A simple model matching for the stabilization of an inverted pendulum cart system

We present a simple model matching controller for the stabilization of the inverted pendulum cart system, assuming that the pendulum is initialized above the horizontal plane. The control strategy consists of forcing the closed‐loop system to behave as another nonlinear target system with some stability properties. To this end, we solve two matching conditions that allow us to shape a suitable target system. Having satisfied both matching conditions, the stabilizing controller is directly obtained from the proposed target system. The obtained close‐loop system is locally asymptotically exponentially stable, with a very large domain of attraction. Copyright © 2007 John Wiley & Sons, Ltd.     

Passivity‐based control of a magnetically levitated flexible beam

This paper solves the asymptotic stabilization problem for a magnetically levitated flexible beam using a nested‐loop passivity‐based controller design. Passivity analyses reveal that the system can be decomposed into two passive subsystems: a mechanical subsystem that consists of a flexible beam with both ends free and that defines a passive map from external forces to the velocity of the points on the flexible beam at which the external forces act; and an electrical subsystem that consists of a pair of electromagnets and that defines a strictly output‐passive map from voltages applied across the electromagnets to magnetic fluxes. The standard method for designing passivity‐based controllers leads to a nonlinear feed‐forward controller for the electrical subsystem, which enables the electrical subsystem to generate given desired magnetic forces, and an output feedback compensator for the mechanical subsystem, which computes the desired forces required to regulate the position and vibration of the beam. The asymptotic stability of each controller may be proven using Lyapunov's stability theory and LaSalle's invariant set theorem. Numerical simulations confirm the asymptotic stability of the equilibrium configuration of the closed‐loop system formed by the magnetically levitated flexible beam together with the proposed controllers. Copyright © 2008 John Wiley & Sons, Ltd.     

Robust stabilization of the angular velocity of a rigid body

The problem of robust control for the angular velocity of a rigid body subject to external disturbances is addressed. It is shown that if the disturbances are matched there exists a control law attenuating the effect of the disturbances, whereas in the case of non-matched disturbances no such a feedback law generically exists. Hence, a new concept of disturbance attenuation is introduced and it is proved that the aforementioned problem is solvable in this weaker sense. Explicit expressions of the control laws solving the proposed robust stabilization problems are given.     

Stabilization of the inverted spherical pendulum via Lyapunov approach

In this paper a nonlinear controller is presented for the stabilization of the spherical inverted pendulum system. The control strategy is based on the Lyapunov approach in conjunction with LaSalle's invariance principle. The proposed controller is able to bring the pendulum to the unstable upright equilibrium point with the position of the movable base at the origin. The obtained closed‐loop system has a very large domain of attraction, that can be as large as desired, for any initial position of the pendulum which lies above the horizontal plane. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Finite‐time consensus of Euler‐Lagrange agents without velocity measurements via energy shaping

Motivated by the energy‐shaping framework and the properties of homogeneous systems, this paper deals with the problem of achieving consensus of multiple Euler‐Lagrange (EL) systems using the energy shaping plus damping injection principles of passivity‐based control. We propose a method to derive a novel family of decentralized controllers that is capable of solving the leaderless and the leader‐follower consensus problems in finite‐time in networks of fully actuated EL systems without employing velocity measurements. As in the energy‐shaping methodology, the controller is another EL system and the plant‐controller interconnection is the gradient of a suitable defined potential function. The potential energy and dissipation functions, of the controller, are provided with some homogeneous properties in order to achieve finite‐time convergence. This paper provides several simulations that corroborate the performance of different controllers.     

A subspace approach to balanced truncation for model reduction of nonlinear control systems

In this paper, we introduce a new method of model reduction for nonlinear control systems. Our approach is to construct an approximately balanced realization. The method requires only standard matrix computations, and we show that when it is applied to linear systems it results in the usual balanced truncation. For nonlinear systems, the method makes use of data from either simulation or experiment to identify the dynamics relevant to the input–output map of the system. An important feature of this approach is that the resulting reduced‐order model is nonlinear, and has inputs and outputs suitable for control. We perform an example reduction for a nonlinear mechanical system. Copyright © 2002 John Wiley & Sons, Ltd.     

Output feedback stabilization of the angular velocity of a rigid body

The problem of stabilization of the angular velocity of a rigid body using only two control signals and partial state information is addressed. It is shown that if any two (out of three) states are measured the system is not asymptotically stabilizable with (continuous) dynamic output feedback. Nevertheless, we prove that practical stability is achievable if the measurable states fulfill a certain structural property, and that, under the same structural condition, a hybrid control law yielding exponential convergence can be constructed. Finally, we also study some geometric features of the Euler’s equations and the connection between local strong accessibility and local observability.     

Energy-based balance control approach to the ball and beam system

This article investigates the set-point balance control problem of the ball and beam system acting on the ball which is an underactuated system. The control law is designed, based on the total mechanical energy and the passivity properties of the system. According to the desired set-point, the zero equilibrium point and non-zero equilibrium points are studied respectively. For the zero case, it is proved that a single PD feedback controller is sufficient to bring the state to zero from any initial condition, on condition that the control parameters satisfy an inequality. For the non-zero case, the control problem is much more complicated. Unlike the previous energy-based control laws, a new form of Lyapunov function candidate is constructed. A complete analysis of the convergence of the energy and the dynamics is given, and the characteristics of the closed-loop system with the proposed feedback control law are illustrated. Moreover, it is proved that with the parameter choice rules proposed in this article, the ball and beam system will eventually converge exactly to the desired non-zero equilibrium point. Furthermore, since the length of the beam is not unlimited, the trajectory of the ball should be restricted within a limited range. The balance control laws are modified to avoid the ball running beyond the joint range limitation for the zero case and the non-zero case respectively. Simulation results show that the control laws proposed in this article are effective for the set-point balance control problem of the ball and beam system.     

Distributed model‐independent consensus of Euler–Lagrange agents on directed networks

This paper proposes a distributed model‐independent algorithm to achieve leaderless consensus on a directed network where each fully‐actuated agent has self‐dynamics described by Euler–Lagrange equations of motion. Specifically, we aim to achieve consensus of the generalised coordinates with zero generalised velocity. We show that on a strongly connected graph, a model‐independent algorithm can achieve the consensus objective at an exponential rate if an upper bound on the initial conditions is known a priori. By model‐independent, we mean that each agent can execute the algorithm with no knowledge of the equations describing the self‐dynamics of any agent. For design of the control laws which achieve consensus, a control gain scalar and a control gain matrix are required to satisfy several inequalities involving bounds on the matrices of the agent dynamic model, bounds on the Laplacian matrix describing the network topology and the set of initial conditions; design of the algorithm therefore requires some knowledge on the bounds of the agent dynamical parameters. Because only bounds are required, the proposed algorithm offers robustness to uncertainty in the parameters of the multiagent system. We systematically show that additional relative velocity information improves the performance of the controller. Numerical simulations are provided to show the effectiveness of the algorithm. Copyright © 2016 John Wiley & Sons, Ltd.     

Tracking control of uncertain Euler–Lagrange systems with finite‐time convergence

A unified solution is presented to the tracking control problem of Euler–Lagrange systems with finite‐time convergence. A reconstruction module is designed to estimate the overall of unmodeled dynamics, disturbance, actuator misalignment, and multiple actuator faults. That reconstruction is accomplished in finite time with zero error. A nonsingular terminal sliding mode controller is then synthesized, and the resultant closed‐loop system is also shown to be finite‐time stable with the reference trajectory followed in finite time. Unlike most sliding mode control methods to handle system uncertainties, the designed control has less conservativeness and stronger fault tolerant capability. A rigid spacecraft system is used to demonstrate the effectiveness and potential of the proposed scheme. Copyright © 2014 John Wiley & Sons, Ltd.     

Distributed finite‐time tracking of multiple Euler–Lagrange systems without velocity measurements

This paper investigates the distributed finite‐time tracking problem of networked agents with multiple Euler–Lagrange dynamics. To achieve finite‐time tracking, a distributed finite‐time protocol is first proposed on the basis of both relative position and relative velocity measurements. By using tools from homogeneous theory, it is theoretically shown that the proposed protocol can guarantee finite‐time tracking in the presence of control input constraints. On the basis of the state feedback analysis and with the aid of second‐order sliding‐mode observer approach, a new class of finite‐time tracking protocols based only on the relative position measurements is developed and employed. It is proved that the multiple agents equipped with the designed protocols can track the target location in finite time. Furthermore, a decentralized finite‐time protocol based on a distributed estimator is proposed to solve the finite‐time tracking problems with a dynamic leader. The effectiveness of the theoretical results is finally illustrated by numerical simulations. Copyright © 2014 John Wiley & Sons, Ltd.     

Combined passivity based control and optimal control for energy management of fuel cell/battery hybrid system

The energy management of hybrid electric vehicles is becoming an interesting topic for many researchers. Furthermore, the wise choice of the energy management strategy allows not only the best distribution of the power between the used sources, but also it allows reduction of consumption, increase in the lifetime of the sources, and improves the autonomy of the hybrid electric vehicle. The autonomy is guaranteed by the optimization of the embedded sources. In this study, the hybrid system consists of combining the fuel cell as the main source with the battery as the auxiliary source. The novelty of the proposed energy management strategy for the studied hybrid system is the combination between interconnection and damping assignment‐passivity based control and the Hamiltonian Jacobi Bellman method. The stability proof is given and the efficiency of the proposed strategy is proved by the experimental work, where the obtained results show the good and adequate results to the proposed scenario.     

A factorization principle for stabilization of linear control systems

By introducing a fictitious signal y⁰ if necessary we define a transform which generalizes the passage from the scattering to the chain formalism in circuit theory. Given a factorization ??? = ?R of ??? where R is a block matrix function with a certain key block equal to a minimal phase (or outer) matrix function, we show that a given compensator u = Ky is internally stabilizing for the system ?? if and only if a related compensator K′ is stabilizing for ?. Factorizations ??? = ?R with ? having a certain block upper triangular form lead to an alternative derivation of the Youla parametrization of stabilizing compensators. Factorizations with ? equal to a J-inner matrix function (in a precise weak sense) lead to a parametrization of all solutions K of the H^∞ problem associated with ??. This gives a new solution of the H^∞ problem completely in the transfer function domain. Computation of the needed factorization ??? = ?R in terms of a state-space realization of ?? leads to the state-space formulas for the solution of the H^∞ problem recently obtained in the literature.     

Consensus for constrained multi‐agent systems with input saturation

The consensus problem for multi‐agent systems with input saturation is addressed in this paper. For agents with double‐integrator dynamics, we first propose two consensus algorithms, respectively, for the cases with and without velocity measurements. Based on graph theory, homogeneous method and the Lyapunov stability theory, it is proved that the proposed algorithms can guarantee not only the state agreement in finite time for all the agents but also the input saturation requirement. Then, the obtained results and techniques are extended to the finite‐time consensus problem for multiple mechanical systems. Numerical simulations are finally provided to verify the effectiveness of the theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.     

On the globally defined sensorless control of induction motors

The sensorless control problem of induction motors imposes a current challenge since the nonlinear model of this kind of machines does not exhibit global observability properties, i.e. there are some operation regimes for which speed observability is lost. One way for dealing with this unavoidable limitation, and at the same time provide globally defined controllers, is to consider that the rotor variables are estimated via an open‐loop observer. In this paper a globally defined passivity‐based speed controller that belongs to the aforementioned class is presented. It is shown that the structure of previously reported passivity‐based controllers, developed under the assumption that the mechanical variables are available for measurement, can be extended to operate under sensorless conditions if a speed observer is included in the control scheme. Since the controller design methodology leads to inherent drawbacks regarding robustness issues, to evaluate the usefulness of the proposed scheme a numerically based study is included that cover topics such as parameters and disturbance (load torque) uncertainty. The advantages and limitations of the proposed scheme are established with respect to other globally defined sensorless controllers. Copyright © 2008 John Wiley & Sons, Ltd.     

Passivity‐based asymptotic stabilization for switched nonlinear systems using the sampled integral stabilization technique

The asymptotic stabilization problem is studied for a cascade connection of passive switched nonlinear systems and a passive switched nonlinear system in this paper. When each subsystem is asymptotically zero state detectable and passive on active time intervals, asymptotic stabilization is achieved via co‐design of switching laws and controllers without damping injection. First, an output‐feedback controller is designed to asymptotically stabilize a cascade connection of two passive switched systems if outputs are measurable. Second, when the output of the first switched system is noisy or unmeasurable, a sampled integral stabilization (SIS) technique is employed to investigate asymptotical stabilization of a cascade connection by measuring only the storage function of the second switched system. Finally, as a special case of a cascade connection, the SIS technique is used to stabilize a passive switched system without damping injection. Under this circumstance, the controller is designed by sampling the integral of the passive output. The two‐link robot manipulator is provided to illustrate the effectiveness of the SIS technique.     

A noncausal approach for PID control

A new approach for the improvement of the set-point following performances achieved by a PID controller is presented. Basically, it consists of applying a suitable command signal to the closed-loop control system in order to achieve a desired transient response when the process output is required to assume a new value. This command signal is determined by means of a stable input–output procedure for which a closed-form solution is presented, thus making the technique suitable to implement in the industrial context. Simulation and experimental results show that high performances are obtained despite the presence of model uncertainties and, above all, almost independently on the PID tuning. Thus, the PID gains can be selected in order to guarantee good load rejection performances without impairing the set-point transient responses.     

An innovative eigenvalue problem solver for free vibration of Euler–Bernoulli beam by using the Adomian decomposition method

This paper deals with free vibration problems of Euler–Bernoulli beam under various supporting conditions. The technique we have used is based on applying the Adomian decomposition method (ADM) to our vibration problems. Doing some simple mathematical operations on the method, we can obtain ith natural frequencies and mode shapes one at a time. The computed results agree well with those analytical and numerical results given in the literature. These results indicate that the present analysis is accurate, and provides a unified and systematic procedure which is simple and more straightforward than the other modal analysis.     

A limit set stabilization by means of the Port Hamiltonian system approach

A solution to the stabilization problem of a compact set by means of the Interconnection and Damping Assignment Passivity‐Based Control methodology, for an affine nonlinear system, was introduced. To this end, we expressed the closed‐loop system as a Port Hamiltonian system, having the property of almost all their trajectories asymptotically converge to a convenient limit set, except for a set of measure zero. It was carried out by solving a partial differential equation (PDE) or single matching condition, which allows the desired energy level or limit set E to be shaped explicitly. The control strategy was tested using the magnetic beam balance system and the pendulum actuated by a direct current motor (DC‐motor), having obtained satisfactory results. Copyright © 2014 John Wiley & Sons, Ltd.     

Sliding mode PI control with backstepping approach for MIMO nonlinear cross‐coupled tank systems

The control of tank systems in industrial applications is an important issue for monitoring the chemical processes involved in the manufacture and delivery of product. The most important reason to control the tank systems is to keep the liquid level in the tanks constant and at the desired level for a specified period of time. In this study, the sliding mode control (SMC) with a repetitive approach called backstepping that is insensitive to uncertainties in system parameters and input disturbances is proposed and experimentally applied to a quadruple, cross‐coupled, uncertain, nonlinear, and multiple‐input/multiple‐output tank system. A proportional‐integral (PI) control is used to reduce the steady‐state error caused by the parameter variations and external noises. The traditional way of introducing PI usually leads to sliding surfaces. In this paper, the PI action is introduced to the control signal. The proposed backstepping sliding mode PI control (BSMPIC) is applied to such a complex tank system for the first time. The experimental results are compared with those of the SMC, sliding mode PI control, and backstepping sliding mode control to see the effect of the proposed BSMPIC on the system. As a result of the comparison, it is observed that less overshoot and tracking error, better tracking performance, and faster rise time in the transient regime is obtained by the BSMPIC.