Elisabetta Punta

A survey of applications of second-order sliding mode control to mechanical systems

The effective application of sliding mode control to mechanical systems is not straightforward because of the sensitivity of these systems to chattering. Higher-order sliding modes can counteract this phenomenon by confining the switching control to the higher derivatives of the mechanical control variable, so that the latter results are continuous. Generally, this approach requires the availability of a number of time derivatives of the sliding variable, and, in the presence of noise, this requirement could be a practical limitation. A class of second-order sliding mode controllers, guaranteeing finite-time convergence for systems with relative degree two between the sliding variable and the switching control, could be helpful both in reducing the number of differentiator stages in the controller and in dealing with unmodelled actuator dynamics. In this paper different second-order sliding mode controllers, previously presented in the literature, are shown to belong to the above cited class, and some challenging control problems involving mechanical systems are addressed and solved. Simulations and experimental results are provided throughout the paper.

Simplex methods for nonlinear uncertain sliding-mode control

We develop a new analysis of the behavior of simplex control methods applied to multiple-input-multiple-output nonlinear control systems under uncertainties. According to such sliding-mode control methods the control vector is constrained to belong to a finite set of (fixed or varying) vectors, with an appropriate switching logic to guarantee the specified sliding condition. Bounded uncertainties acting on the nominal system are allowed. The proposed sliding control methodology relies on the knowledge of the nominal system only. We prove rigorously the convergence of these methods to the sliding manifold in a finite time under explicit quantitative conditions on the system parameters and the available bounds of the uncertainty. Application to a robotic problem is discussed and a nonlinear example is presented.

Multi-input second-order sliding-mode hybrid control of constrained manipulators

This paper deals withthe hybrid position/force control of constrainedmanipulators subjected to uncertainties. A solution is proposedthat is based on sliding-mode control theory, which proved tobe highly effective in counteracting uncertainties for some classesof nonlinear systems. Specific problems involved in this techniqueare chattering elimination and the algebraic coupling betweenconstraint forces and possibly discontinuous control signals.Both the problems are addressed in this paper by exploiting therobustness properties of a second-order sliding-mode controlalgorithm. This algorithm, recently proposed by the authors forsolving the single-input hybrid control problem, is generalizedin this paper to deal with the class of multi-input differentialalgebraic systems describing the behaviour of constrained mechanicalsystems.

Adaptation of sliding modes

Adaptive sliding mode strategies of first and second sliding orders are developed for single-input single- output systems of the first and second relative degrees respectively. Since the only concrete known uncertainty bounds are assumed to be the upper bounds of logarithmic derivatives, no standard sliding- mode technique can solve the problem

Reduced-Order Observer in the Sliding-Mode Control of Nonlinear Nonaffine Systems

The note considers the variable-structure control of nonlinear known nonaffine systems when the state vector is not completely available and the use of observers is required. The strategy of introducing integrators in the input channel is exploited to enlarge the class of tractable control systems. A new reduced-order observer is proposed and conditions are found under which it is proven the convergence to the unique ideal solution of both system and observer. The control problem is solved by forcing a sliding regime for the observer, while satisfying an exponential stability criterion for the observation error state equation.

Approximability Properties for Second-Order Sliding Mode Control Systems

New definitions of approximability are presented for nonlinear second-order sliding mode control systems. Such robustness properties are compared with those already known for first-order methods. Sufficient conditions are obtained for second-order regularization, a sliding error estimate is derived, and some relevant examples are discussed.

Analysis of a second-order sliding-mode algorithm in presence of input delays

In this note, a double integrator system under the action of a second-order sliding-mode control algorithm is considered, and the resulting closed-loop behavior in presence of an input delay is analysed. Due to the delay, in the limit the system trajectories are periodic. Whenever the control modulus is chosen to be constant, the amplitude and period of the resulting oscillations are fixed for any initial value. If the control behaves asymmetrically, it is shown that this is no more true, since the overall dynamical system can admit diverse limit cycles

Brief paper: Simplex sliding mode methods for the chattering reduction control of multi-input nonlinear uncertain systems

The simplex sliding mode control method is further developed by considering uncertain control systems non-affine in the control law. In order to reduce chattering effects, a set of integrators is added in the input channels. The augmented system is then controlled by a switching logic based on the simplex control method. As a result, the original control vector turns out to be continuous. A second order sliding mode observer is used when the sliding output is not available. Explicit conditions are identified about systems uncertainties and the simplex geometry in order to guarantee the convergence of the proposed methodology.

Brief paper: Simplex sliding mode control of multi-input systems with chattering reduction and mono-directional actuators

This paper analyzes features and problems related to the application of the simplex sliding mode control to systems with mono-directional actuators and integrators in the input channel. The plain use of the method formulated in previous contributions results in unacceptable behaviors, such as control laws (the output of the mono-directional actuators), which increase without bounds. A nontrivial modification of the original algorithm is proposed; the new simplex strategy allows the fulfillment of the control objectives by means of bounded inputs from the actuators.

Sliding mode output-feedback stabilization of uncertain nonlinear nonaffine systems

The paper considers the variable-structure output-feedback stabilization of nonlinear uncertain nonaffine systems when the state vector is not completely available and the use of observers is required. The strategy of introducing integrators in the input channel is exploited to enlarge the class of tractable control systems. A full-order observer is designed and the control problem is solved by forcing a sliding regime for the observer. Depending on the type of uncertainties, conditions are found under which convergence to the unique ideal solution of both the system and the observer, either exponentially or with a bounded error, is proven.