2. Problem Definition

In the case of rigid dynamical systems, SMC algorithm establishment is possible even for rather complicated mechanisms. However, for continuum mechanics systems, the equations of motions are often extracted based on the calculation of complicated system energy expressions such as Lagrange or Hamilton principle. The obtained equations generally include complicated terms constituting spatial derivatives resulted from transverse vibrations of continuum subsystems which render the expressions difficult to work within control applications.

In order to derive dynamical equations of motion regarding continuum mechanics system, modal analysis is required to demonstrate transverse vibrations. To this end, corresponding partial differential equations should be transformed to set of ordinary differential equations [Reference Rao2] using Eq. (1):

(1) \begin{align} w\left( {,t} \right) = \mathop \sum_{j = 1}^{{m_\phi }} {\Phi _j}\left( \xi\right){q_j}(t)\end{align}

where ${\textrm{w}}\!\left( {{{\xi}},{\textrm{t}}} \right)$ denotes discretized expression of transverse vibration, ${{{\Phi}}_{\textrm{j}}}(\xi)$ are potentially unknown spatial functions regarding mode shapes in certain boundary conditions, ${{\textrm{q}}_{\textrm{j}}}({\textrm{t}})$ are corresponding temporal state variables and $${m_\phi }$$ is the total number of mode shapes used for discretization of transverse vibrations. In order to mitigate calculation burden during discretization, only a limited number of vibrational modes are taken into account and usually, high-frequency modes are ignored due to their relatively minor effects. This approximation is not valid when the system is stimulated by high-frequency disturbances that resonate with high-frequency modes and amplify their already ignored effect. The issue demands more attention when switching occurs and alters system dynamics in an unknown pattern.

In this section, discrete form of state-space realization of MIMO continuum mechanics systems is expressed in Eq. (2). The obtained formulations are subsequently used in the construction of control algorithm. The procedure of obtaining dynamical expressions of Eq. (2) is detailed in the authors’ previous work [Reference Homaeinezhad20]. In this study, a novel control algorithm is proposed considering this class of dynamical systems under arbitrarily generated switching effects. In addition, the significant effect of control input gains uncertainty which was ignored previously will be included here. The considered class of dynamical systems without switching effects is expressed as

(2) $$\matrix{ {{{\underline x }_{\rm{r}}}\left( {k + 1} \right)} & { = \underline {{f_{r}}} [\underline x (k)] + {{\underline {\rm{f}} }_{\rm{q}}}[{{\underline {\bf{x}} }_q}(k)] + {\bf{B}}(k)\underline u (k) + \underline v (k){\rm{ }}} \hfill \cr \quad \quad {\underline \nu (k)} & { = [{\nu _1}(k),\; \cdots ,\;{\nu _{{m_u}}}(k)]{\rm{ }}} \hfill \cr \quad \quad {{\nu _i}(k)} & { = \underline b _{i,k - 1}^T(k)\underline u (k - 1) + \cdots + \underline b _{i,k - {m_c}}^T(k)\underline u (k - {m_c})} \hfill \cr } $$

where ${\underline{\textbf{x}}} _{{\textrm{r}}}({{k}}) = [{{{x}}_1}({{k}}),\ \cdots,\ {{{x}}_{{{{m}}_{{r}}}}}({{k}}) ]^{\textrm{T}}$ denotes position vectors corresponding to rigid states and ${{\textrm{m}}_{\textrm{r}}}$ represents system total degrees of freedom (DOF). ${{\textbf{x}}_{\textrm{k}}}$ denotes states corresponding to a rigid model of the system and ${\underline {\textbf{x}}} _{{\textrm{q}}}({\textrm{k}}) = [ \underline {\textbf{x}} (\textrm{k}),\ {{{\underline{\textbf{q}}}_{\textrm{f}}}(\textrm{k})} ]$ includes both system states relating to rigid dynamics along with continuum mechanics vibrational terms ${{\textrm{q}}_{\textrm{i}}}(\textrm{k})$ . ${\underline{\boldsymbol{\nu}}}(\textrm{k})$ contains terms corresponding to past values of control inputs resulted from discretization order of ${{\textrm{m}}_{\textrm{c}}}$ . ${{\textbf{f}}_{\textrm{r}}}[ {\underline {\textbf{x}} (\textrm{k})} ]$ includes dynamical terms related to a rigid model of system and ${{\textbf{f}}_{\textrm{q}}}[ {{{\underline {\textbf{x}} }_{\textrm{q}}}(\textrm{k})} ]$ represents dynamical effects resulted from continuum mechanics vibrations which is difficult to accurately ascertain and will be approximated in this study. ${\textbf{B}}(\textrm{k})$ denotes control input gains including both parametric and unstructured uncertainties, ${{\textrm{m}}_{\textrm{u}}}$ is the total number of control inputs and ${\underline{\textbf{u}} (\textrm{k})}$ stands for control input vector for MIMO system.

Considering multiple switching configurations, Eq. (2) for each individual subsystem can be written like:

(3) $${{x_{{r_{ij}}}}\left( {k + 1} \right) = {f_{{r_{ij}}}}\left[ {\underline x (k)} \right] + {f_{{q_{ij}}}}\left[ {{{\underline x }_q}(k)} \right] + \underline b _{ij}^T(k)\underline u (k) + {\nu _{ij}}(k)}$$

where index ${{j}}$ stands for arbitrary switching configuration for each term.

In conclusion, for MIMO continuous vibrational systems, the existence of spatial derivative terms of transverse vibrations, nonlinear terms, and boundary conditions in addition to various interactions between subsystems tend to render the task of obtaining a dynamical model almost inaccessible, resource demanding, and potentially inaccurate. Naturally, dynamical and numerical analysis of obtained models is likewise impossible. The fact that few studies have been conducted to address tracking control of MIMO flexible system indicates the aforementioned issue. In this paper, overcoming complications arising from the difficulty of dynamical analysis in control of switched vibrational systems is discussed and FEA is applied to approximate modeling uncertainties caused by the continuum mechanics vibrations.

3 Main results

Considering the proposed model for each individual subsystem in Eq.(3), in this section, discrete sliding mode control (DSMC) algorithm will be developed such that closed-loop stability and tracking performance of continuum mechanics MIMO system in presence of both parametric and unstructured uncertainties with unknown switching configuration is ensured. To this end, the sliding function ${{\textrm{s}}_{{\textrm{ij}}}}(\textrm{k})$ for each individual subsystem ${\textrm{i}}$ at given switching configuration ${\textrm{j}}$ is defined as

(4) \begin{align}{s_{ij}}( k ) \buildrel \Delta \over = \tilde{x_r}_{ij}( k ) + {\lambda _{ij}}\tilde{x_r}_{ij}\left( {k - 1} \right)\end{align}

Where $\tilde{\textrm{x}}_{{\textrm{r}}_{{\textrm{ij}}}}( {\textrm{k}}) = {\textrm{x}}_{{\textrm{r}}_{{\textrm{ij}}}}( {\textrm{k}}) - {\textrm{x}}_{{\textrm{id}}}( {\textrm{k}})$ and ${{\textrm{x}}_{{\textrm{id}}}}( {\textrm{k}})$ is the desired reference for ${{\textrm{x}}_{\textrm{i}}}( {\textrm{k}})$ . ${{{\lambda }}_{{\textrm{ij}}}}$ s are control parameters set such that ${{\textrm{s}}_{{\textrm{ij}}}}( {\textrm{k}}) = 0$ would result in ${{\textrm{x}}_{\textrm{r}}}_{{\textrm{ij}}}( {\textrm{k}}) = {{\textrm{x}}_{{\textrm{id}}}}( {\textrm{k}})$ .

(5) \begin{align}\hat{s}_{ij}( k ) = \frac{{2 - {\omega _{0s}}T}}{{2 + {\omega _{0s}}T}}\hat{s}_{{ij}}\left( {k - 1} \right) + \frac{{{\omega _{0s}}T}}{{2 + {\omega _{0s}}T}}\left[ {{s_{ij}}( k ) + {s_{ij}}\left( {k - 1} \right)} \right]\end{align}

In addition, high-frequency signals included in control inputs that are generated due to measurement noise are normally ignored due to the limited frequency bandwidth of physical actuators. In other words, physical actuator dynamics oftentimes behave as a first-order low-pass filter in the presence of high-frequency signals. In such cases, it is necessary to similarly incorporate actuator dynamics in the construction of control law.

(6) \begin{align}{u_i}( k ) = \frac{{2 - {\omega _{0a}}T}}{{2 + {\omega _{0a}}T}}{u_i}\left( {k - 1} \right) + \frac{{{\omega _{0a}}T}}{{2 + {\omega _{0a}}T}}\left[ {{u_{ic}}( k ) + {u_{ic}}\left( {k - 1} \right)} \right]\end{align}

In Eqs. (5), and (6), ${{{\omega }}_{0{\textrm{s}}}}$ and ${{{\omega }}_{0{\textrm{a}}}}$ are bandwidth frequencies corresponding to low-pass filters applied, respectively, to sliding functions and control input signals. Replacing ${{\textrm{u}}_{\textrm{i}}}( {\textrm{k}})$ from Eq. (6) into convex inequality of Eq. (14) which is being discussed further, $\underline{\textbf{U}}_{\textrm{c}}(\textrm{K}-1)=[\textrm{U}_{1\textrm{c}}(\textrm{k}),\ \cdots\ \textrm{U}_{\textrm{nc}}(\textrm{k})]^{\textrm{T}}$ is obtained. Hence, the control signal is calculated considering actuators’ frequency bandwidth and measurement noise effects [Reference Homaeinezhad, FotoohiNia and Gholyan15].

Substituting ${{\textrm{x}}_{\textrm{r}}}_{{\textrm{ij}}}( {\textrm{k}})$ from Eq. (3) into Eq. (4) and shifting the equation for one sample ahead results in:

(7) $${{s_{ij}}\left( {k + 1} \right) = {f_{{r_{ij}}}}\left[ {\underline{\textbf {x}} (k)} \right] + {f_{{q_{ij}}}}\left[ {{{\underline x }_q}(k)} \right] + {\nu _{ij}}(k) + {\lambda _{ij}}{x_r}_{ij}(k) - {x_{id}}\left( {k + 1} \right) - {\lambda _{ij}}{x_{id}}(k) + \underline b _{ij}^T(k)\underline u (k)}$$

Eq. (7) is further summarized to Eq. (8):

(8) \begin{align} {s_{ij}}(k + 1) &= {f_{rij}}[\underline{\textbf{x}} (k)] + {f_{{q_{ij}}}}[{\underline{\textbf{x}} _q}(k)] + \underline{{b}} _{ij}^T(k)\underline{{u}} (k) + {v_{ij}}(k) + {\xi_{ij}}(k), \nonumber\\[3pt] &\quad {{{\xi}}_{{{ij}}}}({{k}}) \buildrel \Delta \over = {{{\lambda }}_{{{ij}}}}{{{x}}_{{r}}}_{{{ij}}}( {{k}}) - {{{x}}_{{{id}}}}\left( {{{k}} + 1} \right) - {{{\lambda }}_{{{ij}}}}{{{x}}_{{{id}}}}( {{k}}) \end{align}

Theorem 1. Discrete-Time MIMO flexible system response expressed in Eq. (2) under unknown switching signal and both parametric and unstructured uncertainties will follow the desired sliding set $${\Omega _{\bf{i}}} = \left\{ {{{\rm{s}}_{\rm{i}}}({\rm{k}}){\mkern 1mu} :{\mkern 1mu} \left| {{{\rm{s}}_{\rm{i}}}({\rm{k}})} \right| \le {\rho _{\rm{i}}}} \right\}$$ in finite number of samples under Assumption 1. In case control input is selected within calculated bounds, it is as follows:

(9) $$\matrix{ {} \hfill & {\underline u (k) = \left[ {1 - \alpha (k)} \right]{{\bf{B}}^{ - 1}}(k){{\underline Q }_L}(k) + \alpha (k){B^{ - 1}}(k){{\underline Q }_H}(k){\rm{ }}} \hfill \cr {} \hfill & {\alpha (k) \in (0,1)} \hfill \cr } $$

${\underline {\textbf{Q}} _{\textrm{L}}}( {\textrm{k}})$ and ${\underline {\textbf{Q}} _{\textrm{H}}}( {\textrm{k}})$ are calculated as lower and upper control input bounds and ${{\alpha }}( {\textrm{k}})$ is the control input proximity factor which is set by the control designer.

Proof. Candidate Lyapunov function ${{\textrm{V}}_{\textrm{i}}}( {\textrm{k}})$ for each subsystem is defined in Eq. (10).

(10) \begin{align}{V_i}( k ) \buildrel \Delta \over = {\left[ {\mathop \sum \limits_{j = 1}^{{m_\sigma }} {\eta _{ij}}{s_{ij}}( k )} \right]^2}\end{align}

In Eq. (10), ${{\textrm{m}}_{{\sigma }}}$ is the number of total possible switching configurations and ${{{\eta }}_{{\textrm{ij}}}} \gt 0$ is an arbitrary weighting factor set by the designer in accordance with the possibility and severity of each switching configuration. The control law corresponding to the dynamical model presented in Eq. (2) is derived such that candidate Lyapunov function is monotonically decreasing. In order to attain finite-time convergence, the Lyapunov function should satisfy Eq. (11) with finite-time convergence factor of $${\mu _{\rm{i}}} \in \left( {0,1} \right)$$

(11) \begin{align}{V_i}\left( {k + 1} \right) \lt \mu _i^2{V_i}( k )\end{align}

As a result;

(12) \begin{align}{\left[ {\mathop \sum \limits_{j = 1}^{{m_\sigma }} {\eta _{ij}}{s_{ij}}\left( {k + 1} \right)} \right]^2} \lt \mu _i^2{\left[ {\mathop \sum \limits_{j = 1}^{{m_\sigma }} {\eta _{ij}}{s_{ij}}( k )} \right]^2}\end{align}

Thus,

(13) \begin{align} - \mu \left| {\mathop \sum \limits_{j = 1}^{{m_\sigma }} {\eta _{ij}}{s_{ij}}( k )} \right| \lt \mathop \sum \limits_{j = 1}^{{m_\sigma }} {\eta _{ij}}{s_{ij}}\left( {k + 1} \right) \lt + {\mu _i}\left| {\mathop \sum \limits_{j = 1}^{{m_\sigma }} {\eta _{ij}}{s_{ij}}( k )} \right|\end{align}

Substituting ${{\textrm{s}}_{{\textrm{ij}}}}( {{\textrm{k}} + 1})$ from Eq. (8) to Eq. (13) leads to:

(14) $$\matrix{ { - \mu \left| {\sum\limits_{j = 1}^{{m_\sigma }} {{\eta _{ij}}} {s_{ij}}(k)} \right|} \hfill & { < \sum\limits_{j = 1}^{{m_\sigma }} {{\eta _{ij}}} \left\{ {{f_{{r_{ij}}}}\left[ {\underline x (k)} \right] + {f_{{q_{ij}}}}\left[ {{{\underline x }_q}(k)} \right] + \underline b _{ij}^T(k)\underline u (k) + {\nu _{ij}}(k) + {\xi _{ij}}(k)} \right\}} \hfill \cr {} \hfill & {{\rm{ < }} + {\mu _i}\left| {\sum\limits_{j = 1}^{{m_\sigma }} {{\eta _{ij}}} {s_{ij}}(k)} \right|} \hfill \cr } $$

Since real system parameters are oftentimes difficult or even impossible to accurately ascertain, parametric uncertainties are inevitable in dynamical analysis and should be considered when obtaining the control algorithm. Therefore, the nominal value of system parameters should be used in the construction of control algorithm. Defining $\textrm{g}_{\textrm{ij}}(\textrm{k}) {\triangleq } \textrm{f}_{\textrm{rij}}[\underline{\textbf{x}}(\textrm{k})]+\nu_{\textrm{ij}}(\textrm{k})$ and ${{\hat{g}}_{{\textrm{ij}}}}( {\textrm{k}})$ and ${{\hat{f}}_{{\textrm{qij}}}}[\underline{\textbf{x}}_{\textrm{q}}(\textrm{k})]$ as approximations (with known nominal parameters) of rigid model and flexible terms, respectively, Eq. (14) turns to Eq. (15) by adding and subtracting the approximated terms $\hat{\textrm{g}}_{\textrm{ij}}(\textrm{k})+{{\hat{\textrm{f}}}_{{\textrm{qij}}}}[\underline{\textbf{x}}_{\textrm{q}}(\textrm{k})]$ .

(15) $$\matrix{ { - \mu \left| {\sum\limits_{j = 1}^{{m_\sigma }} {{\eta _{ij}}} {s_{ij}}(k)} \right|} \hfill & {{\rm{ < }}\sum\limits_{j = 1}^{{m_\sigma }} {{\eta _{ij}}} \{ \left[ {{g_{ij}}(k) + {f_{{q_{ij}}}}\left[ {{{\underline x }_q}(k)} \right]} \right] - \left[ {{{\hat g}_{ij}}(k) + {{\hat f}_{{q_{ij}}}}\left[ {{{\underline x }_q}(k)} \right]} \right] + {{\hat g}_{ij}}(k){\rm{ }}} \hfill \cr {} \hfill & {\quad + {{\hat f}_{{q_{ij}}}}\left[ {{{\underline x }_q}(k)} \right] + \underline b _{ij}^T(k)\underline u (k) + {\xi _{ij}}(k)\} {\rm{ < }} + {\mu _i}\left| {\sum\limits_{j = 1}^{{m_\sigma }} {{\eta _{ij}}} {s_{ij}}(k)} \right|} \hfill \cr } $$

By subtracting $\sum _{\textrm{j}=1}^{{\textrm{m}}_{{\sigma }}}{{\eta }}_{\textrm{i}\textrm{j}}\{[{\textrm{g}}_{\textrm{i}\textrm{j}}(\textrm{k})+{\textrm{f}}_{{\textrm{q}}_{\textrm{i}\textrm{j}}}[{\underline{\mathbf{x}}}_{\textrm{q}}(\textrm{k})]]-[{\hat{\textrm{g}}}_{\textrm{i}\textrm{j}}(\textrm{k})+{\hat{\textrm{f}}}_{{\textrm{q}}_{\textrm{i}\textrm{j}}}[{\underline{\mathbf{x}}}_{\textrm{q}}(\textrm{k})]]+{\hat{\textrm{g}}}_{\textrm{i}\textrm{j}}(\textrm{k})+{\hat{\textrm{f}}}_{{\textrm{q}}_{\textrm{i}\textrm{j}}}[{\underline{\mathbf{x}}}_{\textrm{q}}(\textrm{k})]+{{\xi}}_{\textrm{i}\textrm{j}}(\textrm{k})\} $ from all sides of convex inequality of (15), it is obtained that:

(16) $$\matrix{ {} \hfill & { - \sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {{\eta _{{\rm{ij}}}}} \left\{ {\left[ {{{\rm{g}}_{{\rm{ij}}}}({\rm{k}}) + {{\rm{f}}_{{{\rm{q}}_{{\rm{ij}}}}}}\left[ {{{\underline {\bf{x}} }_{\rm{q}}}({\rm{k}})} \right]} \right] - \left[ {{{\widehat{\rm{g}}}_{{\rm{ij}}}}({\rm{k}}) + {{\widehat{\rm{f}}}_{{{\rm{q}}_{{\rm{ij}}}}}}\left[ {{{\underline {\bf{x}} }_{\rm{q}}}({\rm{k}})} \right]} \right] + {{\widehat{\rm{g}}}_{{\rm{ij}}}}({\rm{k}}) + {{\widehat{\rm{f}}}_{{{\rm{q}}_{{\rm{ij}}}}}}\left[ {{{\underline {\bf{x}} }_{\rm{q}}}({\rm{k}})} \right] + {\xi _{{\rm{ij}}}}({\rm{k}})} \right\}{\rm{ }}} \hfill \cr {} \hfill & {\quad - {\mu _{\rm{i}}}\left| {\sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {{\eta _{{\rm{ij}}}}} {{\rm{s}}_{{\rm{ij}}}}({\rm{k}})} \right|{\rm{ < }}\sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {{\eta _{{\rm{ij}}}}} \left[ {\underline {\bf{b}} _{{\rm{ij}}}^{\rm{T}}({\rm{k}})\underline {\bf{u}} ({\rm{k}})} \right]{\rm{ < }} - \sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {{\eta _{{\rm{ij}}}}} \{ \left[ {{{\rm{g}}_{{\rm{ij}}}}({\rm{k}}) + {{\rm{f}}_{{{\rm{q}}_{{\rm{ij}}}}}}\left[ {{{\underline {\bf{x}} }_{\rm{q}}}({\rm{k}})} \right]} \right]{\rm{ }}} \hfill \cr {} \hfill & {\quad - \left[ {{{\widehat{\rm{g}}}_{{\rm{ij}}}}({\rm{k}}) + {{\widehat{\rm{f}}}_{{{\rm{q}}_{{\rm{ij}}}}}}\left[ {{{\underline {\bf{x}} }_{\rm{q}}}({\rm{k}})} \right]} \right] + {{\widehat{\rm{g}}}_{{\rm{ij}}}}({\rm{k}}) + {{\widehat{\rm{f}}}_{{{\rm{q}}_{{\rm{ij}}}}}}\left[ {{{\underline {\bf{x}} }_{\rm{q}}}({\rm{k}})} \right] + {\xi _{{\rm{ij}}}}({\rm{k}})\} + {\mu _{\rm{i}}}\left| {\sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {{\eta _{{\rm{ij}}}}} {{\rm{s}}_{{\rm{ij}}}}({\rm{k}})} \right|} \hfill \cr } $$

(17) $$\matrix{ {{g_i}(k) + {f_{qij}}[{{\underline x }_q}(k)]} \hfill & { \approx {{\rm{x}}_{rij}}(k) - \underline b _{ij}^T(k)\underline u (k){\rm{ }}} \hfill \cr \quad \qquad {{{\hat f}_{qij}}[{{\underline x }_q}(k)]} \hfill & { \approx {{\hat x}_{rij}}(k) - \underline b _{ij}^T(k)\underline u (k) - {{\hat g}_{ij}}(k)} \hfill \cr } $$

System states for dummy models are expressed as ${{\hat{\textrm{x}}}_{{\textrm{ij}}}}( {\textrm{k}})$ at given switching configuration of ${\textrm{j}}$ . Considering approximations in Eq. (17), Eq. (16) yield to Eq.(18):

(18) $$\matrix{ {} \hfill & { - \sum\limits_{j = 1}^{{m_\sigma }} {{\eta _{ij}}} \left\{ {{x_r}_{ij}(k) - {{\hat x}_{{r_{ij}}}}(k) + {{\hat g}_{ij}}(k) + {{\hat f}_{{q_{ij}}}}\left[ {{{\underline x }_q}(k)} \right] + {\xi _{ij}}(k)} \right\}{\rm{ }}} \hfill \cr {} \hfill & { - \sum\limits_{j = 1}^{{m_\sigma }} {{\eta _{ij}}} \left\{ { - \left[ {\underline {\bf{b}} _{ij}^T\left( {k - 1} \right) - \underline {\widehat{\bf{b}}} _{ij}^T\left( {k - 1} \right)} \right]\underline {\bf{u}} \left( {k - 1} \right)} \right\} - {\mu _i}\left| {\sum\limits_{j = 1}^{{m_\sigma }} {{\eta _{ij}}} {s_{ij}}(k)} \right|{\rm{ }}} \hfill \cr {} \hfill & {{\rm{ < }}\sum\limits_{j = 1}^{{m_\sigma }} {{\eta _{ij}}} \left[ {\underline {\bf{b}} _{ij}^T(k)\underline {\bf{u}} (k)} \right]{\rm{ < }} + {\mu _i}\left| {\sum\limits_{j = 1}^{{m_\sigma }} {{\eta _{ij}}} {s_{ij}}(k)} \right| - \sum\limits_{j = 1}^{{m_\sigma }} {{\eta _{ij}}} \left\{ { - \left[ {\underline {\bf{b}} _{ij}^T\left( {k - 1} \right) - \underline {\widehat{\bf{b}}} _{ij}^T\left( {k - 1} \right)} \right]\underline {\bf{u}} \left( {k - 1} \right)} \right\}{\rm{ }}} \hfill \cr {} \hfill & { - \sum\limits_{j = 1}^{{m_\sigma }} {{\eta _{ij}}} \left\{ {{x_{{r_{ij}}}}(k) - {{\hat x}_{{r_{ij}}}}(k) + {{\hat g}_{ij}}(k) + {{\hat f}_{{q_{ij}}}}\left[ {{{\underline x }_q}(k)} \right] + {\xi _{ij}}(k)} \right\}} \hfill \cr } $$

(19) \begin{align}{{\textbf{B}}_{\textrm{j}}}( {\textrm{k}}) \buildrel \Delta \over = \left[ {\textbf{I}} + {{\boldsymbol{\Gamma }}_{\textrm{j}}}( {\textrm{k}}) \right]{{\hat{\textbf{B}}}_{\textrm{j}}}( {\textrm{k}})\end{align}

In which ${{\boldsymbol{\Gamma }}_{\textrm{j}}}( {\textrm{k}})$ denotes the uncertainty matrix with known bounds and ${{\hat{\textbf{B}}}_{\textrm{j}}}( {\textrm{k}})$ is the approximated controller signal multiplier matrix with nominal parameters. Furthermore, it is assumed that ${{\textbf{B}}_{\textrm{j}}}( {\textrm{k}}){{\boldsymbol{\Gamma }}_{\textrm{j}}}( {\textrm{k}})$ is nonsingular for all possible cases of model uncertainty and switching signals.

The terms at the sides of inequality expressed in Eq. (18) determine stability bounds of a control input signal in the form of $\textrm{q}_{\textrm{l,i}}(\textrm{k})\le \sum_{\textrm{j}=1}^{\textrm{m}\sigma}\eta_{\textrm{ij}} [\underline{\textbf{b}}_{\textrm{ij}}^{\textrm{T}}(\textrm{k})\underline{\textbf{u}}(\textrm{k})]\le\textrm{q}_{\textrm{h,i}}(\textrm{k})$ for every individual subsystem considering all switching signals ${\textrm{j}}$ in which ${{\textrm{q}}_{{\textrm{l}},{\textrm{i}}}}( {\textrm{k}})$ and ${{\textrm{q}}_{{\textrm{h}},{\textrm{i}}}}( {\textrm{k}})$ denote lower and upper stability bounds, respectively. Similar expressions could be obtained for all rigid dynamical states ${\textrm{i}} = 1, \ldots ,{{\textrm{m}}_{\textrm{r}}}$ therefore, the terms corresponding to control inputs bounds are merged into a set of convex inequalities according to Assumption 2.

(20) \begin{align}{\underline {\textbf{q}} _{\textrm{L}}}({\textrm{k}}) \lt \left\{ {\mathop \sum \limits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} \left[ \begin{array}{c@{\quad}c@{\quad}c} {{{{\eta }}_{1{\textrm{j}}}}} & \cdots & 0 \\ \\[-7pt] \vdots & \ddots & \vdots \\ \\[-7pt] 0 & \cdots & {{{{\eta }}_{{{\textrm{m}}_{\textrm{x}}},{\textrm{j}}}}} \end{array} \right]\left[ {{{\textbf{B}}_{\textrm{j}}}( {\textrm{k}})} \right]} \right\}\underline {\textbf{u}} ( {\textrm{k}}) \lt {\underline {\textbf{q}} _{\textrm{H}}}({\textrm{k}})\end{align}

Where $\underline{\textbf{q}}_{\textrm{L}}(\textrm{k})=[\textrm{q}_{\textrm{l,1}}(\textrm{k})\ \cdots\ \textrm{q}_{\textrm{l,m}_{\textrm{u}}}(\textrm{k})]^{\textrm{T}}$ and $\underline{\textbf{q}}_{\textrm{H}}(\textrm{k})=[\textrm{q}_{\textrm{h,1}}(\textrm{k})\ \cdots\ \textrm{q}_{\textrm{h,m}_{\textrm{u}}}(\textrm{k})]^{\textrm{T}}$ .

According to Assumption 2, control input gains uncertainties equation $[ {{{\textbf{B}}_{\textrm{j}}}( {{\textrm{k}} - 1}) - {{{\hat{\textbf{B}}}}_{\textrm{j}}}( {{\textrm{k}} - 1})}] = {{\boldsymbol{\Gamma }}_{\textrm{j}}}( {{\textrm{k}} - 1}){{\hat{\textbf{B}}}_{\textrm{j}}}( {{\textrm{k}} - 1})$ is obtained. Adapting the introduced format in Eq. (20), stability bounds on either side of Eq. (18) could be rewritten as Eqs. (21–22) such that they include all subsystems.

(21) $$\matrix{ {} \hfill & {{{\underline {\bf{q}} }_{\rm{L}}}({\rm{k}}) \buildrel \Delta \over = \left\{ {\sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {\left[ {\matrix{ {{\eta _{1{\rm{j}}}}} & \cdots & 0 \cr {} & {} & {} \cr { \vdots } & \ddots & \vdots \cr {} & {} & {} \cr {0} & \cdots & {{\eta _{{{\rm{m}}_{\rm{x}}},{\rm{j}}}}} \cr } } \right]} \left[ {{\Gamma _{\rm{j}}}({\rm{k}} - 1){{\widehat{\bf{B}}}_{\rm{j}}}\left( {{\rm{k}} - 1} \right)} \right]} \right\}\underline {\bf{u}} \left( {{\rm{k}} - 1} \right)} \hfill \cr {} \hfill & {\quad + \left[ {\matrix{ { - \sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {\eta _1^{\rm{j}}} \left\{ {{\rm{x}}_{{{\rm{r}}_{\rm{i}}}}^{\rm{j}}({\rm{k}}) - {{\widehat{\rm{x}}}_{{{\rm{r}}_{\rm{i}}}}}{\rm{j}}({\rm{k}}) + {{\widehat{\rm{g}}}_{1{\rm{j}}}}({\rm{k}}) + {{\widehat{\rm{f}}}_{{{\rm{q}}_{1{\rm{j}}}}}}\left[ {{{\underline {\bf{x}} }_{\rm{q}}}({\rm{k}})} \right] + {\xi _{1{\rm{j}}}}({\rm{k}})} \right\} - {\mu _1}\left| {\sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {{\eta _{1{\rm{j}}}}} {{\rm{s}}_{1{\rm{j}}}}({\rm{k}})} \right|} \cr \vdots \cr { - \sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {{\eta _{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}} \left\{ {{{\rm{x}}_{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}({\rm{k}}) - {{\widehat{\rm{x}}}_{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}({\rm{k}}) + {{\widehat{\rm{g}}}_{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}({\rm{k}}) + {{\widehat{\rm{f}}}_{{{\rm{q}}_{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}}}\left[ {{{\underline {\bf{x}} }_{\rm{q}}}({\rm{k}})} \right] + {\xi _{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}({\rm{k}})} \right\} - {\mu _{{{\rm{m}}_{\rm{x}}}}}\left| {\sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {{\eta _{{{\rm{m}}_{\rm{x}}},{\rm{j}}}}} {{\rm{s}}_{{{\rm{m}}_{\rm{x}}},{\rm{j}}}}({\rm{k}})} \right|} \cr } } \right]} \hfill \cr } $$

(22) $$\matrix{ {} \hfill & {{{\underline {\bf{q}} }_{\rm{H}}}({\rm{k}}) \buildrel \Delta \over = {\rm{ }}} \hfill \cr {} \hfill & { \times \left[ {\matrix{ { - \sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {{\eta _{1{\rm{j}}}}} \left\{ {{{\rm{x}}_{{{\rm{r}}_{{\rm{ij}}}}}}({\rm{k}}) - {{\widehat{\rm{x}}}_{{{\rm{r}}_{{\rm{ij}}}}}}({\rm{k}}) + {{\widehat{\rm{g}}}_{1{\rm{j}}}}({\rm{k}}) + {{\widehat{\rm{f}}}_{{{\rm{q}}_{1{\rm{j}}}}}}\left[ {{{\underline {\bf{x}} }_{\rm{q}}}({\rm{k}})} \right] + {\xi _{1{\rm{j}}}}({\rm{k}})} \right\} + {\mu _1}\left| {\sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {{\eta _{1{\rm{j}}}}} {{\rm{s}}_{1{\rm{j}}}}({\rm{k}})} \right|} \cr \vdots \cr { - \sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {{\eta _{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}} \left\{ {{{\rm{x}}_{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}({\rm{k}}) - {{\widehat{\rm{x}}}_{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}({\rm{k}}) + {{\widehat{\rm{g}}}_{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}({\rm{k}}) + {{\widehat{\rm{f}}}_{{{\rm{q}}_{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}}}\left[ {{{\underline {\bf{x}} }_{\rm{q}}}({\rm{k}})} \right] + {\xi _{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}({\rm{k}})} \right\} + {\mu _{{{\rm{m}}_{\rm{x}}}}}\left| {\sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {{\eta _{{{\rm{m}}_{\rm{x}}},{\rm{j}}}}} {{\rm{s}}_{{{\rm{m}}_{\rm{x}}},{\rm{j}}}}({\rm{k}})} \right|} \cr } } \right]{\rm{ }}} \hfill \cr {} \hfill & {\quad + \left\{ {\sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {\left[ {\matrix{ {{\eta _{1{\rm{j}}}}} & \cdots & 0 \cr {} & {} & {} \cr { \vdots } & \ddots & \vdots \cr {} & {} & {} \cr {0} & \cdots & {{\eta _{{{\rm{m}}_{\rm{x}}},{\rm{j}}}}} \cr } } \right]} \left[ {{\Gamma _{\rm{j}}}({\rm{k}} - 1){{\widehat{\bf{B}}}_{\rm{j}}}\left( {{\rm{k}} - 1} \right)} \right]} \right\}\underline {\bf{u}} \left( {{\rm{k}} - 1} \right)} \hfill \cr } $$

Then, using $ 0\lt{\alpha }(\textrm{k})\textrm{}\lt1$ as proximity factor, control signals are obtained as

\begin{align*} \underline{\textbf{u}} ( {\textrm{k}}) &= \left[ {1 - {{\alpha }}( {\textrm{k}})} \right]{{\textbf{B}}^{ - 1}}( {\textrm{k}})\underline {\textbf{Q}} _{\textrm{L}}( {\textrm{k}}) + {{\alpha }}({\textrm{k}}){{\textbf{B}}^{ - 1}}( {\textrm{k}})\underline {\textbf{Q}} _{\textrm{H}}( {\textrm{k}})\end{align*}

(23) \begin{align} \underline{\textbf{Q}} _L({\textrm{k}}) &= \left\{ \sum\nolimits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} \left[ \begin{array}{c@{\quad}c@{\quad}c} {{{\eta }}_{1{\textrm{j}}}} & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & {{{\eta }}_{{{\textrm{m}}_{\textrm{x}}},{\textrm{j}}}} \end{array} \right]\left[ {{\textbf{I}} + {\boldsymbol\Gamma _{\textrm{j}}}( {\textrm{k}})} \right]\!{{{\hat{\textbf{B}}}}_{\textrm{j}}}( {\textrm{k}}) \right\}^{ - 1}{\underline {\textbf{q}} _{\textrm{L}}}({\textrm{k}}) \nonumber\\[3pt] \underline{\textbf{Q}}_{\textrm{H}}({\textrm{k}}) &= \left\{ \sum\nolimits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{\boldsymbol{\sigma }}}} \left[ \begin{array}{c@{\quad}c@{\quad}c} {{\eta }}_{1{\textrm{j}}} & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & {{\eta }}_{{{\textrm{m}}_{\textrm{x}}},{\textrm{j}}} \end{array} \right] \left[ {\textbf{I}} + \boldsymbol\Gamma _{\textrm{j}}( {\textrm{k}}) \right]\!{{{\hat{\textbf{B}}}}_{\textrm{j}}}( {\textrm{k}}) \right\}^{ - 1}\underline{\textbf{q}}_{\textrm{H}}({\textrm{k}}) \end{align}

Proof of Theorem 1 ends here.

Proposition 1 (Consideration of time delay). In case the system dynamics is being affected by time delay, the dynamical equations of motion and corresponding sliding functions should be modified accordingly. To this end, the dynamical discrete model Eq. (3) should include terms corresponding to the effect of time delay. In order to consider the difficulty of locating the exact point in which time delay affects system dynamic, the time delay is considered unknown with limited bounds such that ${{\textrm{m}}_{{\textrm{d}},{\textrm{ij}}}} \in \left[ {1,{{{{\bar{\textrm{m}}}}}_{{\textrm{dij}}}}} \right]$ where ${{{\bar{\textrm{m}}}}_{{\textrm{dij}}}}$ is the maximum delay corresponding to individual subsystem ${\textrm{i}}$ at given switching configuration of ${\textrm{j}}$ . The procedure of obtaining a control algorithm in the presence of time delay is briefly discussed as followed.

(24) \begin{align}{{\textrm{x}}_{{\textrm{rij}}}}( {{\textrm{k}} + 1}) &= {{\textrm{f}}_{{\textrm{rij}}}}\!\left[ {\underline {\textbf{x}} ( {\textrm{k}})} \right] + {{\textrm{f}}_{{\textrm{qij}}}}\!\left[ {{{\underline {\textbf{x}} }_{\textrm{q}}}( {\textrm{k}})} \right] + {{\textrm{h}}_{{\textrm{rij}}}}\!\left[ {\underline {\textbf{x}} \left( {{\textrm{k}} - {{\textrm{m}}_{{\textrm{d}},{\textrm{ij}}}}} \right)} \right] + {{\textrm{h}}_{{\textrm{qij}}}}\!\left[ {{{\underline {\textbf{x}} }_{\textrm{q}}}\left( {{\textrm{k}} - {{\textrm{m}}_{{\textrm{d}},{\textrm{ij}}}}} \right)} \right] \nonumber\\[3pt] &\quad + \underline {\textbf{b}} _{{\textrm{ij}}}^{\textrm{T}}( {\textrm{k}})\underline {\textbf{u}} ( {\textrm{k}}) + {{{\nu }}_{{\textrm{ij}}}}({\textrm{k}})\end{align}

In Eq. (24), ${{\textrm{h}}_{{\textrm{rij}}}},{{\textrm{h}}_{{\textrm{qij}}}}:{\mathbb{R}^{{{\textrm{m}}_{\textrm{x}}}}} \to \mathbb{R}$ are switched functions corresponding to sections of system dynamics that include time delay. Consequently, sliding functions Eq. (8) are required to be rewritten as Eq. (25).

(25) \begin{align}{{\textrm{s}}_{{\textrm{ij}}}}( {{\textrm{k}} + 1}) &= {{\textrm{f}}_{{\textrm{rij}}}}\!\left[ {\underline {\textbf{x}} ( {\textrm{k}})} \right] + {{\textrm{f}}_{{{\textrm{q}}_{{\textrm{ij}}}}}}\!\left[ {{{\underline {\textbf{x}} }_{\textrm{q}}}( {\textrm{k}})} \right] + {{\textrm{h}}_{{\textrm{rij}}}}\!\left[ {\underline {\textbf{x}} \left( {{\textrm{k}} - {{\textrm{m}}_{{\textrm{d}},{\textrm{ij}}}}} \right)} \right] + {{\textrm{h}}_{{{\textrm{q}}_{{\textrm{ij}}}}}}\!\left[ {{{\underline {\textbf{x}} }_{\textrm{q}}}\left( {{\textrm{k}} - {{\textrm{m}}_{{\textrm{d}},{\textrm{ij}}}}} \right)} \right]\nonumber\\[3pt] &\quad+ \underline {\textrm{b}} _{{\textrm{ij}}}^{\textrm{T}}( {\textrm{k}})\underline {\textrm{u}} ( {\textrm{k}}) + {{{\nu }}_{{\textrm{ij}}}}({\textrm{k}}) + {{{\xi}}_{{\textrm{ij}}}}({\textrm{k}})\end{align}

In addition, the candidate Lyapunov function introduced in theorem 1 is replaced by Lyapunov–Krasovskii function (LKF) including sliding functions correspond to the effect of time delay.

(26) \begin{align}{{\textrm{V}}_{\textrm{i}}}( {\textrm{k}}) \buildrel \Delta \over = {\left[ {\mathop \sum \limits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} {{{\eta }}_{{\textrm{ij}}}}{{\textrm{s}}_{{\textrm{ij}}}}( {\textrm{k}})} \right]^2} + \mathop \sum \limits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} \mathop \sum \limits_{{\textrm{l}} = {\textrm{k}} - {{\textrm{m}}_{{\textrm{d}},{\textrm{ij}}}}}^{{\textrm{k}} - 1} {{{\beta }}_{{\textrm{ij}}}}{\textrm{s}}_{{\textrm{ij}}}^2({\textrm{l}})\end{align}

In Eq. (26), ${{{\eta }}_{{\textrm{ij}}}}$ and ${{{\beta }}_{{\textrm{ij}}}}$ are design parameters, respectively. assigned to current sliding functions and the sliding functions defined over the uncertain time delay horizon. Similar to the procedure followed in Theorem 1, to ensure attaining a monotonically decreasing LKF and attaining finite-time convergence, it is considered that ${{\textrm{V}}_{\textrm{i}}}( {{\textrm{k}} + 1}) - {{\textrm{V}}_{\textrm{i}}}( {\textrm{k}}) \lt 0$ .

(27) \begin{align} &{{\textrm{V}}_{\textrm{i}}}( {{\textrm{k}} + 1}) - {{\textrm{V}}_{\textrm{i}}}( {\textrm{k}}) \lt 0 \nonumber\\ &\quad \Rightarrow {\left[ {\mathop \sum \limits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} {{{\eta }}_{{\textrm{ij}}}}{{\textrm{s}}_{{\textrm{ij}}}}( {{\textrm{k}} + 1})} \right]^2} + \mathop \sum \limits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} \mathop \sum \limits_{{\textrm{l}} = {\textrm{k}} - {{\textrm{m}}_{{\textrm{d}},{\textrm{ij}}}} + 1}^{\textrm{k}} {{{\beta }}_{{\textrm{ij}}}}{\textrm{s}}_{{\textrm{ij}}}^2({\textrm{l}}) - {\left[ {\mathop \sum \limits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} {{{\eta }}_{{\textrm{ij}}}}{{\textrm{s}}_{{\textrm{ij}}}}( {\textrm{k}})} \right]^2} - \mathop \sum \limits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} \mathop \sum \limits_{{\textrm{l}} = {\textrm{k}} - {{\textrm{m}}_{{\textrm{d}},{\textrm{ij}}}}}^{{\textrm{k}} - 1} {{{\beta }}_{{\textrm{ij}}}}{\textrm{s}}_{{\textrm{ij}}}^2({\textrm{l}}) \lt 0\end{align}

Design parameters $${\eta _{{\rm{ij}}}}$$ and ${{{\beta }}_{{\textrm{ij}}}}$ should be allocated such that ${\big[ {\mathop \sum \nolimits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} {{{\eta }}_{{\textrm{ij}}}}{{\textrm{s}}_{{\textrm{ij}}}}( {\textrm{k}})} \big]^2} - \mathop \sum \nolimits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} {{{\beta }}_{{\textrm{ij}}}}$ $\left[ {{\textrm{s}}_{{\textrm{ij}}}^2( {\textrm{k}}) - {\textrm{s}}_{{\textrm{ij}}}^2\left( {{\textrm{k}} - {{\textrm{m}}_{{\textrm{d}},{\textrm{ij}}}}} \right)} \right] \gt 0$ . Hence, the conditions for Krasovskii weights are expressed through investigations over the term $\mathcal{P}^{2}_{\textrm{i}}(\textrm{k})\triangleq{\big[ {\mathop \sum \nolimits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} {{{\eta }}_{{\textrm{ij}}}}{{\textrm{s}}_{{\textrm{ij}}}}( {\textrm{k}})} \big]^2} - \mathop \sum \nolimits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} {{{\beta }}_{{\textrm{ij}}}} \left[ {{\textrm{s}}_{{\textrm{ij}}}^2( {\textrm{k}}) - {\textrm{s}}_{{\textrm{ij}}}^2\left( {{\textrm{k}} - {{\textrm{m}}_{{\textrm{d}},{\textrm{ij}}}}} \right)} \right]$ which is required to be positive.

(28) \begin{align} {\mathcal{P}}_{\textrm{i}}^2( {\textrm{k}}) \gt \bar{\mathcal{P}}_{\textrm{i}}^2( {\textrm{k}}) \buildrel \Delta \over = \mathop \sum \limits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} {{\eta }}_{{\textrm{ij}}}^2{\textrm{s}}_{{\textrm{ij}}}^2( {\textrm{k}}) + \mathop \sum \limits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} \mathop \sum \limits_{{\textrm{l}} = 1}^{{{\textrm{m}}_{{\sigma }}}} {{{\eta }}_{{\textrm{ij}}}}{{{\eta }}_{{\textrm{il}}}}{{\textrm{s}}_{{\textrm{ij}}}}( {\textrm{k}}){{\textrm{s}}_{{\textrm{il}}}}( {\textrm{k}}) - \mathop \sum \limits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} {{{\beta }}_{{\textrm{ij}}}}{\textrm{s}}_{{\textrm{ij}}}^2( {\textrm{k}})\end{align}

A sufficient condition for $\bar{\mathcal{P}}^{2}_{\textrm{i}}(\textrm{k})\ge 0$ can be expressed as

(29) \begin{align}{{\eta }}_{{\textrm{ij}}}^2{\textrm{s}}_{{\textrm{ij}}}^2( {\textrm{k}}) + \mathop \sum \limits_{{\textrm{l}} = 1}^{{{\textrm{m}}_{{\sigma }}}} {{{\eta }}_{{\textrm{ij}}}}{{{\eta }}_{{\textrm{il}}}}{{\textrm{s}}_{{\textrm{ij}}}}( {\textrm{k}}){{\textrm{s}}_{{\textrm{il}}}}( {\textrm{k}}) - {{{\beta }}_{{\textrm{ij}}}}{\textrm{s}}_{{\textrm{ij}}}^2( {\textrm{k}}) \gt 0;{\textrm{j}} = 1, \ldots ,.{{\textrm{m}}_{{\sigma }}}\end{align}

Then, the feasibility conditions for Krasovkii weights are obtained as

(30) \begin{align}{{{\beta }}_{{\textrm{ij}}}} \lt {{\eta }}_{{\textrm{ij}}}^2 + {{{\eta }}_{{\textrm{ij}}}}\frac{{\mathop \sum \nolimits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} {{{\eta }}_{{\textrm{il}}}}{{\textrm{s}}_{{\textrm{il}}}}( {\textrm{k}})}}{{{{\textrm{s}}_{{\textrm{ij}}}}( {\textrm{k}})}}\end{align}

More details are left to be seen in [Reference Yaqubi and Homaeinezhad22]. Proposition 1 ends here.

Proposition 2 (Finite-time convergence). In order to ensure closed-loop response convergence to sliding surface in finite number of steps, $| { \sum \nolimits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} {{{\eta }}_{{\textrm{ij}}}}{{\textrm{s}}_{{\textrm{ij}}}}( {{\textrm{k}} + 1})} | \lt {{{\mu }}_{\textrm{i}}}| { \sum \nolimits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} {{{\eta }}_{{\textrm{ij}}}}{{\textrm{s}}_{{\textrm{ij}}}}( {\textrm{k}})} |$ was defined as Lyapunov asymptotic stability condition with ${{{\mu }}_{\textrm{i}}}$ as finite-time convergence factor. Therefore;

(31) \begin{align} &\left|{{{\eta }}_{\textrm{i}1}\textrm{s}}_{\textrm{i}1}\left(1\right)+{{\eta }}_{\textrm{i}2}{\textrm{s}}_{\textrm{i}2}\left(1\right)+\cdots +{{\eta }}_{\textrm{i}{\textrm{m}}_{{\sigma }}}{\textrm{s}}_{\textrm{i}{\textrm{m}}_{{\sigma }}}\left(1\right)\right| \lt{{\mu }}_{\textrm{i}}\left|{{{\eta }}_{\textrm{i}1}\textrm{s}}_{\textrm{i}1}\left(0\right)+{{\eta }}_{\textrm{i}2}{\textrm{s}}_{\textrm{i}2}\left(0\right)+\cdots +{{\eta }}_{\textrm{i}{\textrm{m}}_{{\sigma }}}{\textrm{s}}_{\textrm{i}{\textrm{m}}_{{\sigma }}}\left(0\right)\right|\nonumber\\[3pt] &\quad \Rightarrow \left|{{{\eta }}_{\textrm{i}1}\textrm{s}}_{\textrm{i}1}\left(2\right)+{{\eta }}_{\textrm{i}2}{\textrm{s}}_{\textrm{i}2}\left(2\right)+\cdots +{{\eta }}_{\textrm{i}{\textrm{m}}_{{\sigma }}}{\textrm{s}}_{\textrm{i}{\textrm{m}}_{{\sigma }}}\left(2\right)\right|\lt{{\mu }}_{\textrm{i}}\left|{{{\eta }}_{\textrm{i}1}\textrm{s}}_{\textrm{i}1}\left(1\right)+{{\eta }}_{\textrm{i}2}{\textrm{s}}_{\textrm{i}2}\left(1\right)+\cdots +{{\eta }}_{\textrm{i}{\textrm{m}}_{{\sigma }}}{\textrm{s}}_{\textrm{i}{\textrm{m}}_{{\sigma }}}\left(1\right)\right|\nonumber\\[3pt] &\quad \lt{{\mu }}_{\textrm{i}}^{2}\left|{{{\eta }}_{\textrm{i}1}\textrm{s}}_{\textrm{i}1}\left(0\right)+{{\eta }}_{\textrm{i}2}{\textrm{s}}_{\textrm{i}2}\left(0\right)+\cdots +{{\eta }}_{\textrm{i}{\textrm{m}}_{{\sigma }}}{\textrm{s}}_{\textrm{i}{\textrm{m}}_{{\sigma }}}\left(0\right)\right| \Rightarrow \cdots \Rightarrow \left|{{{\eta }}_{\textrm{i}1}\textrm{s}}_{\textrm{i}1}(\textrm{k})+{{\eta }}_{\textrm{i}2}{\textrm{s}}_{\textrm{i}2}(\textrm{k})+\cdots +{{\eta }}_{\textrm{i}{\textrm{m}}_{{\sigma }}}{\textrm{s}}_{\textrm{i}{\textrm{m}}_{{\sigma }}}(\textrm{k})\right|\nonumber\\[3pt] &\quad \lt{{\mu }}_{\textrm{i}}^{\textrm{k}}\left|{{{\eta }}_{\textrm{i}1}\textrm{s}}_{\textrm{i}1}\left(0\right)+{{\eta }}_{\textrm{i}2}{\textrm{s}}_{\textrm{i}2}\left(0\right)+\cdots +{{\eta }}_{\textrm{i}{\textrm{m}}_{{\sigma }}}{\textrm{s}}_{\textrm{i}{\textrm{m}}_{{\sigma }}}\left(0\right)\right|\end{align}

Sliding set is defined as ${{\boldsymbol{\Omega }}_{\textrm{i}}} = \{ {{{\textrm{s}}_{{\textrm{ij}}}}( {\textrm{k}}):\left| {\mathop \sum \nolimits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} {{{\eta }}_{{\textrm{ij}}}}{{\textrm{s}}_{{\textrm{ij}}}}( {\textrm{k}})} \right| \le {{{\rho }}_{\textrm{i}}}} \}$ . In which, ${{{\rho }}_{\textrm{i}}} \gt 0$ is half the width of the sliding band. Solving the equation ${{\mu }}_{\textrm{i}}^{\textrm{k}}\left| {{{{\eta }}_{{\textrm{i}}1}}{{\textrm{s}}_{{\textrm{i}}1}}(0) + {{{\eta }}_{{\textrm{i}}2}}{{\textrm{s}}_{{\textrm{i}}2}}(0) + \cdots + {{{\eta }}_{{\textrm{i}}{{\textrm{m}}_{{\sigma }}}}}{{\textrm{s}}_{{\textrm{i}}{{\textrm{m}}_{{\sigma }}}}}(0)} \right| = {{{\rho }}_{\textrm{i}}}$ for ${\textrm{k}}$ , it is concluded that system response trajectory converges to ${{\boldsymbol{\Omega }}_{\textrm{i}}}$ in ${{\textrm{n}}_{{{\Omega i}}}} = {\log _{{{{\mu }}_{\textrm{i}}}}}( {\frac{{{{{\rho }}_{\textrm{i}}}}}{{\left| {{{{\eta }}_{{\textrm{i}}1}}{{\textrm{s}}_{{\textrm{i}}1}}(0) + {{{\eta }}_{{\textrm{i}}2}}{{\textrm{s}}_{{\textrm{i}}2}}(0) + \cdots + {{{\eta }}_{{\textrm{i}}{{\textrm{m}}_{{\sigma }}}}}{{\textrm{s}}_{{\textrm{i}}{{\textrm{m}}_{{\sigma }}}}}(0)} \right|}}} )$ samples.

Proposition 3 (Sliding mode). When the closed-loop system enters sliding manifold neighborhood with radius of ${{{\rho }}_{\textrm{i}}},$ that is, $| {\mathop \sum \limits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} {{{\eta }}_{{\textrm{ij}}}}{{\textrm{s}}_{{\textrm{ij}}}}( {\textrm{k}})} | \lt {{{\rho }}_{\textrm{i}}}$ , the stability conditions will be slightly different from Eq. (12) for reaching mode. In fact, the ideal control input should be obtained such that it forces the system dynamics to reside within sliding band.

(32) \begin{align}{\left[ {\mathop \sum \limits_{{\textrm{j}} = 1}^{{{\textrm{m}}_{{\sigma }}}} {{{\eta }}_{{\textrm{ij}}}}{{\textrm{s}}_{{\textrm{ij}}}}( {{\textrm{k}} + 1})} \right]^2} \lt {{\rho }}_{\textrm{i}}^2\end{align}

Following the procedure similar to the proof of Theorem 1, it is obtained that:

(33) $$\matrix{ {{{\underline q }_{\bf{L}}}({\rm{k}}){\rm{ < }}\left\{ {\sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {\left[ {\matrix{ {{\eta _{1{\rm{j}}}}} & \cdots & 0 \cr {} & {} & {} \cr { \vdots } & \ddots & \vdots \cr {} & {} & {} \cr {0} & \cdots & {{\eta _{{{\rm{m}}_{\rm{x}}},{\rm{j}}}}} \cr } } \right]} \left[ {{\bf{I}} + {\Gamma _{\rm{j}}}({\rm{k}})} \right]{{\widehat{\bf{B}}}_{\rm{j}}}({\rm{k}})} \right\}\underline {\bf{u}} ({\rm{k}}){\rm{ < }}{{\underline q }_{\bf{H}}}({\rm{k}})} \hfill \cr } $$

where ${\underline{\textbf{q}}}_({\textrm{k}})$ and ${\underline{\textbf{q}}}_H({\textrm{k}})$ are defined as boundary dynamical terms in sliding mode.

(34) $$\matrix{ {} \hfill & {{{\underline q }_{\bf{L}}}({\rm{k}}) < \left\{ {\sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {\left[ {\matrix{ {{\eta _{1{\rm{j}}}}} & \cdots & 0 \cr \vdots & \ddots & \vdots \cr 0 & \cdots & {{\eta _{{{\rm{m}}_{\rm{x}}},{\rm{j}}}}} \cr } } \right]} \left[ {{\Gamma _{\rm{j}}}({\rm{k}} - 1){{\widehat{\bf{B}}}_{\rm{j}}}\left( {{\rm{k}} - 1} \right)} \right]} \right\}\underline {\bf{u}} \left( {{\rm{k}} - 1} \right){\rm{ }}} \hfill \cr {} \hfill & {\quad + \left[ {\matrix{ { - \sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {{\eta _{1{\rm{j}}}}} \left\{ {{{\rm{x}}_{\rm{r}}}_{1{\rm{j}}}({\rm{k}}) - {{\widehat{{{\rm{x}}_{\rm{r}}}}}_{1{\rm{j}}}}({\rm{k}}) + {{\widehat{\rm{g}}}_{1{\rm{j}}}}({\rm{k}}) + {{\widehat{\rm{f}}}_{{{\rm{q}}_{1{\rm{j}}}}}}\left[ {{{\underline {\bf{x}} }_{\rm{q}}}({\rm{k}})} \right] + {\xi _{1{\rm{j}}}}({\rm{k}})} \right\} - {\rho _i}} \cr \vdots \cr { - \sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {{\eta _{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}} \left\{ {{{\rm{x}}_{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}({\rm{k}}) - {{\widehat{\rm{x}}}_{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}({\rm{k}}) + {{\widehat{\rm{g}}}_{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}({\rm{k}}) + {{\widehat{\rm{f}}}_{{{\rm{q}}_{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}}}\left[ {{{\underline {\bf{x}} }_{\rm{q}}}({\rm{k}})} \right] + {\xi _{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}({\rm{k}})} \right\} - {\rho _{{{\rm{m}}_{\rm{x}}}}}} \cr } } \right]} \hfill \cr } $$

(35) $$\matrix{ {{{\underline {\bf{Q}} }_{\bf{H}}}({\rm{k}})} \hfill & { \buildrel \Delta \over = \left[ {\matrix{ { - \sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {{\eta _{1{\rm{j}}}}} \left\{ {{{\rm{x}}_{{{\rm{r}}_{1{\rm{j}}}}}}({\rm{k}}) - {{\widehat{\rm{x}}}_{{r_{1{\rm{j}}}}}}({\rm{k}}) + {{\widehat{\rm{g}}}_{1{\rm{j}}}}({\rm{k}}) + {{\widehat{\rm{f}}}_{{{\rm{q}}_{1{\rm{j}}}}}}\left[ {{{\underline {\bf{x}} }_{\rm{q}}}({\rm{k}})} \right] + {\xi _{1{\rm{j}}}}({\rm{k}})} \right\} + {\rho _i}} \cr \vdots \cr { - \sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {{\eta _{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}} \left\{ {{{\rm{x}}_{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}({\rm{k}}) - {{\widehat{\rm{x}}}_{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}({\rm{k}}) + {{\widehat{\rm{g}}}_{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}({\rm{k}}) + {{\widehat{\rm{f}}}_{{{\rm{q}}_{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}}}\left[ {{{\underline {\bf{x}} }_{\rm{q}}}({\rm{k}})} \right] + {\xi _{{{\rm{m}}_{\rm{x}}}{\rm{j}}}}({\rm{k}})} \right\} + {\rho _{{{\rm{m}}_{\rm{x}}}}}} \cr } } \right]{\rm{ }}} \hfill \cr {} \hfill & {\quad + \left\{ {\sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {\left[ {\matrix{ {{\eta _{1{\rm{j}}}}} & \cdots & 0 \cr \vdots & \ddots & \vdots \cr 0 & \cdots & {{\eta _{{{\rm{m}}_{\rm{x}}},{\rm{j}}}}} \cr } } \right]} \left[ {{\Gamma _{\rm{j}}}({\rm{k}} - 1){{\widehat{\bf{B}}}_{\rm{j}}}\left( {{\rm{k}} - 1} \right)} \right]} \right\}\underline {\bf{u}} \left( {{\rm{k}} - 1} \right)} \hfill \cr } $$

Hence, control input is calculated according to Eq. (36).

(36) $$\matrix{ {\underline {\bf{u}} ({\rm{k}})} \hfill & { = \left[ {1 - \alpha ({\rm{k}})} \right]{{\bf{B}}^{ - 1}}({\rm{k}}){{\underline {\cal Q} }_{\rm{L}}}({\rm{k}}) + \alpha ({\rm{k}}){{\bf{B}}^{ - 1}}({\rm{k}}){{\underline {\cal Q} }_{\rm{H}}}({\rm{k}}){\rm{ }}} \hfill \cr {{{\underline {\cal Q} }_{\rm{L}}}({\rm{k}})} \hfill & { = {{\left\{ {\sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {\left[ {\matrix{ {{\eta _{1{\rm{j}}}}} & \cdots & 0 \cr \vdots & \ddots & \vdots \cr 0 & \cdots & {{\eta _{{{\rm{m}}_{\rm{x}}},{\rm{j}}}}} \cr } } \right]} \left[ {{\bf{I}} + {\Gamma _{\rm{j}}}({\rm{k}})} \right]{{\widehat{\bf{B}}}_{\rm{j}}}({\rm{k}})} \right\}}^{ - 1}}{{\underline q }_{\rm{L}}}({\rm{k}}){\rm{ }}} \hfill \cr {{{\underline {\cal Q} }_{\rm{H}}}({\rm{k}})} \hfill & { = {{\left\{ {\sum\limits_{{\rm{j}} = 1}^{{{\rm{m}}_\sigma }} {\left[ {\matrix{ {{\eta _{1{\rm{j}}}}} & \cdots & 0 \cr \vdots & \ddots & \vdots \cr 0 & \cdots & {{\eta _{{{\rm{m}}_{\rm{x}}},{\rm{j}}}}} \cr } } \right]} \left[ {{\bf{I}} + {\Gamma _{\rm{j}}}({\rm{k}})} \right]{{\widehat{\bf{B}}}_{\rm{j}}}({\rm{k}})} \right\}}^{ - 1}}{{\underline q }_{\rm{H}}}({\rm{k}})} \hfill \cr } $$

Proposition 3 ends here.

It is observed that control input bounds are calculated based on the rigid model of system, desired reference values, and approximated uncertainties from FEA. In the reaching phase, Lyapunov stability condition is based on the selection of inputs in appropriate bounds according to Eq. (23). After reaching sliding mode, control input bounds are selected such that closed-loop system will reside within the sliding band according to Eq. (36). In other words, by following the detailed procedure, it is ensured that the system will reach sliding surface in finite time and remains there afterward. Therefore, the control signal is not calculated through a switching signal like the conventional continuous-time SMC approach and the effects of input chattering will no longer exist.

Summarizing the previous steps, the algorithm for Discrete-Time Sliding mode Control of Uncertain Continuum Mechanics Switched MIMO systems (DSCMS) is listed as follows:

1. (a) Deriving mathematical rigid model of the real system using nominal parameters for each switching configuration.

2. (b) Discretizing the rigid model to obtain ${{\hat{g}}_{{\textrm{ij}}}}( {\textrm{k}})$ and ${{\hat{\textbf{B}}}_{\textrm{j}}}( {\textrm{k}})$ .

3. (c) Setting up the dummy models for each switching configuration to approximate parametric and unstructured uncertainties.

4. (d) Adjusting initial control weighting parameters ${{{\eta }}_{{\textrm{ij}}}}$ , finite-time convergence factor ${{{\mu }}_{\textrm{i}}}$ , proximity factor ${{\alpha }}( {\textrm{k}})$ and sliding function parameters ${{{\lambda }}_{{\textrm{ij}}}}$ through investigation over acceptable reference error ranges, system response behavior over different DOFs, and tracking priorities according to Remark 6.

5. (e) Calculating stability bounds according to Theorem 1, Propositions 1 and 3.

6. (f) Applying DSCMS control input through appropriate selection of control proximity factor ${{\alpha }}( {\textrm{k}})$ .

In Fig. 1, the general block-diagram of the proposed DSCMS algorithm illustrating the operation logic of the feedback system in cases where flexibility of switching system is excited by control commands, is depicted.

4. Numerical Example

This section demonstrates controller performance and effectiveness through FEA Simulation for a continuum mechanics switched MIMO system. The selected flexible mechanism which is located on a horizontal plane consists of two flexible beams fixed to each other in the center and carries four concentrated masses attached to either end of each beam according to Fig. 2. A nonlinear van der pol oscillator is also attached to the mechanism Center of Gravity (COG) to increase system nonlinearity for control system evaluation. The proposed mechanism could be represented as a simplified model of a quadrature, space station, or other similar applications. Nevertheless, its capability of demonstrating the deteriorating effect of continuum mechanics vibration in the control process was the main motivation to use it for the evaluation of DSCMS.

Fig. 1.

Flowchart of DSCMS control algorithm.

Fig. 2.

Schematic description of the investigated dynamical system. Systems parameters and geometry are similar to the model of study [Reference Homaeinezhad20].

The control objective is to guide the system such that its COG follows a predefined trajectory. Since the entire motion takes place in a horizontal plane, the mechanism would have three DOFs and three subsequent subsystems would be available throughout the development of DSCMS. To obtain the squared system described in Assumption 1, three concentrated forces as control inputs are applied in arbitrary positions over beam lengths and friction forces are exerted in the location of point masses. Proposed control system verification regarding switching dynamics could be investigated from various aspects. However, without loss of generality, switching is assumed to take place between force exertion points. Two different configurations are simulated such that in the first configuration, control input forces are exerted at the very end of beams close to concentrated masses and in the second configuration, exertion points will be chosen near COG.

The closed-loop system is modeled in the FEA framework in a multitude of details to accurately simulate physical implementation. These details include mechanical strength properties such as elasticity and Poisson ratio, geometric sections, Coulomb friction, and other simulation preliminaries. The whole process necessitates the transformation of control algorithm codes into ANSYS^® mechanical APDL macros for use in transient analysis. Auxiliary dummy models are also constructed in FEA and run alongside the system sharing identical control reference values, initial position for each solution time step, and controller design parameters with the exception that their geometry and mechanical properties are defined according to known nominal values. Table I defines parameters used for both real system (within parentheses) and dummy models in simulation setup:

Table I.

Parameters for simulation setup.

Dynamical equations regarding the mathematical rigid model of proposed mechanism are comprehensively expressed in the author’s previous study [Reference Homaeinezhad20] and only simulation results of the closed-loop system are analyzed here. Two dummy models corresponding to each switching configuration of force exertion point are constructed to estimate model uncertainties according to Remark 3. It should be noted that dummy models are stimulated with the same control input forces applied on the real system and their initial position is set exactly similar to the system initial position for every solution time step. Figure 3 shows different system behavior for either dummy model responding to identical initial condition and input force though with different force exertion points. In Fig. 3(a), both models are coincided in the center to clearly demonstrate the differences in magnitude and orientation of each node displacement vector.

For better graphical representation of dummy models regarding to each switching configuration, Fig. 3(b) shows the separated velocity vector fields of the dummy models computed by ANSYS.

Fig. 3.

Nodal displacement vector analysis for dummy models corresponding to each switching configuration. (a) Dummy models coincided in the center. (b) Dummy models regarding each switching configuration.

Switching signal is depicted in Fig. 4 in which dwell time is set to 2.5 s for the current simulation.

Fig. 4.

Switching signal that alters force exertion points throughout the flexible arms.

Fig. 5.

Time history plots of the response comparison of flexible system (a) Horizontal position of COG. (b) Horizontal velocity of COG. (c) Vertical position of COG. (d) Vertical velocity of COG. (e) Angular displacement. (f) Angular velocity.

Figure 5 depicts closed-loop system response comparison in tracking sinusoidal references ${{\textrm{X}}_{\textrm{d}}} = 2.0\sin 0.6{{\pi t}}$ , ${{\textrm{Y}}_{\textrm{d}}} = 2\sin 0.4{{\pi t}}$ and ${{{\theta }}_{\textrm{d}}} = 0.1{{\pi }}\sin 0.2{{\pi t}}$ and alongside with their respective derivatives for horizontal (Fig. 5(a), (b)), vertical (Fig. 5(c), (d)), and angular (Fig. 5(e), (f)) degrees of freedom. Simulation results are compared with those of FEA-based controllers (Discrete Sliding Mode Controller for MIMO Vibrational Systems abbreviated as DSCMV) presented in [Reference Homaeinezhad20], which does not include switching effects in the construction of control algorithm.

Fig. 6.

Deviation of sliding functions comparison (a) ${{\textrm{s}}_{11}}$ . (b) ${{\textrm{s}}_{12}}$ . (c) ${{\textrm{s}}_{21}}$

Fig. 7.

Path tracking of COG for MIMO flexible system.

Fig. 8.

Control input forces required to track desired references.

It is observed that the reference values are successfully tracked for DSCMS closed-loop system despite frequent switches altering control input force exertion points. It is also notable that system response undergoes a brief disturbance right after switching moment but the controller immediately drives the system into reference trajectories. Since switching alters the exertion point of input forces and does not influence their magnitude, system horizontal and vertical motion is less affected after switching. In this comparison, the DSCMV controller algorithm is constructed based on the worst switching case (when input forces are closer to origin) to satisfy stability conditions for less critical cases. Nevertheless, the closed-loop system becomes unstable due to the fact that adapting controller inputs stability bounds to more critical switching cases, typically results in severe deformations which rule out system linearity. These nonlinear behaviors are oftentimes too quick to be detectable by the FEA estimator. In theory, the issue could be addressed by the increasing sampling rate. However, smaller sampling period results in further complications such that increasing data reading rate involves high-frequency modes and inaccurate control signal calculation.

Figure 6 illustrates sliding surface fluctuations around origin in period of simulation for horizontal (Fig. 6(a)), vertical (Fig. 6(b)), and angular (Fig. 6(c)) motions. Clearly, there is no sign of high-frequency zigzag motion around the sliding surface. In fact, the selection of control inputs between stability bounds totally removes the chattering effect (which is normally seen in SMC techniques) without notable performance compromise. Presence of model uncertainties justifies the deviation of sliding functions from the origin which is notably reduced in DSMCS closed-loop response.

Path tracking of system COG is depicted in Fig. 7. In this figure, it is observed that planar motion of the mechanism follows the desired path, which was expected based on precise trajectory tracking of state values illustrated in Fig. 5.

Figure 8 represents controller inputs that is required to stabilize the close-loop system under switching signal is depicted in Fig. 4. The superiority of DSCMS controller performance stands out regarding both chattering effects and control effort. Minor fluctuations are observed after each switching period which is quickly compensated through the controller effort to reach the sliding surface.