Quadratic stabilization of bilinear systems: Linear dynamical output feedback

We consider stabilization of bilinear control systems by means of linear output dynamical controllers. Using the linear matrix inequality technique, quadratic Lyapunov functions, and a special iterative method, we propose a regular approach to the construction of the stabilizability ellipsoid having the property that the trajectories of the system emanating from the points of this ellipsoid asymptotically tend to zero. The developed approach enables for an efficient construction of nonconvex inner approximations of domains of stabilizability of bilinear control systems.     

Quadratic cost output feedback control for bilinear systems

This paper considers output feedback control problems for bilinear systems with a quadratic cost function and develops a robust control theory approach. A key point of the approach is to regard once the bilinear term as a term with unknown parameters and then use some robust control synthesis methods. This approach derives two types of nonlinear output feedback controller that guarantee an upper bound for the cost function. Efficiencies of the proposed approach are also demonstrated through a numerical example.     

Output stabilization of distributed bilinear systems

This paper studies regional stabilization of a distributed bilinear system evolving on a spatial domain $varOmega$. Sufficient conditions for regional weak, strong and exponential stabilization are given. Also we discuss a regional optimal stabilization problem. The obtained results are illustrated by examples and simulations.     

Quadratic stabilization of switched systems

We consider quadratic stabilization of uncertain switched systems when a switching rule is imposed on state feedback controllers of subsystems. A method is proposed to constructively design switching rules for continuous and discrete-time switched systems with norm-bounded time-varying uncertainties. The switching rules designed via this method do not rely on uncertainties, and the switched system is quadratically stabilizable via switched state feedback for all uncertainties.     

The output stabilization of SISO bilinear systems

A quite general approach to the problem of stabilizing a SISO (single-input single-output) bilinear system by output feedback is presented. Following a Lyapunov-like method, a family of admissible controls is defined and an easy way for deriving its general features is proposed. Previous results can be obtained as special cases of this approach. Numerical examples are included to show the feasibility of this algorithm in determining simple control laws     

Quadratic stabilization of switched nonlinear systems

In this paper, the problem of quadratic stabilization of multi-input multi-output switched nonlinear systems under an arbitrary switching law is investigated. When switched nonlinear systems have uniform normal form and the zero dynamics of uniform normal form is asymptotically stable under an arbitrary switching law, state feedbacks are designed and a common quadratic Lyapunov function of all the closed-loop subsystems is constructed to realize quadratic stabilizability of the class of switched nonlinear systems under an arbitrary switching law. The results of this paper are also applied to switched linear systems. Supported partially by the National Natural Science Foundation of China (Grant No. 50525721)     

Quadratic stabilization for uncertain stochastic systems

1IntroductionInthelast decades ,manyauthors studiedrobust quadraticstabilization control of deterministic linear systems withparameter uncertainty or structured uncertainty, see[1 ~8] . Robust quadratic stability and stabilization ofdeterministic systems were first introduced by [1] ,bymeans of a common Lyapunovfunction.Most earlier resultson robust quadratic stabilization,including some necessaryand sufficient conditions , were expressed in terms ofRiccati_type equations or inequalities , wh…     

Quadratic stabilization for uncertain stochastic systems

This paper discusses the robust quadratic stabilization control problem for stochastic uncertain systems,where the uncertain matrix is norm bounded,and the external disturbance is a stochastic process.Two kinds of controllers are designed,which include state feedback case and output feedback case.The conditions for the robust quadratic stabilization of stochastic uncertain systems are given via linear matrix inequalities.The detailed design methods are presented.Numerical examples show the effectiveness of our results.     

On the stabilization of the homogeneous bilinear systems

The problem of stabilizing the single-input homogeneous bilinear systems is considered. A special class of these systems is considered: the matrix of the linear (i.e., control independent) term of the right-hand side is semitable while the matrix of the bilinear term is skew-symmetric. Two types of stabilization are investigated: constant feedback global asymptotic stabilization and practical stabilization by a family of the linear feedbacks. It will be shown for the planar case that the above system is always practically stabilizable by a family of the linear feedbacks. For n = 3, the solution of both stabilization problems depends on the mutual position of the expanding direction of the linear part and the direction of the infinitesimal rotation defined by the bilinear term.     

Quadratic control systems

We outline a geometric theory for a class of homogeneous polynomial control systems called quadratic systems. We describe an algorithm to compute a minimal realization and study the feedback classification problem. Feedback invariants are related to the singularities of the input-output mapping and canonical forms are exhibited.     

Quadratic stabilization of sampled-data systems with quantization

A design method of memoryless quantizers in sampled-data systems is proposed. The design objective is quadratic stability in the continuous-time domain, and thus the decay rate between sampling times is guaranteed. Our general treatment enables us to look for quantizers efficient in terms of data rate.     

Global feedback stabilization of new class of bilinear systems

In this paper, we consider some new classes of bilinear systems. We give necessary and sufficient conditions for the asymptotic stabilization by using homogeneous feedbacks.     

On the stabilization of a class of bilinear systems

In this paper the problem of the stabilization of a class of continuous time bilinear systems, containing the class of the homogeneous ones, is studied, and simple stabiliza-bility conditions are derived, expressed in terms of the stability or stabilizability of some linear or homogeneous bilinear subsystems of the given system. On this basis, it is shown that this problem can be decomposed into subproblems and reduced to the stabilization of homogeneous subsystems of the system, obtained through successive decompositions of it. A sufficient condition for the stabilization of a homogeneous bilinear system is also given.     

Exponential stabilization of a class of unstable bilinear systems

Considers the control design of bilinear systems with multiplicative control inputs. Previous control designs for such systems normally assume that the open-loop bilinear system is (neutrally) stable. In this paper, a nonlinear control design is proposed for open-loop unstable bilinear systems. The new control stabilizes the bilinear system globally and exponentially if a sufficient stability condition, which can be checked by off-line computer simulations in advance of the control, is satisfied     

Quadratic stabilization of linear networked control systems via simultaneous protocol and controller design

We derive conditions for quadratic stabilizability of linear networked control systems by dynamic output feedback and communication protocols. These conditions are used to develop a simultaneous design of controllers and protocols in terms of matrix inequalities. The obtained protocols do not require knowledge of controller and plant states but only of the discrepancies between current and the most recently transmitted values of nodes’ signals, and are implementable on controller area networks. We demonstrate on a batch reactor example that our design guarantees quadratic stability with a significantly smaller network throughput than previously available designs.     

Quadratic stabilization of a collection of linear systems

The paper considers the problem of simultaneous quadratic (SQ) stablizability of a finite collection linear time-invariant (LTI) system by a static state feedback controller. At first, the solvability conditions for SQ stablization are derived in terms of the solution of certain reduced order matrix inequalities. The existence conditions for the solution of such matrix inequalities are then investigated. Based on that, two new classes of systems are characterized for which a SQ stabilization problem is solvable. These class of systems are (1) partially commutative systems and (2) partially normal systems. These systems are shown to be different from matched uncertain systems and also do not possess any generalised antisymmetric configurations. Both existence and computational algorithm for designing a state feedback controller are given.     

Quadratic stability and stabilization of dynamic interval systems

This note presents necessary and sufficient conditions for the quadratic stability and stabilization of dynamic interval systems. The results are obtained in terms of linear matrix inequalities (LMIs) and extended to the quadratic stability and stabilization of linear systems with uncertain parameters. With the powerful LMI toolbox, it is very convenient to solve these problems. The illustrative examples show that this method is effective and less conservative to check the robust stability and to design the stabilizing controller for dynamic interval systems.     

Constrained control of SISO bilinear systems

Relative degree and nonminimum phase difficulties limit the applicability of input-output feedback linearization; hence the need for approximations. Earlier work on predictive control of bilinear systems overcame these problems by means of interpolation between feedback linearization and state feedback, the former providing optimality and the latter guaranteeing feasibility and stability through the use of invariant/feasible polytopes. The current work also makes use of polytopes in preference to ellipsoids but achieves distinctly different objectives. First, it is shown that feedback linearization can be used over particular polytopes without needing to resort to either approximation or interpolation. Then, it is shown that invariant polytopes based on bilinear controllers can be much larger. These two approaches are combined in an algorithm that guarantees stability over much larger initial condition sets and gives much improved closed-loop performance.     

Optimal control of distributed bilinear systems

The optimal control problem for a bilinear distributed parameter system subject to a quadratic cost functional is solved. It is shown that the optimal control is given by a convergent power series in the state with tensor coefficients.     

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.