Differential-algebraic Representation Research Articles

Semi-analytical adaptive guidance computation for autonomous planetary landing

A novel algorithm for autonomous landing guidance computation is presented. The trajectory is expressed in polynomial form of minimum order to satisfy a set of 17 boundary constraints, depending on 2 parameters: time-of-flight and initial thrust magnitude. The consequent control acceleration is expressed in terms of differential algebraic (DA) variables, expanded around the point of the domain along the nominal trajectory followed at the retargeting epoch. The DA representation of the objective and constraints gives additional information about their sensitivity to variations of the optimization variables, which is exploited to find the desired fuel minimum solution (if existing) robustly and with a very light computational effort.

A convex optimization based solution for the robotic manipulator control design problem subject to input saturation

This paper proposes a state-feedback design procedure for robotic manipulator systems with saturating actuators, a solution which is based on convex optimization subject to constraints in the form of linear matrix inequalities. Our fundamental idea is to express the system dynamics in a novel differential-algebraic representation with state-derivative components. This approach allows us to provide a systematic control design framework with formal theoretical guarantees, such as the asymptotic stabilization of the manipulator attitude reference error within a prescribed exponential decay-rate. Our method is capable of dealing with the nonlinearities of a mechanical manipulator system, including the input saturation effect, without relying on any kind of linearization or approximation. A two-link planar robotic manipulator example is employed in order to illustrate the proposed approach.

Reduced linear fractional representation of nonlinear systems for stability analysis

Based on symbolic and numeric manipulations, a model simplification technique is proposed in this paper for the linear fractional representation (LFR) and for the differential algebraic representation introduced by Trofino and Dezuo (2013). This representation is needed for computational Lyapunov stability analysis of uncertain rational nonlinear systems. The structure of the parameterized rational Lyapunov function is generated from the linear fractional representation (LFR) of the system model. The developed method is briefly compared to the n-D order reduction technique known from the literature. The proposed model transformations does not affect the structure of Lyapunov function candidate, preserves the well-posedness of the LFR and guarantees that the resulting uncertainty block is at most the same dimensional as the initial one. The applicability of the proposed method is illustrated on two examples.

Semidefinite Programming Solution to the Spacecraft Analysis and Control Problem

This paper provides a semidefinite optimization solution to the analysis and control of a rigid rotating spacecraft. Due to the quaternion representation of a rotating body, the nonlinearities that describe the system are polynomial in nature and may be expressed in the Differential Algebraic Representation (DAR). Once the system is in this form, it is possible to apply linear-based tools for the analysis and control design of the closed loop dynamics. The task is then reduced to that of a semidefinite optimization problem subject to constraints in the form of Linear Matrix Inequalities (LMIs).

Event-triggered Control for Nonlinear Rational Systems

In this work, the problem of designing emulation based stabilizing event-triggered state-feedback controllers for nonlinear rational systems is addressed by means of a weighted relative error measure in the trigger function. It is shown that the proposed event generator does not lead to a Zeno behavior for the class of nonlinear rational systems. Considering a differential-algebraic representation of the nonlinear system and applying the Lyapunov theory, a convex optimization problem in terms of LMI constraints is derived to ensure regional asymptotic stability of the origin while designing the trigger function parameters in order to minimize the control updates. Numerical experiments illustrate the approach showing its potentials as a tool for designing event trigger functions.

The order completion method: a differential-algebraic representation

The order completion method: a differential-algebraic representation

Regional robust stabilisation and domain-of-attraction estimation for MIMO uncertain nonlinear systems with input saturation

ABSTRACTThe regional robust stabilisation by means of linear time-invariant state feedback control for a class of uncertain MIMO nonlinear systems with parametric uncertainties and control input saturation is investigated. The nonlinear systems are described in a differential algebraic representation and the regional stability is handled considering the largest ellipsoidal domain-of-attraction (DOA) inside a given polytopic region in the state space. A novel set of sufficient Linear Matrix Inequality (LMI) conditions with new auxiliary decision variables are developed aiming to design less conservative linear state feedback controllers with corresponding larger DOAs, by considering the polytopic description of the saturated inputs. A few examples are presented showing favourable comparisons with recently published similar control design methodologies.

Dynamic anti-windup design for a class of nonlinear systems

ABSTRACTThis work addresses the dynamic anti-windup design problem for a class of nonlinear systems subject to actuator saturation. The considered class regards the systems that can be cast in a differential algebraic representation (DAR), such as rational and polynomial systems. Considering that a nonlinear dynamic output feedback controller, which can also be written as a DAR, has been designed to stabilise the nonlinear system disregarding the saturation effects, LMI-based conditions are proposed to compute a stabilising dynamic anti-windup compensator. Two problems are considered: the first one regards the regional internal stabilisation; and the second one, assuming that an energy-bounded disturbance acts on the system, concerns the input-to-state stability. The derived conditions can therefore be incorporated in convex optimisation problems that allow to compute the anti-windup compensator in order to maximise the region of guaranteed asymptotic stability, to maximise a bound on the admissible -norm of the disturbance or even to minimise a bound on the -gain between the disturbance and the output. Numerical examples illustrate the proposed methods.

Robust nonlinear state estimation of bioreactors based on H∞ hybrid observers

This paper proposes a robust nonlinear observer for bioreactors combining the classical asymptotic observer and a nonlinear Luenberger-like observer. The resulting hybrid observer considers a new definition of the hybridization parameter which reflects the kinetic model confidence. The nonlinear observer is tuned on the basis of robust H-infinity approach and the differential-algebraic representation (DAR) of nonlinear systems. A simulated case study concerning fed-batch animal cell cultures is presented to demonstrate the potentials and advantages of the proposed approach for state estimation of bioreactors.

Design of Anti-Windup Compensators for a Class of Nonlinear Control Systems with Actuator Saturation

This paper addresses the design of static anti-windup compensators for a class of nonlinear systems subject to actuator saturation. Considering that a (possibly nonlinear) dynamic output feedback control law has been previously designed disregarding the saturation, a method to compute a static anti-windup gain which ensures the local stability of the closed-loop system is proposed. The results are based on a differential algebraic representation of rational systems and on a modified version of the generalized sector condition to deal with the saturation effects. From these elements, an algorithm based on the solution of LMIs is proposed to calculate an anti-windup gain to enlarge the region of attraction of the closed-loop system. Moreover, extensions of the results are presented to deal with parameter uncertainty and the presence of energy bounded disturbances. A numerical example is presented to illustrate the proposed method.

Static anti‐windup design for a class of nonlinear systems

SUMMARYThis paper focuses on the problem of static anti‐windup design for a class of multivariable nonlinear systems subject to actuator saturation. The considered class regards all systems that are rational on the states or that can be conveniently represented by a rational system with algebraic constraints considering some variable changes. More precisely, a method is proposed to compute a static anti‐windup gain which ensures regional stability for the closed‐loop system assuming that a dynamic output feedback controller is previously designed to stabilize the nonlinear system. The results are based on a differential algebraic representation of rational systems. The control saturation effects are taken into account by the application of a generalized sector bound condition. From these elements, LMI‐based conditions are devised to compute an anti‐windup gain with the aim of enlarging the closed‐loop region of attraction. Several numerical examples are provided to illustrate the application of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.

Anti-windup Design for a Class of Nonlinear Control Systems

AbstractThis paper focuses on the problem of static anti-windup design for a class of nonlinear systems subject to actuator saturation. Considering that a nonlinear dynamic output feedback controller has been designed to stabilize the nonlinear system, a method is proposed to compute a static anti-windup gain which ensures the local stability of the the closed-loop system while providing an estimate of the region of attraction. The results are based on a differential-algebraic representation of rational systems and a modified sector bound condition to deal with the saturation effects. From these elements, LMI based conditions are proposed to compute an anti-windup gain for enlarging the size of the closed-loop region of attraction. A numerical example is given to illustrate the proposed method.

Modelling, simulation and control of a redundant parallel robotic manipulator based on invariant manifolds

In this article a systematic approach of modelling and control for a parallel robotic manipulator is presented. Regarding the framework of structured analysis of dynamical systems the derivation of a differential-algebraic model of the mechanical system is straightforward. Using some differential-geometric considerations based on invariant manifolds and the definition of fictitious additional input and output variables a suitable state feedback can be constructed which transforms the differential-algebraic representation into a state-space model for the robotic manipulator. On this basis a classical two-degree-of-freedom (2-DOF) control structure has been designed using the well-known input–output linearization and a linear time-variant Kalman filter-based output feedback. Finally, the control structure including a friction compensation is applied to the robotic system in the laboratory which shows the practical applicability of the proposed procedure.

On modeling and simulation of tethered systems with sharp, rough contacts

We extend the analysis of a reeled, tethered system in a constant gravitational field by (i) proposing a differential–algebraic representation when the tether is inextensible and (ii) examining the interaction between the tether and a sharp, fixed guide when contact is rough. The governing equations are derived using an approach that clearly illustrates how singular supplies of linear momentum, such as friction at the fixed guide, feature in the equations of motion. After semidiscretization via finite differences, we formulate the system of differential equations and algebraic constraints as a differential–algebraic equation and solve for various motions of the reeled tether. In addition, we show that, by comparison to a rounded contact analysis with arbitrarily small radius, the reaction force acting on the tether at the guide does not converge to the value of this force when the guide is modeled as a sharp point of contact.

Computing estimates of the region of attraction for rational control systems with saturating actuators

This study focuses on the problem of regional stability analysis of rational control systems with saturating actuators. Estimates of the region of attraction are computed by means of invariant domains associated to rational Lyapunov functions. Conditions for computing these invariant domains (regions of stability) are proposed in the form of linear matrix inequalities (LMIs). These conditions are derived considering a differential-algebraic representation of the rational system dynamics. The saturation effects are taken into account by means of a generalised sector condition for deadzone non-linearities. The obtained conditions are cast in convex optimisation schemes in order to compute a Lyapunov function which leads to a maximised estimate of the region of attraction.

Robust Linear $H_{\infty}$ Filter Design for a Class of Uncertain Nonlinear Systems: An LMI Approach

This paper addresses the design of robust $H_{\infty}$ filters for a class of nonlinear systems subject to uncertain, possibly time-varying, parameters. The nonlinear system is described by a differential-algebraic representation, which can model the whole class of systems with rational functions of the state and uncertain parameters, as well as some trigonometric nonlinearities. The admissible values of the uncertain parameters and their rate of variation are assumed to belong to a given polytope. A linear matrix inequality methodology is proposed for designing full-order robust linear filters with a minimized upper-bound on the $\mathcal{L}_2$-gain of the noise-to-estimation error operator for all admissible uncertainty. In order to improve the performance of the latter filter, the methodology is then extended to the design of high-order linear filters. Numerical examples are presented to demonstrate the effectiveness of the proposed filter designs.

MULTICRITERIA ANALYSIS OF VISUAL SERVOS VIA BIQUADRATIC LYAPUNOV FUNCTIONS

This paper attempts to propose a generic approach to the muticriteria analysis of position-based and image-based visual servocontrol schemes in robotics. Several criteria can simultaneously be taken into account, such as convergence, avoidance of actuators’ and sensors’ saturations, and 3D constraints. Position- based and image-based servos are unified into a rational systems framework, which is turned into a differential algebraic representation. The constraints are incorporated to the definition of a biquadratic Lyapunov function through the S-procedure and the Finsler lemma, leading to Linear Matrix Inequalities. The method is illustrated on a case study.

ON THE DESIGN OF HIGH-ORDER ROBUST LINEAR H∞ FILTERS FOR A CLASS OF UNCERTAIN NONLINEAR SYSTEMS

This paper addresses the design of robust linear H∞ filters for a class of nonlinear systems subject to uncertain, possibly time-varying, parameters. The nonlinear system is described by a differential-algebraic representation, which can model the whole class of systems with rational functions of the state and uncertain parameters, as well as some trigonometric nonlinearities. The admissible values of the uncertain parameters and their rate of variation are assumed to belong to a given polytope. A linear matrix inequality (LMI) method is presented to design full-order robust linear filters with a minimized upper-bound on the &ℒ2-gain of the noise-to-estimation error operator for all admissible uncertainty. An LMI procedure based on the latter method is then developed to obtain high-order robust linear filters, which are shown, via an example, to achieve significant performance improvement over the full-order filter.