Unconstrained Parameterization of Stable LPV Input-Output Models: with Application to System Identification

TL;DR

An unconstrained DT-LPV-IO parameterization is developed which gives a stable model for any choice of model parameters, and identification using the stable DT-LPV-IO model with neural network coefficient functions is demonstrated on a simulation example of a parameter-varying mass-damper-spring system.

Abstract

Ensuring stability of discrete-time (DT) linear parameter-varying (LPV) input-output (IO) models estimated via system identification methods is a challenging problem as known stability constraints can only be numerically verified, e.g., through solving Linear Matrix Inequalities. In this paper, an unconstrained DT-LPV-IO parameterization is developed which gives a stable model for any choice of model parameters. To achieve this, it is shown that all quadratically stable DT-LPV-IO models can be generated by a mapping of transformed coefficient functions that are constrained to the unit ball, i.e., a small-gain condition. The unit ball is then reparameterized through a Cayley transformation, resulting in an unconstrained parameterization of all quadratically stable DT-LPV-IO models. As a special case, an unconstrained parameterization of all stable DT linear time-invariant transfer functions is obtained. Identification using the stable DT-LPV-IO model with neural network coefficient functions is demonstrated on a simulation example of a position-varying mass-damper-spring system.

Figures (4)

• Figure 1: Graphic representation of the stable DT-LPV-IO model class. For any value of $X_W$, and for any parameterized functions $X_M(\rho),Z_M(\rho),L(\rho)$, the coefficient functions $a_i(\rho),b_i(\rho)$ result in a stable DT-LPV-IO model.
• Figure 2: Bode plot of frozen dynamics of $\mathcal{G}(\delta,\rho)$, displaying a resonance with scheduling-dependent damping.
• Figure 3: Train (top) and validation (bottom) prediction residuals $y-\hat{y}_\phi$ ( ) and noise realization $y - \tilde{y}$ ( ). The prediction residuals coincide with the noise, i.e., the LPV-IO model \ref{['eq:LPV_IO']} with neural network coefficient functions is able to learn all dynamics while simultaneously guaranteeing stability.
• Figure 4: Coefficient set $K_P = \{K(\rho) = a_1(\rho),a_2(\rho) \ | \ P \succ 0, (F-GK(\rho))^\top\! P (F-GK(\rho))- P \prec 0 \ \forall \rho \in \mathbb{P} \}$, i.e., the set of all possible values $K(\rho)$ such that LPV IO-model \ref{['eq:maximum_ss_short']} is stable with Lyapunov certificate $P$ at iteration 1 (), 10 (), and 100 () of the optimization. Lyapunov certificate $P$ is optimized such that the true coefficient functions () are contained within the stable coefficient set. Optimizing $P$ thus corresponds to rotating, scaling and translating this set. All stable coefficient sets are included in the stability triangle that describes the stable coefficients for the LTI case ().

Theorems & Definitions (17)

• Remark 1
• Definition 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Corollary 6
• proof