Discrete-time Bilinear Systems Research Articles

Ideal observability for bilinear discrete-time systems with and without delays in observation

Ideal observability for bilinear discrete-time systems with and without delays in observation

Time domain model order reduction of discrete-time bilinear systems by discrete Walsh functions

In this paper, the discrete Walsh functions are first used for time domain model order reduction (MOR) of discrete-time bilinear systems. Firstly, in terms of the discrete Walsh functions, the reduced order system is generated by means of the explicit solution formula for the discrete-time bilinear system. Meanwhile, we expand the discrete-time bilinear system under space spanned by the discrete Walsh functions. Then the reduced order system is produced by the orthogonal projection matrix that is defined by discrete Walsh coefficient matrix. Moreover, MOR method via discrete Walsh functions is investigated for discrete-time bilinear systems with nonzero initial conditions, where nonzero initial condition is also utilized to create the orthogonal projection matrix. Theoretical analyses show that the outputs of the resulting reduced order systems can match the first several outputs of the original system in finite intervals. Finally, an example is given to illustrate the effectiveness of the proposed methods.

Positive-Controllability, Positive-Near-Controllability, and Canonical Forms of Driftless Discrete-Time Bilinear Systems

Positive-Controllability, Positive-Near-Controllability, and Canonical Forms of Driftless Discrete-Time Bilinear Systems

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.

On the Maximal Output Admissible Set for a Class of Bilinear Discrete-time Systems.

Given a discrete-time controlled bilinear systems with initial state x0 and output function yi, we investigate the maximal output set Θ(Ω) = {x0 ∈ ℝn, yi ∈ Ω, ∀ i ≥ 0} where Ω is a given constraint set and is a subset of ℝp. Using some stability hypothesis, we show that Θ(Ω) can be determined via a finite number of inequations. Also, we give an algorithmic process to generate the set Θ(Ω). To illustrate our theoretical approach, we present some examples and numerical simulations. Moreover, to demonstrate the effectiveness of our approach in real-life problems, we provide an application to the SI epidemic model and the SIR model.

Time domain model order reduction of discrete-time bilinear systems with Charlier polynomials

This paper investigates time domain model order reduction of discrete-time bilinear systems with inhomogeneous initial conditions. The state of the system is approximated by the power series associated with the Charlier polynomials and the recurrence relation of the expansion coefficients is derived. The expansion coefficients are orthogonalized to construct the projection matrix by the modified multi-order Arnoldi method. The output of the resulting reduced order system maintains a certain number of expansion coefficients of the original output, and the error estimation of the reduced order system is briefly discussed. Due to the fact that the projection matrix involves the information of initial conditions, the proposed method can well reduce discrete-time bilinear systems with inhomogeneous initial conditions. Two numerical examples are employed to illustrate the effectiveness of the proposed method.

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.

Structural controllability and structural near-controllability of a class of discrete-time bilinear systems

In this paper, we consider the structural controllability problems of a class of driftless discrete-time bilinear systems which have nearly the same structure as linear systems. Algebraic as well as graph-theoretic conditions for structural controllability of such systems with single-input and multi-input are, respectively, obtained. Furthermore, we introduce a new notion, called structural near controllability, for those structured systems which cannot be structurally controllable but structurally nearly-controllable. It is shown that the bilinear systems can be structurally nearly-controllable although their linear counterparts can never have a large controllable region no matter what the parameters are chosen. Necessary and sufficient conditions for structural near-controllability of the bilinear systems are thus derived. We also consider the structural controllability problem in the continuous-time case. Examples and graphs are given to illustrate the obtained results of this paper.

Zonotopic reachable set estimation for bilinear systems with time-varying delays

ABSTRACT This paper studies reachable set estimation for bilinear discrete-time systems with time-varying delays considering state feedback and unknown-but-bounded disturbance. A novel reachable set estimation method based on the zonotopic technique is proposed for the considered systems under non-zero initial conditions. A series of zonotopes can be obtained in the derived theorem for bounding real-time states of the considered systems. The effectiveness of the proposed method is illustrated by a numerical example.

Data-based stabilization of unknown bilinear systems with guaranteed basin of attraction

Motivated by the goal of having a building block in the direct design of data-driven controllers for nonlinear systems, we show how, for an unknown discrete-time bilinear system, the data collected in an offline open-loop experiment enable us to design a feedback controller and provide a guaranteed under-approximation of its basin of attraction. Both can be obtained by solving a linear matrix inequality for a fixed scalar parameter, and possibly iterating on different values of that parameter. The results of this data-based approach are compared with the ideal case when the model is known perfectly.

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.

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.

Output-Controllability and Output-Near-Controllability of Driftless Discrete-Time Bilinear Systems

Unlike output-controllability of linear systems which is a consequence of controllability, the output-controllability problems of nonlinear systems can be much more difficult than the controllabili...

Asynchronous control of discrete-time stochastic bilinear systems with Markovian switchings

This paper is concerned with the problem of asynchronous control for a class of discrete-time Markov systems with multiplicative stochastic white noises. Based on a stability analysis scheme developed from mode-dependent Lyapunov function method, we first derive testable conditions in linear matrix inequality (LMI) setting to ensure the robust stability of the closed-loop system. We then recast the proposed stability conditions into equivalent forms that are later utilised to design a multi-mode asynchronous state-feedback controller (ASFC) that makes the closed-loop system stable. An extension to the case of deficient mode information (i.e. transition rates of the system and the controller are not fully accessible) is also presented. Finally, a model of networked control with DC devices is given to demonstrate the efficacy of the proposed design scheme.

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.

Stabilisation of discrete-time bilinear systems by constant control inputs

ABSTRACTIn this paper, the stabilisation problem of discrete-time bilinear systems by using constant control inputs is considered. It is shown that the spectrum of a matrix derived from the system matrices plays a key role in the stabilisation design. Sufficient conditions for the systems to be stabilizable by constant control inputs are presented in the cases when the derived matrix has no complex eigenvalue as well as in the cases when it has complex eigenvalues. Particularly, it is proved that, if the derived matrix has only nonzero real eigenvalues, then the systems can always be stabilised by constant control inputs. Finally, simulations are given to demonstrate the obtained results.

Reachable set estimation for discrete-time bilinear systems with time-varying delays

The problem of reachable set estimation is studied for discrete-time bilinear system in this paper. Time-varying delays and bounded input disturbances are both considered in bilinear system. The aim is to find reachable set that converges from all the states of system with initial conditions. By constructing Lyapunov–Krasovskii functional, sufficient delay-dependent less conservative stable conditions of reachable set estimation are obtained for bilinear delay system via the reciprocally convex combination and delay partition approaches. The derived theorem can guarantee that all the states of system with initial conditions from some domain are bounded in an ellipsoid and all the states from other domain are converged exponentially within a ball. One simulation example is presented to illustrate the correctness of the derived result in this paper.

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.

$H_{2}$ model order reduction based on gramians of discrete-time bilinear systems

Abstract In this paper, on the basis of gramians, $H_{2}$-optimal model order reduction of discrete-time bilinear dynamic systems is investigated. First, based on the reachability gramian and the observability gramian, the $H_{2}$-optimal necessary conditions of the discrete-time bilinear system are derived and the corresponding algorithm is proposed. When the algorithm is convergent, the resulting reduced system satisfies the necessary conditions. Then, for the single-input and single-output system, the $H_2$-optimal necessary conditions based on the generalized cross gramian are presented and the related algorithm is accordingly generated. Similarly, the convergent algorithm can make the reduced system satisfy the necessary conditions. Finally, two numerical examples are employed to illustrate the effectiveness of the proposed methods.

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.