Implicit systems

Sliding-mode control of implicit systems

Implicit systems are described by mathematical models in which the state's derivative is determined implicitly by a function whose arguments are: the state, the derivative itself, and the control. The class of implicit (or descriptor) models contains the class of state-space models, in which the state's derivative is given explicitly as a function of the state and the control (i.e., vector fields). Implicit models can be used, e.g., to describe systems that perform a derivative action on the input, a feature that cannot be captured by state-space models. In [CHZF14], we apply integral sliding-mode control to linear descriptor systems to reject matched perturbations. Eigenvalue assignment using conventional sliding-mode control is considered in [HZCF16]. In contrast with classical state-space systems, one has to take special care of the differentiability of the control action (otherwise, a solution to the system equations might not exist).

Control of implicit port-Hamiltonian systems

For a restricted class of systems, it is possible to suitably eliminate some of the state variables and transform a given implicit model into an explicit one. Even when possible, in principle, it might be preferable to work directly with an implicit model in some cases. In the specific context of Hamiltonian systems, going from an implicit representation to an explicit one entails an effective reduction in the number of state variables but complicates the Hamiltonian (energy) function substantially. Implicit representations are of higher dimension and, depending on the particular application, may require us to solve for variables defined implicitly. On the other hand, the corresponding Hamiltonian functions have simpler expressions. More precisely, it is possible to split the Hamiltonian functions into two terms: one depending on the momenta only (the kinetic energy) and one depending on the positions only (the potential energy), i.e., the Hamiltonians are separable. This property has been largely exploited in numerical simulations, resulting in self-correcting numerical simulation algorithms and discrete-time sampled-data models. When applying the interconnection and damping assignment passivity-based control methodology to a Hamiltonian system, the resulting partial differential equations are simplified considerably if the kinetic energy does not depend on the positions. Thus, there are possible advantages to using implicit representations in a control context as well.

The relationship between implicit and explicit port-Hamiltonian models can be understood in terms of commutative diagrams. See [CGHM13] for details and, in particular, the explicit maps for going from one representation to the other. Discrete-time models for port-Hamiltonian systems are provided in [CMGH15]. The interconnection and damping assignment (IDA) control technique is developed for implicit port-Hamiltonian models in [CG16]. A linear matrix inequality framework for under-actuated mechanical systems is presented in [CCR19].

State observers for linear time-varying implicit systems

The problem of state observation for a large class of systems can be recast as a parameter-estimation problem, namely, the problem of estimating the initial condition. This is the approach undertaken when designing generalized parameter estimation-based observers (GPEBOs). The approach is particularly useful for the state observation of state-affine systems, which can be easily written as linear time-varying (LTV) systems. Furthermore, it can be shown that the state of LTV systems can be reconstructed using a GPEBO by only imposing the (necessary) condition of observability. Such a condition stands in contrast with the usual stronger requirement of uniform observability. In [OBCN26], GEPEBOs are extended to descriptor systems.