Linear Multivariable Systems Research Articles

Zonotopic guaranteed state estimation for uncertain systems

This paper presents a new approach for guaranteed state estimation based on zonotopes for linear discrete-time multivariable systems with interval multiplicative uncertainties, in the presence of bounded state perturbations and noises. At each sample time, the presented approach computes a zonotope which contains the real system state. A P-radius-based criterion is minimized in order to decrease the size of the zonotope at each sample time and to obtain an increasingly accurate state estimation. The proposed approach allows one to efficiently handle the trade-off between the complexity of the computation and the accuracy of the estimation. An illustrative example is analyzed in order to highlight the advantages of the proposed state estimation technique.

Assigning the set of zeros in control systems with parallel compensation

The problem of ensuring a given set of zeros in a linear multivariable dynamical system with the equal number of inputs and outputs that includes a parallel compensator and feedback loop is considered. Methods reducing this problem to the control of eigenvalues of a certain matrix are proposed; the simultaneous assignment of poles and zeros is reduced to the control of poles of two plants using a single controller. The calculations are based on well-known and well-tested modal control techniques.

Consistency of the extended gradient identification algorithm for multi-input multi-output systems with moving average noises

The consistency of identification algorithms for systems with colored noises is a main topic in system identification. This paper focuses on the extended stochastic gradient (ESG) identification algorithm for the multivariable linear systems with moving average noises. By integrating the noise regression terms and the noise model parameters into the information matrix and the parameter vector, and based on the gradient search principle, the ESG algorithm is presented. The unknown noise terms in the information matrix are replaced with their estimates. The convergence analysis shows that the parameter estimation error converges to zero under a persistent excitation condition. Two simulation examples are given to illustrate the effectiveness of the algorithm.

Reduced Order Observer for Linear Time-Invariant Multivariable Systems with Unknown Inputs

This paper presents the design of a new reduced order observer to estimate the state for a class of linear time-invariant multivariable systems with unknown inputs. The proposed design approach is a combination of the approaches proposed by Hou and Muller (IEEE Trans. Autom. Control 37:871–875, 1992) and Boubaker (Int. J. Autom. Control Syst. Eng. 5:45–51, 2005); matrix decompositions, state transformations, and substitutions based on coordinate changes are used. It is shown that the problem of reduced order observers for linear systems with unknown inputs can be reduced to a standard one (the unknown input vector will not interfere in the observer equations). The effectiveness of the suggested design algorithm is illustrated by a numerical example (aircraft lateral motion), and, for the same aircraft dynamics, we compare our new observer with other already existing observers from the existence conditions and dynamic characteristics point of view; the superiority of the new designed observer is demonstrated.

The Stability Analysis and Planning for Multivariable Linear Time-Delay Systems and its Application in Supervision Oilfield

Based on Lyapunovs theory of stability, analyze on a control process of oil-water separation with time-delay, coupling and uncertainty in the unite station. a equation of sufficient condition for multi-variable Linear Delayed Systems with delayed independent stabilization is derived and another forms are also given. Based on this, several simple criterions for judging independent stabilization from linear delayed system are presented. We also discuss the exponential stability for delayed system and also provide the sufficient condition of exponential stability with any appointed convergent rate and its corresponding deductions. Using these conditions, we can choose a set of suitable parameters to reduce the conservation. By calculating and being compared with methods of the literature, the results show that our methods have less conservation.

Inverse Rendering of Faces with a 3D Morphable Model

In this paper, we present a complete framework to inverse render faces with a 3D Morphable Model (3DMM). By decomposing the image formation process into geometric and photometric parts, we are able to state the problem as a multilinear system which can be solved accurately and efficiently. As we treat each contribution as independent, the objective function is convex in the parameters and a global solution is guaranteed. We start by recovering 3D shape using a novel algorithm which incorporates generalization error of the model obtained from empirical measurements. We then describe two methods to recover facial texture, diffuse lighting, specular reflectance, and camera properties from a single image. The methods make increasingly weak assumptions and can be solved in a linear fashion. We evaluate our findings on a publicly available database, where we are able to outperform an existing state-of-the-art algorithm. We demonstrate the usability of the recovered parameters in a recognition experiment conducted on the CMU-PIE database.

Parametric Jordan Form Assignment Revisited

AbstractThe Jordan form assignment problem (complete and partial) for linear multivariable control system is formulated and completely solved by the unified Sylvester matrix equation approach. A new explicit parametrization of all state feedbacks assigning an admissible Jordan form through the minimum number of parameters is provided. The main advantage of this parametrization, unlike previous ones, is that it is explicit; thus, it can be used, not only for numeric, but also symbolic computations. This fact is demonstrated on the problem of parametrization of all output feedbacks assigning an admissible Jordan form, where Gröbner basis method is applied. Developed algorithms for Jordan form assignment by state and output feedback are implemented in the freely available Maple package.

On the Existence of Controlled Maximal Contractive Sets for Multi Input Linear Systems

In this work, the existence of maximal contractive sets for unstable multi input linear systems is discussed. It is shown that, under suitable algebraic conditions, linear feedback laws can be designed such that the set of values satisfying the saturation constraints is an invariant set for the closed-loop system, which is asymptotically stable.

Characteristic equation for a multivariable linear periodic system with delay

Characteristic equation for a multivariable linear periodic system with delay

Double monitoring of common and specific features for multimode process

ABSTRACTA novel monitoring framework for multimode processes is proposed in this article. Generally, although the different modes have different process characteristics, they still share some common features, and the specific information is the unique characteristic that is not shared by all modes. On the basis of principal component analysis (PCA), a new statistical method, termed as local principal component analysis (LPCA), is developed for modeling multimode process data. LPCA is a modified version of PCA with the idea of maximizing local data variance, which aims to discover some directions that data, from all modes, share comparatively similar variance structure. In this low‐dimensional representation, the common features are captured and the residual corresponding to each mode can be treated as specific features. With the obtained LPCA model, both the common and specific features are utilized for monitoring. Moreover, a Bayesian inference strategy is further adopted to derive global indices of specific features for fault detection. Its feasibility and validity of the proposed method are illustrated through a simple multivariate linear system and the Tennessee Eastman challenge problem. © 2013 Curtin University of Technology and John Wiley & Sons, Ltd.

Extensions in adaptive model tracking with mitigated passivity conditions

Feasibility of nonlinear and adaptive control methodologies in multivariable linear time-invariant systems with state space realization {A, B, C} has apparently been limited by the standard strict passivity (or positive realness) conditions that imply that the product CB must be positive definite symmetric. More recently the symmetry condition has been mitigated, requiring instead that the not necessarily symmetric matrix CB be diagonalizable and with positive real eigenvalues. However, although the mitigated conditions are useful in proving pure stabilizability with Adaptive Controllers, the Model Tracking question has remained open and counterexamples seem to demonstrate total divergence of standard model reference adaptive controllers when the regular passivity conditions are not fully satisfied. Therefore, this paper further extends the previous results, showing that the new passivity conditions do guarantee stability with adaptive model tracking. Examples show how the new conditions solve the case of flexible structures with unknown parameters when perfect collocation is not possible. Also, the so-called counterexamples become simple, well-behaved, examples.

Solving Multilinear Systems via Tensor Inversion

Higher order tensor inversion is possible for even order. This is due to the fact that a tensor group endowed with the contracted product is isomorphic to the general linear group of degree $n$. With these isomorphic group structures, we derive a tensor SVD which we have shown to be equivalent to well-known canonical polyadic decomposition and multilinear SVD provided that some constraints are satisfied. Moreover, within this group structure framework, multilinear systems are derived and solved for problems of high-dimensional PDEs and large discrete quantum models. We also address multilinear systems which do not fit the framework in the least-squares sense. These are cases when there is an odd number of modes or when each mode has distinct dimension. Numerically we solve multilinear systems using iterative techniques, namely, biconjugate gradient and Jacobi methods.

A Design Method of Pole Placement and Observer for Non-Lexicographically-Fixed Linear Time-Varying Discrete MIMO Systems

The linear time-varying multivariable system whose controllability indices or observability indices are not constant is called the non-lexicographically-fixed system. Valasek proposed the pole placement design method for such a continuous system by augmenting the system equation so that the augmented system is lexicographically-fixed. This paper concerns the pole placement and the observer design method for linear time-varying discrete non-lexicographically-fixed system. Using the Valasek's idea, the procedure to augment a discrete non-lexicographically-fixed system to a lexicographically-fixed augmented system will be presented. Then, the simple pole placement technique can be applied to the augmented system without transforming the system into any canonical form. Further, using the property of the anti-causal dual system, it will be shown that the pole placement design method can be used for the augmented observer for non-lexicographically-fixed systems. Finally, the stability and the separation principle of the total closed loop system are also shown for the case where both of the augmented pole placement controller and the augmented observer are used.

Online Identification of Multivariable Discrete Time Delay Systems Using a Recursive Least Square Algorithm

This paper addresses the problem of simultaneous identification of linear discrete time delay multivariable systems. This problem involves both the estimation of the time delays and the dynamic parameters matrices. In fact, we suggest a new formulation of this problem allowing defining the time delay and the dynamic parameters in the same estimated vector and building the corresponding observation vector. Then, we use this formulation to propose a new method to identify the time delays and the parameters of these systems using the least square approach. Convergence conditions and statistics properties of the proposed method are also developed. Simulation results are presented to illustrate the performance of the proposed method. An application of the developed approach to compact disc player arm is also suggested in order to validate simulation results.

Finite Time Model Matching for Multivariable Linear Systems

In this paper, we consider an approximated model matching problem for multivariable linear systems. In this problem, the error on the model tracking converges to zero in a finite time. This means that exact model matching can be recovered after a delay, which is a parameter of our procedure. This solution allows us to extend classical results on model matching, since a solution can always be achieved.

A gametheoretic approach for non-uniform pole shifting and pole homothety

This paper investigates the problems of non-uniform pole shifting or pole homothety in multivariable linear systems by using a gametheoretic approach and more specifically a particular Nash strategy with an open-loop information structure for a game including a time preference rate in the quadratic criteria. Such a result is possible due to properties of non-symmetric Algebraic Riccati equations associated with an open-loop Nash strategy. Numerical examples allow to illustrate the efficiency of the proposed approach.

Invasive weed optimization for model order reduction of linear MIMO systems

In this work, a model order reduction (MOR) technique for a linear multivariable system is proposed using invasive weed optimization (IWO). This technique is applied with the combined advantages of retaining the dominant poles and the error minimization. The state space matrices of the reduced order system are chosen such that the dominant eigenvalues of the full order system are unchanged. The other system parameters are chosen using the invasive weed optimization with objective function to minimize the mean squared errors between the outputs of the full order system and the outputs of the reduced order model when the inputs are unit step. The proposed algorithm has been applied successfully, a 10th order Multiple-Input–Multiple-Output (MIMO) linear model for a practical power system was reduced to a 3rd order and compared with recently published work.

Performance analysis for uncertain multivariable systems obtained by system identification

This article provides a framework for robust performance analysis of linear time-invariant uncertain multivariable systems obtained by classical system identification methods. The performance measures are in terms of H 2 or H ∞ norm of a closed-loop transfer matrix. An upper bound for the performance analysis criterion is computed via an LMI-based optimisation problem. The LFT description is used as a tool for uncertainty modelling. The proposed performance conditions are derived based on using the parameter-dependent Lyapunov functions and are deduced via a parametrisation for the set of multipliers corresponding to the ellipsoidal uncertainty set delivered by system identification procedure. The effectiveness of the proposed analysis approach is demonstrated by a numerical example.

Parameterization of Decoupling Controllers in the Generalized Plant Model

A diagonal decoupling problem in linear multi-variable systems is formulated for the most general setting. A necessary and sufficient condition for the existence of a decoupling controller is presented and, when a solution exists, the class of all decoupling controllers is characterized in terms of free parameters of stable rational vectors.

Norm-bounded integrity conditions of uncertain multivariable linear time-invariant systems under decentralized control with integral action

Decentralized controllers with integral action represent a large portion of multivariable controllers currently employed in the process industry. The integrity of these systems with respect to loop failures has been relatively well researched. In particular, important results have been found with respect to the relative gain array (RGA). However, less has been done on the treatment of the integrity of uncertain systems. This paper considers the integrity conditions of uncertain systems with norm-bounded uncertainties, and extends four existing integrity conditions to these uncertain systems. These are: integral stabilizable (IS), integral controllable (IC), integral controllable with integrity (ICI), and decentralized integral controllability (DIC). For the first three, necessary and sufficient conditions are derived. For the DIC condition, a sufficient condition has been derived. These conditions impose a limit on the norm of the allowed perturbations, below which the closed-loop system is guaranteed to maintain integrity.