Digital Control of a Continuous Stirred Tank Reactor

Abstract

We project a novel digital control law for continuous stirred tank reactors, based on sampled measures of temperatures and reactant concentration, as it happens in practice. The methodology of relative degree preservation under sampling is used. It is proved that a suitably approximated sampled system, obtained by Taylor series expansion and truncation, in closed loop with the projected control law, is asymptotically stable, provided that a condition on the sampling period is verified. Such condition allows for values of the sampling period larger than necessary in practical implementation with usual technology. Many simulations show the high performance of the proposed digital control law.

1. Introduction

The control of the operation of chemical reactors has attracted the attention of researchers for a long time. The underlying motivation relies on the fact that industrial chemical reactors are frequently operated at unstable operating conditions, which often corresponds to optimal process performance [1–19]. Polymerization processes [5] and bioreactors fermentors [18] are important examples of large-scale chemical reactors operated at unstable conditions. As well known, the measures of the reactant concentration cannot be achieved on continuous time. In practice, these measures are available at certain time intervals. It is therefore very important to project control laws which are piecewise constant and are updated at each measures update. That is, for practical use, a digital control law has to be projected. Applications of nonlinear digital control theory to some chemical reactors can be found in [20, 21]. In [11] a discrete-time controller is obtained by linear approximations successively executed at each sampling time, and an application to a continuous stirred tank reactor is shown.

In this paper we deal with a novel digital controller design for a continuous stirred tank reactor, where an irreversible, exothermic, liquid-phase reaction 𝐴→𝐵 evolves. The controlled output is the reactor temperature. The jacket temperature dynamics is considered. The control law is actuated by means of a pneumatic valve which regulates the cooling water flow rate. This actuation process is easier to implement than the ones based on the use of the inlet reactor concentration or of the jacket temperature as manipulated inputs. The model consists of three nonlinear ordinary differential equations. We develop a digital control law by means of a suitably approximated sampled model, obtained by Taylor series expansions (see [21, 22]). The preservation of relative degree methodology shown in [22] is here used. For the underlying continuous time model, the relative degree is not full. The relative degree is preserved for the approximated sampled model by introducing a dummy output. Then a feedback control law is projected, by which the approximated sampled system becomes asymptotically stable. Many performed computer simulations show the high performance of the proposed digital controller. Saturation effects have been taken into account in simulations. The convergence of the state variables to the arbitrarily chosen operating point is always obtained with the many considered system initial conditions (even start-up initial conditions).

2. The Model of the CSTR

Here we study a CSTR with jacket cooling in which a first-order irreversible exothermic reaction takes place [23]: 𝐴→𝐵 (Figure 1).

Figure 1 

Schematic reactor type CSTR.

The liquids in the reactor and in the jacket are assumed perfectly mixed, that is, with no radial, axial, or angular gradients in properties (temperature, concentrations). This allows to consider reactor and jacket temperatures and reactant concentrations as space invariant variables. If physical properties are assumed constant (densities and heat capacities), the reactor volume is constant and the jacket volume is constant, the mathematical model of the CSTR is given by the following set of nonlinear functional differential equations:𝑥̇₁(𝑡)=𝐹𝐶𝑎₀−𝑥₁(𝑡)𝑉_𝑟−𝑥₁(𝑡)𝐾₀e^{−𝐸/𝑅𝑥2(𝑡)},𝑥̇₂(𝑡)=𝐹𝑇₀−𝑥₂(𝑡)𝑉_𝑟−𝜆𝑥₁(𝑡)𝐾₀e^{−𝐸/𝑅𝑥2(𝑡)}𝜌⁻¹𝐶_𝑝⁻¹−𝑈𝐴_𝑗𝑥₂(𝑡)−𝑥₃(𝑡)𝑉_𝑟𝜌𝐶_𝑝,𝑥̇₃(𝑡)=𝑢(𝑡)𝑇_cin−𝑥₃(𝑡)𝑉_𝑗+𝑈𝐴_𝑗𝑥₂(𝑡)−𝑥₃(𝑡)𝑉_𝑗𝜌_𝑗𝐶_𝑗, (2.1) where 𝑥₁=𝐶𝑎 = reactant concentration (kmol/m³), 𝑥₂=𝑇_𝑅 = reactor temperature (K), 𝑥₃=𝑇_𝐽 = jacket temperature (K), 𝑢 = flow rate of coolant (control input) (m³/s), 𝐾₀ = pre-exponential factor (s⁻¹), 𝐸 = activation energy (J/kmol), 𝑅 = universal gas constant, 8314 Jkmol⁻¹K⁻¹, 𝐹 = flow rate of feed and product (m³/s), 𝜌 = density of product stream (kg/m³), 𝐶𝑎₀ = concentration of reactant 𝐴 in feed (kmol/m³), 𝑉_𝑅 = volumetric holdup of liquid in reactor (m³), 𝑇₀ = temperature of feed (K), 𝐶_𝑝 = heat capacity of product (Jkg⁻¹K⁻¹), 𝜆 = heat of reaction (J/kmol), 𝑈 = overall heat transfer coefficient (WK⁻¹m⁻²), 𝐴_𝑗  =  jacket heat transfer area (m²), 𝜌_𝑗 = density of coolant (kg/m³), 𝐶_𝑗 = heat capacity of coolant (Jkg⁻¹K⁻¹), and 𝑇_cin = supply temperature of cooling medium (K).

Let 𝜘∈[0,1] be the reaction conversion𝜘=𝐶𝑎₀−𝐶𝑎𝐶𝑎₀. (2.2) Let 𝜘_eq be the chosen operating point reaction conversion. By (2.2) it follows that the operating point concentration of reactant in the reactor 𝐶𝑎_eq is given by𝐶𝑎_eq=𝐶𝑎₀1−𝜘_eq. (2.3) From equations in (2.1), setting zero the left-hand side, it follows that the operating point reactor temperature (𝑇_𝑅eq), the operating point jacket temperature (𝑇_𝑗eq), the necessary coolant water flow rate (𝑢_eq) for the chosen operating point are given by 𝑇_𝑅eq−𝐹=−𝐸ln−𝐶𝑎₀+𝐶𝑎_eq𝐶𝑎_eq𝐾₀𝑉_𝑟⁻¹𝑅⁻¹,𝑇_𝐽eq=𝜆𝐶𝑎₀𝐹𝜘_eq𝑈𝐴_𝑗−𝐹𝜌𝐶_𝑝𝑇₀𝑈𝐴_𝑗−𝐸ln𝐹𝜘_eq1−𝜘_eq𝐾₀𝑉_𝑟⁻¹𝑅⁻¹−𝐹𝜌𝐶_𝑝𝐸ln𝐹𝜘_eq1−𝜘_eq𝐾₀𝑉_𝑟⁻¹𝑅⁻¹𝑈⁻¹𝐴_𝑗⁻¹,𝑢_eq=𝑈𝐴_𝑗𝐸𝜌_𝑗⁻¹𝐶_𝑗⁻¹𝑇_cin−𝑇_𝐽eq⁻¹ln𝐹𝜘_eq1−𝜘_eq𝐾₀𝑉_𝑟⁻¹𝑅⁻¹+𝑈𝐴_𝑗𝑇_𝐽eq𝜌_𝑗𝐶_𝑗𝑇_cin−𝑇_𝐽eq. (2.4)

3. Sampled System and Digital Control Law

The system (2.1) is in the form𝑥̇(𝑡)=𝑓(𝑥(𝑡))+𝑔(𝑥(𝑡))⋅𝑢(𝑡), (3.1) where the functions 𝑓, 𝑔 can be easily defined by (2.1) as𝑓(𝑥(𝑡=𝑓))¹(𝑥(𝑡𝑓))₂(𝑥(𝑡𝑓))₃(𝑥(𝑡=𝐹))𝐶𝑎₀−𝑥₁(𝑡)𝑉_𝑟−𝑥₁(𝑡)𝐾₀e^{−𝐸/𝑅𝑥2(𝑡)}𝐹𝑇₀−𝑥₂(𝑡)𝑉_𝑟−𝜆𝑥₁(𝑡)𝐾₀e^{−𝐸/𝑅𝑥2(𝑡)}𝜌⁻¹𝐶_𝑝⁻¹−𝑈𝐴_𝑗𝑥₂(𝑡)−𝑥₃(𝑡)𝑉_𝑟𝜌𝐶_𝑝𝑈𝐴_𝑗𝑥₂(𝑡)−𝑥₃(𝑡)𝑉_𝑗𝜌_𝑗𝐶_𝑗,𝑔(𝑥(𝑡=𝑔))¹(𝑥(𝑡𝑔))₂(𝑥(𝑡𝑔))₃(𝑥(𝑡=00𝑇))^cin−𝑥₃(𝑡)𝑉_𝑗. (3.2) In order to rewrite (3.1) in normal form, as required in [22], let us now consider the following variables:𝜉(𝑡)=𝜉₁(𝑡)𝜉₂(𝑡)𝜉₃(𝑡)=𝑥₂(𝑡)−𝑇_𝑅eq𝑓₂(𝑥(𝑡𝑥))₁(𝑡)−𝐶𝑎_eq=𝜙(𝑥(𝑡)), (3.3) from which the easy inverse relation holds𝑥(𝑡)=Φ⁻¹(𝜉(𝑡=𝜉))³(𝑡)+𝐶𝑎_eq𝜉₁(𝑡)+𝑇_𝑅eq−𝐹𝜌𝐶_𝑝𝑈𝐴_𝑗𝑇₀−𝜉₁(𝑡)−𝑇_𝑅eq+𝜆𝑉_𝑟𝜉₃(𝑡)+𝐶𝑎_eq𝐾₀e^{−𝐸/𝑅(𝜉1(𝑡)+𝑇𝑅eq)}𝑈𝐴_𝑗+𝜉₂(𝑡)𝑉_𝑟𝜌𝐶_𝑝𝑈𝐴_𝑗+𝜉₁(𝑡)+𝑇_𝑅eq. (3.4) Let the control input be equal to𝑢(𝑡)=𝑢_eq+𝑣(𝑡). (3.5) Let the controlled output be𝑦(𝑡)=ℎ(𝜉(𝑡))=𝜉₁(𝑡). (3.6) We want to control to zero this output, which corresponds to drive the reactor temperature to the reactor temperature operating point 𝑇_𝑅eq.

We can write the system equations by using the new variables 𝜉(𝑡) as follows:𝜉̇(𝑡)=𝑝₁(𝜉(𝑡))+𝑝₂(𝜉(𝑡))𝑣(𝑡), (3.7) where 𝑝₁, 𝑝₂ are given by 𝑝₁(𝜉(𝑡=𝑝))¹¹(𝜉(𝑡𝑝))₂₁(𝜉(𝑡𝑝))₃₁(𝜉(𝑡,𝑝))₁₁(𝜉(𝑡))=𝜉₂(𝑡),𝑝₂₁(𝜉(𝑡))=−𝜆𝐾₀𝐹𝐶𝑎₀𝑉_𝑟⁻¹𝜌⁻¹𝐶_𝑝⁻¹e^{𝐸/𝑅(𝜉1(𝑡)+𝑇𝑅eq)−1}−𝑢_eq𝑈𝐴_𝑗𝜉₁(𝑡)𝑉_𝑗𝑉_𝑟𝜌𝐶_𝑝+𝑢_eq𝐹𝑇₀𝑉_𝑗𝑉_𝑟+𝜆𝜉₃(𝑡)𝐾₀𝐹𝑉_𝑟⁻¹𝜌⁻¹𝐶_𝑝⁻¹e^{𝐸/𝑅(𝜉1(𝑡)+𝑇𝑅eq)−1}−𝑢_eq𝑈𝐴_𝑗𝑇_𝑅eq𝑉_𝑗𝑉_𝑟𝜌𝐶_𝑝−𝑢_eq𝐹𝜉₁(𝑡)𝑉_𝑗𝑉_𝑟+𝜆𝐾₀𝐶𝑎_eq𝐹𝑉_𝑟⁻¹𝜌⁻¹𝐶_𝑝⁻¹e^{𝐸/𝑅(𝜉1(𝑡)+𝑇𝑅eq)−1}−𝑢_eq𝜉₂(𝑡)𝑉_𝑗+𝑢_eq𝑈𝐴_𝑗𝑇_cin𝑉_𝑗𝑉_𝑟𝜌𝐶_𝑝+𝜆𝐾₀²𝜉₃(𝑡)e^{𝐸/𝑅(𝜉1(𝑡)+𝑇𝑅eq)−2}𝜌⁻¹𝐶_𝑝⁻¹−𝑢_eq𝐹𝑇_𝑅eq𝑉_𝑗𝑉_𝑟−𝑈𝐴_𝑗𝜉₂(𝑡)𝑉_𝑗𝜌_𝑗𝐶_𝑗+𝜆𝐾₀²𝐶𝑎_eqe^{𝐸/𝑅(𝜉1(𝑡)+𝑇𝑅eq)−2}𝜌⁻¹𝐶_𝑝⁻¹−𝜉₂(𝑡)𝐹𝑉_𝑟−𝜉₂(𝑡)𝜆𝐾₀𝐸𝜉₃(𝑡)𝜌⁻¹𝐶_𝑝⁻¹𝑅⁻¹𝜉₁(𝑡)+𝑇_𝑅eq⁻²e^{𝐸/𝑅(𝜉1(𝑡)+𝑇𝑅eq)−1}−𝜉₂(𝑡)𝜆𝐾₀𝐸𝐶𝑎_eq𝜌⁻¹𝐶_𝑝⁻¹𝑅⁻¹𝜉₁(𝑡)+𝑇_𝑅eq⁻²e^{𝐸/𝑅(𝜉1(𝑡)+𝑇𝑅eq)−1}−𝑈𝐴_𝑗𝜉₂(𝑡)𝑉_𝑟𝜌𝐶_𝑝+𝑈𝐴_𝑗𝐹𝑇₀𝑉_𝑗𝜌_𝑗𝐶_𝑗𝑉_𝑟−𝑈𝐴_𝑗𝐹𝜉₁(𝑡)𝑉_𝑗𝜌_𝑗𝐶_𝑗𝑉_𝑟−𝑈𝐴_𝑗𝐹𝑇_𝑅eq𝑉_𝑗𝜌_𝑗𝐶_𝑗𝑉_𝑟−𝑈𝐴_𝑗𝜆𝐾₀𝜉₃(𝑡)𝑉_𝑗⁻¹𝜌_𝑗⁻¹𝐶_𝑗⁻¹𝜌⁻¹𝐶_𝑝⁻¹e^{𝐸/𝑅(𝜉1(𝑡)+𝑇𝑅eq)−1}−𝑈𝐴_𝑗𝜆𝐾₀𝐶𝑎_eq𝑉_𝑗⁻¹𝜌_𝑗⁻¹𝐶_𝑗⁻¹𝜌⁻¹𝐶_𝑝⁻¹e^{𝐸/𝑅(𝜉1(𝑡)+𝑇𝑅eq)−1}−𝑢_eq𝜆𝐾₀𝜉₃(𝑡)𝑉_𝑗⁻¹𝜌⁻¹𝐶_𝑝⁻¹e^{𝐸/𝑅(𝜉1(𝑡)+𝑇𝑅eq)−1}−𝑢_eq𝜆𝐾₀𝐶𝑎_eq𝑉_𝑗⁻¹𝜌⁻¹𝐶_𝑝⁻¹e^{𝐸/𝑅(𝜉1(𝑡)+𝑇𝑅eq)−1},𝑝₃₁(𝜉(𝑡=𝐹))𝐶𝑎₀−𝜉₃(𝑡)−𝐶𝑎_eq𝑉_𝑟−𝜉₃(𝑡)+𝐶𝑎_eq𝐾₀e^{−𝐸/𝑅(𝜉1(𝑡)+𝑇𝑅eq)},𝑝₂(𝜉(𝑡=𝑝))¹²(𝜉(𝑡𝑝))₂₂(𝜉(𝑡𝑝))₃₂(𝜉(𝑡,𝑝))₁₂(𝜉(𝑡𝑝))=0,₂₂(𝜉(𝑡=𝑇))^cin𝑈𝐴_𝑗𝑉_𝑗𝑉_𝑟𝜌𝐶_𝑝+𝐹𝑇₀𝑉_𝑗𝑉_𝑟−𝐹𝜉₁(𝑡)𝑉_𝑗𝑉_𝑟−𝐹𝑇_𝑅eq𝑉_𝑗𝑉_𝑟−𝑈𝐴_𝑗𝜉₁(𝑡)𝑉_𝑗𝑉_𝑟𝜌𝐶_𝑝−𝑇_𝑅eq𝑈𝐴_𝑗𝑉_𝑗𝑉_𝑟𝜌𝐶_𝑝−𝜉₂(𝑡)𝑉_𝑗−𝜆𝐾₀𝜉₃(𝑡)𝑉_𝑗⁻¹𝜌⁻¹𝐶_𝑝⁻¹e^{𝐸/𝑅(𝜉1(𝑡)+𝑇𝑅eq)−1}−𝜆𝐾₀𝐶𝑎_eq𝑉_𝑗⁻¹𝜌⁻¹𝐶_𝑝⁻¹e^{𝐸/𝑅(𝜉1(𝑡)+𝑇𝑅eq)−1},𝑝₃₂(𝜉(𝑡))=0. (3.8)

Since 𝜉=0 corresponds to 𝑥=^{𝐶𝑎eq𝑇𝑅}_eq𝑇𝐽eq, it follows that 𝑝₁(0)=0. The introduction of the new variables allows to obtain the CSTR equations in normal form [22], and the origin is an equilibrium point. Therefore, the stability of the origin for the system described by (3.7) is equivalent to the stability of the chosen operating point for the system described by (2.1). Let us now sample and approximate, by Taylor series expansion and truncation, the system (3.7) as follows: setting 𝑣(𝑡)=𝑣(𝑘𝛿)=𝑣_𝑘, 𝑘𝛿≤𝑡<(𝑘+1)𝛿, 𝑘=0,1,2,…,𝜉̃₁𝜉̃₂)𝛿)=𝜉̃((𝑘+11𝛿01𝜉̃¹₂()+𝛿𝑘𝛿2²𝛿𝑅₀(),𝜉̃𝑘𝛿₃((𝑘+1)𝛿)=𝜉̃₃(𝑘𝛿)+𝛿𝑝₃₁𝜉̃()+𝛿𝑘𝛿²2𝑞𝜉̃(𝑘𝛿), (3.9) where 𝜉̃(𝑘𝛿) is the approximation of 𝜉(𝑘𝛿); 𝑅₀ is given by (Lie derivatives are used), 𝑅₀(𝑘𝛿)=𝑙²_𝑝1ℎ𝜉̃(𝑘𝛿)+𝑙_𝑝2𝑙_𝑝1ℎ𝜉̃(𝑣();𝑙𝑘𝛿)𝑘𝛿_2𝑝1ℎ𝜉̃(𝑘𝛿)=𝜕𝑙¹_𝑝1ℎ𝜉̃(𝑘𝛿)𝜕𝜉̃()𝑝𝑘𝛿¹𝜉̃(𝑘𝛿);𝑙¹_𝑝1ℎ𝜉̃()=𝜉̃()𝑘𝛿𝜕ℎ𝑘𝛿𝜕𝜉̃()𝑝𝑘𝛿¹𝜉̃();𝑙𝑘𝛿_𝑝2𝑙_𝑝1ℎ𝜉̃(𝑘𝛿)=𝜕𝑙_𝑝1ℎ𝜉̃(𝑘𝛿)𝜕𝜉̃()𝑝𝑘𝛿²𝜉̃(𝑘𝛿); (3.10)𝑞(𝜉̃(𝑘𝛿)) is given by 𝑞𝜉̃(𝑘𝛿)=𝜕𝑝₃₁𝜉̃(𝑘𝛿)𝜕𝜉̃()𝑝𝑘𝛿¹𝜉̃(),𝑞𝜉̃()𝑘𝛿𝑘𝛿=−𝐾₀𝐸𝜉̃₂(𝑘𝛿)𝜉̃₃𝑅⁻¹𝜉̃₁(𝑘𝛿)+𝑇_𝑅eq⁻²e^𝐸/𝑅(𝜉̃_{1(𝑘𝛿)+𝑇𝑅eq)}⁻¹−𝐾₀𝐸𝜉̃₂(𝑘𝛿)𝐶𝑎_eq𝑅⁻¹𝜉̃₁(𝑘𝛿)+𝑇_𝑅eq⁻²e^𝐸/𝑅(𝜉̃_{1(𝑘𝛿)+𝑇𝑅eq)}⁻¹−𝐹²𝐶𝑎₀𝑉_𝑟²+𝐹²𝜉̃₃𝑉_𝑟²+𝐹²𝐶𝑎_eq𝑉_𝑟²+2𝐹𝐾₀𝜉̃₃𝑉_𝑟⁻¹e^𝐸/𝑅(𝜉̃_{1(𝑘𝛿)+𝑇𝑅eq)}⁻¹+2𝐹𝐾₀𝐶𝑎_eq𝑉_𝑟⁻¹e^𝐸/𝑅(𝜉̃_{1(𝑘𝛿)+𝑇𝑅eq)}⁻¹+𝐾₀²𝐶𝑎_eqe^𝐸/𝑅(𝜉̃_{1(𝑘𝛿)+𝑇𝑅eq)}⁻²−𝐹𝐾₀𝐶𝑎₀𝑉_𝑟⁻¹e^𝐸/𝑅(𝜉̃_{1(𝑘𝛿)+𝑇𝑅eq)}⁻¹+𝜉̃₃(𝑘𝛿)𝐾₀²e^𝐸/𝑅(𝜉̃_{1(𝑘𝛿)+𝑇𝑅eq)}⁻². (3.11)

The approximated sampled system (3.9) is obtained by exploiting the normal form of (3.7) (see [22]) and by neglecting, in the Taylor series expansion, the terms which are 0(𝛿³) in the expression of 𝜉₁((𝑘+1)𝛿) and in the expression of 𝜉₃((𝑘+1)𝛿), and the terms which are 0(𝛿²) in the expression of 𝜉₂((𝑘+1)𝛿) (see terms 𝑁²_𝑟,𝛿 in [22, Proposition 3.2]). Notice that, since (𝜕𝑝₃₁(𝜉̃(𝑘𝛿))/𝜕𝜉̃(𝑘𝛿))𝑝₂(𝜉̃(𝑘𝛿))=0, the inclusion of a term which is 0(𝛿²) in the approximation of 𝜉₃((𝑘+1)𝛿) does not cause a new presence (besides the one in 𝑅₀) of the control input 𝑣_𝑘. As well, neglecting terms which are 0(𝛿²) in the expression of 𝜉₂((𝑘+1)𝛿) avoids to have further terms (besides 𝑅₀) which involve the control input. This fact is instrumental for building the digital control law here proposed, as will be shown below.

The approximated sampled system (3.9) admits relative degree equal to 1, thus not preserving the relative degree equal to 2 of the original continuous time system (2.1) with respect to the output 𝑥₂(𝑡) (reactor temperature). Thus, a digital input/output linearizing feedback control law built up on the basis of the given output (3.6) yields a further internal dynamics. In order to preserve the relative degree, let us introduce the dummy output as proposed in [22],𝑦_𝛿()=𝛿𝑘𝛿1−2𝜉̃₁()𝜉̃𝑘𝛿₂(𝑘𝛿), (3.12) and let us define the variables𝜒(𝑘)=𝜒₁(𝑘)𝜒₂(𝑘)𝜒₃(𝑘)=𝑦_𝛿()𝑦𝑘𝛿_𝛿)𝛿)𝜉̃((𝑘+1₃()𝜉̃()𝑘𝛿=𝑀𝑘𝛿, (3.13) where the nonsingular matrix 𝑀 is given by𝛿𝑀=1−201𝛿20001. (3.14) Let 𝑀₁ be the upleft submatrix of 𝑀𝑀₁=𝛿1−21𝛿2. (3.15) We obtain, from (3.9),𝜒₁()𝜒𝑘+1₂(𝑘+1)=𝑀₁𝑀1𝛿01¹⁻¹𝜒₁(𝑘)𝜒₂(𝑘)+𝑀₁𝛿²2𝛿𝑅₀(),𝜒𝑘𝛿₃(𝑘+1)=𝜒₃(𝑘)+𝛿𝑝₃₁𝑀⁻¹𝜒(𝑘)+𝛿²2𝑞𝑀⁻¹𝜒(𝑘), (3.16) where 𝑅₀, rewritten with the variables 𝜒, becomes𝑅₀(𝑘𝛿)=𝑙²_𝑝1ℎ𝑀⁻¹𝜒(𝑘)+𝑙_𝑝2𝑙_𝑝1ℎ𝑀⁻¹𝜒(𝑘)𝑣_𝑘. (3.17)

Theorem 3.1. Let Γ=Γ₁Γ₂∈𝑅^1𝑥2 (3.18) be chosen such that the matrix Γ01₁Γ₂ has eigenvalues inside the open complex unitary circle. Let the sampling period 𝛿>0 satisfy the following inequality: 𝛿<𝛿_max=2𝐹/𝑉_𝑟+𝐾₀e^{−𝐸/𝑅𝑇𝑅𝑒𝑞}. (3.19) Then, the following discrete time feedback control law 𝑣_𝑘=−𝑙²_𝑝1ℎ𝑀⁻¹𝜒(𝑘)𝑙_𝑝2𝑙_𝑝1ℎ𝑀⁻¹𝜒(𝑘+[)¹⁻²]^𝜒1(𝑘)_𝜒2(𝑘)+Γ^𝜒1(𝑘)_𝜒2(𝑘)𝛿²𝑙_𝑝2𝑙_𝑝1ℎ𝑀⁻¹𝜒(𝑘) (3.20) is such that the trivial solution of the closed-loop system (3.16)–(3.20) is asymptotically stable. Moreover, the eigenvalues of the Jacobian (evaluated at 0) of the right-hand function describing the closed-loop system (3.16)–(3.20) are given by the eigenvalues of the matrix Γ01₁Γ₂ and by the eigenvalue 𝐹1−𝛿𝑉_𝑟+𝐾₀e^{−𝐸/𝑅𝑇𝑅eq}+𝛿²2𝐹𝑉_𝑟+𝐾₀e^{−𝐸/𝑅𝑇𝑅eq2}, (3.21) which takes its minimum, equal to 1/2, at 𝛿=𝛿_(1/2)=1/((𝐹/𝑉_𝑟)+𝐾₀e^{−𝐸/𝑅𝑇𝑅eq}).

Proof. By the form of the matrix 𝑀₁, the following equation holds for 𝜒₁ and for 𝜒₂, in (3.16) 𝜒₁()𝜒𝑘+1₂()=𝜒𝑘+101−12¹(𝑘)𝜒₂(𝑘)+0𝛿²𝑅₀(𝑘𝛿). (3.22) Notice that, by construction, such subsystem (and thus the overall system (3.16)) admits relative degree equal to 2, in a neighborhood of the origin, with respect to the dummy output 𝜒₁(𝑘). Indeed, taking into account of the form of 𝑅₀(𝑘𝛿), we have 𝜒₁(𝑘+1)=𝜒₂(),𝜒𝑘+1₁(𝑘+2)=𝜒₂(𝑘+1)=−12𝜒(𝑘)+𝛿²𝑙²_𝑝1ℎ𝑀⁻¹𝜒(𝑘)+𝑙_𝑝2𝑙_𝑝1ℎ𝑀⁻¹𝜒(𝑘)𝑣(𝑘𝛿), (3.23) and the term multiplying 𝑣(𝑘𝛿) is nonzero in a neighborhood of the origin, since 𝑙_𝑝2𝑙_𝑝1ℎ(𝑀⁻¹𝜒(𝑘)) is smooth in a neighborhood of the origin and 𝑙_𝑝2𝑙_𝑝1ℎ(0)≠0. Therefore, taking again into account the form of 𝑅₀, the proposed discrete-time feedback control law yields the following equations for the closed-loop-system (3.16), (3.20) 𝜒₁()𝜒𝑘+1₂()=𝑘+1Γ01₁Γ₂𝜒₁(𝑘)𝜒₂(𝑘),𝜒₃(𝑘+1)=𝜒₃(𝑘)+𝛿𝑝₃₁𝑀⁻¹𝜒(𝑘+𝛿)²2𝑞𝑀⁻¹𝜒(𝑘). (3.24) In order to show the asymptotic stability of the nonlinear system (3.24), it is sufficient to show that the linear system obtained by linearizing (3.24) is asymptotically stable. The Jacobian, at zero, of the function in the right-hand side of (3.24) is given, taking into account of 𝑝₃₁ and of 𝑞, by 𝐽=Γ010₁Γ₂0𝛼𝛽𝛾, (3.25) where 𝐹𝛾=1−𝛿𝑉_𝑟+𝐾₀e^{−𝐸/𝑅𝑇𝑅eq}+𝛿²2𝐹𝑉_𝑟+𝐾₀e^{−𝐸/𝑅𝑇𝑅eq2} (3.26) and 𝛼,𝛽 are suitable reals. Given the triangular structure of the Jacobian 𝐽, the eigenvalues are given by the eigenvalues of the matrix Γ01₁Γ₂ and by 𝛾. The eigenvalues of the matrix Γ01₁Γ₂ are inside the open unitary circle. The eigenvalue 𝛾 is always positive (for positive sampling period), and is inside the open unitary circle if the inequality (3.19) is satisfied. Moreover, by using the derivative of 𝛾 with respect to 𝛿, we obtain that 𝛾 takes the minimum value, 0.5, at 𝛿_(1/2).

Now, going back to initial variables 𝑥, by means of the (approximated) relation 𝜒(𝑘)≊𝑀𝜙⁻¹(𝑥(𝑘𝛿)), the digital feedback control law here proposed is given, for 𝑘𝛿≤𝑡<(𝑘+1)𝛿, 𝑘=0,1,2,…, by𝑢(𝑡)=4𝑉_𝑗𝜆𝐾₀𝐹𝐶𝑎₀𝑇_cin−𝑥₃(𝑘𝛿)⁻¹𝑈⁻¹𝐴_𝑗⁻¹e^{𝐸/𝑅𝑥2(𝑘𝛿)−1}𝑉+2^𝑗𝑉_𝑟𝜌𝐶_𝑝𝑇_𝑅eqΓ₁𝛿²𝑇_cin−𝑥₃(𝑘𝛿)𝑈𝐴_𝑗𝑉−2^𝑗𝐹𝑥₂(𝑘𝛿)𝑇_cin−𝑥₃(𝑘𝛿)𝑉_𝑟+𝑉_𝑗𝜌𝐶_𝑝𝐹²𝑇₀𝑇_cin−𝑥₃(𝑘𝛿)𝑈𝐴_𝑗𝑉_𝑟𝑉−2^𝑗𝑉_𝑟𝜌𝐶_𝑝𝑥₂(𝑘𝛿)𝛿²𝑇_cin−𝑥₃(𝑘𝛿)𝑈𝐴_𝑗𝑉+2^𝑗𝑥₂(𝑘𝛿)𝛿𝑇_cin−𝑥₃(𝑘𝛿)−2𝑉_𝑗𝜆𝐾₀𝐹𝑥₁(𝑘𝛿)𝑇_cin−𝑥₃(𝑘𝛿)⁻¹𝑈⁻¹𝐴_𝑗⁻¹e^{𝐸/𝑅𝑥2(𝑘𝛿)−1}−𝑉_𝑟𝑉_𝑗𝜆𝐾₀²𝑥₁(𝑘𝛿)𝑇_cin−𝑥₃(𝑘𝛿)⁻¹𝑈⁻¹𝐴_𝑗⁻¹e^{𝐸/𝑅𝑥2(𝑘𝛿)−2}+𝑉_𝑗𝐹𝑇₀𝜆𝑥₁(𝑘𝛿)𝐾₀𝐸𝑇_cin−𝑥₃(𝑘𝛿)⁻¹𝑈⁻¹𝐴_𝑗⁻¹𝑅⁻¹𝑥₂(𝑘𝛿)⁻²e^{𝐸/𝑅𝑥2(𝑘𝛿)−1}−𝑉_𝑗𝜆𝐾₀𝐹𝑥₁(𝑘𝛿)𝐸𝑇_cin−𝑥₃(𝑘𝛿)⁻¹𝑈⁻¹𝐴_𝑗⁻¹𝑥₂(𝑘𝛿)⁻¹𝑅⁻¹e^{𝐸/𝑅𝑥2(𝑘𝛿)−1}−𝑉_𝑟𝑉_𝑗𝜆²𝑥₁(𝑘𝛿)²𝐾₀²𝐸𝑇_cin−𝑥₃(𝑘𝛿)⁻¹𝑈⁻¹𝐴_𝑗⁻¹𝜌⁻¹𝐶_𝑝⁻¹e^{𝐸/𝑅𝑥2(𝑘𝛿)−2}1𝑅𝑥₂(𝑘𝛿)−𝑉_𝑗𝜆𝑥₁(𝑘𝛿)𝐾₀𝑇_cin−𝑥₃(𝑘𝛿)⁻¹𝜌⁻¹𝐶_𝑝⁻¹e^{𝐸/𝑅𝑥2(𝑘𝛿)−1}𝑉−2^𝑗𝑉_𝑟𝜌𝐶_𝑝Γ₁𝑥₂(𝑘𝛿)𝛿²𝑇_cin−𝑥₃(𝑘𝛿)𝑈𝐴_𝑗−𝑉_𝑗𝜆𝑥₁(𝑘𝛿)𝐾₀𝐸𝑇_cin−𝑥₃(𝑘𝛿)⁻¹𝜌⁻¹𝐶_𝑝⁻¹𝑥₂(𝑘𝛿)⁻¹𝑅⁻¹e^{𝐸/𝑅𝑥2(𝑘𝛿)−1}−𝑉_𝑗𝑈𝐴_𝑗𝑥₂(𝑘𝛿)𝑇_cin−𝑥₃(𝑘𝛿)𝑉_𝑟𝜌𝐶_𝑝+𝑉_𝑗𝑥₃(𝑘𝛿)𝐹𝑇_cin−𝑥₃(𝑘𝛿)𝑉_𝑟+𝑉_𝑗𝐹𝑇₀𝑇_cin−𝑥₃(𝑘𝛿)𝑉_𝑟−𝑉_𝑗𝜌𝐶_𝑝𝐹²𝑥₂(𝑘𝛿)𝑇_cin−𝑥₃(𝑘𝛿)𝑈𝐴_𝑗𝑉_𝑟+𝑉_𝑗𝑥₃(𝑘𝛿)𝜆𝑥₁(𝑘𝛿)𝐾₀𝐸𝑇_cin−𝑥₃(𝑘𝛿)⁻¹𝜌⁻¹𝐶_𝑝⁻¹𝑅⁻¹𝑥₂(𝑘𝛿)⁻²e^{𝐸/𝑅𝑥2(𝑘𝛿)−1}+𝑉_𝑗𝑈𝐴_𝑗𝑥₃(𝑘𝛿)𝑇_cin−𝑥₃(𝑘𝛿)𝑉_𝑟𝜌𝐶_𝑝−𝑈𝐴_𝑗𝑥₂(𝑘𝛿)𝑇_cin−𝑥₃(𝑘𝛿)𝜌_𝑗𝐶_𝑗+𝑈𝐴_𝑗𝑥₃(𝑘𝛿)𝑇_cin−𝑥₃(𝑘𝛿)𝜌_𝑗𝐶_𝑗𝑉−2^𝑗𝑥₃(𝑘𝛿)𝛿𝑇_cin−𝑥₃(𝑉𝑘𝛿)+2^𝑗𝑉_𝑟𝜌𝐶_𝑝𝑇_𝑅eq𝛿²𝑇_cin−𝑥₃(𝑘𝛿)𝑈𝐴_𝑗𝑉−2^𝑗𝜌𝐶_𝑝𝐹𝑇₀𝛿𝑇_cin−𝑥₃(𝑘𝛿)𝑈𝐴_𝑗𝑉+2^𝑗𝜌𝐶_𝑝𝐹𝑥₂(𝑘𝛿)𝛿𝑇_cin−𝑥₃(𝑘𝛿)𝑈𝐴_𝑗+2𝑉_𝑟𝑉_𝑗𝜆𝑥₁(𝑘𝛿)𝐾₀𝛿⁻¹𝑇_cin−𝑥₃(𝑘𝛿)⁻¹𝑈⁻¹𝐴_𝑗⁻¹e^{𝐸/𝑅𝑥2(𝑘𝛿)−1}. (3.27)

The digital control law (3.27) is applied to the CSTR system, described by the differential equations (2.1), by means of a zero-order hold device. Such control law can be easily implemented on a microprocessor. It is a static feedback control law, that is, no dynamics are involved. The control law has to be computed at every sampling period which is very much larger than the computation time (many minutes versus some seconds at maximum). The great advantage of this digital control law is also given by the fact that only sampled-data measures, provided by a suitable device, are necessary. Take into account that, in practice, continuous time measures are difficult to obtain, or even impossible. The feedback control law here presented is therefore easily realizable. The closed-loop system (2.1), (3.27) (with the zero-order hold) is simulated in the following section.

4. Simulation Results

Simulations, using MatLab (The MathWorks, Inc.) Software Package, have been carried out to verify the effectiveness of the proposed method. The values of the model parameters used in simulation are given in Table 1 (taken from [23]). In all the carried out simulations, a minimum and a maximum value for the input 𝑢, specifically the volumetric flow rate of the refrigerant, is imposed, to take into account of the actuator saturation. It is considered: 𝐹_𝐽min≤𝑢(𝑡)≤𝐹_𝐽max (4.1) with the following acceptable physical values for a control valve 𝐹_𝐽min=0.002m³/s=7.2m³𝐹/h,_𝐽max=0.08m³/s=288m³/h. (4.2) With these parameters, 𝛿_max in condition (3.19) is equal to 0.57 h, which corresponds to about 34 min. The value of 𝛿_(1/2) is equal to about 0.28 h which corresponds to about 16 min. We performed several sets of simulation runs and all showed the high performance of the digital control law here proposed. We choose the reactor conversion 𝜘_eq=0.85, to which the following unstable operating point corresponds: 𝐶𝑎_eq=1.2015 kmol/m³, 𝑇_𝑅eq=350 K, 𝑇_𝐽eq=312 K, 𝑢_eq=69.86 m³/h. In the simulations here presented, the process is initially assumed to be at the start up, that is 𝑥₁(0)=𝐶𝑎₀ (concentration of reactant in feed) = 8.01 kmol/m³, 𝑥₂(0)=𝑇₀ (temperature of feed) = 294 K, 𝑥₃(0)=𝑇_cin (supply temperature of cooling medium) = 294 K. This means that the reactor is very far from the operating point.

Table 1 

Reactor and controller parameters values.

In Simulation 1 (see Table 2, Figure 2) we consider the reactor at nominal parameters values (see Table 1), without any uncertainty. Figure 2 shows the evolution of the reactant concentration, of the reactor temperature, of the jacket temperature, and of the piecewise constant control signal, respectively, with sampling period equal to 10⁻¹ h (6 min).

Table 2 

Process parameters of Simulation 1.

(a)

(b)

(c)

(d)

Figure 2 

Results of Simulation 1.

As can be seen, all the state variables converge to the operating point ones in a very good fashion, without dangerous oscillations. The reactor temperature settling time is estimated from plots to be about 6 h, the steady-state error is equal to 0.

In Simulation 2 (see Table 3, Figure 3) we consider again the reactor at nominal parameters values (see Table 1), without any parameters mismatch, and a sampling period equal to 0.25 h (15 min) is considered. The same desirable convergence to the operating point is achieved, with a reactor temperature settling time of about 8 h. Again no dangerous oscillations appear and the reactor temperature is driven in a very good fashion to the operating point, with steady-state error equal to 0.

Table 3 

Process parameters of Simulation 2.

(a)

(b)

(c)

(d)

Figure 3 

Results of Simulation 2.

In Simulations 3, 4 (see Table 4, Figure 4, and Table 5, Figure 5, resp.) we consider the reactor whose parameters are uncertain, with sampling time again equal to 0.1 h and 0.25 h, respectively. We apply the controller obtained using the nominal values of the parameters (see Table 1), to a reactor whose parameter values are different with respect to the nominal ones as reported in Table 5. In this case we observe a reactor temperature steady-state error of about 5 K. Take into account that the parameters mismatch has been deliberately chosen very critical, in order to show the robustness of the proposed controller.

Table 4 

Process parameters of Simulation 3.

Table 5 

Process parameters of Simulation 4.

(a)

(b)

(c)

(d)

Figure 4 

Results of Simulation 3.

(a)

(b)

(c)

(d)

Figure 5 

Results of Simulation 4.

The performed simulations show the high performance of the proposed controller, as evidenced, in the case the reactor parameters are exactly known, by zero steady state error, no dangerous oscillations, good settling time. When parameters mismatch occurs, the proposed controller yields a very good behavior for the reactor, evidenced by small steady-state errors, no dangerous oscillations, and good settling time. We stress the fact that the proposed controller allows for sampled-data measures of the state variables, which is very useful in practical applications.

5. Conclusion

In this paper we have presented a digital control law for a continuous stirred tank reactor, constructed on the basis of an approximated sampled model and of the preservation of the relative degree methodology. The proposed feedback control law is easy to implement and simulations show high performance. A condition on the sampling period is found, by which the asymptotic stability of the closed-loop approximated sampled system is proved. Such condition is verified by sampling periods belonging to feasible sets, allowing for values larger than necessary in practical implementation. Future work will concern the construction of a discrete-time observer, in order to avoid reactant concentration measures, which are difficult to do and expensive (see [24–27]). Another topic which will be investigated is the problem arised by the recycle time-delay (see [21, 28–31]), on the basis of the results in [32].

Acknowledgments

This paper is supported by the University of L’Aquila Atheneum RIA Project and by the Center of Excellence for Research DEWS, L’Aquila, Italy.