Controllability Of Discrete-time Bilinear Systems Research Articles

Constrained controllability of discrete-time bilinear systems

In real systems, control inputs are often constrained, and the constrained controllability problems are far more difficult to deal with than the unconstrained controllability ones. In this paper, we consider the constrained controllability problem of discrete-time bilinear systems. Specifically, we study the bounds of the control inputs for the systems to be controllable. Our approach is to first prove constrained near-controllability of a special class of bilinear system and then use constrained near-controllability to approximate constrained controllability of general bilinear systems. Based on this approach, we derive conditions for constrained controllability of the systems, where the upper bound of the control inputs can be easily computed. Algorithms and examples are provided to demonstrate the results of this paper.

Algebraic Criteria on Controllability and Positive-Controllability of Discrete-Time Bilinear Systems: Seeking Nonzero Entries

In this article, we consider controllability of discrete-time bilinear systems. A classical sufficient condition was obtained in this journal in 1973, which, unfortunately, is not algebraically verifiable, and the required control inputs to achieve state transition are unknown. However, based on a recently established notion, near-controllability, we propose a seeking nonzero entries approach to derive fully algebraically verifiable controllability criteria for discrete-time bilinear systems. More specifically, by our approach, the verification of controllability of the systems is transformed into the problem of seeking nonzero entries of a sequence of matrices generated by the two system matrices. In addition, the control inputs steering the controllable systems between any given pair of states can be easily computed by proposing two algorithms based on near-controllability. As applications, the seeking nonzero entries approach is generalized to construct arbitrary-finite-dimensional controllable discrete-time bilinear systems and also to prove positive-controllability of discrete-time bilinear systems, where algebraic criteria are obtained by the root locus theory and positive-near-controllability that is a new notion introduced in this article. Examples are provided to illustrate the obtained algebraic controllability criteria and algorithms.

Quadratic Stabilization of Discrete-Time Bilinear Control Systems Subjected to Exogenous Disturbances

We consider the design problem for discrete-time bilinear control systems subjected to arbitrary bounded exogenous disturbances. A procedure for the construction of the stabiliz-ability ellipsoids and stabilizability domain for discrete-time bilinear control systems is proposed and its efficiency is proved. The main tools are the linear matrix inequality technique and the apparatus of quadratic Lyapunov functions. This simple yet general approach is of great potential; for instance, it can be generalized to the various robust statements of the problem.

On the Properties of Controllability of Discrete-Time Bilinear Systems Associated with Block-Triangular and Permutation Matrices

The new results associated with properties of the controllability of discrete-time bilinear systems are presented. The properties of the controllability of a class of bilinear systems with block-triangular matrices are discussed. As a consequence, systems with permutation matrices and scalar bounded control are investigated. A series of necessary controllability conditions related to the dimension of the state vector and spectral properties of systems’ matrices are established and proved.

Invariant sets and controllability of discrete‐time bilinear systems

In this study, the author investigates controllability of discrete-time bilinear systems by using invariant sets. An invariant set defined in this study is a set on which the system is invariant, namely for any initial state that belongs to the set, the system cannot be steered away from this set. If a system is controllable, then it has no invariant set in its state space other than the state space itself. Invariant sets are thus closely related to controllability. By studying the invariant sets of discrete-time bilinear systems, the author first presents a necessary and sufficient condition for controllability of the systems in dimension two, which covers a classical result under the same condition. Then the author considers the high-dimensional systems and improve the recent results on controllability by relaxing a condition concerning invariant sets, where more general controllability criteria are derived. Finally, examples are given to illustrate the obtained results.

Quadratic Stabilization of Discrete-Time Bilinear Systems

We consider the problem of stabilization of discrete-time bilinear control systems. Using the linear matrix inequality technique and quadratic Lyapunov functions, we formulate a method for the construction of the so-called stabilizability ellipsoid having the property that the trajectories of the closed-loop system emanating from the points in the ellipsoid asymptotically tend to the origin. The proposed approach allows for an efficient construction of nonconvex domains of stabilizability of discrete-time bilinear control systems. The results are extended to the robust statement of the problem where the system matrix is subjected to structured uncertainties.

On controllability of discrete-time bilinear systems by near-controllability

In this paper, controllability of discrete-time bilinear systems is studied. By applying a recent result on near-controllability, a new sufficient condition for controllability of the systems is presented, where controllability is proved by approximation with near-controllability. The new condition is algebraically verifiable and is hence easy to apply compared with a classical result on controllability of discrete-time bilinear systems, which can be effective even when the classical result does not work. Furthermore, the control inputs to achieve the transition of the systems between any given pair of states are approximately computable according to near-controllability. Therefore, near-controllability can be used to not only better characterize the system properties, but also prove controllability with computable control inputs. The new condition is then generalized to derive similar results on controllability and near-controllability of the systems. Finally, examples are given to illustrate the results of this paper.

Optimal control of discrete-time bilinear systems with applications to switched linear stochastic systems

This paper aims at characterizing the most destabilizing switching law for discrete-time switched systems governed by a set of bounded linear operators. The switched system is embedded in a special class of discrete-time bilinear control systems. This allows us to apply the variational approach to the bilinear control system associated with a Mayer-type optimal control problem, and a second-order necessary optimality condition is derived. Optimal equivalence between the bilinear system and the switched system is analyzed, which shows that any optimal control law can be equivalently expressed as a switching law. This specific switching law is most unstable for the switched system, and thus can be used to determine stability under arbitrary switching. Based on the second-order moment of the state, the proposed approach is applied to analyze uniform mean-square stability of discrete-time switched linear stochastic systems. Numerical simulations are presented to verify the usefulness of the theoretic results.

Model reduction of discrete-time bilinear systems by a Laguerre expansion technique

In this paper, we investigate model reduction of discrete-time bilinear control systems using a Laguerre expansion technique. The transfer function of systems is expanded under the Laguerre functions basis and we show that Laguerre coefficients take on a recursion relationship, which allows us to execute the reduction elegantly based on the recursion, instead of the cumbersome expression of Laguerre coefficients. Then reduced models are produced in the framework of two-sided projection methods and a certain number of Laguerre coefficients of the original systems are preserved implicitly, thereby resulting in a good approximation. Noting that the order of reduced models may rise rapidly, a terminal reduction method is introduced to modify our algorithm. The modified algorithm is achieved by properly defining an auxiliary system, significantly shrinking the order of reduced models. Theoretical analysis and numerical experiments show the efficiency of our approach.

Approximately nearly controllability of discrete-time bilinear control systems with control input characteristic

In this paper we study a class of discrete-time bilinear control systems that can be used to model the production storage management problems. A weaker but more applicable concept of controllability, approximately nearly controllability, is proposed under the framework of hypercyclicity from the theory of dynamical systems. The complete characteristic of the system structure for being approximately nearly controllable is provided. Necessary and sufficient conditions for the system to be approximately nearly controllable are presented. Moreover, the required number of different input values for the system to achieve the approximately nearly controllability during the evolutionary process is obtained.

On necessary conditions and sufficient conditions for controllability of discrete‐time bilinear systems

In this study, controllability of discrete-time bilinear systems is studied. Necessary conditions and sufficient conditions for the systems to be controllable are presented. In particular, the sufficient conditions improve some existing results and the necessary conditions are new and easy to verify. Examples and simulations are provided to demonstrate the results of the paper.

Generalised tangential interpolation for model reduction of discrete-time MIMO bilinear systems

In this article, we discuss a model order reduction method for multiple-input and multiple-output discrete-time bilinear control systems. Similar to the continuous-time case, we will show that a system can be characterised by a series of generalised transfer functions. This will be achieved by a multivariate Z-transform of kernels corresponding to an explicit solution formula for discrete-time systems. We will further address the problem of generalised tangential interpolation which naturally comes along with this approach. We will introduce a reasonable generalisation of the linear ℋ2-norm. Based on this concept, we discuss the choice of interpolation points. Furthermore, an efficient discretisation of continuous-time systems is provided. The performance of the proposed method is evaluated in some numerical examples and compared with the method of balanced truncation for bilinear systems.

On controllability of discrete-time bilinear systems

In this paper, necessary and sufficient conditions for the controllability of a class of discrete-time bilinear systems are proposed, which extend the existing results and show that the controllability counterexample in [1] that is derived by Euler discretization of an uncontrollable bilinear system, is a special case of the necessary and sufficient conditions.

A second-order maximum principle for discrete-time bilinear control systems with applications to discrete-time linear switched systems

A powerful approach for analyzing the stability of continuous-time switched systems is based on using optimal control theory to characterize the “most unstable” switching law. This reduces the problem of determining stability under arbitrary switching to analyzing stability for the specific “most unstable” switching law. For discrete-time switched systems, the variational approach received considerably less attention. This approach is based on using a first-order necessary optimality condition in the form of a maximum principle (MP), and typically this is not enough to completely characterize the “most unstable” switching law. In this paper, we provide a simple and self-contained derivation of a second-order necessary optimality condition for discrete-time bilinear control systems. This provides new information that cannot be derived using the first-order MP. We demonstrate several applications of this second-order MP to the stability analysis of discrete-time linear switched systems.

Regular Elements and Global Controllability in SL(d, R)

We introduce a generalized concept of split-regular elements and study the dynamics of such elements on the flag manifolds corresponding to the Lie group SL(d, R). Based on these concepts, we obtain the stable and unstable manifolds, the Morse decomposition, and obtain the limit sets inside the corresponding control sets. These dynamical aspects are studied with the purpose of applying them in order to extend a result of global controllability of discrete-time bilinear systems defined by elements of sl(d, R).

Nonsingular and Stable Adaptive Control of Discrete-Time Bilinear Systems

A certainty equivalence adaptive control algorithm for bilinear systems is studied. Achieving a well-defined control law is seen to be a nontrivial aspect of the algorithm. A scheme for resolving this problem and a stability result are presented.

Dead-beat control of discrete-time bilinear systems †

The subject, of this paper is the dead-beat control of discrete-time bilinear systems with independent additive and multiplicative controls. The first, issue is the proof that controllability to the origin is a necessary and sufficient existence condition, and that the dead-beat controller can be chosen, without loss of generality, with a two-layer structure. Then, the controllability to the origin of the system is shown to be equivalent to that of homogeneous subsystems, obtained through successive decompositions of it. Thus, the problem of finding a dead-beat controller is reduced to that of zeroing the state of these subsystems. Lastly, some conditions for the controllability to the origin of a special type of homogeneous bilinear system are given.

Deterministic and stochastic control of discrete-time bilinear systems

Optimal control of a class of time invariant single-input, discrete bilinear systems is investigated in this paper. Both deterministic and stochastic problems are considered. In the deterministic problem, for the initial state in a certain set ∑ 0, the solution is the same as the solution to the associated linear system. The optimal path may be a regular path or a singular path. The stochastic control problem is considered with perfect state observation, and additive and multiplicative noise in the state equation. It is demonstrated that the presence of noise simplifies the analysis compared to that in the determinstic case.

Controllability of discrete bilinear systems with bounded control

Controllability of time-invariant discrete-time bilinear systems is discussed. Bilinear systems are classified into two categories: homogeneous and inhomogeneous. Sufficient conditions which ensure the global controllability of discrete-time bilinear systems are obtained by localized analysis in control variables.