Adaptive Control for a Biological Process under Input Saturation and Unknown Control Gain via Dead Zone Lyapunov Functions

Round 1

Reviewer 1 Report

The paper concerns the stability of a model for hydroponics formed by a system of ordinary differential equations consisting of three equations. The article is well written and provides all the calculations in detail. There are two main theorems and they prove that closed-loop control is bounded and that asymptotic convergence is obtained.

It would be in my opinion useful to add some considerations on the practical meaning of the law of control stated in equation (54).

In assumption 2.4, in x3 there is noise, while x1 and x2 have been assumed known: are the results the same if instead x1 and x2 were noisy?

Author Response

RESPONSE TO FIRST REVIEWER’S COMMENTS

We thank the reviewer for its valuable comments. The changes performed in the manuscript are enhanced by means of red color font. 

First comment

Comment. It would be in my opinion useful to add some considerations on the practical meaning of the law of control stated in equation (54).

Response. A discussion on the practical meaning of the control law has been incorporated in the manuscript; see the new Remark 3.2 and Remark 3.1, lines 267 to 276.   

Second comment

Comment. In assumption 2.4, in x₃ there is noise, while x₁ and x₂ have been assumed known: are the results the same if instead x₁ and x₂ were noisy?

Response. The noise in x₃ has been considered in order to provide a case of unaccurate knowledge on the control gain b. However, other possibility of unaccurate knowledge on b is by considering unaccurate knowledge on P_ad.

In the developed controller design, no measurement noise is considered in x₁ and x₂. Therefore, the case with noise would imply modifying the whole controller design procedure, by addition of robustness terms to tackle the effect of these measurement noises.

Author Response File: Author Response.pdf

Reviewer 2 Report

Attached

Comments for author File: Comments.pdf

Author Response

RESPONSE TO SECOND REVIEWER’S COMMENTS

We thank the reviewer for his valuable comments. The changes performed in the manuscript are enhanced by means of red color font. 

First comment

Comment. My main observation refers to the fundamental contribution of the work and is the design of the adaptive control. The control design made in section 3 is very confusing for me, they perform a series of mathematical manipulations that are not properly motivated.

• For example, authors introduce in formula (9) a truncated Lyapunov function inspired in a previous work but authors do not motivate this inspiration and the properties that it has.
• In formula (12) authors introduce the term K_1b|f_z1| in order to provide robustness but they do not clarify the need for this term or what it contributes to robustness.
• In formula (54) they introduce a control law for u without motivation, etc.

The entire section must be rewritten so that the mathematical process underlying the definition of the control is understood.

Response. The motivations for the main mathematical manipulations have been incorporated:

-Formula (9): a motivation for using the Lyapunov function (9) has been provided, and its main properties are presented. See lines 160 to 165.

-Formula (12): a motivation on the introduction of the term K_1b|f_z1| has been incorporated in order to clarify its importance. See lines 212, 213, 215-221.    

-Formula (54): A motivation for the choice of the control law for u has been incorporated. See lines 261 to 266, including equation (57).

Also, the motivations for other important tasks of the adaptive control design have been created and incorporated, including:

• motivation for the general controller design procedure: see lines 132 to 138
• motivation for the definition of z₁: see lines 148 to 155
• motivation for the use of parameter updating error and updated parameter for Ѳ₁: see lines 172 to 176 and 178,179.
• motivation for the use of parameter updating error for Ѳ_ub: see lines 227 to 232.
• motivation for the use of other parameter updating errors: see lines 244 to 247
• motivation for the definition of the state z₂: see lines 183 to 187
• motivation for the definition of ψ₁: see lines 193 to 196. Also, additional motivation for the definition of the input W _ψ of the auxiliary system is presented in lines 234 to 237
• motivation for the definition of V_z2: see lines 201 to 203
• motivation for expressing v in terms of u and the input error △u: see lines 223 to 225
• motivation for the choice of the update laws: see lines 286 to 289 and 295.

First specific comment

Comment. Assumption 2.1. Authors assume that the state variables x₁, x₂ and x₃ are bounded for a bounded control v. They also assume that the state variables are non negative. The question is, can this assumption be deduced from the state system under more restrictive hypothesis? Is it a realistic assumption?

Response. Recall that x₁=P_e is the nutrient concentration in the upper CSTR; x₂=P_i is the nutrient concentration in the lower CSTR; x₃ is the liquid volume in the lower CSTR. In real life operation of the considered bioprocess, x₁, x₂, x₃ are neither negative or zero, and they are bounded provided the flow v = Q_ad is bounded. This is verified from data in Lee (2017).

In addition, it is possible to prove that x₁, x₂, x₃ are bounded for input v bounded, by using Lyapunov theory and model (1), (2), (3). A stability analysis of this type is performed for bioreaction systems in Meadows (2019), de Battista (2018) and Mairet (2019). Such study comprises:

• defining an overall Lyapunov function, and also defining a Lyapunov function for each state variable
• determining the time derivative of the Lyapunov function of the state variables, and the time derivative of the overall Lyapunov function
• integrating the resulting time derivative of the overall Lyapunov function

Nevertheless, developing such study for model (1), (2), (3) would be complex and it is beyond the scope of this paper.

348. De Battista, M. Jamilis, F. Garelli, J. Picó. (2018). Global stabilisation of continuous bioreactors: Tools for analysis and design of feeding laws. Automatica, 89, 340-348.

468. Mairet, H. Ramírez, A. Rojas-Palma. (2017). Modeling and stability analysis of a microalgal pond with nitrification. Applied Mathematical Modelling, 51, 448–468.

689. Meadows, M. Weedermann, G.S.K. Wolkowicz. (2019). Global analysis of a simplified model of anaerobic digestion and a new result for the chemostat. SIAM J. Appl. Math., 79(2), 668-689.

Second specific comment

Comment. In formula (5) it is necessary to clarify how the diferential operator works. In principle, as the formula is written, it is not clear how this diferential operator works.

Response. A more understandable presentation of the reference model (5) has been added, using derivatives of y_d instead of the differential operator p = d/dt. See the new three expressions after equation (5), lines 113 to 115.    

Third specific comment

Comment. Line 337 wwere.

Response. This mistake has been corrected. See line 432.

Fourth specific comment

Comment. Authors say in line 338 that The procedure is clear and detailed so that it can be applied to other second order nonlinear systems. This claim is not conveniently justified.

Response. The developed control design procedure involves the required description of the mathematical manipulations, even if all the motivations are not provided. This has been clarified in the paper. See lines 434, 435. In addition, several motivations have been incorporated as stated in the response to the first comment of the reviewer.     

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report