Where G A, G B are the transfer functions of the chemical reactor, Z A, Z B are the system gains, T A, T B are the time constants, and D A, D B are the time delays. The superscript s denotes the steady-state values that correspond with the operation point: q A s = q A s = 5 ml s − 1 and pH s = 7. The measurable system output was y( k) =. The vector of manipulated variables was defined as u( k) = ⊤ and the vector of unmeasurable system states was x( k) = ⊤. The novelty of this experimental case study originates in using two actuators, i.e., the pump feeding the solution of acid and the pump dosing the solution of base into the reactor. The steady-state values were moved into the origin. The system in (1) was normalized for the robust MPC design. Function convhull denotes the convex hull that maps original set to the smallest-volume convex set that includes the original set. The matrix superscript ( v) denotes the v-th vertex system of A. Parameter n v represents the total number of uncertain system vertices. Where k ≥ 0 is an element of the discrete-time domain, x k ∈ ℝ n x is the real-valued vector of system states, u k ∈ ℝ n u are control inputs, y k ∈ ℝ n y are system outputs, x 0 is the measured or estimated vector of system initial conditions, A ∈ ℝ n x × n x denotes the system-state matrix, B ∈ ℝ n x × n u is the matrix of system inputs, C ∈ ℝ n y × n x is the matrix of system outputs and v = 1,…, n v. (1b) A v B v ∈ A, A = convhull A v B v ∀ v , The set of uncertain controlled systems A was represented by all systems lying in the convex hull of the vertices given by: Vertices of the controlled system were the linear state-space systems derived for all combinations of limit values of uncertain parameters. For robust MPC design, the discrete-time domain of the uncertain controlled system was considered. The detail non-linear model of CSTR was derived in Holaza et al. The volumetric flow rates of inlet streams were within ml s − 1. Concentrations of input solutions were c A = 0.01 mol dm − 3 for acid, and c B = 0.01 mol dm − 3 for base. Pump A ensured the required volumetric flow rate of acid and pump B dosed base into the reactor. Two peristaltic pumps delivered solutions into the reaction vessel. Two retention tanks were used to store acid and base, and their volumes were V A, V B = 100 dm 3, respectively.

The actual pH value was measured using a pH probe. The controlled variable was pH (potential of hydrogen) of the outlet flow from the reaction vessel. The chemical reaction is described as follows: NaOH(aq) + CH 3COOH(aq) → CH 3COONa(aq) + H 2O (l). The products of neutralization were sodium acetate (CH 3COONa) and water (H 2O). In the reaction vessel neutralization of sodium hydroxide (NaOH) and acetic acid (CH 3COOH) ran. The main part of the chemical reactor is a reaction vessel that has volume V = 1.5 dm 3. The controlled plant was a laboratory continuous stirred-tank reactor (CSTR) of Armfield PCT40.

Reaction vessel meaning