Discrete-time Bilinear Systems Research Articles

Frisch scheme based identification of dynamic errors-in-variables bilinear systems

An approach for the identification of dynamic single-input single-output bilinear discrete-time system models within the errors-in-variables framework for the case of white input and output noise sequences is presented. The method is based on the combination of the instrumental variables technique and the Frisch scheme. Whilst the instrumental variables technique is utilised to avoid correlations of the noise sequences corresponding to the bilinear part of the noise covariance matrix, the Frisch scheme allows to obtain, subsequently, the model parameters together with the variances of the input and output noise sequences. The concept gives rise to two novel algorithms, which are compared with other errors-in-variables approaches via a Monte-Carlo simulation.

Bilinear black-box identification and MPC of the activated sludge process

In this paper the activated sludge process, which is a process for biological nitrogen removal in municipal wastewater treatment plants, is modeled as a discrete-time bilinear system by application of a recursive prediction error method system identification technique. A novel bilinear model predictive control algorithm is also derived and applied on a simulation model of the activated sludge process. For discrete-time bilinear systems, a quadratic cost on the predicted outputs and inputs, together with input/state constraints, results in a nonlinear non-convex optimization problem. An investigation is performed where the suggested control algorithm is compared with a linear counterpart. The results reveals that even though the identified bilinear black-box model describes the dynamics of the activated sludge process better than linear black-box models, bilinear model predictive control only gives moderate improvements of the control performance compared to linear model predictive control laws.

Constrained Stabilization of Bilinear Discrete-Time Systems Using Polyhedral Lyapunov Functions

In this paper, the stabilization problem of discrete-time bilinear systems by linear state-feedback control is investigated. First, conditions guaranteeing the positive invariance of polyhedral sets with respect to nonlinear systems with second order polynomial nonlinearity are established. Then these results are used for the determination of linear state-feedback unconstrained and constrained control laws making a prespecified polyhedral set a domain of attraction of the resulting closed-loop system.

Saturated Feedback Stabilization of Discrete-Time Descriptor Bilinear Systems

This note addresses global asymptotic stabilization of a class of discrete-time descriptor bilinear systems via a saturated state feedback controller. Based on LaSalle invariance principle and the implicit function theorem, sufficient conditions are presented to guarantee the uniqueness and existence of solution and the global asymptotic stability of the resulting closed-loop systems simultaneously. It is shown that the condition is independent of the state transformation for the given descriptor systems. Furthermore, the effectiveness of the proposed design method is illustrated by a numerical example.

Successive-approximation approach of optimal control for bilinear discrete-time systems

A successive-approximation approach designing optimal controllers is developed for bilinear discrete-time systems with a quadratic performance index. By using the successive-approximation approach, the original optimal control problem is transformed into a sequence of nonhomogeneous linear two-point boundary-value problems. The optimal control law consists of an accurate linear term and a nonlinear compensating term that is the limit of the adjoint vector sequence. Through a finite-step iteration process of a nonlinear compensating sequence a suboptimal control law is obtained. Simulation results show that the algorithm is easily implemented and has a fast convergence rate.

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).

On gramians and balanced truncation of discrete-time bilinear systems

This paper obtains a balanced truncation model reduction method for discrete-time bilinear systems. After deriving the sufficient conditions for the discrete-time bilinear systems to lead to bounded outputs for some bounded inputs, we study the properties of the reachability and observability gramians. The gramians can be computed by solving two generalized Lyapunov equations. By balancing the reachability and observability gramians, a model reduction method is obtained. Some propositions of the reduced order models are given. A numerical example is given to illustrate the effectiveness of the proposed method.

STABILIZING CONTROL FOR DISCRETE TIME MULTI-INPUT BILINEAR SYSTEMS

This paper presents two stabilizing control methods for discrete time bilinear systems. One is a stabilizing controller for single input systems, and the other is a generalized stabilizing control method for discrete time multi-input bilinear systems. The proposed control methods guarantee the stability for each closed loop systems with single-input case and multi-input case, respectively. This paper introduces a lemma used to extend the stabilizing control method which is useful to represent any bilinear systems as pure differential (difference) matrix equations. The proposed algorithms are verified by a numerical example.

Closed-Loop Stabilizing MPC for Discrete-Time Bilinear Systems

Model-based predictive control (MPC) strategies for discrete-time bilinear systems are developed, which incorporate the guarantee of nominal closed-loop stability. It is demonstrated how the structure of the bilinear model can be exploited, both for a MPC strategy which is based on the combined use of an end-point weighting and end-point inequality constraint, and for a MPC strategy which uses the closed-loop paradigm, deploying perturbations on an off-line designed control law. These two MPC algorithms are compared by means of a numerical example.

Identification of multivariable bilinear state space systems based on subspace techniques and separable least squares optimization

We discuss identification of discrete-time bilinear state space systems with multiple inputs and multiple outputs. Subspace identification methods for bilinear systems suffer from the curse of dimensionality. Already for relatively low order systems, the matrices involved become so large that the method cannot be used in practice. We have modified the subspace algorithm such that it reduces the dimension of the matrices involved. Only the rows that have the largest influence on the model are selected; the remaining rows are discarded. This obviously leads to an approximation error. The initial model that we get from the subspace method is optimized using the principle of separable least squares. According to this principle, we can first solve for the matrices that enter non-linearly in the output error criterion and then obtain the others by solving a linear least squares problem.

Bilinear State Space Systems for Nonlinear Dynamical Modelling

We discuss the identification of multiple input, multiple output, discrete-time bilinear state space systems. We consider two identification problems. In the first case, the input to the system is a measurable white noise sequence. We show that it is possible to identify the system by solving a nonlinear optimization problem. The number of parameters in this optimization problem can be reduced by exploiting the principle of separable least squares. A subspace-based algorithm can be used to generate initial estimates for this nonlinear identification procedure. In the second case, the input to the system is not measurable. This makes it a much more difficult identification problem than the case with known inputs. At present, we can only solve this problem for a certain class of single input, single output bilinear state space systems, namely bilinear systems in phase variable form.

Strong consistency and convergence rate of parameter identification for bilinear systems

This paper discusses the parameter identification problem of discrete-time stochastic bilinear systems, when the dynamic system noise is coloured and its variance function may be unbounded in time. It will be proved, under weak conditions, that the extended least squares estimate of the system parameter has strong consistency and a fast convergence rate.

BIBO Stability of the Discrete Bilinear System

The discrete-time bilinear system is investigated for bounded input bounded output (BIBO) stability. The general state-space model is considered and two different sufficient conditions are established for BIBO stability. The bound on the state vector is also found for each case. The direct-form realization of the bilinear system is considered as a special case and a sufficient condition is derived for its stability. The necessary and sufficient condition is established for the stability of the second order system. The derived conditions are useful for finding the range of the input sequence in order to guarantee a bounded output sequence. The results are illustrated with several examples.

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.

Non-linear control design and robustness of discrete-time bilinear systems

We present a non-linear state feedback approach to solving the robust stability problem for a discrete bilinear system with non-linear perturbation. In addition, according to the Lyapunov stability theorem and numerical method, the tolerable perturbation bound on the norms is also obtained. The main result of Yaz and Niu (1989) is a special case of the result of this paper. In the illustrated examples we give a comparison between the results of this paper and that of Yaz and Niu (1989).

Estimation and control of stochastic bilinear systems with prescribed degree of stability

The design is described of exponentially data-weighting state estimators and feedback controllers with a prescribed degree of stability for stochastic bilinear (stochastic-parameter) discrete-time systems. Not only stability but also stability robustness in the presence of deterministic and stochastic parameter perturbations/modelling uncertainties are considered. The existence is shown of robustness properties similar to those encountered in the theory of linear deterministic systems.

Walsh function analysis of linear and bilinear discrete time systems

A technique is presented for analysing linear and bilinear discrete systems in terms of Walsh functions. The method replaces the solution of a dynamical difference equation with the solution of an algebraic equation. In the special case of a linear system, a numerically stable recursive technique is derived to solve this algebraic equation. The technique is illustrated with an example.

Suboptimal regulation of a class of bilinear interconnected systems on finite time-sliding planning horizons

This paper focuses on the suboptimization of a class of multivariable discrete-time bilinear systems consisting of interconnected bilinear subsystems with respect to a linear quadratic optimal regulation criterion which involves the use of state weighting terms only. Conditions which ensure the controllability of the overall system are given as a previous requirement for optimization. Three transformations of variables are made on the system equations in order to implement the scheme on an equivalent linear system. This allows another equivalent representation of the used quadratic performance index that involves the appearance of quadratic weighting terms related to both transformed input and state variables. In this way, a Riccati matrix-sequence, which allows a standard feedback control law implementation, is obtained.

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.

非同次離散時間双線形制御系の同定と最小実現

非同次離散時間双線形制御系の同定と最小実現