Continuous-time Bilinear Systems Research Articles

Controllability of Continuous-Time Bilinear Systems Without Using the Lie-Algebraic Conditions: A New Perspective

In this paper, we propose a new perspective to study controllability of continuous-time bilinear systems without using the Lie-algebraic conditions. Specifically, we first consider controllability of the bilinear systems in the single-input case under a commutativity condition. We show that, although the Lie algebra rank condition, which is necessary for a classical controllability result to work, does not fit such bilinear systems, they can still be controllable. Our approach to prove controllability is using controllability of the discrete-time counterparts of the continuous-time systems, and we derive a necessary and sufficient controllability criterion without the Lie algebra rank condition, which is algebraically verifiable for any finite dimension. More importantly, through this controllability study, we propose a new perspective to deal with the controllability problems of continuous-time bilinear systems by changing the verification of the Lie algebra rank condition to solving two linear algebra problems. Examples are given to illustrate the obtained controllability results of this paper.

Balanced Truncation for a Special Class of Bilinear Descriptor Systems

In this letter, we investigate balanced truncation for a class of continuous-time bilinear descriptor systems, appearing, e.g., in constraint circuit analysis or resulting from semi-discretization of the Navier–Stokes equations. For this, as widely done in the literature, we first aim at transforming the bilinear descriptor system into an equivalent ODE system by means of projectors for which the standard balanced truncation method for bilinear systems can be applied. Subsequently, we discuss how to solve the arising generalized Lyapunov equations corresponding to the equivalent ODE, without requiring the explicit computation of the projectors. The efficiency of the proposed algorithm is illustrated through a numerical example, which also shows its competitiveness with an iterative ${\mathcal {H}}_{2}$ -optimal model order reduction scheme.

Necessary and Sufficient Condition for Controlled Distinguishability of Continuous-Time Bilinear Systems

Controlled distinguishability of two dynamical systems is the property of the systems that guarantees the existence of a control input generating different outputs of the systems regardless of their initial state vectors. These inputs are referred in the literature as discerning control inputs. In this paper, a necessary and sufficient condition for controlled distinguishability of continuous-time bilinear systems is established. It generalizes the classic one provided in the literature for the class of linear systems and a method for designing discerning inputs of bilinear systems that stabilize the systems is also proposed.

Continuous-time bilinear system identification using single experiment with multiple pulses

A novel method is presented for the identification of a continuous-time bilinear system from the input–output data generated by a single experiment with multiple pulses. In contrast to the conventional approach utilizing multiple experiments, the current work documents the advantage of using a single experiment and sets up a procedure to obtain bilinear system models. The special pulse inputs employed by earlier research can be avoided and accurate identification of the continuous-time system model is possible by performing a single experiment incorporating a class of control input sequences combining pulses with free-decay response. The algorithm presented herein is more attractive in practice for the identification of bilinear systems. Numerical examples presented demonstrate the methodology developed in the paper.

A Maximum Principle for the Stability Analysis of Positive Bilinear Control Systems with Applications to Positive Linear Switched Systems

We consider a continuous-time bilinear control system (BCS) with Metzler matrices. Each entry in the transition matrix of such a system is nonnegative, making the positive orthant an invariant set of the dynamics. Motivated by the stability analysis of positive linear switched systems (PLSSs), we define a control as optimal if, for a fixed final time, it maximizes the spectral radius of the transition matrix. Our main result is a first-order necessary condition for optimality in the form of a maximum principle (MP). The proof of this MP combines the standard needle variation with a basic result from the Perron--Frobenius theory of nonnegative matrices. We describe several applications of this MP to the stability analysis of PLSSs under arbitrary switching.

Generalized Bilinear System Identification

A novel method is presented for identification of a bilinear system generalized to include higher-order input coupling terms. It derives from an existing method for identification of a continuous-time bilinear system. The method first generates a set of pulse responses from a constant input over a sample period for identification of the state matrix, the output matrix, and the direct transmission matrix. The method then produces another set of pulse responses with the same constant input over varying sample periods for identification of the input matrix and the coefficient matrices associated with the coupling terms between the state and the inputs. A simple example is given to illustrate the concept of the identification method.

Parameter estimation of continuous-time bilinear systems based on numerical integration and separable non-linear least-squares

This paper considers the problem of estimating the parameters in continuous-time bilinear systems. The system identification approach is based on numerical integration and separable non-linear least-squares. The situation where the output signal is contaminated with noise is also discussed. The suggested estimation method uses a bias compensating approach and the model parameters together with parameters associated with the noise are estimated by solving an overdetermined system of equation in a least-squares sense.

Robust H∞ state feedback control methods for bilinear systems

Robust H∞ state feedback control methods for continuous-time bilinear systems with an additive disturbance input are presented. The robust H∞ control for bilinear systems requires a solution of the state dependent algebraic Riccati equation (SDARE). However, the proposed robust H∞ control techniques based on the successive approximation method solve the SDARE by converting the bilinear system into the time varying linear system and solving a sequence of algebraic Riccati and Lyapunov equations, respectively. Numerical examples are considered to verify the efficiency of the proposed algorithms.

Continuous-Time Bilinear System Identification

The objective of this paper is to describe a new method for identification of a continuous-time multi-input and multi-output bilinear system. The approach is to make judicious use of the linear-model properties of the bilinear system when subjected to a constant input. Two steps are required in the identification process. The first step is to use a set of pulse responses resulting from a constant input of one sample period to identify the state matrix, the output matrix, and the direct transmission matrix. The second step is to use another set of pulse responses with the same constant input over multiple sample periods to identify the input matrix and the coefficient matrices associated with the coupling terms between the state and the inputs. Numerical examples are given to illustrate the concept and the computational algorithm for the identification method.

On H2 model reduction of bilinear systems

The H 2 model reduction problem for continuous-time bilinear systems is studied in this paper. By defining the H 2 norm of bilinear systems in terms of the state-space matrices, the H 2 model reduction error is computed via the reachability or observability gramian. Necessary conditions for the reduced order bilinear models to be H 2 optimal are given. The gradient flow approach is used to obtain the solution of the H 2 model reduction problem. The formulation allows certain properties of the original models to be preserved in the reduced order models. The model reduction procedure developed can also be applied to finite-dimensional linear time-invariant systems. A numerical example is employed to illustrate the effectiveness of the proposed method.

Robust Stabilization of Constrained Bilinear Systems

The problem of asymptotic stabilization of uncertain bilinear systems containing saturating actuators is discussed. Using a state-feedback approach, sufficient conditions are derived to guarantee the stability of the saturating system in the presence of a norm-bounded, possibly nonlinear uncertainty. The time-domain robust stability conditions presented for continuous-time bilinear systems make use of the measure of a matrix and depend on bounds of the relevant matrices and the initial state. The region of attraction for the system is characterized.

On global controllability of discrete-time control systems

Let xn+1=exp(A+uB)xn be a discrete-time control system which is obtained by discretizing a continuous-time bilinear system. This paper gives sufficient conditions on the matricesA andB for global controllability of this system. These conditions are similar to those of Jurdjevic and Kupka [JK1] for the continuous-time case.

KYP lemma, state feedback and dynamic output feedback in discrete-time bilinear systems

This paper shows how a generalization of the Kalman-Yacubovitch-Popov (KYP) lemma can be built for discrete-time bilinear passive systems, and how the concepts of passivity and feedback equivalence can be applied to solve the problem of global stabilization via smooth state feedback, bounded state feedback and dynamic output feedback for a class of discrete-time bilinear systems. The results generalize some recent advances due to Gauthier and Kupka (1992) in continuous-time bilinear systems to their discrete-time counterparts.

A Hybrid Method for Real-Time Simulation of Continuous-Time Bilinear Systems

In this paper we present a method for the simulation of continuous-time bilinear systems ẋ ( t ) = Ax ( t ) + Nx ( t ) u (t) + bu ( t ), that can be used to realize bilinear observers for real-time control systems. Our method attempts to approximate the state variable x ( t ) in the bilinear term x ( t ) ut ) with an interpolation polynomial and treats this term as an independent input signal, then solves the bilinear state space equation like a linear system equation. The method yields, especially for large integrating step sizes, a more exact solution than a 4th order Runge-Kutta method. Even if the Runge-Kutta method becomes unstable the method works well.

On the stabilization of a class of bilinear systems

Abstract 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.

State Determination of Bilinear Systems and its Dependence on Input Class

The state determinability of continuous-time bilinear systems and its dependence on the class of admissible inputs are investigeted. The input which makes possible the determination of the initial state is shwon to be constructed as a piecewise constant function. Moreover, the piecewise constant function may be a bang-bang with arbitrary binary values. The upper bound on the minimal number of switchings of these inputs are explicitly given. If analytic functions or polynomials is admitted, one analytic function or one polynomial input is sufficient to determine the state instantaneously.

An observable canonical form of discrete-time bilinear systems

A canonical form of discrete-time bilinear systems is investigated. A new concept of sequential observability is introduced and it is shown that the discrete bilinear system is sequentially observable if and only if it can be transformed into some canonical form. One of the distinctions between discrete-time and continuous-time bilinear systems is discussed from uniform observability and sequential observability.