Robustness analysis by a $\mu$-analysis based approach

Please report any problems to Clément Roos (croos@onera.fr). I will try to fix them as quickly as possible!

Overview

The SMART library (Skew Mu Analysis based Robustness Tools) of the SMAC toolbox contains a set of $\mu$-analysis based tools to evaluate the robustness properties of high-dimensional LTI plants subject to numerous LTI uncertainties. These tools allow to compute both upper and lower bounds on the (skewed) robust stability margin, the worst-case $\mathcal{H}_\infty$ performance level, as well as the worst-case gain, phase, modulus and time-delay margins. The key idea is to solve the problem on just a coarse frequency grid and to perform a fast validation on the whole frequency range, which results in guaranteed but conservative bounds on the aforementioned quantities. Some heuristics are then applied to determine a set of worst-case parametric configurations leading to over-optimistic bounds. A branch and bound scheme is finally implemented, so as to tighten these bounds with the desired accuracy, while still guaranteeing a reasonable computational complexity. A short summary of all these algorithms can be found in [1].

Problem statement

Let us consider the standard interconnections of Figure 1. $M(s)$ is a continuous-time stable and proper real-rational transfer function representing the nominal closed-loop system. $\Delta(s)$ is a continuous-time block-diagonal LTI operator:
$$\Delta(s)=\mbox{diag}(\Delta_1(s),\dots,\Delta_N(s))$$

which gathers all model uncertainties. Each $\Delta_i(s)$ can be:

• either a time-invariant diagonal matrix $\Delta_i(s)=\delta_iI_{n_i}$, where $\delta_i$ denotes a real or complex parametric uncertainty,
• or a stable and proper real-rational unstructured transfer function of size $n_i\times n_i$ usually representing neglected dynamics.

Figure 1: Standard interconnections for robustness analysis

Let $n=\sum_{i=1}^{N}n_i$. The set of all $n\times n$ matrices with the same block-diagonal structure and the same nature (real or complex) as $\Delta(j\omega)$ is denoted by $\bf\Delta$. The notation $\Delta(s)\in\bf\Delta$ is then introduced to specify that $\Delta(j\omega)\in\bf\Delta$ for all $\omega\in\Omega$, where $\Omega$ denotes the frequency range of interest (usually equal to $\mathbb{R}_+$). Finally, let $k\mathcal{B}_{{\bf\Delta}}= \{\Delta\in{\bf\Delta}\,:\,\overline{\sigma}(\Delta) < k\}$, where $\overline{\sigma}\left(.\right)$ denotes the largest singular value.

$\mu$-analysis [2] is probably the most efficient technique to analyze the robustness properties of the interconnections of Figure 1, especially when high-dimensional systems are considered. The underlying theory [3,4] is not broached as such in this paper due to space limitations, but a few useful definitions are recalled below.

Definition 1. Let $\omega\in\mathbb{R}_+$ be a given frequency. If no matrix $\Delta\in{\bf\Delta}$ makes $I-M(j\omega)\Delta$ singular, then the structured singular value $\mu_{\bf\Delta}(M(j\omega))$ is equal to zero. Otherwise:
\begin{equation}
\mu_{\bf\Delta}(M(j\omega))=\Big[\min_{\Delta\in{\bf\Delta}}\left\{\overline{\sigma}(\Delta),\,\textrm{det}(I-M(j\omega)\Delta)=0\right\}\Big]^{-1}
\end{equation}

Lemma 1. The interconnection of Figure 1 (left) is stable $\forall\Delta(s)\in k_r\mathcal{B}_{\bf\Delta}$, where the robust stability margin $k_r$ is defined as the inverse of the largest value of $\mu_{\bf\Delta}(M(j\omega))$ over the frequency range of interest:
\begin{equation}
k_r = \Big[\max_{\omega \in\Omega}\,\mu_{\bf\Delta}(M(j\omega))\Big]^{-1}
\end{equation}

Assume now that ${\bf\Delta}$ is split into two distinct block structures, i.e. ${\bf\Delta}=\mbox{diag}({\bf\Delta_f},{\bf\Delta_u})$. Let ${\bf\Delta_s}=\mbox{diag}(\mathcal{B}_{\bf\Delta_f},{\bf\Delta_u})$ and $k\mathcal{B}_{\bf\Delta_s}=\mbox{diag}(\mathcal{B}_{\bf\Delta_f},k\mathcal{B}_{\bf\Delta_u})$. The skewed structured singular value is defined as follows [5].

Definition 2. Let $\omega\in\mathbb{R}_+$ be a given frequency. If no matrix $\Delta=\mbox{diag}(\Delta_f,\Delta_u)\in{\bf\Delta_s}$ makes $I-M(j\omega)\Delta$ singular, then the skewed structured singular value $\nu_{\bf\Delta_s}(M(j\omega))$ is equal to zero. Otherwise:
\begin{equation}
\nu_{\bf\Delta_s}(M(j\omega))=\Big[\min_{\Delta\in{\bf\Delta_s}}\left\{\overline{\sigma}(\Delta_u),\,\textrm{det}(I-M(j\omega)\Delta)=0\right\}\Big]^{-1}
\end{equation}

Lemma 2. The interconnection of Figure 1 (left) is stable $\forall\Delta(s)\in k_s\mathcal{B}_{\bf\Delta_s}$, where the skewed robust stability margin $k_s$ is defined as the inverse of the largest value of $\nu_{\bf\Delta_s}(M(j\omega))$ over the frequency range of interest:
\begin{equation}
k_s = \Big[\max_{\omega \in\Omega}\,\nu_{\bf\Delta_s}(M(j\omega))\Big]^{-1}
\end{equation}

In other words, $k_s$ is the $H_\infty$ norm of the smallest uncertainty $\Delta_u(s)\in{\bf\Delta_u}$ such that there exists $\Delta_f(s)\in\mathcal{B}_{\bf\Delta_f}$ for which the interconnection between $M(s)$ and $\Delta(s)=\mbox{diag}(\Delta_f(s),\Delta_u(s))\in{\bf\Delta_s}$ is unstable. $\Delta_f(s)$ and $\Delta_u(s)$ thus correspond to fixed range and unbounded uncertainties respectively. $\nu$-analysis is a more general framework than $\mu$-analysis, since the classical structured singular value is recovered in Definition 2 if ${\bf\Delta_f}$ is empty. It also proves to be very useful in practice. Indeed, it allows to consider a wide class of analysis problems, such as computing the worst-case $H_\infty$ performance level, the maximal allowable amount of parametric uncertainties in the presence of neglected dynamics, and the worst-case gain, phase, modulus and delay margins.

The exact computation of $k_r$ or $k_s$ is known to be NP hard in the general case [6], so both lower and upper bounds are computed instead. But even computing these bounds is a challenging problem with an infinite number of frequency-domain constraints. It is usually solved on a finite frequency grid $(\omega_i)_{i\in [1,m]}$ and estimates of the robust stability margins are then obtained as:
\begin{equation*}
\frac{1}{\displaystyle\max_{i\in [1,m]}(\overline{\mu}_{\bf\Delta}(M(j\omega_i)))}\le\ k_r\le \frac{1}{\displaystyle\max_{i\in [1,m]}(\underline{\mu}_{\bf\Delta}(M(j\omega_i)))}
\end{equation*}
\begin{equation*}
\frac{1}{\displaystyle\max_{i\in [1,m]}(\overline{\nu}_{\bf\Delta_s}(M(j\omega_i)))}\le\ k_s\le \frac{1}{\displaystyle\max_{i\in [1,m]}(\underline{\nu}_{\bf\Delta_s}(M(j\omega_i)))}
\end{equation*}
where lower and upper bounds $\underline{\mu}_{\bf\Delta}$ and $\overline{\mu}_{\bf\Delta}$ on $\mu_{\bf\Delta}$ at a given frequency $\omega_i$ can be computed using the methods of [7,8] and [9] respectively, while lower and upper bounds $\underline{\nu}_{\bf\Delta_s}$ and $\overline{\nu}_{\bf\Delta_s}$ on $\nu_{\bf\Delta_s}$ at a given frequency $\omega_i$ can be computed using the methods of [10] and [11] respectively.

However, a crucial problem appears in this procedure: the grid must contain the most critical frequency point for which the maximal value of $\mu_{\bf\Delta}$ or $\nu_{\bf\Delta_s}$ is reached. If not, the upper bound on $k_r$ or $k_s$ can be very poor, notably in the case of flexible systems, whose $\mu_{\bf\Delta}$ or $\nu_{\bf\Delta_s}$ plot often exhibits very high and narrow peaks. Even worse, the lower bound can be over-evaluated, i.e. be larger than the real value of $k_r$ or $k_s$. Unfortunately, the aforementioned critical frequency is usually unknown! In this context, the considered frequency grid must be sufficiently dense, which can lead to a prohibitive computational cost. But even so, it is still possible to miss a critical frequency (see for example [12]).

To overcome this issue and compute both tight and reliable bounds on $k_r$ and $k_s$, some efficient methods are implemented in the SMART library of the SMAC toolbox. Their key features are summarized below:

• An upper bound on $\mu_{\bf\Delta}$ or $\nu_{\bf\Delta_s}$ is computed at some frequency, for which nothing has been assessed yet. This bound is slightly increased, and a frequency interval on which it remains valid is computed. Such a strategy is repeated until the whole frequency range is investigated, leading to a lower bound $\underline{k}_r$ or $\underline{k}_s$ on $k_r$ or $k_s$ [13,14]. The latter is guaranteed on the whole frequency range, and not only on a frequency grid as it is the case of most existing methods. See the routines muub and muub_mixed.
• Some heuristics are applied to determine the smallest possible perturbation $\tilde{\Delta}(s)\in{\bf\Delta}$ or ${\bf\Delta_s}$ such that $\mbox{det}(I-M(j\tilde{\omega})\tilde{\Delta}(j\tilde{\omega}))=0$ for some frequency $\tilde{\omega}\in\Omega$, i.e. such that the interconnection between $M(s)$ and $\tilde{\Delta}(s)$ is unstable. A lower bound on $\mu_{\bf\Delta}$ or $\nu_{\bf\Delta_s}$, and thus an upper bound $\overline{k}_r$ or $\overline{k}_s$ on $k_r$ or $k_s$, is then obtained [15,12]. Unlike most existing methods, frequency is an optimization parameter, which allows to detect critical frequencies and usually leads to more accurate bounds. See the routines mulb, mulb_mixed, mulb_nreal, mulb_1real and hinflb_real.
• A branch and bound algorithm is implemented. It allows to tighten the gap between the bounds on $k_r$ or $k_s$ to the desired accuracy at a reasonable computational cost [1]. The $\mu$-sensitivities are used to select the most critical uncertainties [18,19]. See the routines mubb and mubb_mixed.

In this context, four main issues can be addressed by the SMART library of the SMAC toolbox:

• robust stability margin - With reference to Figure 1 (left), compute the robust stability margin $k_r$ for a given block structure ${\bf\Delta}$.
• skewed robust stability margin - With reference to Figure 1 (left), compute the skewed robust stability margin $k_s$ for a given block structure ${\bf\Delta_s}$.
• worst-case $\mathcal{H}_\infty$ performance level - With reference to Figure 1 (right) and assuming that $k_r>1$, compute the highest value $k_\infty$ of the $\mathcal{H}_\infty$ norm of the transfer matrix from $e$ to $y$ when $\Delta(s)$ takes all possible values in $\mathcal{B}_{\bf\Delta}$.
• worst-case input-output margins - With reference to Figure 1 (right) and assuming that $k_r>1$, compute the worst-case gain, modulus, phase and delay margins $k_g$, $k_m$, $k_p$ and $k_d$, i.e. the highest value of the (real or complex) gain, the phase shift or the time delay that can be inserted between $y$ and $e$ without destabilizing the system when $\Delta(s)$ takes all possible values in $\mathcal{B}_{\bf\Delta}$.

Both lower and upper bounds on $k_r$, $k_s$, $k_\infty$, $k_g$, $k_m$, $k_p$ and $k_d$ can be computed with the desired accuracy:

• robust stability margin computation is a classical $\mu$ problem, which can be solved by computing lower and upper bounds on $\mu_{\bf\Delta}$ on the whole frequency range,
• skewed robust stability margin, worst-case $\mathcal{H}_\infty$ performance [12] and input-output margins [1,16] computation are skew-$\mu$ (or $\nu$) problems, which can be solved by computing lower and upper bounds on $\nu_{\bf\Delta_s}$ on the whole frequency range.

References