Model Matching and Passivation of MIMO Linear Systems via Dynamic Output Feedback and Feedforward

A model matching and passivating control architecture for multi-input/multi-output linear systems, comprising dynamic feedback and feedforward, is proposed. The approach—essentially without any restriction on the relative degree and the zeros of the underlying system and by relying only on input/output measurements—provides a closed-loop system, the transfer matrix of which matches any desired matrix of rational functions. An alternative implementation of the above design allows to achieve an arbitrary approximation accuracy of a desired transfer matrix while also preserving structural properties—in particular observability—of the overall interconnected system. Such a construction can be then specialized to provide input/output decoupling or a system that is passive from a novel control input to a modified output. The result is achieved by arbitrarily assigning the relative degree and location of the poles and zeros on the complex plane of the interconnected system in a systematic way. It is also shown that similar ideas can be employed to enforce a desired, arbitrarily small, $\mathcal {L}_2$-gain from an unknown disturbance input to a modified output, while preserving the corresponding gain from the control input to the same output. The article is concluded with applications and further discussions on the results.

gradually gained a central role in control theory. In this respect, the ability of assessing, or even imposing, desirable energetic properties for each node of the overall network is of paramount importance. Such properties are typically provided in terms of dissipation requirements for the input/output behavior of each subsystem, thus leading to the concept of passivity or more general dissipativity notions, such as finite L 2 -gain from a specific input to the interconnecting output; (see, e.g., [38] for a comprehensive discussion on the topic).
Such properties, in addition to the input/output energy characterization, are rendered even more desirable by their intrinsic relation with (asymptotic) stability-see e.g. [7], [25], [28], [34] for applications to the stabilization task for composite or interconnected systems-and robustness [14]. As a result, intensive research effort has been devoted to envision control architectures that enforce passivity properties of the resulting closed-loop system (see [1], [3], [6], [10], [12], [15], [21], [23], [26], [27], [41], [44] and references therein) in those cases in which they are not originally possessed by the underlying system. Interestingly, in the case of linear time-invariant systems, these concepts are intimately related to the notion of positive real (PR) or strictly positive real (SPR) systems, which has been deeply explored not only in control but also in circuit theory [2], [31], [32], [40]. In [18], a generalization of the feedback positive real (FPR) design, introduced in [25], is discussed and subsequently extended to the setting of switching systems in [19]. The construction relies upon an observer-based state feedback control and on the circle criterion, thus increasing the realm of applicability of the solution proposed in [30], which yields a synthesis strategy in terms of a state feedback under the assumption that the underlying system is minimum-phase and possesses relative degree one.
Several alternative techniques, such as series, feedback, or feedforward interconnections, are discussed in [21]-with the aim of extending passivity-based techniques to systems that are not passive-for which an LMI characterization is given in [22]. In particular, the control architectures discussed in [1], [5], [13], [15], [27], [37] rely only on (state or output) feedback and, as a consequence, allow to impose passivity for specially structured classes of systems, while, on the other hand, preserving the original output function as a desirable side effect. In fact, for linear time invariant systems, passivity requires the properties of minimum-phaseness and relative degree one, which are not affected by feedback alone, and hence the role of feedforward action has been immediately recognized and emphasized (see [3], [7] and [4]) in the context of adaptive control. Therefore, a passivating technique consisting of static feedback/feedforward actions is proposed in [41], while [10], based on the characterization of SPR systems in terms of the Kalman-Yakubovich-Popov lemma [9], [33], envisions the use of observers, thus leading to a dynamic, rather than static, construction and yielding an LMI-based approach for passivation of multi-input/multi-output (MIMO) systems. The design introduced in [17], instead, is based on reduced-order observers for the original linear system. A suitably designed input-output transformation matrix is proposed in [43]: this allows to guarantee positive passivity indices. Similar ideas are employed in [45] in the case of event-triggered feedback systems, whereas the main objective of [42] consists in assessing passivity properties of a certain system based only on approximate models of the plant itself.
Differently from the former constructions hitherto based on feedforward actions, herein, a dynamic, instead of static, control scheme is proposed, which permits, differently from the methods available in the literature, the closed-form and exact placement of the resulting closed-loop zeros on the complex plane. Finally, it is worth mentioning that existing techniques based on feedforward compensation (see e.g. [10], [24]) typically do not allow to systematically preserve observability of the overall extended system comprising the original plant and the observer, and the designer may wonder how much information on the plant is retained from the auxiliary output or if the resulting passivity property is essentially possessed only by the dynamic (feedforward) compensator itself. This crucial aspect is discussed, and tackled, in detail in the following sections.
The main contribution of the article consists in the definition of a control architecture-comprising dynamic feedback as well as dynamic feedforward-that allows to enforce desired dissipativity properties to the closed-loop system by relying only on output measurement. More precisely, the main aspect of interest of the methodology is that such an approach, despite its intrinsic simplicity-in terms of design procedure and use of available degrees of freedom-allows to achieve, essentially without any restriction, desired passivity (relative degree, minimum phaseness) and more general dissipativity (e.g., L 2 -gain) properties. Moreover, such objectives are obtained in a systematic, though flexible, manner. A preliminary version of the results discussed here has appeared in [35]. With respect to [35], here we provide the extension to MIMO plants, together with several additional results-concerning, for instance, the ability of preserving structural properties in the closed-loop system as well as the (nongeneric) case of plants that are not controllable-and the proof of the main results. Moreover, a more general result concerning the adaptive control of nonminimum phase systems is considered.
The rest of the article is organized as follows. Useful notation and preliminary results on matrix transfer functions are recalled in Section II, while the problem under investigation is introduced and discussed in Section III, together with notation and the main standing assumptions. The results are specialized to the case of single-input/single-output (SISO) systems in Section IV, providing additional interesting insights on the construction. The application of the design methodology to the adaptive control problem for nonminimum phase systems is the topic of Section V. Finally, conclusions are drawn in Section VI.

II. NOTATION AND PRELIMINARIES
Consider a MIMO matrix transfer function with real coefficients described by with s ∈ C, W (s) ∈ C p×m and where d(s) = s n + a 1 s n−1 + · · · + a n−1 s + a n denotes the least common multiple of the denominators of the entries of the transfer matrix W (s), hence describing the minimal polynomial, and N (s) = N 1 s n−1 + N 2 s n−2 + · · · + N n−1 s + N n , with N i ∈ R p×m , for i = 1, . . . , n. Clearly, n denotes the highest degree of the polynominal d(s), whereas p and m represent the number of outputs and inputs, respectively. Note that, differently from the previous definition, in [8,Definition 7.1], the degree of the transfer matrix is defined in terms of the least common denominators of all minors of W (s), and hence associated to the characteristic polynomial. Remark 1: It may be possible to encompass also nonstrictly proper functions in (1), i.e., in the presence of a feed-through term. This can be achieved with minor modifications of the following discussions, which however unnecessarily increase the notational burden, and hence are omitted (see also Example 4 in Section IV).
It is well known that MIMO systems possess in general minimal realizations, namely controllable and observable state-space descriptions, with state dimension greater than the degree n of d(s) (see, e.g., [20,Chapter 6]). Similarly to SISO systems, several alternative state-space representations of minimal dimension can be considered. Among such descriptions, one of particular interest is the so-called Gilbert's Diagonal Realization [20], obtained for systems in which the denominator possesses distinct roots, namely such that d(s) = Π n i=1 (s − λ i ), with λ i = λ j for all i = j. To this end, consider the expansion of W (s) into partial fractions and let i denote the rank of the residual R i of dimension p × m. Then, n min = n i=1 i and a minimal realization, of order 1 + · · · + n n, is obtained by considering the matrices 1 1 Note that, in general, (3) may provide a state-space complex realization. Nonetheless, since the coefficients of W (s) are real, if there is λ i ∈ C, then there must be λ j ∈ C such that λ j = λ i , namely the complex-conjugate of λ i . As a consequence, also the corresponding individual realizations are complexconjugate and an equivalent real realization that preserves the parallel blockdiagonal structure of (3) can be derived by combining, with standard techniques, the realizations of λ i and λ j . and C = [ C 1 C 2 · · · C n ], where the matrices B i and C i , of dimensions i × m and p × i , respectively, are such that Remark 2: Since the set of matrices that have n distinct eigenvalues is open and dense in the set of square matrices of dimension n, it follows that the dimension of the minimal realization n min is generically-namely with probability one for randomly selected problem data-equal to n i=1 i , with the i 's defined above. Moreover, since also the residual matrices R i are generically full-rank, given a randomly selected transfer matrix W (s) as in (1), it follows that, generically, n min = min{np, nm}.
As a consequence of the discussions in Remark 2 and in the case m p (namely systems possessing at least the same number of control inputs as outputs), which is a scenario of specific interest for the model matching and passivation problems dealt with below, it follows that also the so-called Block Observer Form, of dimension np, is, generically, a minimal realization of a given transfer matrix W (s). More precisely, such state-space description of the transfer matrix (1) is given by an LTI system defined by the equations where a = [a 1 , . . . , a n ] ∈ R n , x(t) ∈ R np , with ⊗ denoting the Kronecker product between two matrices, and the matrices A o , B o , and C o are defined by and C o = [I p 0 · · · 0], where the coefficients a i and the matrices N i have been introduced in the discussion below (1). In the following sections, σ(A) denotes the spectrum of the matrix A ∈ R n×n .

III. MODEL MATCHING, INPUT/OUTPUT DECOUPLING AND PASSIVATION
In this section, we discuss the main results concerning model matching and passivation of LTI MIMO systems by output feedback and feedforward on the measured output. The objective is initially achieved under generic assumptions, namely for almost all transfer matrices, whereas specialized cases are discussed in what follows. For clarity, in the case in which the plant Σ p possesses at least the same number of inputs as outputs, such assumptions are explicitly stated, despite the fact that, by the arguments in Remark 2 and the discussions therein, they hold generically.
Assumption 1: The state-space realization Σ p of the transfer matrix W (s) in (1) is minimal, hence Σ p is controllable, as well as observable by construction.
• The control objectives are formalized in the following statement.
Problem 1: Consider a matrix transfer function W (s) as in (1) and suppose that Assumption 1 holds. Find, if possible, an integer ν 0 and a dynamic compensator Σ c described in statespace form by the equationṡ ∈ R m and with matrices of appropriate dimensions, such that the interconnected system Σ e obtained settingŷ = y + y ξ (see Fig. 1 for the control architecture) is asymptotically stable and possesses one of the following properties. G1) Given a desired asymptotically stable rational matrix W d (s) ∈ C p×m , the behavior of Σ e from the input v to the outputŷ = y + y ξ , described by the transfer matrix W e (s), matches that prescribed by W d (s), namely W e (s) = W d (s) for all s ∈ C. G2) Assume m = p, Σ e is such thatŷ i (t) is affected only by v i (t), for all t 0 and i = 1, . . . , m. G3) Assume m = p, Σ e is passive from the input v to the outputŷ, namely there exists a constant β such that 2 τ 0ŷ (t) v(t)dt β for all functions 3 v ∈ L 2 (0, τ) and all τ 0. • The three tasks stated above formalize the problems of model matching, input/output decoupling, and passivation, respectively.

A. Model Matching
The following statement provides a solution to the first of such objectives, under the (generic) Assumption 1. The nongeneric 2 See, e.g., [11] for the definition of passivity and related discussions. 3 case, namely the case of transfer matrices (1) for which the state-space realization (4) is not controllable, is discussed below.
Proposition 1: Consider a matrix transfer function W (s) as in (1) and suppose that Assumption 1 holds. Let W d (s) ∈ C p×m be any desired proper, asymptotically stable rational matrix. Let the compensator Σ c be described by the equationṡ and consider the feedback/feedforward interconnection of Σ p and Σ c , namely, Σ e with input v and outputŷ y + y ξ , described by the matrix transfer function W e (s). Then, there exist α ∈ R n ,N i ∈ R p×m , i = 1, . . . , n, and D ∈ R p×m such that W e (s) = W d (s) for all s ∈ C.
Proof: To begin with, consider the change of coordinates z = x + ξ, with x denoting the state of the block observer realization (4) of W (s), which is controllable and observable by Assumption 1. The corresponding dynamics in the transformed coordinates are given bẏ where the last equation is obtained by the dynamics ofξ as in (7), recalling that u = Kξ and definingŷ = y + C o ξ + Dv = C o z + Dv. Then, the closed-loop system is described by the equationṡ that is, it possesses the closed-loop matrices Note that ξ describes an unobservable component of the extended (z, ξ)-state. It is now straightforward to note that the resulting W e (s) from the input v to the outputŷ is described by the transfer matrix with the coefficients α i , the matricesN i , for i = 1, . . . , n, and the feed-through term D arbitrarily assignable to match any desired proper rational matrix W d (s).
Interestingly, the proof of the result above suggests that the role of the control input u consists essentially in stabilizing the compensator Σ c , rather than the plant. Moreover, the statement of Proposition 1 entails that the arbitrary shaping of the closedloop transfer matrix is achieved at the price of observability of the extended (plant and compensator) system. More precisely, the feature that, in the transformed (z, ξ)-coordinates, only the state z is observable implies that in the closed-loop system, the behavior of the state of the original plant x becomes essentially indistinguishable from the time evolution of the internal state of the compensator ξ. This issue is resolved in the following result, in which it is shown that the resulting closed-loop transfer matrix may be shaped to approximate, with an arbitrary accuracy, a desired matrix while preserving structural properties, specifically observability, in the interconnected plant. As a by-product, such a requirement additionally rules out the somewhat trivial solution in which a direct feed-through from y is exploited in Σ c to define the output as y ξ = −y +ỹ ξ , thus completely canceling, in the modified outputŷ, the contribution of the plant Σ p , i.e., making the plant unobservable.
Proposition 2: Consider a matrix transfer function W (s) as in (1) and suppose that Assumption 1 holds. Let W d (s) ∈ C p×m be any proper, asymptotically stable rational matrix. Let the compensator Σ c,δ be described by the equationṡ and consider the feedback/feedforward interconnection of Σ p and Σ c,δ , namely Σ e,δ with input v and outputŷ y + y ξ , described by the matrix transfer function W e,δ (s). Then, there exist α ∈ R n andN i ∈ R p×m , i = 1, . . . , n, such that for any ε > 0, there ii) the interconnected system Σ e,δ is observable.
Proof: The claim is shown by employing arguments similar to those in the proof of Proposition 1. More precisely, the main difference consists in the fact that, compared to Σ e , the closedloop system Σ e,δ with the compensator defined as in (12) is described by the matrices Recall now that the observability properties of a pair (A, C) are invariant with respect to static output feedback, i.e., such properties coincide with those of the pair (A + ΓC, C) for any matrix Γ of appropriate dimension. Therefore, the system Σ e,δ described by (13) is observable if and only if the pair , is observable, namely the pair comprising C z and Observability of the latter is then implied by observability of , guaranteed for any δ = 0, and the fact that C z = [C o , 0], hence proving item ii) of the statement. Item i) follows from the continuity of the entries of A z (δ) with respect to δ, the property that W e,0 (s) = W d (s), for all s ∈ C and the fact that W d (s) does not have poles on the imaginary axis. Example 1: Consider a plant Σ p described in state-space representation by matrices as in (4) with which is passive from the input v to the outputŷ. Fig. 2 shows the Nyquist plots of the closed-loop transfer functions obtained by relying on the constructions of Proposition 2, for various values of the parameter δ, which remain passive and for which observability of the overall interconnected system is preserved for δ = 0. Remark 3: By defining the transfer matrices C u,v (s), C u,y (s), C y ξ ,v (s) and C y ξ ,y (s) as Then, it can be easily shown that the Laplace transform of the input is u(s) . As a consequence, the Laplace transform of the auxiliary outputŷ is provided bŷ Since the design of the compensator Σ c as in (7) is such that the matrix transfer function between v(s) andŷ(s) is given by W d (s), it follows that the selection of the matrices in (17) ensures stability of the interconnected system together with the property that Such a frequency-domain interpretation of the design provides the following interesting insight. The existence of the trivial solution to (20) in which essentially the compensator entirely bypasses the original plant via the feedforward action, corroborates the importance of the results stated in Proposition 2, which suggests a systematic design that allows to retain observability of the original plant, with respect to the existing literature (see, e.g., [10]). It may be of interest to assess, in the context of Proposition 1, the relative contribution of the original output y and that of the feedforward output y ξ to the outputŷ. The following construction and subsequent formal result provide a preliminary quantitative answer to the above issue, at least in terms of the static gain of the resulting transfer matrices. To provide a concise statement, for a given choice of α, is controllable and consider first the change of coordinatesξ = T ξ such that the corresponding system matrices are described bŷ for some constant coefficients a jk i . Moreover, the input matrix becomes with the nonzero elements corresponding to the rows n 1 , n 1 + n 2 , ..., n, while the entries of C o T −1 are arbitrary. Consider then, in the transformed coordinates, a feedback gain matrix defined as Proposition 3: Consider a transfer matrix W (s) as in (1) and suppose that Assumption 1 holds. Let W d (s) ∈ C p×m be any desired strictly proper rational matrix. Let the compensator Σ c be described by (7) and, in addition, such that the pair (Â, B o ) is controllable; ii) for any ∈ R + there existsP =P > 0 such that Then, for any μ > 0, there exists K such that W e (s) = W d (s) for all s ∈ C and C y ξ ,v (0) < μ.
Proof: The first part of the claim is clearly shown in the proof of Proposition 1. As for the second point, this is achieved by assigning arbitrarily fast eigenvalues to the matrixÂ − B o K, which is always possible by assumption i) of the statement while preserving asymptotic stability of the compensator by relying on assumption ii). In particular, the static gain which is an input/output property, is determined in the transformed coordinatesξ. It then follows that α i d,1 = (−1) n n i j=1 λ d,j . Straightforward derivations then allow computing the columns of the matrix inverse where d denotes the determinant of the matrix Remark 4: The result of Proposition 3 entails that the effect of the input v on the output y ξ , with y(s) ≡ 0, can be rendered, at least for constant or, by continuity, low-frequency inputs, arbitrarily small with respect to the overall behavior between v and the outputŷ, which can be assigned via the choice of the desired transfer function W d (s). Note, however, that the objective of completely reducing the contribution of the controller Σ c and preserving the role of the original output y is achieved, instead, by solving the more challenging problem which will be addressed in future works, namely minimizing the H ∞ -norm of the controller's transfer matrix, while simultaneously enforcing the primary task of model matching.
By inspecting the structure of the internal transfer matrices of the interconnected system given in (17), the following statement is straightforward, and hence its proof is omitted.
Proposition 4: Suppose that the assumptions of Proposition 1 hold. Then, the closed-loop system Σ e is internally stable provided the gain K is such that the matrix A ξ =Â − B o K is Hurwitz.
Corollary 1: Consider a transfer matrix W (s) as in (1) and suppose that Assumption 1 holds. Let W d (s) ∈ C p×m be any desired proper rational matrix. Let the compensator Σ c be described by (7) with ξ(t) ∈ R np , α = [α 1 , . . . , α n ] , B = [N 1 , . . . ,N n ] , and K such that and consider the feedback/feedforward interconnection of Σ p and Σ c . Then, the zero equilibrium of the closed-loop system Σ e , with v = 0, is GAS, provided the coefficients α i , i = 1, . . . , n, are selected such that the polynomial s n + α 1 s n−1 + · · · + α n−1 s + α n is Hurwitz.
Proof: The claim is a straightforward consequence of the block lower triangular structure of the system (9), with the overall eigenvalues corresponding to those of (− Remark 5: The output feedback stabilizer Σ c is a nonidentity observer according to the definition in the seminal paper by Luenberger [29]. In fact, letting the exogenous input v be equal to zero, the state ξ asymptotically reproduces the quantity −x or, in other words, defining the error variable e = x + ξ, one obtains, by following steps identical to those in the proof of Proposition 1, thatė = (−α ⊗ C o + A o )e. Such dynamics entail that the subspace of R n × R n defined by the equation e = x + ξ = 0 is invariant and externally attractive. An interesting difference with respect to the classical output feedback stabilization based on identity observers is that the observable modes of the transfer function from the input v to the outputŷ coincide with the eigenvalues of the error dynamics and not with those assigned by the control input u = Kξ. This is essentially due to the fact that the additional input v enters the closed-loop system from a different channel with respect to u, thus guaranteeing the property of passivity from such additional input channel (compare the block diagram in Fig. 1 with that of Fig. 3 displaying the classical output feedback stabilization scheme by means of an identity observer).

B. The "Uncontrollable" Case
Consider now the case in which Assumption 1 does not hold, namely suppose that the state-space realization in observer block form (4) is not controllable. As discussed in Remark 2, this case represents a nongeneric (i.e., roughly speaking, a particularly unfortunate) scenario. It is worth stressing that the nongeneric property is in fact the existence of plants not satisfying Assumption 1, whereas the properties, e.g., passivity, obtained via the proposed control architecture, are indeed robust with respect to small perturbations. Two alternative approaches can be then pursued to carry out constructions similar to the one described in Proposition 1. The first essentially consists in prestabilizing the plant by means of classical output feedback techniques and then applying a feedforward compensation to shape the transfer function as desired, hence leading to a higher dimensional construction. The second approach, on the other hand, hinges upon a restriction on the achievable desired transfer matrices.
To provide a concise statement of the former construction, let A m , B m , and C m denote a minimal state-space description of W (s) in (1), namely such that A m ∈ R n min ×n min , B m ∈ R n min ×m , and C m ∈ R p×n min , with n min denoting the dimension of a minimal realization, and the pairs (A m , B m ) and (A m , C m ) controllable and observable, respectively.
Proposition 5: Consider a transfer matrix W (s) as in (1) and suppose that n min < np. Let W d (s) ∈ C p×m be any desired proper rational matrix. Let the compensator Σ c be described by the equationṡ and a e ∈ R n e is such that d e (s) = s n e + a e,1 s n e −1 + · · · + a e,n e . Consider the feedback/feedforward interconnection of Σ p and Σ c , namely, Σ e with input v and outputŷ y + y ξ , described by the transfer matrix W e (s). Then, there exist α e ∈ R n e andN i ∈ R p×m , i = 1, . . . , n e , such that W e (s) = W d (s) for all s ∈ C. Proof: The construction is carried out in two sequential steps based on the combination of a standard output feedback scheme and a dynamic feedforward borrowed from Proposition 1. More precisely, by partitioning the state ξ of (24) as ξ(t) = [ξ 1 (t) , ξ 2 (t) ] ∈ R n min × R n e p , it can be shown that the component ξ 1 behaves as a Luenberger observer for a minimal realization of W (s), i.e.,ξ 1 = A m ξ 1 + B m u + G(y − C m ξ 1 ), together with the stabilizing feedback u = Kξ 1 . The second component ξ 2 , on the other hand, provides the feedforward action required to assign the desired zeros to the closedloop system by following arguments similar to those employed in the proof of Proposition 1, specialized to the extended system of the plant and the stabilizing observer, defined with respect to the extended state x e = [x m , ξ 1 ] ∈ R 2n min , dynamic matrix A e as in (25), which is Hurwitz by the selection of K 1 and G. Once the interconnection of the plant and the observer with state ξ 1 has been stabilized, the design of the dynamic feedforward that assigns the desired transfer matrix must be carried out. To this end, the transfer matrix resulting from the interconnection of Σ p and ξ 1 -subsystem is computed as C e (sI − A e ) −1 B e , with the matrices defined in the statement of the proposition. Such transfer matrix is then realized in state-space by considering the block observer form (5). While the latter may not be controllable, since the eigenvalues of −a e ⊗ C o + A o all lie on the open lefthand side of the complex plane, the construction of Proposition 1 may be carried out by setting the matrix K mentioned in the statement of Proposition 1 equal to zero, hence motivating the structure of the dynamics of ξ 2 .
Example 2: Consider the transfer matrix namely as in (1) with n = 2, p = 2, m = 2, and d(s) = (s 2 − s − 2), and suppose that the objective is to decouple the two input/output channels and impose a desired frequency response on each channel characterized by 1/(s + τ ), τ > 0. By relying on the technicalities of Gilbert's diagonal realization, it can be shown that n min = 3 < 4 = np, and hence the block observer form is not minimal. A minimal realization is instead obtained by considering A m = diag(−1, −1, 2), together with Note that the system is nonminimum phase since it possesses a zero at s = 0.5. By letting now andN i = I p , i = 1, 2. Therefore, the overall dynamic feedback/feedforward controlleṙ for a sufficiently small δ = 0 in the spirit of Proposition 2, with u = [K 1 0] ξ, is such that the closed-loop transfer matrix and the interconnected system (27), (29) is observable fromŷ for any δ = 0.
Before giving the formal statement concerning the second alternative approach to Proposition 1, whenever Assumption 1 fails to be satisfied, the definition of reachability matrix is recalled.
Definition 1: Consider a linear time-invariant system described by the equationsẋ = Ax + Bu, with x(t) ∈ R n and u(t) ∈ R m . Then, the reachability matrix, denoted as R (A, B), associated to the system is defined as • To streamline the following statement, recall thatĀ = −a ⊗ Proposition 6: Consider a transfer matrix W (s) as in (1) and suppose that n min < np. Let W d (s) ∈ C p×m be any desired proper rational matrix, with denominator given by the polynomial s n + α 1 s n−1 + · · · + α n−1 s + α n and such that 5 Let the compensator Σ c be described by the equationṡ with ξ(t) ∈ R np , and K such that the eigenvalues of the reachable subsystem, denoted (A r , B r ), of (Ā, B o ) belong to C − , and consider the feedback/feedforward interconnection of Σ p and Σ c , namely Σ e with input v and outputŷ y + y ξ , described by the transfer matrix W e (s). Then, W e (s) = W d (s) for all s ∈ C.
Proof: The claim is proved by first considering the construction discussed in the proof of Proposition 1, hence essentially defining the dynamics of the compensator as in (7), without however requiring u = Kξ to asymptotically stabilize the pair (−a ⊗ C o + A o , −B o ). In fact, since Assumption 1 does not hold, hence n min < np, it follows that the observer block state-space realization is not controllable. To overcome this issue, the inclusion (32), which poses restrictions on the feasible selections of the matricesN i , ensures that the behavior of the compensator between the inputs v and y and the output u is not affected by the uncontrollable subsystem of the pair Thus, an internally stable compensator that preserves the required input/output behavior can be obtained from (7) by reducing it to the pair (A r , B r ), which can be indeed stabilized by a selection of K.

C. Input/Output Decoupling
The aim of this section consists in the specialization of the model matching machinery derived in the previous section to the case in which the desired transfer matrix possesses a special structure, namely a diagonal W d (s).

Corollary 2. (Input/Output Decoupling of MIMO Plants):
Consider the LTI plant Σ p in (4) and suppose that p = m and that Assumption 1 holds. Let the compensator Σ c be described by (7). Then, the closed-loop transfer matrix W e (s) is diagonal for any s ∈ C provided thatN i = β i I p , for all β i ∈ R, i = 1, . . . , n, and D is diagonal, and hence the outputŷ i (t) depends only on v i (t), for all t ∈ R.
• The proof of the above claim is obtained by a straightforward adaptation of the arguments employed for the proof of Proposition 1, and hence it is omitted. Note that, in the nongeneric case discussed in Section III-B in which Assumption 1 does not hold, a statement similar to Proposition 5 can be provided to generalize Corollary 2 to the context of intput/output decoupling, as also illustrated by Example 2.

D. Passivation
The results discussed in this section provide the solution to the objective (G3) of Problem 1. The issue of passivating a given linear time-invariant system, which may not have vector relative degree one or even not be minimum-phase originally, has become increasingly crucial in control theory, due to the wide diffusion of passivity-based techniques for stabilization.
• The proof of the first three claims is a straightforward adaptation of the proof of the previous statements, and hence it is omitted. To render the above proposition constructive, note that objective (G3) is practically achieved by requiring that the desired transfer matrix satisfies

IV. SINGLE-INPUT/SINGLE-OUTPUT CASE
The aim of this section consists in specializing the results discussed above to the case of SISO systems. First, the theory introduced in Section III is adapted to the SISO setting and illustrated by means of a few numerical examples, and then slightly modified control objectives are stated for the SISO case by allowing the presence of unmeasured disturbance inputs.
Example 3: Consider the LTI system in state-space form described by the equationṡ with x(t) ∈ R 3 , u(t) ∈ R, and y(t) ∈ R. System (34) is controllable and observable. Its transfer function is , from which it can be deduced that system (34) possesses unstable poles and is nonminimum phase. In addition, note that the transfer function (35) does not satisfy the parity interlacing property (see [39]), i.e., it does not have an even number of real poles between every pair of real zeros in Re(s) 0. Let now α = [12, 47, 60] and B = [1,3,2] , and consider the dynamic feedback/feedforward compensatoṙ with K = [−6.375, −2.875, −1.5417]. Then, the interconnected system Σ e with input v and output y + y ξ described by (34)-(36) is passive.
Example 4: Consider the system described by the proper transfer function which is not passive, being unstable and nonminimum phase. The model may be described in state-space form by the dynamic equation in (4), with a = [−1, −2] and B o = [2,0] , and with output defined by y = x 1 + u, namely with a feed-through term. The architecture in (7) is consequently adapted aṡ Consider now the objective of enforcing a different dissipativity property to the system, namely finite, and possibly arbitrarily small, L 2 -gain γ between an unknown disturbance input w and a possibly modified outputŷ (see, e.g., [36]). Therefore, consider the linear time-invariant system Σ p defined in state-space form byẋ where x(t) ∈ R n denotes the state of Σ p , while u(t) ∈ R, y(t) ∈ R and w(t) ∈ R q denote the control input, the measured output, and an unknown disturbance signal, respectively, and suppose that the following assumption holds. Assumption 2: The pairs (A p , B p ) and (C p , A p ) are controllable and observable, respectively.
• Problem 2: (L 2 -gain via Dynamic Feedforward and Feedback). Consider the system Σ p defined in (39). Find, if possible, a dynamic compensator Σ c described by the equationṡ with ξ(t) ∈ R n , v(t) ∈ R, such that the interconnected system Σ e (see Fig. 4 for the control architecture) is internally stable and the following holds. G1') Given γ > 0, the L 2 -gain of Σ e from the disturbance w to the outputŷ = y + y ξ is smaller than γ, namely for any w ∈ L 2 (0, τ) and any τ 0.
• To streamline the statement of the following result, we suppose that the roots of the polynomial μ α (s) = s n + α 1 s n−1 + · · · + α n−1 s + α n are all selected with zero imaginary part and negative real part, and that, for a generic choice of the vector α, they are denoted by the (ordered) set {λ 1 , λ 2 . . . , λ n } with |λ 1 | > |λ 2 | > · · · > |λ n |. Finally, to provide a concise statement of the following results, let T denote the change of coordinates that transforms system (39), with w = 0, into canonical observability form Proposition 7: Consider the LTI plant Σ p in (39) and suppose that Assumption 2 holds. Let the compensator Σ c be described by the equationṡ with ξ(t) ∈ R n ,ŷ = y + C o ξ, α = [α 1 , . . . , α n ] , and K such that σ(−aC o + A o − B o K) ⊂ C − , and consider the feedback/feedforward interconnection of Σ p and Σ c , namely Σ e with disturbance input w and outputŷ. For any γ > 0, let the smallest root of the polynomial μ α (s) be such that 6 Then, the L 2 -gain from the disturbance input w to the outputŷ is smaller than γ, namely for all disturbances w ∈ L 2 (0, τ) and for any τ 0.
• Proof: To begin with, note that the closed-loop system Σ e is described, in the (z, ξ) coordinates, with z = T x + ξ, by the equationsż in which the eigenvalues of (−αC o + A o ) can be arbitrarily assigned by selecting the vector α ∈ R n , while the input channel provided by B o is employed to asymptotically stabilize the ξ component of the extended closed-loop system. By performing an additional diagonalizing change of coordinates, defined in terms of the nonsingular matrixT ∈ R n×n , namelyz =T z, system (44a) can be written aṡ with Δ α a diagonal matrix having the eigenvalues of the matrix (−αC o + A o ) as diagonal elements andP =T P T T P p . Consider now the algebraic matrix Riccati inequality in the unknown symmetric, positive semidefinite matrix X ∈ R n×n . It can be easily shown that if the inequality (46) is satisfied, then by letting V (z) =z Xz, one obtainsV γ 2 w w − z T − T −1z , for any disturbance input w. Hence, by integrating both sides between zero and τ and since V is nonnegative and z(0) = 0, it follows that: Therefore, to conclude the proof of the claim, it remains to show that inequality (46) holds. To this end, let X =T − T −1 and note that the inequality is satisfied by the choice of the eigenvalues of (−αC o + A o ) dictated by (42) and by recalling thatσ(T P p P p T + I) σ(T P p P p T ) + σ(I) =σ(T P p P p T ) + 1.
Remark 6: It is evident that the arbitrarily small gain γ from the disturbance w to the modified outputŷ is enforced by means of high gain-like techniques in the selection of the vector α. It seems then that the proposed approach may be equivalently replaced by means of an output feedback scheme, comprising a classical observer, in which the eigenvalues of the original state and those of the error dynamics are pushed arbitrarily to the left-hand side of the complex plane. However, the key difference lies in the fact that the choice of the gain matrix K is required only to stabilize the dynamics in (44b), and it should not be taken arbitrarily large to obtain a desired L 2 -gain. This aspect is due to the fact that the resulting transfer function from the input w to the outputŷ is given by Wŷ w (s) = C o (sI + αC o − A o ) −1 T P p , while adopting the alternative observer-based scheme mentioned above, one obtains W yw (s) = C(sI − A − BK) −1 P p + C(sI − A − BK) −1 BK(sI − A + LC) −1 P p . This is the result of the fact that the dynamics of ξ are independent of w, while in the observer-based scheme, the error dynamics are indeed affected by the disturbance.
From the construction described in Proposition 7, it can be immediately deduced that, in addition to arbitrarily reducing the L 2 -gain between the disturbance and the modified output, also the L 2 -gain between an additional control input v and the output y is decreased. This is an undesirable property in general, since the control effort, e.g., to steerŷ to track a reference signal is proportionally increased. This issue is solved by the following proposition.
Proposition 8: Consider the LTI plant Σ p in (39) and suppose that Assumption 2 holds. Let the compensator Σ c be described by the equationṡ Then, the L 2 -gain from the disturbance input w to the outputŷ is smaller than γ, for all disturbances w ∈ L 2 , and the L 2 -gain from the control input v to the outputŷ is equal to 1.
• Proof: By taking advantage of the derivations carried out in the previous proof, it can be seen that the closed-loop system is described, for the z component, by the equationṡ Let nowv = η + v, then the transfer function from the inputv to the outputŷ = C o z is defined by Wŷv = β 1 s n−1 + β 2 s n−2 + · · · + β n−1 s + β n s n + α 1 s n−1 + α 2 s n−2 + · · · + α n−1 s + α n where the second equality is obtained by considering the stable zero-pole cancellations performed by the choice of the coefficients of the matrix B, which assigns the zeros of the resulting transfer matrix. The proof is concluded by noting that the transfer function from the input v to the outputv = η + v is then given by hence, WŷvWv v = 1/(s + 1), concluding the proof.

V. APPLICATION TO ADAPTIVE CONTROL OF NONMINIMUM PHASE SYSTEMS
Consider uncertain SISO plants described by transfer functions of the form where k p is referred to as the high-frequency gain and Z(s) and R(s) are monic polynomials in the variable s. It is well known, see, e.g., [16], that the output-feedback model reference adaptive control (MRAC) can be applied to W (s) provided the following assumptions are satisfied: C1) the order n and the relative degree r 1 are known; C2) the sign of the high-frequency gain k p is known; C3) Z(s) and R(s) are coprime and Z(s) is Hurwitz. The most critical assumption to satisfy is typically (C3), which requires the plant to be minimum phase with respect to the measured output. Therefore, in the following, we suppose that (C1) and (C2) are indeed satisfied, while (C3) may not hold. Thus, in this section, we employ the construction introduced above to remove such an assumption or replace it with weaker requirements. Toward this end, consider an uncertain transfer function defined bỹ W (s) = θ 2,1 s n−1 + θ 2,2 s n−2 + · · · + θ 2,n−1 s + θ 2,n s n + θ 1,1 s n−1 + · · · + θ 1,n−1 s + θ 1,n which satisfies the following standing assumption. Assumption 3: The vectors ϑ 1 = [θ 1,1 , . . . , θ 1,n ] and ϑ 2 = [θ 2,1 , . . . , θ 2,n ] are unknown and belong to compact sets Θ i ⊂ R n , respectively. Moreover, the numerator and the denominator of (53) are coprime for any ϑ i ∈ Θ i , i = 1, 2.
• Note that, apart from the inclusion ϑ 2 ∈ Θ 2 , Assumption 3 does not impose any constraint on the location of the resulting zeros of the plant induced by the values of ϑ 2 . A state-space representation of the above uncertain transfer function (53) can be given in terms of a linear system in observable canonical form, namelyẋ together with the output function y = x 1 = C o x, and the matrices A o and C o defined as in (5). Proposition 9: Consider the uncertain plant Σ p defined by (54). Let the compensator Σ c be described bẏ with ξ(t) ∈ R n , α = [α 1 , . . . , α n ] , B = [β 1 , . . . , β n ] such that the system (−αC o + A o , B, C o ) is strictly passive, and hence for any Q = Q > 0, there exists P = P > 0 such that Moreover, letθ 1 ,θ 2 , and K be such that , and a positive scalar ε such that H1) the linear matrix inequality (56) shown at the bottom of this page, holds for any Then, the closed-loop system Σ p , Σ c as in Fig. 1 from the input v to the outputŷ = y + y ξ satisfies Assumptions (C1)-(C3) of the MRAC scheme. Moreover, letting v = −κŷ = −κ(y + y ξ ) with κ > max all the trajectories of the uncertain closed-loop system globally exponentially converge to the origin of R n × R n .
• Proof: Constructions borrowed from the proofs of the previous results lead immediately to the extended closed-loop system in the z = x + ξ and ξ coordinates described bẏ with η = [z , ξ ] . By mimicking the constructions in the nominal case discussed above, it can be easily shown that the z-subsystem (A α , B, C o ) is passive. Then, it can be shown that, lettingP the item (H1) implies thatPĀ +Ā P < −εI, whereas items (H2) and (H3) together guarantee thatPB =C , withC = [C o , 0], hence showing passivity, from the input v to the outputŷ, of the extended system characterized by the corresponding triple (Ā,B,C), namelyη =Āη +Bv,ŷ =Cη. Moreover, the latter property is in turn instrumental to conclude stability properties of the uncertain interconnected system (58). To this end, consider the candidate Lyapunov function V (η) = (1/2)η P η. Then, where the first inequality follows by passivity of the triple (Ā,B,C), whereas the last inequality is obtained by (57).

Remark 7:
The requirements (H1)-(H3) are rather mild. Note, to begin with, that items (H1) and (H2) alone are satisfied by the trivial choice X = 0 and Y selected as any Lyapunov function for the matrix A ξ , which is guaranteed to exist, by construction. Focusing instead on the conditions (H2) and (H3), it can be immediately seen that they consist in 2n linear equations in n 2 + n(n + 1)/2 unknowns, and hence generically solvable for n > 1.
To partially specialize the above results to specific cases in which closed-form selections can be proposed, consider an uncertain double integrator described by the equationṡ with the parameters θ i unknown and such that θ 1 ∈ [θ 1,min , θ 1,max ] and θ 2 ∈ [θ 2,min , θ 2,max ]. We suppose that the plant (61) is not minimum-phase, hence we assume that θ 1,min > 0 and θ 2,max < 0. Proposition 10: Consider the LTI plant Σ p defined by (61). Let the compensator Σ c be described by (55) Then, the interconnected system (55)-(61) from the input v to the outputŷ = x 1 + ξ 1 satisfies Assumptions (C1)-(C3) of the MRAC scheme.
• Proof: The characteristic polynomial associated to the extended plant becomes which is then Hurwitz, and hence the interconnected plant is minimum-phase, if and only if 1 + θ 1 κ 2 > 0, κ 2 θ 2 + κ 1 θ 1 > 0 and κ 1 θ 2 > 0 for any θ ∈ Θ. The last inequality implies that κ 1 < 0, and hence the need for the right inequality in (62a). This, in turn, implies that also κ 2 must be negative to satisfy the second inequality. Finally, rewriting the first two inequalities above in terms of the worst case value for θ 1 and θ 2 , one obtains the constraints in (62). Note that the choice of the gain K as suggested in the statement of Proposition 10 is such that indeed the interconnected plant is minimum-phase, but nothing can be a priori concluded about the location of the closed-loop poles.
As a numerical example, consider the case in which θ 1 ∈ [1, 10] and θ 2 ∈ [−10, −1]. Since the numerator of the transfer function from u to y is defined by θ 1 s + θ 2 , the ranges of variation of the two unknown parameters imply that the system is always nonminimum phase, with the unstable zero ranging from 0.1 to 10. The compensator Σ c described by the equationṡ is such that the interconnected system (61)-(64) is minimumphase, from the input v to the modified outputŷ, for any admissible uncertain parameter θ, provided κ 1 and κ 2 are selected according to −0.01 < κ 1 < 0 and −0.1 < κ 2 < 10κ 1 .
In the following simulations, the parameters are selected as κ 1 = −0.005 and κ 2 = −0.075. The control objective consists in steering the output of the original nonminimum phase plant (61) to track a sinusoidal function of time. To this end, by following steps of the classical MRAC design [16], we introduce the standard desired model reference and, in addition, the dynamics required to compensate at steady state the fact that the output y = y + C o ξ, in place of y alone, is employed for feedback, namelyẋ

VI. CONCLUSION
A model matching and passivating control architecture for MIMO linear systems, comprising dynamic feedback and feedforward, has been proposed. The design methodology does not rely on assumptions concerning the relative degree or minimum phaseness of the plant and employs only input/output measurements. The construction provides a closed-loop system that from suitably modified inputs and outputs matches any desired transfer matrix. This is achieved, in its basic formulation, at the price of observability of the overall interconnected system. Therefore, an alternative implementation of the above design has been proposed that allows to achieve an arbitrary approximation accuracy of the desired matrix while also preserving structural properties-in particular observability-of the overall interconnected system. Such a construction can then be specialized to provide input/output decoupling or a system that is passive from a novel control input to a modified output. The result is achieved by arbitrarily assigning the relative degree and the location of poles and zeros on the complex plane of the interconnected system in a systematic way. It is also shown that similar ideas can be employed to enforce a desired, arbitrarily small, L 2 -gain from an unknown disturbance input to a modified output, while preserving the corresponding gain from the control input to the same output. Finally, similar constructions are employed to study the adaptive control problem for uncertain linear systems that are potentially not minimum-phase.