Decentralized Robust Control of Nonlinear Uncertain Multivariable Systems

Abstract

In this paper, we propose the development of a new strategy for the synthesis of a decentralized robust control for multiple-input–multiple-output (MIMO) nonlinear systems. The proposed method takes into account system’s loop interactions to synthesize a MIMO robust control stabilizing the system facing parametric uncertainties. The use of RGA theory is proposed to define the structure of the robust controller, synthesized by the loop shaping design procedure (LSDP), with the most effective input/output pairing. Then, an extension of an existing transition technique to the nonlinear multivariable case is proposed to decide the equilibrium points. The transition from an equilibrium point to another is ensured by the variation in the scheduling parameter (SP) while monitoring the variation in the gap metric compared to the maximum stability margin of the LSDP. We note that the new developed transition approach exploits during its execution the RGA theory whose purpose is to impose a common input/output pairing. Furthermore, for each equilibrium point a local full-order robust controller is obtained based on the LSDP approach, whose configuration is simplified also by exploiting RGA theory by proposing a selection matrix to deduce a local final robust controller. In this case, the overall decentralized robust control system is obtained by switching between local final robust controllers, where the maximum stability margin is used as an indicator of robustness in terms of stability and precision against parametric uncertainties. The proposed algorithm, for the synthesis of the decentralized robust multivariable control of uncertain nonlinear MIMO systems, is validated for the regulation of a benchmark of type continuous stirred tank reactor in the presence of parametric uncertainties. Moreover, the efficiency of the proposed decentralized robust control is validated on an experimental communicating two tank system.

Appendices

A Local submodels parameters the CSTR

Table 5 Equilibrium points $({\varvec{x}}_{eq}^i,{\varvec{u}}_{eq}^i )$ and their corresponding transfer matrices ${\varvec{G}}_{eq}^i$ and RGA metric $\varvec{\varLambda }({\varvec{G}}_{eq}^i)$

Table 6 Shaped systems ${\varvec{G}}_{eq,sh}^i$ and and their corresponding maximum stability margins $\varepsilon _{max}^{i}$

B Robust Controllers

$$\begin{aligned} \begin{matrix} K_{f,12}^1(s)=\dfrac{170.6 s^3 + 1.407e^4 s^2 + 1.642e^5 s + 9.141e^4}{s^5 + 160.8 s^4 + 7632 s^3 + 9.287e^4 s^2 + 2.23e^5 s + 223}~\\ K_{f,21}^1(s)=\dfrac{3940 s^3 + 3.203e^5 s^2 + 1.746e^6 s + 2.287e^6}{s^5 + 161.3 s^4 + 7712 s^3 + 9.668e^4 s^2 + 2.694e^5 s + 1.115e^5} \\ \end{matrix} \end{aligned}$$

(87)

$$\begin{aligned} \begin{matrix} K_{f,12}^2(s)=\dfrac{138.6 s^3 + 1.215e^4 s^2 + 1.394e^5 s + 6.034e^4}{s^5 + 150.6 s^4 + 6578 s^3 + 7.344e^4 s^2 + 1.428e^5 s + 142.7}\\ K_{f,21}^2(s)=\dfrac{4419 s^3 + 2.823e^5 s^2 + 1.299e^6 s + 1.51e^6}{s^5 + 151.1 s^4 + 6654 s^3 + 7.672e^4 s^2 + 1.795e^5 s + 7.137e^4} \\ \end{matrix} \end{aligned}$$

(88)

$$\begin{aligned} \begin{matrix} K_{f,12}^3(s)=\dfrac{186.9 s^3 + 1.26e^4 s^2 + 1.423e^5 s + 7.786e^4}{s^5 + 149.4 s^4 + 6476 s^3 + 7.659e^4 s^2 + 1.87e^5 s + 187}\\ K_{f,21}^3(s)=\dfrac{3283 s^3 + 2.75e^5 s^2 + 1.458e^6 s + 1.951e^6}{s^5 + 149.9 s^4 + 6550 s^3 + 7.982e^4 s^2 + 2.253e^5 s + 9.348e^4} \\ \end{matrix} \end{aligned}$$

(89)

$$\begin{aligned} \begin{matrix} K_{f,12}^4(s)=\dfrac{130.8 s^3 + 8502 s^2 + 9.43e^4 s + 3.622e^4}{s^5 + 123.5 s^4 + 4533 s^3 + 4.779e^4 s^2 + 8.706e^4 s + 87.01}\\ K_{f,21}^4(s)=\dfrac{3251 s^3 + 1.872e^5 s^2 + 8.069e^5 s + 9.083e^5}{s^5 + 124 s^4 + 4594 s^3 + 5.005e^4 s^2 + 1.109e^5 s + 4.351e^4} \\ \end{matrix} \end{aligned}$$

(90)

$$\begin{aligned} \begin{matrix} K_{f,12}^5(s)=\dfrac{109.1 s^3 + 1.142e^4 s^2 + 1.497e^5 s + 7.407e^4}{s^5 + 165.3 s^4 + 7968 s^3 + 1.008e^5 s^2 + 2.286e^5 s + 228.5}\\ K_{f,21}^5(s)=\dfrac{4041 s^3 + 2.69e^5 s^2 + 1.382e^6 s + 1.854e^6}{s^5 + 165.8 s^4 + 8051 s^3 + 1.048e^5 s^2 + 2.789e^5 s + 1.142e^5} \\ \end{matrix} \end{aligned}$$

(91)

$$\begin{aligned} \begin{matrix} K_{f,12}^6(s)=\dfrac{97.83 s^3 + 8271 s^2 + 9.667e^4 s + 3.426e^4}{s^5 + 131.8 s^4 + 4930 s^3 + 5.176e^4 s^2 + 8.763e^4 s + 87.58}\\ K_{f,21}^6(s)=\dfrac{4064 s^3 + 1.934e^5 s^2 + 7.868e^5 s + 8.584e^5}{s^5 + 132.3 s^4 + 4996 s^3 + 5.422e^4 s^2 + 1.135e^5 s + 4.379e^4} \\ \end{matrix} \end{aligned}$$

(92)

C Local submodels parameters of the CTTS

Table 7 Equilibrium points $({\varvec{x}}_{eq}^i,{\varvec{u}}_{eq}^i )$ and their corresponding transfer matrices ${\varvec{G}}_{eq}^i$

Table 8 Shaped systems ${\varvec{G}}_{sh,eq}^i$ and their corresponding maximum stability margins $\varepsilon _{max}^{i}$