof is given to Wonham's theorem that a time invariant multi-input linear dynamical system is controllable only if its poles can arbitrarily be reassigned in a closed-loop system by means of a constant (state variable) feedback law. The construction provided in the proof is directly applicable as an effective algorithm for this pole assignment.

The pole assignment problems of linear systems by decentralized static output feed-back are considered in this paper. A compactification of decentralized static feedback space, product Grassmannians, is introduced in this paper. Its degree under Plücker–Segre embedding is computed. Sufficient conditions for arbitrary and almost arbitrary pole assignability are given. It is also proved that the generic $m \times p$ system of McMillan degree n has arbitrary pole assignability by r-channel decentralized static output with $m_i $ inputs and $p_i $ outputs on the ith channel if $\sum_{i = 1}^r {m_i p_i } \geq n$ when the degree of the product Grassmannians is odd, or if a $\sum_{i = 1}^r {m_i p_i } > n$ and the local channels have either the same numbers of inputs or the same numbers of outputs when the degree of the product Grassmannians is even.

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 method is presented to place a number of system eigenvalues by means of output feedback, and to use any remaining design freedom to minimize a specified performance functional. The solution for this combination of optimal control and pole assignment is shown to be similar to that for the complete state feedback case and uses a novel pole assignment algorithm.

In this study, we propose a class of dynamic compensators based on linear function observers which achieve arbitrary pole assignment of the closed loop. The construction procedure directly utilizes Wang's algorithm (1992, 1996) for the static output feedback pole assignment problem, and the same technique can be applied to dual observers also. As a result, assuming the structure of either observer, the sufficient condition of the compensator's order is naturally derived. The condition is more transparent than the one proposed by Rosenthal and Wang (1996), and it is shown that both give the same minimal order under some mild assumptions.

It is shown that controllability of an open-loop system is equivalent to the possibility of assigning an arbitrary set of poles to the transfer matrix of the closed-loop system, formed by means of suitable linear feedback of the state. As an application of this result, it is shown that an open-loop system can be stabilized by linear feedback if and only if the unstable modes of its system matrix are controllable. A dual of this criterion is shown to be equivalent to the existence of an observer of Luenberger's type for asymptotic state identification.

The problem of pole assignment in linear, multivariable systems using an unconstrained-rank feedback matrix is considered. The effect of output feedback of unspecified rank on the characteristic polynomial of a multivariable system is first studied. The results are then used to derive a recursive algorithm for pole assignment without any restrictions on the rank of the output feedback matrix used. The method is based on the pseudoinverse concept for obtaining least-squares solutions of sets of linear equations, and is computationally efficient.

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 .

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.

This paper considers a new algorithm for discrete time adaptive pole assignment which incorporates the internal model principle and hence allows for rejection of deterministically modelled disturbances and tracking of simple reference signals. A complete global stability analysis is given.

A computational method for designing controllers which attempt to place the roots of the characteristic polynomial of an uncertain system inside some prescribed regions is presented. The analysis is based on transfer functions of characteristic polynomials, and the problem is formulated as one of semi-infinite programming. An example of an application is given to illustrate this approach. >

A sufficient rank condition for arbitrary pole assignability by static output feedback of m-input, p-output linear systems of McMillan degree n<mp is given. This condition is easy to check and is satisfied by the generic system. An effective computational method to compute the feedback laws assigning any self-conjugate set of closed-loop poles is also given.

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. >