Arbitrary pole assignment using dynamic compensators based on linear function observers

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 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. >

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 .

An extended Howze-Pearson compensator avoiding pole-zero cancellations in a block decoupling system is considered. The basic relation that the compensator satisfies in order to realize block decoupling and arbitrary pole assignment is investigated. A design method to obtain the compensator is derived according to the basic relation. The dynamical order of the compensator is explicitly given by using the coprime fraction description of the system transfer matrix. The results are extended to the case for which partial pole-zero cancellations are allowed.

It is important to know how far non-dominant poles can be located away from the dominant poles since the closed-loop system transient response may not be as expected, if the non-dominant poles are not placed far enough from the dominant pole pair. Even if the arbitrary pole assignment is possible with a full state feedback or an output feedback controller with the order of plant order minus one, low order controllers and especially PID controllers are used in most practical applications. Therefore, the farthest possible non-dominant pole locations with PID controllers are found for all pole systems in this study. Proposed calculation method is demonstrated on two different examples and the closed-loop system performance is investigated with the designed PID controllers.

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.

Regional pole assignment is studied for a class of linear systems via static output feedback. The shortage of the optimization problem subject to a set of bilinear matrix inequalities(BMIs) is analyzed. Then the algorithm of static output feedback pole assignment based on the regional attractors is proposed. A set of attractors in the satisfactory region is determined firstly. After that, the optimization direction of the output feedback matrix can be obtained via the one-dimensional searching method of variable polling. And a feasible solution can be attained by iterative computation. Computational results are presented demonstrating the effectiveness of the provided algorithm.

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.

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 note analyzes a new algorithm presented in [1] for designing a linear function observer with a minimum number of arbitrary poles. It shows that the maximum order of the observer of [1] is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\nu + ... + \nu_{p} - p</tex> for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p \leq q</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n - q</tex> for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p \geq q</tex> , instead of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p(\nu_{1} - 1)</tex> as suggested in [1], where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\nu{i}, i = 1</tex> to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> , are the descending ordered observability indexes of system ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A, C</tex> ) and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n,p</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> are the order of the system, the number of the functions, and the number of the system outputs, respectively. This note also shows the significance of this result. For presentational purposes, only a special case of [1] is considered here. However, the technical properties as proved in this note are general.

Insensitive Pole Assignment Design

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 considers dynamical compensators design for purpose of pole assignment for discrete-time linear periodic systems. Similar to linear time-invariant systems, it is pointed out that the design of a periodic dynamical compensator can be converted into the design of a periodic output feedback controller for an augmented system. Utilizing the recent result on output feedback pole assignment, parametric solutions for this problem are obtained. The design approach can be used as a basis for the robust dynamical compensator design for this type of systems. Combined with a robustness index presented in this paper, robust dynamical compensator design problem is converted into a constrainted optimization problem. A numerical example is employed to illustrate the validity and feasibility of the methods.

The problem of trim point transitions is investigated for 6DOF quadrotors, using I/O decoupling controllers. The linear approximant of the quadrotor is derived in parametric form with respect to the physical parameters of the nonlinear model and the trim points. Based upon the linear approximant, a static state feedback control law achieving I/O decoupling, command following and arbitrary pole assignment for the linear approximant of the closed loop system, is derived. The controller is expressed in terms of the physical parameters and the closed loop characteristic polynomial arbitrary coefficients. To increase the range of accurate transition among different trim points of the nonlinear model, appropriate cost criteria are used and a trim point transition algorithm is proposed. The performance of the overall scheme is demonstrated through simulation results.

This paper discusses the design of a single linear functional observer (SLFO) for a linear time-invariant system. The SLFO is used to estimate a scalar-valued linear function of the system state, and is generally of lower order than the Luenberger observer. Two cases are considered: (i) feedback gain included as a design parameter and designed at the same time as the SLFO (the SLFO with unfixed feedback gain), and (ii) feedback gain is previously specified, before the design of the SLFO (the SLFO with fixed feedback gain). A standard form, a canonical form and the minimal order are obtained and they are used to obtain the optimal SLFO for the single-input linear-quadratic (LQ) problem. The SLFO with unfixed feedback gain is more difficult to calculate but has the following advantages over the SLFO with fixed feedback gain: the order is generally lower, the cost for the LQ problem is smaller and the responses are improved.