Adaptive Control for a Biological Process under Input Saturation and Unknown Control Gain via Dead Zone Lyapunov Functions
Abstract
:1. Introduction
- The modified tracking error converges to a compact set whose width is user-defined, so that it does not depend on the bounds of either external disturbances, model terms, system states, or model parameters. This is in contrast to common adaptive backstepping control designs (see in [12,15]) and also those that use the Nussbaum gain strategy (see in [14]) where the width of the convergence region of the modified tracking error depends on such kind of bounds.
- The convergence region of the tracking error is determined for the closed loop system under the formulated controller with the proposed auxiliary system.
2. Model Description, Reference Model and Control Goal
2.1. Model Description
2.2. Reference Model
2.3. Control Goal
3. Control Design and Stability Analysis
3.1. Controller Design
- a new auxiliary system of second order is proposed, whose input includes the control signal error , which is the difference between the constrained and the unconstrained control signals;
- a modified tracking error is defined as the sum of the regular tracking error and the state of the auxiliary system;
- the definition the states is based on the adaptive state backstepping method;
- dead zone radially unbounded quadratic forms are used instead of current quadratic forms; and
- a new treatment of the term is proposed, including a new parameterization of the unknown model parameters, and the formulation of a new auxiliary system.
3.2. Boundedness and Convergence Analysis
4. Simulation Example
- the signal is near at initial time , it enters at days and it remains inside until days (Figure 1d).
- the updated parameters remain bounded, and its change is not excessive; , change when , and remain constant otherwise (Figure 3).
- input signal v: for days it exhibits reiterated saturation at its lower bound, with only one moment of saturation at its upper bound (at days approx); during other moments it exhibits changing behavior (Figure 2c,d).
- the signal is inside at days , it leaves, it enters at days approx. and it remains inside afterwards (Figure 1d).
- the updated parameters remain bounded, , are constant when , and the other updated parameters are constant when (Figure 3).
- input signal v: for days, it remains saturated at its upper bound; for days, it exhibits saturation at its lower bound with some few saturation at its upper bounds; for days, it exhibits reiterated saturation at both its upper and lower bounds (Figure 2c,d).
5. Conclusions
- It tackles the combined effect of constrained control input and unknown varying control gain with unknown bounds. To this end, a new auxiliary system is proposed.
- The modified tracking error asymptotically converges to a compact set whose width is user-defined and it does not depend on bounds of either external disturbances, model terms or parameters. Recall that in common robust backstepping designs, the tracking error converges to a compact set whose width depends on such kind of bounds, so that these bounds are required in order to obtain the expected width.
- the model coefficients, and upper and lower bounds of model terms are not required to be known, except ;
- the exact value of the reaction rate term is not required to be known;
- the control gain b is varying and unknown, although it can be expressed as , where is known and is unknown;
- discontinuous functions are not used in the control law, update laws and auxiliary system; instead, saturation type functions are used; and
- the boundedness of the updated parameters is ensured in the presence of input saturation, so that excessive parameter increase is avoided.
Author Contributions
Funding
Conflicts of Interest
Appendix A. Hydroponic System and Formulation of the Mass Balance Model
- The upper CSTR corresponds to the nutrient solution in the cultivation beds. The nutrient concentration is denoted as , the water volume is denoted as , the rate of nutrient removal is denoted as , and the evapotranspiration rate is denoted as . Nutrient removal occurs via sorption and plant uptake. We assume that the water volume is constant.
- The lower CSTR corresponds to the nutrient solution in the mixing tank. The nutrient concentration is denoted as and the water volume is denoted as . The nutrient solution mixes with the incoming flow, which is in turn the flow leaving the upper CSTR. We assume that is varying because of water evaporation losses and varying nature of flow .
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Rincón, A.; Hoyos, F.E.; Candelo-Becerra, J.E. Adaptive Control for a Biological Process under Input Saturation and Unknown Control Gain via Dead Zone Lyapunov Functions. Appl. Sci. 2021, 11, 251. https://doi.org/10.3390/app11010251
Rincón A, Hoyos FE, Candelo-Becerra JE. Adaptive Control for a Biological Process under Input Saturation and Unknown Control Gain via Dead Zone Lyapunov Functions. Applied Sciences. 2021; 11(1):251. https://doi.org/10.3390/app11010251
Chicago/Turabian StyleRincón, Alejandro, Fredy E. Hoyos, and John E. Candelo-Becerra. 2021. "Adaptive Control for a Biological Process under Input Saturation and Unknown Control Gain via Dead Zone Lyapunov Functions" Applied Sciences 11, no. 1: 251. https://doi.org/10.3390/app11010251
APA StyleRincón, A., Hoyos, F. E., & Candelo-Becerra, J. E. (2021). Adaptive Control for a Biological Process under Input Saturation and Unknown Control Gain via Dead Zone Lyapunov Functions. Applied Sciences, 11(1), 251. https://doi.org/10.3390/app11010251