Abstract
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.
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Abbreviations
- \(\mathbf{A}_{0}\) :
-
Nominal dynamic matrix
- \(C_{0}\) :
-
Nominal capacity of the system
- \(\hat{C}_0\) :
-
Estimated capacity of the system
- d(t):
-
Voltage disturbance
- \(\mathbf{e}(t)\) :
-
Error vector
- \(f_{\mathrm {m}}\) :
-
Maximal available frequency
- \(f_{\mathrm {M}}\) :
-
Maximal value of the bandwidth
- \(\mathbf {G}\) :
-
Observer matrix
- h :
-
Exponential scaling factor
- \(\mathbf {H}\) :
-
Output observer matrix
- \({H_{u}}\) :
-
Scaling factor of the input signal
- \({H_{y}}\) :
-
Scaling factor of the output signal
- i(t):
-
Current of the system
- \(\hat{i}(t)\) :
-
Observed current of the system
- \({K_{\mathrm {s}}}\) :
-
Steady-state factor
- \(L_0\) :
-
Nominal inductance of the system
- \(\hat{L}_0\) :
-
Estimated inductance of the system
- \(\mathbf{L}_{\mathrm {s}}(k)\) :
-
Discrete gain matrix
- \(\mathbf{P}_{\mathrm {s}}(k)\) :
-
Discrete gain matrix
- \(R_0\) :
-
Nominal resistance of the system
- \(\hat{R}_0\) :
-
Estimated resistance of the system
- s(t):
-
Sliding surface
- \(t_{\mathrm {s}}\) :
-
Sampling rate
- \(t_{\mathrm {s_m}}\) :
-
Scaled sampling rate
- T :
-
Calculate factor
- \({u}_{\mathrm {s}}(k)\) :
-
Discrete scaled input voltage of the model
- \({u}_C(t)\) :
-
Capacitive voltage
- \(u_{\mathrm {in}}(t)\) :
-
Input voltage
- \(u_L(t)\) :
-
Inductance voltage
- \(\hat{u}_L(t)\) :
-
Observed inductance voltage
- \(\hat{u}_{L_{\mathrm {max}}}\) :
-
Maximal output voltage of the system
- \(u_{\mathrm {out}}(t)\) :
-
Output voltage of the system
- \({x}_{e}(t)\) :
-
Magnetic flux error
- \({\hat{x}}_{2}(t)\) :
-
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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Schimmack, M., Mercorelli, P. (2016). A Sliding Mode Control with a Bang–Bang Observer for Detection of Particle Pollution. In: Rauh, A., Senkel, L. (eds) Variable-Structure Approaches. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-31539-3_5
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DOI: https://doi.org/10.1007/978-3-319-31539-3_5
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