A Characterization of All Passivizing Input-Output Transformations of a Passive-Short System
AA Characterization of All Passivizing Input-OutputTransformations of a Passive-Short System
Miel Sharf a Daniel Zelazo a a Technion - Israel Institute of Technology, Haifa, Israel (e-mail: [email protected],[email protected]).
Abstract
Passivity theory is one of the cornerstones of control theory, as it allows one to prove stability of a large-scale system whiletreating each component separately. In practice, many systems are not passive, and must be passivized in order to be included inthe framework of passivity theory. Input-output transformations are the most general tool for passivizing systems, generalizingoutput-feedback and input-feedthrough. In this paper, we classify all possible input-output transformations that map a systemwith given shortage of passivity to a system with prescribed excess of passivity. We do so by using the connection betweenpassivity theory and cones for SISO systems, and using the S-lemma for MIMO systems. We also present several possibleapplications of our results.
Key words:
Passivity-based Control, Passive-short Systems, Nonlinear Systems, Transformation
Over the last few decades, many technological systemshave become much more complex, as networked systemsand large-scale systems turned common, and “system-of-systems” evolved into a leading design methodology. Toaddress the ever-growing complexity of systems, manyresearchers suggested various component-level tools thatguarantee system-level properties, e.g. input-output sta-bility. One important example of such a notion is passiv-ity, which can be informally stated as “energy-based con-trol” [Khalil (2002),Pavlov and Marconi (2008)]. It hasbeen used to solve different problems in control theory inmany areas, including networked systems [Arcak (2007);Bai et al. (2011)], cyber-physical systems [Antsaklis et al.(2013)], robotics [Hatanaka et al. (2015)], power systems[De Persis and Monshizadeh (2018)] and space struc-tures [Benhabib et al. (1981)].In practice, however, many systems are not passive. Ex-amples include systems with input/output delays (suchas chemical processes), human operators, generators,power-system networks, vehicle networks, and electroniccircuits with non-linear components [Trip and De Per-sis (2018); Xia et al. (2014); Harvey and Qu (2016); At-man et al. (2018); Jain et al. (2018); Sharf and Zelazo(2019b)]. A system’s lack of passivity is often quanti-fied using passivity indices. In order to use passivity-based design techniques, one needs to passivize the sys-tem under consideration. The most common methods for passivation (also known as passification) include gains,output-feedback, input-feedthrough, or a combinationthereof [Byrnes et al. (1991); Zhu et al. (2014); Jain et al.(2018); Sharf and Zelazo (2019b)].More generally, a passivation method relying on aninput-output transformation was suggested in [Xia et al.(2018); Sharf et al. (2019)]. An input-output transfor-mation is a concise formulation for the aggregation ofoutput-feedback, input-feedthrough, and gains. Namely,Xia et al. (2018) showed that any system with finite L -gain can be passivized using this formulation, where theinput-output transformation is found using an algebraicapproach. More recently, Sharf et al. (2019) used a geo-metric approach to prescribe a passivizing input-outputtransformation for SISO systems. More precisely, oneconstructs an input-output transformation, mapping asystem with known passivity indices to a system withprescribed passivity indices. This was achieved using aconnection between passivity and cones through the no-tion of projective quadratic inequalities. We note thatthe connection between passivity and cones, stemmingfrom sector-bounded nonlinearities, has been previouslyexplored [Khalil (2002); Zames (1966); McCourt andAntsaklis (2009)].In this paper, we use the geometric approach of Sharfet al. (2019) to give a full description of all passivizinginput-output transformations of a given SISO system.More precisely, we give a concise description of all input- Preprint submitted to Automatica 7 April 2020 a r X i v : . [ ee ss . S Y ] A p r utput transformations that map a system with knownpassivity indices to a system with prescribed excess ofpassivity. This is done by understanding the action ofthe group of (invertible) input-output transformationson the collection of cones in the plane. We show that anytransformation mapping a system with known passivityindices to a system with prescribed excess of passivitycan be written (up to a scalar) as a product of threematrices - one depending on the original passivity in-dices, one depending on the desired excess of passivity,and a non-negative matrix, i.e., a matrix whose entriesare all non-negative. We then use similar mechanismsto give an analogous result for MIMO systems, wherethe non-negative matrix is replaced by a matrix satis-fying a certain generalized algebraic Riccati inequality.Our results can be seen as an analogue of the Youla pa-rameterization [Kuˇcera (2011)], dealing with passivizinginput-output transformations instead of stabilizing con-trollers. We also consider multiple application domainsof our results. First, we consider the problem of find-ing a transformation that achieves multiple objectives,e.g. passivation of more than one system. We exploreapplications of this problem for controller synthesis offaulty systems, as well as synthesizing a single controllerfor multiple systems, which is important when the spe-cific plant we need to control is not known ahead oftime. Second, we consider the problem of passivizing aplant with respect to multiple equilibria, which relatesto different notions of equilibrium-independent passivity[Hines et al. (2011); B¨urger et al. (2014); Sharf and Ze-lazo (2019a)]. Lastly, we consider the problem of findinga passivizing transformation T which optimizes a cer-tain cost function, e.g. the L -gain of the transformedsystem, or the distance in the operator norm betweenthe original system and the transformed system.The rest of the paper is as follows. Section 2 briefs overthe geometric approach of Sharf et al. (2019), and for-mulates the problem at hand. Section 3 characterizesall passivizing transformations of a given passive-shortSISO system, and Section 4 generalizes the characteriza-tion to MIMO systems. Section 5 provides three possibleapplication domains of the achieved results. Notation:
We denote the unit circle inside R by S .We also use various notions from linear algebra [Hornand Johnson (2012)]. We denote the group of all invert-ible matrices T ∈ R d × d as GL d ( R ). Such matrices in-duce invertible linear transformations R d → R d . Givena linear transformation S : R d → R d and a basis B for R d , we denote the representing matrix of S in the ba-sis B as [ S ] B . Furthermore, given two bases B , B of R d , we denote the change-of-base matrix from B to B by I B →B ∈ GL d ( R ). We note that I − B →B = I B →B .Moreover, for any linear transformation S : R d → R d ,we have that [ S ] B = I B →B [ S ] B I B →B . Lastly, we de-note the Kronecker product by ⊗ , and the d × d identity matrix as Id d . We consider dynamical systems given by the state-spacerepresentation ˙ x = f ( x, u ) , y = h ( x, u ) , where u ∈ R n u is the input, y ∈ R n y is the output,and x ∈ R n x is the state of the system. We recall thedefinition of passivity: Definition 2.1
Let Σ be a dynamical system with equalinput and output dimensions. Assume that u = 0 , y = 0 is an equilibrium of the system. We say that the systemis passive if there exists a positive-definite C -smoothfunction (i.e., a storage function) S of the state x suchthat the inequality, dS ( x ) dt = ∇ S ( x ) ˙ x ≤ u ( t ) (cid:62) y ( t ) , (1) holds for any trajectory ( u ( t ) , y ( t )) of the system. The notion of passivity stems from energy-based con-trol, as S ( x ) can be thought of as the amount of po-tential energy stored inside the system, so (1) impliesthat the change in the energy stored in the system can-not be greater than the power supplied to the system.Passivity has been used to solve problems in various ap-plication domains, including multi-agent networks [Ar-cak (2007); Bai et al. (2011)], cyber-physical systems[Antsaklis et al. (2013)] and robotics [Chopra and Spong(2006); Hatanaka et al. (2015)]. We can expand the no-tion of passivity to consider both the case of total en-ergy dissipation, and the case of (bounded) total energygain, by adding either a negative or a positive term tothe right-hand side of (1): Definition 2.2
Let Σ be a dynamical system with equalinput and output dimensions. Assume that u = 0 , y = 0 is an equilibrium of the system. Let ρ, ν ∈ R .i) We say that the system is output ρ -passive if thereexists a storage function S such that the inequality, dS ( x ) dt ≤ u ( t ) (cid:62) y ( t ) − ρ (cid:107) y ( t ) (cid:107) , (2) holds for any trajectory ( u ( t ) , y ( t )) of the system.ii) We say that the system is input ν -passive if there existsa storage function S such that the inequality, dS ( x ) dt ≤ u ( t ) (cid:62) y ( t ) − ν (cid:107) u ( t ) (cid:107) , (3) holds for any trajectory ( u ( t ) , y ( t )) of the system. ii) We say that the system is input-output ( ρ, ν )-passive if ρν < / and there’s a storage function S such that, dS ( x ) dt ≤ u ( t ) (cid:62) y ( t ) − ρ (cid:107) y ( t ) (cid:107) − ν (cid:107) u ( t ) (cid:107) , (4) holds for any trajectory ( u ( t ) , y ( t )) of the system. Remark 1
The demand ρν < / is made to assurethat the right-hand side of (4) is not always positive, noralways negative, as it would either imply that all staticnonlinearities are input-output ( ρ, ν ) -passive, or that nosystem is input-output ( ρ, ν ) -passive, both of which areabsurd. The case in which ρ, ν > strict passivity (or “excess of passivity”), and the casein which ρ, ν < passive short (or“shortage of passivity”). The definition above allows usto consider both cases in a united framework. Passive-short and non-passive systems appear in many practi-cal applications [Atman et al. (2018); Trip and De Per-sis (2018); Xia et al. (2014); Harvey and Qu (2016)].In order to incorporate them into passivity-based con-trol schemes, one usually passivizes them using a trans-formation [Byrnes et al. (1991); Zhu et al. (2014); Xiaet al. (2014); Jain et al. (2018); Sharf and Zelazo (2019b);Sharf et al. (2019)]. Common transformations includeoutput-feedback, input-feedthrough, and gains. We com-bine them and consider a transformed plant ˜Σ with newinput ˜ u and output ˜ y , which are connected to u, y via (cid:34) ˜ u ˜ y (cid:35) = T (cid:34) uy (cid:35) , (5)for some invertible matrix T . The action of the trans-formation T on the a system Σ as a block diagram canbe seen in Figure 1. This transformation can be seen asan amalgamation of a constant gain input-feedthrough,constant gain output-feedback, and and cascade with aconstant gain (see [(Sharf et al., 2019, Section IV)] fordetails). This can be visualized in Figure 2. We wish tounderstand the effect of these input-output transforma-tions on the passivity of the transformed system. For-mally, the problem we consider is the following: Fig. 1. A block diagram describing the transformed system˜Σ after the linear transformation T . Text
Fig. 2. A different block diagram describing the transformedsystem ˜Σ after the linear transformation T , as in Figure 1. IfΣ is SISO and T = [ a bc d ], then δ A = b/a, δ B = d − ba c, δ C = c and δ D = a . Problem 2.1
Let Σ be any dynamical system with equalinput and output dimensions, which is input-output ( ρ, ν ) -passive, and let ρ (cid:63) , ν (cid:63) be any numbers such that ρ (cid:63) ν (cid:63) < / . Characterize all linear input-output trans-formations R → R of the form (5) such that thetransformed system ˜Σ is input-output ( ρ (cid:63) , ν (cid:63) ) -passive. In order to address this problem, we consider the geo-metric approach to passivity of [Sharf et al. (2019)]. Theapproach begins by considering an abstraction of the in-equalities appearing in Definition 2.2, known as projec-tive quadratic inequalities:
Definition 2.3 (PQI) A d -dimensional projectivequadratic inequality (PQI) is an inequality in the vari-ables ξ, χ ∈ R d of the form ≤ a (cid:107) ξ (cid:107) + bξ (cid:62) χ + c (cid:107) χ (cid:107) := f ( a,b,c ) ( ξ, χ ) , (6) for some numbers a, b, c , not all zero. The inequality iscalled non-trivial if b − ac > . The associated solu-tion set of the d -dimensional PQI is the set of all points ( ξ, χ ) ∈ R d satisfying the inequality. If d = 1 , we’ll omitthe dimension and call the inequality a PQI. We note that the d -dimensional PQI resembles the in-equality corresponding to input-output ( ρ, ν )-passivity,in which ξ, χ, a, b, c are replaced by u, y, − ν, , − ρ re-spectively. For this reason, we denote f ( a,b,c ) ( ξ, χ ) as ϕ ρ,ν ( ξ, χ ), and the corresponding solution set as C ρ,ν,d .Namely, we define: ϕ ρ,ν ( ξ, χ ) = − ν (cid:107) ξ (cid:107) + ξ (cid:62) χ − ρ (cid:107) χ (cid:107) , (7)and C ρ,ν,d = { ( ξ, χ ) ∈ R d × R d : ϕ ρ,ν ( ξ, χ ) ≥ } . Forconvenience, we denote the latter as C ρ,ν when d = 1. Remark 2.1
The definition of d -dimensional PQIs al-lows an abstraction of the inequality defining passivity,and it encapsulates more sophisticated variants of pas-sivity, such as shifted passivity, incremental passivity[Pavlov and Marconi (2008)], equilibrium-independentpassivity [Hines et al. (2011)] and maximal equilibrium-independent passivity [B¨urger et al. (2014); Sharf andZelazo (2019a)]. Hence, the results of the paper also ap-ply to these variants.
3s noted in Sharf et al. (2019), input-output transfor-mations give rise to an action of the group of 2 × GL ( R ), on the collection of PQIs, whichin turn induces an action of the same group on the solu-tion sets. In particular, let A be the solution set of a ( d -dimensional) PQI, A = { ( ξ, χ ) : 0 ≤ f ( a,b,c ) ( ξ, χ ) } . Forany invertible matrix T ∈ GL d ( R ), the solution set ofthe transformed ( d -dimensional) PQI is given by T ( A ),the image of A under the linear transformation inducedby T . In fact, one can show that an input-output trans-formation maps an input-output ( ρ, ν )-passive systemto an input-output ( ρ (cid:63) , ν (cid:63) )-passive system if and onlyif it maps the d -dimensional PQI, 0 ≤ ϕ ρ,ν ( ξ, χ ) to the d -dimensional PQI, 0 ≤ ϕ ρ (cid:63) ,ν (cid:63) ( ξ, χ ) (or to a stricter in-equality).Following Sharf et al. (2019), we first focus on the caseof SISO systems. Definition 2.4 A symmetric section S on the unit circle S ⊆ R is defined as the union of two closed disjoint sec-tions that are opposite to each other, i.e., S = B ∪ ( − B ) ,where B is a closed section of angle < π . A symmetricdouble cone is defined as A = { λs : λ > , s ∈ S } forsome symmetric section S . The connection between cones and passivity theory isintricate, stemming from the notion of sector-boundednonlinearities [Khalil (2002); Zames (1966); McCourtand Antsaklis (2009)]. An example of a symmetric sec-tion and the associated symmetric double-cone can beseen in Figure 3. These are of interest due to their closerelationship with (1-dimensional) PQIs. Namely,
Theorem 2.1 (Sharf et al. (2019))
The solution setof any non-trivial PQI is a symmetric double cone. More-over, any symmetric double-cone is the solution set ofsome non-trivial PQI, which is unique up to a multiplica-tive positive constant.
As a corollary, we conclude that a map transforms aninput-output ( ρ, ν )-passive system to an input-output( ρ (cid:63) , ν (cid:63) )-passive system if and only if it sends C ρ,ν into C ρ (cid:63) ,ν (cid:63) , which we denote by C ρ,ν (cid:44) → C ρ (cid:63) ,ν (cid:63) Thus, wewish to characterize maps C ρ,ν (cid:44) → C ρ (cid:63) ,ν (cid:63) . One possiblesolution is given through the following theorem: Theorem 2.2 (Sharf et al. (2019))
Let ρ , ν , ρ (cid:63) and ν (cid:63) be any numbers such that ρν, ρ (cid:63) ν (cid:63) < / . Let ( ξ , χ ) and ( ξ , χ ) be two non-colinear solutions to ϕ ρ,ν ( ξ, χ ) = 0 . Moreover, let ( ξ , χ ) and ( ξ , χ ) be twonon-colinear solutions to ϕ ρ (cid:63) ,ν (cid:63) ( ξ, χ ) = 0 . Define T = (cid:34) ξ ξ χ χ (cid:35) (cid:34) ξ ξ χ χ (cid:35) − , T = (cid:34) ξ − ξ χ − χ (cid:35) (cid:34) ξ ξ χ χ (cid:35) − . Let α be equal to if ϕ ρ,ν ( ξ + ξ , χ + χ ) ≥ and zero -1 -0.5 0 0.5 1-1-0.8-0.6-0.4-0.200.20.40.60.81 Fig. 3. A double cone (in blue), and the associated symmetricsection (in solid red). The parts of S outside the symmetricsection are presented by the dashed red line. otherwise. Moreover let α be equal to if ϕ ρ (cid:63) ,ν (cid:63) ( ξ + ξ , χ + χ ) ≥ and zero otherwise.i) If α = α , then T is C ρ,ν (cid:44) → C ρ (cid:63) ,ν (cid:63) .ii) If α (cid:54) = α , then T is C ρ,ν (cid:44) → C ρ (cid:63) ,ν (cid:63) .In other words, the transformation maps the 1-dimensional PQI ϕ ρ,ν ( ξ, χ ) ≥ to the 1-dimensionalPQI ϕ ρ (cid:63) ,ν (cid:63) ( ξ, χ ) ≥ . We wish to understand how to characterize all input-output transformations mapping an arbitrary dynam-ical system Σ to an input-output ( ρ (cid:63) , ν (cid:63) )-passive sys-tem. Namely, we assume that the given system is input-output ( ρ, ν )-passive (for some known ρ, ν ), and seek alltransformations that force the transformed system tobe input-output ( ρ (cid:63) , ν (cid:63) )-passive. We do so by finding alltransformations that map a given cone C ρ,ν into C ρ (cid:63) ,ν (cid:63) .Theorem 2.2 provides one way to build a map from anarbitrary cone into another arbitrary cone, but does notprescribe a general method to find all such maps. How-ever, we can use Theorem 2.2 to show that all mapsfrom an arbitrary cone into another arbitrary cone canbe built using maps from C , into itself. Proposition 3.1
Let ρ, ν, ρ (cid:63) , ν (cid:63) be any four numberssuch that ρν, ρ (cid:63) ν (cid:63) < / , and let T be any matrix C ρ,ν (cid:44) → C ρ (cid:63) ,ν (cid:63) . Let S ρ,ν , S ρ (cid:63) ,ν (cid:63) be the invertible matrices C , (cid:44) → C ρ,ν , C , (cid:44) → C ρ (cid:63) ,ν (cid:63) respectively, as built using Theorem2.2. Then there exists a matrix Q , which is C , (cid:44) → C , ,such that T = S ρ (cid:63) ,ν (cid:63) QS − ρ,ν holds. PROOF.
We note that Theorem 2.2 shows that S − ρ,ν , S − ρ (cid:63) ,ν (cid:63) map C ρ,ν and C ρ (cid:63) ,ν (cid:63) into C , , respectively.Define Q = S − ρ (cid:63) ,ν (cid:63) T S ρ,ν . Then Q is an invertible matrixas a product of invertible matrices. Moreover, it maps C , into itself as C , S ρ,ν (cid:44) → C ρ,ν T (cid:44) → C ρ (cid:63) ,ν (cid:63) S − ρ(cid:63),ν(cid:63) (cid:44) → C , . Proposition 3.1 gives a prescription for finding all matri-ces mapping C ρ,ν into C ρ (cid:63) ,ν (cid:63) . It contains two main in-gredients, namely the matrices S α,β , and matrices map-ping C , into itself. We therefore wish to understandboth better. We start by finding all matrices in GL ( R )mapping C , into itself: Proposition 3.2
A matrix T ∈ GL ( R ) sends C , intoitself if and only if all of the entries of T have the samesign, i.e. T ij T kl ≥ for every i, j, k, l ∈ { , } . PROOF.
We first show that if T sends C , into itself,then all of the entries of T = ( T ij ) i,j have the same sign.We recall that C , contains all points ( ξ, χ ) such that ξχ ≥
0, i.e., C , is a union of { } , the first quadrant,and the third quadrant. We note that e = (1 , (cid:62) and e = (0 , (cid:62) are in C , , hence T e (cid:62) = ( T , T ) (cid:62) and T e (cid:62) = ( T , T ) (cid:62) are also in C , . This implies that T , T have the same sign, and that T , T have thesame sign, and in each pair not both elements are zero(as T is invertible). We note that by switching between T and − T , we may assume without loss of generalitythat T , T are both non-negative. We want to showthat T , T are also both non-negative.Assume the contrary, that is, that T , T are both non-positive. Moreover, as T e , T e (cid:54) = 0, we conclude that T e lies in the first quadrant of R , and that T e liesin the third quadrant. We note that the line between e , e lies inside C , , so the same is true for the line be-tween T e , T e , as T is linear and maps C , into itself.However, as T e is in the first quadrant and T e is inthe third, the straight line between them passes eitherthrough zero, the second quadrant or the fourth quad-rant. The latter two cases are impossible, as C , con-tains no points from these quadrants, and the formercase is impossible as it would imply that the invertibletransformation T maps a non-zero point to zero. As wearrived at a contradiction, we conclude that all entriesof T have the same sign.Conversely, assume that all of the entries of T havethe same sign. By switching between − T and T , wecan assume without loss of generality that T ij ≥ i, j ∈ { , } , so that T e , T e are both in the firstquadrant. Take any point x ∈ C , . If x = 0 then T x = 0 ∈ C , . If x is in the first quadrant, then it isa linear combination of e , e with non-negative coeffi-cients, not both zero. Thus T x is a linear combination of
T e , T e with non-negative coefficients (not both zero),so because T e , T e are in the first quadrant, we con-clude the same for T x . If x is in the third quadrant, then − x is in the first quadrant, so T ( − x ) = − T x is in thefirst quadrant, hence
T x is in the third quadrant. As we showed that
T x ∈ C , for all x ∈ C , , this concludesthe proof. (cid:3) Remark 2
More generally, given some ρ, ν , one couldask for a characterization of all matrices T ∈ GL ( R ) that are C ρ,ν (cid:44) → C ρ,ν . Mimicking the proof above, one canshow that a map T ∈ GL ( R ) is C ρ,ν (cid:44) → C ρ,ν if and onlyif all of the elements of the matrix I e →B T I − e →B possessthe same sign, where B is composed of the non-colinearsolutions to the equation − νξ + ξχ − ρχ = 0 and e isthe standard basis. A more explicit form for the basis B can be achieved by taking the columns of the matrix S ρ,ν ,as seen in Proposition 3.3 below. We now clarify the second component appearing inProposition 3.1, namely the matrices S µ,τ : Proposition 3.3
Let µ, τ be any two numbers such that µτ < / . Recall that S µ,τ is a map C , (cid:44) → C µ,τ , asconstructed in Theorem 2.2.i) If τ < , we can choose S µ,τ = 12 τ (cid:34) − − √ − τ µ − √ − τ µ − τ τ (cid:35) . ii) If τ > ,, we can choose S µ,τ = 12 τ (cid:34) √ − τ µ − √ − τ µ τ τ (cid:35) . iii) If τ = 0 , we can choose S µ,τ = (cid:34) µ (cid:35) . PROOF.
We use Theorem 2.2 to build S µ,τ . As we con-sider a map C , (cid:44) → C µ,τ , we take ( ξ , χ ) = (1 ,
0) and( ξ , χ ) = (0 , ξ + ξ , χ + χ ) = (1 ,
1) satisfiesthe PQI ξχ ≥
0, we choose: S µ,τ = (cid:20) ξ ξ χ χ (cid:21) α = 1 (cid:20) ξ − ξ χ − χ (cid:21) α (cid:54) = 1 , where we recall that ( ξ , χ ) , ( ξ , χ ) are two non-colinear solutions to − τ ξ + ξχ − µχ = 0, and α = 1if and only if ( ξ + ξ , χ + χ ) satisfies the PQI − τ ξ + ξχ − µχ ≥ τ (cid:54) = 0. Then we can write − τ ξ + ξχ − µχ = 0 as − τ ( ξ − a χ )( ξ − a χ ) = 0, where a , a ig. 4. A general transformation mapping a SISO in-put-output ( ρ, ν )-passive system to a SISO input-output( ρ (cid:63) , ν (cid:63) )-passive system. The entries of the matrix Q ∈ R × all have the same sign. are given by a = − √ − τ µ − τ = 1 − √ − τ µ τa = − − √ − τ µ − τ = 1 + √ − τ µ τ , where we note that a (cid:54) = a as µτ < . Choose ( ξ , χ ) =( − a , − , ( ξ , χ ) = ( a , ξ + χ , ξ + χ ) =( a − a ,
0) satisfies the PQI − τ ξ + ξχ − µχ ≥ τ <
0, where we recall that we assumed τ (cid:54) = 0.We therefore conclude the desired result for τ (cid:54) = 0 fromTheorem 2.2.Suppose now that τ = 0. We note that ( ξ , χ ) = (1 , ξ , χ ) = ( µ,
1) are two non-colinear solutions to ξχ − µχ = 0, and that ( ξ + ξ , χ + χ ) = (1 + µ, − τ ξ + ξχ − µχ ≥
0, as ξχ − µχ = χ ( ξ − µχ ). This completes the proof of the proposition. (cid:3) We now conclude with the following theorem:
Theorem 3.1
Let Σ be a SISO input-output ( ρ, ν ) -passive system, and let T ∈ GL ( R ) be an invertiblematrix inducing an input-output transformation of theform (5) . The transformed system ˜Σ is input-output ( ρ (cid:63) , ν (cid:63) ) -passive if and only if there exists a matrix M ∈ GL ( R ) such that M ij ≥ for all i, j ∈ { , } andsome θ ∈ {± } such that T = θS ρ (cid:63) ,ν (cid:63) M S − ρ,ν , where thematrices S ρ,ν , S ρ (cid:63) ,ν (cid:63) are as given in Proposition 3.3. Inother words, the transformed system ˜Σ is input-output ( ρ (cid:63) , ν (cid:63) ) -passive if and only if the all of the entries of thematrix S − ρ (cid:63) ,ν (cid:63) T S ρ,ν have the same sign.
A block diagram visualizing Theorem 3.1 can be seen inFigure 4.
PROOF.
Proposition 3.1 implies that for an invert-ible matrix T ∈ GL ( R ), the transformed system ˜Σ isinput-output ( ρ (cid:63) , ν (cid:63) )-passive if and only if there exists an invertible matrix Q ∈ GL ( R ) which is C , (cid:44) → C , such that T = S ρ (cid:63) ,ν (cid:63) QS − ρ,ν . By Proposition 3.2, a ma-trix Q is C , (cid:44) → C , if and only if all of its entries pos-sess the same sign. By letting θ ∈ {± } be that sign,we can write any matrix Q sending C , into itself as Q = θM , where M ∈ GL ( R ) and M ij ≥ i, j ∈ { , } . Thus, the transformed system ˜Σ is input-output ( ρ (cid:63) , ν (cid:63) )-passive if and only if there exists some θ ∈ {± } and M ∈ GL ( R ) with non-negative entriessuch that T = θS ρ (cid:63) ,ν (cid:63) M S − ρ,ν . (cid:3) Up to now, we gave an explicit description of all input-output transformations mapping input-output ( ρ, ν )-passive
SISO systems to input-output ( ρ (cid:63) , ν (cid:63) )-passive SISO systems. One could try and generalize this ideato MIMO systems, but a few problems arise. The cor-nerstone in the characterization for SISO systems wasTheorem 2.2, whose proof uses the fact that for SISOsystems, the solution sets of PQIs are two-dimensional,and their boundary is the union of two straight lines[Sharf et al. (2019)]. For d × d MIMO systems, thesolution set of a PQI lies in R d , and its boundary, ingeneral, is of dimension 2 d − d × d invert-ible linear transformations, GL d ( R ), on the collectionof d -dimensional PQIs. As before, we use the notion ofsolution sets. We recall that we denoted the solution setof the d -dimensional PQI ϕ ρ,ν ( ξ, χ ) ≥ C ρ,ν,d . Asbefore, T maps one d -dimensional PQI to another if andonly if it maps the associated solution sets to the an-other. We start with the following proposition: Proposition 4.1
Let ρ, ν, ρ (cid:63) , ν (cid:63) be any real num-bers, and let S ∈ GL ( R ) be any matrix mapping the1-dimensional PQI ≤ − νξ + ξχ − ρχ to the 1-dimensional PQI ≤ − ν (cid:63) ξ + ξχ − ρ (cid:63) χ . Then S ⊗ Id d maps the d -dimensional PQI − ν (cid:107) ξ (cid:107) + ξ (cid:62) χ − ρ (cid:107) χ (cid:107) tothe d -dimensional PQI − ν (cid:63) (cid:107) ξ (cid:107) + ξ (cid:62) χ − ρ (cid:63) (cid:107) χ (cid:107) . Thus, the MIMO analogue of the transformations S ρ,ν are S ρ,ν ⊗ Id d . We now prove the proposition. PROOF.
We define A = (cid:104) − ρ − ν (cid:105) , B = (cid:104) − ρ (cid:63) − ν (cid:63) (cid:105) . The 1-dimensional PQI 0 ≤ − ν (cid:63) ξ + ξχ − ρ (cid:63) χ can bewritten as Ξ (cid:62) A Ξ ≥
0, where Ξ = [ χ, ξ ] (cid:62) ∈ R , and the1-dimensional PQI 0 ≤ − ν (cid:63) ξ + ξχ − ρ (cid:63) χ is writtenas Ξ (cid:62) B Ξ ≥
0. By setting ˜Ξ = S Ξ, we see that S mapsthe first 1-dimensional PQI to the second if and only if6 S − ) (cid:62) AS − = B , and the latter condition implies(( S ⊗ Id d ) − ) (cid:62) ( A ⊗ Id d )( S ⊗ Id d ) − = B ⊗ Id d . The proof in now complete, where we note that the d -dimensional PQIs can be written as Ξ (cid:62) d ( A ⊗ Id d )Ξ d ≥ (cid:62) d ( B ⊗ Id d )Ξ d ≥
0, where Ξ d = [ ξ (cid:62) , χ (cid:62) ] (cid:62) ∈ R d . (cid:3) Remark 3
Proposition 4.1 does not claim that all mapsbetween d -dimensional PQIs stem from maps between -dimensional PQIs. We now search for a MIMO analogue for the second com-ponent we had, namely non-negative matrices. Before,non-negative matrices stemmed from maps C , (cid:44) → C , . Proposition 4.2
An invertible matrix T ∈ GL d ( R ) maps C , ,d into itself if and only if there exists some λ > such that T (cid:62) JT − λJ ≥ , where J = (cid:104) Id d Id d (cid:105) . PROOF.
As before, we denote the stacked variable vec-tor as Ξ d = [ ξ (cid:62) , χ (cid:62) ] (cid:62) ∈ R d . The set C , ,d is the col-lection of all vectors Ξ d satisfying Ξ (cid:62) d J Ξ d ≥
0. The im-age of C , ,d under T consists of all vectors ˜Ξ d such that˜Ξ (cid:62) d ( T − ) (cid:62) JT − ˜Ξ d ≥
0. Thus, T maps C , ,d inside it-self if and only if the following implication holds:˜Ξ (cid:62) d ( T − ) (cid:62) JT − ˜Ξ d ≥ ⇒ ˜Ξ (cid:62) d J ˜Ξ d ≥ , ∀ ˜Ξ d ∈ R d . By the S-lemma, or S-procedure, [(Boyd and Vanden-berghe, 2004, Appendix B)], the above implication isequivalent to the existence of some µ > T − ) (cid:62) JT − − µJ ≤
0. By multiplying the inequality by T (cid:62) on the left and by µ − T on the right, the inequality isequivalent to T (cid:62) JT − λJ ≥
0, where λ = µ − > (cid:3) Remark 4
The inequality T (cid:62) JT − λJ ≥ can be seenas a certain generalized version of an algebraic Riccatiinequality. Indeed, the algebraic Riccati equation is givenby A (cid:62) P + P A − P XP + Q = 0 , where X, Q are positive-definite matrices, and P is a symmetric matrix vari-able [Doyle et al. (1989)]. The corresponding inequality, A (cid:62) P + P A − P XP + Q ≤ , has also been consideredin literature [Willems (1971)]. Choosing Q = λJ, A = 0 and X = J , and not restricting the matrix P to be sym-metric, results in the inequality P (cid:62) JP − λJ ≥ . As Q, J we chose are not positive definite, this is a general-ized version of an algebraic Riccati inequality.
Combining Propositions 4.1 and 4.2, we conclude withthe following theorem: We have to use P (cid:62) XP instead of P XP to guarantee thatthe matrix is symmetric.
Theorem 4.1
Let Σ be an input-output ( ρ, ν ) -passivesystem with input and output dimension equal to d , andlet T ∈ GL d ( R ) be an invertible matrix inducing aninput-output transformation of the form (5) . The trans-formed system ˜Σ is input-output ( ρ (cid:63) , ν (cid:63) ) -passive if andonly if there exists a matrix M ∈ GL d ( R ) and somepositive λ > such that: T = ( S ρ (cid:63) ,ν (cid:63) ⊗ Id d ) M ( S − ρ,ν ⊗ Id d ) , M (cid:62) JM − λJ ≥ , where J = (cid:104) Id d Id d (cid:105) . , i.e., ˜Σ is input-output ( ρ (cid:63) , ν (cid:63) ) -passive if and only if there exists λ > such that X =( S − ρ (cid:63) ,ν (cid:63) ⊗ Id d ) T ( S ρ,ν ⊗ Id d ) satisfies X (cid:62) JX − λJ ≥ . PROOF.
Write X = (cid:2) a bc d (cid:3) . The matrix X (cid:62) JX − λJ can be computed as: X (cid:62) JX − λJ = 12 (cid:34) ac ad + bc − λad + bc − λ bd (cid:35) . By Sylvester’s criterion, X (cid:62) JX − λJ is positive semi-definite if and only if all of its principal minors are non-negative, i.e. ac ≥ bd ≥ X (cid:62) JX − λJ ) ≥ a, c possess the same sign, and the same holds for b, d . Byswitching between X, − X , we may assume without lossof generality that a, c are non-negative. If b, d are alsonon-negative, the proof is complete. Thus, it’s enoughto show that if b, d are non-positive (and not both zero),then for any λ >
0, 4 det( X (cid:62) JX − λJ ) <
0. By defini-tion, we have:4 det( X (cid:62) JX − λJ ) = − ( ad + bc − λ ) + 4 abcd Moreover, if b, d are non-positive then abcd ≤
0. If abcd <
0, then 4 det( X (cid:62) JX − λJ ) must be nega-tive. Otherwise, the determinant can be non-positiveonly at λ = ad + bc , but because a, c ≥
0, if b, d ≤ ad + bc is non-positive. In particular, if b, d ≤ λ >
0. Thus,det( X (cid:62) JX − λJ ) ≥ a, b, c, d ≥
0. Thisconcludes the proof. (cid:3)
The theorem can be seen as a generalization of Theorem3.1, as one can verify that for d = 1, X (cid:62) JX − λJ ≥ λ > X -s entries possessthe same sign. Indeed, Proposition 4.3
Let X ∈ GL ( R ) , and let J = (cid:104) (cid:105) .There exists some λ > such that X (cid:62) JX − λJ ≥ ifand only if all of the entries of X possess the same sign. ROOF.
Write X = (cid:2) a bc d (cid:3) . The matrix X (cid:62) JX − λJ can be computed as: X (cid:62) JX − λJ = 0 . (cid:2) ac ad + bc − λad + bc − λ bd (cid:3) . By Sylvester’s criterion, X (cid:62) JX − λJ is positive semi-definite if and only if all of its principal minors are non-negative, i.e. ac ≥ bd ≥ X (cid:62) JX − λJ ) ≥ a, c possess the same sign, and the same holds for b, d . Byswitching between X, − X , we may assume without lossof generality that a, c are non-negative. If b, d are alsonon-negative, the proof is complete. Thus, it’s enoughto show that if b, d are non-positive (and not both zero),then for any λ >
0, 4 det( X (cid:62) JX − λJ ) <
0. By defini-tion, we have,4 det( X (cid:62) JX − λJ ) = − ( ad + bc − λ ) + 4 abcd. Moreover, if b, d are non-positive then abcd ≤
0. If abcd <
0, then 4 det( X (cid:62) JX − λJ ) must be nega-tive. Otherwise, the determinant can be non-positiveonly at λ = ad + bc , but because a, c ≥
0, if b, d ≤ ad + bc is non-positive. In particular, if b, d ≤ λ >
0. Thus,det( X (cid:62) JX − λJ ) ≥ a, b, c, d ≥
0. Thisconcludes the proof. (cid:3)
In this section, we consider possible applications of theachieved characterization for synthesis. We explore threepossible applications, including multi-purpose transfor-mations, passivation with respect to multiple equilibria,and optimal passifying transformations.
We consider the following problem of a “one size fits all”transformation. We consider different systems { Σ i } i ∈ I which are input-output ( ρ i , ν i )-passive. Our goal is todesign a transformation T mapping each system Σ i to aninput-output ( ρ (cid:63)i , ν (cid:63)i )-passive system (for i ∈ I ). The set I is a set of indices which can be either finite or infinite.This problem arises in two real-world occasions. First,we can consider a problem where we need to design acontroller which stabilizes a collection of (possibly non-linear) plants. The specific system at hand is unknownuntil the moment the controller is connected. One cancompute dissipation inequalities for all possible plants,find a transformation which passivizes all of them, andthen implement any negative output feedback, whichwill stabilize them. Second, we consider the problem ofa system which can fault. We want to find a transforma-tion which makes the faultless system as strictly passiveas possible (to improve the convergence rate when con-nected with a constant gain output feedback), but also passivizes any faulty version of the system (to make sureit stays stable when connected with a constant gain out-put feedback). Proposition 5.1
Consider the MIMO systems { Σ } i ∈ I with input- and output-dimension equal to d , which areinput-output ( ρ i , ν i ) -passive. Let ( ρ (cid:63)i , ν (cid:63)i ) be real num-bers such that for each i , ρ (cid:63)i , ν (cid:63)i < / . Consider a gen-eral input-output transformation T of the form (5) . Thetransformed systems { ˜Σ } i ∈ I are input-output ( ρ (cid:63)i , ν (cid:63)i ) -passive for all i , if and only if there exists matrices T i ∈ GL d ( R ) , and numbers λ i ≥ such that the following setof constraints holds: (cid:26) T = ( S ρ (cid:63)i ,ν (cid:63)i ⊗ Id d ) T i ( S − ρ i ,ν i ⊗ Id d ) T (cid:62) i JT i − λ i J ≥ . (8)The proof of the proposition follows immediately fromProposition 4.1. We note that when we wish to passivizethe system Σ with respect to all equilibria (i.e., ρ (cid:63)i , ν (cid:63)i =0), we get the following set of equations and inequalities: (cid:26) T = T i ( S − ρ i ,ν i ⊗ Id d ) T (cid:62) i JT i − λ i J ≥ . (9) Remark 5.1
If the set I is finite, or that the set { ( ρ i , ν i , ρ (cid:63)i , ν (cid:63)i ) } i ∈ I contains only finitely many distinctelements, one can find a solution to (8) by stating anoptimization problem with an arbitrary cost functionand constraints of the form (8) . If the cost function ischosen as a coercive function, one can show that theproblem has a solution whenever it is feasible, as the setof constraints is closed. Thus, a solution can be foundusing optimization software. See Section 5.3 for more onoptimization problems and passivizing transformations. Example 5.1
Consider the SISO system Σ which is theparallel interconnection of two linear and time-invariantSISO systems Σ , Σ given by the transfer functions G ( s ) = s +1) and G ( s ) = s +6 s +5 s +1 s +5 s +9 s +7 s +2 . The sys-tem Σ is linear and time-invariant, and its transfer func-tion can be computed to be G ( s ) = s +3 s +3 s +2 = s +2 + s +1 .It is easy to verify that Σ is passive, and actually output-strictly passive with a parameter ρ = . However,the component corresponding to Σ is unreliable, andmay fault. When it does, the transfer function changesto G ( s ) , which is not passive as it is non-minimumphase. However, it does have a finite L -gain equal to max Re( s ) > (cid:107) G ( s ) (cid:107) = 1 . It is shown in (Sharf et al.,2019, Section VI) that a system with a finite L -gainequal to β is input ν -passive for ν = − β − . . Thus,the faulty system is input ( ν ) -passive for ν = − . . Weget that the faultless system is output -passive, and thefaulty system is input ( − . -passive.Suppose we want to find a transformation T that mapsthe faultless system to an output -passive system, and he faulty system to a passive system. By (8) , if we define T = S − , T S , and T = S − , T S , − , then we want allof the entries of both T and T have the same sign. Wechoose T = [ . . ] . A simple computation shows that: T = S − , T S , = (cid:34) . .
215 1915 (cid:35) T = S − , T S , − = (cid:34) . .
12 1 . (cid:35) , meaning both T , T have entries which have the samesign. Thus, the map T satisfies the requirements we es-tablished, which is a fact we now verify independentlyof the computation above. First, we note that the map T can be written as the product [ . ] [ ] , meaningit operates in the following way - it first implementsa constant output feedback with gain equal to , andthen implements a constant input feed-through with gain . . Thus, it transforms the faultless transfer function G to ˜ G ( s ) G +2 + 0 . . s +2 . s +3 . s +7 s +8 , and simultaneouslytransforms the fault transfer function G ( s ) to ˜ G ( s ) = G +2 +0 . . s +0 . s +1 . s +2 s +3 . One could easily check (e.g.,using the MATLAB command “getPassiveIndex”) that ˜ G ( s ) is output-strictly passive with index ρ ≈ . , andthat ˜ G ( s ) is passive, and actually output-strictly passivewith index ρ ≈ . . Thus, the transformation T mapsthe faultless system to an output -passive system (andactually to an output . -passive system). Moreover,it maps the faulty system to a passive system (and actu-ally an output . -passive system), as required.5.2 Passivation with Respect to Multiple Equilibria andEquilibrium-independent Passivity In several occasions, one wishes to study the behavior ofa system around more than one equilibrium. One exam-ple includes consumer products that have multiple oper-ation settings e.g., a food processor with a low, middle,and high options, or a refrigerator with multiple cool-ing levels possible. Another example consists of systemswhich need to operate under a wide range of inputs, e.g.egg-sorting machines which need to lift eggs of differentsizes without breaking them, or warehouse robots whichneed to manipulate goods or crates of different shapesand sizes without breaking them or dropping them. Athird example includes multi-agent networks, for whichthe steady-state limit can be hard to impossible to guessbefore running the network due to the agents having dif-ferent models, goals and restrictions.In this direction, one can consider passivity (or short-age thereof) with respect to an arbitrary steady-stateinput-output pair (u , y). Indeed, the notions of output ρ -passivity, input ν -passivity and input-output ( ρ, ν )-passivity can be extended to other steady-states by re-placing y ( t ) by y ( t ) − y and u ( t ) by u ( t ) − u in (2),(3), and (4) respectively. When designing controllersfor systems which can operate around more than oneequilibrium, we need to consider passivity (or input-output ( ρ, ν )-passivity) with respect to each equilibrium.The same system can behave differently around differ-ent equilibria. e.g. the static nonlinearity y = uu +1 ispassive around the steady-state input-output pair (0 , , . ρ, ν such that the system is input-output ( ρ, ν )-passive with respect to all equilibria, and in that casewe can use the method of Sharf et al. (2019) to passivizethe system with respect to all equilibria. However, wecan consider a more general case, where different equilib-ria will have different corresponding dissipation inequal-ities.Before moving forward, we note that the inequalitiesdefining input-output ( ρ, ν )-passivity with respect toany steady-state can also be written as ( d -dimensional)PQIs in exactly the same way used for input-output( ρ, ν )-passivity with respect to the origin. Thus, our re-sults characterize all transformations T that passivize aplant with respect to any fixed equilibrium. In that direc-tion, we consider a system Σ and a collection of steady-state input-output pairs { (u i , y i ) } . Our goal is to find atransformation T that passivizes the system Σ with re-spect to all (transformed) steady-state pairs simultane-ously. Remark 5.2
Intuitively, one might try to keep the samesteady-state input-output pairs for the transformed sys-tem. However, this might be impossible if the transformedsystem must be passive. For example, if we have a SISOsystem Σ with two steady-state input-output pairs (0 , and (1 , − , then the system cannot be passive, as thecorresponding steady-state relation is non-monotone. Unsurprisingly, this problem is very similar to the mul-tiple objective transformation considered in Section 5.1.We can prove the following proposition:
Proposition 5.2
Consider a MIMO system Σ withinput- and output-dimension equal to d . Let { (u i , y i ) } be a (possibly infinite) collection of steady-state input-output pairs of Σ , and let ( ρ i , ν i ) , ( ρ (cid:63)i , ν (cid:63)i ) be real numberssuch that for each i , ρ i ν i , ρ (cid:63)i , ν (cid:63)i < / . Suppose thatfor each i , the system Σ is input-output ( ρ i , ν i ) -passivewith respect to the steady-state input-output pair (u i , y i ) . onsider a general input-output transformation T ofthe form (5) , and consider the new system ˜Σ and thenew steady-state pairs { T (u i , y i ) } . ˜Σ is input-output ( ρ (cid:63)i , ν (cid:63)i ) -passive with respect to T (u i , y i ) , for all i , if andonly if there exists matrices T i ∈ GL d ( R ) , and numbers λ i ≥ such that the following set of constraints holds: (cid:26) T = ( S ρ (cid:63)i ,ν (cid:63)i ⊗ Id d ) T i ( S − ρ i ,ν i ⊗ Id d ) T (cid:62) i JT i − λ i J ≥ . (10)As before, the proof of the proposition follows immedi-ately from Proposition 4.1. We again note that when wewish to passivize the system Σ with respect to all equi-libria (i.e., ρ (cid:63)i , ν (cid:63)i = 0), we get the following set of equa-tions and inequalities: (cid:26) T = T i ( S − ρ i ,ν i ⊗ Id d ) T (cid:62) i JT i − λ i J ≥ . In the previous sections of the paper, we characterized alltransformations that passivize a given system Σ. Thus,it is natural to ask questions such as “which passivizingtransformation minimizes (or maximizes) a given quan-tity?” One class of quantities of interest can either besystem-theoretic properties of the transformed system,e.g. the L -gain or tracking error for a given input anda desired output. Another class of interesting quantitiesto optimize consists of properties of T . These include,for example, the distance of the transformation T froma nominal transformation T , e.g. the identity.Generally, one could consider a transformation T thatmaps a given input-output ( ρ, ν )-system to an input-output ( ρ (cid:63) , ν (cid:63) ) system. The quantity we wish to mini-mize can be written as a function Φ( T ) of T . The asso-ciated optimization problem reads:min T Φ( T )s . t . T maps an input-output ( ρ, ν ) systemto an input-output ( ρ (cid:63) , ν (cid:63) )-system.One could use Theorem 4.1 to restate the optimizationproblem in a tractable form:min T,λ,M Φ( T ) (11)s . t . M = ( S ρ (cid:63) ,ν (cid:63) ⊗ Id d ) − T ( S ρ,ν ⊗ Id d ) (12) M (cid:62) JM − λJ ≥ λ ≥ , (14) where J = (cid:104) / / (cid:105) . This optimization problem can beeasily defined for any cost function Φ, whether it is ex-plicitly defined using a formula involving T , or implicitlydefined by a characteristic of the transformed system ˜Σ.However, solving the optimization problem can be hard.First, the function Φ might not be explicitly given, ornon-convex. Second, even if the function Φ was convex,the constraint M (cid:62) JM − λJ ≥ J is not positive semi-definite. We should note,however, that the latter problem can be easily reme-died for SISO systems. Indeed, by Proposition 4.3, theconstraint can be replaced by the following pair of con-straints: i) T = (cid:34) a bc d (cid:35) ii) a, b, c, d have the same sign . This constraint is still non-convex, but can be convex-ified by separating the problem into two sub-problems,one with the constraint a, b, c, d ≥
0, and one with theconstraint a, b, c, d ≤ Remark 5.3
Returning to the MIMO case, one canprove that the matrix M = (cid:104) α Id d β Id d γ Id d δ Id d (cid:105) ∈ R d × d sat-isfies the inequality M (cid:62) JM − λJ ≥ for some λ > whenever α, β, γ, δ have the same sign, Similarly toProposition 4.3. Thus, one can consider a tractable re-laxation of the optimization problem (11) by similarlyreplacing the constraints M (cid:62) JM − λJ ≥ , λ ≥ by theconstraint M = (cid:104) α Id d β Id d γ Id d δ Id d (cid:105) ∈ R d × d and demandingthat α, β, γ, δ have the same sign. We now give examples of two tractable optimizationproblems:
Example 5.2
Suppose we wish to choose a passivizingtransformation T in order to minimize the L -gain ofthe transformed system. Taking any passivizing trans-formation T , mapping Σ to ˜Σ and any number k > ,and define the transformation T = [ k ] T . It’s easy tosee that T maps the system Σ to the system ˜Σ k , whichcorresponds to applying the transformation T , and thencascading the transformed system with a gain of size k .Thus, the L -gain of ˜Σ k is equal to k times the L -gain of ˜Σ . Thus, by taking k → , we can force the transformedsystem to have an arbitrarily small L -gain. Example 5.3
Consider the problem of transforming aninput-output ( ρ, ν ) -passive system with an input-output ( ρ (cid:63) , ν (cid:63) ) -passive system. We wish to find such a trans-formation which is closest to a given transformation T ,i.e., minimizes the operator norm (cid:107) T − T (cid:107) . By the dis- ussion above, we can write the problem as: min T,M,λ (cid:107) T − T (cid:107) s . t . M = ( S ρ (cid:63) ,ν (cid:63) ⊗ Id d ) − T ( S ρ,ν ⊗ Id d ) M (cid:62) JM − λJ ≥ λ ≥ However, minimizing (cid:107) T − T (cid:107) directly is hard. Instead,we introduce a new variable γ and demand that ( T − T )( T − T ) (cid:62) ≤ γ Id , so that minimizing on γ gives thedesired result (and the operator norm (cid:107) T − T (cid:107) is givenby √ γ . One can rewrite the last inequality as a linearmatrix inequality using the Schur complement, giving thefollowing equivalent optimization problem: min T,M,λ,γ γ (15)s . t . M = ( S ρ (cid:63) ,ν (cid:63) ⊗ Id d ) − T ( S ρ,ν ⊗ Id d ) M (cid:62) JM − λJ ≥ (cid:34) Id T − T T (cid:62) − T (cid:62) γ Id (cid:35) ≥ λ ≥ . As a concrete example, we take ( ρ, ν ) = (0 , − and ( ρ (cid:63) , ν s tar ) = (1 , . We thus seek a transformation ofthe form (5) mapping an input passive-short SISO sys-tem with parameter ν = − to an output strictly-passivesystem with ρ = 1 . Classically, one would first use feed-through to passivize the system, and then implement afeedback to increase its output passivity. This results inthe transformation T = [ ] [ ] = [ ] . We wish tofind such a transformation which is closest to the iden-tity transformation, i.e., minimizes the operator norm (cid:107) T − Id (cid:107) . Using Proposition 4.3, we can rewrite theproblem (15) as: min T,γ,a,b,c,d γ s . t . a, b, c, d have the same sign T = (cid:34) (cid:35) (cid:34) a bc d (cid:35) (cid:34) (cid:35) , (cid:34) Id T − Id T (cid:62) − Id γ Id (cid:35) ≥ . This problem is non-convex, but it can be written as theminimum of cone programming problems, one with theconstraint a, b, c, d ≥ , and one with the constraints a, b, c, d ≤ . We can thus solve the problem by computerand find the optimal transformation T = [ . . . . ] , whichcorresponds to a = 1 , b = c = 0 , d = 0 . . The operatornorm in this case is √ γ = √ ≈ . . As a comparison, the operator norm (cid:107) T − I (cid:107) for T = [ ] is √ √ √ ≈ . . The paper considers the notion of ( ρ, ν )-passivity, whichcontains both shortage and excess of passivity. We char-acterized all input-output transformations mapping aninput-output ( ρ, ν )-passive system to an input-output( ρ (cid:63) , ν (cid:63) )-passive system. Starting with the SISO case, weused the geometric approach of Sharf et al. (2019) toconvert the problem into characterizing all linear trans-formations that map a given symmetric double-cone toa desired symmetric double-cone. We then solved thelatter problem by studying the action of the collectionof invertible 2 × GL ( R ), on the collectionof symmetric double-cones. This culminated in a resultshowing that any input-output transformation mappingan input-output ( ρ, ν )-passive system to an input-output( ρ (cid:63) , ν (cid:63) )-passive system can be written (up to a sign) asthe product of three matrices, S ρ,ν , S − ρ (cid:63) ,ν (cid:63) and a non-negative matrix. We then shifted our focus to the MIMOcase, where we showed a similar result, in which the non-negative matrix is replaced by a matrix satisfying a cer-tain generalized version of an algebraic Riccati inequal-ity. We presented three possible applications our results.Future work can try and better characterize the collec-tion of matrices satisfying the generalized algebraic Ric-cati inequality, giving a more explicit characterizationfor the MIMO case. Another avenue for future researchcan use the achieved parameterization to study variousoptimization problems, similar to the one discussed inSection 5. References
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