On the Applicability of SISO and MIMO Impedance-Based Stability Assessment of DC–DC Interlinking Converters

This article presents a formal mathematical correlation between the standardly used port-level (terminated) single-input single-output (SISO), and the recently acknowledged device-level (unterminated) multiple-input multiple-output (MIMO) impedance-based method for the stability assessment of dc–dc interlinking converters. Based on this, the conditions that must be met to ensure the correct stability assessment by the SISO method applied to a single port-pair are derived. It is shown that without prior knowledge on whether these conditions are met, the SISO method must be applied to every port-pair to account for possible port-level hidden dynamics. Alternatively, the MIMO method can be used, which is revealed to inherently account for any port-level hidden dynamics. It is further analyzed which method is advantageous in terms of computational complexity, intuitiveness, and simplicity for applications featuring meshed grids or multiport interlinking converters, as well as in terms of interpreting the resulting stability margins. Finally, suitability of the MIMO method for termination-independent stability-oriented controller design and stability assessment based on measurements is highlighted. The presented methodology is illustrated for a simplified dc system with a current-controlled buck converter. Analytical stability predictions are validated using hardware-in-the-loop simulations and also experimentally, using a laboratory hardware prototype.

Although widely used in the literature [2], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], when applied to a single port only, this method may lead to a wrong prediction of stability [5], [8].This occurs when at the considered port the system features unstable port-level hidden dynamics, i.e., when the unstable poles do not appear in the transfer functions corresponding to that port [15].Some examples of this are reported in [5], [8], [16], and [17] for radial (nonmeshed) systems.Along the line of unstable port-level hidden dynamics, further limitations of the port-level impedance-based method may arise when, to enhance reliability and flexibility, instead of radial, the system features a meshed structure [8], [18].Moreover, for such systems, even in absence of any unstable port-level hidden dynamics, this method loses intuitiveness as the definition of the terminated immittances provides little physical insight when the ports are fully coupled.As a more general method, the device-level impedance-based method can be used, which is hereafter also referred to as multiple-input multiple-output (MIMO) one.The method relies on the all-port unterminated MIMO representation [6], [19], [20], [21], [22], [23] of both subsystems of interest: the converter under study and the rest of the system it interconnects.The subsystems' immittances within the Norton/Thevenin equivalents are in this case transfer function matrices, and the generalized Nyquist criterion (GNC) [24], [25] is applied to the resulting MIMO minor loop gain.Other than briefly outlining the idea in [26] and [27], the application of such MIMO impedance-based method, to the best of authors' knowledge, has not been thoroughly analyzed for dc systems.As for ac systems, a variant of the MIMO impedance-based method that additionally involves the passive/active component grouping was used in [21], [22], and [23], to avoid appearance of the termination-caused right half plane (RHP) poles in the minor loop gain.Nevertheless, application of the device-level MIMO method in light of accounting for the port-level hidden dynamics, which, as mentioned above, may be disregarded by the port-level impedance-based method, has not been previously explored.In addition, the potential of the device-level method to handle meshed systems and/or systems with multiport converters has not been discussed either.Finally, its relationship with the portlevel impedance-based method has not been derived so far.
To fill in these gaps, this article presents a formal mathematical correlation between the standardly used SISO, and the recently acknowledged MIMO impedance-based method for stability assessment of dc-dc interlinking converters.Based on this, the conditions under which the methods account for the (unstable) port-level hidden dynamics are for the first time derived and the computational complexity required for this is discussed.It is revealed that the MIMO method is advantageous in these aspects.Furthermore, the suitability of the SISO and the MIMO methods for stability assessment in systems with meshed structures and/or multiport converters, as well as for defining various stability margins is explored.Finally, the unterminated (black-box) representation-related assets of the MIMO method are underlined, which are relevant when the stability analysis is to be performed based on measurements, rather than analytical models, as well as when robust termination-independent stabilization methods are to be developed.
The rest of this article is organized as follows.Section II explains unterminated small-signal modeling and the MIMO impedance-based method.Section III recalls different loopgain-based approaches for determining stability of a MIMO feedback system and provides relationship between the MIMO and the SISO impedance-based methods.Section IV illustrates the use of the presented methodology on a simplified dc system, where a current-controlled buck converter is used as an interlinking converter.Hardware-in-the-loop (HIL) and experimental validations are also presented.Advantages and disadvantages of the MIMO and the SISO impedance-based method are summarized in Section V. Section VI concludes this article.

A. System Under Study
This article considers dc power electronic systems, such as the one illustrated in Fig. 1(a).It features an interlinking (intermediate-bus) dc-dc converter, encircled in red in Fig. 1(a).For the purpose of keeping the presentation clear, the considered converter has only two ports.Nevertheless, the analysis is applicable to multiport interlinking converters as well.The proposed methodology also accounts for the case when, in parallel to the considered interlinking converter, additional interlinking converters exist, such as one shown in gray in Fig. 1(a), or when instead of radial, the network features meshed configuration.Since in either of these cases the ports of the converter under study are not solely interconnected by the converter itself, but also by other components of the system, such structure is referred to as a meshed system/grid.The goal of the sections that follow is to present a general impedance-based method for determining small-signal stability properties of an interlinking converter, such as the one from Fig. 1(a).For this, the system is first split into twosubsystems: the converter and the grid, as shown in Fig. 1(b).The subsystems are then represented by their small-signal Norton/Thevenin equivalent circuits, and the properties of the resulting impedance/admittance network are used to determine the small-signal stability of the interconnected system.

B. Small-Signal Representation
In this article, the Norton equivalent small-signal s-domain representation is used for the converter, while the Thevenin one is used for the grid. 1 This representation is shown in Fig. 2, 1 Discussion about which equivalent representation of each of the subsystems should be used to avoid appearance of the RHP poles in the minor-loop gain is out of the scope of this article [7], [21], [28].
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.where ˆis used to denote the corresponding small-signal perturbation components of the quantities from Fig. 1(b).The Norton/Thevenin equivalents are determined considering the MIMO nature of the system, which relies on the unterminated modeling approach [19].Along this line, the converter is represented via the Norton current sources îc1 (s) and îc2 (s) (that model the converter's response to reference perturbation), and the unterminated MIMO admittance matrix 2 Y(s) where s is the complex variable of the Laplace transform, and , 2 In this article, a bold notation is used for matrices and vectors, while an italic one is used for scalars.
For a given converter topology and control system structure, the expressions for Y 11 (s), Y 12 (s), Y 21 (s), and Y 22 (s) can be derived based on the small-signal model of the converter and its control system [26], [29].
Similarly, the grid is represented by the Thevenin voltage sources vg1 (s) and vg2 (s), and the unterminated MIMO impedance matrix Z g (s) where Z g11 , Z g12 , Z g21 , and Z g22 are defined analogously to Y 11 , Y 12 , Y 21 , and Y 22 in (2).In case the system is not meshed, i.e., the two buses are interconnected only via the considered converter, the grid impedance matrix becomes diagonal

C. Device-Level (MIMO) Impedance-Based Method
As a general method for assessing stability of an interlinking converter, such as the one from Fig. 1(b), the MIMO impedancebased method can be used, as explained as follows.According Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
to Fig. 2(a), the following holds: where I is the identity matrix, v(s) = [v 1 (s), v2 (s)] T , vg (s) = [v g1 (s), vg2 (s)] T , îc (s) = [ îc1 (s), îc2 (s)] T and T is the transpose operator.Subsequently, the circuit from Fig. 2(a) can be represented in a compact form shown in Fig. 2(b).Then, assuming that the grid and the converter are standalone stable ( îc (s) and vg (s) are stable), stability of the interconnected system is determined by the stability of the MIMO closed-loop system from Fig. 2(e), where is the corresponding (minor) loop gain.Consequently, as explained in Section III-A, the generalized NC [25], [24] can be applied to (7) and used to determine stability of the interconnected system.This and alternative approaches for determining the system stability are discussed in the following section.

III. LOOP-GAIN-BASED STABILITY ASSESSMENT OF A MIMO FEEDBACK SYSTEM
With the goal of providing a correlation between the MIMO and the SISO impedance-based methods, this section first recalls different loop-gain-based approaches for determining the stability of a linear time-invariant MIMO feedback system Σ T , such as the one from Fig. 2(e).Its loop-gain transfer function matrix L(s) is given by (7), while the corresponding closed-loop transfer function matrix (from v g (s) to v(s), which, according to (5) and (6), is the same as the transfer function matrix from i c (s) to i(s)) is given by where As any transfer function-based methodology, the subsequent analysis can, in a general case, be used only to determine the bounded-input bounded-output (BIBO) stability [24], [30] of the considered MIMO system Σ T .However, BIBO stability may not be sufficient to ensure internal stability, if Σ T is not observable or not controllable [24], [30].Nevertheless, if Σ T is detectable and stabilizable, its internal stability is equivalent to its BIBO stability [24], [30].Hence, stability assessment of the MIMO transfer function matrix T(s) is valid for determining also the internal stability of Σ T , as long as the below stated Condition 1 is satisfied, which is assumed in this article.Condition 1: A state-space representation of the considered MIMO system Σ T (which corresponds to T(s) from ( 8)) features no unobservable or uncontrollable modes (eigenvalues) in the RHP, i.e., Σ T is detectable and stabilizable (it features no unstable device-level hidden dynamics). 3

A. MIMO Loop-Gain-Based Approach
The first stability assessment approach to be recalled is based on the MIMO loop-gain and is stated as follows.
Approach 1: Stability of the closed-loop MIMO system T(s) from ( 8) and Fig. 2(e) can be determined by applying the GNC to the corresponding loop-gain transfer function matrix L(s) from (7) [24].
For this, either the determinant-based GNC or the eigenlocibased GNC can be used [24], [25].Determinant-based GNC involves applying the NC to and, with L m being a scalar value, always relies on a single Nyquist plot (NP) [24].On the other hand, the eigenlocibased GNC involves applying NC to all eigenloci (also called characteristic loci) λ i , which are the eigenvalues of L(s) obtained from det(λ i (s)I − L(s)) = 0 [24], [25].In this case, the number of the required NPs is equal to the number of systems inputs N , which for the system from Fig. 2(a) is equal to two.Note that, since det(I + L(s)) = 1 + i λ i (s), the determinant-based GNC and the eigenloci-based GNC are equivalent in terms of evaluating whether the system is stable or not.However, they yield different stability margins, which brings ambiguity in assessing the system's robustness [24], [31].
Finally, it should be emphasized that, as long as Condition 1 is satisfied, Approach 1 is always valid for determining internal stability, i.e., its applicability is not conditioned by other properties of the system under study.

B. SISO Loop-Gain(s)-Based Approach
The second stability assessment approach to be recalled is based on SISO loop-gains and is stated as follows: Approach 2a: Stability of the closed-loop MIMO system T(s) from ( 8) and Fig. 2(e) can be determined by applying the NC to every loop-at-a-time (LAAT) loop-gain transfer-function [24], [30], [32] where, i, j ∈ {1, 2, . .., N} and, according to ( 9) The idea behind this approach relies on breaking, one at a time, the SISO loops (paths from a single input to the single output) within the MIMO closed-loop system [32], as illustrated in Fig. 3. Stability of the closed-loop SISO system corresponding to the broken loop is then checked by evaluating its loop-gain [32].By repeating evaluation of every SISO loop-gain obtained in this way, stability of the closed-loop MIMO system can be determined.Given that the number of these SISO loop-gains NPs.This approach can be simplified to using a single L LAAT ij (s) in case the Condition 2, stated as follows, is satisfied.
Condition 2: A state-space representation of the SISO system Σ T ij [which corresponds to T ij (s) in ( 8) and ( 9)] features no unobservable or uncontrollable modes (eigenvalues) in the RHP, i.e., Σ T ij is detectable and stabilizable.In this case, the MIMO system Σ T is detectable and stabilizable for the SISO variation, 4i.e., port-pair (i, j), 5 which means it features no unstable portlevel hidden dynamics 6 for that port-pair (i, j).This simplified approach, which relies on a single NP, is referred to as Approach 2b and is formally stated below.
Approach 2b: If for port-pair (i, j) Condition 2 holds, stability of the closed-loop MIMO system T(s) from ( 8) and Fig. 2(e), can be determined by applying the NC to the single LAAT loopgain transfer function L LAAT ij (s) from ( 11) [24], [30].Along this line, several interesting remarks can be made.First, Approach 1 and Approach 2a always yield the same stability assessment result and thus, for this purpose, can be used indistinguishably [32].Nevertheless, Approach 1 may be favorable since it always requires NC to be applied less times, as outlined in Table I and discussed in Section V. Second, Approaches 1, 2a (or 2b) yield different stability margins 7 [31], [32].Finally, it shall be noted that in case Condition 2 is not satisfied for every LAAT SISO variation, i.e., port-pair (i, j), depending on which LAAT SISO variation is chosen, Approach 2b may result in inaccurate stability predictions.Thus, without prior knowledge on whether, and for which port-pair, Condition 2 holds, either Approach 2a or Approach 1 must be used to ensure correct stability assessment result.

C. Relationship With the Standardly Used Port-Level (SISO) Impedance-Based Method
When applying the above discussed approaches for impedance-based stability assessment, given the reasoning from Section II, the loop gain L(s) of the MIMO feedback system under consideration is the product of the converter's unterminated MIMO admittance matrix and grid's unterminated MIMO impedance matrix, as described by (7).Thereby, the MIMO (device-level) impedance-based method from Section II-C relies on directly assessing properties of the MIMO loop gain defined in this way, by using Approach 1.On the other hand, the SISO (port-level) impedance-based method, which is, by far, the mostly used approach in the literature [2], [4], [6], [10], [11], is founded on a different principle.The subsequent analysis will show that, when applied in its standardly used form, this method involves assessing properties of the corresponding SISO LAAT loop-gain(s) L LAAT ii (s). 7Stability margins, such as phase margin, gain margin or vector (disk) margin, are not univocal parameters in MIMO systems [24], [31], as briefly discussed in Section V. Detailed discussion about merits/demerits of different approaches from the stability margin point-of-view is left for future work, since the most adequate way to define stability margins of a MIMO system is still an open topic also in control systems theory [31].
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Recalling ( 5)-( 9), we can write where i, j, k, l ∈ {1, 2, . .., N} and k = j.Based on this and (11), each SISO LAAT loop-gain L LAAT ij (s) can be expressed as a product of an impedance and an admittance where i, j, k, l ∈ {1, 2, . .., N} and k = j, is termed converter's terminated admittance for port-pair (i, j) and is termed grid's terminated impedance for port-pair (i, j).Thus, relying on Approach 2a or Approach 2b, the SISO impedancebased method involves evaluating LAAT minor-loop gain(s) given by (11).Thereby, depending on whether Approach 2a or Approach 2b is used, stability check is performed for, respectively, every possible or a single port-pair.
Still, in the existing literature on the SISO impedance based stability assessment [2], [4], [6], [10], [11], only Approach 2b is typically used, applied to one of the ports, i.e., a single ii (s) is assessed.However, this is not generally applicable, since it does not always guarantee accurate BIBO stability prediction (as elaborated in the previous subsection), due to possible, unstable port-level hidden dynamics. 8s a partial generalization, evaluation of all L LAAT ii (s) is sometimes performed (which corresponds to applying the SISO method at every port) [5], [6], [8], [16].For example, in [5], stability of the voltage source converter in nonmeshed HVDC system featuring unstable port-level hidden dynamics at the dc port could not have been accurately predicted by applying the SISO impedance based method to that port.Rather, the SISO method had to be applied to the ac port.
Still, even the evaluation of all L LAAT ii (s) may not always be sufficient.Namely, even though for i = j Z t gij (s) and Y t ij (s) provide little physical insight, also all L LAAT ij (s) = Z t gij (s)Y t ij (s) must be evaluated (Approach 2a must be used) to account for possible unstable port-level hidden dynamics, which, in a general case, may arise for any port-pair.Such scenarios can be considered likely to appear in modern power electronics systems with meshed structures.
For meshed systems, even in absence of any unstable portlevel hidden dynamics, though mathematically-wise correct, the SISO impedance-based stability assessment has no intuitive physical interpretation.This is because for such systems, where both Y(s) and Z g (s) are nondiagonal, as one can derive from ( 14) and ( 15), Z t gij (s) and Y t ij (s) depend on all elements of both Y(s) and Z g (s), i.e., they are all coupled.This is probably the reason why, to the best of authors' knowledge, the SISO impedance-based stability assessment has so far been used only in nonmeshed systems.Thus, to provide a clear understanding and the correlation of the above presented with the way SISO impedance-based method is standardly used [2], [6], [10], [11], a nonmeshed variant of the system from Fig. 2(a) is considered and the application of the SISO method to one of the ports is illustrated as follows.
When applied to port i (i = 1 or i = 2), the SISO impedancebased method relies on the converter's terminated admittance Y t ii (s) and the grid's terminated impedance Z t gii (s), seen at that port.For a nonmeshed system with a diagonal Z g (s) [given by (4)] and a nondiagonal Y(s) [given by ( 1)], which is hereafter considered, it can be derived from ( 15) that Z t g11 (s) = Z g11 and Z t g22 (s) = Z g22 .Similarly, ( 14) yields and Subsequently, depending on whether the SISO impedance-based method is applied to port 1 or to port 2, the circuit from Fig. 2(a) can be simplified to the one in Fig. 2(c) or (d).According to Fig. 2(c), the following holds: where according to Fig. 2(a), ît c1 (s) is for the considered nonmeshed system given by . (20) Similarly, according to Fig. 2(d), the following holds: where, according to Fig. 2(a), ît c2 (s) is for the considered nonmeshed system given by The expressions such as ( 18)-( 19) and ( 21)- (22), in fact, motivated the development and wide application of the SISO impedance-based method at a port of interest [2], [6], [10], [11].For this, similarly as in Section II-C, the grid and the converter are assumed to be standalone stable.Then, according to (18) and Fig. 4. Circuit representing a simple example of a DC system from Fig. 1, which is used to illustrate the presented methodology analytically, in HIL simulations and experimentally.The system consists of a nonmeshed grid and the current-controlled buck converter, which is used as an example of a two-port interlinking converter.The circuit parameters are provided in Table II.(19) [or ( 21) and ( 22)] stability of the interconnected system can be determined by checking the stability of the SISO closed-loop system from Fig. 2(f) [or Fig. 2(g)], where are the corresponding, so-called minor, loop-gains for the SISO impedance-based stability assessment at port 1 and at port 2, respectively.Nevertheless, given the theory from Section III-B, it shall be underlined once again that stability assessment of solely L LAAT 11 (s) or L LAAT 22 (s) is sufficient to determine stability of an interconnected nonmeshed two-port system only if, for the considered port, Condition 2 holds.Otherwise, along with the more intuitive handling of multiport and meshed systems, etc. the use of the MIMO impedance-based method is recommended, as discussed in Section V.

A. Considered Test-Case
To illustrate the use of the previously discussed stability assessment approaches, a simple nonmeshed system resembling the one from Fig. 1(b) is considered, which is shown in Fig. 4 and features parameters from Table II, which were adopted as an example.Note that the applicability of the methodology presented in this article is not limited to the specific grid/converter parameter choice; rather, the methodology is of general use and remains valid for any parameter values.The system under study consists of a digital pulsewidth modulated current-controlled two-level buck converter and a grid.The grid's dc bus voltages are formed by a constant voltage source and a constant voltage load.The grid's impedances are realized by passive LC elements. 9The grid's impedances thus feature resonances which threaten to endanger system stability.
First, it was of interest to analytically predict stability properties of such a system, by applying the SISO impedance-based 9 The impedances Z ps = R ps ||(sL ps ) and Z el (s) = R el ||(1/(sC el )||(sL el ), which are also contributing to grid impedances, are included in Fig. 4 to match the experimentally tested circuit, discussed in Section IV-C.More precisely, these impedances account for the non-ideal dynamics of the electronic source and load used for experimental validation.

TABLE II
PARAMETERS OF THE TESTED CONVERTER AND THE GRID method (Approach 2b) 10 for the port-pairs (1,1) and (2,2), i.e., ports 1 and 2, as well as the MIMO impedance-based method (Approach 1).For this, the elements of the grid impedance matrix, Z g11 (s), Z g22 (s), and the converter's admittance matrix Y 11 (s), Y 22 (s), Y 21 (s), Y 12 (s) are at first calculated, based on Fig. 4, Table II and the small-signal s-domain model of the converter and its closed-loop control system. 11 7), (24), and (25), and the NC is 10 The considered system does not feature unstable port-level hidden dynamics for port-pairs (1,1) and (2,2) and thus it was sufficient to apply Approach 2b to either of these port-pairs (ports). 11In the considered case, which includes a current-controlled buck converter, the expressions for Y 11 (s), Y 22 (s), Y 21 (s), Y 12 (s) can be found in [26].
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply., and L m feature such encirclement.Thus, all three considered approaches predict a stable response.On the other hand, when I ref = 20 A, instability is predicted by all three approaches, since the corresponding Nyquist curves all encircle (−1,0).Note that, since the considered system does not feature unstable port-level hidden dynamics for neither port 1 nor port 2, all three considered approaches, Approach 1, Approach 2b at port 1, and Approach 2b at port 2, in each considered scenario (Figs. 7 and 8), yielded the same stability prediction result.Thus, for solely determining whether the system will be stable or unstable, they can be used indistinguishably in the considered simple example.However, the three methods yield different stability margins, as seen from Fig. 7. Furthermore, as discussed in Section V, Approach 1 may be favorable in systems with meshed structures, multiport converters or unstable port-level hidden dynamics, as well as for

B. HIL Validation
To verify the analytical stability predictions from previous subsection, real-time control-HIL (C-HIL) simulations of the system from Fig. 4 are performed.For this, Typhoon HIL 402 is used to emulate the converter and the grid, with the circuit solver time step set to 0.5 μs.The inductor current is acquired from the HIL's analog output.Analog-to-digital-conversion (ADC), the current control and DPWM are realized in the digital signal processor (DSP) within Imperix B-Board PRO control platform.The same parameters (provided in Table II) are used as for the results in Figs.5-8.
To validate that the realized system faithfully models dynamics of the one used for analytical predictions, frequency response measurements are performed to obtain Z g11 (jω), Z g22 (jω) and Y 11 (jω), Y 12 (jω), Y 21 (jω), Y 22 (jω) for I ref = 5 A and I ref = 20 A. For this, the series perturbation injection circuits were emulated in HIL, and the HIL's dedicated SCADA widget was used to obtain the frequency responses of interest, which are plotted (with dots) in Figs. 5 and 6.As seen, the results obtained from HIL measurements are excellently matching those obtained from analytical models.Thus, the analytical stability predictions are expected to match the stability properties of the system realized using HIL.

C. Experimental Validation
To further validate analytical predictions of different stability assessment approaches discussed in previous sections, the laboratory prototype of the system from Fig. 4 is built, featuring, same as previously, parameters from Table II.The picture of the test setup is shown in Fig. 10.The converter is realized using PEB8024 SiC half bridge modules from Imperix, and passive LC elements are used for the converter's inductive filter and the grid impedances.The dc voltages at the converter's input and output ports are provided by the dc power supply Chroma 62050P-100-100 and the electronic load EA-EL 9750-120 B. The inductor current is sensed by the built-in LEM-based current sensor from Imperix.The realization of the digital current control is the same as the one used for HIL simulations, compiled to another control platform (B-Box RCP from Imperix), which is compatible with the hardware used for the power stage and features the same DSP as the one used for HIL validations.Time domain stability test is performed, in the same way as before.The ramp change of the inductor current reference from A is imposed and the converter's input and output voltage and current waveforms are captured by an oscilloscope.The captured responses are shown in Fig. 11.As predicted in Figs.7 and 8, and previously verified using HIL, the experimentally tested system achieves stable operation for  I ref = 5 A, while, due to interactions between the converter and the grid, the system gets destabilized for I ref = 20 A.

V. MERITS AND LIMITATIONS OF SISO AND MIMO IMPEDANCE-BASED METHODS
Given all of the above presented, it is of interest to comment on the strengths and weaknesses of the MIMO (device-level), and the SISO (port-level) impedance-based method for the stability assessment of dc-dc interlinking converters in grid-connecting scenarios.The methods are compared based on several different indicators: 1) computational complexity required to account for the unstable port-level hidden dynamics; 2) suitability for multiport interlinking converters; 3) applicability in meshed grids; 4) potentials from the stability margins point of view; 5) appropriateness for termination-independent stabilityoriented controller design; 6) applicability for the stability assessment based on measurements.
When it is of interest to determine stability without prior knowledge on whether (and for which port-pair) there could be unstable port-level hidden dynamics, either the MIMO method (which corresponds to Approach 1) or the SISO method applied to every possible port-pair (which corresponds to Approach 2a) Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.can be used.Thereby, the MIMO method may be more convenient to use than the SISO one, as it is able to account for unstable port-level hidden dynamics with a single NP (by using determinant-based GNC).This advantage, as illustrated in Table I, gets further emphasized as the number of ports (inputs/outputs) increases.
When it is known in advance that for a certain port-pair, the unstable port-level hidden dynamics do not appear, as in the example from Section IV, both the MIMO method (Approach 1) and the SISO method applied to that single port-pair (Approach 2b) can be used to accurately assess stability.Consequently, if the determinant-based GNC is used for the former, both methods rely on a single NP.Still, the MIMO method may be favorable for analytical stability predictions due to its modularity and scalability.For example, its extension to systems with higher number of ports, such as multiport converters, is straightforward.On the contrary, the complexity of expressions for the SISO method, more specifically, for terminated immittances (such as ( 16) and ( 17)) significantly increases as the number of ports increases [33].
As for the stability assessment of an interlinking converter in meshed grids, the MIMO impedance-based method inherently and intuitively handles this scenario, since with this approach, the subsystems' immittances to be evaluated are defined in the all-port MIMO unterminated sense.On the contrary, though theoretically wise possible to use for analytical stability assessment, the SISO impedance based-approach loses intuitiveness and provides little physical insight in meshed systems.This is because the interconnected system gets fully coupled and, consequently, the subsystems' terminated immittances become dependent on all of both subsystems' unterminated immittances.
Regarding the stability margins, provided that no port-level hidden dynamics appear, both SISO and MIMO methods may be useful, depending on the information of interest.The SISO method allows to determine stability margins of a single output when exposed to a perturbation at a single input, whereas the MIMO method allows to determine stability margins of all outputs when exposed to simultaneous perturbations at all inputs [24], [31].Compared to the MIMO method using determinant-based GNC, the MIMO method using eigenlocibased GNC, as well as the SISO method, may provide more insight on where the risk for destabilization comes from [32].Nevertheless, with the determinant-based MIMO method, definition of a single stability margin is straightforward [31].Though interpretation of all these stability margins is different, it is important to emphasize that they all reflect properties of the same converter's control system.Thus, it may be difficult to control them independently in practice.Research along this line is left for future studies.
Next, it is of interest to comment on the applicability and suitability of the SISO and MIMO impedance-based methods for the robust stability-oriented controller design.The goal of such design strategies is to ensure stability even when termination varies, which, as explained below, may be difficult using converter's terminated immittances.As one example of such strategies, admittance passivity-oriented controller design [12], [14], [26] is considered below.Stemming from the SISO impedance-based method, the conventional admittance passivity-oriented design concepts strive for passivizing the converter's (terminated) SISO admittance at a connection port.However, this requires assumptions about termination at all other converter's ports, limiting thereby the applicability of such a concept for preventing port-coupling induced instability [26].On the contrary, by aiming to passivize the converter's unterminated MIMO admittance matrix, stability can be ensured for an arbitrary (even-meshed) passive termination [26].This concept, which was for the first time proposed by Cvetanovic et al. [26], naturally comes to one's mind if, for evaluating stability, instead of the standardly used SISO, the MIMO impedance-based method is considered.More details along this line can be found in [26].
Finally, when the stability assessment of an N -port interlinking converter is to be performed based on measurements, rather than analytical models, the MIMO method may again be advantageous.Namely, for such stability assessment, frequency responses of the immittances that appear in the minor-loop gain, which is to be evaluated, must be first measured.To avoid potential instability that may arise when the considered converter is, even at a single port, connected to the grid of interest, such measurements (and subsequent stability assessments) must be first performed under the termination for which the converter's control system is known to be stable, which typically corresponds to zero grid impedances (ideal termination).Frequency responses of the unterminated immittances that can be obtained in this way [19] are sufficient to apply the MIMO method.However, as elaborated in Section III, application of the SISO method relies on the specific terminated immittances.Measurement of these immittances at the port of interest requires terminating the converter at all other ports by the grid impedances that correspond to the scenario for which stability is to be assessed.Thus, prior to measuring the terminated immittances required for the SISO stability assessment at the port of interest, the SISO stability assessments at all other ports must be performed, using the corresponding unterminated immittances.Consequently, for the stability assessment based on measurements, even in absence of any unstable port-level hidden dynamics, the SISO method would have to be applied more than once (N times).Thus, though in this case, the number of the required frequency response measurements, i.e., independent perturbation injections, remains the same for the SISO and the MIMO method, the MIMO method may be preferable as it allows NC to be applied only once (if the determinant-based GNC is used).In addition, in case the stability assessment is to be repeated once termination changes, the MIMO method would require less additional measurements (and assessments).Furthermore, if the system is meshed, using the SISO method for the stability assessment based on measurements is feasible only in case the converter remains stable when connected to the grid of interest.This is because, contrary to nonmeshed system, measurement of terminated immittances in meshed system requires the converter to be operated under the conditions for which stability assessment is to be performed, since, as explained in Section III-C, terminated immittances are in this case fully coupled.

VI. CONCLUSION
By relying on the formal control systems theory principles, this article provides the correlation and the comparison between the standardly used port-level (terminated) SISO, and the recently acknowledged device-level (unterminated) MIMO impedance-based method for the stability assessment of dc-dc interlinking converters.The capability of these methods to account for (unstable) port-level hidden dynamics is for the first time discussed and the MIMO method is revealed to be preferable for this purpose.The suitability of the SISO and the MIMO methods for stability assessment in systems with meshed structures and/or multiport converters, as well as for defining various stability margins is also addressed, again showing that the MIMO method is advantageous.Moreover, the unterminated (black-box) representation-related assets of the MIMO method are highlighted, which are relevant when the stability is to be assessed based on measurements, as well as when robust stability-oriented termination-independent control strategies are to be designed.The presented methodology is validated in HIL simulations and experimentally, using the laboratory prototype of a current-controlled buck converter.Future studies will focus on extending the presented methodology to ac-dc systems.

APPENDIX A
Consider an arbitrary continuous LTI system Σ G whose statespace representation is given by where x(t) ∈ R N x , u(t) ∈ R N u , and y(t) ∈ R N y denote, respectively, system states, inputs, and outputs, the number of each being N x , N u , and N y , and A G , B G , C G , and D G are appropriately dimensioned real constant matrices [30].The corresponding transfer function (matrix) from u to y is given by [30] G The system Σ G is internally stable if all the eigenvalues (modes)  [30].
The possible difference between the two stabilities (internal and external) stems from the fact that not all the eigenvalues of A G are necessarily the poles [34] of G(s), due to possible pole-zero cancellations [30].This occurs in case the system Σ G is not controllable or observable, which are the properties defined as follows. 12The system Σ G is called controllable if the below stated Condition A.1 holds.
Condition A.1: The matrix for all λ = {λ i ∈ λ A G } has full row rank [30]. 12All the subsequent definitions apply regardless of whether the system Σ G is SISO (N u = N y = 1) or MIMO (N u > 1, N y > 1) system.Similarly, the system Σ G is called observable if the below stated Condition A.2 holds.
If the system Σ G is not observable or controllable (Condition A.1 or Condition A.2 does not hold), it is, in this article, said to feature hidden dynamics.In this case its internal stability may be different from BIBO stability, i.e., instability may arise that can not be predicted by evaluating poles of G(s).Still, even when the system features hidden dynamics, internal stability can be accurately predicted from G(s) [30]  Based on the definitions above, it is now of interest to distinguish between the unstable device-level hidden dynamics and the unstable port-level hidden dynamics, the properties being mentioned in Section III.For this, the LTI MIMO system Σ T (such as one from Section III) is considered hereafter, which is characterized by the state-space representation from (26), where The corresponding transfer function matrix T(s) can be obtained from (27).The elements of T(s) are T ij (s), where i ∈ {1, 2, . ..N u T } and j ∈ {1, 2, . ..N y T }.Each T ij (s) is the transfer function of the SISO system Σ T ij , which is characterized by the state space representation from (26), where 13 If the considered MIMO system Σ T satisfies Condition A.3 and Condition A.4, this system is said not to feature unstable device-level hidden dynamics.As such, it is detectable and stabilizable in a MIMO sense, i.e., its internal stability can be accurately predicted by evaluating the poles of its transfer function matrix T(s).If the SISO system Σ T ij satisfies Condition A.3 and Condition A.4, the considered MIMO system Σ T is said not to feature unstable port-level hidden dynamics for a SISO variation, i.e., port-pair (i, j).In this case, the considered MIMO system Σ T is detectable and stabilizable for a single SISO variation (i, j) (in a SISO sense).Consequently, the BIBO stability of the MIMO system Σ T is equivalent to the BIBO stability of the SISO system Σ T ij , which can be accurately predicted by evaluating the poles of the transfer function T ij .Nevertheless, without prior knowledge on whether the considered MIMO system Σ T features any unstable port-level hidden dynamics, the poles of all T ij must be evaluated to accurately determine (BIBO) stability of the MIMO system Σ T .As explained in Section III-B, this is very important if it is of interest to use the SISO-based tools to determine (BIBO) stability of the MIMO system [30], which is often called LAAT approach.
To illustrate the above outlined properties, some examples are provided as follows.
Such system features two eigenvalues λ 1 = −3 and λ 2 = 2.According to (27), the corresponding input-output transfer function matrix is given by It features one pole p 1 = −3.By checking the above outlined conditions, it can be shown that the system Σ T is observable, and thus detectable, but neither controllable nor stabilizable.As such, it features unstable device-level hidden dynamics, and hence, though BIBO stable (Re{p 1 } < 0) it is internally unstable (λ 2 > 0).It features two poles p 1 = −3 and p 2 = 1.By checking the above outlined conditions, it can be shown that the system Σ T is both controllable (and thus stabilizable) and observable (and thus detectable).As such, it does not feature device-level hidden dynamics, and its internal stability is equivalent to BIBO stability, which can be determined by evaluating poles of T(s).Therefore, since Re{p 2 } > 0, the system is unstable.
Along this line, it is of interest to check whether the stability of the considered MIMO system Σ T can also be accurately predicted by evaluating poles of a single element of T(s).For this, as previously explained, the above outlined conditions should be checked for the SISO subsystems Σ T ij .Given the state-space representation of the considered MIMO system Σ T , the state-space representation of each SISO subsystems Σ T ij can be determined.Since in the considered example T(s) is diagonal matrix, only two "nonzero" SISO subsystems exist: By checking the above outlined conditions it can be shown that Σ T 11 , though being neither controllable nor observable, it is both stabilizable and detectable.On the contrary, Σ T 22 is neither stabilizable nor detectable.Thus, for the port-pair (2,2) the MIMO system Σ T features unstable port-level hidden dynamics, while for the port-pair (1,1) though present, the port-level hidden dynamics is not unstable.Accordingly, the stability of the MIMO system Σ T can be accurately predicted by evaluating poles of T 11 (s) = 1 s−1 , but not by evaluating poles of T 22 (s) = 1 s+3 .Thus, without prior knowledge on whether (and for which port-pair) the considered MIMO system Σ T features unstable port-level hidden dynamics, both14 T 11 (s) and T 22 (s) must be evaluated to accurately determine (BIBO) stability of Σ T .

Fig. 1 .
Fig. 1.(a) DC power electronic system under study.The DC-DC converter in gray illustrates the possibility of forming a meshed system structure.(b) The two-subsystem equivalent representation, featuring the grid and the two-port interlinking DC-DC converter, encircled in red.

Fig. 3 .
Fig. 3. LAAT representation of the MIMO (N = 2) feedback system from Fig. 2(e), obtained by breaking (a) only the loop from the first input vg1 to the first output v1 (sw 1 is open and sw 2 is closed); (b) only the loop from the second input vg2 to the second output v2 (sw 1 is closed and sw 2 is open).The corresponding LAAT SISO loop gains L LAAT 11 (s) and L LAAT 22 (s) can be obtained from (a) and (b), respectively.Though less intuitive, similar representation can be made for obtaining L LAAT 12 (s) and L LAAT 21 (s).
The resulting frequency responses of the converter's admittances are obtained for two different values of the current reference I ref = 5 A and I ref = 20 A. Due to space limitations, only the results for I ref = 20 A are shown (with full-lines) in Fig. 5.The resulting frequency responses of the grid's impedances are shown (with full lines) in Fig. 6.Then, the minor loop-gains L LAAT 11 , L LAAT 22 , L are calculated based on (

Fig. 5 .
Fig. 5. Frequency responses of the converter's unterminated MIMO impedance matrix elements, corresponding to the system from Fig. 4 with I ref = 20 A. Comparison between the results obtained using analytical model (full lines) and HIL simulations (dots).

Fig. 6 .
Fig. 6.Frequency responses of the grid's unterminated MIMO impedance matrix elements, corresponding to the system from Fig. 4. Comparison between the results obtained using analytical model (full lines) and HIL simulations (dots).

Fig. 7 .Fig. 8 .
Fig. 7. NPs used for the stability assessment of the system from Fig. 4 with the parameters from Table II and I ref = 5 A. The blue, green and purple plots, denoted by L LAAT 11 , L LAAT 22 , and L m , correspond to, respectively.Approach 2b applied to port 1, Approach 2b applied to port 2, and Approach 1. Stable response is predicted by all three approaches.

Fig. 9 .
Fig. 9. Response of the circuit from Fig. 4 to the reference ramp change from I ref = 5 A to I ref = 20 A, obtained using HIL simulations.As predicted in Fig. 8, instability arises for I ref = 20 A.

Fig. 11 .
Fig. 11.Experimentally measured response of the circuit from Fig. 4 (realized using the prototype from Fig. 10) to the current reference ramp change from I ref = 5 A to I ref = 20 A. As predicted in Fig. 8, instability arises for I ref = 20 A.

Example 2 :
Consider an LTI MIMO system Σ T , whose matrices describing the state-space representation are Such system features two eigenvalues λ 1 = −3 and λ 2 = 1.According to(27), the corresponding input-output transfer function matrix is given by

TABLE I OVERVIEW
OF DIFFERENT STABILITY ASSESSMENT APPROACHES FOR THE MIMO FEEDBACK SYSTEM WITH N INPUTS(OUTPUTS), SUCH AS THE ONE FROM FIG.2(e), WHICH CORRESPONDS TO AN N -PORT INTERLINKING DC-DC CONVERTER in the open left-half plane, while it is externally (input-output or BIBO) stable if all the poles p G = {p 1 , p 2 , . ..} of G are in the open left-half plane in case the system Σ G is stabilizable and detectable, which are the properties defined as follows.The system Σ G is called stabilizable if the below stated Condition A.3 holds.Condition A.3: The matrix (28) for all λ = {λ i ∈ λ A G | Re{λ i ≥ 0}}, has full row rank.Similarly, the system Σ G is called detectable if the below stated Condition A.4 holds.Condition A.4: The matrix (29), for all λ = {λ i ∈ λ A G | Re{λ i ≥ 0}}, has full column rank.