THE INFLUENCE OF PARAMETERS ON STABILIZATION FOR HOMOGENEOUS POLYNOMIAL DYNAMICAL SYSTEMS IN THE PLANE

In this paper the robust positivity of polynomials under coefficient perturbation is investigated. This robust positivity of polynomials can be used for polynomial systems to solve a large number of problems in robust control, nonlinear contol, convex and non-convex optimization such as determining the robust asymptotic stability of a polynomial system. In this article it is assumed that the polynomials under investigation depend linearly on some parameters. The aim is to determine the whole parameter perturbation region, for which the polynomial is globally positive. The theorem of Ehlich and Zeller is used to achieve this aim. This theorem enables us to give conditions in the parameter space for global positivity. These conditions are linear inequalities. By means of these inequalities an inner and an outer approximation are calculated to the relevant perturbation region. Two nontrivial examples conclude the paper and show the effectiveness of the presented method

It is well known that most analysis and design problems in robust and nonlinear control can be formulated as global optimization problems with polynomial objective functions and constraints. The objective of this article is to show how GloptiPoly can solve challenging nonconvex optimization problems in robust nonlinear control. GloptiPoly is a general-purpose software with a user-friendly interface. First, we describe several nonconvex optimization problems arising in control system analysis and design. These problems involve multivariable polynomial objective functions and constraints. We then review the theoretical background behind GloptiPoly. Next, an example is presented to illustrate the successive linear matrix inequality relaxations and general features of the GloptiPoly software. Finally, we apply GloptiPoly to several control-related problems.

The problem of robust tracking control of electromechanical systems has been studied and solved by many different approaches within the robot control community (see e.g. Sage et al., 1999; Kelly et al., 2005 and references therein) in order to ensure accurate motion in typical industrial tasks (painting, navigation, cutting, etc). In the last decade, the homogeneity approach attracted considerable interest from the research and engineering communities (see e.g. Lebastard et al., 2006; Ferrara et al., 2006; Bartolini et al., 2006) because it was demonstrated that homogeneous systems with homogeneity degree 0 < η exhibit robustness and finite-time convergence properties (Bhat & Bernstein, 1997; Hong et al., 2001; Orlov, 2005). Control laws based on the homogeneity approach (Bhat & Bernstein, 1997; Hermes, 1995; Orlov, 2003a; Rosier, 1992) are attractive in robotic applications because they can cope with many mechanical perturbations, including external vibrations, contact forces, and nonlinear internal phenomena such us Coulomb and viscous friction, dead zone and backlash, while it is possible to ensure exact tracking to continuously differentiable desired trajectories. Several homogeneous controllers and studies have been proposed in the literature. For example, Rosier (1992) constructed a homogeneous Lyapunov function associated with homogeneous dynamic systems. Hermes (1995) addressed the homogeneous stabilization control problem for homogeneous systems. Bhat and Bernstein (1997) examined the finite time stability of homogeneous systems. Levant (2005a, 2005b) developed robust output-feedback high-order sliding mode controllers that demonstrate finite-time convergence (see also (Fridman & Levant, 1996; Fridman & Levant, 2002)) where the controller design is based on homogeneity reasoning while the accuracy is improved in the presence of switching delay, and the chattering effect is treated by increasing the relative degree. Orlov et al. (2003a, 2003b) proposed applying homogeneous controller to solve the set-point problem dealing with mechanical imperfections such as Coulomb friction, viscous friction, and backlash. Orlov et al., (2005) extended the finite time stability analysis to nonlinear nonautonomous switched systems.

Robust nonlinear feedback control for uncertain linear systems with nonquadratic performance criteria

One of the main applications of smart structiires is in the vibration control of flexible structures using smart sensors and actuators. A commonly used smart sensors and actuators are piezoelectric senors and actuators. The control schemes developed so far for this application have assumed a perfect structure with out any flaws as the plant model. But in general, there are discrepancies between the mathematical model and the physical system. Such discrepancies are due to damages or flaws. The controller designed for a perfect structure can not be used for the damaged structure. The result will be a poor performance or a system instability. Hence a robust controller is required to account for flaws or damages . Here delamination is assumed to be the damage in a composite beam The effects of delaminations are considered as uncertainities. p synthesis is used to develop a robust controller. *Graduate Student, Student Member AIAA tProfessor,Member AIAA '@copyright c 1991 by the American Institute of Aeronautics and Astronautics,Inc. All rights reserved INTRODUCTION During the past several years, there has been a considerable amount of research activity to use bonded or embedded piezoceramic (or PVDF) sensors and actuators to control vibration in light weight structures. Bonded piezoceramic sensors and detection circuits can be designed such that the rate of deformation of a beam structure will result in a signal that is proportional to the difference of the slope rate a t the two ends of the transducer1. Similar results can be derived for voltage time histories as a function of the deformation rates of other types of structures. The detected signal can be conditioned by operations such as filtering, phase shift and amplification. The conditioned signals are used as inputs to bonded or embedded piezoceramic actuators to transmit energy to the structure. The objective of operations of sensing, conditioning and feed back to selected actuators is to design a vibration controller to the structure. An early application of piezoelectric transducers to active control of structures has been attributed to 01sen2 . Other reported early applications of piezoceramic transducers to vibration control, that followed Olsen's work are due to orw ward^, Forward and Liu4, Forward and Swigert5, Forward,Swigert and 0ba16,Hanagud and 0ba17~8, Crawley and ~ e L u i s @ and ~ansonl ' . Similarly, early work in the area of vibration control by using PVDF film is by Bailey and Hubbard''. Since then, there has been an explosion in the research activity on using piezoceramic transducers, PVDF films, Shape memory alloys, electrostricive transducers, voice coil actuators and electro rheological fluids to control vibrations. Jet propulsion laboratory has demonstrated the use of piezoceramic transducer in the control of vibrations in a space truss like structure. Hanagud and his colleagues have used artificial intelligence techniques to control vibrations in a time varying adaptive structures l2%l3and health monitoring of structures14. In a recent work Hanagud15 and ~ a n s o n l ~ have discussed robust vibration control by the use of piezoceramic sensors and actuators. Robustness has been addressed to account for unmodelled dynamics15116 and debonded sensors 15. However , none of the reported work has addressed the problem of robust control that is needed to account for inaccurate sensor output due to imperfections and flaws in structures. This problem has been addressed in this paper. ROBUST CONTROL PROBLEM SETTING We are considering the problem of designing a vibration controller for a flexible structure. In particular, we are focussing our attention on the problem of vibration control of layered composite beams subjected to a prescribed set of disturbances. The problem of vibration suppression is being accomplished using a set of piezoceramic sensors and actuators which are bonded to the structure a t seFigure 1: Effect of delamination on sensor sig-

Robust stability of uncertain dynamic systems has major importance when real world system models are considered. A realistic approach has to consider uncertainties of various kinds in the system model. Uncertainties due to inherent modelling/identification inaccuracies in any physical plant model specify a certain uncertainty domain, e.g. as a set of linearized models obtained in different working points of the plant considered. Thus, a basic required property of the system is its stability within the whole uncertainty domain denoted as robust stability. Robust control theory provides analysis and synthesis approaches and tools applicable for various kinds of processes, including multi input – multi output (MIMO) dynamic systems. To reduce multivariable control problem complexity, MIMO systems are often considered as interconnection of a finite number of subsystems. This approach enables to employ decentralized control structure with subsystems having their local control loops. Compared with centralized MIMO controller systems, decentralized control structure brings about certain performance deterioration, however weighted against by important benefits, such as design simplicity, hardware, operation and reliability improvement. Robustness is one of attractive qualities of a decentralized control scheme, since such control structure can be inherently resistant to a wide range of uncertainties both in subsystems and interconnections. Considerable effort has been made to enhance robustness in decentralized control structure and decentralized control design schemes and various approaches have been developed in this field both in time and frequency domains (Gyurkovics & Takacs, 2000; Zecevic & Siljak, 2004; Stankovic et al., 2007). Recently, the algebraic approach has gained considerable interest in robust control, (Boyd et al., 1994; Crusius & Trofino, 1999; de Oliveira et al., 1999; Ming Ge et al., 2002; Grman et al., 2005; Henrion et al., 2002). Algebraic approach is based on the fact that many different problems in control reduce to an equivalent linear algebra problem (Skelton et al., 1998). By algebraic approach, robust control problem is formulated in algebraic framework and solved as an optimization problem, preferably in the form of Linear Matrix Inequalities (LMI). LMI techniques enable to solve a large set of convex problems in polynomial time (see Boyd et al., 1994). This approach is directly applicable when control problems for linear uncertain systems with a convex uncertainty domain are solved. Still, many important control problems even for linear systems have been proven as NP hard, including structured linear control problems such as decentralized control and simultaneous static output feedback (SOF) designs. In these cases the prescribed structure of control feedback matrix (block diagonal for decentralized control) results in nonconvex problem formulation. There

The paper deals with the problem of robust speed control of electrical servodrives. A robust controller is developed using a nonlinear IP controller. The controller nonlinear characteristic is obtained due to fuzzy logic technique application. An original method of controller settings adjustment is presented. The use of this adjustment procedure ensures robust speed control against the variations of the moment of inertia and a step change of load torque. Simulations and laboratory results validate the robustness of the servodrive with permanent magnet synchronous motor.

In this article, the robust Stackelberg controllability (RSC) problem is studied for a nonlinear fourth-order parabolic equation, namely the Kuramoto–Sivashinsky equation. When three external sources are acting into the system, the RSC problem consists essentially in combining two subproblems: the first one is a saddle point problem among two sources. Such sources are called the “follower control” and its associated “disturbance signal.” This procedure corresponds to a robust control problem. The second one is a hierarchic control problem (Stackelberg strategy), which involves the third force, so-called leader control. The RSC problem establishes a simultaneous game for these forces in the sense that the leader control has as objective to verify a controllability property, while the follower control and perturbation solve a robust control problem. In this paper, the leader control obeys to the exact controllability to the trajectories. Additionally, iterative algorithms to approximate the robust control problem as well as the robust Stackelberg strategy for the nonlinear Kuramoto–Sivashinsky equation are developed and implemented.

In this paper we develop a unified framework to address the problem of optimal nonlinear robust control. Specifically, we transform a given robust control problem into an optimal control problem by properly modifying the cost functional to account for the system uncertainty. As a consequence, the resulting solution to the modified optimal control problem guarantees robust stability and performance for a class of nonlinear uncertain systems. The overall framework generalizes the classical Hamilton-Jacobi-Bellman conditions to address the design of robust optimal controllers for uncertain nonlinear systems.

This paper deals with the problem of robust speed control of electrical servodrives. A robust controller is developed using a nonlinear PI controller. The controller nonlinear characteristics is obtained due to fuzzy logic technique application. An original method of controller settings adjustment is presented. The use of this adjustment procedure ensures robust speed control against the variations of the moment of inertia. Simulations and laboratory results validate the robustness of the servodrive with Permanent Magnet Synchronous Motor

We consider the robust (H???) control problem for linear time-invariant systems over unreliable communication channels that are subject to packet losses from sensors to the controller and from the controller to the actuators. Moreover, the acknowledgments of control packet transmissions are also sent over reverse unreliable communication channels, which is known as the quasi-TCP model. The robust control problem is formulated within a stochastic zero-sum dynamic game framework, from which we obtain an H??? controller that minimizes the worst-case cost function in the presence of an adversary in the dynamical system. We characterize a set of existence conditions of the H??? controller in terms of the disturbance attenuation parameter ?? and packet loss rates. For the least disturbance attenuation scenario (as ?? ??? 8), we show that the H??? control system converges to the corresponding LQG system. We also show that the H??? controller for the TCP- and UDP-cases can be arrived at as appropriate limits of the solution for the quasi-TCP problem. Numerical examples are presented to illustrate the theoretical results.

In this work we consider robust stabilization of uncertain dynamical systems and show that this can be achieved by solving a non-classically constrained analytic interpolation problem. In particular, this non-classical constraint confines the range of the interpolant, when evaluated on the imaginary axis, to a frequency-dependent set. By considering a sufficient condition for when this interpolation problem has a solution, we derive an approximate solution algorithm that can also be used for controller synthesis. The conservativeness of the method is reduced by introducing a shift, which can be tuned by the user. Finally, the theory is illustrated on a numerical example with a plant with uncertain gain, phase, and output delay.

This paper considers robust control problems for a 3D space robot of two rigid bodies connected by a universal joint with an initial angular momentum. It is particularly difficult to measure an initial angular momentum in parameters of space robots since the value of an initial angular momentum depends on the situations. Hence, the main purpose of this paper is to develop a robust controller with respect to initial angular momenta for the 3D space robot. First, a mathematical model, some characteristics, and two types of control problems for the 3D space robot are presented. Next, for the robust attitude stabilization control problem of the 3D space robot, a numerical simulation is performed by using the nonlinear model predictive control method. Then, for the robust trajectory tracking control problem of the 3D space robot, another numerical simulation is carried out. As a result, it turns out that this approach can realize robust control on initial angular momenta for the two control problems. In addition, computation amount is reduced by this approach and real-time control of the 3D space robot can be achieved.

In this paper, we consider robust control problems which are governed by a two-dimensional phase-field model of Warren—Boettinger, which describes the solidification of a binary alloy, in order to take into account the influence of noises in the data. Firstly, the robust control problems are formulated. Afterwards, the existence and uniqueness of the solution of the perturbation (of the nonlinear governing system of equations) is proved as well as stability under mild assumptions. We prove the existence of an optimal solution, and we also find necessary optimality conditions.

Robust and adaptive control: fidelity or an open relationship?