A tool for generating a self-excited oscillations for an inertia wheel pendulum by means of a variable structure controller is proposed. The original system is transformed into the normal form for exact linearization. The design procedure, based on Describing Function (DF) method, allows for finding the explicit expressions of the two-relays controller gain parameters in terms of the desired frequency and amplitude. Necessary condition for orbital asymptotic stability of the output of the exactly linearized system is derived. Performance issues of the system with self-excited oscillations are validated with experiments.
The problem of generating oscillations of the inertia wheel pendulum is considered. We combine exact feedback linearization with two-relay controller, tuned using frequency-domain tools, such as computing the locus of a perturbed relay system. Explicit expressions for the parameters of the controller in terms of the desired frequency and amplitude are derived. Sufficient conditions for orbital asymptotic stability of the closed-loop system are obtained with the help of the Poincare map. Performance is validated via experiments. The approach can be easily applied for a minimum phase system, provided the behavior of the states of the zero dynamics is of no concern. Copyright (C) 2011 John Wiley & Sons, Ltd.
Two models of an elastic controller are studied. A control rule with feedback for members and rotor velocities is proposed. The global stability of the equilibrium of a controlled system is proved within the frame-work of a simplified model. Numerical experiments demonstrate that the stability area includes all reasonable initial data, even taking into account discarded small factors.
The design of robust output and state feedback controllers for practical track-ing and stabilization for nonlinear systems with large-scale parametric uncertainty is considered. We consider a class of single-input single-output systems that can be transformed into a special form, where unavailable states (if any) are derivatives of the measured outputs. We consider a large class of nonlinear systems that could be stabilized via a Lyapunov-based technique if the level of parametric uncertainty was much lower and if the unmeasured states were available for feedback. We propose and investigate a new approach based on subdividing the set of parameters into smaller subsets, designing candidate controllers for each subset, and implementing a logic-based switching between them. We use the output of an extended-order high-gain observer to substitute the unavailable states in the candidate controllers, check the inequality for the derivative of the Lyapunov function and switch if it is not satisfied, and to approximately identify the parameters. We discuss the issues of digital implementation and measurement noise and illustrate our design procedure on several examples.
В диссертации получены следующие новые научные результаты.
Для общей нелинейной Лагранжевой системы прямого управления и, в частности, для робототехнического манипулятора, податливостью элементов конструкции которого можно пренебречь, впервые показано:
пропорционально-интегрально-дифференциальный (ПИД) регулятор, широко распространённый в промышленности, гарантирует асимптотическую устойчивость "в целом" желаемого положения равновесия при естественных ограничениях на ПД-коэффициенты усиления и при достаточной малости коэффициента усиления интегральной (И) обратной связи;
тот же результат справедлив при замене дифференциальной (Д) - составляющей обратной связью по выходу некоторой линейной системы дифференциальных уравнений, вносящей диссипацию энергии и идейно близкой к простейшему и очень неточному оценивателю скоростей;
при удовлетворении некоторых несложных неравенств для матриц ПИД - коэффициентов имеет место экспоненциальная устойчивость "в большом", то есть удается оценить границы области притяжения и гарантированную скорость сходимости процесса;
тот же результат справедлив при замене Д--составляющей обратной связью по выходу некоторой линейной системы дифференциальных уравнений (решаемых параллельно с движением и вносящих диссипацию энергии);
ПИД - регулятор удовлетворительно решает задачу слежения, если движение по желаемой траектории происходит достаточно медленно;
Для математической модели многозвенного пространственного робототехнического манипулятора, учитывающей нежёсткость конструкции (путём введения в модель линейных податливых элементов в сочленениях), показано:
регулятор, состоящий из пропорционально-дифференциальной обратной связи по положению роторов управляющих двигателей и постоянной добавки компенсирующей статический прогиб ("ПД+"), обеспечивает асимптотическую устойчивость в целом замкнутой системы даже при учёте наличия насыщения в характеристиках усилителей обратных связей;
тот же результат справедлив при замене Д - составляющей с насыщением обратной связью по выходу некоторой нелинейной системы дифференциальных уравнений (с нелинейностью в точности совпадающей с нелинейностью характеристик усилителей), вносящей диссипацию энергии;
для псевдо "ПД+" регулятора, а именно, для случая наиболее распространённой и легче всего реализуемой на практике обратной связи по положениям звеньев и скоростям роторов двигателей, имеет место асимптотическая устойчивость замкнутой системы при достаточно естественных условиях;
некоторые робастные линейные законы управления, основывающиеся на той же наиболее разумной комбинации измерений (датчиков), и, в частности, два варианта реализации "псевдо ПИД" - регуляторов гарантируют асимптотическую устойчивость в целом при выполнении некоторых условий на матрицы коэффициентов усиления;
вместо интегральной обратной связи можно использовать также несложную итеративную процедуру обучения управления нужной компенсационной добавке.
Некоторые, но далеко не все, идеи по поводу управления роботами могут быть обобщены на случай общих нелинейных систем управления. В частности, удалось ввести известное из линейной теории систем понятие астатизма и проанализировать свойства астатических систем, такие как ограниченность реакции на линейно растущее возмущение.
We consider a tracking problem for a partially feedback linearizable nonlinear system with stable zero dynamics. The system is uncertain and only the output is measured. We use an extended high-gain observer of dimension n+1, where n is the relative degree. The observer estimates n derivatives of the tracking error, of which the first (n-1) derivatives are states of the plant in the normal form and the $n$th derivative estimates the perturbation due to model uncertainty and disturbance. The controller cancels the perturbation estimate and implements a feedback control law, designed for the nominal linear model that would have been obtained by feedback linearization had all the nonlinearities been known and the signals been available. We prove that the closed-loop system under the observer-based controller recovers the performance of the nominal linear model as the observer gain becomes sufficiently high. Moreover, we prove that the controller has an integral action property in that it ensures regulation of the tracking error to zero in the presence of constant nonvanishing perturbation.
We consider a partially feedback linearizable system with stable zero dynamics. The system is uncertain and only the output is measured. Consequently, exact feedback linearization is not applicable. We propose to design an extended high-gain observer to recover unmeasured derivatives of the output and an extra one, which contains information about the uncertainty. The observer can be stabilized via feedback linearization followed by a linear control design, such as pole placement or LQR. After a short peaking period, a partial state vector, which includes the output and its derivatives, will be in a small neighborhood of the state of the observer; therefore, the performance achievable under exact feedback linearization will be recovered.
We consider an underactuated two-link robot called the inertia wheel pendulum. The system consists of a free planar rotational pendulum and a symmetric disk attached to its end, which is directly controlled by a DC-motor. The goal is to create stable oscillations of the pendulum, which is not directly actuated. We exploit a recently proposed feedback-control design strategy based on motion planning via virtual holonomic constraints. This strategy is shown to be useful for design of regulators for achieving orbitally exponentially stable oscillatory motions. The main contribution is a step-by-step procedure on how to achieve oscillations with pre-specified amplitude from a given range and an arbitrary independently chosen period. The theoretical results are verified via experiments with a real hardware setup.
Recently, a new technique for generating periodic motions in mechanical systems which have less actuators than degrees of freedom has been proposed. A motivating example for studying such motions is a dynamically stabilized walking robot, where the target trajectory is periodic, and one of the joints - the ankle joint - is unactuated, or weakly actuated. In this paper, the technique is implemented on the Furuta pendulum, an experimental testbed that is simpler than a walking robot but retains many of the key challenges - it is underactuated, open-loop unstable, and practical problems such as friction and velocity estimation must be overcome. We present a detailed description of the practical implementation of the controller. The experiments show that the technique is sufficiently robust to be useful in practice.
The paper by Chaillet, Loría, and Kelly is devoted to study robustness of mechanical systems controlled by proportional integral-differential (PID) regulators. These control strategies are classical and are the most frequently used in industrial applications of robotic manipulators despite various other available techniques. There is a number of results on properties of PID-controlled mechanical systems, see references in the paper and [1,2,5–7,11–13] to mention a few.
We consider output feedback stabilization of uniformly observable uncertain nonlinear systems when the uncertain parameters belong to a known but comparably large compact set. In a previous paper, we proposed a new logic-based switching control to improve the performance of continuous high-gain-observer-based sliding mode controllers. Our main goal here is to show that similar techniques can be exploited for solving challenging control problems for a more general class of uncertain nonlinear systems. We require neither the sign of the high-frequency gain to be known nor the system to be minimum-phase. The key idea is to split the set of parameters into smaller subsets, design a controller for each of them, and switch the controller if, after a dwell-time period, the derivative of the Lyapunov function does not satisfy a certain inequality. A high-gain observer is used to estimate the derivatives of the output as well as the derivative of the Lyapunov function. Another goal of this paper is to introduce a switching strategy that uses on-line information to decide on the controller to switch to, instead of using a pre-sorted list as in our previous work. The new strategy can improve the transient performance of the system.
We consider output feedback stabilization of uniformly observable uncertain nonlinear systems when the uncertain parameters belong to a known but comparably large compact set. In a previous paper, we proposed a new logic-based switching control to improve the performance of continuous high-gain-observer-based sliding mode controllers. Our main goal here is to show that similar techniques can be exploited for solving challenging control problems for a more general class of uncertain nonlinear systems. We require neither the sign of the high-frequency gain to be known nor the system to be minimum-phase. The key idea is to split the set of parameters into smaller subsets, design a controller for each of them, and switch the controller if, after a dwell-time period, the derivative of the Lyapunov function does not satisfy a certain inequality. A high-gain observer is used to estimate the derivatives of the output as well as the derivative of the Lyapunov function. Another on-line information to decide on the controller to switch to, instead of using a goal of this paper is to introduce a switching strategy that uses on pre-sorted list as in our previous work. The new strategy can improve the transient performance of the system. (c) 2006 Elsevier Ltd. All rights reserved.
A planar compass-like biped on a shallow slope is the simplest model of a passive walker. It is a two-degrees-of-freedom impulsive mechanical system known to possess periodic solutions reminiscent to human walking. Finding such solutions is a challenging task. We propose a new approach to obtain stable as well as unstable hybrid limit cycles without integrating the full set of differential equations. The procedure is based on exploring the idea of parameterizing a possible periodic solution via virtual holonomic constraints. We also show that a 2-dimensional manifold, defining the hybrid zero dynamics associated with a stable hybrid cycle, in general, is not invariant for the dynamics of the model of the compass-gait walker.
A planar compass-like biped on a shallow slope is one of the simplest models of a passive walker. It is a 2-degree-of-freedom (DOF) impulsive mechanical system that is known to possess periodic solutions reminiscent of human walking. Finding such solutions is a challenging computational task that has attracted many researchers who are motivated by various aspects of passive and active dynamic walking. We propose a new approach to find stable as well as unstable hybrid limit cycles without integrating the full set of differential equations and, at the same time, without approximating the dynamics. The procedure exploits a time-independent representation of a possible periodic solution via a virtual holonomic constraint. The description of the limit cycle obtained in this way is useful for the analysis and characterization of passive gaits as well as for design of regulators to achieve gaits with the smallest required control efforts. Some insights into the notion of hybrid zero dynamics, which are related to such a description, are presented as well.
We consider a tracking problem for mechanical systems. It is assumed that feedback controller is designed neglecting some disturbances, which could be approximately modeled by a dynamic LuGre friction model. We are interested to derive an additive observer-based compensator to annihilate or reduce the influence of such a disturbance. We exploit a recently suggested approach for observer design for LuGre-friction-model-based compensation. In order to follow this technique, it is necessary to know the Lyapunov function for the unperturbed system, parameters of the dynamic friction model, and to have certain structural property satisfied. The case when this property is passivity with respect to the matching disturbance related to the given Lyapunov function is illustrated in the paper with an example of a DC-motor. The main contribution is some new insights into numerical real time implementation of friction compensators for various LuGre-type models. The other contribution, built upon the main one, is experimental verification of the suggested observer design procedure.
A tracking problem for a mechanical system is considered. We start with a feedback controller that is designed without attention to disturbances, which are assumed to be adequately described by a dynamic LuGre friction model. We are interested in deriving a superimposed observer-based compensator to annihilate or reduce the influence of such a disturbance. We exploit a recently suggested approach for observer design for LuGre-friction-model-based compensation. In order to apply this technique, it is necessary to know the Lyapunov function for the unperturbed system, as well as the parameters of the dynamic friction model, and to verify that a certain structural property satisfied. The case when the system is passive with respect to the matching disturbance related to the given Lyapunov function is illustrated in this brief with a DC-motor example. The main contribution is some new insights into the numerical real-time implementation of a compensator for disturbances describable by one of various LuGre-type models. The other contribution, which is built upon the main one, is experimental verification of the suggested model-based observer design procedure.
This paper presents a new control strategy for an underactuated two-link robot, called the Pendubot. The goal is to create stable oscillations of the outer link of the Pendubot, which is not directly actuated. We exploit a recently proposed feedback control design strategy, based on motion planning via virtual holonomic constraints. This strategy is shown to be useful for design of regulators for achieving: stable oscillatory motions, a closed-loop-design-based swing-up, and propeller motions. The theoretical results are verified via successful experimental implementation.
We consider the problem of creating oscillations of the Furuta pendulum around the open-loop unstable equilibrium. We start with a control transformation shaping the energy of the passive link. Then, a dissipativity-based controller is designed to create oscillations, neglecting the possibility of unbounded motion of the directly actuated link. After that, an auxiliary linear feedback action is added to the control law stabilizing a desired level of the reshaped energy. Parameters of the controller are tuned to approximately keep the originally created oscillations but ensuring bounded motion of both links. The analysis is valid only for oscillations of sufficiently high frequency and is based on higher order averaging technique. The performance of the designed controller is verified using numerical simulations as well as experimentally.
We consider the challenging problem of creating oscillations in underactuated mechanical systems. Target oscillatory motions of the indirectly actuated degree of freedom of a mechanical system can often be achieved via a straightforward to design feedback transformation. Moreover, the corresponding part of the dynamics can be forced to match a desired second-order system possessing the target periodic solution (Aracil, J., Gordillo, F., and Acosta, J.A. (2002), 'Stabilization of Oscillations in the Inverted Pendulum', in Proceedings of the 15th IFAC World Congress, Barcelona, Spain; Canudas-de-Wit, C., Espiau, B., and Urrea, C. (2002), 'Orbital Stabilisation of Underactuated Mechanical Systems', in Proceedings of the 15th IFAC World Congress, Barcelona, Spain). Sometimes, it is possible to establish the presence of periodic or bounded motions for the remaining degrees of freedom in the transformed system. However, typically this motion planning procedure leads to an open-loop unstable orbit and by necessity should be followed by a feedback control design. We propose a new approach for synthesis of a (practically) stabilising feedback controller, which ensures convergence of the solutions of the closed-loop system into a narrow tube around the preplanned orbit. The method is illustrated in detail by shaping oscillations in the inverted pendulum on a cart around its upright equilibrium. The complete analysis is based on application of a non-standard higher-order averaging technique assuming sufficiently high frequency of oscillations and is presented for this particular example.
The problem is to create a hybrid periodic motion, reminiscent of walking, for amodel of an underactuated biped robot. We show how to construct a transverse linearization analytically and how to use it for stability analysis and for design of an exponentially orbitally stabilizing controller. In doing so, we extend a technique recently developed for continuous-time controlled mechanical systems with degree of underactuation one. All derivations are shown on an example of a three-link walking robot, modeled as a system with impulse effects.
Abstract: The paper presents a new method for numerical solution of matrixRiccati equation with periodic coeﬃcients. The method is based on approximationof stabilizing solution of the Riccati equation by trigonometric polynomials.
The problem of swinging up inverted pendulums has often been solved by stabilizing a particular class of homoclinic structures present in the dynamics of a physical pendulum. Here, new arguments are suggested to show how other homoclinic curves can be preplanned for dynamics of the passive-link of the robot. This is done by reparameterizing the motions according to geometrical relations among the generalized coordinates, which are known as virtual holonomic constraints. After that, conditions that guarantee the existence of periodic solutions surrounding the planned homoclinic orbits are derived. The corresponding trajectories, in contrast to homoclinic curves, admit efficient design of feedback control laws ensuring exponential orbital stabilization. The method is illustrated by simulations and supported by experimental studies on the Furuta pendulum. The implementation issues are discussed in detail.
We consider a 3-link planar walker with two legs and an upper body. An actuator is introduced between the legs, and the torso is kept upright by torsional springs. The model is a 3-DOF impulsive mechanical system, and the aim is to induce stable limit-cycle walking in level ground. To solve the problem, the ideas of the virtual holonomic constraints approach are explored, used and extended. The contribution is a novel systematic motion planning procedure for solving the problem of gait synthesis, which is challenging for non-feedback linearizable mechanical systems with two or more passive degrees of freedom. For a preplanned gait we compute an impulsive linear system that approximates dynamics transversal to the periodic solution. This linear system is used for the design of a stabilizing feedback controller. Results of numerical simulations are presented to illustrate the performance of the closed loop system.
The problem of swinging up inverted pendulums has often been solved by stabilization of a particular class of homoclinic structures present in the dynamics of the standard pendulum. In this article new arguments are suggested to show how different homoclinic curves can be preplanned for dynamics of the passive-link of the robot. This is done by reparameterizing the motion according to geometrical relations among the generalized coordinates. It is also shown that under certain conditions there exist periodic solutions surrounding such homoclinic orbits. These trajectories admit designing feedback controllers to ensure exponential orbital stabilization. The method is illustrated by simulations and supported by experimental studies.
We consider a benchmark example of a three-link planar biped walker with torso, which is actuated in between the legs. The torso is thought to be kept upright by two identical torsional springs. The mathematical model reflects a three-degree-of-freedom mechanical system with impulse effects, which describe the impacts of the swing leg with the ground, and the aim is to induce stable limit-cycle walking on level ground. The main contribution is a novel systematic trajectory planning procedure for solving the problem of gait synthesis. The key idea is to find a system of ordinary differential equations for the functions describing a synchronization pattern for the time evolution of the generalized coordinates along a periodic motion. These functions, which are known as virtual holonomic constraints, are also used to compute an impulsive linear system that approximates the time evolution of the subset of coordinates that are transverse to the orbit of the continuous part of the periodic solution. This auxiliary system, which is known as transverse linearization, is used to design a nonlinear exponentially orbitally stabilizing feedback controller. The performance of the closed-loop system and its robustness with respect to various perturbations and uncertainties are illustrated via numerical simulations.
In the field of robotics there is a great interest in developing strategies and algorithms to reproduce human-like behavior. In this paper, we consider motion planning and generation for humanoid robots based on the concept of virtual holonomic constraints. For this purpose, recorded kinematic data from a particular human motion are analyzed in order to extract geometric relations among various joint angles defining the instantaneous postures. The analysis of a simplified human body representation leads to dynamics of a corresponding underactuated mechanical system with parameters based on anthropometric data of an average person. The motion planning is realized by considering solutions of reduced system dynamics assuming the virtual holonomic constraints are kept invariant The relevance of such a mathematical model in accordance to the real human motion under study is shown. An appropriate controller design procedure is presented together with simulation results of a feedback-controlled robot.
In the field of robotics there is a great interest in developing strategies and algorithms to reproduce human-like behavior. In this paper, we consider motion planning for humanoid robots based on the concept of virtual holonomic constraints. At first, recorded kinematic data of particular human motions are analyzed in order to extract consistent geometric relations among various joint angles defining the instantaneous postures. Second, a simplified human body representation leads to dynamics of an underactuated mechanical system with parameters based on anthropometric data. Motion planning for humanoid robots of similar structure can be carried out by considering solutions of reduced dynamics obtained by imposing the virtual holonomic constraints that are found in human movements. The relevance of such a reduced mathematical model in accordance with the real human motions under study is shown. Since the virtual constraints must be imposed on the robot dynamics by feedback control, the design procedure for a suitable controller is briefly discussed.
A lack of sufficient actuation power as well as the presence of passive degrees of freedom are often serious constraints for feasible motions of a robot. Installing passive elastic mechanisms in parallel with the original actuators is one of a few alternatives that allows for large modifications of the range of external forces or torques that can be applied to the mechanical system. If some motions are planned that require a nominal control input above the actuator limitations, then we can search for auxiliary spring-like mechanisms complementing the control scheme in order to overcome the constraints. The intuitive idea of parallel elastic actuation is that spring-like elements generate most of the nominal torque required along a desired trajectory, so the control efforts of the original actuators can be mainly spent in stabilizing the motion. Such attractive arguments are, however, challenging for robots with non-feedback linearizable non-minimum phase dynamics that have one or several passive degrees of freedom. We suggest an approach to resolve the apparent difficulties and illustrate the method with an example of an underactuated planar double pendulum. The results are tested both in simulations and through experimental studies.
In this paper we consider the problem of motion planning and control of a kinematically redundant manipulator, which is used on forestry machines for logging. Once a desired path is specified in the 3D world frame, a trajectory can be planned and executed such that all joints are synchronized and constrained to the Cartesian path. We introduce an optimization procedure that takes advantage of the kinematic redundancy so that time-efficient joint and velocity profiles along the path can be obtained. Differential constraints imposed by the manipulator dynamics are accounted for by employing a phase-plane technique for admissible path timings. In hydraulic manipulators, such as considered here, the velocity constraints of the individual joints are particularly restrictive. We suggest a time-independent control scheme for the planned trajectory which is built upon the standard reference tracking controllers. Experimental tests underline the benefits and efficiency of the model-based trajectory planning and show success of the proposed control strategy.
A new approach for solving an optimal motion planning problem for a simplified 2-degrees-of-freedom model of a human arm is proposed. The motion of interest resembles ball pitching. The model of a planar two-link robot is used with actuation only at the shoulder joint and a passive spring at the elbow joint representing the stiffness of the arm. The goal is formulated as finding a trajectory and the associated torque of the active joint that maximizes the velocity of the end effector in horizontal direction at the moment of crossing a vertical ball-release line. The basic idea is to search for an optimal motion parametrized by the horizontal displacement of the end-effector from the start point to the release point. The suggested procedure leads to analytical expressions for the coefficients of a nonlinear differential equation that governs the geometric relation between the links along an optimal motion. The motion planning task is reformulated to a finite-dimensional search for the corresponding initial conditions.
Nonlinear H-infinity synthesis is developed to solve the tracking control problem into a 3-DOF helicopter prototype. Planning of periodic motions under a virtual constraints approach is considered prior the controller design in order to achieve our goal. A local H-infinity controller is derived by means of a certain perturbation of the differential Riccati equations that appear while solving the corresponding H-infinity control problem for the linearised system. Stabilisability and detectability properties of the control system are thus ensured by the existence of the proper solutions of the unperturbed differential Riccati equations, and hence the proposed synthesis procedure obviates an extra verification work of these properties. Due to the nature of the approach, the resulting controller additionally yields the desired robustness properties against unknown but bounded external disturbances. Convergence and robustness of the proposed design are supported by simulation results.
In the forest industry, trees are logged and harvested by human-operated hydraulic manipulators. Eventually, these tasks are expected to be automated with optimal performance. However, with todays technology the main problem is implementation. While prototypes may have rich sensing information, real cranes lack certain sensing devices, such as encoders for position sensing. Automating these machines requires unconventional solutions. In this paper, we consider the motion planning problem, which involves a redesign of optimal trajectories, so that open loop control strategies can be applied using feed-forward control signals whenever sensing information is not available.
A short term goal in the forest industry is semi-automation of existing machines for the tasks of logging and harvesting. One way to assist drivers is to provide a set of predefined trajectories that can be used repeatedly in the process. In recent years much effort has been directed to the design of control strategies and task planning as part of this solution. However, commercialization of such automatic schemes requires the installation of various sensing devices, computers and most of all a redesign of the machine itself, which is currently undesired by manufacturers. Here we present an approach of implementing predefined trajectories in an open-loop fashion, which avoids the complexity of sensor and computer integration. The experimental results are carried out on a commercial hydraulic crane to demonstrate that this solution is feasible in practice.
We discuss a constructive procedure for planning human-like motions of humanoid robots on finite-time intervals. Besides complying with various constraints present in the robot dynamics, it allows us to reflect certain features of recorded and analyzed motions of a test person, even though the dimensions, the mass distribution, and the actuation of the robot are different. So, the steps in motion planning are complemented by a novel algorithm for control design to ensure contraction of the orbits of various perturbed closed-loop motions to the orbit of the prescribed target trajectory.
A class of (directly and non-directly) controlled Lagrangian systems is considered. After a brief overview of known control laws, the main attention is paid to the most simple and the most popular PD, PID or PID-like controllers broadly used in robotics which are analyzed in details. New results demonstrating in particular robustness of closed-loop systems with respect to uncertainty in model description are shown. The efficiency of PID-like controllers for flexible joint robotic manipulators based on measurements of rotors velocities and links positions is investigated in detail also.
This paper is based on a new procedure for dynamic output feedback design for systems with nonlinearities satisfying quadratic constraints. The new procedure is motivated by the challenges of output feedback control design for the 3-state Moore-Greitzer compressor model. First, we use conditions for stability of a transformed system and the associated matching conditions to find the data of the stabilizing controllers for the surge subsystem. Second, using the set of stabilizing controllers satisfying the given constraints for the closed-loop system with the dynamic output feedback controller we made optimization over the parameter set. We present the data of the stabilizing controllers and the new constraints for the corresponding parameters. The contributions in this paper are simplified conditions for the synthesis and optimization over the control parameter set.
A general method for planning and orbitally stabilizing periodic motions for impulsive mechanical systems with underactuation one is proposed. For each such trajectory, we suggest a constructive procedure for defining a sufficient number of nontrivial quantities that vanish on the orbit. After that, we prove that these quantities constitute a possible set of transverse coordinates. Finally, we present analytical steps for computing linearization of dynamics of these coordinates along the motion. As a result, for each such planned periodic trajectory, a hybrid transverse linearization for dynamics of the system is computed in closed form. The derived impulsive linear system can be used for stability analysis and for design of exponentially orbitally stabilizing feedback controllers. A geometrical interpretation of the method is given in terms of a novel concept of a moving Poincare section. The technique is illustrated on a devil stick example.
This paper provides an introduction to several problems and techniques related to controlling periodic motions of dynamical systems. In particular, we consider planning periodic motions and designing feedback controllers for orbital stabilization. We review classical and recent design methods based on the Poincaré first-return map and the transverse linearization. We begin with general nonlinear systems and then specialize to a class of underactuated mechanical systems for which a particularly rich structure allows many of the problems to be solved analytically.
This study examines the mechanical systems with an arbitrary number of passive (non-actuated) degrees of freedom and proposes an analytical method for computing coefficients of a linear controlled system, solutions of which approximate dynamics transverse to a feasible motion. This constructive procedure is based on a particular choice of coordinates and allows explicit introduction of a moving Poincaré section associated with a nontrivial finite-time or periodic motion. In these coordinates, transverse dynamics admits analytical linearization before any control design. If the forced motion of an underactuated mechanical system is periodic, then this linearization is an indispensable and constructive tool for stabilizing the cycle and for analyzing its orbital (in)stability. The technique is illustrated with two challenging examples. The first one is stabilization of a circular motion of a spherical pendulum on a puck around its upright equilibrium. The other one is creating stable synchronous oscillations of an arbitrary number of planar pendulums on carts around their unstable equilibria.
This study examines the mechanical systems with an arbitrary number of passive (non-actuated) degrees of freedom and proposes an analytical method for computing coefficients of a linear controlled system, solutions of which approximate dynamics transverse to a feasible motion. This constructive procedure is based on a particular choice of coordinates and allows explicit introduction of a moving Poincaré section associated with a nontrivial finite-time or periodic motion. In these coordinates, transverse dynamics admits analytical linearization before any control design. If the forced motion of an underactuated mechanical system is periodic, then this linearization is an indispensable and constructive tool for stabilizing the cycle and for analyzing its orbital (in)stability. The technique is illustrated with two challenging examples. The first one is stabilization of a circular motions of a spherical pendulum on a puck around its upright equilibrium. The other one is creating stable synchronous oscillations of an arbitrary number of planar pendula on carts around their unstable equilibria.
We propose here a new procedure for output feedback design for systems with nonlinearities satisfying quadratic constraints. It provides an alternative for the classical observer-based design and relies on transformation of the closed-loop system with a dynamic controller of particular structure into a special block form. We present two sets of sufficient conditions for stability of the transformed block system and derive matching conditions allowing such a representation for a particular challenging example. The two new tests for global stability proposed for a class of nonlinear systems extend the famous Circle criterion applied for infinite sector quadratic constraints. The study is motivated and illustrated by the problem of output feedback control design for the well-known finite dimensional nonlinear model qualitatively describing surge instabilities in compressors. Assuming that the only available measurement is the pressure rise, we suggest a constructive procedure for synthesis of a family of robustly globally stabilizing feedback controllers. The solution relies on structural properties of the nonlinearity of the model describing a compressor characteristic, which includes earlier known static quadratic constraints and a newly found integral quadratic constraint. Performance of the closed-loop system is discussed and illustrated by simulations.
We consider a special benchmark mechanical system with two degrees of freedom and underactuation degree one. Furuta Pendulum consists of an arm, rotating in the horizontal plane, and a pendulum, attached to the end of the arm freely and moving in the vertical plane. Our goal is to create and partially shape stable oscillations of the pendulum. The approach is based on the idea of stabilization of a particular virtual holonomic constraint, imposed on the configuration coordinates. We describe here a step-by-step design procedure in details. Our approach is illustrated through successful experimental tests.
The Furuta pendulum consists of an arm rotating in the horizontal plane and a pendulum attached to its end. Rotation of the arm is controlled by a DC motor, while the pendulum is moving freely in the plane, orthogonal to the arm. Motivated, in particular, by possible applications for walking/running/balancing robots, we consider the Furuta pendulum as a system for which synchronized periodic motions of all the generalized coordinates are to be created and stabilized. The goal is to achieve, via appropriate feedback control action, orbitally exponentially stable oscillations of the pendulum of various shapes around its upright and downward positions, accompanied with oscillations of the arm. Our approach is based on the idea of stabilization of a particular virtual holonomic constraint imposed on the configuration coordinates, which has been theoretically developed recently. Here, we elaborate on the complete design procedure. The results are illustrated not only through numerical simulations but also through successful experimental tests.
The paper suggests conditions for presence of quadratic Lyapunov functions for nonlinear observer based feedback systems with an ’input nonlinearity’ in the feedback path. Provided that the system using state feedback satisfies the circle criterion (i.e., when all states can be measured), we show that stability of the extended system with output feedback control from a (full state) Luenberger-type observer may be concluded using the circle criterion. As another result, we state a separation principle for a class of feedback systems with an input nonlinearity.When only local stability results can be stated, our method provides an estimate of the region of attraction.
A class of mechanical systems with many unactuated degrees of freedom is studied. An analytical method for computing coefficients of a linear controlled system, solutions of which approximate dynamics of transverse part of coordinates of an underactuated mechanical system along a feasible motion, is proposed. The procedure is constructive and is based on a particular choice of coordinates in a vicinity of the motion. It allows explicit introduction of the so-called moving Poincare section associated with a finite-time or periodic motion. It is shown that the coordinates admit analytical linearization of transverse part of the system dynamics prior to any controller design. If the motion is periodic, then these coordinates are used for developing feedback controllers. Necessary and sufficient conditions for exponential orbital stabilization of a cycle for underactuated mechanical systems are derived. Two illustrative examples are elaborated in details.
We study the problem of motion planning for underactuated mechanical systems. The idea is to reduce complexity by imposing via feedback a sufficient number of invariants and then to compute a projection of the dynamics onto an induced invariant sub-manifold of the closed-loop system. The inspiration comes from two quite distant methods, namely the method of virtual holonomic constraints, originally invented for planning and orbital stabilization of gaits of walking machines, and the method of controlled Lagrangians, primarily invented as a nonlinear technique for stabilization of (relative) equilibria of controlled mechanical systems. Both of these techniques enforce the presence of particular invariants that can be described as level sets of conserved quantities induced in the closed-loop system. We link this structural feature of both methods to a procedure to transform a Lagrangian system with control inputs via a feedback action into a control-free Lagrangian system with a sufficient number of first integrals for the full state space or an invariant sub-manifold. In both cases, this transformation allows efficient (analytical) description of a new class of trajectories of forced mechanical systems appropriate for further orbital stabilization. For illustration purposes, we approach the challenging problem for a controlled mechanical system with two passive degrees of freedom: planning periodic (or bounded) forced upperhemisphere trajectories of the spherical pendulum on a puck. Another example of the technique is separately reported in [21].
The well-known and commonly accepted finite dimensional model qualitatively describing surge instabilities in centrifugal (and axial) compressors is considered. The problem of global output feedback stabilization for it is solved. The solution relies on two new criteria for global stability proposed for a class of nonlinear systems exploiting quadratic constraints for infinite sector nonlinearities. The constructive steps in developing a family of output feedback controllers based on these stability tests are presented. Performance of the closed-loop systems are illustrated by simulations.