Pole assignment by gain output feedback

It is shown that the problem of pole assignment by gain output feedback in linear time-invariant multivariable systems may be reduced to an algebraic optimisation problem, using the concept of virtual model following. The method requires no transformation of the open-loop system to a canonical form. A number of related problems involving combinations of pole placement with a secondary design objective are also discussed.

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 pole assignment of linear systems by gain output feedback

A new result of the problem of pole assignment by output feedback is presented. A simple condition is given under which min (n, m + r − 1) closed-loop poles can be assigned arbitrarily by a gain output feedback, where n, mand r are the dimensions of the state, output and input vectors, respectively. Under the same condition, a new lower bound for the number of poles assignable by a fixed-order dynamic compensator is given which improves considerably on the results of Breach and Pearson (1970) and Ahmari and Vacroux (1973). A design algorithm and illustrative examples are also given.

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.

This paper highlights the importance of the co-efficient matrix of the numerator transfer function matrix in pole placement and stabilization assessment of multivariable control systems. The problem of pole assignment with non-unity-rank output feedback, obtained as the sum of unity rank output feedback in two stages, is discussed. The design equations of Seraji [5] formulated in terms of the appropriately partitioned co-efficient matrix of the system numerator transfer function matrix. The pole placement and stabilization assessment procedures are illustrated with numerical examples.

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.

ABSTRACTThe paper considers the pole assignment and the stabilisation problem under the practically essential constraint, that the designed controllers have to be causal, because only in that case they will be realisable in real time. It is shown that in case of a strictly proper process, when the set of solutions of the pole assignment or stabilisation problem is not empty, then it always contains a non-empty subset of non-causal controllers. Moreover, according to the set of causal controllers, we have one of the three situations: (1) the set is empty, (2) the set has actually one element and (3) there exists a subset of causal solutions. The paper provides complete solutions for both problems in the form of instructions for building the corresponding sets of causal controllers. Drawing on examples of a double integrator with delay, the procedures are illustrated.

MotivationParameter estimation methods for ordinary differential equation (ODE) models of biological processes can exploit gradients and Hessians of objective functions to achieve convergence and computational efficiency. However, the computational complexity of established methods to evaluate the Hessian scales linearly with the number of state variables and quadratically with the number of parameters. This limits their application to low-dimensional problems.ResultsWe introduce second order adjoint sensitivity analysis for the computation of Hessians and a hybrid optimization-integration-based approach for profile likelihood computation. Second order adjoint sensitivity analysis scales linearly with the number of parameters and state variables. The Hessians are effectively exploited by the proposed profile likelihood computation approach. We evaluate our approaches on published biological models with real measurement data. Our study reveals an improved computational efficiency and robustness of optimization compared to established approaches, when using Hessians computed with adjoint sensitivity analysis. The hybrid computation method was more than 2-fold faster than the best competitor. Thus, the proposed methods and implemented algorithms allow for the improvement of parameter estimation for medium and large scale ODE models.Availability and implementationThe algorithms for second order adjoint sensitivity analysis are implemented in the Advanced MATLAB Interface to CVODES and IDAS (AMICI, https://github.com/ICB-DCM/AMICI/). The algorithm for hybrid profile likelihood computation is implemented in the parameter estimation toolbox (PESTO, https://github.com/ICB-DCM/PESTO/). Both toolboxes are freely available under the BSD license.Supplementary informationSupplementary data are available at Bioinformatics online.

A computational approach is proposed to solve the minimum norm and/or robust pole assignment problem for linear periodic discrete-time systems. The proposed approach uses a periodic Sylvester equation based parametrization of the periodic pole assignment problem and exploits the non-uniqueness of the problem by imposing conditions on the norm of the resulting periodic state feedback and/or on the condition numbers of the periodic eigenvector matrices of the closed-loop system. The solution method relies on using gradient search methods on suitably defined cost functions. Explicit expression of the gradients of cost functions are derived and the efficient evaluation of the cost functions and gradients is discussed. Numerical examples illustrate the effectiveness of the proposed approach.

In this paper, we study the pole assignment problem for 3D systems. We transform the denominator of transfer functions of the closed-loop system into the product of three stable 1D polynomials, by performing 2D dynamical feedback and input transformation on the given 3D systems. Next, we consider the possibility that these 2D dynamic compensators are realizable thoroughly, and propose the counter-measure in a case that they are not realizable. We also obtain the conditions so that the closed-loop 3D systems are stable. Moreover, we calculate the dynamical dimension which is necessary for the pole assignment, and suggest a pole assignment method with the lowest dynamical dimension. >

Robust partial pole assignment problem for high order control systems

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.

In this paper, a theory of polynomial and rational matrix interpolation is introduced and applied to problems in systems and control. The polynomial matrix interpolation theory is first outlined and then applied to solving matrix equations; it is also used in pole assignment and other control problems. Rational matrix interpolation is also discussed and it is used to solve rational matrix equations including the model matching problem.

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.