An approach to nonlinear optimisation via set stabilisation of dynamical systems with an application to synchronous machines

Abstract

An approach to constraint nonlinear optimisation is proposed, where the variables of the objective function and of the constraint function belong to the state of a nonlinear dynamical system. A nonlinear closed-loop feedback system is designed, whose solutions converge to a target set consisting of the optimal points of the underlying optimisation problem. The feedback law for the control input of the nonlinear dynamical system is designed by exact linearisation of a dynamical formulation of a first-order optimality condition. For the problem formulation and the stability analysis of the nonlinear closed-loop feedback system, the notion of V-detectability is considered. An application to energy-optimal feed-forward torque control for different types of synchronous machines is presented, which demonstrates the usefulness of the method. In this regard, anisotropic permanent magnet synchronous machines and electrically excited synchronous machines are covered.

Keywords:

• Nonlinear systems
• set stabilisation
• v-detectability
• nonlinear optimisation
• synchronous machine
• maximum torque per ampere (MTPA)

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The author confirms that the data supporting the findings of this study are available within the article.

Notes

1 See e.g. (Braun et al., 2023; Eldeeb et al., 2018; Monzen & Hackl, 2023 and the references therein).

2 See e.g. (Slotine & Li, 1991, p. 225).

3 For works in the context of set stabilisation see e.g. (Böck & Kugi, 2014; Böck & Kugi, 2016; Nielsen, 2009; Nielsen & Maggiore, 2008; Shiriaev, 2000).

4 For a definition of the relative degree for non-square multivariable systems, see e.g. (Kolavennu et al., 2001).

5 This is possible when ($\mathrm{B}2$) in Assumption 3.2 is satisfied, see e.g. (Nielsen, 2009).

6 See (Shiriaev & Fradkov, 1999, Definition 2.1), see also (Shiriaev & Fradkov, 2001).

7 See (Shiriaev, 2000, Definition 2.2).

8 See (Shiriaev, 2000, Proposition 2.9).

9 See (Shiriaev, 2000, Theorem 2.3).

10 This is the case, when at least one of the terms $L_{{\mathit{g}}_k}{L}_{\mathit{f}}^{r-1}h(\mathit{x})$, $k=1,2,\dots ,m$, is nonzero, where r is defined as the relative degree (Kolavennu et al., 2001), see also (McLain et al., 1996).

11 See e.g. (Consoli et al., 2010; Gallegos-López et al., 2005; Meyer & Böcker, 2006).

12 For different existing realtime-capable operation strategies for anisotropic PMSMs see e.g. (Bolognani et al., 2014; Bonifacio & Kennel, 2018; Braun et al., 2023; Eldeeb et al., 2018; Hackl et al., 2021; Jung et al., 2013b; Morimoto et al., 1994). EESMs are much less considered in the literature, see e.g. (Monzen & Hackl, 2023) and the references therein.

13 It should be noted that the comparison with the exact offline solution was chosen in order to evaluate the accuracy of the proposed method. A comparison with state-of-the-art real-time capable methods is reasonable regarding computational aspects. In this respect, it is referred to Braun et al. (2023), where a comparison of the feedback-systems-based method with Newton's method in Bonifacio and Kennel (2018) and Ferrari's method in Jung et al. (2013b) is made.

14 Compare for a similar discussion about exponential convergence on the subject of set stabilisation in El-Hawwary and Maggiore (2011).

15 See e.g. the convergence lemma in Haddad and Chellaboina (2008, p. 201).

16 Compare for the analysis of converge time estimation in Antonello et al. (2014).

17 For the sake of readability, the speed dependence is neglected without loss of generality as in present example only the constant torque region is considered (Braun et al., 2023).

18 The stator-referred parameters $R_{\mathrm{f}}$, $L_{\mathrm{f}}$ and variables $x_3$, $u_3$ are related to the real rotor quantities by the transformation ratio $a_r=\frac{L_{\mathrm{hd}}}{L_{\mathrm{df}}}$, where $L_{\mathrm{df}}$ is the mutual inductance between rotor and stator d-axis. The transformations are as follows (Grune, 2013; Müller et al., 2023): $x_3=x_{3,\mathrm{rotor}}/a_r$, $u_3=\frac{2}{3}{a}_r{u}_{3,\mathrm{rotor}}$, $R_{\mathrm{f}}=\frac{2}{3}{a}_r^2{R}_{\mathrm{f},\mathrm{rotor}}$, and $L_{\mathrm{f}}=\frac{2}{3}{a}_r^2{L}_{\mathrm{f},\mathrm{rotor}}$.