A Sliding Mode Control with a Bang–Bang Observer for Detection of Particle Pollution

  • Manuel SchimmackEmail author
  • Paolo MercorelliEmail author
Part of the Mathematical Engineering book series (MATHENGIN)


This chapter presents a single-input single-output (SISO) adaptive sliding mode control combined with an adaptive bang–bang observer to improve a metal–polymer composite sensor system. The proposed techniques improve the disturbance rejection of a sensor system and thus their reliability in an industrial environment. The industrial application is based on the workplace particulate pollution of welding fumes. Breathing welding fumes is extremely detrimental to human health and exposes the lungs to great hazards, therefore an effective ventilation system is essential. Typically, sliding mode control is applied in actuator control. In this sense, the proposed application is an innovative one. It seeks to improve the performance of sensors in terms of robustness with respect to parametric uncertainties and in terms of insensibility with respect to disturbances. In particular, a sufficient condition to obtain an asymptotic robustness of the estimation of the proposed bang–bang observer is designed and substantiated. The whole control scheme is designed using the well-known Lyapunov approach. A particular sliding surface is defined to obtain the inductive voltage as a controlled output. The adaptation is performed using scalar factors of the input–output data with the assistance of an output error model. A general identification technique is obtained through scaling data. To obtain this data, recursive least squares (RLS) methods are used to estimate the parameters of a linear model using input–output scaling factors. In order to estimate the parametric values in the small-scale range, the input signal requires a high frequency and thus a high sampling rate is needed. Through this proposed technique, a broader sampling rate and input signal with low frequency can be used to identify the small-scale parameters that characterise the linear model. The results indicate that the proposed algorithm is practical and robust.


Sliding Mode Controller Welding Fumes Inductive Voltage Small Scale Parameter Adaptive Sliding Mode Control Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Nominal dynamic matrix


Nominal capacity of the system


Estimated capacity of the system


Voltage disturbance


Error vector

\(f_{\mathrm {m}}\)

Maximal available frequency

\(f_{\mathrm {M}}\)

Maximal value of the bandwidth

\(\mathbf {G}\)

Observer matrix


Exponential scaling factor

\(\mathbf {H}\)

Output observer matrix


Scaling factor of the input signal


Scaling factor of the output signal


Current of the system


Observed current of the system

\({K_{\mathrm {s}}}\)

Steady-state factor


Nominal inductance of the system


Estimated inductance of the system

\(\mathbf{L}_{\mathrm {s}}(k)\)

Discrete gain matrix

\(\mathbf{P}_{\mathrm {s}}(k)\)

Discrete gain matrix


Nominal resistance of the system


Estimated resistance of the system


Sliding surface

\(t_{\mathrm {s}}\)

Sampling rate

\(t_{\mathrm {s_m}}\)

Scaled sampling rate


Calculate factor

\({u}_{\mathrm {s}}(k)\)

Discrete scaled input voltage of the model


Capacitive voltage

\(u_{\mathrm {in}}(t)\)

Input voltage


Inductance voltage


Observed inductance voltage

\(\hat{u}_{L_{\mathrm {max}}}\)

Maximal output voltage of the system

\(u_{\mathrm {out}}(t)\)

Output voltage of the system


Magnetic flux error


Observed current

\({x}_{\mathrm {2d}}(t)\)

Desired current

\({y}_{\mathrm {s}}(k)\)

Discrete scaled current of the model

\(\lambda _{\mathrm {f}}\)

Forgetting factor

\({\theta }_{\mathrm {s}}(k)\)

Discrete parameter vector of scaled system

\({\theta }_{u_{\mathrm {s}}}(k)\)

Discrete parameter vector of scaled input signal

\({\theta }_{y_{\mathrm {s}}}(k)\)

Discrete parameter vector of scaled output signal


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institute of Product and Process InnovationLeuphana University of LueneburgLueneburgGermany

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