On Iterative Closed-Loop Identification Using Affine Takagi–Sugeno Models and Controllers

Abstract

Often models are used for controller design that was identified under the objective to well approximate the system under study. In this paper, a scheme for identifying discrete-time locally affine Takagi–Sugeno (TS) models is presented, which better reflects the dedicated model use for designing a TS controller. For this purpose, after an initial open-loop experiment and controller design step, additional experiments are carried out in closed loop, each followed by an identification and controller design step. The deployed TS controllers are of parallel distributed compensator type but augmented by parallel drift and steady-state error compensation. The focus in this work is on a complete method that is simple and usable for real-world applications. To illustrate the practicality of the method, it is demonstrated on a laboratory-scale three-tank system.

Appendix

Considering the equation of the controlled model

$$\begin{aligned} \varvec{x}(k+1) =&\sum _{i=1}^{c} \phi _{i}(\varvec{\alpha }(k)) \left( \varvec{x}_i(k+1) \right) \end{aligned}$$

(6.1)

$$\begin{aligned} =\sum _{i=1}^{c} \phi _{i}(\varvec{\alpha }(k)) \left( {\Delta }\varvec{x}_{i}(k+1) + {\varvec{x}}_{\mathrm{AP},i} \right) \end{aligned}$$

(6.2)

$$\begin{aligned} =\sum _{i=1}^{c} \phi _{i}(\varvec{\alpha }(k)) \bigl ( \varvec{A}_i {\Delta }\varvec{x}_i(k) + \varvec{B}_i {\Delta }\varvec{u}_i(k) \nonumber \\+\,{\varvec{f}}_{0,i} + {\varvec{x}}_{\mathrm{AP},i} \bigr ) \end{aligned}$$

(6.3)

and taking into account that the input of each local system is a global controller output, cf. (2.8),

$$\begin{aligned} \varvec{u}(k) =&\sum _{j=1}^{c} \phi _{j}(\varvec{\alpha }(k))\ \varvec{u}_j(k) \end{aligned}$$

(6.4)

$$\begin{aligned} =\sum _{j=1}^{c} \phi _{j}(\varvec{\alpha }(k))\ ({\Delta }\varvec{u}_j(k) + {\varvec{u}}_{\mathrm{AP},j}) \end{aligned}$$

(6.5)

$$\begin{aligned} =\sum _{j=1}^{c} \phi _{j}(\varvec{\alpha }(k))\ \bigl ( -{\varvec{K}_{j}} {\Delta }\varvec{x}_{j}^{}(k) + {\varvec{V}_{j}}^{} {{\Delta }\varvec{w}_{j}}^{}(k) \nonumber \\- \varvec{B}_j^{\dagger } {\varvec{f}}_{0,j}^{} + {\varvec{u}}_{\mathrm{AP},j}^{} \bigr ), \end{aligned}$$

(6.6)

we obtain

$$\begin{aligned} \varvec{x}(k+1)= {} \sum _{i=1}^{c} \phi _{i}(\varvec{\alpha }(k))\ \Biggl [ \varvec{A}_i {\Delta }\varvec{x}_i(k) + \varvec{B}_i \Biggl ( \sum _{j=1}^{c} \phi _{j}(\varvec{\alpha }(k)) \nonumber \\\times \Bigl (-{\varvec{K}_{j}} {\Delta }\varvec{x}_{j}^{}(k) + {\varvec{V}_{j}}^{} {{\Delta }\varvec{w}_{j}}^{}(k) \underbrace{-\varvec{B}_j^{\dagger } {\varvec{f}}_{0,j}^{}}_{{\varvec{u}_{\mathrm {aff}, j}}} + {\varvec{u}}_{\mathrm{AP},j}^{} \Bigr ) \nonumber \\-{\varvec{u}}_{\mathrm{AP},i} \Biggr ) + {\varvec{f}}_{0,i} + {\varvec{x}}_{\mathrm{AP},i} \Biggr ]. \end{aligned}$$

(6.7)

Rewriting (6.7) results in (2.15).