A Schur method for 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.

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.

The problem of pole assignment in linear multivariable systems by means of unity rank output feedback is studied in this paper. It is shown that when the number of specified poles is equal to the number of inputs or outputs of the system, exact pole assignment can be achieved and the feedback matrix is calculated from a set of simple linear equations. When the number of specified poles exceeds this value, the pole assignment problem becomes non-linear. Three different formulations of the problem are considered and three numerical techniques are described for the calculation of the feedback matrix. Numerical examples are given for illustration.

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.

This paper presents a new algorithm for computing the feedback matrix of finite spectrum assignment of multivariable systems with time-delays under spectral controllability. The key is the Smith-form of the controllability matrix. That yields a weakly R-controllable system. The feedback matrix over rational functions in delay operator for a weakly R-controllable system is transformed to the feasible one with polynomials and finite Laplace transforms.

We develop a general class of stochastic optimal control problems for which the problem of designing optimal linear feedback gains is convex. The class of problems includes arbitrary time varying linear systems and costs that are mixtures of exponentiated quadratics. This allows us to model problems with quadratic state costs and linear constraints on states and state transitions. Further, convexity in the feedback gains lets us impose arbitrary convex constraints or penalties on the feedback matrix: Thus we can model problems like distributed control (by imposing a sparsity structure on the feedback matrix) and variable-stiffness control (by applying time-varying penalties to feedback gain matrices). We show that the convex optimization problem can be solved efficiently by using the structure of the matrices involved. Finally, we present an application of these ideas to a practical problem arising in distributed control of power systems.

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.

FEEDBACK CONTROL FOR LINEAR CHAOTIC SYSTEMS

Pole assignment with optimal spectral conditioning

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.

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.

The theory of pole assignment in linear time invariant systems with incomplete state measurements is generalized to include the case in which a controllable observable plant of order n is augmented by a compensator of fixed dimensionality p (0≤p<n). The maximum sot of eigenvalues that can be preassigned arbitrarily is specified. The results previously available are shown to be particular cases of the theory proposed in the present paper, and Davison's (1970) algorithm is extended so as to apply to fixed order compensators design.

This article introduces a new notion of stability for internal autonomous system descriptions that is referred to as ‘strong stability’ and characterises the case of avoiding overshoots for all initial conditions within a given sphere. This is a stronger notion of stability compared with alternative definitions (asymptotic, Lyapunov), which allows the analysis and design of control systems described by natural coordinates to have no overshooting response for arbitrary initial conditions. For the case of linear time-invariant systems necessary and sufficient conditions for strong stability are established in terms of the negative definiteness of the symmetric part of the state matrix. The invariance of strong stability under orthogonal transformations is established and this enables the characterisation of the property in terms of the invariants of the Schur form of the state matrix. Some interesting relations between strong stability and special forms of coordinate frames are examined and links between the skewness of the eigenframe and the violation of strong stability are derived. The characterisation of properties of strong stability given here provides the basis for the development of feedback theory for strong stabilisation.

Excitation control of turbogenerators with output feedback

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.