1 Introduction↩︎

Numerous methods have been proposed for small-signal stability analysis of inverter-base resources (IBRs) in the recent years. A significant number of papers have focused on applying frequency-domain stability analysis to multi-input multi-output (MIMO) systems. While mainstream control theory has largely moved away from classical frequency-domain tools, such as Nyquist plots or gain and phase margins, for MIMO systems, many recent studies on IBRs revisit these tools in their analyses. A review of this body of work, comprising more than one hundred papers, reveals some common mistakes:

• failing to clearly and formally define the objective of the analysis

• selecting inappropriate MIMO stability analysis techniques for the specific application,

• incorrect implementation or misapplication of the chosen methods,

This paper provides an in-depth discussion of established MIMO stability methods and explains their proper use in this context. Regarding the first point, the papers often state that their main objective is to check the stability of a closed-loop linear system with a known transfer matrix. However, this is trivial to check. It seems that many of these papers have secondary aims. The papers check properties like gain and phase margin. But this is not needed if the sole purpose is to check stability so one must assume that the authors want to guarantee stability given model inaccuracies but this is not clarified.

The rest of this paper is organized as follows. Section 2 presents a brief discussion of SISO stability analysis, which provides background and conceptual grounding for the MIMO methods discussed in later sections. Section 3, 4, 5, 6 and 7 discuss different aspects of MIMO stability analysis. Finally, conclusions are presented in section 8,

2 Single-input, single-output (SISO) systems↩︎

In many papers on IBRs stability analysis such as [1]–[3], the stated purpose of the paper is to verify the stability of a particular system often given in the form of a feedback loop as in Fig.1

where \(P\) is a linear system. In the case of single-input, single-output (SISO) systems the system can be described by a transfer function \(P(s)\) and the transfer function of the intersection is then given by \[\frac{P(s)}{1+P(s)}=1-\frac{1}{1+P(s)}\] If \[P(s)=\frac{n(s)}{d(s)}\] with \(n\) and \(d\) the numerator and denominator polynomials then stability of this interconnection basically amounts to checking that the polynomial \(n(s)+d(s)\) has all zeros in the open left half plane which is easily verified by, for instance, the Routh-Hurwitz test, see [4]. However, in the IBRs stability analysis literature commnly use the Nyquist criterion or the Bode plot. In the Nyquist criterion one examines whether the closed curve \(P(j\omega)\) with \(\omega\in \mathbb{R}\) with increasing \(\omega\) encircles the point \(-1\). The conclusion, that the Nyquist curve does not encircle or intersect with the point \(-1\) and hence the system is stable is often made. However, this is only true if the system \(P(s)\) is stable. In general the number of counterclockwise encirclements should be equal to the number of unstable poles of the system \(P(s)\) (and checking the stability of \(P(s)\) is as much work as checking the stability of the closed loop system). If \(P(s)\) has poles on the imaginary axis, one has to be a bit more careful as clearly explained in for instance [5]. Some methods only check whether the Nyquist curve does not intersect the the line segment \((-\infty,-1)\) (for instance easy to check using the Bode plot). However, this needs some words of caution:

• This is not a necessary condition. There are systems which yields a stable closed loop system even though the Nyquist curve intersects the line segment \((-\infty,-1)\) on the real axis. An example of this phenomenon is given by: \[P(s)=\tfrac{5(s+10)^2}{(s+0.3)^3}\]

• One has to be careful what is meant by intersection. The system \[P(s)=\tfrac{1-2s}{1+s}\] yields an unstable closed loop system but the Nyquist curve only intersects \((-\infty,-1)\) in the limit, i.e. the following holds: \[\lim_{\omega\rightarrow \infty} P(j\omega)=-2\] is on the interval \((-\infty,-1)\).

Also there are papers that only plot the Nyquist curve for positive frequencies which is a bad idea because one has no closed curves making it hard to see the encirclements.

Given all the above, why do many studies on inverter‑based resource (IBR) stability analysis employ the Nyquist criterion instead of just checking the stability of the closed-loop transfer function? Many papers, do not explicitly give a reason but stability of a system is clearly only one condition that want to impose on a closed loop system. Disturbances may affect the system. It is desirable to have a system with limited overshoot and quick response to command signals. The model might not be perfect due to uncertain parameters, ignoring fast dynamics or nonlinearities and the like. For SISO systems, the Bode plot (amplitude and phase) completely describes the system. The Nyquist curve almost completely describes the system (one cannot determine bandwidth from the Nyquist contour). By checking properties like gain and phase margin (which will be discussed in more detail in the next section) but also high frequency rolloff and the like one can easily analyse the properties of the system. These properties are easily checked using the Nyquist curve or the Bode plot. One can generally say that a decent gain and phase margin implies that the system behaves decently. It can handle model uncertainty, it does not have a huge overshoot and the like. There are some exceptions (often mathematically constructed) and clearly more analysis can be done but this is a decent rule of thumb. Hence for SISO systems Nyquist curve is commonly used to check gain/phase margin and analyse the Bode plot to investigate the frequency behavior. More importantly it also is a beautiful tool for controller design through loopshaping. For SISO systems this frequency domain analysis is still standard and is taught in almost all undergraduate control classes.

3 Multiple-input, multiple-output (MIMO) systems↩︎

In the 1980s control theory moved to multi-input, multi-output systems and controller design. The advances in computing made this possible and it was required to achieve objectives in many applications. Therefore, there was an urgency felt to expand loopshaping and other frequency-domain techniques to the multi-input, multi-output case. This was not easy. For instance, the concept of zeros becomes much more difficult because of the directionaliy issues introduced in the MIMO context. Many other issues arose, some of which outlined below, but essentially the conclusion in the area of control was that to properly handle MIMO systems one has to move to state space models and use different tools. For instance, gain and phase margins can not be extended to the MIMO context in a reasonable and effective way as will be explained later. Effectively, since around year 2000, the mainstream control community has abandoned studying MIMO systems in the frequency-domain using tools like Nyquist or gain/phase margin. This was intially resisted by many because it required some new specific mathematical skills not present in every control engineer.

Now that papers on IBRs stability analysis are moving toward the MIMO stability analysis such as [6]–[11] one can see that they also focus on frequency domain techniques referring to control papers and books from before 2000. The following sections outline some of the issues encountered in the attempts to use frequency domain techniques for MIMO stability analysis of IBRs.

Note, like for SISO systems, if the sole objective is to check the stability of an interconnection such as in 1 then one can simply check whether the location of the poles of \[\label{eq1aa} P(I+P)^{-1}=I-(I+P)^{-1}\tag{1}\] are in the open left half plane. In the context of MIMO systems, for an \(n\times n\) transfer matrix one would have to check each of the \(n^2\) transfer functions from each input to each output but this is still quite trivial unless dimensions really blow up. It may appear that the analysis can be simplified by looking at the determinant of \[\label{eq1a} \det (I+P)\tag{2}\] However, stability of the inverse of 2 does not always guarantee stability of the closed loop system.

In this definition, one has to be careful to count the number of unstable poles of \(P\). This has to be done based on the Smith-McMillan form presented in [12]. Clearly if \(P\) is stable, one does not have to worry about this. It is sometimes stated that the system should have no hidden unstable modes (without a clear definition). In the above, a formal definition is given by what it means to not have hidden unstable modes: the multiplicity of the unstable poles in \(P\) should be equal to the multiplicity of the unstable poles in the determinant of \(I+P\).

The fact that determining the multiplicity is not as trivial as one might think is illustrated via an example.

4 Generalized Nyquist↩︎

For SISO systems, a single Nyquist curve or a Bode plot completely describes the system. In contrast, for \(n\times n\) MIMO systems, there are essentially \(n^2\) transfer functions and hence there are \(n^2\) possible Nyquist curves or Bode plots that one could investigate. On the other hand, it is obvious that while looking at \(4\) Bode plots for a \(2\times 2\) system it is really hard to fully grasp the systems behavior.

Therefore the \(\det(I+P)\) is considered. As stated before (with some limitations) one can assess the stability of the system based on the determinant. However, one looses all directionality information and it is basically impossible to deduce anything regarding the behavior of the system by only looking at this determinant.

If one looks at stability tests for MIMO systems, then checking whether \(\det(I+P)\) has unstable zeros can be evaluated using a version of the Nyquist criterion. If the Nyquist curve of \(\det(I+P)\) is obtained, then one should verify whether the number of counterclockwise encirclements of the point \(0\) equals the number of unstable poles of \(P\) (counting multiplicities). Note the fact that this time one must check encircling \(0\) not \(1\). In principle correct but one needs to check the multiplicity of unstable poles of \(P\) carefully (using the Smith-McMillan form) and it is easier to simply check the pole locations of \((I+P)^{-1}\) directly.

The reason why initially the Nyquist criterion was investigated for MIMO systems was to extend concepts such as gain/phase margin and loop shaping (which were extremely powerful for SISO systems) to the MIMO domain.

If one thinks of expanding the gain margin to the MIMO domain the logical first step is to look at \(\det(I+kP)\) and check for which \(k\) the system is stable. However, due to the determinant, this function depends nonlinearly on the parameter \(k\) and the gain margin can no longer be obtained from the Nyquist curve of \(\det(I+P)\). The proposed solution was to look at the eigenvalues of \(P(j\omega)\): \[\lambda_1\left[ P(j\omega) \right],\ldots, \lambda_n\left[ P(j\omega) \right]\] After all, the following holds: \[\det(I+P)=\prod_{i=1}^n (1+\lambda_i [P])\] This makes it possible to relate the encirclements of \(0\) for the determinant to encirclements of \(-1\) for the eigenvalues. However, if one wants to count encirclements of \(-1\) then one needs closed curves. There is one obvious reason and one very subtle reason why one might not obtain closed curves.

The obvious reason is the way one numbers the eigenvalues. It is needed that \(\lambda_i \left[ P(j\omega) \right]\) are continous functions of \(\omega\) for each \(i\). A simple ordering that \(\lambda_1\) is the largest eigenvalue in amplitude (as done in some paper in the power literature) does not work and can create discontinuities. However, it is known that a continuous ordering exists.

The second potential issue is illustrated in the following example.

This phenomenon was originally discovered by [13] and it was noted that the issue could be handled by pairwise combining these \(n\) eigenvalue graphs to create \(q\) closed curves (with \(q\leqslant n\)). Then, one should count the sum of the number of encirclements in these \(q\) graphs, and this should be equal to the number of open-loop poles.

Gain and phase margin can be defined based on the feedback loop given in Fig.5. For SISO systems, if one chooses \(U=1\) then the interconnection of Fig.1 is obtained. If this system is stable then the question can be posed how sensitive this conclusion is when small perturbations are introduced. A logical step is to set \(U=k\) and check for which values of \(k\) the closed loop system remains stable. From the Nyquist curve one can easily obtain an interval \([k_1,k_2]\) for \(k\) with \(k_1<1<k_2\) where stability is preserved. If \(k_1\) and \(k_2\) are “far” away from \(1\) then one can conclude that stability is preserved given variations in the feedback gain. If one chooses \(U=e^{i\theta}\) then from the Nyquist curve one can easily obtain an interval \([-\theta_1,\theta_1]\) for \(\theta\) with \(\theta_1>0\) where stability is preserved. If \(\theta_1\) is “far” away from \(0\) then one can conclude that stability is preserved given small delays in the feedback gain (recall that delays yield a \(e^{\tau s}\) term in the frequency domain).

The question is whether the above can be used to obtain a form of gain- and phase-margin for MIMO systems. Note that in Fig. 5 the \(U\) is no longer scalar but matrix valued. For \(U=I\) closed loop interconnection of Fig.1 can be obtained. Then the gain margin can be defined as the largest interval \((k_1,k_2)\) with \(k_1<1<k_2\) such that the closed loop system is stable for all \(U=kI\) with \(k\in (k_1,k_2)\). Similarly, the phase margin is defined as the largest \(\theta_1>0\) such that the closed loop system is stable for all \(U=e^{j\theta} I\) with \(\theta\in (-\theta_1,\theta_1)\). This is a direct extension of the SISO case. Since \[kI=\begin{pmatrix} k & 0 & \cdots & 0 \\ 0 & k & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & k \end{pmatrix}\] (and similarly for the phase margin) it can be observed that the analysis assumes all channels have the same gain variation \(k\). That is why this is sometimes called uniform uncertainty.

The following result can be obtained:

However, using generalized Nyquist to obtain gain and phase margin only verifies the effect of uniform changes in all channels. In most cases, it makes no sense to only look at uniform changes in each channel. However, as noted below the Nyquist curve of the eigenvalues misses essential information about the system.

To first natural step for a more reasonanle definition of gain and phase margins in the MIMO case, is not to look at uniform disturbances but look at potentially different perturbations in each channel: \[\label{nonuni} U = \begin{pmatrix} k_1 & 0 & \cdots & 0 \\ 0 & k_2 & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & k_n \end{pmatrix}\tag{3}\] Here \(k_i\in \mathbb{R}\) if one is looking for gain margin while \(k_i=e^{i\theta_i}\) if one is looking for phase margin.

However, it should first be noted that having satisfactory gain and phase margins under uniform perturbations does not guarantee robustness with respect to perturbations of the form 3 .

Whether satisfactory gain and phase margins exist for nonuniform perturbations can inherently not be checked using the generalized Nyquist criterion. Another example illustrates that the issue is even more involved since even if satisfactory phase margins exist for nonuniform perturbations of the form 3 (note that there is no efficient tool to effectively compute gain and phase margins for nonuniform perturbations) then one still cannot be sure that the system is robust since nondiagonal perturbations might have dramatic effect as illustrated by the following example.

Note that some papers use tools such as Gershgorin circles to find estimates for the eigenvalues and thus obtain simpler but often very conservative methods to assess stability. This raises two questions.

• Why introduce conservative tools to check stability if very efficient tools exist which are not conservative (actually necessary and sufficient).

• These tools still only investigate the eigenvalues of \(P\) and hence they miss intrinsic information about the system as is outlined in the above.

In conclusion, generalized Nyquist does not provide any insight on the robustness of the closed loop stability. It intrinsically misses information about the system to properly assess the behavior of the closed loop system. If one is careful one can properly evaluate whether the system is stable using generalized Nyquist. On the other hand, if one wants to only check stability then one can much more efficiently check the poles of the closed-loop system directly. There is no additional insight obtained from generalized Nyquist and hence it has not been used in the control area for the past 30 years.

5 Small-gain theorem↩︎

As noted in the previous section, generalized Nyquist does not give insights in robustness of the closed loop stability. Therefore, different tools are needed. As noted in the literature the use of eigenvalues is inherently insuitable in this context. For instance, the classical reference for nonuniform perturbations as discussed in the previous section is the paper [14] which uses singular values instead of eigenvalues. The basic tool for many results in this context is the small gain theorem.

The \(H_\infty\) norm is defined as follows: \[\| P \|_\infty = \sup_{\omega} \| P(j\omega) \|.\] for a stable transfer function \(P\) where \(\| A \|\) for a matrix \(A\) is equal to the largest singular value. An important property of the \(H_\infty\) norm is that \[\| P_1P_2 \|_\infty \leqslant\| P_1 \|_\infty \| P_2 \|_\infty\] i.e. the norm is submultiplicative. The small gain theorem basically states that the interconnection in Fig.6 is stable for two stable systems \(G_1\) and \(G_2\) if \(\| G_1 G_2 \|_\infty <1\).

The two classical models for uncertainty are additive and multiplicative uncertainty as presented in Fig.7 and 8 respectively. They can be viewed as measuring absolute and relative errors in the model. The uncertainty structure presented in the previous section basically sets \(\Delta=U-I\) and using the multiplicative model of Fig.8. The small gain theorem basically yields the following:

The interconnection in Fig.7 can be rewritten in the form of Fig.6 by choosing \(G_1=\Delta\) and \(G_2=(I+P)^{-1}\). The small gain theorem then guarantees the interconnection is stable provided \(\| G_1 G_2\|_{\infty}<1\). Since \[\label{submulti} \| G_1 G_2 \|_{\infty} \leqslant\| G_1 \|_{\infty} \| G_2 \|_{\infty}\tag{4}\] The results for multiplicative uncertainty can be obtained using similar arguments using that the interconnection in Fig.8 can be rewritten in the form of Fig.6 by choosing \(G_1=\Delta\) and \(G_2=P(I+P)^{-1}\).

For instance, if one considers the intersection in Fig.5 in the previous section with \(U\) given by 3 then [14] obtained estimates for the nonuniform gain and phasse margin based on the above result with \(\Delta=U-I\).

In the literature, different norms have been used. For instance, in the IBRs stability analysis literature the \(G\)-norm is defined in [15] which does not satisfy the crucial submultiplicative property of 4 and hence those results do not follow from a standard application of the small gain theorem (which requires norm that are submultiplicative). These results using the \(G\)-norm are actually quite conservative.

However, besides the two models (multiplicative/additive uncertainty) there is much more that can incorporated into the analysis:

• Sometimes, one has frequency information about the uncertainty such as: \[\| \Delta(j\omega) \| \leqslant f(j\omega)\] for \(\omega\in \mathbb{R}\) with \(f\) a known function can be incorporated into the analysis. This can, for instance, be used to incorporate information that there is more model uncertainty at high frequency.

• There might be uncertain parameters in the model and the objective is to guarantee that the system remains stable for all parameter values in a certain range.

• There might be inaccuracies in the model because of ignoring certain dynamics in the modeling. For instance, one might have a simple model \(y_1=P_1u_1\) and \(y_2=P_2u_2\) but might have ignored small interactions in the model meaning that \(u_1\) does have some effect on \(y_2\). These kinds of structured knowledge about the model uncertainty can be incorporated in the analysis.

Robust control can analyse the stability subject to all kind of model uncertainty. The important issue is to model the uncertainty in the sense that you incorporate into the analysis as much prior information as one possibly can. Although much more could be discussed on this topic, further details can be found in [16], [17].

Finally, it should be noted that in the context of IBRs stability analysis, \(P=YZ\) with admittance \(Y\) and a grid modelled by an infinite bus with impedance \(Z\). In that case, one clearly can rewrite Fig.1 into Fig.6 with \(G_1=Z\) and \(G_2=Y\). Then, in order to test for stability, one could use the small gain theorem to conclude that the system is stable provided \(\| YZ \|_{\infty}<1\). This is a very conservative result but has been used in the IBRs stability analysis literature with different types of norms. Note that in the beginning of this section \(G_1=\Delta\) was used which is unknown. In the current context, both \(G_1\) and \(G_2\) are known and one can just check the stability of the closed loop system directly without invoking the small gain theorem. The small gain theorem is conservative if you check for one specific \(G_1\). But it becomes necessary and sufficient if you want the check for stability for all possible \(G_1\) with norm less than \(1\).

6 Passivity↩︎

An intrinsically different approach to analyse the stability of closed loop systems is to use the concept of passivity. This has also been studied in the IBRs stability analysis literature, see [18]–[21].

Passivity is based on the concept of energy. It is very easy to conclude that a standard RLC circuit will always be stable because this system does not contain any energy sources. Therefore, it is easy to conclude that signals remain bounded unless external energy is supplied and, because of resistors, energy dissipates. Therefore, signals will converge to zero. This concept is not limited to linear systems and these arguments equally apply to nonlinear circuits where one, for instance, adds a diode or another nonlinear element. Also passivity is used extensively in the context of robotics and other mechanical structures were similar arguments can be applied.

First a formal definition and introduction to this concept is provided which is based on [22], [23]. Note that these definitions apply equally well to SISO and MIMO systems.

A standard misconception is that the interconnection of two passive systems yields a stable interconnection. However, if in the interconnection in Fig.9, the, possibly nonlinear, systems \(\Sigma_1\) and \(\Sigma_2\) are both passive then the interconnection might still be unstable.

For that purpose the concepts of strict input- and output passivity have been introduced:

This result is very general and applies to nonlinear systems as well as linear systems. The following simple example illustrates that more than passivity of each system is needed to guarantee the stability of the closed loop

For linear systems passivity can be connected to the positive real property of the transfer system. Positive real has a very different history and is intrinsically related to linear systems. If one has a given transfer system \(G\), one can ask whether it is possible to construct an RLC circuit such that the transfer matrix of this RLC circuit is equal to the given transfer matrix. The answer turned out to be that this is possible when the transfer matrix is positive real. In the literature there are different versions of positive real being used:

Let us relate these different concepts:

The following theorem connects the properties of positive real and passivity for linear systems:

Note that the following results exist regarding the stability of the interconnection in Fig.9.

The key strength of passivity is that one can arbitrarily connect strict passive systems and preserve passivity and stability. For instance, if one ensure a set of IBRs is strictly passive and they are connected through an arbitrary network then passivity is preserved assuming the network itself is passive. The latter is actually quite natural since, for instance, a power line logically cannot be an energy source.

7 Mixed small gain theorem and positive real↩︎

In the paper [24] the concepts of positive real and the small-gain theorem are mixed which considers a system with transfer matrix \(G\) for which there exists a \(c>0\) such that:

• \(G(j\omega)+G^{*}(j\omega) >0\) for \(|\omega|\leqslant c\)

• \(\sup_{\omega\in\mathbb{R}, |\omega|>c} \| G(j\omega) \| < 1\)

The limitation of this theorem is that the components must satisfy these conditions for the same bandwidth \(c\).

If all inverters in a network are strictly passive then the interconnection of all these inverters through a network is always stable since obviously the network is passive.

When one tries to achieve the same type of results here, it should be noted that even if all the inverters satisfy this mixed property for the same \(c\), their interconnection through the network might still yield an unstable system. After all, even though the network is passive, this does not imply that it also satisfies such a mixed property.

8 Conclusion↩︎

This paper provided a critical review of the commonly used stability analysis methods for IBRs. It presented authors’ collective findings, derived from a comprehensive examination of a large body of literature on the stability of IBRs. Specifically, the paper clarified the intended applications of the stability analysis methods and discussed potential sources of misinterpretation or improper use. It underscored the necessity of employing modern control theory for MIMO systems in the stability analysis of IBRs. This necessity arises because classical frequency‑domain techniques from the 1980s and 1990s, developed as extensions of SISO approaches to MIMO systems, now broadly regarded as inadequate for such analyses. Several of their key limitations were discussed and illustrated in this paper.

References↩︎

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1. Anton A. Stoorvogel is with the Department of Electrical Engineering, Mathematics and Computer Science, University of Twente, P.O. Box 217, Enschede, The Netherlands (e-mail: A.A.Stoorvogel@utwente.nl)↩︎

2. Saeed Lotfifard and Ali Saberi are with the School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA 99164, USA (e-mail: s.lotfifard@wsu.edu,saberi@wsu.edu)↩︎