An Enhanced Smith-Predictor-Based Control Law for a MIMO CSTR with Multiple Time-Delays

1. Introdction

Many dynamic systems witness dead-time or delay and are categorised as time-delay systems [1,2,3,4,5]. These include chemical processes, engines, manufacturing and telecommunication systems, etc. [6,7,8,9,10,11,12,13] . Time-delay is largely regarded as a source of instability and poor performance in control systems [14,15]. There are two categories of time-delay systems, widely known as network-induced and input-delay [16]. Network-induced delay is caused by the time needed to exchange data between devices in electrical/communication networks. In this category, states of a system affect their time derivatives with a delay. Control of system with network-induced time-delay is addressed using Lyapunov-Krasovskii and Razumikhin theorems [17,18] and is out of the scope of process control and this paper. On the other hand, in input-delay category, the control inputs affect the time derivatives of the system states with a delay, i.e. the actuator(s) affect the system after a non-negligible time [19,20]. This category of time-delay is a serious concern in process control [21,22] and is the focus of this paper.

Two main approaches have been employed to design model-based control systems for processes with input-delay: (i) integrating the delay into the model prior to control system design and (ii) the use of predictors/predictive models. Recently, the first approach has been used to design discrete/digital control but only for electromechanical systems with very short (<10 ms) delays [23,24] and not yet for process control. Alternatively, in the first approach, the term presenting the delay within the model (i.e. an exponential term in Laplace domain) is commonly replaced by its approximate transfer function, an all pass filter in Laplace domain, using Pade technique [22,25]. Then, the resultant model is used for control system design. Pade approximation leads to model inaccuracy and transforms a stable minimum-phase system to a non-minimum-phase one with a higher order [22]. These disadvantages, however, do not exist in the predictive methods, such as Smith predictors [26]. As a result, they are anticipated to exhibit higher performance.

Smith Predictors were originally developed for stable SISO systems, but their application was later extended to MIMO systems with multiple delays [27,28], unstable systems[29], long delays [30] and disturbance [31]. Consequently, Smith predictors were suggested to be a part of control solution for any stable or unstable time-delay system [19,32], and even branded as the best approach to deal with time delays [33].

Despite all the advantages and advances of SPCSs, there are reports in the literature about the superiority of feedback control systems based on Pade approximation over SPCSs. For example, in control of linear systems with identifiable time-varying delays [34,35]. The reasons for such an unexpected underperformance of SPCSs were investigated for single-input single output (SISO) systems in [16] and an enhanced SPCS, with an improved performance, was introduced, detailed in section 2 . This research extended enhanced SPCS to MIMO systems. The case study of this paper is a full-scale tracking control of a CSTR with three inputs, two outputs and three time delays. The employed CSTR model is unprecedented in its inclusion of system details.

2. Enhanced SISO Smith-Predictor-Based Control Systems- A Brief Review

Figure 1 shows a generic SISO SPCS, where $G\left(s\right)$ and $\widehat{G}\left(s\right)$ are the plant true and approximated transfer functions without the delay, respectively:

$$y\left(s\right)= u\left(s\right) e^{-s{t}_1}G\left(s\right),$$

(1)

where t₁ is the time delay in seconds, and $e^{-s{t}_1 }$represents the delay in Laplace domain. u, y and y_d are the input, output and its reference. In Figure 1, Figure 2 and Figure 3, C(s) is a linear classical controller designed for G(s), and hat ^ indicates an approximation.

Figure 2 is the re-arranged form of Figure 1. As mentioned earlier, the control architecture presented in Figure 1 has two inherent shortcomings: (i) the use of a classical feedback controller, C(s), intrinsically designed based on trade-off, and (ii) the reference y_d is compared with the approximated output at a time in the future. Figure 2 clealrly demonstrates the asynchrony between signals generating the error in a conventional SPCS.

Detailed analysis in [16] concludes that the architecture presented in Figure 3 eliminates both aforementioned shortcomings. If G(s) is a first order transfer function, K and u^* will be real numbers and stabilise the system [14]. y_d (s) $e^{s{t}_1}$ is the future value of the reference t₁ seconds ahead of current time.

3. Enhanced MIMO Smith-Predictor-Based Control Systems

This section aims to develop an enhanced SPCS for linear MIMO time-delay processes, presented in (2):

$$\left[\begin{array}{c}{y}_1\left(s\right)\\ ⋮\\ y_{n_y}\left(s\right)\end{array}\right]=\left[\begin{array}{ccc}{G}_{11}\left(s\right)e^{-s{t}_{11}}& \dots & G_{1n_u}\left(s\right)e^{-{st}_{1n_u}}\\ ⋮& \ddots & ⋮\\ G_{n_{y}1}\left(s\right)e^{-s{t}_{n_{y}1}}& \dots & G_{n_y{n}_u}\left(s\right)e^{-s{t}_{n_y{n}_u}}\end{array}\right]\left[\begin{array}{c}{u}_1\left(s\right)\\ ⋮\\ u_n\left(s\right)\end{array}\right],$$

(2)

where n_u and n_y are the number of inputs and outputs respectively, and t stand for time delay. Ogunnaike and Ray extended original SPCS to MIMO linear systems[36]. Their works leads to a control law of (3) for (2):

$$u_j\left(s\right)={\displaystyle \sum _{i=1}^{n_y}}{C}_{ij}\left(s\right)\left(y_{d_i}\left(s\right)-y_{p_i}\left(s\right)\right),$$

(3)

where all C_ij controllers are classical controllers, and indices p and d stand for predicted and desired. Either (4) or (5) yield the predicted output components, $y_{p_i}$s:

$$y_{p_i}\left(s\right)={\displaystyle \sum _{j=1}^{n_u}}{\widehat{G}}_{ij}\left(s\right)u_j\left(s\right)\left(1-e^{-s{t}_{ij}}\right)+y_i\left(s\right),$$

(4)

$$\left[\begin{array}{c}{y}_{p1}\left(s\right)\\ ⋮\\ y_{p{n}_y}\left(s\right)\end{array}\right]=\stackrel{predictormatrix}{\overbrace{\left[\begin{array}{ccc}{\widehat{G}}_{11}\left(s\right)\left(1-e^{-s{t}_{11}}\right)& \dots & {\widehat{G}}_{1n_u}\left(s\right)\left(1-e^{-s{t}_{1n_u}}\right)\\ ⋮& \ddots & ⋮\\ {\widehat{G}}_{n_{y}1}\left(z\right)(1-e^{-s{t}_{n_{y}1}})& \dots & {\widehat{G}}_{n_y{n}_u}\left(z\right)(1-e^{-s{t}_{n_y{n}_u}})\end{array}\right]}}\left[\begin{array}{c}{u}_1\left(s\right)\\ ⋮\\ u_{n_u}\left(s\right)\end{array}\right]+\left[\begin{array}{c}{y}_1\left(s\right)\\ ⋮\\ y_{n_y}\left(s\right)\end{array}\right],$$

(5)

Use of an array of classical controllers, alike (3), is still the dominant approach for SPCS of MIMO systems[37]. For the system of (2) and the control law of (3), [36] showed that delays do not influence the closed loop stability, if discrepancy of G_ij(s) and $\widehat{G}$_ij(s) is negligible for all values of i and j. In other words, if C_ij controllers can stabilise the delay-less system, they can stabilise (2) too.

The control law of (3), as a widely accepted form of an MIMO SPCS, has the following shortcomings similar to its SISO counterpart:

(i)

Classical controllers of C_ij suffer from trade-off and/or windup. The design of classical controllers involves a trade-off between the performance and the steady-state error. Moreover, windup phenomenon (the influence of actuator’s saturation on integrals [38]) happens to classical controllers. Both trade-off-based design and windup (as well as known anti-windup components) may negatively affect the control response and push it away from optimal behaviour.

(ii)

y_pi and y_di are asynchronous in (3) . In order to illustrate this asynchrony, let us extend (4):

$$y_{pi}\left(s\right)={\displaystyle \sum _{j=1}^{n_u}}{\widehat{G}}_{ij}\left(s\right)u_j\left(s\right)\left(1-e^{-s{t}_{ij}}\right)+\stackrel{y_i\left(s\right)}{\overbrace{{\displaystyle \sum _{j=1}^{n_u}}{G}_{ij}\left(s\right)u_j\left(s\right)e^{-s{t}_{ij}}}}\stackrel{G_{ij}\approx {\widehat{G}}_{ij}}{\Rightarrow }{y}_{pi}\left(s\right)\approx {\displaystyle \sum _{j=1}^{n_u}}\stackrel{y_{pij}\left(s\right)}{\overbrace{{\widehat{G}}_{ij}\left(s\right)u_j\left(s\right)}}={\displaystyle \sum _{j=1}^{n_u}}{y}_{pij}\left(s\right)$$

Therefore,

$${ y_{pij}\left(s\right)=G}_{ij}\left(s\right)u_j\left(s\right).$$

(6)

where y_pij is a constituent of y_pi influenced by u_j. In addition, from (2),

$$y_i\left(s\right)={\displaystyle \sum _{j=1}^{n_u}}\stackrel{y_{ij}\left(s\right)}{\overbrace{G_{ij}\left(s\right)u_j\left(s\right)e^{-s{t}_{ij}}}}={\displaystyle \sum _{j=1}^{n_u}}{y}_{ij}\left(s\right)\Rightarrow y_{ij}\left(s\right)=G_{ij}\left(s\right)u_j\left(s\right)e^{-s{t}_{ij}},$$

(7)

where y_ij is the constituent of y_i influenced by u_j.

Comparison of (6) and (7) shows that, in conventional MIMO SPCS, demonstrated by (3) and (4), the constituents of predicted output (y_pij s) do not belong to the same time as the constituents of output, y_ij. Indeed, y_ij is t_ij seconds behind y_pij. As a result, in conventional MIMO SPCS, two asynchronous variables (y_di and y_pi) are compared to find the control error, fed to the controller.

In order to address the aforementioned dual shortcomings, two enhancements are proposed for MIMO systems. Enhancement 1 removes the need for classical feedback controllers (shortcoming 1), and Enhancement 2 tackles asynchrony (shortcoming 2).

Enhancement 1 MIMO: A feedback-feedforward control law of (8) replaces the feedback controller of C_ij in (3):

$$u_j={u_j}^{*}+\sum _{\iota =1}^{n_x}{K}_{\iota j}\left(x_{d\iota }-x_{p\iota }\right),j=1:n_u.$$

(8)

where K_ιj is an element of K found through pole placement. x_dι and x_pι are the states calculated based on y_di s and y_pi s. u^*_j is an input at the desired status.

Justification for Enhancement 1 MIMO: As to [36], providing that the discrepancy of G_{i j} and ${\widehat{G}}_{ij}$ is negligible, a control system which stabilises the delay-free process, i.e. (2) when Ɐ r_ij=0, will stabilise the original time-delay process, i.e. (2) with any values for r_ij . Let us assume (9) is a continuous state space model of the delay-free system:

$$\left\{\begin{array}{l}\dot{X}=AX+BU\\ Y=CX\end{array}\right..$$

(9)

With the desired status of (10):

$$\left\{\begin{array}{l}{\dot{X}}_d=A{X}_d+B{U}^{*}\\ Y_d=C{X}_d\end{array}\right..$$

(10)

By subtracting (9) from (10):

$$\left\{\begin{array}{l}{\dot{X}}_d-\dot{X}=A\left(X_d-X\right)+B\left(U^{*}-U\right)\\ Y_d-Y=C\left(X_d-X\right)\end{array}\right..$$

(11)

A state vector feedback control law of (12), designed to place stable poles for (11), stabilises (9), the delay-free system [39].

$$U={\rm K}(X_d-X)+U^{*},$$

(12)

which pushes the error, Y_d - Y, towards zero. (12) is an equivalent of (8).

Enhancement 2 MIMO: $y_{di}(t+\max t_{ij}\left|j=1:n_u\right.)$ is used to derive x_dι and u_j for control law of (8).

As explained in the shortcoming (ii), y_ij is t_ij seconds behind y_pij, or y_pi is the predicted output for a future time. However, in conventional MIMO SPCSs, y_pi is compared with the present time reference, y_di, to produce the control error. Enhancement 2 addresses this issue with the use of future values of reference upfront to produce the control error.

4. Case Study, a MIMO CSTR

The case study is a complex exothermic continuous stirred tank reactor (CSTR) with two control outputs, three control inputs and three various delays. This system is modelled as (13), composed of molar equilibrium and energy equilibrium equations. In fact, (13) is the combination of a single-delay two-output model presented in [40] and a double-input model of [41]; it also includes all possible time delays, neglected in previous research. Two outputs of CSTR model of (13) are effluent concentration C_A(t) [mol/L] and temperature T(t) [K]. The control inputs are feed flow rate q_f (t) [L/min],water flow rate q_w(t) [L/min] and auxiliary temperature T_x(t) [K]. θ₁, θ₂ and θ₃ [min] present time-delays.

$$\left\{\begin{array}{c}V{\displaystyle \frac{d{C}_A\left(t\right)}{dt}}=q_f\left(t-{\theta }_1\right)\left(C_f-C_A\left(t\right)\right)+q_w\left(t-{\theta }_1\right)\left(0-C_A\left(t\right)\right)-V\kappa \mathit{exp}\left({\displaystyle \frac{-E}{RT\left(t\right)}}\right)C_A\left(t\right), \\ V\rho c{\displaystyle \frac{dT\left(t\right)}{dt}}=\left(q_f+q_w\right)\left(t-{\theta }_2\right)\rho c\left(T_f-T\left(t\right)\right)-VH\kappa \mathit{exp}\left({\displaystyle \frac{-E}{RT\left(t\right)}}\right)C_A\left(t\right)-U\left(T\left(t\right)-T_x\left(t-{\theta }_3\right)\right). \end{array}\right.$$

(13)

V, C_f, κ, E, R, c, U, H and ρ stand for reactor volume [L], feed concentration [mol/L], reaction velocity constant [min^-1], Arrhenius activation energy [J/ mol], gas constant [J/(mol.K)], specific heat capacitance [g/(L.K)], overall heat transfer coefficient [J/(min.K)], heat of reaction [J/min] and density [g/L], respectively. Water and feed temperature are assumed equal, T_f. All listed parameters are considered to be time-invariant.

4.1. Linearisation of the Model

This subsection explains how (13), with three inputs, is converted into dual models for control purposes, each with two inputs: (i) model (14) when the concentration is intended to increase and (ii) model (15) when the concentration is intended to decrease. These dual models were approximated by linear state equations of (16) and (17) for state vector feedback control system design.

Plausibly, feed and water valves, which are in charge of increasing and decreasing the concentration, should not work simultaneously. Therefore, two different models of (14) and (15) are proposed for situations where the concentration is intended to increase (water valve is closed) or to decrease (feed valve is closed), respectively:

$$\left\{\begin{array}{c}V{\displaystyle \frac{d{C}_A\left(t\right)}{dt}}=q_f\left(t-{\theta }_1\right)\left(C_f-C_A\left(t\right)\right)-V\kappa \mathit{exp}\left({\displaystyle \frac{-E}{RT\left(t\right)}}\right)C_A\left(t\right), \\ V\rho c{\displaystyle \frac{dT\left(t\right)}{dt}}=q_f\left(t-{\theta }_2\right)\rho c\left(T_f-T\left(t\right)\right)-VH\kappa \mathit{exp}\left({\displaystyle \frac{-E}{RT\left(t\right)}}\right)C_A\left(t\right)-U\left(T\left(t\right)-T_x\left(t-{\theta }_3\right)\right). \end{array}\right.$$

(14)

$$\left\{\begin{array}{c}V{\displaystyle \frac{d{C}_A\left(t\right)}{dt}}=-q_w\left(t-{\theta }_1\right)C_A\left(t\right)-V\kappa \mathit{exp}\left({\displaystyle \frac{-E}{RT\left(t\right)}}\right)C_A\left(t\right), \\ V\rho c{\displaystyle \frac{dT\left(t\right)}{dt}}=q_w\left(t-{\theta }_2\right)\rho c\left(T_f-T\left(t\right)\right)-VH\kappa \mathit{exp}\left({\displaystyle \frac{-E}{RT\left(t\right)}}\right)C_A\left(t\right)-U\left(T\left(t\right)-T_x\left(t-{\theta }_3\right)\right). \end{array}\right.$$

(15)

Compared to (13), $q_w$ and $q_f$ equal zero in (14) and (15), respectively.

Let us consider C_A, T, $q_f$,$q_w$and T_x as y₁, y₂, u₁, u₂ and u₃, respectively. With the use of nominal values of parameters listed in [40], $T_f$=350 K and reasonable delays, (14) and (15) are converted to (16) and (17), respectively.

$$\left\{\begin{array}{c}{\dot{y}}_1\left(t\right)=u_1(t-0.05)\left(0.01-0.01y_1\left(t\right)\right)-7.2\times 10^{10}\mathit{exp}\left({\displaystyle \frac{-8750}{y_2\left(t\right)}}\right)y_1\left(t\right), \\ {\dot{y}}_2\left(t\right)=u_1\left(t-0.1\right)\left(3.5-0.01y_2\left(t\right)\right)+1.506\times 10^{13}{y}_1\left(t\right)\mathit{exp}\left({\displaystyle \frac{-8750}{y_2\left(t\right)}}\right)-2.092\left(y_2\left(t\right)-u_3\left(t-0.2\right)\right). \end{array}\right.$$

(16)

$$\left\{\begin{array}{c}{\dot{y}}_1\left(t\right)=-u_2\left(t-0.05\right)\left(0.01y_1\left(t\right)\right)-7.2\times 10^{10}\mathit{exp}\left({\displaystyle \frac{-8750}{y_2\left(t\right)}}\right)y_1\left(t\right), \\ {\dot{y}}_2\left(t\right)=u_2\left(t-0.1\right)\left(3.5-0.01y_2\left(t\right)\right)+1.506\times 10^{13}{y}_1\left(t\right)\mathit{exp}\left({\displaystyle \frac{-8750}{y_2\left(t\right)}}\right)-2.092\left(y_2\left(t\right)-u_3\left(t-0.2\right)\right). \end{array} \right.$$

(17)

For control system design purposes, (16) was linearized around the operating point of${\overline{y}}_1=0.2$ mol/L and ${\bar{y}}_2=300K$, to produce an approximate linear model when the concentration is intended to rise:

$$\left[\begin{array}{c}{\dot{y}}_1(t)\\ {\dot{y}}_2\left(t\right)\end{array}\right]=\left[\begin{array}{cc}-0.016& 0\\ 3.243& -2.102\end{array}\right]\left[\begin{array}{c}{y}_1\left(t\right)\\ y_2\left(t\right)\end{array}\right]+\left[\begin{array}{cc}0.008e^{-0.05s}& 0\\ 0.500e^{-0.1s}& 2.092e^{-0.2s}\end{array}\right]\left[\begin{array}{c}{u}_1\left(t\right)\\ u_3\left(t\right)\end{array}\right].$$

(18)

Similarly, if concentration is intended to decline, the approximate linear model of (17), is

$$\left[\begin{array}{c}{\dot{y}}_1(t)\\ {\dot{y}}_2\left(t\right)\end{array}\right]=\left[\begin{array}{cc}-0.016& 0\\ 3.243& -2.102\end{array}\right]\left[\begin{array}{c}{y}_1\left(t\right)\\ y_2\left(t\right)\end{array}\right]+\left[\begin{array}{cc}-0.002e^{-0.05s}& 0\\ 0.500e^{-0.1s}& 2.092e^{-0.2s}\end{array}\right]\left[\begin{array}{c}{u}_2\left(t\right)\\ u_3\left(t\right)\end{array}\right].$$

(19)

4.2. Control

In this research, the proposed enhanced MIMO SPCS of (8) was employed to simultaneously control concentration and temperature. Sole control of temperature [42] or concentration [40] have been reported for time-delay CSTR models. However, none of several double-input double-output control solutions for CSTRs, e.g. [41,43,44,45,46,47,48,49], consider and address time-delay(s).

In order to develop the proposed enhanced SPCS, presented in (8), the following should be defined:

(i)

feedback control gains, K_ij s in (8), which are the elements of K in (12),

(ii)

a feedforward control law defining u_j^* in (8), or practically u₁^*, u₂^* and u₃^* in this problem,

(iii)

Smith predictor to estimate $x_{p\iota } s in \left(8\right)$ or practically predicted y₁ and y₂ in this case study,

(iv)

values of maximum t_ij, which are practically max (t₁₁ and t₁₂) to be used with u_1, max (t₂₁ and t₂₂) to be used with u₂ and max (t₃₁ and t₃₂) to be used with u₃.

Feedback Control Gains

In the investigated case study, the states, represented by x in (8), are same as the outputs, represented by y. Delay-free form of state space models, (18) and (19), were used to obtain feedback control gains. Closed loop poles of -5 and -2 were opted for concentration and temperature, y₁ and y₂, respectively. The pole related to concentration was chosen further from zero due to the higher importance of this output. The matrices of feedback control gains, when concentration is intended to be increased or decreased, is shown as K in (20) and (21), respectively:

$$\left[\begin{array}{c}{u}_1\\ u_3\end{array}\right]=\stackrel{\mathbf{K}}{\overbrace{\left[\begin{array}{c}623.0622 0\\ -147.3616 -0.0488\end{array}\right]}}\left[\begin{array}{c}{y}_{d1}{-y}_1\\ y_{d2}{-y}_2\end{array}\right]+\left[\begin{array}{c}{u^{*}}_1\\ {u^{*}}_3\end{array}\right].$$

(20)

$$\left[\begin{array}{c}{u}_2\\ u_3\end{array}\right]=\stackrel{\mathbf{K}}{\overbrace{\left[\begin{array}{c} -2492.2486 0\\ 597.1977 -0.0488\end{array}\right]}}\left[\begin{array}{c}{y}_{d1}{-y}_1\\ y_{d2}{-y}_2\end{array}\right]+\left[\begin{array}{c}{u^{*}}_2\\ {u^{*}}_3\end{array}\right].$$

(21)

4.3. Feedforward Control Law Regardless of Time Delays

At the desired status of ${\dot{y}}_1(t)={\dot{y}}_2(t)=0$, $y_1=y_{d1}$ and $y_2=y_{d2}$, delay-free form of (16 and 17) lead to (22 and 23):

$$\left\{\begin{array}{c}0={u^{*}}_1\left(0.01-0.01y_{d1}\right)-7.2\times 10^{10}{y}_{d1}\mathit{exp}\left({\displaystyle \frac{-8750}{y_{d2}}}\right), \\ 0={u^{*}}_1\left(3.5-0.01y_{d2}\right)+1.506\times 10^{13}{y}_{d1}\mathit{exp}\left({\displaystyle \frac{-8750}{y_{d2}}}\right)-2.092\left(y_{d2}-{u^{*}}_3\right). \end{array}\right.$$

(22)

$$\left\{\begin{array}{c}0=-{u^{*}}_2\left(0.01y_{d1}\right)-7.2\times 10^{10}{y}_{d1}\mathit{exp}\left({\displaystyle \frac{-8750}{y_{d2}}}\right), \\ 0={u^{*}}_2\left(3.5-0.01y_{d2}\right)+1.506\times 10^{13}{y}_{d1}\mathit{exp}\left({\displaystyle \frac{-8750}{y_{d2}}}\right)-2.092\left(y_{d2}-{u^{*}}_3\right). \end{array} \right.$$

(23)

The combination of the second equations in (22 and 23) leads to (24):

$$0=0.5\left({{u^{*}}_1+u^{*}}_2\right)\left(3.5-0.01y_{d2}\right)+1.506\times 10^{13}{y}_{d1}\mathit{exp}\left({\displaystyle \frac{-8750}{y_{d2}}}\right)-2.092\left(y_{d2}-{u^{*}}_3\right)\Rightarrow$$

$${u^{*}}_3={\displaystyle \frac{1}{-2.092}}\left(0.5\left({{u^{*}}_1+u^{*}}_2\right)\left(3.5-0.01y_{d2}\right)+1.506\times 10^{13}{y}_{d1}\mathit{exp}\left({\displaystyle \frac{-8750}{y_{d2}}}\right)-2.092\left(y_{d2}\right)\right).$$

(24)

The following feedforward control law is derived out of (22, 23 and 24) :

$$\left\{\begin{array}{c}\begin{array}{c}{u^{*}}_1=y_{d1}\left(1+7.2\times 10^{12}\mathit{exp}\left({\displaystyle \frac{-8750}{y_{d2}}}\right)\right),\mathrm{i}\mathrm{f}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{e},\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}0\\ {u^{*}}_2=-7.2\times 10^{12}\mathit{exp}\left({\displaystyle \frac{-8750}{y_{d2}}}\right),\mathrm{i}\mathrm{f}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{e},\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e} 0 \end{array} \\ {u^{*}}_3=-0.2390\left({{u^{*}}_1+u^{*}}_2\right)\left(3.5-0.01y_{d2}\right)-7.2\times 10^{12}{y}_{d1}\mathit{exp}\left({\displaystyle \frac{-8750}{y_{d2}}}\right)+y_{d2}. \end{array}\right.$$

(25)

4.3.1. Smith Predictor

Equations (18 and 19) can be converted to (26 and 27) to produce $G_{ij}\left(s\right)e^{-s{t}_{ij}}$s of (7):

$$\left[\begin{array}{c}{y}_1\left(s\right)\\ y_2\left(s\right)\end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{ 0.008 s + 0.0168}{s^2+ 2.118 s + 0.0326}}{e}^{-0.05s}& 0\\ {\displaystyle \frac{ 0.5 s + 0.0337}{s^2+ 2.118 s + 0.0326}}{e}^{-0.1s}& {\displaystyle \frac{ 2.092 s + 0.0324}{s^2+ 2.118 s + 0.0326}}{e}^{-0.2s}\end{array}\right]\left[\begin{array}{c}{u}_1\left(s\right)\\ u_3\left(s\right)\end{array}\right].$$

(26)

$$\left[\begin{array}{c}{y}_1\left(s\right)\\ y_2\left(s\right)\end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{-0.002 s-0.0042}{s^2+2.118 s+0.0326}}{e}^{-0.05s}& 0\\ {\displaystyle \frac{ 0.5 s+0.0013}{s^2+2.118 s+0.0326}}{e}^{-0.1s}& {\displaystyle \frac{ 2.092 s+0.0324}{s^2+2.118 s+0.0326}}{e}^{-0.2s}\end{array}\right]\left[\begin{array}{c}{u}_2\left(s\right)\\ u_3\left(s\right)\end{array}\right].$$

(27)

4.3.2. Maximum t_ij

(26 and 27) evidently show max t_1j is 0.05 s and max t_2j is 0.2 s. With these delays, in the control law of (8), $x_{d1}=y_{d1}(t+0.05)$ and $x_{d2}=y_{d2}(t+0.2)$. Consequently, in practice, $y_{d1}(t+0.05)$ and $y_{d2}(t+0.2)$ were used instead of $y_{d1}$ and $y_{d2}$ in (20), (21) and (25).

5. Results and Discussion

Figure 4 and Figure 5 compare the simulation results of the proposed Enhanced SPCS with a conventional SPCS with two arrays of PIDs, four each, carefully turned for delay-less form of transfer functions in (26) and (27), as detailed in the Appendix A. A conventional SPCS was also designed for the case study with PIDs tuned using decentralised and centralised multivariable approaches. However, it failed to exhibit a presentable performance showing the challenging nature of this control problem. The original nonlinear system of (13) has been used in simulations with a sampling frequency of 1Hz, which is easily achievable in practice. The range of auxiliary temperature is [280350] K, and maximum flow rate of valves is 60 L/min. No measurement noise was considered in simulations presented in Figure 4 and Figure 5.

The enhanced SPCS evidently outperforms conventional SPCS with PIDs in concentration control and catches temperature reference much faster. The results also show that changes of concentration reference affect the temperature control for the proposed control system. The reason is that the proposed control system priorities concentration control, as a design requirement, through selection of faster closed loop poles for the concentration. Interestingly, upper graph of Figure 5 shows that the proposed control system uses the capacity of valves in full to compensate the error where the reference changes. The conventional SPCS with PIDs does not use this capacity and is consequently slower. This behaviour is rooted in inherent trade-off in PID design that slows down the system to avoid concentration overshoots. Moreover, the effect of integral windup is evident in bottom graph of Figure 5 as another weak point of the conventional SPCS with PIDs.

Figure 5 also indicates that, in the proposed control system, significant and fast changes of control inputs only happened at the beginning of operation or in the case of a change in the reference; while, conventional SPCS with PIDs turns water/feed valves ON/OFF very frequently e.g. in time periods of 3-10 minutes and 18-23 minutes and struggles to maintain the concentration on the reference, after it is reached. This is a great advantage for the proposed enhanced SPCS in terms of implementation and energy consumption. This advantage is influenced by the feedforward component of the enhanced SPCS. In order to elucidate such an influence, it should be noted that the CSTR (like many other processes) matches with the definition of generalised type zero (GTZ) systems proposed in [50]. In a GTZ system, the control equilibrium point (CEP) of the system is maintained only by continuous exertion of a control input; CEP is the situation where the error and its derivatives are zero [51]. The feedforward component in (8), $u_j^{*},$ detailed in (25) for the case study, has been designed to maintain the desired status of (10), when it is reached. The desired status equals CEP in the case study of this paper as detailed in section 3-2. In the absence of such a feedforward component in the control systems merely relying feedback PID controllers, when the reference is reached, the control input drops to zero shortly, which cannot keep a GTZ on the reference, this leads to the rise of error and a jump in control input to compensate it, e.g. the jumps in the middle graph of Figure 5.

The area outlined with an ellipse in Figure 5 shows the effect of Enhancement 2, where output of the Smith predictors is compared with its asynchronous value of reference in the future to produce the predicted error; please be notes that the reference changes at 10.05 minutes or 10 minutes and 3 second, not exactly at 10 min. The predicted error is fed to (20)/(21) for a timely action and improved performance.

Figure 6 is similar to Figure 4 but with an addition: the influence of random measurement noises of ±0.4 K and ±2% for temperature and concentration, respectively. Such noises are considered moderate for the case study [52,53]. Both control systems can handle noises very well.

6. Conclusions

In this paper, first, two shortcomings of conventional Smith-Predictor-based control systems were identified: (i) use of classical feedback controllers with their inherent constraints and (ii) asynchrony between the reference and the predicted output. The latter particularly causes the response to have a lag in tracking the reference. On this basis, two enhancements were proposed for SPCSs designed for MIMO time-delay systems: (i) replacing the array of classical feedback controllers by a feedback-feedforward arrangement, and (ii) supplying some future reference value(s) upfront to the control system. Section 2 demonstrates the proposed control system in detail.

The proposed enhanced SPCS was examined to control a nonlinear exothermic MIMO CSTR with multiple time delays and outperformed a conventional SPCS with PID feedback controllers manifestly. The employed CSTR model includes unprecedented level of details and comprehensiveness in terms of inputs, outputs, time-delays, actuators’ limits and sensor noises to assure that the proposed results are realistic, and the proposed method is implementable. The proposed method not only presented an excellent tracking performance; but also, it witnessed much fewer undesirable rapid changes of control inputs.