Homogeneous Bilinear Systems Research Articles

Practical stabilizability of a class of homogeneous bilinear systems

Practical stabilizability of a class of homogeneous bilinear systems

Control sets for bilinear and affine systems

For homogeneous bilinear control systems, the control sets are characterized using a Lie algebra rank condition for the induced systems on projective space. This is based on a classical Diophantine approximation result. For affine control systems, the control sets around the equilibria for constant controls are characterized with particular attention to the question when the control sets are unbounded.

Controllability and near-controllability of a class of discrete-time inhomogeneous bilinear systems

In this paper, a class of discrete-time inhomogeneous bilinear systems whose controllability cannot be checked by the classical results is considered. By using an algebraic technique, it is shown that the systems are never controllable. Near-controllability of the systems is thus studied and, by defining a novel transition operator, an algebraic necessary and sufficient criterion is established, where the control inputs to achieve state transition can also be computed. Furthermore, the established near-controllability result is applied to study nearly controllable subspaces of the discrete-time homogeneous bilinear systems. Although such systems were investigated in the previous work, a new and larger class of nearly controllable subspaces of the homogeneous systems is derived, which cannot be found by the previous work due to different algebraic structures. Finally, an algorithm for solving the control inputs to achieve state transition as well as examples are given to demonstrate the results of this paper.

Weak and strong stabilisation of bilinear systems in a Banach space

ABSTRACTIn this paper, we study the feedback stabilisation of homogeneous bilinear systems on a Banach state-space. Sufficient conditions for weak and strong stabilisation are given. A decay rate of the stabilised state is explicitly estimated. Applications to heat and transport equations are provided.

Uniform global asymptotic stabilisation of bilinear non-homogeneous periodic systems with stable free dynamics

ABSTRACTWe study the problem of uniform global asymptotic stabilisation of the origin for bilinear non-homogeneous control systems with periodic coefficients by means of state feedback. We assume that the free dynamic system is Lyapunov stable. We obtain new controllability-like rank conditions, which are sufficient for uniform global asymptotic stabilisation of the periodic bilinear systems. The proof is based on the use of the Krasovsky–La Salle invariance principle for periodic systems. A stabilising feedback control law is quadratic in the state variable and periodic in the time variable. Consequences derived for linear and homogeneous bilinear systems. Under some assumptions, the controllability-like rank condition for linear and homogeneous bilinear systems coincides correspondingly with the property of complete controllability and consistency.

New Results for Robust Stability of Discrete Bilinear Uncertain Time-Delay Systems

In this paper, new results for the problem of the robust stability of discrete homogeneous bilinear time-delay systems subjected to nonlinear or parametric uncertainties are addressed. Applying a concise upper bound to the Lyapunov equation approach and then associating some linear algebraic techniques, several delay-independent conditions are presented to assure the robust stability of the aforementioned systems. One of the features of the present criteria is that they are independent of any Lyapunov equation, although the Lyapunov equation approach is adopted. Comparing to an existing work, it is shown that the obtained results are sharper. Finally, all obtained results are applied to solve the same problem of discrete bilinear uncertain systems and discrete time-delay systems with/without uncertainties. Concise criteria for the robust stability of the mentioned systems are developed, and comparisons between the obtained results and those appeared in the literature are also made. It is shown that all obtained results in this work are either tighter or more concise.

Stabilizing multimodel sliding mode control for homogeneous TS-bilinear systems

The sliding mode control of uncertain homogeneous bilinear systems is the main interest of this paper. Modelling is based on TS-bilinear approach. A sliding mode multimodel control is suggested and a stability study is conducted leading to sufficient conditions of global stabilization. Simulations on numerical examples are presented to prove the effectiveness of the proposed approach.

Global feedback stabilization of multi-input bilinear systems

This paper mainly investigates the problem of stabilization of homogeneous bilinear systems with multiple inputs. Explicit state feedback laws are given to stabilize the bilinear systems. Meanwhile, an estimate of the convergence speed is obtained under the given feedback laws. Besides, sufficient conditions, which are easy to be verified, are presented for the stabilization of the bilinear systems.

Further results for robust stability of homogeneous large-scale bilinear systems with time delays and uncertainties

In this paper, the problem of stability analysis for homogeneous large-scale time-delay bilinear systems subjected to uncertainties is considered. Two classes of uncertainty are treated: nonlinear uncertainties and parametric uncertainties. By making use of the Lyapunov stability approach associated with solution bounds of the Lyapunov equation, two delay-independent criteria are presented to guarantee the robust stability of the overall systems. The stability condition for the mentioned system with nonlinear uncertainties is sharper than that of a previous work. The main feature of the schemes presented is that they do not involve any Lyapunov equation which may be unsolvable although the Lyapunov stability theorem is used.

Robust Stability of Discrete Bilinear Uncertain Time-Delay Systems

This paper addresses the problem of robust stability for discrete homogeneous bilinear time-delay systems subjected to uncertainties. Two kinds of uncertainties are treated: (1) nonlinear uncertainties and (2) parametric uncertainties. For parametric uncertainties, we also discuss both unstructured uncertainties and interval matrices. By using the Lyapunov stability theorem associated with some linear algebraic techniques, several delay-independent criteria are developed to guarantee the robust stability of the overall system. One of the features of the newly developed criteria is its independence from the Lyapunov equation, although the Lyapunov approach is adopted. Furthermore, the transient response and the decay rate of the resulting systems are also estimated. In particular, the transient responses for the aforementioned systems with parametric uncertainties also do not involve any Lyapunov equation which remains unsolved. All the results obtained are also applied to solve the stability analysis of uncertain time-delay systems.

On controllability of a class of discrete‐time homogeneous bilinear systems with solvable controls

SUMMARYThis paper studies the controllability of a class of discrete‐time homogeneous bilinear systems. Necessary and sufficient conditions for the controllability are obtained. In particular, an algorithm for computing the controls, which achieve given states transition, for the controllable systems is given with examples. Furthermore, the necessary and sufficient conditions are applied to present four cases of the change of the controllability for Euler discretization, which show that Euler discretization may and may not change the controllability of nonlinear systems. Copyright © 2011 John Wiley & Sons, Ltd.

Practical stabilizability of dyadic homogeneous bilinear systems

This paper considers a class of homogeneous bilinear systems in which the rank of the B matrix is equal to one. It uses a transformation to convert the system into a form similar to the companion from. It then considers feedback control laws in the form of variable structure. Results indicate that it is possible to obtain practical stability for a class of dyadic systems. The results are particularly useful for low dimensional systems. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Parametrization of attainability sets for a class of homogeneous quasicommutative bilinear systems

The attainability set for a homogeneous quasicommutative bilinear system ẋ = (A + uB)x with a scalar limited control is considered. Under certain assumptions, each point of the attainability set is associated with a piecewise-constant control, for which the estimate of the number of switchings is related to the dimension of the phase space of the system. This fact makes it possible to use the switchings of these controls for description of the boundary and the interior of the attainability set of the system under consideration. A class of quasicommutative bilinear systems for which the numbers of switchings needed for the description of the boundary and the interior of the attainability set differ only slightly is established. Examples of homogeneous bilinear systems for which this result is valid are given.

On the stability of uncertain homogeneous bilinear systems subjected to time‐delay and constrained inputs

This paper addresses the stability test problem for homogeneous bilinear uncertain time‐delay systems with constrained inputs. By using differential inequality, some sufficient criteria are presented to guarantee the asymptotic stability of overall systems. Furthermore, the transient behaviors of the mentioned systems are also estimated.

Stabilization of continuous‐time homogeneous bilinear systems by constant inputs through nonlinear minimization with applications to the static output feedback stabilization problem

AbstractAn algorithm to provide constant‐input stabilizing control inputs for multi‐input continuous‐time bilinear systems is proposed in this paper. The algorithm is based on the formal discrete‐time approximation of the system and the unconstrained nonlinear minimization. The key features of the new algorithm are as follows. First, the formal discrete‐time approximation makes the set of stabilizing control inputs star‐shaped centred at the origin, hence the minimization is to be performed only in a neighbourhood of the origin selected by the designer. Second, the algorithm is always capable of finding a solution if one exists, as long as the minimization inside the neighbourhood is successful. Third, by a slight modification, the algorithm permits us to place all the eigenvalues of the system inside a rectangular region in the complex plane, as long as it is feasible. The algorithm is also applicable to the static output feedback stabilization problem of linear time‐invariant systems. Copyright © 2007 John Wiley & Sons, Ltd.

Design of sliding mode controllers for bilinear systems with time varying uncertainties.

Sliding mode controllers for the bilinear systems with time varying uncertainties are developed in this paper. The bilinear coefficient matching condition which is similar to the traditional matching condition for linear system is defined for the homogeneous bilinear systems. It can be seen that the bilinear coefficient matching condition is very limited and is not generally applicable to the nonhomogeneous bilinear system. Thus, the sliding coefficient matching condition is also considered for the bilinear systems with time varying uncertainties. Then, the sufficient conditions are provided for the reaching mode of the time varying uncertain bilinear systems to be guaranteed by the designed sliding mode controllers. Moreover, the stability of the uncertain bilinear systems with the sliding mode controller is discussed. Simulation results are included to illustrate the effectiveness of the proposed sliding mode controllers.

Division controllers for homogeneous dyadic bilinear systems

When a homogeneous bilinear system is dyadic, it permits the use of the so-called division controller for global stabilization. However, due to singularities in the control law, a dead-zone must be cascaded with the division controller in practical implementation. The contribution of this note is to show that the division controller, when cascaded with a dead-zone, can asymptotically drive the bilinear system state into a bounded set around the origin with the size of the bounded set proportional to the size of dead-zone. This note further proposes a C/sup 0/ modification and a C/sup /spl infin// modification of the conventional discontinuous dead-zone so that the control signals from the modified division controllers are continuous and smooth, respectively.

Remark on local and global stabilization of homogeneous bilinear systems

In this paper, we deal with the problem of stabilization of homogeneous bilinear systems. The aim is to clarify some results on stabilizability of these systems.

Reachable sets and controllability of bilinear time-invariant systems: a qualitative approach

This paper is concerned with a qualitative method for the study of reachable sets and controllability of bilinear time-invariant systems. A nonlinear decomposition technique which reduces the solution of the problem to the analysis of the qualitative behavior of a finite number of linear uncontrolled systems is introduced. Special consideration is given to the homogeneous bilinear systems with commutative matrices. Two- and three-dimensional examples are discussed.

Stabilization of homogeneous bilinear systems

In this paper, we investigate the problem of stabilization of homogeneous bilinear systems. We show that if a bilinear system is locally asymptotically stabilize by a state feedback law, then it becomes globally asymptotically stabilizable by some feedback law with the same upper bound.