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Globally monotonic tracking control of multivariable systems

In this paper we present a method for designing a linear time invariant (LTI) state-feedback controller to monotonically track a constant step reference at any desired rate of convergence for any arbitrarily assigned initial condition. Necessary and sufficient constructive conditions are given to deliver a monotonic step response from all initial conditions. This method is developed for multi-input multi-output (MIMO) systems, and can be applied to square and non-square systems, strictly proper and bi-proper systems, and, importantly, also minimum and non-minimum phase systems. The control methods proposed here show that for MIMO LTI systems the objectives of achieving a rapid settling time, while at the same time avoiding overshoot and/or undershoot, are not necessarily competing objectives.


I. INTRODUCTION
The problem of improving the shape of the step response curve for linear time invariant (LTI) systems is as old as control theory. Its relevance is seen in countless applications such as heating/cooling systems, elevator and satellite positioning, automobile cruise control and the positioning of a CD disk read/write head. The common element in these problems involves designing a control input for the system to make the output take a certain desired target value, and then keep it there.
A fundamental issue in classical feedback control is the design of control laws that provide good performance both at steady state and during the transient. The steady state performance is typically assumed to be satisfactory if, once the transient vanishes, the output of the system is constant and equal (or very close) to the desired value. When dealing with the transient respectively, the logical not, and and or. Given a set V and a logical proposition p that depends on an object v ∈ V , we use the symbol {v ∈ V | p} to indicate the subset in V of the elements v for which p is true.

II. PROBLEM FORMULATION
In what follows, whether the underlying system evolves in continuous or discrete time makes only minor differences and, accordingly, the time index set of any signal is denoted by T, on the understanding that this represents either R + in the continuous time or N in the discrete time.
The symbol C g denotes either the open left-half complex plane C − in the continuous time or the open unit disc C • in the discrete time. A matrix M ∈ R n×n is said to be asymptotically stable if σ (M) ⊂ C g , i.e., if it is Hurwitz in the continuous time and Schur in the discrete case. Finally, we say that λ ∈ C is stable if λ ∈ C g . Consider the LTI system Σ governed by where, for all t ∈ T, x(t) ∈ X = R n is the state, u(t) ∈ U = R m is the control input, y(t) ∈ Y = R p is the output, and A, B, C and D are appropriate dimensional constant matrices. The operator  1 We recall that the Rosenbrock system matrix is defined as the matrix pencil C D (2) in the indeterminate λ ∈ C, see e.g. [20]. The invariant zeros of Σ are the values of λ ∈ C for which the rank of P Σ (λ ) is strictly smaller than its normal rank 2 . More precisely, the invariant zeros are the roots of the non-zero polynomials on the principal diagonal of the Smith form of P Σ (λ ), see e.g. [1]. Given an invariant zero λ = z ∈ C, the rank deficiency of P Σ (λ ) at the value λ = z is the geometric multiplicity of the invariant zero z, and is equal to the number of elementary divisors (invariant polynomials) of P Σ (λ ) associated with the complex frequency λ = z. The degree of the product of the elementary divisors of P Σ (λ ) corresponding to the invariant zero z is the algebraic multiplicity of z, see [14]. The set of invariant zeros of Σ is denoted by Z , and the set of minimum-phase invariant zeros of Σ is Z g def = Z ∩ C g .
We denote by V ⋆ the largest output-nulling subspace of Σ, i.e., the largest subspace V of X for which a matrix F ∈ R m × n exists such that (A + B F) V ⊆ V ⊆ ker(C + D F). Any real matrix where σ (A + B F | V ⋆ ) is the spectrum of A + B F restricted to V ⋆ , and its elements are the inner eigenvalues of the closed-loop with respect to V ⋆ , and σ (A +B F | X /V ⋆ ) is the spectrum of the mapping induced by A + B F on the quotient space X /V ⋆ . Its elements are referred to as outer eigenvalues of the closed-loop with respect to V ⋆ . We say that V ⋆ is inner stabilisable if a friend F of V ⋆ exists such that σ (A + B F | V ⋆ ) ⊂ C g , and that V ⋆ is outer stabilisable if a friend F of V ⋆ exists such that σ (A +B F | X /V ⋆ ) ⊂ C g . The eigenvalues of A +B F restricted to V ⋆ can be further split into two disjoint sets: the eigenvalues of σ (A + B F|R ⋆ ), are all freely assignable 3 with a suitable choice of F in F(V ⋆ ). The eigenvalues of σ (A +B F|V ⋆ /R ⋆ ) are fixed for all the 1 If B D has non-trivial kernel, a subspace U 0 of the input space exists that does not influence the state dynamics. By performing a suitable (orthogonal) change of basis in the input space, we may eliminate U 0 and obtain an equivalent system for which this condition is satisfied. Likewise, if [C D ] is not surjective, there are some outputs that result as linear combinations of the remaining ones, and these can be eliminated using a dual argument using a change of coordinates in the output space. 2 The normal rank of a rational matrix M(λ ) is defined as normrank M(λ ) def = max λ ∈C rankM(λ ). The rank of M(λ ) is equal to its normal rank for all but finitely many λ ∈ C. choices of F in F(V ⋆ ) and coincide with the invariant zeros of Σ. Thus, V ⋆ is inner stabilisable if and only if Z ⊂ C g . Finally, we use the symbol V ⋆ g to denote the largest stabilisability outputnulling subspace of Σ, i.e., the largest subspace V of X for which a matrix F ∈ R m × n exists

A. The tracking control problem
In this paper, we are concerned with the problem of the design of a state-feedback control law for (1) such that for all initial conditions the output y tracks a step reference r ∈ Y with zero steady-state error and is monotonic in all components. If y asymptotically tracks the constant reference r and is monotonic, then it is also both non-overshooting and non-undershooting. The converse is obviously not true in general. The following standing assumption is standard for tracking control problems (see e.g. [8]), and ensures that any given constant reference target r can be tracked from any initial condition x 0 ∈ X : Assumption 2.1: System Σ is right invertible and stabilisable. We also assume that Σ has no invariant zeros at the origin in the continuous time case, or at 1 in the discrete case.
Right invertibility and the absence of invariant zeros at the origin in the continuous time (or at 1 in the discrete case) are natural assumptions for a tracking control problem, as they ensure that any given constant reference target r can be tracked from any given initial condition, see [8]. These two assumptions generically hold when m ≥ p. Under Assumption 2.1, the standard method for designing a tracking controller for a step reference signal is carried out as follows.
Given the step reference r ∈ Y to track, choose a feedback gain matrix F such that A + B F is asymptotically stable: this is always possible since the pair (A, B) is assumed to be stabilisable.
Let us choose two vectors x ss ∈ X and u ss ∈ U that, for any r ∈ Y , satisfy in the continuous and in the discrete case, respectively. Such pair of vectors x ss ∈ X and u ss ∈ U exist since (i) right invertibility ensures that the system matrix pencil P Σ (λ ) is of full row-rank for all but finitely many λ ∈ C, see [27, Theorem 8.13], and, as already recalled, the values λ ∈ C for which P Σ (λ ) loses rank are invariant zeros of Σ; (ii) in the continuous (resp. discrete) time case, the absence of invariant zeros at the origin (resp. at 1) guarantees that the matrix P Σ (0) (resp. P Σ (1)) is of full row-rank. As such, Assumption 2.1 guarantees that the linear system  (4) to (1) and using the change of variable ξ def = x − x ss gives the closed-loop homogeneous system Since A + B F is asymptotically stable, x converges to x ss , ξ converges to zero and y converges to r as t goes to infinity. We shall refer to ξ as the error state coordinates.
B. Achieving a globally monotonic response with any desired convergence rate In this paper we are concerned with the general problem of finding a gain matrix F such that the closed-loop system obtained using (4) in (1) achieves a monotonic response at any desired rate of convergence, from all initial conditions. We shall describe this property as global monotonicity.
We describe the problem as follows in terms of the tracking error ε(t) .
Problem 1: (Global Monotonicity) Let ρ ∈ R, such that ρ < 0 in the continuous time and ρ ∈ (0, 1) in the discrete time. Find a state-feedback matrix F such that applying (4) with this F to Σ yields an asymptotically stable closed-loop system Σ hom for which the tracking error term ε(t) converges monotonically to 0 at a rate at least ρ in all outputs, from all initial conditions. Specifically, we require that in the continuous time where ε k (t) is strictly monotonic in t, and in the discrete time where, again, ε k (t) is strictly monotonic in t.
If we are able to obtain a tracking error ε(t) that consists of a single exponential per component in the continuous time or a single power per component in the discrete time, i.e., respectively, and we are able to choose each λ k in such a way that λ k ≤ ρ < 0 in the continuous time and 0 ≤ λ k ≤ ρ in the discrete time, then we obtain a solution to Problem 1. Indeed, February 24, 2014 DRAFT DRAFT 8 asymptotically stable exponentials of λ k or powers of λ k are monotonic functions. This suggests that a possible way of solving Problem 1 consists in the solution of the following problem.
Problem 2: (Single mode outputs). Let ρ ∈ R, such that ρ < 0 in the continuous time and ρ ∈ (0, 1) in the discrete time. Find a feedback matrix F and a p-tuple of distinct values L p = (λ 1 , λ 2 , · · · , λ p ) such that, in the continuous time λ k ≤ ρ < 0, and in the discrete time 0 ≤ λ k ≤ ρ such that applying (4) with this F to Σ yields an asymptotically stable closed-loop system Σ hom for which, from all initial conditions, the tracking error term is given by (8) for some real coefficients {β k } p k=1 depending only upon ξ (0). If we can guarantee that F yields (8) for any initial condition and any r ∈ Y , and such that λ k ≤ ρ < 0 in the continuous time and 0 ≤ λ k ≤ ρ in the discrete time, then obviously such feedback also solves Problem 1. However, the following result shows that the converse is true as well, i.e., the only way to obtain a feedback that ensures global monotonic tracking with arbitrary rate of convergence of the tracking error is to obtain a tracking error in the form (8).
Lemma 1: Problem 1 is equivalent to Problem 2.
Proof: Let us consider the continuous time case, the discrete case being entirely equivalent.
where each α i is a column matrix of dimension m i associated with V i . For the case of λ i complex, we have V i = V i+1 and hence also α i = α i+1 . The tracking error arising from ξ 0 is given by where γ i, j is a function of the entries of V i and α i . Thus, each component of the tracking error is comprised of functions of the form with 0 ≤ j − 1 ≤ m i , and where for complex λ i we have σ i = Re{λ i } and ω i = Im{λ i }. By (6), we conclude that Re{λ i } ≤ ρ for all λ i ∈ L . Further, by assumption the response is monotonic, which implies that only components of the form exp(λ i t) for real λ i can appear in each output (notice that, for any real λ i < 0, the function t j−1 exp(λ i t) is monotonic only if j = 1). Thus for each output ε k (t) we have real coefficientsβ k,i , depending on ξ 0 , such that Moreover each response is monotonic from all initial conditions. From Lemma A.1 of [21], if ε k (t) is the sum of two or more negative real exponential functions, it will change sign (and hence not be monotonic) for some values of the coefficientsβ k,i . Since the mapping ξ 0 → α is surjective, the response can only be globally monotonic if each output component is comprised of a single real exponential function. Thus, we must have ε k (t) =β k,i exp(λ i t) for some λ i ∈ L and some real coefficient β k def =β k,i . If we select the k-th element of the p-tuple L p to equal the element of the set L that appears in the tracking error component ε k , Problem 2 is solved.
Another important and useful problem is one in which the requirements include a specified choice of the closed-loop modes that are visible in each component of the tracking error: Problem 3: (Single mode outputs). Let L p = (λ 1 , λ 2 , · · · , λ p ) be a p-tuple of real numbers such that, in the continuous time λ i < 0 and in the discrete time λ i ∈ (0, 1) for all i ∈ {1, . . ., p}.
Find a feedback matrix F such that applying (4) to Σ yields an asymptotically stable closed-loop system Σ hom for which, from all initial conditions and for all step references, the tracking error term is given by (8).

III. GLOBAL MONOTONICITY: THE INTUITIVE IDEA
In the previous section, we observed that in order for the problem of global monotonic tracking to be solvable, we need to render at least n − p closed-loop modes invisible from the tracking error and distribute the remaining p modes evenly into the tracking error with one mode per error component. If this is possible, then the step response is guaranteed to be monotonic for any initial condition, and therefore also non-overshooting and non-undershooting. The converse is true as well, as shown in Lemma 1. If we are able to render more than n − p modes invisible at ε(t), one or more components of the tracking error can be rendered identically zero, and therefore for those components instantaneous tracking can also be achieved, in which the output component immediately takes the desired reference value. The aim of the next part of the paper is to find conditions under which a gain matrix F can be obtained to deliver the single mode structure (8) for any initial condition. Consider Σ hom in (5), which can be re-written as where ω(t) = F ξ (t). Clearly, Σ hom can be identified with the quadruple (A, B,C, D). The task is now to find a feedback matrix F such that the new control ω(t) = F ξ (t) guarantees that for every initial condition ξ 0 ∈ X the tracking error ε(t) is characterised by a single stable real mode per component. Let j ∈ {1, . . ., p}. Let λ j be real, stable and not coincident with any of the invariant zeros of Σ. Consider a solution v j and w j of the linear equation where β j = 0 and e j is the j-th vector of the canonical basis of Y . Notice that (13) always has a solution in view of the right-invertibility of Σ. By choosing F such that F v j = w j , we find Hence, from (9) we know that for any initial error where γ j depends on the particular initial state ξ 0 . Considering L p = (λ 1 , . . ., λ p ) with each λ j real and stable and different from the invariant zeros, by applying this argument for all components of the tracking error, we obtain a set of solutions (13). If v 1 , . . ., v p are linearly independent, we can choose F to be such that F v i = w i for all i ∈ {1, . . ., p}. Then, for every ξ 0 ∈ span{v 1 , v 2 , . . . , v p }, by superposition we find . . .
However, this result only holds when ξ 0 ∈ span{v 1 , v 2 , . . . , v p }. In order for this response to be achievable from any initial condition, we also need to render the remaining n − p closed-loop modes invisible at ε(t). This task can be accomplished by exploiting the supremal stabilisability output-nulling subspace V ⋆ g of the system, which is defined as the largest subspace of X for which a friend F exists such that, for every initial state lying on it, the corresponding state feedback generates a state trajectory that asymptotically converges to zero while the corresponding output (the tracking error in the present case) remains at zero. We will see in Section IV that a basis for V ⋆ g can always be obtained as the image of a matrix sume for the moment that they are real and distinct). If the dimension of V ⋆ g +span{v 1 , v 2 , . . ., v p } is equal to n, every initial state ξ 0 ∈ X can be decomposed as the sum ξ v +ξ r , where ξ v ∈ V ⋆ g and ξ r ∈ span{v 1 , v 2 , . . . , v p }. If for the sake of argument we have dim V ⋆ g = n − p, and we can find a set of linearly independent column vectors {v p+1 , . . ., v n } from the columns of that is linearly independent of {v 1 , . . . , v p }, we can take w p+1 , . . ., w n to be the columns of W g that correspond to v p+1 , . . . , v n , and construct the feedback control , the response associated with ξ v is identically zero, while the one associated with ξ r is still given by (15). Hence, the tracking error can be written as in (8) We now provide some more intuition on our design method by using an example that embodies all those system theoretic characteristics that are perceived as the major difficulties in achieving monotonic tracking. This system is MIMO, bi-proper, uncontrollable (although obviously stabilisable) and is characterised by 3 non-minimum phase zeros. To the best of the authors' knowledge, there are no methods available in the literature that can solve the tracking problem for MIMO systems with a guaranteed monotonic response under such assumptions, and especially in the presence of three non-minimum phase invariant zeros. We also want to stress that this problem is even solved in closed form.

Example 3.1: Consider the bi-proper continuous-time LTI system Σ in (1) with
We want to find a feedback matrix F such that the output of this system monotonically tracks a unit step in all output components, and the assignable closed-loop eigenvalues are equal to λ 1 = −1, λ 2 = −2 and λ 3 = −1 for the corresponding error components. This system is not square since the number of inputs exceeds the number of outputs. It is seen to be right invertible but not left invertible. This system is not reachable but it is stabilisable, since the only uncontrollable eigenvalue is −6, and is equal to the only minimum-phase invariant zero µ 1 = z 1 = −6 of the system. The other three invariant zeros of the system are non-minimum-phase, and their values are z 2 = 2, z 3 = 3 and z 4 = 5. The first p = 3 modes must be evenly distributed among the 3 components of the tracking error. The subspace of initial conditions for which the closed-loop mode governed by the eigenvalue λ j appears in the j-th output-component is given by the span of v j , where v j solves (13) for a suitable w j ∈ U . The linear equation (13) can be solved by pseudo-inversion with λ 1 = −1, λ 2 = −2 and λ 3 = −1, and gives v 1 The subspace V ⋆ g is spanned by a matrix V g obtained from a basis matrix In this specific case we find V g = Indeed, the subspace of X spanned by any basis matrix of the nullspace of P Σ (λ ) for any other value λ is linearly dependent of imV g , and therefore does not contribute to the construction of a basis for V ⋆ g . Since the dimension of V ⋆ g is equal to 2, two closed-loop modes can be rendered invisible at the tracking error. The columns of the square The response from any arbitrarily assigned initial error state ξ 0 ∈ X is given by the sum of the responses obtained by projecting the initial state on the subspaces imV g = V ⋆ g , span{v 1 }, span{v 2 } and span{v 3 }. The first projection gives a null contribution, because the corresponding state trajectory is invisible at the tracking error by definition of V ⋆ g . The projection of the initial state on span{v j } is such that only the j-th component of the response is affected and is governed by the mode exp(λ j t) as in (14), while all other components are equal to zero.
By superposition, the overall state trajectory originating from ξ 0 yields a tracking error with one mode per component. The feedback matrix is given in closed form by    (17), yields the tracking which has the single mode form of (15). Therefore, the system exhibits a globally monotonic step response. The tracking errors of the closed-loop system are shown in Figure 1 for two different initial conditions.
Since the solvability condition for global monotonicity is given in terms of the dimension of the subspace V ⋆ g + span{v 1 , . . ., v p }, and that v 1 , . . . , v p depend on the choice of the closed-loop eigenvalues λ 1 , . . ., λ p , the solvability condition given in the previous example seems to depend on the particular choice of the closed-loop eigenvalues. The question at this point is: how does the choice of the closed-loop eigenvalues affect the dimension of V ⋆ g + span{v 1 , . . . , v p }? Are there good and bad choices of the closed-loop eigenvalues? More generally, can we find alternative solvability conditions given solely in terms of the system structure and not in terms of a choice of eigenvalues? These are the crucial points that will be addressed in the sequel.

IV. MATHEMATICAL BACKGROUND
In the previous section we identified the basic tools that can be used to obtain a gain matrix F such that p of the n closed-loop modes are evenly distributed into the p components of the tracking error as in (15), and the remaining n − p modes are rendered invisible at the tracking error. The first of these tools is the subspace V ⋆ g , which is made up of the sum of two parts. The first is the subspace R ⋆ , and the second is, loosely, the subspace spanned by the directions of the minimum-phase invariant zeros of Σ. In this section, we recall some important results concerning the relations between these subspaces and the null-space of the Rosenbrock system matrix pencil P Σ . Given µ ∈ C, we use the symbol N Σ (µ) to denote a basis matrix for the null-space of P Σ (µ), and we denote by d(µ) the dimension of this null-space. Let d are respectively 1-, 2-and 0-conformably ordered.

A. Computation of a basis of R ⋆
The following result, see [18] and [19], presents a procedure for the computation of a basis matrix for R ⋆ and, simultaneously, for the parameterisation of all the friends of R ⋆ that place the eigenvalues of the closed-loop restricted to R ⋆ at arbitrary locations. This procedure aims at constructing a basis for R ⋆ starting from basis matrices N Σ (µ i ) of the null-spaces of the Rosenbrock matrix relative to an σ -conformably ordered set L = {µ 1 , . . ., µ r }, where r def = dim R ⋆ , which will result as closed-loop eigenvalues. No generality is lost by assuming that for every where V K ∈ R n×r and W K ∈ R m×r . Then, (i) Matrix V K is generically full column-rank with respect to K, i.e., rankV K = r for every K except for those lying in a set of Lebesgue measure zero; (ii) For all K such that rankV K = r, we have R ⋆ = imV K ; (iii) The set of all friends of R ⋆ such that Lemma 2 permits us to write a spanning set of R ⋆ in terms of the selection of at most r real numbers, as we now show. For any µ ∈ R \ Z , let us define From this definition, a decomposition of R ⋆ can be obtained from Lemma 2.
Corollary 1: Given any distinct set {µ 1 , · · · , µ r } ⊂ R \ Z , there holds Proof: Let be a basis matrix of ker P Σ (µ i ), and let it be partitioned conformably with Since B D is assumed to be of full column-rank, we conclude that ω ∈ kerW i , so that ω ∈ ker B. Computation of a basis of V ⋆ g We now turn our attention to the computation of V ⋆ g . From now on, we will assume that the minimum-phase invariant zeros are all distinct (i.e., their algebraic multiplicity is one): This assumption does not lead to a significant loss of generality. In fact, the case of coincident zeros can be dealt with by using a procedure that is similar in spirit to that outlined in [25].
where v K,H,i are n-dimensional and w K,H,i are m-dimensional for all i. Finally, let Thus, (i) For almost every choice of K = diag{k 1 , . . ., k r } and H = diag{h r+1 , . . . , h r+t } we have rankV K,H = r + t; (ii) If K and H are such that rankV K,H = r + t, the matrix V K,H is a basis matrix where K, H are such that rankV K,H = r + t. Remark 1: We now show how we can build a basis for V ⋆ g using the result of Lemma 3. Let {µ 1 , µ 2 , . . ., µ g } ⊂ C g include the minimum-phase invariant zeros. Let this set be sconformably ordered. In view of Lemma 3, for all i ∈ {1, . . ., g} we can find vectors . . , g} such that, by defining v i = Re{φ i } when i ≤ 2 s is odd, v i = Im{φ i } when i ≤ 2 s is even, and v i = φ i when i ∈ {2 s + 1, . . ., g}, the matrix [ v 1 . . . v g ] is of full column-rank. We can define w 1 , . . . , w g similarly using ψ i instead of φ i .
Then, by virtue of Lemma 3 we have span{v 1 , . . ., v g } = V ⋆ g . Corollary 2: Let r = dim R ⋆ and assume that Assumption 4.1 holds. Let Z g = {z 1 , . . . , z t } be the s z -conformably ordered set of minimum-phase invariant zeros of Σ. When µ ∈ C, let us V. SOLUTION TO PROBLEM 3 Our aim in this section is to provide tractable and constructive necessary and sufficient conditions for the existence of a solution to the problem of global monotonicity.

A. A first necessary and sufficient condition for Problem 3
As explained above, in order to achieve a globally monotonic step response we need to find the feedback matrix F that evenly distributes p of the n closed-loop modes into the p components of the tracking error, and renders the remaining n − p modes invisible at the tracking error. The number of closed-loop modes that can be made invisible by state feedback equals the dimension of the subspace V ⋆ g . Thus, for the tracking control problem with global monotonicity to be solvable we need the condition dim V ⋆ g ≥ n − p to be satisfied. This condition is only necessary, because we need also the linearly independent vectors v 1 , . . . , v p obtained with the procedure indicated above to be linearly independent of V ⋆ g . In the case in which dim V ⋆ g > n − p holds, if it is possible to find linearly independent vectors v 1 , . . . , v p that are independent of V ⋆ g , then not only is the monotonic tracking control problem solvable, but we are potentially able to also obtain a response that achieves instantaneous tracking in some outputs.
We now make a simplifying technical assumption, which in view of the discussion above amounts to putting ourselves in a "worse-case scenario" of all the possible situations in which the tracking problem is solvable. This assumption is made for the sake of simplicity: It is easy to see that, given λ ∈ R, the setR j (λ ) is not a subspace of X .
February 24, 2014 DRAFT The following lemma provides a necessary and sufficient condition for Problem 3 to admit solutions in terms of the setsR j (λ ) defined above.
, so that each component of ε(t) is given by a single exponential, and is therefore monotonic.
Let us now consider necessity. If Problem 3 admits the solution F, by Lemma 1, the tracking error has a single closed-loop mode per component, i.e., it is in the form given by (15). This implies that the remaining n − p closed-loop modes (which are asymptotically stable because F is stabilising) must disappear from the tracking error. Hence, dim V ⋆ g ≥ n − p (and in particular dim V ⋆ g = n − p in view of Assumption 5.1). Let us define, as in Lemma 1, by the eigenvector matrix of A + B F, so that V is invertible (recall that the closed-loop eigenvalues are distinct as established in the proof of Lemma 1). No generality is lost by assuming that {v p+1 , . . ., v n } is a basis for V ⋆ g . Let us define ξ 0 ∈ X and α = V −1 ξ 0 as in Lemma 1. Then . . . where e j is the j-th canonical basis vector of X . It is easy to see that ξ 0, j spans an outputnulling subspace of the system (A, B,C ( j) , D ( j) ) obtained by removing the j-th output because all components of ε j (t) except the j-th are zero. Since in ε j (t) there is only one mode, we have (A + B F) ξ 0, j = λ j ξ 0, j and (C ( j) + D ( j) F) ξ 0, j = 0. The latter implies (C + D F) ξ 0, j = γ j e j for a certain γ j , which cannot be zero, because this would imply dim V ⋆ g > n − p against Assumption 5.1. Thus, we may define v j = 1 γ j ξ 0, j and w j = 1 γ j F ξ 0, j , which satisfy (13). By superposition, we need {v 1 , . . ., v p } to span a p-dimensional subspace of X independent of V ⋆ g . Remark 2: Whenever (24) is satisfied, Problem 3 can be solved with an arbitrary convergence rate. At first glance, this property seems to be in contrast with the fact that the pair (A, B) has not been assumed to be completely reachable, but only stabilisable. In other words, one may argue that the uncontrollable modes (which are asymptotically stable), may limit the convergence rate.
However, it is easy to see that this is not the case. Indeed, from the right invertibility of the quadruple (A, B,C, D), one can conclude that every uncontrollable eigenvalue of the pair (A, B) is also an invariant zero of Σ. 4 Hence, every uncontrollable eigenvalue of the pair (A, B) is rendered invisible at the tracking error, and therefore it does not limit the rate of convergence. It is also worth observing that there is freedom in the choice of the closed-loop eigenvalues associated with R ⋆ , when computing a basis matrix for V ⋆ g . Even though these eigenvalues are invisible at the tracking error (and hence any choice will be correct as long as they are asymptotically stable and distinct from the minimum-phase invariant zeros) this freedom may be important for the designer, since the selection of closed-loop eigenvalues affects other considerations like control amplitude/energy. Thus, it is worth emphasising that the designer has complete freedom to chose any set of stable eigenvalues provided the minimum-phase invariant zeros are included, and provided at least p of these meet the desired convergence rate.
Lemma 4 already provides a set of necessary and sufficient conditions for the solvability of the globally monotonic tracking control problem. However, such conditions are not easy to test, because they are given in terms of the setsR j (λ j ) which are not, in general, subspaces of X .
The tools that we now present are aimed at replacingR j (λ j ) in condition (24) with particular reachability subspaces of the state-space, which we now define. As in the proof of Lemma 4, for each output j ∈ {1, . . . , p} we introduce Σ j = (A, B,C ( j) , D ( j) ) as the quadruple in which C ( j) ∈ R (p−1)×n and D ( j) ∈ R (p−1)×m are obtained by eliminating the j-th row from C and D, respectively. We observe that the right invertibility of the quadruple (A, B,C, D) guarantees that the set of invariant zeros of Σ contains the set of invariant zeros of Σ j for any j ∈ {1, . . ., p}.
The largest output nulling reachability subspace of Σ j is denoted by R ⋆ j . Similarly to what was done for R ⋆ in Corollary 1, for any distinct set {µ 1 , · · · , µ r j } ⊂ R \ Z , we decompose R ⋆ j as where r j = dim R ⋆ j and Remark 3: As established in Corollary 1, a spanning set for R ⋆ j (µ i ) is given by the columns of V i , where V i is the upper part of a basis matrix . However, differently from R ⋆ (µ), this time it is not guaranteed that V i obtained in this way is of full column-rank, because the matrix B D ( j) may very well have a non-trivial kernel. The relationship betweenR j (µ) and R ⋆ j (µ) is stated through the two following results. Proposition 1: Let µ ∈ R \ Z . For all j ∈ {1, · · · , p}, there holds To this end, we first show that Let v be an element of R ⋆ j (µ). Then, there exists a w ∈ U such that Proposition 2: Let µ ∈ R \ Z . For all j ∈ {1, · · · , p}, there holds Proof: Since Σ is right invertible and µ is not an invariant zero, the inclusion R ⋆ (µ) ⊆ Indeed, in such a case, [C j D j ] is linearly independent from every row of , which in general does not hold as an equality since R ⋆ (µ) andR j (µ) may very well have non-zero intersection. Thus, (28) follows readily.
Roughly speaking, this result, together with Proposition 1, implies that R ⋆ j (µ) is coincident withR j (µ) modulo a set of points belonging to a proper algebraic variety within R ⋆ j (µ). This essential step justifies the fact that from now on we will use R ⋆ j (µ), instead ofR j (µ), to establish constructive necessary and sufficient condition for our tracking problem.

B. A tractable condition for the solution of Problem 3
LetL p be the set of all p-tuples (λ 1 , · · · , λ p ) ∈ R p such that for all i ∈ {1, · · · , p} we have λ i / ∈ Z , and λ i ∈ R − or λ i ∈ (0, 1) in the continuous or in the discrete time, respectively.

C. Computation of the gain feedback
Let V g be a basis matrix for V ⋆ g . We first consider the case in which V g has h = n − p columns. Let V g = v g,1 v g,2 . . . v g,h and W g = w g,1 w g,2 . . . w g,h , which satisfy A−µ i I n B C D v g,i w g,i = 0 (here we assume for the sake of simplicity that all the µ i are real, but in the case of complex conjugate minimum-phase invariant zeros, one can apply the construction of Corollaries 1 and 2 with the obvious modifications). The necessary and sufficient condition V ⋆ g +span{v 1 , · · · , v p } = X is satisfied with (v 1 , . . . , v p ) ∈ R ⋆ 1 (λ 1 ) ×· · ·×R ⋆ p (λ p ) and, hence, there exists {w 1 , . . . , w p } such that Since this condition is equivalent to the condition rank[ v 1 . . . v p V g ] = n, we can compute F = W V −1 . From Proposition 2, (v 1 , . . . , v p ) ∈R 1 (λ 1 ) × · · · ×R p (λ p ) gener- for some γ 1 , . . . , γ p ∈ R as required. We now consider the case where Assumption 5.1 does not hold. In other words, since we know that dim V ⋆ g ≥ n − p is a necessary solvability condition, we now assume dim V ⋆ g = h > n − p.
Proposition 3: Let L p = (λ 1 , . . . , λ p ) ∈L p . Problem 3 admits solution if and only if there exists a set δ ⊆ {1, · · · , p} satisfying card δ = n − h and Proof: Using the same argument of the proof of Lemma 4, Problem 3 is seen to admit solutions if and only if there exists δ ⊆ {1, · · · , p} with card δ = n − h and a bijective map  components of the tracking error are identically equal to zero, which means that in those components the output is identically equal to the corresponding component of the reference signal for any initial condition (and this obviously can only happen whenever the corresponding row of the feedthrough matrix D is non-zero). Although fully tractable, the necessary and sufficient condition proposed in Proposition 3 requires to test each p n−h possible injective map β : {1, · · · , n − h} −→ δ . A necessary and sufficient condition for Problem 3 is which clearly reduces to (29) when h = n − p. We omit the proof.

VI. SOLUTION TO PROBLEM 1
In this section, the role played by the eigenvalues L p = (λ 1 , λ 2 , · · · , λ p ) in the existence of solutions to Problem 3 is investigated.

Theorem 2: Problem 1 admits solution if and only if
Proof: Suppose that (34) is not satisfied. This means that there exists S ∈ 2 {1,...,p} such that for any (λ 1 , · · · , λ p ) ∈L p , since by (25) there holds R ⋆ j (λ j ) ⊆ R ⋆ j for all j ∈ {1, · · · , p} and λ j ∈ R\Z . In view of Theorem 1, this shows that Problem 3 is never solvable, which implies that Problem 1 does not admit solution.
Observe that P c+1 can be decomposed as (P c+1 ∩ H c+1 ) ∪ (P c+1 \H c+1 ). Thus, to prove that P c+1 has empty interior, it suffices to prove that both P c+1 ∩ H c+1 and P c+1 \H c+1 have empty interior. Corollary 3 in Appendix B ensures that this is true for the latter. To prove that this also holds for the former, we first show that H c+1 has empty interior. Let and consider the condition for all j ∈ {1, · · · , c + 1}. By means of a simple reindexing, it is seen that (IH) is equivalent to which is now valid for all j ∈ {1, . . ., c + 1}. From the trivial identities we see that H c+1 can be written as which leads to the decomposition In view of (43), if the condition is satisfied, then (41) is satisfied for all j = {1, · · · , c + 1}. In this paper, the problem of achieving a monotonic step response from any initial condition has been addressed for the first time in the literature for LTI MIMO systems. This new approach opens the door to a range of developments that for the sake of conciseness cannot be addressed in this paper, but that we briefly discuss: • In the case that global monotonicity cannot be achieved, it is important to find structural conditions ensuring that every component of the tracking error consists of the sum of at most two, three, or more closed-loop modes. In such case, even if the response is not globally monotonic, it is still monotonic starting from suitable initial conditions. Thus, an important issue is the characterisation of the regions of the state space where the initial state must belong to guarantee that the system response can be made monotonic; • A second relevant problem involves the use of the method in [19] and [18] to the end of computing the state feedback that achieves a globally monotonic step response and which at the same time delivers a robust closed-loop eigenstructure, by ensuring that the closed-loop eigenvalues are rendered insensitive to perturbations in the state matrices. This task can be accomplished by obtaining a feedback matrix that minimises the Frobenius condition number of the matrix of closed-loop eigenvectors, which is a commonly used robustness measure. The problem of obtaining a feedback matrix with minimum gain measure can be handled in a similar way, by minimising the Frobenius norm of the feedback matrix.
• Another important extension is the one of time-varying references, along the same lines of the extension that was proposed in [24] of the procedure introduced in [21].
• Using the same approach of [23], this method can be extended to the case of multivariable dynamic output feedback tracking controllers.

APPENDIX A
Let h be the dimension of V ⋆ g . Throughout this Appendix, we consider an arbitrary integer k ∈ {1, . . ., n − h} and a set of k non-zero subspaces of X denoted by {M 1 , · · · , M k }.
Definition 1: Let us introduce the following proposition: The proposition ¬p k 1 corresponds to the condition dim(V ⋆ g + span{v 1 , · · · , v k }) ≥ h + k. Since clearly h + k vectors cannot span a subspace of dimension strictly greater than h + k, with a slight abuse for the sake of simplicity we will consider that Using those equations, we define the sets The following statements hold true: 1) If (51) is satisfied, the set K k has measure zero; 2) If (51) is not satisfied, then (48) holds for all (v 1 , · · · , v k ) ∈ M 1 × · · · × M k .
As a preliminary step toward the proof of Lemma 5, consider the following definition.

Proof of Lemma 5:
We first prove the second point. Suppose that (51) is not satisfied, i.e., there exists S ∈ 2 {1,...,k} such that dim(V ⋆ g + ∑ j∈S M j ) < h + card(S). This implies that, for every collection of vectors v j such that v j ∈ M j for all j ∈ S, we have dim(V ⋆ g + ∑ j∈S span{v j }) < h + card(S) since V ⋆ g + ∑ j∈S span{v j } ⊆ V ⋆ g + ∑ j∈S M j . This means that there exists a linear dependence among vectors v j and any basis of V ⋆ g . Consequently, p 1 is satisfied for all (v 1 , · · · , v k ) ∈ M 1 × · · · × M k .
We now assum that (51) holds. For brevity, letM j def = M 1 × · · · × M j . Let us prove that mM k (K k ) = 0 by induction on i ∈ {1, . . ., k}. The Inductive Hypothesis (IH) for i reads: mM i (K i ) = 0. We first prove (IH) for i = 1. Observe that (51) implies dim(V ⋆ g + M 1 ) ≥ h + 1 and hence Consequently, a generic vector v 1 ∈ M 1 satisfies v 1 / ∈ V ⋆ g . This is equivalent to saying that m M 1 (K 1 ) = 0. Let i ∈ {1, . . . , k − 1} and assume that mM i (K i ) = 0; we now show that mM i+1 (K i+1 ) = 0. Let us first introduce the following proposition and define In the rest of the proof, we use the chain of implications To prove that mM i (S i \ K i ) = 0, it suffices to prove that S c i is non-empty. This follows from Lemma 4 by observing that S c Suppose by contradiction that S c i is empty. First note that K c i is non-empty because (IH) holds together with the decomposition M 1 ×· · ·×M i = K c i ∪K i , where every M j is a non-zero subspace. Hence, the set S c i -where ¬p i 1 ∧ ¬p 4 holds -is empty by assumption, whereas K c i -where only ¬p i 1 holds -is not. Thus, every element of K c i satisfies (65), which is equivalent to saying that M i+1 ⊆ A where A is given by (52). Using this inclusion, observing that V ⋆ g ⊆ A and using the definition of γ given by (53), we get By Lemma 5, the dimension of the subspace on the left hand-side of (69) is smaller or equal to h + l. On the other hand, (51) ensures that -for S = γ ∪ {k + 1} -this particular dimension is greater or equal to h + l + 1, leading to a contradiction. 5 Consequently, S c i is non-empty and mM i (S i \ K i ) = 0. The equality mM i (S i ) = 0 follows from (i) gives mM i (S i ) ≤ mM i (S i \ K i ) + mM i (S i ∩ K i ), and (ii) mM i (S i ∩ K i ) = 0. Indeed, mM i (S i ∩ K i ) ≤ mM i (K i ) = 0 as (IH) holds. Since mM i (S i ) = 0, a generic (v 1 , · · · , v i ) ∈ M 1 × · · · × M i = S i ∪S c i belongs to S c i and in turn verifies both ¬p i 1 and ¬p 4 . This easily gives mM i+1 (K i+1 ) = 0. Thus (IH) is valid for i + 1 and hence for all i ∈ {1, . . ., k}. 5 At this stage, we can even conclude that A = V ⋆ g + ∑ j∈γ M j . Indeed, (51) implies that dim(V ⋆ g + ∑ j∈γ M j ) ≥ h + l which, together with V ⋆ g + ∑ j∈γ M j ⊆ A -which is deduced from (69) -and (56), allows to write h + l ≤ dim(V ⋆ g + ∑ j∈γ M j ) ≤ dim (A ) ≤ h + l. This leads to dim(V ⋆ g + ∑ j∈γ M j ) = dim (A ) = h + l and A = V ⋆ g + ∑ j∈γ M j .

APPENDIX B
In this Appendix, we present a set of results that are used in the proof of Theorem 2.
Definition 3: For any strictly positive integer c, let L 1 , . . . , L c be finite non-empty sets of real numbers containing l 1 , . . . , l c distinct elements, respectively. We say that a set G c ⊆ R c is a grid in R c of dimension (l 1 , . . . , l c ) if G c = L 1 × · · · × L c . For each j ∈ {1, . . ., c}, we use to denote an indexing of the elements of each L j . We define Λ = (λ 1 , . . . , λ c ) as a node of the grid G c , where each λ j = λ t j j , for some 1 ≤ t j ≤ l j . Lemma 6: For any c ∈ {1, . . ., p−1}, let L 1 , . . ., L c+1 denote arbitrary sets containing r 1 , . . ., r c+1 distinct real numbers, respectively, not coincident with the invariant zeros of Σ. Assume the sets L j are indexed as in (70), and let G c+1 be a grid inL p \ H c+1 of dimension (r 1 , . . . , r p ). Then, there exists (at least) one node Λ ∈ G c+1 such that Λ / ∈ P c+1 .