Systems In Descriptor Form Research Articles

On the SPRification of Linear Descriptor Systems via Output Feedback

We consider input–output systems in descriptor form and ask when such systems can be rendered strictly positive real (SPR) via output feedback. Time- and frequency-domain conditions are given to determine when and how this is possible. In addition, a synthesis procedure for controller design is also derived. Together, the results provide a complete answer to when a linear descriptor system can be made SPR via output feedback, and provide a recipe for design of a feedback controller when one exists.

Adaptive control design with S-variable LMI approach for robustness and L2 performance

ABSTRACTMatrix inequality-based results are proposed for the design of direct adaptive control. The focus is on guaranteeing robustness of the closed-loop when facing both structured parametric uncertainties and input disturbances. Results apply to systems in descriptor form with polytopic uncertainties and readily extend to systems rational in the uncertain parameters. The exposed methodology is an extension of previous studies started in cooperation with Alexander L. Fradkov. These studies aim at applying up-to-date linear robust control tools for passivity-based adaptive control. The paper contributes to this framework with new parameter-dependent Lyapunov functions and new degrees of freedom in the adaptive laws offering the possibility to design controllers with improved closed-loop performances.

Robust adaptive magnetic control of satellites with uncertain parameters

Recently developed results for direct adaptive control design in the context of uncertain systems in descriptor form are discussed and applied to a satellite attitude control problem. Thanks to descriptor model handling the methodology allows to address with convexity arguments uncertainties on the inertia matrix. The design of the adaptive law is done with semi-definite programming tools (LMI solvers) and is applicable with the sole knowledge of a controller stabilizing the nominal system (no passivity assumptions). The tackled satellite model is linear (assuming small pointing errors), full 3D rotations are considered (axes are coupled), and mixed reaction wheel and magnetotorquer actuation is adopted. Compared to the provided initial LTI control law, adaptive control increases robustness margins by a factor 262%.

Stabilization for Damping Multimachine Power System with Time-Varying Delays and Sector Saturating Actuator

This paper studies the stabilization problem for damping multimachine power system with time-varying delays and sector saturating actuator. The multivariable proportional plus derivative (PD) type stabilizer is designed by transforming the problem of PD controller design to that of state feedback stabilizer design for a system in descriptor form. A new sufficient condition of closed-loop multimachine power system asymptomatic stability is derived based on the Lyapunov theory. Computer simulation of a two-machine power system has verified the effectiveness and efficiency of the proposed approach.

Approximate Bisimulation-Based Reduction of Power System Dynamic Models

In this paper we propose approximate bisimulation relations and functions for reduction of power system dynamic models in differential-algebraic (descriptor) form. The full-size dynamic model is obtained by linearization of the nonlinear transient stability model. We generalize theoretical results on approximate bisimulation relations and bisimulation functions, originally derived for a class of constrained linear systems, to linear systems in descriptor form. An algorithm for transient stability assessment is proposed and used to determine whether the power system is able to maintain the synchronism after a large disturbance. Two benchmark power systems are used to illustrate the proposed algorithm and to evaluate the applicability of approximate bisimulation relations and bisimulation functions for reduction of the power system dynamic models.

Port Hamiltonian System in Descriptor Form for Balanced Reduction: Application to a Nanotweezer

This paper proposes a method of balanced model reduction for constrained linear port Hamiltonian systems. Constrained linear port Hamiltonian systems are first written in a canonical descriptor form such that the Hamiltonian structure is preserved. The computations of the controllability and observability Gramians are then used to derive the balanced port Hamiltonian representation of the system. The method of flow constraint is applied to reduce the system. Finally, numerical simulations for the reduction of a micro mechanical actuator model is given to illustrate the effectiveness of the proposed method.

Observability analysis and sensor location study for structured linear systems in descriptor form with unknown inputs

This paper deals with the state and input right-hand side observability and the sensor location problem to ensure this property for structured linear systems in the descriptor form. The proposed method is based on a graph-theoretic approach. It is applicable to a large class of descriptor systems including regular and nonregular systems even if they are underdetermined, overdetermined or square. After defining a bipartite graph which represents such systems, we deduce some graphic necessary and sufficient conditions to achieve the right-hand side observability of the state and the input. Next, we propose a sensor placement procedure to recover this property. More precisely, we provide the minimal number of the sensors required and either their pertinent location or a necessary and sufficient condition to check if a given sensor configuration is adequate or not to ensure state and input right-hand side observability. Our approach uses algorithms with polynomial complexity orders and assumes only the knowledge of the system’s structure and so is suited for large scale systems.

ℋ∞ Observers design for a class of nonlinear time-delay systems in descriptor form

In this article, we address the problem of designing an observer for a class of nonlinear time delay systems in descriptor form, in both discrete-time and continuous-time cases. We propose an observer design method using a particular Lyapunov function that leads to solve a linear matrix inequality condition ensuring the ℋ∞ convergence of the estimation error. Indeed, we use a Lyapunov function depending on the disturbances in order to avoid a specific obstacle in the stability analysis. Finally, three numerical examples are provided in order to show the validity of the proposed method.

Multivariable fractional order PID controller design via LMI approach

AbstractIn this paper, we study the design problem of multivariable fractional-order (FO) PID controllers which guarantee the stability of the FO closed-loop systems. The idea is to transform the problem of FO PID controller design to that of static output feedback (SOF) controller design for an FO system in descriptor form. In terms of linear matrix inequalities, the PID gains can be found once the SOF matrix is obtained. Finally, a numerical example is provided to illustrate the effectiveness of the proposed method.

Robust H2 perfomance of discrete-time periodic systems: LMIs with reduced dimensions

Recent papers in the field of LMI-based robust control have provided extensions of known results for linear time-invariant systems to the case of periodically time varying linear systems. These results, theoretically satisfactory because formulated in terms of optimization problems of polynomial complexity, may still have limited applications in practice because the number of variables and constraints is very large. The present paper proposes a new formulation of these results that allows to reduce the computational burden both by reducing the number of decision variables and the size of the constraints. Along with this numerical improvement, the paper produces a new modeling of periodic discrete-time systems in descriptor form that is believed promising for future research.

Observability of structured linear systems in descriptor form: A graph-theoretic approach

This paper deals with the right-hand side observability of structured linear systems in descriptor form. The method we propose is based on a graph-theoretic approach. It is applicable to a large class of descriptor systems including regular and nonregular systems even if they are square, underdetermined or overdetermined. After defining digraphs which represent such systems, we express in graphic terms necessary and sufficient conditions to achieve the observability of the whole or of a given part of the state. These conditions are easy to check because they are based on comparison of integers and on finding edges and subgraphs in a digraph. Therefore, our approach is suited to study large scale and/or uncertain systems.

INPUT AND STATE FUNCTIONAL OBSERVABILITY FOR DESCRIPTOR SYSTEMS

This paper deals with the right-hand side observability of a linear function of the state and the unknown inputs for linear systems in descriptor form. For such systems, necessary and sufficient conditions ensuring the ability of reconstructing any given functional of the state and unknown inputs are determined by means of geometrical tools like subspace sequences.

An improvement on multivariable PID controller design via iterative LMI approach

This note improves the results of Zheng, Wang, and Lee (Automatica 38 (2002) 517) on the design of multivariable PID controllers via LMI approach. The idea of the improvement is to transform the problem of PID controller design to that of static output feedback (SOF) controller design for a system in descriptor form. The merits of the present result are: (1) The present condition removes the invertibility constraint on the controllers in Zheng et al. (2002) and incorporates it to the matrix inequality to be solved. (2) Once the SOF matrix is obtained, the PID gains are ready in hand and no additional computation is needed. (3) The present approach can deal with the case of some system parameters subject to perturbations.

Aspects of Modeling Dynamical Systems by Differential-Algebraic Equations

In recent years the analysis and synthesis of control systems in descriptor form has been established. The general description of dynamical systems by differential-algebraic equations (DAE) is important for many applications in mechanics and mechatronics, in electrical and electronic engineering, and in chemical engineering as well. In this contribution the pros and cons of system modelling by differential-algebraic equations are discussed and an actual state of the art of descriptor systems is presented. Firstly, the advantages of modelling are touched in general and illustrated in detail by Lagranges equations of first kind, by subsystem modeling and by the statement of the tracking control problem. Secondly, the development of tools for numerical integration is discussed resulting in the comment that today stable and efficient DAE solvers exist and that the simulation of descriptor systems is not a problem any longer. Thirdly, the methods of analyzing and designing descriptor systems are considered. Here, linear and nonlinear systems have to be distinguished. For linear descriptor systems more or less the required methods to solve usual control tasks are available in principal. But actually a related program package for fast and reliable application of these methods is still missed. However, in the near future such a toolbox is expected. Main difficulties arise for nonlinear problems. A few results on stability and optimal control are known only and still a lot of research work has to be effected. In spite of these deficiencies, all over the descriptor system approach is very attractive for modeling and simulation, and will become attractive more and more for analysis and design.

Descriptor systems: pros and cons of system modelling by differential-algebraic equations

In recent years the analysis and synthesis of control systems in descriptor form has been established. The general description of dynamical systems by differential-algebraic equations (DAE) is important for many applications in mechanics and mechatronics, in electrical and electronic engineering, and in chemical engineering as well. In this contribution the pros and cons of system modelling by differential-algebraic equations and an actual state of the art of simulation, analysis and design of descriptor systems is presented.

Linear-quadratic optimal control of descriptor systems

In recent years the analysis and synthesis of (mechanical) control systems in descriptor form has been established. This general description of dynamical systems is important for many applications in mechanics and mechatronics, in electrical and electronic engineering, and in chemical engineering as well. This contribution deals with linear mechanical descriptor systems and its control design with respect to a quadratic performance criterion. Here, the notion of properness plays an important role whether the standard Riccati approach can be applied as usual or not. Properness and non-properness distinguish between the cases if the descriptor system is exclusively governed by the control input or by its higher-order time-derivatives additionally. In the unusual case of non-proper systems a quite different problem of optimal control design has to be considered. Both cases will be solved completely.

Linear control design of linear descriptor systems

In recent years the analysis and synthesis of control systems in descriptor form has been established. The general description of dynamical systems by differential algebraic equations is important for many applications in mechanics and mechatronics, in electrical and electronic engineering, and in chemical engineering as well. This contribution deals with linear descriptor systems and its control design with respect to a quadratic performance criterion and based on a generalized Lu-enberger observer. Here, properness plays an important role whether usual design algorithms can be applied or not. The distinction of proper and non-proper systems is necessary to distinguish descriptor systems which are exclusively governed by the control input or by its higher-order time-derivatives additionally. In the unusual case of non-proper systems a quite different problem of optimal control design and observer realization has to be considered. Both cases will be solved completely.

Augmented lagrangian and mass-orthogonal projection methods for constrained multibody dynamics

This paper presents a new method for the integration of the equations of motion of constrained multibody systems in descriptor form. The method is based on the penalty-Augmented Lagrangian formulation and uses massorthogonal projections for the solution to satisfy the kinematic constraint conditions. The number of equations being solved is equal to the number of states, and does not depend on the number of constraint conditions. Therefore, the method is particularly suitable for systems with redundant constraints, singular configurations or topology changes. The major advantage of the new method relies on the fact that for a low computational cost, the constraints in positions, velocities and accelerations are satisfied to machine precision during the numerical integration. This process is efficiently done by means of a mass-orthogonal projection without the need for coordinate partitioning or reduction to a minimum set of coordinates. The projection scheme allows for a more accurate and robust integration of the equations of motion since constraint violations constitute one of the primary sources of numerical errors and instabilities during the integration process. The proposed projection is also applied to the classical Lagrangian approach, thus eliminating the need for further stabilization as well as the selection of parameters in Baumgarte's method.

Decentralized Control of Mechatronic Systems in Descriptor Form

A decentralized controller design procedure is developed for a special class of mecha-tronic systems in descriptor form using linear quadratic methods. The decentralized system consists of mechanical subsystems with constraint equations up to index 3. Interconnections between the subsystems are limited to be of index 1. Under the assumption of regularity and causality decomposition of the global descriptor system into its Kronecker-canonical form leads to the well known decentralized LQ control problem for normal state-space systems. The advantage of having direct influence on the interconnection variables in the performance index is demonstrated in an example.

Graph-theoretic Approach to the Generic Structure of Zeros and Poles of Large-scale Systems in Descriptor form

For large-scale systems, the state-space description in standard form may not be regarded as an appropriate system representation in most cases. System representations in descriptor form (1), (2) are better suited. However, there may arise new phenomena such as poles and decoupling zeros at infinity. This paper deals with a graphtheoretic approach for investigating poles and zeros of systems in descriptor form. A definition of poles and zeros in terms of the originally given system matrices is a prerequisite of their graphical interpretation (Section 2). Then, the concept of structure matrices and a few graphtheoretic conventions are explained. Finally, important structural properties of zeros and poles are graph-theoretically characterized.