Inverse interactorizing and triangularizing with an arbitrary pole assignment using the state feedback

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.

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 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.

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.

This study proposes less conservative conditions for robust linear quadratic regulator controllers using state-derivative feedback (SDF). The algebraic Ricatti equation was formulated using the SDF, and its solution was obtained by linear matrix inequalities. SDF was chosen owing to the presence of accelerometers as sensors. Since accelerometers are the main sensors in mechanical systems, the proposed technique may be used to control/attenuate their vibrations/oscillations. Moreover, to formulate the less conservative conditions, some methods in the specialised literature were used, such as, for example, slack variables by Finler's Lemma. The paper also offers necessary and sufficient conditions for an arbitrary convex combination of square real matrices to be a nonsingular matrix, and thus an invertible one: must be nonsingular and all the real eigenvalues of must be positive. This result is important in the formulation of the proposed less conservative conditions since it was assumed that a given convex combination is nonsingular. A feasibility analysis demonstrates that the proposed conditions reduce the conservatism. Thereby, it is possible to stabilise a higher number of systems and to reduce the guaranteed cost. Furthermore, a practical implementation illustrated the application of the proposed conditions.

Using the Lyapunov function method, this study investigates both state and output feedback stabilisation control design problems for fractional order non‐linear systems in the lower triangular form, and presents a number of new results. First, some new properties for Caputo fractional derivative are presented. Second, by introducing appropriate transformations of coordinates, the feedback stabiliser design problem is converted into the determination of finding some parameters, which can be obtained by solving the Lyapunov equation and relevant matrix inequalities. Finally, based on the Lyapunov function method, both state and output feedback stabilisers are explicitly designed to make the closed‐loop system asymptotically stable. The study of an illustrative example shows that the obtained results are effective in designing feedback stabilisers for fractional order non‐linear systems in the lower triangular form.

This paper presents a systematic approach to design observer-based output feedback controllers for hybrid dynamical systems arising from bipedal walking. We consider a class of parameterized observer-based output feedback controllers for local exponential stabilization of hybrid periodic orbits. The properties of the Poincare map are investigated to show that the Jacobian linearization of the Poincare map takes a triangular form. This demonstrates the nonlinear separation principle for periodic orbits. In particular, the exponential stabilization of hybrid periodic orbits under dynamic output feedback control can be achieved by solving separate eigenvalue placement problems for the nonlinear state feedback and the observer. The paper then solves the state feedback and observer design problems by employing an iterative algorithm based on a sequence of optimization problems involving bilinear and linear matrix inequalities. The theoretical results are confirmed by designing a nonlinear observer-based output feedback controller for underactuated walking of a 3D humanoid model with 18 state variables, 54 state feedback parameters, and 271 observer parameters.

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 .

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.

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 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.

Multivariable nonlinear controller synthesis in discrete-time

It is shown that in uncontrollable linear system ẋ = Ax + Bu it is possible to assign arbitrarily the eigenvalues of the closed-loop system with state feedbacks u = Kx, K ∈ ℜn⨉m if rank [A B] = n. The design procedure consists in two steps. In the step 1 a nonsingular matrix&amp;nbsp; M ∈ ℜn⨉m is chosen so that the pair (MA,MB) is controllable. In step 2 the feedback matrix K is chosen so that the closed-loop matrix Ac = A&amp;nbsp; −&amp;nbsp;BK has the desired eigenvalues. The procedure is illustrated by simple example.

The aim of the paper is the output feedback stabilization of a class of single-input single-output (SISO) nonlinear systems. We deal with a class of systems, so-called triangular form systems, that may have unobservable and unstabilizable approximate linearization about the origin. This kind of systems, referred to as singular triangular forms, may neither be stabilized by a smooth state feedback nor smoothly detectable. However, under certain simple conditions, it is known to be stabilizable by a static state feedback. A solution to continuous dynamic output feedback stabilization is proposed. This is based on the construction of the continuous observer and nonlinear analogue of the well known separation principle. The resulting observer may be computed and tuned in a straightforward constructive way. Numerical simulation of selected examples are performed to underline the applicability of the presented approach and to illustrate good convergence of both observer and controller system.

The group of automorphisms of a zero-divisor graph based on rank one upper triangular matrices