Fernando Castaños

Analysis and design of integral sliding manifolds for systems with unmatched perturbations

The robustness properties of integral sliding-mode controllers are studied. This note shows how to select the projection matrix in such a way that the euclidean norm of the resulting perturbation is minimal. It is also shown that when the minimum is attained, the resulting perturbation is not amplified. This selection is particularly useful if integral sliding-mode control is to be combined with other methods to further robustify against unmatched perturbations. H/sub /spl infin// is taken as a special case. Simulations support the general analysis and show the effectiveness of this particular combination.

Control by Interconnection and Standard Passivity-Based Control of Port-Hamiltonian Systems

The dynamics of many physical processes can be suitably described by Port-Hamiltonian (PH) models, where the importance of the energy function, the interconnection pattern and the dissipation of the system is underscored. To regulate the behavior of PH systems it is natural to adopt a Passivity-Based Control (PBC) perspective, where the control objectives are achieved shaping the energy function and adding dissipation. In this paper we consider the PBC techniques of Control by Interconnection (CbI) and Standard PBC. In CbI the controller is another PH system connected to the plant (through a power-preserving interconnection) to add up their energy functions, while in Standard PBC energy shaping is achieved via static state feedback. In spite of the conceptual appeal of formulating the control problem as the interaction of dynamical systems, the current version of CbI imposes a severe restriction on the plant dissipation structure that stymies its practical application. On the other hand, Standard PBC, which is usually derived from a uninspiring and non-intuitive ldquopassive output generationrdquo viewpoint, is one of the most successful controller design techniques. The main objectives of this paper are: (1) To extend the CbI method to make it more widely applicable-in particular, to overcome the aforementioned dissipation obstacle. (2) To show that various popular variants of Standard PBC can be derived proceeding from a unified perspective. (3) To establish the connections between CbI and Standard PBC proving that the latter is obtained restricting the former to a suitable subset-providing a nice geometric interpretation to Standard PBC-and comparing the size of the set of PH plants for which they are applicable.

Passivity of Nonlinear Incremental Systems: Application to PI Stabilization of Nonlinear RLC Circuits

It is well known that if the linear time invariant system xdot = Ax + Bu, y = Cx is passive the associated incremental system xtilde = Axtilde + Butilde, ytilde = Cxtilde, with (middot) = (middot) - (middot)*, u*, y*, the constant input and output associated to an equilibrium state x* , is also passive. In this paper, we identify a class of nonlinear passive systems of the form x = f(x) +gu, y = h(x) whose incremental model is also passive. Using this result we then prove that general nonlinear RLC circuits with convex and proper electric and magnetic energy functions and passive resistors with monotonic characteristic functions are globally stabilizable with linear PI control

Integral Sliding Mode Control for Nonlinear Systems With Matched and Unmatched Perturbations

We consider the problem of designing an integral sliding mode controller to reduce the disturbance terms that act on nonlinear systems with state-dependent drift and input matrix. The general case of both, matched and unmatched disturbances affecting the system is addressed. It is proved that the definition of a suitable sliding manifold and the generation of sliding modes upon it guarantees the minimization of the effect of the disturbance terms, which takes place when the matched disturbances are completely rejected and the unmatched ones are not amplified. A simulation of the proposed technique, applied to a dynamically feedback linearized unicycle, illustrates its effectiveness, even in presence of nonholonomic constraints.

Passivity of nonlinear incremental systems : Application to PI stabilization of nonlinear RLC circuits

It is well known that if the linear time invariant system xdot = Ax + Bu, y = Cx is passive the associated incremental system xtilde = Axtilde + Butilde, ytilde = Cxtilde, with (middot) = (middot) - (middot)*, u*, y*, the constant input and output associated to an equilibrium state x* , is also passive. In this paper, we identify a class of nonlinear passive systems of the form x = f(x) +gu, y = h(x) whose incremental model is also passive. Using this result we then prove that general nonlinear RLC circuits with convex and proper electric and magnetic energy functions and passive resistors with monotonic characteristic functions are globally stabilizable with linear PI control

Robust Semiglobal Stabilization of the Second Order System by Relay Feedback with an Uncertain Variable Time Delay

We present sufficient conditions for robust relay-delayed semiglobal stabilization of second order systems, which relate the upper bound to an uncertain time delay and the parameters of the plant. We also suggest an algorithm of delayed relay control gain adaptation for semiglobal stabilization, which is based on delayed information about the sign of the controlled variable only. The proposed algorithm suppresses bounded uncertainties in the time delay; that is, being designed for the upper bound of uncertainty in the time delay, the control law ensures semiglobal stabilization independently of any variable time delay obeying the given upper bound.

Integral sliding-mode control for linear time-invariant implicit systems

We propose an integral sliding surface for linear time-invariant implicit systems (descriptor systems). We show that, under reasonable assumptions (regularity, stabilizability) it is possible to design a stabilizing controller that compensates the matched perturbations exactly. Higher-order sliding motions are required since, for the solutions of the implicit system to be well defined, special care must be taken on the degree of smoothness of the controller and the perturbations. The algorithm is tested on a system where the perturbation enters through an algebraic equation.

Dynamic switching surfaces for output sliding mode control

The robustness properties of sliding-mode and H ∞ controllers are exploited to produce a dynamic output feedback controller that is insensitive to matched perturbations and attenuates the unmatched ones. The assumptions on the plant differ from the standard assumptions of the Riccati state-space approach to H ∞ control. The sliding-mode controller drives the state into a reduced-order manifold for which the equivalent system does satisfy the standard assumptions and hence the standard theory can be applied. The resulting Riccati equations are of reduced order.

Energy-balancing passivity-based control is equivalent to dissipation and output invariance

Passivity-based controllers (PBCs) achieve stabilization of nonlinear systems, rendering the closed-loop passive with a desired energy (storage) function. A natural question is under which conditions is it possible to make this function equal to the difference between the plant and controller energies-when the controller is said to be energy-balancing. In this paper we prove that a necessary and sufficient condition for energy-balancing is that the open and the closed-loop systems have the same dissipation functions and passive outputs. A second contribution of our work is the identification of a new passive output for port-Hamiltonian systems, which is invariant to the action of PBCs that modify only the energy function-so-called basic interconnection and damping assignment PBCs-proving that they are energy-balancing. To establish these results a new algebraic framework for analysis and design of PBCs, centered around the principles of output and dissipation invariance, is developed. Using this framework several PBC schemes reported in the literature are compared. Also, we present a systematic procedure to generate new passive outputs, this result is of interest on its own, since it allows to extend the applicability of PBC to systems that are non-minimum phase and/or have relative degree larger than one.

Asymptotic stabilization via control by interconnection of port-Hamiltonian systems

We study the asymptotic properties of control by interconnection, a passivity-based controller design methodology for stabilization of port-Hamiltonian systems. It is well-known that the method, in its basic form, imposes some unnatural controller initialization to yield asymptotic stability of the desired equilibrium. We propose two different ways to overcome this restriction, one based on adaptation ideas, and the other one adding an extra damping injection to the controller. The analysis and design principles are illustrated through an academic example.