Nonlinear Constraint Optimization Based Robust Decentralized PID Controller for a Benchmark CSTR System Using Kharitonov–Hurwitz Stability Analysis

Abstract

This paper exploits the design of a decentralized proportional integral derivative (PID) controller based on nonlinear optimization for a continuously stirred tank reactor system. The basic objective is to attain the design specifications by maintaining both the temperature and concentration. The continuously stirred tank reactor is modeled to a first-order plus dead time system by designing a decoupler. The proposed PID controller is designed on the basis of fundamentals of nonlinear optimization. Further, the overshoot is bounded with constraints on the maximum closed-loop amplitude ratio. The control algorithm is designed for decoupled systems to reduce the loop interactions and attain the servo response. The robust stability is analyzed by considering multiplicative input as well as output uncertainties, while stability is verified with the Kharitonov–Hurwitz theorem. A concise comparison is made between the proposed technique with existing methods. It is envisaged that the proposed control algorithm exhibits better servo and regulatory responses compared to the existing techniques. Furthermore, the efficacious nature of the proposed control scheme is validated by considering a wide range of closed-loop amplitude ratios.

Appendices

Appendix A

The open-loop transfer function is given by Eq. (23)

$$\begin{aligned} {\mathcal {G}}(s){{\mathcal {C}}}(s)=\frac{{\mathcal {K}}_{\mathcal{C}\mathcal{L}} {\mathcal {K}}_{\mathcal{P}\mathcal{L}}({\mathcal {T}}_{{\mathcal {I}}}{\mathcal {T}}_{{\mathcal {D}}}s^{2} +{\mathcal {T}}_{{\mathcal {I}}}s+1)e^{-{\Phi }s}}{{\mathcal {T}}_{{\mathcal {I}}}s({\mathcal {T}}s+1)} \end{aligned}$$

(74)

As solving a third-order system is quite complicated, the model is reduced by the Routh stability criterion model reduction method. The transfer function of the reduced-order model can be obtained from the Routh array as reported in [42] and is given as

$$\begin{aligned} {\mathcal {G}}(s){{\mathcal {C}}}(s)=\frac{{\mathcal {K}}_1s+2{\mathcal {K}}_{\mathcal{C}\mathcal{L}} {\mathcal {K}}_{\mathcal{P}\mathcal{L}}(2{\mathcal {T}}_D-\Phi )}{{\mathcal {K}}_2s} \end{aligned}$$

(75)

where

$$\begin{aligned} {\mathcal {K}}_1= & {} 4{\mathcal {K}}_{\mathcal{C}\mathcal{L}}{\mathcal {K}}_{\mathcal{P}\mathcal{L}} {\mathcal {T}}_{{\mathcal {I}}}{\mathcal {T}}_{{\mathcal {D}}}-2{\mathcal {K}}_{\mathcal{C}\mathcal{L}} {\mathcal {K}}_{\mathcal{P}\mathcal{L}}\Phi {\mathcal {T}}_{{\mathcal {I}}}\nonumber \\{} & {} +{\mathcal {K}}_{\mathcal{C}\mathcal{L}}{\mathcal {K}}_{\mathcal{P}\mathcal{L}}\Phi ^2 \nonumber \\ {\mathcal {K}}_2= & {} (2{\mathcal {T}}_{{\mathcal {D}}}-\Phi )(2{\mathcal {T}} {\mathcal {T}}_{{\mathcal {I}}}+\Phi {\mathcal {T}}_{{\mathcal {I}}})s^2 +2{\mathcal {T}}_{{\mathcal {I}}}(2{\mathcal {T}}_{{\mathcal {D}}}-\Phi ) \nonumber \\{} & {} 1+{\mathcal {G}}(s){\mathcal {C}}(s)=0 \end{aligned}$$

(76)

Substituting Eq. (81) in (76)

$$\begin{aligned} s^2+(\beta )s+\frac{2{\mathcal {K}}_{\mathcal{C}\mathcal{L}} {\mathcal {K}}_{\mathcal{P}\mathcal{L}}(2{\mathcal {T}}_{{\mathcal {D}}}-\Phi )}{(2{\mathcal {T}}_{{\mathcal {D}}}-\Phi )(2{\mathcal {T}}{\mathcal {T}}_{{\mathcal {I}}} +\Phi {\mathcal {T}}_{{\mathcal {I}}})}=0 \end{aligned}$$

(77)

where

$$\begin{aligned} \beta =\frac{(2{\mathcal {T}}_{{\mathcal {I}}}(2{\mathcal {T}}_{{\mathcal {D}}}-\Phi ) +4{\mathcal {K}}_{\mathcal{C}\mathcal{L}}{\mathcal {K}}_{\mathcal{P}\mathcal{L}} {\mathcal {T}}_{{\mathcal {I}}}{\mathcal {T}}_{{\mathcal {D}}} -2{\mathcal {K}}_{\mathcal{C}\mathcal{L}}{\mathcal {K}}_{\mathcal{P}\mathcal{L}} \Phi {\mathcal {T}}_{{\mathcal {I}}}+{\mathcal {K}}_{\mathcal{C}\mathcal{L}} {\mathcal {K}}_{\mathcal{P}\mathcal{L}}\Phi ^2)}{(2{\mathcal {T}}_{{\mathcal {D}}} -\Phi )(2{\mathcal {T}}{\mathcal {T}}_{{\mathcal {I}}}+\Phi {\mathcal {T}}_{{\mathcal {I}}})} \end{aligned}$$

As Eq. (82) is of the form $s^2+2\zeta \omega _n s+\omega _n ^2$, it can be simplified as

$$\begin{aligned} \omega _n= & {} \sqrt{\frac{2{\mathcal {K}}_{\mathcal{C}\mathcal{L}}{\mathcal {K}}_{\mathcal{P}\mathcal{L}}}{(2{\mathcal {T}}{\mathcal {T}}_{{\mathcal {I}}}+\Phi {\mathcal {T}}_{{\mathcal {I}}})}} \end{aligned}$$

(78)

$$\begin{aligned} \zeta= & {} \frac{2{\mathcal {T}}_{{\mathcal {I}}}(2{\mathcal {T}}_{{\mathcal {D}}}-\Phi )+(4{\mathcal {K}}_{\mathcal{C}\mathcal{L}}{\mathcal {K}}_{\mathcal{P}\mathcal{L}}{\mathcal {T}} _{{\mathcal {I}}}{\mathcal {T}}_{{\mathcal {D}}}-2{\mathcal {K}}_{\mathcal{C}\mathcal{L}}{\mathcal {K}}_{\mathcal{P}\mathcal{L}}\Phi {\mathcal {T}}_{{\mathcal {I}}}+{\mathcal {K}} _{\mathcal{C}\mathcal{L}}{\mathcal {K}}_{\mathcal{P}\mathcal{L}}\Phi ^2)}{2(2{\mathcal {T}}_{{\mathcal {D}}}-\Phi )\sqrt{(2{\mathcal {T}}{\mathcal {T}}_{{\mathcal {I}}} +\Phi {\mathcal {T}}_{{\mathcal {I}}})}\sqrt{2{\mathcal {K}}_{\mathcal{C}\mathcal{L}}{\mathcal {K}}_{\mathcal{P}\mathcal{L}}} } \end{aligned}$$

(79)

The model specified in Eq. (81) can be approximated as second-order model. Hence, closed-loop resonant frequency, maximum overshoot, resonant peak and bandwidth are given by

$$\begin{aligned} {\mathcal {M}}_r= & {} \frac{1}{2\zeta \sqrt{1-\zeta ^2}} \nonumber \\ {\mathcal {M}}_p= & {} e^{-\frac{\zeta \pi }{\sqrt{1-\zeta ^2}}} \omega _c=\sqrt{1-2\zeta ^2+\sqrt{2-4\zeta ^2+4\zeta ^4}} \end{aligned}$$

(80)

The relation between damping ratio and phase margin is given as

$$\begin{aligned} \sigma _{{\mathcal {M}}}=tan^{-1}\frac{2\zeta }{\sqrt{-2\zeta ^2+\sqrt{1+4\zeta ^4}}} \end{aligned}$$

The frequency domain specifications of the second-order system is shown in Fig. 29.

Fig. 29

Frequency domain specifications

Appendix B: Stability Analysis

1.1 Lower Region

Table 6 Hurwitz matrix and row reduced Hurwitz matrix—Lower region

The FOPDT model of the plant

$$\begin{aligned} {\mathcal {A}}(s)=\begin{bmatrix} \frac{1.183}{0.315s+1}e^{-0.6412s} &{} 1\\ 1 &{} \frac{0.0016}{0.2856s+1}e^{-2.357s} \end{bmatrix} \end{aligned}$$

(81)

The designed PID controller is

$$\begin{aligned} {\mathcal {C}}(s)=\begin{bmatrix} 0.048+\frac{0.33}{s}+0.0575s &{} 0\\ 0 &{} 0.033+\frac{45.8}{s}+0.095s \end{bmatrix}\nonumber \\ \end{aligned}$$

(82)

Hence, as per Eq. (45), the open-loop transfer function can be obtained as

$$\begin{aligned} {\mathcal {W}}_{0}(s)= & {} {\mathcal {C}}(s)* {\mathcal {A}}(s) \nonumber \\= & {} \begin{bmatrix} 0.048+\frac{0.33}{s}+0.0575s &{} 0\\ 0 &{} 0.033+\frac{45.8}{s}+0.095s \end{bmatrix} * \nonumber \\{} & {} \begin{bmatrix} \frac{1.183}{0.315s+1}e^{-0.6412s} &{} 1\\ 1 &{} \frac{0.0016}{0.2856s+1}e^{-2.357s} \end{bmatrix} \end{aligned}$$

(83)

Table 7 Hurwitz matrix and row reduced Hurwitz matrix—Middle region

Table 8 Hurwitz matrix and row reduced Hurwitz matrix—Higher region

where the time delays can be expressed by neglecting the higher order terms as $e^{-0.6412s}=1-0.641269s$ and $e^{-2.357s} = 1-2.357s$. Further, Eqs. (81) and (83) can be rearranged as

$$\begin{aligned} {\mathcal {A}}(s)= & {} \begin{bmatrix} \frac{1.183-0.758s}{0.315s+1} &{} 1 \\ 1 &{}\frac{0.0016-0.0038s}{0.2856s+1} \end{bmatrix} \nonumber \\ {\mathcal {W}}_{0}(s)= & {} \begin{bmatrix} \frac{0.0575s^2+0.048s+0.33}{s} &{} 0 \\ 0 &{} \frac{0.095s^2+0.033s+45.8}{s} \end{bmatrix} \nonumber \\{} & {} * \begin{bmatrix} \frac{1.183-0.758s}{0.315s+1} &{} 1 \\ 1 &{}\frac{0.0016-0.0038s}{0.2856s+1} \end{bmatrix} \end{aligned}$$

(84)

$$\begin{aligned} {\mathcal {W}}_{0}(s)= & {} \begin{bmatrix} \frac{-0.043s^3+0.032s^2-0.193s+0.4}{0.315s^2+s} &{} \frac{0.0575s^2+0.048s+0.33}{s} \\ \frac{0.095s^2+0.033s+45.8}{s} &{} \frac{-0.00035s^3+0.0014s^2-0.17s+0.073}{0.2856s^2+s} \end{bmatrix}\nonumber \\ \end{aligned}$$

(85)

As described in Eq. (46),

$$\begin{aligned} det[I+{\mathcal {C}}(s){\mathcal {A}}(s)]=0 \end{aligned}$$

(86)

On substituting, Equations (85) into (86), the characteristics Equation can be obtained as

$$\begin{aligned}{} & {} 0.0004764s^6+0.01632s^5+0.01856s^4+1.29s^3\nonumber \\{} & {} \quad +4.54s^2+10.9s+15.08=0 \end{aligned}$$

(87)

The four Kharitonov polynomials are derived from Eqs. (505152)–(53) as

$$\begin{aligned} {\mathcal {K}}^1(s)= & {} s^6+0.6s^5+0.25s^4+0.08s^3\nonumber \\{} & {} +0.013s^2+0.009s+0.003 \end{aligned}$$

(88)

$$\begin{aligned} {\mathcal {K}}^2(s)= & {} s^6+0.88s^5+0.37s^4+0.085s^3 \nonumber \\{} & {} +0.012s^2+0.005s+0.0038 \end{aligned}$$

(89)

$$\begin{aligned} {\mathcal {K}}^3(s)= & {} s^6+0.72s^5+0.37s^4+0.1s^3\nonumber \\{} & {} +0.013s^2+0.001s+0.0038 \end{aligned}$$

(90)

$$\begin{aligned} {\mathcal {K}}^4(s)= & {} s^6+0.72s^5+0.25s^4+0.07s^3\nonumber \\{} & {} +0.013s^2+0.01s+0.000026 \end{aligned}$$

(91)

The Hurwitz matrix and the row reduced Hurwitz matrices for the Kharitonov polynomials described in Eqs. (888990)–(91) are given by Table 6. The elements in the matrix is positive, and there is no sign change. The positive values of the diagonal elements are verified, and the proposed controller is robust stable.

1.1.1 Middle Region

Similarly, Table 7 presents the Hurwitz matrix and the row reduced Hurwitz matrix for the Kharitonov polynomials for middle region. The positive values of the diagonal elements are verified, and the proposed controller is robust stable.

1.1.2 Higher Region

Subsequently, the Hurwitz matrix and the row reduced Hurwitz matrix for the Kharitonov polynomials for higher region is presented in Table 8. The positive values of the diagonal elements are verified and the proposed controller is robust stable.