Performance Analysis for Discrete-Time Linear Systems with Saturated Linear Feedback: a Nonlinear Saturation-Dependent Gain-Scheduling Approach

In this paper, we propose a new technique for the performance analysis of discrete-time linear systems controlled by a saturated linear control law. Two performance indices, the computation of invariant sets and the L2 performance, are considered. The main contributions of the paper are the following: i) a new linear parameter varying system framework is presented to model the saturated system, ii) a nonlinear saturation-dependent auxiliary feedback matrix is considered, iii) new sufficient conditions for the performance analysis are proposed. It is shown that the conditions can be expressed as a set of linear matrix inequalities. Furthermore, it is shown that the conditions are guaranteed to be less conservative than existing solutions in the literature. Three numerical examples are presented to illustrate the effectiveness of the proposed method. <br>

Performance Analysis for Discrete-Time Linear Systems with Saturated Linear Feedback: a Nonlinear Saturation-Dependent Gain-Scheduling Approach

In this paper, we propose a new technique for the performance analysis of discrete-time linear systems controlled by a saturated linear control law. Two performance indices, the computation of invariant sets and the L2 performance, are considered. The main contributions of the paper are the following: i) a new linear parameter varying system framework is presented to model the saturated system, ii) a nonlinear saturation-dependent auxiliary feedback matrix is considered, iii) new sufficient conditions for the performance analysis are proposed. It is shown that the conditions can be expressed as a set of linear matrix inequalities. Furthermore, it is shown that the conditions are guaranteed to be less conservative than existing solutions in the literature. Three numerical examples are presented to illustrate the effectiveness of the proposed method. <br>

Adaptive Kalman Filter with L2 Feedback Control for Active Suspension Using a Novel 9-DOF Semi-Vehicle Model

In order to further improve driving comfort, this paper takes the semi-vehicle active suspension as the research object. Furthermore, combined with a 5-DOF driver-seat model, a new 9-DOF driver seat-active suspension model is proposed. The adaptive Kalman filter combined with L2 feedback control algorithm is used to improve the controller. First, a discrete 9-DOF driver seat-active suspension model is established. Then, the L2 feedback algorithm is used to solve the optimal feedback matrix of the model, and the adaptive Kalman filter algorithm is used to replace the linear Kalman filter. Finally, the improved active suspension model and algorithm are verified through simulation and test. The results show that the new algorithm and model not only significantly improve the driver comfort, but also comprehensively optimize the other performance of the vehicle. Compared with the traditional LQG control algorithm, the RMS value of the acceleration experienced by the driver’s limb are, respectively, decreased by 10.9%, 15.9%, 6.4%, and 7.5%. The RMS value of pitch angle acceleration experienced by the driver decreased by 6.4%, and the RMS value of the dynamic tire deflection of front and rear tire decreased by 32.6% and 12.1%, respectively.

Spectral analysis of Euler–Bernoulli beam model with distributed damping and fully non-conservative boundary feedback matrix

The distribution of natural frequencies of the Euler–Bernoulli beam resting on elastic foundation and subject to an axial force in the presence of several damping mechanisms is investigated. The damping mechanisms are: ( i ) an external or viscous damping with damping coefficient ( − a 0 ( x )), ( ii ) a damping proportional to the bending rate with the damping coefficient a 1 ( x ). The beam is clamped at the left end and equipped with a four-parameter (α, β, κ 1 , κ 2 ) linear boundary feedback law at the right end. The 2 × 2 boundary feedback matrix relates the control input (a vector of velocity and its spacial derivative at the right end) to the output (a vector of shear and moment at the right end). The initial boundary value problem describing the dynamics of the beam has been reduced to the first order in time evolution equation in the state Hilbert space of the system. The dynamics generator has a purely discrete spectrum (the vibrational modes). Explicit asymptotic formula for the eigenvalues as the number of an eigenvalue tends to infinity have been obtained. It is shown that the boundary control parameters and the distributed damping play different roles in the asymptotical formulas for the eigenvalues of the dynamics generator. Namely, the damping coefficient a 1 and the boundary controls κ 1 and κ 2 enter the leading asymptotical term explicitly, while damping coefficient a 0 appears in the lower order terms.

SVD-Krylov based Sparsity-preserving Techniques for Riccati-based Feedback Stabilization of Unstable Power System Models

We propose an efficient sparsity-preserving reduced-order modelling approach for index-1 descriptor systems extracted from large-scale power system models through two-sided projection techniques. The projectors are configured by utilizing Gramian based singular value decomposition (SVD) and Krylov subspace-based reduced-order modelling. The left projector is attained from the observability Gramian of the system by the low-rank alternating direction implicit (LR-ADI) technique and the right projector is attained by the iterative rational Krylov algorithm (IRKA). The classical LR-ADI technique is not suitable for solving Riccati equations and it demands high computation time for convergence. Besides, in most of the cases, reduced-order models achieved by the basic IRKA are not stable and the Riccati equations connected to them have no finite solution. Moreover, the conventional LR-ADI and IRKA approach not preserves the sparse form of the index-1 descriptor systems, which is an essential requirement for feasible simulations. To overcome those drawbacks, the fitting of LR-ADI and IRKA based projectors from left and right sides, respectively, desired reduced-order systems attained. So that, finite solution of low-rank Riccati equations, and corresponding feedback matrix can be executed. Using the mechanism of inverse projection, the Riccati-based optimal feedback matrix can be computed to stabilize the unstable power system models. The proposed approach will maintain minimized ℌ2 -norm of the error system for reduced-order models of the target models.

Virtual Sensors in the Fault Diagnosis Problem

The paper is devoted to the problem of fault diagnosis (isolation and identification) in linear dynamic systems under disturbances. The performances of fault diagnosis depend on the sensors which are in the system under diagnosis. To improve the performances, additional sensors can be applied. But sometimes it is impossible to use such sensors; besides they have low reliability. In this paper, we suggest to use so-called virtual sensors instead of additional ones. To obtain such sensors,Luenberger observers can be used. Such an observer is designed in two steps. On the first step, the model of minimal dimension invariant with respect to the disturbances and estimating a predetermined component of the system state vector and some other components of the system state vector is designed. The second components are necessary to provide stability of the observer by means of generating residual and using feedback. Such components are determined during t he process of the problem solution which is based on the canonical form of matrices describing the model. On the second step, the feedback matrix is found based on the required quality of transient. To obtain this matrix, eigenvalues are selected and coefficients of the characteristic equation are calculated. The rule to find the predetermined component of the system state vector to be estimated by vir tual obser ver is suggested. Theoretical results are illustrated by practical example of well known three tank system.

Method for the transformation of complex automatic control systems to integrable form

The article considers a class of automatic control systems that is described by a multi- dimensional system of ordinary dil'erential equations. The right hand-side of the system additively contains a linear part and the product of a control matrix by a vector that is the sum of a control vector and an external perturbation vector. The control vector is defined by a nonlinear function dependent on the product of a feedback matrix by a vector of current coordinates. The authors solve the problem of constructing a matrix of a nonsingular transformation, which leads the matrix of the linear part of the system to the Jordan normal form or the first natural normal form. The variables included in this transformation allow us to vary the system settings, which are the parameters of both the control matrix and the feedback matrix, as well as to convert the system to an integrable form. Integrable form is understood as a form in which the system can be integrated in a final form or reduced to a set of subsystems of lower orders. Furthermore, the sum of the subsystem orders is equal to the order of the original system. In the article, particular attention is paid to cases when the matrix of the linear part has complex conjugate eigenvalues, including multiple ones.

Feedback control of chaotic systems using multiple shooting shadowing and application to Kuramoto–Sivashinsky equation

We propose an iterative method to evaluate the feedback control kernel of a chaotic system directly from the system’s attractor. Such kernels are currently computed using standard linear optimal control theory, known as linear quadratic regulator theory. This is however applicable only to linear systems, which are obtained by linearizing the system governing equations around a target state. In the present paper, we employ the preconditioned multiple shooting shadowing (PMSS) algorithm to compute the kernel directly from the nonlinear dynamics, thereby bypassing the linear approximation. Using the adjoint version of the PMSS algorithm, we show that we can compute the kernel at any point of the domain in a single computation. The algorithm replaces the standard adjoint equation (that is ill-conditioned for chaotic systems) with a well-conditioned adjoint, producing reliable sensitivities which are used to evaluate the feedback matrix elements. We apply the idea to the Kuramoto–Sivashinsky equation. We compare the computed kernel with that produced by the standard linear quadratic regulator algorithm and note similarities and differences. Both kernels are stabilizing, have compact support and similar shape. We explain the shape using two-point spatial correlations that capture the streaky structure of the solution of the uncontrolled system.