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# IMA Journal of Mathematical Control and Information Current Issue

Published: Wed, 27 Jul 2016 00:00:00 GMT

Last Build Date: Sat, 16 Dec 2017 09:47:54 GMT

An ℋ∞ approach to data-driven simultaneous fault detection and control

Wed, 27 Jul 2016 00:00:00 GMT

Abstract
This paper proposes a data-driven ℋ∞ approach to simultaneous fault detection and control problem. This problem is formulated as a time domain ℋ∞ optimization problem and its solution produces a data-driven ℋ∞ controller/detector unit of modest complexity which is capable of achieving some control and fault detection objectives. The mathematical model is assumed to be unknown and only input/output data are used for controller/detector designing. The tradeoff between these objectives is established by tuning a scalar parameter and some weighting matrices. An easily implementable algorithm summarizes the methodology presented in the paper. The proposed algorithm is applied to a numerical example in order to illustrate its effectiveness.

On the controllability of hyperbolic linear systems with constraints on the gradient

Tue, 26 Jul 2016 00:00:00 GMT

Abstract
We present some results concerning the controllability of a hyperbolic linear equation with constraints on the gradient, we're interested in an internal subregion of the system evolution domain Ω. We analyse the controllability problem with constraints (also called the enlarged controllability) with distributed control that ensures the transfer of our system to a final desired gradient between two prescribed profiles f1 and f2. The proofs use a sub-differential approach, the Lagrangian one, and finally numerical results are illustrated.

Deadlock Control Based on Capacity Restrictions for FMS

Tue, 26 Jul 2016 00:00:00 GMT

Abstract
In flexible manufacturing systems (FMS), deadlock prevention is an important problem to be solved. Petri nets are a powerful tool and wildly used to model and control FMS. Based on the capacity restrictions for activity places in the Petri net model of an FMS, a new approach is presented to implement the liveness of the controlled net system in this research. First, given a Petri net prone to deadlocks, its initial capacity vector is decided. Then, an algorithm is proposed to compute its final capacity vector iteratively. At each iteration, the capacity of one activity place is decreased by one. The objective is to prevent the excessive occupancy of one resource by one production process, which is an important reason causing a deadlock. This process is carried out until the net model becomes live. With the constraints by the final capacity vector, the controlled net is live. It is proved the iterative algorithm is convergent. Considering the redundancy of capacity restrictions, another algorithm is given to eliminate it. The presented method for FMS control guarantees its live operation and high performance in terms of resource utilization. Generally, the proposed method is applicable, easy to use, effective and straightforward. Finally, two parameterized examples are used to demonstrate the applicability and effectiveness of the proposed approach.

H∞ control for a class of non-linear discrete-time networked systems with limited information

Tue, 26 Jul 2016 00:00:00 GMT

Abstract
In this article, the problems of stabilization and H∞ control are investigated for a class of non-linear networked systems with packet dropouts, network-induced delays and sensor faults. We construct a novel model by taking packet dropouts, distributed delays and sensor faults into account in a unified way. The packet dropouts process is modelled as a Markov chain taking values in a finite state space. Network-induced delays with distributed characteristics are divided into two intervals satisfying the Bernoulli random distribution. Sensor faults are described as stochastic variables and each sensor has different fault rate and independent of others. The resulting closed-loop system is converted into a Markov switching system. A mode-dependent controller is designed such that the closed-loop system is stochastically stable and satisfies H∞ disturbance attenuation level in terms of certain linear matrix inequalities. Finally, a numerical example is given to illustrate the usefulness of the developed method in this article.

$\mathscr{H}_{2}$ model order reduction for bilinear systems based on the cross gramian

Fri, 22 Jul 2016 00:00:00 GMT

Abstract
The $\mathscr{H}_{2}$ optimal model order reduction on the basis of the cross gramian for a given single-input and single-output (SISO) bilinear system is discussed in this article. According to the cross gramian, the expression of $\mathscr{H}_{2}$ norm for the SISO bilinear system can be represented. Then, taking advantage of the partial derivative, we can derive the necessary conditions of the $\mathscr{H}_{2}$ optimality for the reduced order bilinear system. From the necessary conditions, the coefficient matrices of the $\mathscr{H}_{2}$ optimal reduced order system can be constructed. After that, we propose an iterative algorithm based on the cross gramian. When the algorithm is convergent, the reduced order system can satisfy the necessary conditions. Finally, the numerical examples exhibit the effectiveness of our algorithm.

The solution bounds and fixed point iterative algorithm for the discrete coupled algebraic Riccati equation applied to automatic control

Tue, 19 Jul 2016 00:00:00 GMT

Abstract
In this article, applying the special properties of nonnegative matrices and symmetric matrices, we consider the coupled term as a whole and derive the solution bounds of the discrete coupled algebraic Riccati equation. This reduces the error of taking apart the coupled term separately when applying inequality techniques to get the bounds in general. Then, using Cauchy–Schwarz inequality, matrix norm inequalities and a fixed point theorem, we discuss the existence uniqueness condition and fixed point iteration for the solution of this equation. Finally, we offer corresponding numerical examples to illustrate the effectiveness of our results.

A collocation method via block-pulse functions for solving delay fractional optimal control problems

Tue, 19 Jul 2016 00:00:00 GMT

Abstract
A collocation scheme is applied to give an approximate solution of the fractional optimal control problems with delays in state and control. The operational matrix of fractional Riemann–Liouville integration, delay operational matrix and direct collocation method are used. The proposed technique is applied to transform the state and control variables into non-linear programming parameters at collocation points. The method is simple and computationally advantageous. Some examples are given to demonstrate the simplicity, clarity and powerfulness of the method.

Stabilization of an Euler–Bernoulli beam system with a tip mass subject to non-uniform bounded disturbance

Tue, 19 Jul 2016 00:00:00 GMT

Abstract
We consider the stabilization problem of an Euler-Bernoulli beam with tip mass, which undergoes non-uniform bounded disturbance. We employ the idea of active disturbance rejection control to design a disturbance estimator that has a time-varying gain of exponential-type, and design a feedback controller, in which the estimate of disturbance is used to cancel the effect of disturbance. We apply the semigroup theory to prove the well-posedness of the resulting closed system. We prove the exponential stability of the closed loop system by the Lyapunov function approach. Some numerical simulations are given to support these results.

Model reduction of discrete-time linear systems using Meixner-like filters

Fri, 15 Jul 2016 00:00:00 GMT

Abstract
In this article, we explore the use of the Meixner-like functions in estimating transfer functions of linear discrete-time systems. By expanding transfer functions with some suitable orthonormal basis functions, it is possible to reduce the estimate parameter number. Thus far, many orthonormal functions have been widely used in this framework such as Laguerre functions and Kautz functions. However, when the system have a slow initial onset or delay, Meixner-like functions, which have a slow start, are more suitable in terms of providing a more accurate approximation to the system. In this article, a minimal state-space realizations for discrete-time linear systems are derived using Meixner-like filters. Further we propose, from input/output measurements, an iterative optimization algorithm for the free parameter (Meixner-like pole) of the Meixner-like filters. The method consists in applying the Newton–Raphson's technique in which their elements are expressed analytically by using the derivative of the Meixner-like functions.

Reinforced gradient-type iterative learning control for discrete linear time-invariant systems with parameters uncertainties and external noises

Mon, 11 Jul 2016 00:00:00 GMT

Abstract
In this article, a reinforced gradient-type iterative learning control algorithm is developed for a type of discrete linear time-invariant systems with parameters uncertainties and external noises. The technique is to construct a symmetric learning gain matrix on basis of the system Markov parameters and an appropriate learning step length. First, for the case when both the model uncertainties and the external noises are absent, sufficient and necessary monotone convergences of the proposed algorithm are derived by means of matrix theory and norm inequality under the assumption that the learning step length is properly chosen. Then, for the cases when the model uncertainties are tolerable and the external noises are bounded, the robust monotone convergence and robustness are respectively analysed. Compared with the conventional gradient-type iterative learning control scheme, the proposed reinforced one is more efficient in speeding up the convergent tracking performance and resisting perturbations. Numerical simulations testify the validity and the effectiveness as well as the feasibility.

Regional optimal control of a class of bilinear systems

Mon, 11 Jul 2016 00:00:00 GMT

Abstract
The purpose of this paper is to study regional optimal control of a class of distributed bilinear systems in both cases of unbounded and bounded controls. Then, we prove the existence of an optimal control that minimizes a functional cost and we give characterization of such a control. Also we give condition that ensures the uniqueness of the optimal control. The obtained results lead to an algorithm that we illustrate by simulations for a bi-dimensional diffusion system excited by a zone actuator.

A new approach to optimal control of delay systems via Haar wavelets

Mon, 11 Jul 2016 00:00:00 GMT

Abstract
In this paper, a numerical method based on Haar wavelet is proposed for solving the optimal control of time-delay system. The optimality conditions of this problem result to a two-point boundary value problem involving both delay and inverse time terms. The operational matrix of inverse time is introduced and applied for linear inverse time system. Then, the operational matrices of inverse time, integration, delay and product are utilized to reduce the two-point boundary value problem to an algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Proof by model: a new knowledge-based reachability analysis methodology for Petri net

Mon, 11 Jul 2016 00:00:00 GMT

Abstract
To solve the state explosion problem in the reachability analysis of Petri nets, Chao recently broke the NP(nondeterministic polynomial time)-complete barrier by developing the first closed-form solution of the number of Control Related States for the kth-order system. In this paper, we propose a new proof methodology known as proof by model, which is based on the validated information of the reverse net, to simplify and accelerate the construction of the closed-form solution for Petri nets. Here, we apply this methodology to the proof procedure of Top-Right systems with one non-sharing resource placed in the top position of the right-side process. The core theoretical and data basis are that any forbidden (resp. live) state in a Petri net is non-reachable (resp. live) in its reverse net; and the validated information of the Bottom-Right system, the reverse net of Top-Right.

Stability and L1-gain control for positive impulsive switched systems with mixed time-varying delays

Mon, 11 Jul 2016 00:00:00 GMT

Abstract
This article addresses the problems of exponential stability and L1-gain control for a class of positive impulsive switched systems with mixed time-varying delays. A new class of hybrid systems, called positive impulsive delayed switched systems, are introduced for the first time. A necessary and sufficient condition for the positivity of such kind of systems is proposed. Moreover, by using the average dwell time approach and the co-positive Lyapunov–Krasovskii function technique, a sufficient condition is presented to ensure the exponential stability with $L_1$-gain performance of the positive impulsive delayed switched system. Based on the obtained results, a sufficient condition for the existence of a desired state feedback controller is derived, and an iterative convex optimization approach is developed to compute the controller gain matrices. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed method.

The cross-motion invariant group and its application to kinematics

Mon, 11 Jul 2016 00:00:00 GMT

Abstract
This article presents the cross-motion invariant group—CMI(3)—whose group operation is defined over unit dual quaternions such that rigid motions are cross-motion invariant; that is, the resultant translation does not depend on rotation and vice-versa. We present the main properties of CMI(3) and the differences between this group and the standard group $\text{Spin}(3)\ltimes\mathbb{R}^{3}$ of unit dual quaternions, as well as the kinematic equations under a sequence of CMI(3) operations. Two numerical examples are presented in order to illustrate the main characteristics of CMI(3).

A numerical scheme for solving two-dimensional fractional optimal control problems by the Ritz method combined with fractional operational matrix

Fri, 08 Jul 2016 00:00:00 GMT

Abstract
In this article, we present a new method to solve a class of two-dimensional fractional optimal control problems based upon the numerical polynomial approximation. In the proposed method, the fractional derivatives are expressed in the Caputo sense. The approach used here is to approximate the state and control functions by the Legendre orthonormal basis by using the Ritz method. To approximate derivative of the basis, the operational matrix of Caputo derivative is taken into account. Then we apply two-dimensional Legendre–Gauss quadrature rule to approximate double integral of the performance index functional. Next, the problem is converted into an equivalent non-linear unconstrained optimization problem. This problem is solved via the Newton’s iterative method. At last, the convergence of the proposed method is extensively investigated and an example is included to illustrate the effectiveness and applicability of the new procedure.

Event-driven model-free control in motion control with comparisons

Fri, 08 Jul 2016 00:00:00 GMT

Abstract
The event-driven model-free control is proposed in this article, which deals with the ‘trade-off’ between computational cost and system performance. Model-free controllers demand low computational resources and have high robustness, which is especially suitable for embedded systems with complex dynamics and/or affected by disturbances. The proposed method is implemented in two distinct nonlinear Multiple Input, Multiple Output (MIMO) motion models: a quadrotor model hovering in different weather circumstances and a vehicular friction control model operating in variable road surface conditions. Under the time- and event-driven schemes, the model-free control is compared with standard control strategies in various realistic scenarios.

Exponential stabilization and motion planning of an overhead crane system

Mon, 04 Jul 2016 00:00:00 GMT

Abstract
Two control problems for an overhead crane system are investigated: exponential stabilization and motion planning. The considered crane system is modelled by a hyperbolic PDE containing spatially varying terms. These terms influence the spectrum location of the open-loop system and therefore affect the analysis of the control problems under investigation. The first problem is solved in C0-semigroups context using an appropriate Lyapunov functional. The suggested stabilizer is explicitly constructed by means of a collocated-type controller of an auxiliary system combined with a term including the kernel of a suitable Volterre transformation. The second problem is solved using the Laplace transform of an adequate input–output function. Further, the explicit expressions of the motion planner and the corresponding full-state are provided. The two pursued methods are systematic and offer an intrinsic insight about the mechanism of the controller’s construction.