Sliding Mode Observers for Set-valued Lur’e Systems with Uncertainties Beyond Observational Range

Differential inclusion serves as an extension of ordinary differential equations, playing a pivotal role as a mathematical model within applied mathematics and control theory. A set-valued Lur’e dynamical system, a form of differential inclusion, incorporates interconnection feedback, broadening its scope to encompass various models frequently used in the analysis of nonsmooth dynamical systems. Such models include evolution variational inequalities, projected dynamical systems, relay systems, and complementarity systems. Set-valued Lur’e dynamical systems have been the subject of thorough investigations in recent decades (we refer, for example, to [1, 2, 3, 7, 8, 9, 12, 14, 20, 22, 27, 28] and the references therein). An observer is a mathematical model used to reconstruct complete information about the state variables from partially measurable data. Observers design for set-valued Lur’e systems represents a significant practical challenge, alongside other fundamental issues like well-posedness and stability analysis, which have received extensive attention in the literature. Numerous references addressing observer design for set-valued Lur’e systems are available, with a predominant focus on two methodological approaches: Luenberger-like techniques [11, 15, 16, 17, 27, 28] and sliding mode methodologies [4, 21].

The Luenberger-like observers for set-valued Lur’e systems was initially explored by Brogliato-Heemels in [11], which is an extension of the work of Arcak-Kokotovic for the nonlinear case [5]. Brogliato-Heemels considered the differential inclusion

and proposed the observer

where $A,B,C,G,F,K$ are matrices with appropriate dimensions, $\mathcal{F}$ is a maximal monotone set-valued operator and $y$ is the partial measurable output. Later, it was extended by Huang et al in [15] to incorporate uncertainty as follows

where $\theta$ is an unknown constant. The proposed observer is

where the range condition $f_{2}^{T}(x,u)P=h(x,u)F$ is satisfied for some symmetric positive definite matrix $P$. The key idea in [15] is to approximate the unknown by using an additional ODE, aiming to achieve observer convergence with an unknown rate. On the other hand, B. K. Le [21] proposed a sliding mode observer for a more general class of set-valued Lur’e systems, which extends the nonlinear case considered by Xiang-Su-Chu [29], see also [26] for a survey. Following this, Adly-Le applied the technique to system (3) in [4], offering clear advantages over Luenberger-like methods (4). These advantages include improved assumptions and achieving exponential convergence for the sliding mode observer without the need to solve additional ordinary differential equations. Moreover, the authors proposed a new, efficient smoothing approximation for the sliding mode technique to reduce the chattering effect using time guiding functions.

Nonetheless, a significant limitation of existing results using Luenberger-like or sliding mode observers (we refer, for example, to [4, 15, 21]) is the necessity of the range condition for uncertainties, meaning that uncertainties must fall within the range of $P^{-1}F^{T}$. An alternative approach involves constructing $H^{\infty}$ observers. However, $H^{\infty}$ observers only offer estimations for the entire process under zero initial conditions, rather than providing estimations at specific time instances (see, for instance, [17]). This limitation prompts us to introduce a sliding mode observer to attain time-specific estimations for the Lur’e system in the presence of uncertainties that do not fall within the observation range, as follows:

where $A\in{\mathbb{R}}^{n\times n},B\in{\mathbb{R}}^{n\times m},C\in{\mathbb{R}}^{m\times n},F\in{\mathbb{R}}^{p\times n}$ are given matrix, $x\in{\mathbb{R}}^{n}$ is the state variable, $\mathcal{F}:{\mathbb{R}}^{m}\rightrightarrows{\mathbb{R}}^{m}$ is a maximal monotone operator, $u\in{\mathbb{R}}^{r}$ is the control input and $y\in{\mathbb{R}}^{p}$ is the measurable output. The nonlinear functions $f$ is Lipschitz continuous and $\xi\in{\mathbb{R}}^{l}$ is unknown. The model in (5) is quite general since in the right-hand side of the ODE, one has the linear $Ax$, the nonlinear $f(x,u)$, the set-valued $B\mathcal{F}(Cx)$ and the general unknown $\xi$. The idea is to decompose $\xi$ into 2 parts: $\xi=\xi_{1}+\xi_{2}$ where $\xi_{1}=P^{-1}F^{T}\xi$ is the projection of $\xi$ onto the observation subspace ${\rm Im}(P^{-1}F^{T})$ and $\xi_{2}$ is in the complement of the observation subspace. We show that if $\xi_{2}$ is bounded, we can obtain a useful estimation for the state of the original system. Interval estimation is practical and unavoidable if the uncertainty $\xi$ is general and not in the range of observation. We refer here some interval observers results for some specific systems [18, 23]. In addition, if $\xi_{2}$ tends to vanish when the time is large, we obtain the exact sliding mode observer, i.e., the observer state converges to the original state asymptotically. Finally, we show that our proposed observer is a $T$- observer, a notion that we introduce to obtain a time instant estimation for the error. We also provide a mild additional condition such that our observer is a strong $H^{\infty}$ observer, an extension of $H^{\infty}$ observers which does not require the zero initial condition. $T$- observers and strong $H^{\infty}$ observers allow us to obtain the total process and time instant estimations of the errors. In practice, there is a preference for the insights provided by the characteristics of strong $H^{\infty}$ observers over those of $H^{\infty}$ observers.

The paper is structured as follows: In Section 2, we revisit established definitions and essential results for our subsequent analyses. Section 3 presents our proposal for a sliding mode observer for the original system, which simultaneously functions as a $T$- observer as well as a strong $H^{\infty}$ observer. In Section 4, we give several numerical examples to validate the theoretical findings. The paper concludes in Section 5, where we offer our closing remarks and explore potential future directions.

We denote the scalar product and the corresponding norm of Euclidean spaces by $\langle\cdot,\cdot\rangle$ and $\|\cdot\|$ respectively. A matrix $P\in{\mathbb{R}}^{n\times n}$ is called positive definite, written $P>0$, if there exists $a>0$ such that

The Sign functions in ${\mathbb{R}}^{n}$ is defined by

where $\mathbb{B}$ denotes the closed unit ball in ${\mathbb{R}}^{n}$.

A set-valued mapping $\mathcal{F}:{\mathbb{R}}^{m}\rightrightarrows{\mathbb{R}}^{m}$ is called $\it{monotone}$ if for all $x,y\in{\mathbb{R}}^{m}$ and $x^{*}\in\mathcal{F}(x),y^{*}\in\mathcal{F}(y)$, one has

Furthermore, $\mathcal{F}$ is called $\it{maximal\;monotone}$ if there is no monotone operator $\mathcal{G}$ such that the graph of $\mathcal{F}$ is contained strictly in the graph of $\mathcal{G}$ (see, e.g, [6, 13]).

In this section, we propose a sliding mode observer for the system (5) under the following assumptions:

Let us recall the definition of $H^{\infty}$ observer (see, e.g., [17]) and introduce the notion of strong $H^{\infty}$ observer as well as $T$-observer.