Comments, with reply, on "Computational algorithms for pole assignment" by N.K. Nichols

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.

A new pole-placement algorithm for multi-input controllable systems is presented. The algorithm can be used to obtain real-valued feedback gains for arbitrarily specified real and complex, distinct or multiple eigenvalues. The multi-input controllable time-invariant system is converted to a single-input system. Pole assignment is achieved by making use of certain properties of the equation TF - AT = D , where A is the open-loop system matrix and F represents a matrix having the desired closed-loop eigenvalues.

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.

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 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.

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 new design for pole assignment in complex plane regions is presented. Vectorial bounds for structured uncertainties such that the assigned poles remain in these regions are given. Application to the linear constrained regulator problem is achieved. This approach makes a link between the pole assignment procedure used for constrained control and sufficient conditions established to check the positions of eigenvalues in the complex plane for a given matrix.

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.

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.

In this paper, the eigenstructure assignment of linear multivariable control systems is studied from a geometric point of view. For the class of systems in which the number of outputs plus the number of inputs exceeds the number of states, genericity properties relative to this problem are derived. It is shown, without any assumption on the genericity of the system, that the pole assignment can be carried out by choosing some closed-loop eigenvectors almost freely. The crucial point is that all the expected degrees of freedom in pole assignment are so described without redundancy, which fully justifies the practical interest of such techniques. Despite the technicality required for the derivation of intermediate results, the main result, which is an eigenstructure assignment algorithm, is very easy to implement because it is only based on the computation of certain sums and intersections of characteristic subspaces. Furthermore, it is shown how the basic tools developed here can be used to tackle the problem of finding a necessary and sufficient condition for exact pole assignment.

Considering that the link states of some practical networks are not measurable, it is impossible to control the links system directly to achieve structural balance. In order to handle this problem, the state observer design and the structural balance tracking control for the links system with complex dynamical networks are investigated. The dynamical model of the links system is molded as the Riccati matrix difference equation, which contains a coupling term associated with external stimulations. For the discrete-time model, the pole assignment algorithm is adopted to calculate the gain matrix for the observer of the links system. It could guarantee that the estimation error of observer for links system is globally asymptotically stable. Further, combining the effective estimation information of observer and external stimulations, an asymptotically tracking controller is proposed, which could achieve the asymptotic tracking of the reference structure balance network to the links system. Finally, the validity of the proposed method is verified by the simulation example.

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.

(1976). Addendum to the paper ‘ On the pole assignment in linear systems with fixed order compensators ’. International Journal of Control: Vol. 23, No. 6, pp. 887-888.

An efficient computational algorithm for pole assignment of linear n-th order m-input systems is proposed. The algorithm requires a small multiple of n3 operations and approximately n2 + (n + m)2 working precision words space. A Fortran program is developed and tested for high-order and ill-conditioned problems.