Sliding-mode control

Sliding-mode control with an H∞ criterion

Master's thesis

It is well known that sliding modes are robust to matched perturbations and model uncertainties. That is, they are robust in the face of perturbations that are coupled to the control action. On the other hand, sliding modes suffer from:

1.  An initial period (called the reaching phase) in which the system's robustness is not guaranteed.

2.  Sensitivity to unmatched perturbations.

3.  Full-state measurement is required.

4.  The so-called chattering phenomenon.

The first problem can be solved using integral sliding modes, which have the additional advantage of furnishing more design parameters. The extra freedom can then be used to minimize, e.g., using an H∞ criterion, the impact of the unmatched perturbations and uncertainties [CF06a, CXF06, REC+11, CHZF14].

It is sometimes possible to enforce the desired sliding motion by constructing a dynamic surface using only partial state information [CF04b], regardless of any matched uncertainty (this addresses the third problem). The input-to-state stability framework can be used to efficiently compute the controller gains [MCF14, MCF15, MCF16]. In the presence of unmatched uncertainties, one can also design the dynamic surface using an H∞ criterion [CF11, MCF13].

The machinery developed fits nicely in a decentralized control scenario [CF05a].

Leonid Fridman supervised the thesis (in Spanish).

Pole-placement in higher-order sliding-mode control

The well-known pole-placement formula by Ackermann was modified by Utkin and Ackermann himself to place the eigenvalues of the sliding dynamics in conventional sliding-mode control. The formula can be further generalized to the case of higher-order sliding modes [HZCF14, CCF16]. The formula can be used as part of a higher-order sliding-mode control design methodology, achieving high accuracy and robustness at the same time.

Extremum seeking

Tracking a varying maximum or minimum (extremum) of a performance (output, cost) function is called extremum-seeking control. It usually has two layers of control. The first layer is an algorithm able to control (stabilize) the system and drive the performance output to its reference. The second layer sets the appropriate reference by seeking an extremum of the performance output.

In some applications, the reference-to-output map has an extremum, and the objective is to select the set-point that keeps the output at the extremum value. The uncertainty in the reference-to-output map makes it necessary to use some sort of adaptation to find the set point that optimizes the output. Usually, the adaptation scheme requires a dither signal to satisfy the so-called persistence of excitation. In [CK15], a nonsmooth ditherless extremum seeker is applied to minimize hydrogen consumption in fuel cells. See also [KC13a, KC13b].

Discrete-time sliding-mode control

From a purely theoretical point of view, the solution of a system that exhibits sliding motions is, in fact, a solution of a differential inclusion set forth by Filippov's theory on differential equations with discontinuous right-hand sides. Ideally, this solution is absolutely continuous and exhibits no chattering. In applications, chattering results from the presence of nonideal components, like actuators with a nonzero time constant, or control laws implemented in discrete time. However, if the control law is discretized using a backward-Euler scheme (also known as the implicit method), the closed-loop system results in a difference inclusion (as opposed to the difference equation obtained with traditional methods), and chattering is substantially suppressed [MBC18].

In [MCB25b], it is argued that first-order sliding-mode control is a particular instance of passivity-based control.  It is further shown that a backward-Euler scheme is necessary for time discretization to preserve passivity. Besides passivity, maximal monotonicity is an essential property of the operators responsible for first-order sliding motions. The proper discretization of systems involving maximal monotone operators results in the proximal-point algorithms thoroughly studied in the optimization literature. 

Sliding-mode controllers are typically valued for their robustness and finite-time convergence. In addition to preserving passivity, backward-Euler discretization methods preserve finite-time stability. A tool that is particularly well suited for proving the finite-time convergence of backward schemes is provided in [MCB25a], where it is also shown that many discrete finite-time stable maps are, in fact, the backward-Euler discretization of a maximal monotone operator.

A discrete-time multivariable twisting-type controller is obtained by applying a backward-Euler discretization to its continuous-time counterpart. A discrete-time Lyapunov analysis motivates structural modifications that improve precision and energy efficiency while recovering global asymptotic stability in the absence of perturbations. Moreover, the monotone structure of the controller components enables the use of efficient operator-splitting algorithms for the online computation of the control input  [MCB26].

Discontinuous switching surfaces

In conventional sliding-mode control, insensitivity to matched disturbances originates from the piecewise-smooth nature of the closed-loop system. It is natural to expect a certain degree of robustness also to unmatched disturbances if the switching surface is itself nonsmooth. One can find results on the properties of systems with sigmoidal switching surfaces, but the limiting case of a true discontinuous switching surface has remained elusive.

A set-valued approach from the outset shows the well-posedness and the limiting behavior of a planar piece-wise smooth system having a discontinuous switching surface [MBC19]. The system is subject to both matched and unmatched disturbances. The approach leads to a single nonsmooth Lyapunov function that decreases along the reaching, sliding, and oscillatory phases of the system trajectories. The implicit Euler discretization of the controller leads to a robust controller with reduced chattering.

Homogeneous control of homogeneous systems with nonholonomic constraints

The main theoretical challenge in the face of a system with nonholonomic constraints is that, even though such systems are controllable, they fail Brockett's test and, hence, are not stabilizable by continuous static feedback laws. Many mechanical systems with nonholonomic constraints can be written in the well-known chained or power forms. When written in these forms:

• the Lie algebras generated by the system vector fields are nilpotent (so easily computed),

• the controllability rank condition is easily verified,

• the system representation is homogeneous,

• and the failure of Brockett's test is readily exposed.

A two-stage homogeneity-preserving controller is presented in [RCM22]. By imposing a negative degree, finite-time stability is attained. The homogeneity of the power form also lends itself to the application of the generalized circle criterion [RMC18], from which absolute stability is achieved. An explicit homogeneous control Lyapunov function is instrumental in the design of the controller.

The design strategy is tried out on a perturbed kinematic model of a car. 

Multivalued Hamiltonian systems with sliding motions

Several second-order sliding-mode controllers can be written as multivalued Hamiltonian systems. These include the twisting, the super-twisting, and the nested algorithms. Being Hamiltonian systems, stability can be easily established using a simple, yet nonsmooth, energy-like function. Asymptotic stability can then be established using the nonsmooth and multivalued version of LaSalle's invariance developed by Bacciotti and Ceragioli. The backward-Euler discretization of such systems is well-posed and preserves the finite-time convergence of the original continuous-time systems [CMB25].

Sliding motions on non-Euclidean state spaces

Filippov's theory of differential equations with discontinuous right-hand sides was developed for systems evolving in Euclidean spaces. In [Cas26]., Filippov's definition is extended to systems defined on smooth manifolds. The main challenge is to describe the notion of solution solely in terms of the manifold’s geometric/topological structure, rather than relying on local (coordinate-based) representations. Also, particular attention must be paid to switching sets, i.e., sets along which the vector field is discontinuous. These sets are required to consist of collections of embedded submanifolds, which ensures that well-defined reduced-order dynamics can be constructed on them.

The regular form can generalized to non-Euclidean settings, which opens the door to the design of first- and second-order sliding-mode controllers that respect the manifold structure. However, particular attention must be given to the role of topological obstructions, as these fundamentally influence the possible sliding dynamics. This is illustrated through examples on the cylinder, Möbius bundle, SO(3), and 2-sphere.