MIMO tracking control of LTI systems: A geometric approach

Systems & Control Letters 126 (2019) 8–20

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Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle

MIMO tracking control of LTI systems: A geometric approach✩ ∗

Fabrizio Padula a , Lorenzo Ntogramatzidis a , , Emanuele Garone b a b

Department of Mathematics and Statistics, Curtin University, Perth, Australia Department SAAS, Université Libre de Bruxelles, Brussels, Belgium

graphical

article

abstract

info

Article history: Received 17 October 2017 Received in revised form 28 September 2018 Accepted 10 February 2019 Available online xxxx

a b s t r a c t This paper addresses the tracking problem of constant references for multiple-input multiple-output linear time-invariant systems. We provide necessary and sufficient conditions for the solvability of the problem under the minimal set of assumptions that guarantee the well-posedness of every control problem requiring stability. Our approach allows to solve tracking problems for systems which are possibly nonright-invertible, simply (not asymptotically) stabilizable, and possibly with invariant zeros at the origin. Our methodology is constructive and results in the design of a stabilizing feedback matrix and a feedforward signal which solve the problem. © 2019 Elsevier B.V. All rights reserved.

1. Introduction The problem of tracking a known reference is one of the cornerstones of control theory. It has countless applications, including flight control, heating/cooling systems, satellite positioning, automobile cruise control and the positioning of a CD disk read/write head to mention a few. The general objective of a tracking problem is the design of a control input for the system to force the output reach a certain desired target value, and then keep it there. A number of methodologies have been presented in different contexts to address different versions of the tracking control problem, involving (i) additional constraints (e.g., limitation of the control effort or the removal of overshoot/undershoot [1–4]), (ii) different control architectures (e.g., state or output feedback ✩ This work was supported by the Australian Research Council under the grant DP160104994. ∗ Corresponding author. E-mail addresses: [email protected] (F. Padula), [email protected] (L. Ntogramatzidis), [email protected] (E. Garone). https://doi.org/10.1016/j.sysconle.2019.02.003 0167-6911/© 2019 Elsevier B.V. All rights reserved.

schemes [5,6]), (iii) further performance criteria (e.g., the minimization of integrated absolute error or of the settling time [7]). With no exception, all the techniques devised to tackle these problems share a fundamental property, independently from the specific performance requirement that they are designed to address: they must stabilize the system. Another fundamental goal in a tracking control problem is the achievement of zero steadystate tracking error, whenever possible, which means that, once the transient vanishes, the output of the system is required to be equal to its desired value. In this paper, we focus our attention on tracking problems for multiple-input multiple-output (MIMO) LTI systems of constant references where the initial state of the system is arbitrary. Among the techniques devised to address these problems, a particularly important method involves the calculation of the steadystate values of the state and of the input signal, which are then used to construct the control input using a feedback matrix that asymptotically stabilizes the system and a feedforward signal, see e.g. [8]. This method is important per se, because it constitutes the prototype of a variety of more complex tracking problems involving e.g. time-varying references, composite nonlinear feedback

F. Padula, L. Ntogramatzidis and E. Garone / Systems & Control Letters 126 (2019) 8–20

controllers and nonovershooting/nonundershooting controllers. However, it is also relevant from a system-theoretic perspective because, when the tracking control problem is solvable, it can always be solved by combining a static state feedback with a feedforward signal. The assumptions that are usually adopted in the literature to guarantee the existence of a stabilizing controller and the tracking of any constant reference with zero steady-state error are asymptotic stabilizability, right invertibility, and absence of invariant zeros at the origin [4,9]. In particular, asymptotic stabilizability is usually regarded as an unavoidable assumption to ensure that the closed-loop system is asymptotically stable, and therefore to guarantee a suitable level of robustness in the pole placement process: slight variations of some parameters of the problem may very well render a critically stable system unstable. On the other hand, some unreachable eigenvalues may happen to be on the imaginary axis for structural reasons. Indeed, while the precise location of poles and zeros in the left or in the right half plane is in most cases affected by uncertainty, the presence and the controllability of eigenvalues on the imaginary axis (and more often at the origin) are normally related to structural characteristics of the system to be controlled. Their real parts are therefore known with absolute precision, and they are independent of parameter uncertainty. For instance, the relationship between (angular) velocity and pose of a rigid body is exactly of integral type, independently from all the possible modeling uncertainties on the body shape, on the friction, on the body mass, etc. If an action cannot be exerted on the rigid body, we obtain a system which is structurally nonasymptotically stable and unreachable, independently from the uncertain parameters. Whenever this is the case, systems with a non-asymptotically stable unreachable part are of practical interest. In fact for those systems it is clearly impossible to achieve closed-loop asymptotic stability, while simple stability, under suitable conditions, is perfectly acceptable. In this context it is essential to make a distinction between the closed-loop eigenvalues on the imaginary axis which have been assigned by the feedback, and those which belong to the unreachable part of the spectrum. It is well-known that placing poles on the imaginary axis leads almost certainly to instability, since the pole placement operation is sensitive to model uncertainties. For this reason, it is essential to avoid placing reachable eigenvalues on the imaginary axis, while simply stable unreachable modes do not represent an issue for the stability of the closed loop. Therefore, the asymptotic stability is not necessary for the well-posedness of the tracking problem, and the only assumption that is needed, in the most general case, is the simple stabilizability of the system, which is also the minimal assumption that guarantees the well-posedness of any control problem requiring stability. In fact, there are systems for which the tracking problem can be fully solved even when some of the classical assumptions are not satisfied. However, in the classical theory, these systems are simply ruled out by assuming asymptotic stabilizability, right invertibility, and absence of invariant zeros at the origin and therefore the results available in the literature cannot be applied whenever one of these assumptions ceases to be satisfied. These include the important class of systems with more inputs that outputs, and those systems where the output map is not surjective. In this paper we address the tracking problem by only assuming the simple stabilizability of the system, which is the minimal assumption for any control problem requiring stability. Under this assumption, not all the references can be tracked in general. Therefore, we first provide a characterization of the references that can be tracked. Then, we find necessary and sufficient conditions, expressed in terms of the system structure,

9

under which the tracking problem can be solved for any initial condition. The proof of the main result is constructive, and can be used to compute the stabilizing feedback matrix that solves the problem. In the classical framework, the asymptotic stabilizability implies the existence of suitable feedback matrices such that the closed-loop system is asymptotically stable, and therefore also bounded-input bounded-output (BIBO) stable as well as boundedinput bounded-state (BIBS) stable. On the contrary, since we only assume stabilizability, the closed-loop system may be stable but not BIBS stable, or vice versa. Accordingly, in the first part of this paper we prove the existence of a feedback matrix such that the system is both stable and BIBS stable, and we show that BIBS stability guarantees that no (reachable) poles are placed on the imaginary axis. This approach is general and goes beyond the particular control problem tackled herein. The second part of the paper hinges on the aforementioned stability framework and focuses on the solution of the tracking problem. As mentioned, traditionally the condition of right invertibility and the absence of invariant zeros at the origin are used, which guarantee the trackability of every point in the output space. Dropping these assumptions requires characterizing the set of feasible references. We show that this set is a subspace and we provide its characterization in geometric terms. Finally, the last part of the paper focuses on the design of a stabilizing feedback matrix such that a feasible reference can be tracked from any initial condition. Indeed, when the system is not right invertible and non-asymptotically stabilizable, we may have situations where even a feasible reference can only be tracked from specific initial conditions. We provide necessary and sufficient conditions for which a stabilizing feedback matrix can be found that guarantees the trackability of every feasible reference from any initial condition. The results presented in this paper have been developed for continuous-time systems. However, they can be easily adapted to discrete-time systems with minor obvious modifications. Notation. The image and the kernel of matrix A are denoted by im A and ker A, respectively. The Moore–Penrose pseudo-inverse of A is denoted by A† . When A is square, we denote by σ (A) the spectrum of A. If A : X −→ Y is a linear map and if J ⊆ X , the restriction of the map A to J is denoted by A |J . If X = Y and J is A-invariant, the eigenstructure of A restricted to J is denoted by σ (A |J ). The symbol ⊕ stands for the direct sum of subspaces. The symbol i R denotes the imaginary axis. Given a polynomial matrix P(λ) in the indeterminate( λ ∈ C, we ) define the normal rank of P as normrank P = max rank P(λ) .1 Finally, we denote λ∈C

by C− the open left-half complex plane. 2. Problem statement Consider the continuous-time LTI system Σ governed by

Σ:

{

x˙ (t) y(t)

= =

A x(t) + B u(t), x(0) = x0 C x(t) + D u(t),

(1)

where, for all t ∈ R+ , x(t) ∈ X = Rn is the state, u(t) ∈ U = Rm is the control input, y(t) ∈ Y = Rp is the output, and A, B, C and D are appropriate dimensional constant matrices.2 We assume that [B] all the columns of D are linearly independent.3 1 Note that normrank P = rank P(λ) for all but finitely many λ ∈ C. 2 Note that we consider systems with a possible algebraic feedthrough (i.e. D ̸ = 0). All the results of this paper clearly also apply to strictly proper systems. [ ] 3 If B has non-trivial kernel, a subspace U of the input space exists D

0

that does not influence the local state dynamics. By performing a suitable

10

F. Padula, L. Ntogramatzidis and E. Garone / Systems & Control Letters 126 (2019) 8–20

Consider a step reference r(t) = r ∈ Y for t ∈ R+ . In this paper, we are concerned with the problem of designing a statefeedback control law for (1) in the form u(t) = F x(t) + v (t) such that

• for all initial conditions the output y tracks r(t) with zero steady-state error; • for all initial conditions and for all bounded exogenous signals the state response remains bounded. Our aim is to provide a solution to this problem under the most general assumptions on the system. The first issue is the requirement on the stability of the closed-loop system. As it will be clear in the sequel, requiring closed-loop stability without any assumption on the asymptotic stabilizability of the system leads to some technical difficulties which will be addressed in the next section. 2.1. Preliminary definitions In this section we recall some definitions and results that will be used throughout the rest of the paper. First, we define the Rosenbrock system matrix pencil as def

P Σ (λ ) =

[

A − λI C

B D

]

,

λ ∈ C.

(2)

We recall that the system is right invertible if the rank of PΣ (λ) is equal to n + p for all but finitely many λ ∈ C, see [11, Theorem 8.13]. The invariant zeros of Σ are the values of λ ∈ C for which the rank of PΣ (λ) is strictly smaller than its normal rank normrank PΣ . We recall that the reachability matrix pencil is defined as def

SA,B (λ) = [A − λ I

B],

λ ∈ C.

The uncontrollable eigenvalues of (A, B) are the values of λ ∈ C for which the rank of SA,B (λ) is strictly smaller than its normal rank normrank SA,B = n. We recall that, if z is an invariant zero, the algebraic multiplicity of z is the degree of the product of the elementary divisors of PΣ (λ) corresponding to z, while the geometric multiplicity of z is the number of elementary divisors (invariant polynomials) of PΣ (λ) associated with λ = z, see [12,13]. In particular, the geometric multiplicity of z can be computed as normrank PΣ − rank PΣ (z). Finally we recall the reachability standard form T

−1

[ AT =

A1,1 0

A1,2 A2,2

]

,

T

−1

[ B=

B1 0

] (3)

obtained with a suitable change of basis T (adapted to the smallest A-invariant subspace of X containing the column-space of B). In the sequel we will denote by

• ηz and ζz the algebraic and the geometric multiplicities of z as an invariant zero of Σ ; • nλ and gλ the algebraic and geometric multiplicities of λ as an eigenvalue of A; • nFλ and gλF the algebraic and geometric multiplicities of λ as an eigenvalue of A + B F ; (orthogonal) change of basis in the input space, we may eliminate U0 and obtain an equivalent system for which this condition is satisfied. By performing this operation, it is obvious that we are eliminating a degree of freedom that might be exploited to satisfy other specifications (reducing the magnitude of the control, etc.) in addition to the pure tracking objective considered here, but it does not impact on the solvability of the problem addressed in this paper. For a discussion on over-actuated systems we refer the readers to [10] and the references cited therein.

• νλ and γλ the algebraic and geometric multiplicities of λ as an eigenvalue of A2,2 . Clearly, with this notation we have nλ = n0λ and gλ = gλ0 . Moreover, notice that

γλ = normrank SA,B − rank SA,B (λ) = n − rank SA,B (λ). We assume that whenever λ (or z) is not an eigenvalue (an invariant zero, respectively), the corresponding algebraic and geometric multiplicities are equal to zero, so that the operators nλ , gλ , νλ , γλ , nFλ , gλF , ηz , and ζz are well defined for any λ, z ∈ C. Recall that, since the pair (A1,1 , B1 ) in (3) is completely reachable, the eigenvalues of A1,1 + B1 F1 , where F = [ F1 F2 ] is partitioned conformably with (3), are all freely assignable, along with their multiplicities. More precisely, the characteristic polynomial of A1,1 + B1 F1 (and thus its eigenvalues) can be made arbitrary. This, however, does not mean that A1,1 + B1 F1 can be made arbitrary. In particular, we do not have complete freedom in assigning the Jordan structure of the closed-loop matrix A1,1 + B1 F1 . The problem of determining the set of all possible Jordan structures that are obtainable is completely solved by the so-called Rosenbrock Theorem [14, Thm 4.2, p. 190] in terms of integer indices called Kronecker invariants (or controllability indices). We conclude this section by recalling a fundamental result for right-invertible systems, see e.g. [11]. Lemma 2.1. Let Σ be right-invertible. Then, every uncontrollable eigenvalue of the pair (A, B) is also an invariant zero of Σ with greater or equal multiplicity. 2.2. Stability In this section we recall some known concepts on the stability of LTI systems, along with some simple properties which, to the best of our knowledge, are not available in the literature, and are functional for the development of the main results of the paper. We recall that a pair (A, B) is said to be BIBS stable if, for any bounded input u, the forced state response (also sometimes referred to as the zero initial state response) is bounded.4 Finally, we say that the pair (A, B) is stabilizable (respectively, asymptotically stabilizable) if there exists a feedback matrix F ∈ Rm×n such that A + B F is stable (respectively, asymptotically stable).5 The following result provides a necessary and sufficient condition for BIBS stability. Lemma 2.2. A pair (A, B) is BIBS stable if and only if νλ = nλ for all λ ∈ σ (A) such that Re{λ} ≥ 0. Proof. The BIBS stability is equivalent to requiring that the reachable part of the spectrum be asymptotically stable, which is in turn equivalent to have all the non-asymptotically stable eigenvalues in the unreachable part of the spectrum, i.e., νλ = nλ for all λ ∈ σ (A) such that Re{λ} ≥ 0. ■ While asymptotic stability implies BIBS stability, the same does not hold for the stability alone. Indeed, if the reachable part of the system is stable but not asymptotically stable, then Lemma 2.2 ensures that the system is not BIBS stable. The opposite implication is also false: the BIBS stability does not impose any condition on the unreachable part of the spectrum of A, and therefore a system can be BIBS stable and unstable at the same time. 4 It is an easily established fact that BIBS stability is stronger than the well-known bounded-input bounded-output (BIBO) stability. 5 Stabilizability is entirely equivalent to having a stable matrix A 2,2 in (3).

F. Padula, L. Ntogramatzidis and E. Garone / Systems & Control Letters 126 (2019) 8–20

11

Fig. 1. The control architecture.

Example 2.1. Consider the pairs (Ai , Bi ) with i ∈ {1, 2, 3} ⋆ ⋆⋆ 0 −2 ⋆ ⋆

[ −1 A1 =

0 0

[ B2 =

0 00 0 00 11

⋆⋆

00 00

]

,

[ B1 =

]

;

[ A3 =

11

⋆⋆

0 0 −1 0 0 0

]

0 0

⋆ ⋆⋆ 0 −2 ⋆ ⋆

[0 A2 =

;

⋆ ⋆⋆ −2 ⋆ ⋆

0 01 0 00

]

,

0 0 00 0 0 00 11

[

B3 =

]

⋆⋆

00 00

]

,

,

where ⋆ denotes an arbitrary real value. The first pair is clearly both stable and BIBS stable. The second pair is clearly stable, but is not BIBS stable because [ ] 3 = n0 ̸= ν0 = 2. For instance, the constant input u¯ = 11 would cause an unbounded growth of the first component of the state variable. Finally, the last pair is BIBS stable but A3 is unstable because of the nontrivial Jordan form of the eigenvalue 0. Note that in all the cases where the system is BIBS stable the reachable part of the spectrum of Ai is asymptotically stable. The next result deals with existence of a suitable feedback matrix under which the closed-loop system exhibits any of the aforementioned properties. Notice that we introduced the notion of stabilizability of the pair (A, B) but not that of BIBS stabilizability. The reason is that the stabilizability also guarantees the possibility of designing a feedback F such that the system is both stable and BIBS stable. Proposition 2.1. Consider a stabilizable pair (A, B). Let F be the set of matrices F such that A + B F is stable and FBIBS be the set of matrices F such that the pair (A + B F , B) is BIBS stable. Then FBIBS is nonempty and FBIBS ⊆ F. Proof. The result follows by noting that the system is stabilizable and therefore matrix A2,2 of the reachability standard form (3) is stable. ■ We must require that the closed-loop system is both stable and BIBS stable. Indeed, the first property guarantees that the system state remains bounded from any initial condition, and the latter ensures that boundedness is preserved regardless of any possible exogenous input. Given a stabilizable pair (A, B), the matrix F may render the closed-loop matrix A + B F stable, but not BIBS stable. Since we are interested in both simple and BIBS stability, the following definition is adopted. Definition 2.1. We say that F is stabilizing if A + B F is stable and (A + B F , B) is BIBS stable. 2.3. The tracking control problem We are now ready to formalize the tracking control problem stated at the beginning of this section. Problem 2.1. Given a constant reference r(t) = r¯ , find a stabilizing matrix F and a bounded signal v (t) such that, for the control law u(t) = F x(t) + v (t), and for all initial conditions x0 , the error signal y(t) − r(t) converges to zero.

The controlled system is obtained by substitution of the control law u(t) = F x(t) + v (t) into (1):

{

x˙ (t) y(t)

= =

(A + B F ) x(t) + B v (t), x(0) = x0 (C + D F ) x(t) + D v (t).

We first show that it is not restrictive to assume that the feedforward component of the control law v (t) in Problem 2.1 is constant. In other words, with no loss of generality we can assume that the control law sought in Problem 2.1 takes the form u(t) = F x(t) + v (t) (with v (t) ) = v ∈ U for all t ≥ 0. Indeed, if a pair F , v (t) , where v (t) is possibly ( )time-varying, solves the problem then there exists a pair F , w (t) with w (t) = w ∈ U constant that also solves the(same problem. In fact, )if the problem is solved we have limt →∞ (C + D F ) x(t) + D v (t) = r from any initial state. Since F is stabilizing the reachable part of the system is) asymptotically stable and this implies that also the ( pair F , w (t) with w (t) = w = limt →∞ v (t) for all t ≥ 0 solves the problem. We show that the two control schemes in Fig. 1 are equivalent, i.e., that a pair (F , v¯ ) which solves the tracking problem exists if and only if there exists a triple (F , xss , uss ) satisfying 0 = A xss + B uss , r = C xss + D uss ,

(4)

which also solves the tracking problem by applying the statefeedback control law u(t) = F x(t) − xss + uss .

(

)

(5)

The ‘‘if’’ part of this statement is obvious: one can choose

v¯ = F xss +uss . Now we consider the opposite implication. Assume that the pair (F , v¯ ) solves the problem for any initial condition. Thus, we can consider the case where∫ x0 is zero, and in view of the t BIBS stability the limit x¯ = limt →∞ 0 e(A+B F )(t −τ ) B v (τ ) dτ exists and it is finite, which implies that the following equations hold:

{

0 = (A + B F ) x¯ + B v¯ r¯ = (C + D F ) x¯ + D v¯ .

It is sufficient to select xss = x¯ and uss = (F x¯ + v¯ to) obtain xss and uss that satisfy (4) and such that u(t) = F x(t)−xss +uss = F x¯ +v¯ . The advantage of using this approach is that it allows us to work with homogeneous systems. Consider the system (1); def changing variable ξ = x − xss and defining the tracking error def ϵ = y − r gives the autonomous closed-loop system

Σaut :

{

ξ˙ (t) = (A + B F ) ξ (t), ξ (0) = ξ0 = x0 − xss ϵ (t) = (C + D F ) ξ (t).

(6)

The tracking problem can be expressed in terms of the autonomous system (6) as follows. Problem 2.2. Find xss and uss satisfying (4) and a stabilizing feedback matrix F such that the output ϵ (t) of the autonomous system Σaut in (6) converges to zero for every initial state x0 .

12

F. Padula, L. Ntogramatzidis and E. Garone / Systems & Control Letters 126 (2019) 8–20

Tracking control problems (see e.g. [4]) are normally solved by assuming the system to be right invertible, with no invariant zeros at the origin and asymptotically stabilizable. Indeed, these assumptions ensure that any given constant reference target r can be tracked from any initial condition. In particular, right invertibility ensures that the system matrix pencil PΣ (λ) is of full row-rank for all but finitely many λ ∈ C, see [11, Theorem 8.13], and, as already recalled, the values λ ∈ C for which PΣ (λ) loses rank with respect to its normal rank are invariant zeros of Σ . Therefore, the absence of invariant [ ] zeros at the origin guarantees that the matrix PΣ (0) = CA DB is of full row-rank. As such, the right invertibility and absence[ of] invariant [zeros ] , at the origin ensure that the linear system 0r = PΣ (0) [ uxss ss ] which is a compact way of writing (4), is always solvable in uxss . ss Moreover, asymptotic stabilizability ensures the existence of a stabilizing feedback matrix F such that not only the error ϵ (t), but also the state ξ (t) of the autonomous system converges to zero. Under the above mentioned assumptions, solving Problem 2.2 is straightforward: the tracking problem is solvable for any initial condition and any reference signal, and every stabilizing F solves the problem. Nevertheless, there are many cases where the tracking problem can be solved even when these assumptions are not satisfied. The goal of this paper is to solve the tracking problem under the weakest set of assumptions, which amounts to finding necessary and sufficient conditions for the solvability of Problem 2.1 (or Problem 2.2) for systems which are not necessarily rightinvertible and/or asymptotically stabilizable, and with possible invariant zeros at the origin. For the well-posedness of the problem we assume the following minimal standing assumption: Assumption 2.1.

The pair (A, B) is stabilizable.

Remark 2.1. We do not require the pair (A, B) to be asymptotically stabilizable. In other words, all the uncontrollable eigenvalues of (A, B) are in the closed left-hand complex plane, and Σ is allowed to have uncontrollable modes on the imaginary axis, provided that their algebraic and geometric multiplicities, as eigenvalues of A2,2 in the standard reachability decomposition (3), coincide. 3. Solution of the tracking problem

Definition 3.1. We define the set of feasible references as def

Yss = {r ∈ Y such that

Eq. (4) admits solution } .

The following theorem provides an explicit characterization of the set of feasible references. Theorem 3.1. Yss = [C

The set Yss is a subspace of Y and

D] ker[A

B].

(7)

Proof. Suppose that r ∈ [C D] ker[A B]. There [exists ] ω ∈ xss A B C D ker [ ] such that [ ] ω = r. Partition ω = . Thus, u ss [ A B ] [ xss ] [ ] = 0r , so that (4) is[solvable. Suppose r] ∈ / [ C D] uss C D ] [ ] [ xss ker[A B]. By contradiction, if [ CA ]DB = 0r has soluuss xss tions then there exists ω = such that [A B] ω = 0 uss and [C D] ω = r, which readily implies that r ∈ [C D] ker[A B]. ■ Remark 3.1. The set of feasible references is invariant with respect to changes of coordinates in the state. Indeed, if T is n × n and invertible, y ∈ [C T D] ker[T −1 A T T −1 B] implies the existence of ξ and ω such that T −1 A T ξ + T −1 B ω = 0 and y = C T ξ + D ω, and defining ζ = T ξ this is equivalent to the existence of ζ , ω such that A ζ + B ω = 0 and y = C ζ + D ω, i.e., y ∈ [C D] ker[A B]. The converse implication follows in the same way. We now study the dimension of the subspace of trackable targets Yss . Theorem 3.2.

def

Let ℓ = n + p − normrank PΣ . There holds

dim Yss = p − ℓ + γ0 − ζ0 .

(8)

(

Proof. From Theorem 3.1, we have dim Yss = dim [C ) ker[A B] , which can be written as6

(

dim Yss = dim [C

D] ker[A

( = dim ker[A ( = dim ker[A

)

B]

D]

) (

B] − dim ker[A

B] ∩ ker[C

)

D]

B] − dim (ker PΣ (0)) .

)

(9)

Considering that the system has a zero at the origin with geometric multiplicity ζ0 we get dim (ker PΣ (0)) = n + m − normrank PΣ + ζ0

This section is devoted to the solution of Problem 2.2 under Assumption 2.1. If the system is not right-invertible and/or has invariant zeros at the origin, the Rosenbrock matrix pencil appearing in (4) is not full row-rank, which implies that the equations may not admit a solution for every reference r. Even when suitable xss and uss exist, Problem 2.2 may still be unsolvable because the system can be nonstabilizable and nonright-invertible, which implies that ξ (t) may not converge to zero and behave in such a way that the reference r is trackable only from specific initial conditions. The original problem can be therefore split into two different subproblems:

• characterization of the space of the feasible references r for which suitable xss and uss that satisfy (4) exist;

• existence of a suitable feedback matrix F that, given a feasible reference, solves the tracking problem. 3.1. Feasible references Given a reference r, the existence of xss and uss is related to the solvability of (4). Accordingly, we adopt the following definition

= m − p + ℓ + ζ0 .

(10)

Moreover, since normrank SA,B = n, we find

(

dim ker[A

B] = n + m − normrank SA,B + γ0 = m + γ0 .

)

Substituting (10) and (11) into (9) we obtain the result.

(11) ■

Corollary 3.1. Let the system be right invertible. Then, Yss = Y if and only if ζ0 = γ0 . Proof. The proof follows from Theorem 3.2 considering that for right-invertible systems ℓ = 0. ■ As a simple consequence of the right invertibility, the result of Lemma 2.1 holds, which implies ζ0 ≥ γ0 . Thus, as expected, dim Yss ≤ p = dim Y . At first glance, Eq. (8) may suggest that when a system is not right invertible, the space of trackable targets increases in dimension, because if the system is not right invertible we may have γ0 > ζ0 . However, when this is the case, 6 Recall that dim(A U ) = dim U − dim(U ∩ ker A), see [15, p. 7].

F. Padula, L. Ntogramatzidis and E. Garone / Systems & Control Letters 126 (2019) 8–20

the tracking problem is not solvable, since the uncontrollable mode at the origin (which is not asymptotically stable) cannot be rendered unobservable from the output. This point will be clarified in Remark 3.4. In general, an uncontrollable eigenvalue at the origin will remain such also for the closed-loop system (A+B F , B, C +D F , D). The state components that are associated with the eigenvalue at the origin will be identically equal to their initial value, which may not be compatible with the tracking of the reference signal, even if (4) admits a solution. for[ example the [ −1 Consider ] [ 1 closed] ] ⋆ 0 1 loop system A + B F = , C = and , B = 0 0 1 0 0 [0] D = 0 . The system is not right invertible, has no invariant zeros and it has an uncontrollable eigenvalue at zero, which is therefore observable from the output. In this case p = 2, ℓ = 1, γ0 = 1, ζ0 = 0, so that Yss = Y . Eq. (4) has solutions for any constant [ x ] reference r. On the other hand, the initial condition x0 = x00,,21 is such that the second component of the output is identically equal to x0,2 , which means that there are no references that can be tracked independently from the initial state. This toy example shows that r ∈ Yss is a necessary but not, in general, a sufficient condition for the solvability of our tracking problem. In general, under the simple (not necessarily asymptotic) stabilizability assumption, some uncontrollable modes may neither converge, nor diverge. Instead, they could remain constantly at their initial value or exhibit a constant oscillation, independently from any possible input signal. If these modes cannot be rendered unobservable from the output, for any constant input the output is not independent from the initial state. This problem is related to the design of a suitable feedback matrix F and will be addressed in the next section. Clearly, a necessary condition is to have a reference signal which is feasible. Accordingly, all the references appearing in the sequel of the paper will be assumed to be feasible. 3.2. Stabilizing feedback If a system is asymptotically stabilizable, the existence of a feasible reference is a sufficient condition for the solvability of the tracking problem from an arbitrary initial state, because the uncontrollable modes are all asymptotically stable, i.e., they go to zero as t → ∞. The transfer function of the system provides the ideal tool to address the tracking problem in the case of an asymptotically stabilizable system. However, if the system is simply stabilizable, the unreachable subsystem may have eigenvalues on the imaginary axis. If these eigenvalues are all unobservable, the tracking problem is still solved, even if the state of the autonomous system Σaut does not converge to zero. If one or more of these eigenvalues are observable, the output will always depend on the initial condition. Therefore, the only viable solution is to render those modes unobservable so that they do not influence the input–output behavior of the closedloop system. This task can be accomplished only via a state-space approach. In this context, the classical right invertibility condition plays a major role: it always guarantees the solvability even in the presence of uncontrollable eigenvalues on the imaginary axis. Indeed, we will show that if the system is right invertible, in view of Lemma 2.1 the uncontrollable modes can be always rendered unobservable via an appropriate state feedback matrix F . On the contrary, if neither of the previous conditions holds, the solvability of the problem is not guaranteed even in the presence of a feasible reference. First notice that if (A, B) is stabilizable and r ∈ Yss is a feasible reference, Problem 2.2 is solvable if there exists a stabilizing matrix F such that lim (C + DF )ξ (t) = 0.

t →∞

13

Note that this condition does not imply that the state of the autonomous system Σaut converges to zero. When we are able to guarantee that not only the output, but also the state converges to zero, we say that the tracking problem is strongly solvable: Definition 3.2. Let (A, B) be stabilizable, and let r ∈ Yss be a feasible reference. We say that the tracking problem is strongly solvable if lim ξ (t) = 0.

t →∞

Remark 3.2. The strong solvability guarantees that not only the output of Σ will converge to r¯ , but also the state will attain the value xss at steady state. If a system is right invertible and asymptotically stabilizable, Problem 2.2 is always strongly solvable and every asymptotically stabilizing feedback matrix can be used. We now introduce the projection operator. We define the projection operator as

⏐ ⏐

{

}

P (W ) = x ∈ X ⏐ ∃ u ∈ U : [ ux ] ∈ W ,

see also [16, Chapter 5], where it is shown that P is linear. For the sake of readability, the next results are stated under the assumption that matrix A2,2 in the reachability standard form (3) is diagonalizable (or, equivalently, that νλ = γλ for any λ ∈ C). Loosely speaking, this condition implies that the uncontrollable eigenvalues are nondefective. We will show in Remark 3.5 how to generalize this result to the case where A2,2 is defective. Let us introduce the following notation. Given a set of h selfconjugate complex numbers L = {λ1 , . . . , λh } containing exactly s complex conjugate pairs, we say that L is s-conformably indexed if 2 s ≤ h and λ1 , . . . , λ2 s are complex, while the remaining are real, and for all odd k ≤ 2 s we have λk+1 = λ¯ k . For example, given L = {λ1 , λ2 , λ3 , λ4 }, we have that L is 1-conformably indexed if λ1,2 = 1 ± i, λ3 = 2, λ4 = 3 and L is 2-conformably indexed if λ1,2 = 2 ± i and λ3,4 = ±3 i. Consider the set Λ of uncontrollable eigenvalues on the imaginary axis. Notice that, when F is a stabilizing feedback matrix, this set coincides with the set of closed-loop uncontrollable eigenval{ } ues on the imaginary axis, i.e. Λ = λ ∈ σ (A + B F ) | Re{λ} = 0 .7 The set Λ is an s-conformably indexed set of ϕ self-conjugate distinct complex numbers λ1 , . . . , λϕ . Let us for all k ∈ {1, . . . , ϕ} denote by Tk a basis matrix for ker PΣ (λk ). Let us denote by Nk a basis matrix for ker(A + B F − λk I). Let us denote by Gk a basis matrix for ker[ A − λk I B ]. Let these bases be chosen in such a way that Tk+1 = Tk∗ , Nk+1 = Nk∗ , and Gk+1 = G∗k when k ≤ 2 s is odd. Let

Re{Tk } Im{Tk } Tk

k ≤ 2 s odd k ≤ 2 s even k > 2 s,

Re{Gk } Im{Gk } Gk

k ≤ 2 s odd k ≤ 2 s even k > 2 s.

{ def

ˆ Tk =

{ def ˆ Gk =

Re{Nk } ˆ Nk = Im{Nk }

{

def

Nk

k ≤ 2 s odd k ≤ 2 s even k > 2 s,

We also define the following subspaces of X ⊕ U : def

NF =

[

I F

]∑ ϕ k=1

im ˆ Nk ,

def

T =

ϕ ∑ k=1

im ˆ Tk ,

def

G=

ϕ ∑

im ˆ Gk .

k=1

The previous subspaces have a precise system-theoretic interpretation. The space NF is obtained by adding the extended (state space and control space) closed-loop eigenspaces associated with the uncontrollable eigenvalues on the imaginary axis. It therefore represents, loosely, the space where the closed-loop trajectories 7 In general we have the inclusion Λ ⊆ λ ∈ σ (A + B F ) | Re{λ} = 0 .

{

}

14

F. Padula, L. Ntogramatzidis and E. Garone / Systems & Control Letters 126 (2019) 8–20

associated with the uncontrollable eigenvalues live, together with the corresponding control trajectories obtained via the statefeedback matrix F . The subspace T is the greatest extended subspace such that there exists a feedback matrix that renders P (T ) the greatest output-nulling, where the closed-loop eigenvalues restricted to P (T ) are Λ [17]. Finally G is the extended subspace that comprises all the directions in the state-space, along with the corresponding directions in the control space, that can be assigned as closed-loop eigenvectors corresponding to the eigenvalues Λ. Let Γ be the set of uncontrollable eigenvalues of the pair (A, B). Consider the s-conformably indexed set Γ \ Λ of ψ self-conjugate distinct complex numbers τ1 , . . . , τψ . Let us for all k ∈ {1, . . . , ψ} denote by Jk a basis matrix for ker[ A − τk I B ]. Let this basis be chosen in such a way that Jk+1 = Jk∗ when k ≤ 2 s is odd. Let

Re{Jk } ˆ Jk = Im{Jk }

{

def

Jk

k ≤ 2 s odd k ≤ 2 s even k > 2 s, def

∑ψ

ˆ and we define the subspace J = k=1 im Jk of X ⊕ U . The subspace J has the same systems-theoretic interpretation as G , but now the set of considered closed-loop eigenvalues is Γ \ Λ instead of Λ. Consider the s-conformably indexed set Υ comprising θ ≤ ρ = dim R self-conjugate distinct complex numbers π1 , . . . , πθ , where R denotes the reachable subspace (i.e., the smallest Ainvariant subspace containing the image of B). Let for all k ∈ {1, . . . , θ } denote by Qk a basis matrix for ker[ A − πk I B ]. Let this basis be chosen in such a way that Qk+1 = Qk∗ and let { def

ˆ Qk =

Re{Qk } Im{Qk } Qk

k ≤ 2 s odd k ≤ 2 s even k > 2 s. def

∑θ

ˆ Finally, we define the subspace Q = k=1 im Qk of X ⊕ U . The subspace Q comprises all the directions in the state-space, along with the corresponding directions in the control space, that can be assigned as closed-loop eigenvectors corresponding to the controllable eigenvalues Υ , with multiplicities compatible with the Rosenbrock Theorem. ∑ϕ Let us choose k=1 γλk vectors from G as ] v λk ,i ∈ im ˆ Gk , k ∈ {1, . . . , ϕ}, i ∈ {1, . . . , γλk } (12) wλk ,i ∑ψ and k=1 γτk vectors from J as [ ] v˜ τk ,i ∈ imˆ Jk , k ∈ {1, . . . , ψ}, i ∈ {1, . . . , γτk }. (13) w ˜ τ k ,i ∑θ Let ℵπ1 , . . . , ℵπθ be integer numbers such that k=1 ℵπk = ρ . Let us choose ρ vectors from Q as [ ] v¯ πk ,i ∈ im ˆ Qk , k ∈ {1, . . . , θ }, i ∈ {1, . . . , ℵπk }. (14) w ¯ πk ,i [

If ℵπ1 , . . . , ℵπθ are such that the conditions of the Rosenbrock Theorem (see Section 2.1) are satisfied, the matrix V = [ vλ1 ,1

···

vλϕ ,γλϕ

v˜ λ1 ,1

···

v˜ λψ ,γλψ

v¯ π1 ,1

v¯ πθ ,ℵπθ ]

···

(15) is invertible for almost all the choices of

[ v¯

πk ,i

w ¯ πk ,i

]

[v

λk ,i

wλk ,i

] [ v˜ ,

τk ,i

w ˜ τk ,i

]

, and

and the feedback matrix computed as F = WV −1 , where

W = [ wλ1 ,1

· · · wλϕ ,γλϕ w ˜ λ 1 ,1 · · · w ˜ λψ ,γλψ w ¯ π 1 ,1 · · · w ¯ πθ ,ℵπθ ]

(16)

is such that σ (A + B F ) = {λ1 , . . . , λϕ , τ1 , . . . , τψ , π1 , . . . , πθ } and the eigenvectors of A + B F are the vectors {vλ1 ,1 , . . . , vλϕ ,γλϕ , v˜ λ1 ,1 , . . . , v˜ λψ ,γλψ , v¯ π1 ,1 , . . . , v¯ πθ ,ℵπθ }, see [18,19]. Provided that ℵπ1 , . . . , ℵπθ satisfy the conditions of the Rosenbrock Theorem, the previous result has three straightforward consequences:

• P (G + J + Q) = X ; • P (Q) ⊇ R. Moreover P (Q) = R if Υ ∩ Γ = ∅, i.e., when Υ does not contain uncontrollable eigenvalues; • P (NF + J + Q) = X . The first property derives directly from the invertibility of V . The second follows from the first by noting that the vectors vλk ,i and v˜ τk ,i are closed-loop eigenvectors and therefore they form a basis for the closed-loop eigenspace associated with the eigenvalues of A2,2 in the reachability standard form (3). The { third property follows from}the first by noting that, since Λ ⊆ λ ∈ σ (A + B F ) | Re{λ} = 0 , then P (NF ) includes the closed-loop eigenspace associated with the eigenvalues of A2,2 on the imaginary axis, and therefore the vectors vλk ,i are contained in such eigenspace. Remark 3.3. The fact that we have to choose the indices ℵπi so as to satisfy the conditions of the Rosenbrock Theorem does not cause any loss of generality in the solvability of the problem. Indeed, it is always possible to select a set Υ comprising exactly θ = ρ distinct eigenvalues so that ℵπk = 1 for all k ∈ {1, . . . , ρ}: this selection of the indices never violates the conditions of the Rosenbrock Theorem. The next result provides the conditions, expressed in terms of subspace inclusions, under which a stabilizing feedback matrix solves Problem 2.2. Lemma 3.1. Consider stabilizing feedback matrix F and a feasible reference. Matrix F solves the tracking problem if and only if NF ⊆ T . Proof. First, from the definitions of P (·) and NF we write P (NF ) =

ϕ ∑

im ˆ Nk .

k=1

The subspace P (NF ) of X is spanned by the eigenvectors associated with the eigenvalues of A + B F on the imaginary axis, i.e., Λ (recall that F is stabilizing). We denote by V the subspace spanned by the (possibly generalized) closed-loop eigenvectors that correspond to closed-loop eigenvalues in σ (A + B F ) \ Λ. The subspaces P (NF ) and V are (A + B F )-invariant since they are given by the sum of eigenspaces of A + B F . Since F is assumed to be stabilizing, the closed loop systems is both stable and BIBS stable. The first property implies that all the closed-loop eigenvalues on the imaginary axis have trivial Jordan structure, i.e., nFλk = gλF k . In view of Lemma 2.2, the BIBS stability implies that all the closed-loop eigenvalues on the imaginary axis are unreachable eigenvalues, i.e., nFλk = νλk . Finally, since the system is assumed to be stabilizable, the matrix A2,2 in the reachability standard form (3) is stable, which implies that all the eigenvalues of A2,2 on the imaginary axis have trivial Jordan structures, i.e., νλk = γλk . Considering the previous equalities together, we obtain nFλk = gλF k = νλk = γλk . ( ) Since F is stabilizing, we have σ A + B F | P (NF ) = Λ ⊂ iR so that P (N ) is stable. This implies that, for almost all ξ¯0 ∈ P (NF ), the limit limt →∞ e(A+BF ) t ξ¯0 is either nonzero (if Λ = {0}) or it does not exist (if Λ includes purely imaginary eigenvalues). Notice that P (NF ) ⊕ V = X because they are spanned by closed-loop eigenvectors associated to distinct sets of eigenvalues

F. Padula, L. Ntogramatzidis and E. Garone / Systems & Control Letters 126 (2019) 8–20

so that P (NF ) ∩ V = ∅. Now, consider an arbitrary initial state x0 and a suitable vector xss . It is always possible to uniquely decompose the initial state ξ0 = x0 − xss of the autonomous system (6) as ξ0 = ξ¯0 + ξˆ0 , where ξ¯0 ∈( P (NF ) and ξˆ0) ∈ V . Since F is stabilizing, we have σ A + B F | V = σ (A + B F ) \ Λ ⊂ C− , so that V is asymptotically stable and therefore limt →∞ e(A+B F ) t ξˆ0 = 0. Since P (NF ) is (A+B F )-invariant, P (NF ) ∋ ξ¯ (t) = e(A+B F ) t ξ¯0 , ∀ t ≥ 0 and the solvability condition reduces to lim (C + DF )ξ¯ (t) = 0.

(17)

t →∞

[ ] ¯ ⊆ T then Fξξ¯(t) ∈ T . Moreover ker PΣ (λ) = (t) ker[A − λ I B] ∩ ker[C D], ∀ λ ∈ C. Hence, [ ]from the definition ξ¯ (t) of T we have T ⊆ ker[C D] so that F ξ¯ (t) ∈ ker[C D], and this also implies that ξ¯ (t) ∈ ker(C + DF ). Hence (C + DF )ξ¯ (t) = 0, ∀ t ≥ 0 and the implication is proven.

15

the largest output nulling subspace of the system where the dynamics are restricted to Λ. Remark 3.4. When Σ is not right-invertible, if γ0 > ζ0 , Problem 2.1 is not solvable. Consider for example the case ζ0 = 0 and γ0 = 1. In this case, ker(A + B F − λ I) = ker(A + B F ) is not contained in R, where R is the reachable subspace of the system, because λ = 0 is uncontrollable, which implies that P (NF ) ⊈ ◁ R. On the other hand, since λ = 0 is not an invariant zero, P (T ) ⊆ R. Thus, P (NF ) is not contained in P (T ). It follows that the condition NF ⊆ T is not satisfied, and the problem is therefore not solvable.

(If). If NF

(Only if). If the tracking problem is solvable, then (17) holds. It follows that for all ϵ > 0 there exists t ⋆ > 0 such that (C + DF )ξ¯ (t) < ϵ for all t ≥ t ⋆ . We want to show that P (NF ) ⊆ ker(C + D F ). To this end, we take ξ˜ ∈ P (NF ), and we show that (C + D F ) ξ˜ < ε for all ε > 0. Let ε > 0. We observe that for all t˜ > 0 there exists ξ¯0 ∈ P (NF ) such that ξ¯ (t˜ ) = ξ˜ . Indeed, as shown above, the (A + B F )-invariance of P (NF ) yields ξ¯ (t˜) = e(A+B F ) t˜ ξ¯0 = ξ˜ . Let ϵ < ε , so that there exists t ⋆ > 0 such that (C + DF ) ξ¯ (t) < ε for all t ≥ t ⋆ . Let us choose t˜ > t ⋆ , so that (C + D F ) ξ¯ (t˜) < ε . Since ξ¯ (t˜ ) = ξ˜ , we obtain [(C ]+ D F ) ξ˜ < ε . Therefore, P (NF ) ⊆ ker(C + D F ). Since NF = FI P (NF ), we find NF ⊆ ker[C D]. Since P (G + J + Q) = X , the family of feedback matrices that assign exactly Λ (whose elements are associated with trivial Jordan blocks8 ) can be exhaustively parameterized using vectors from im ˆ Gk , imˆ Jk , and im ˆ Qk and using (16) (see the proof of exhaustiveness in [19, Theorem 2.1]). In view of the exhaustiveness of the parameterization, there exist vectors {vλk ,1 , . . . , vλk ,νλ } ⊆ k

P im ˆ Gk for all k ∈ {1, . . . , ϕ}, which can be used in (16) to build the feedback matrix F . These will also be closed-loop eigenvectors associated to the uncontrollable eigenvalues λk ∈ Λ, for all λk ∈ Λ. Since P (NF ) is, by construction, a controlled invariant spanned by the closed-loop eigenspaces associated)to Λ, we have P (NF ) = (∑ ϕ ˆ span{vλ1 ,1 , . . . , vλϕ ,νλϕ } ⊆ P k=1 im Gk = P (G ) and, by construction (see (16)), F P (NF ) = F span{vλ1 ,1 , . . . , vλϕ ,νλϕ } = span{wλ1 ,1 , . . . , wλϕ ,νλϕ }, which implies NF ⊆ G . Hence, NF ⊆ G ∩ ker[C D], which immediately implies NF ⊆ T . ■

(

)

The rationale behind the previous theorem is that, to achieve tracking, we need to render the closed-loop non-asymptotically stable modes in Λ unobservable, which correspond to the openloop non-asymptotically stable uncontrollable modes because F is stabilizing. The subspace P (NF ) must be an output-nulling subspace for Σ , and F must be a corresponding friend, i.e., if F solves the tracking problem, it renders the subspace P (NF ) an (A + B F )-invariant subspace contained in ker(C + D F ). The control space associated with P (NF ) is F P (NF ). The family of feedback matrices that assign Λ and render the corresponding eigenspace [output] nulling can be exhaustively parameterized using vectors vλk ,i wλk ,i

taken from im ˆ Tk instead of im ˆ Gk , see the proof of ex-

∑ϕ

ˆ haustiveness in [18]. Therefore we have NF ⊆ T = k=1 im Tk . The latter also implies P (NF ) ⊆ P (T ), which has a clear systemtheoretic interpretation considering that P (T ) is, by construction, 8 Notice that there is no loss of generality when we consider only trivial Jordan block associated with the eigenvalues λk ∈ Λ since F is stabilizing which implies nFλk = gλF k = νλk = γλk .

The previous lemma says under which condition a stabilizing F solves the tracking problem, but it does not provide the condition for the existence of such feedback matrix. Now we need to prove under which conditions a stabilizing F that solves the tracking problem exists. Theorem 3.3. Let Σ be stabilizable and let r be a feasible reference. Assume νλ = γλ for any λ ∈ C. The tracking problem is solvable if and only if P (T + J ) + R = X .

(18)

Proof. (If) Given an s-conformably indexed set Υ of ρ distinct numbers in C− such that Υ ∩ Γ = ∅, the condition P (T + J ) + R = X can be rewritten as

( P

ϕ ∑

im ˆ Tk +

k=1

ψ ∑ k=1

imˆ Jk +

ρ ∑

) im ˆ Qk

= X.

(19)

k=1

The previous condition implies that we can always extract a basis for X by n vectors from P (T + J + Q). We can always ∑taking ϕ choose k=1 γλk vectors from T as

[

v λk ,i wλk ,i

]

∈ im ˆ Tk ,

k ∈ {1, . . . , ϕ}, i ∈ {1, . . . , γλk }

(20)

and the remaining vectors as in (13)–(14). The Rosenbrock Theorem is always satisfied because Υ comprises ρ elements, so that θ = ρ and ℵπk = 1 for k ∈ {1, . . . , ρ}.9 By contradiction, suppose that a basis can be built by taking a different number of vectors from the image of each matrix in (19). By applying (15)–(16) this would lead, by construction, to a feedback matrix such that the closed-loop and the open-loop eigenstructures of the unreachable part are different, which is clearly impossible. Considering that ker PΣ (λ) = ker[ A − λ I B ] ∩ ker[ C D ] for all λ ∈ C, from the definitions of ˆ Tk and ˆ Gk , it is immediate to see that im ˆ Tk ⊆ im ˆ Gk for all k ∈ {1, . . . , ϕ}. Therefore, by selecting linearly independent vectors γλk (= νλk ) from im Tk we select vectors that are also in im Gk , so that we can build a feedback matrix using (15)–(16). Since Γ ∩ Υ = ∅, by construction we have σ (A + B F ) = Υ ∪ Γ . The obtained F is therefore stabilizing, since Υ ⊂ C− and the unreachable eigenvalues are stable and the system is stabilizable. Moreover, the vectors vλk ,i are, by construction, closed-loop eigenvectors; thus P (NF ) = span{vλ1 ,1 , . . . , vλϕ ,γλϕ }. Considering that, by construction, F P (NF ) = F span{vλ1 ,1 , . . . , vλϕ ,νλϕ } = span{wλ1 ,1 , . . . , wλϕ ,νλϕ }, we readily obtain NF ⊆ T . (Only if) If the tracking problem is solvable, by virtue of Lemma 3.1 there exists a stabilizing F such that NF ⊆ T . In

9 Recall that ρ is the dimension of the reachable subspace.

16

F. Padula, L. Ntogramatzidis and E. Garone / Systems & Control Letters 126 (2019) 8–20

general we have P (G + J + Q) = X . Since P (NF ) is the closed-loop eigenspace associated with the uncontrollable modes Λ, we have P (NF + J + Q) = X , which implies P (T + J + Q) = X . Finally, in virtue of the linearity of P (·), we obtain (18) by noting that P (Q) ⊇ R. ■

3.3. Strong solvability

Remark 3.5. The mathematical apparatus of the entire section has been developed under the assumption that the Jordan form associated with the uncontrollable eigenvalues is trivial, i.e., when νλ = γλ . We stress that all the uncontrollable eigenvalues on the imaginary axis, i.e., in Λ, must have trivial Jordan form for the well-posedness of the control problem, i.e., the system stabilizability. The possibility of assigning nontrivial Jordan structures to the reachable part of the spectrum does not change the solvability of the problem. In other words, if the tracking problem is solvable, it is also clearly solvable by assigning ρ distinct stable eigenvalues. Therefore, the only interesting case is the one where the unreachable eigenvalues in Γ \ Λ exhibit nontrivial Jordan structures. In this case, instead of considering ker[A − τk I B] in the construction of Jk , we have to consider the Jordan chain relative to the generalized eigenspace, see [19]. The machinery developed in this section remains essentially unchanged.

Proof. (If) If the tracking problem is solvable, there exists a stabilizing feedback F such that NF ⊆ T . Moreover, if there are no purely imaginary eigenvalues, the limit

The next result shows how the right-invertibility condition simplifies the solution of the tracking problem.

T −1 (A + B F ) T =

Corollary 3.2. Let Σ be stabilizable and let r be a feasible reference. The tracking problem is solvable if Σ is right invertible. Proof. In view of Lemma 2.1, if the system is right invertible, for each uncontrollable mode there is a zero with higher or equal multiplicity, i.e., ζλ ≥ γλ . For each λk ∈ Λ the matrix [A − λk I B] becomes rank] deficient and its rank decreases by [ kI B γλ , and the matrix A−λ becomes rank deficient and its rank C D decreases by ζλ ≥ γλ . Therefore, we can always extract a set of γλk linearly independent vectors from im Gk which are in im Tk , and selecting the remaining vectors as in (13)–(14) we can form a basis for the state-space. Thus, (18) holds. ■ Remark 3.6. The feedback matrix obtained following the constructive proof of Theorem 3.3 is robust in terms of the closedloop stability, because it does not place closed-loop eigenvalues on the imaginary axis. Indeed, the only closed-loop eigenvalues on the imaginary axis are structural non-reachable poles of the open loop system. However, the method presented in this paper hinges upon the construction of a feedback matrix that renders the non-asymptotically stabilizable eigenvalues unobservable. The computation of this matrix uses eigenstructure assignment techniques that require the computation of eigenspaces, see e.g. [20]. For this reason, a certain degree of uncertainty may result in the closed-loop eigenstructure, resulting in a possible steady-state offset between the reference and the output. Note that this issue is common to every technique relying on static state/output feedback. In our case, this undesirable behavior can be minimized by using robust eigenstructure techniques, such as the minimization of the Frobenius norm. A complete discussion on this is out of scope; however, we refer the reader to [18] for a detailed explanation. We only limit ourselves to observing that the techniques proposed in [18] address the robust eigenstructure assignment problem using null-spaces of the Rosenbrock/controllability matrix pencil, and therefore it fits exactly in the framework proposed here, in the sense that it can be applied verbatim, without the necessity of adaptations/modifications, to the machinery proposed in this

Theorem 3.4. Given a feasible reference and a stabilizable system, the tracking problem is strongly solvable if and only if it is solvable and the system has no uncontrollable eigenvalues in i R \ {0}.

ξ¯0 = lim e(A+B F )t ξ0 t →∞

exists and it is finite. In general, uxss = PΣ (0)† 0r + Kz, where ss im K = ker PΣ (0), z is an arbitrary vector of suitable dimension, and ξ0 = − xss . We define Π : Rn+m −→ Rn as a function such ([ xξ0]) that Π ω = ξ where ξ ∈ Rn and ω ∈ Rm . We have xss =

[

]

[ ]

( [ ] ) ( [ ]) Π PΣ (0)† 0r + Kz = Π PΣ (0)† 0r − Π (K z). If there exists z such that ξ¯0 = 0, the tracking problem is strongly solvable. Computing ξ¯0 amounts to computing a matrix A∞ = limt →∞ e(A+B F )t such that ξ¯0 = A∞ ξ0 . Therefore, the tracking problem is strongly solvable if there exists z such that A∞ x0 − Π PΣ (0)†

(

(

[ ]) 0 r

) − Π (K z) = 0.

(21)

Consider the change of basis matrix T = [T1 T2 ], where im T1 = ker(A + B F ) and im [ T2 = ]im(A + B F ), and compute A˜ = A˜ 1,1 A˜ 1,2 0 A˜ 2,2

(

˜

)

. Then A∞ = T limt →∞ eAt T −1 .

The matrix A˜ 2,2 is asymptotically stable; hence, it is nonsingular and invertible, while A˜ 1,1 = 0 because its spectrum comprises the uncontrollable eigenvalue at the origin and, in view of the stabilizability of the system, there are only trivial Jordan forms. Hence, ˜

A˜ ∞ = lim eAt = t →∞

[

I 0

1 −A˜ 1,2 A˜ − 2,2

]

0

(22)

and im A˜ ∞ = ker A˜ so that, changing basis, im A∞ = ker(A + B F ) = P (NF ), where the last equality holds since there are no purely imaginary eigenvalues. Thus, Λ = {0} (or Λ = ∅ if there are no uncontrollable eigenvalues at the origin, in which case the statement is trivial because A∞ is the zero matrix). In view of the solvability of the tracking problem, NF ⊆ T , so that P (NF ) ⊆ P (T ) and P (T ) = P (ker PΣ (0)), since there are no( purely imaginary eigenvalues. Therefore we have im A∞ ⊆ ) ˜ ∞ | ˜ is the P ker PΣ (0) . From (22) we immediately see that A im A∞ identity map, and therefore it is a bijection from im A˜ ∞ to itself. This property is independent from the set of coordinates, so that the same property holds true for A∞ . Eq. (21) can be rewritten as A∞ x0 − A∞ Π PΣ (0)†

(

[ ]) 0 r

= A∞ Π (K z).

(23)

Clearly, the left hand-side of (23) is a vector of im A∞ . Moreover, since im A∞ ⊆ P (ker PΣ (0)) and A∞ |im A∞ is a bijection from im A∞ to itself, the previous equation is always solvable in z and the implication is proven. (Only if) Immediate by noting that if we have pairs of complex conjugate poles in iR \ 0 the limit limt →∞ e(A+B F )t ξ0 does not exist for some initial conditions. ■ Remark 3.7. The fact that the matrix A∞ defines a bijective map from its image to itself has a clear system-theoretic interpretation. Indeed, ξ¯0 = A∞ ξ0 = limt →∞ e(A+B F )t ξ0 represents the steady-state value of the state variable ξ (t). If we consider a new initial value problem where the initial state is ξ¯∞ = ξ¯0 , starting from a steady state the system will remain at the steady state, or, in other words, ξ¯0 = A∞ ξ∞ = A∞ ξ¯0 . In view of the arbitrariness of ξ0 , the vector ξ¯0 can assume any value in im A∞ and therefore we immediately see that im A∞ = A∞ im A∞ .

F. Padula, L. Ntogramatzidis and E. Garone / Systems & Control Letters 126 (2019) 8–20

17

Fig. 2. The considered plant.

Also note that the definition of ξ¯0 is consistent with the definition adopted in the proof of Lemma 3.1, i.e., ξ0 = ξ¯0 + ξˆ0 , where ξ¯0 ∈ P (NF ) and ξˆ0 ∈ V , provided that there are no poles in iR \ 0.

The level of the tanks will therefore be

3.4. Constructive algorithm

where l0,in and l0,out denote the initial levels while sin and sout denote the basis areas. Consider the plant in Fig. 2 in its standard configuration, i.e., when the valves r1 and r3 are closed and the engine cooling system has water suction and discharge directly outboard (solid lines in Fig. 2). Defining the state variable as the vecdef tor whose components are the levels of the tanks i.e., x = ⊤ [l1 l2 l3 l4 ] , and replicating the previous equations for each tank, it is immediate to obtain the following state-transition and input matrices

When the problem is solvable, the user can use the following constructive steps: (i) compute the set of feasible references Yss , and select a feasible reference r¯ ∈ Yss ; (ii) compute xss and uss from (4); [ ]

∑ϕ

vλ ,i

Tk , k ∈ γλk vectors from T as wλkk ,i ∈ im ˆ {1, . . . , ϕ}, i ∈ {1, . . . , γλk }; ∑ψ (iv) choose k=1 γτk vectors from J as in (13); (v) choose an arbitrary stable complex-conjugate set {π1 , . . . , πρ }, select θ = ρ and ℵπ1 = · · · = ℵπθ = 1, and choose θ

(iii) choose

k=1

vectors from Q as in (14); (vi) build V as in (15), W as in (16), and the feedback matrix as F = WV −1 ; (vii) implement the control law (5).

lin = l0,in −

t

∫

1 sin

q(ξ )dξ ,

lout = l0,out +

0

⎡ ⎡0

0 0 0

⎢ ⎢0 A=⎢ ⎢ ⎣

0

0 0

s3 r2

ρ2 ⎥ ⎥ s3 r 2 ⎥ , ⎥ −ρ2 ⎦

s4 r2

s4 r 2

−ρ2

0 0

0 0

⎤

ρ2

⎢ s1 ⎢ ⎢ 1 ⎢ B=⎢ ⎢ s2 ⎢ ⎢ ⎣ 0

Consider the ship in Fig. 2. The rear tanks s1 and s2 are filled with oil, while the front tanks s3 and s4 can be filled with marine water to balance the ship. The pairs of tanks (s1 , s2 ) and (s3 , s4 ) share the same pressure. The pump p1 generates a flow rate of oil from s1 to s2 . When the pump is broken, a bypass valve with hydraulic resistance r1 can be opened. The pump p2 pumps water into s3 . The tank s3 is connected to the tank s4 through a piping system whose hydraulic resistance is r2 . The discharge valve of the tank s4 can be either open or close. When it is open its hydraulic resistance is r3 , and the tank s4 discharges outboard. Finally, we consider the engine cooling system. In normal operational mode (solid lines in Fig. 2) p3 pumps marine water through the engine E and the system discharges directly outboard. We also consider a second operating mode where p3 pumps water from s3 through the engine into s4 (dashed lines). We are concerned with the following control problem: maintain the ship balanced, both longitudinally and laterally by compensating the pitch and the roll torques, and provide a sufficient water flow to cool down the engine. The control variables are the flow rates generated by the three pumps p1 , p2 , and p3 denoted by u1 , u2 , and u3 , respectively. We begin by developing a suitable state-space model for this system. Consider a valve with hydraulic resistance r. Let lin be the level of a tank with constant section connected to the inlet shaft of the valve, lout be the level of a second tank connected to the outlet shaft and let ρ be the density of the fluid. The flow rate q(t) through the hydraulic resistance from the inlet tank to the outlet one is governed, in first approximation, by the following equation: q(t) = ρ

lin − lout r

.

t

∫

sout

−1

0

4. An illustrative example

1

q(ξ )dξ , 0

⎤ 0

0

⎥ ⎥ ⎥ ⎥ 0 0⎥ ⎥, ⎥ 1 ⎥ 0⎦

s3 0

0

where si , for i ∈ {1, . . . , 4}, denotes the area of the basis of each tank.10 From the previous equation it is clear that the third input does not affect the state variable. This reflects the plant configuration, where the engine cooling system is completely decoupled from the tank system. Considering the aforementioned control objectives, the output matrices C and D are

[ ρ1 s 1 y 1 C =

ρ1 s2 y2 ρ2 s3 y3 ρ2 s4 y4 ρ1 s1 x1 ρ1 s2 x2 ρ2 s3 x3 ρ2 s4 x4 0

0

0

0

]

,

D=

[0

0 0 0 0 0 0 0 1

]

,

where ρ1 is the density of the fluid contained in the tanks s1 and s2 , ρ2 is the density of the fluid contained in the tanks s3 and s4 , and xi and yi are the lateral and longitudinal positions, respectively, of the tank si measured from the buoyancy point. The system is right invertible and has an uncontrollable mode and an invariant zero at the origin.11 The open-loop spectrum is σ (A) = {λ, 0}, where n0 = 3, g0 = 3 and λ < 0, for all parameter choices. Therefore we have ℓ = 0, γ0 = 1, and ζ0 = 1, and from Theorem 3.2 we know that the space of feasible references is the entire output space Yss = Y . Moreover, in view of the right invertibility of the system, we can exploit the result in Corollary 3.2 which guarantees the solvability of the tracking problem. In order to build a suitable feedback matrix, we follow the steps in the proof of Theorem 3.3, which is constructive. Let us first assign reasonable values to the parameters of the problem: 10 Note that it is clear from the context whether s is used to denote a specific i tank or its basis area, and we do prefer to avoid the introduction of further symbols for the sake of readability. 11 The system may become nonright-invertible if the tanks are grouped twoby-two in the same quadrant, and may have two invariant zeros at the origin if each pair of connected tanks is distributed on quadrants that are opposite with respect to the origin; obviously, these are meaningless design choices.

18

F. Padula, L. Ntogramatzidis and E. Garone / Systems & Control Letters 126 (2019) 8–20

Fig. 3. System outputs (left) and state (right).

we consider all the tanks to have basis areas of 400 m2 . The tanks s1 and s2 contain oil which has density ρ1 = 870 kg/m3 while the tanks s3 and s4 contain water with density ρ2 = 1000 kg/m3 . All the hydraulic resistances are 0.1 kg s/m5 . The four tanks are symmetrically distributed with respect to the coordinate axes so that −x1 = −x2 = x3 = x4 = 50 m and −y1 = y2 = −y3 = y4 = 10 m. We have Γ = Λ = {0}, so that λ1 = 0, and γλ1 = 1, and dim R = 3. Therefore, three closed-loop modes can be selected arbitrarily, while one closed-loop eigenvalue has to be 0. We choose the arbitrary modes to be Υ = {−0.02, −0.03, −0.04} so that π1 = −0.02, π2 = −0.[03, and ] π3 = −0.04 and vπ ,1 ℵ1 = ℵ2 = ℵ3 = 1. We select wπ1 ,1 ∈ ker[A − π1 I B],

[v

π2 ,1 wπ2 ,1

[v

]

π3 ,1 wπ3 ,1

]1

∈ ker[A − π3 I ∈ ker[A − π2 I B] and [ vλ ,γ ] 1 λ1 also select a vector wλ ,γλ ∈ T = ker PΣ (0) 1 1 ⎤ ⎡

B]. We

100

⎢ 3 ⎥ ⎥ ⎢ [ 1 ] [ 8 ] [0] ⎢ 1 ⎥ ⎥ ⎢ = ⎢ 100 ⎥ , wπ1 ,1 = 0 ; vπ2 ,1 = −11 , wπ2 ,1 = −48 ; 0 0 5 ⎥ ⎢ ⎣ 3 ⎦

vπ1 ,1

31

vπ3 ,1 =

[

31 1 −1 0 0

]

, wπ3 ,1 =

[ 12 ] 0 0

; vλ1 ,γλ1 =

[

1 −1 3 5

]

, wλ1 ,γλ1 =

[ 16 ] 32 0

.

The solvability of the problem guarantees that the vectors vi,j are linearly independent and the friend matrix can be computed in finite terms as F = [wπ1 ,1

wπ2 ,1

× [vπ1 ,1

wπ3 ,1

vπ2 ,1

wλ1 ,γλ1 ]

vπ3 ,1

vλ1 ,γλ1 ]−1 =

[ 6.372 −5.628

] −0.8

0 5.952 5.952 −4 −8.8 0 0 0 0

.

We choose r¯ = [−5000 −40000 10]⊤ and we consider an initial condition where only the tank s1 is filled [ with ] 5 m of crude [ ] oil, i.e., x0 = [5 0 0 0]⊤ . We compute uxss = PΣ (0)† 0r¯ , ss which is solvable because r¯ ∈ Yss is a feasible reference, obtaining

[ −0.1734 ] xss =

1.1707 −0.5362 −0.5362

, uss =

[

0 0 10

]

.

As expected, the triple (F , xss , uss ) solves the tracking problem as show in Fig. 3. Nevertheless the state does not attain the value xss as t tends to ∞. From Theorem 3.4 we know that the problem is also strongly solvable because the system does not have uncontrollable eigenvalues in iR \ {0}. Applying the

machinery in the proof of Theorem 3.4, we first compute A∞ and then we solve (23) obtaining

[ 1.828 ] xss =

3.172 1.325 1.325

, uss =

[

0 0 10

]

,

where, as expected, xss is the steady-state value of the state variable in Fig. 3. Note that we do not assign eigenvalues on the imaginary axis, but despite all the possible model and pole placement uncertainties, the presence of a pole at the origin of the complex plane is unavoidable. Indeed, the unreachable eigenvalue at the origin is structural and connected with the fact that the global amount of oil loaded onto the ship cannot change. Therefore, this does compromise the robustness of the system because the pole cannot migrate into the right half plane. Now let us consider a different situation: suppose that the pump p1 is broken, and the valve r1 is open to avoid an excessive lateral unbalance. We are interested in understanding if we can track some references, i.e., if we can, at least partially, regulate the balance of the ship. In this new configuration we have

⎡ −ρ

1

ρ1

⎤ 0

0

⎢ s1 r1 s1 r1 ⎢ ⎢ −ρ1 ρ1 ⎢ 0 ⎢ s2 r1 s2 r1 0 A=⎢ ⎢ −ρ2 ρ2 ⎢ 0 0 ⎢ s3 r 2 s3 r 2 ⎢ ⎣ ρ2 −ρ2 0

0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎦

⎡

0 0 0 0

⎢ 1

B=⎣

s3

0

⎤ ⎥ ⎦,

D=

[0 0] 00 01

,

0 0

s4 r 2 s4 r 2

while the matrix C remains unchanged. The system is no longer right invertible and we have ℓ = 1, γ0 = 1, and ζ0 = 1, so that dim Yss = 2 and Yss = [C

D] ker[A

B] = im

[0 0] 10 01

.

From the previous equation it is clear the we can only control the longitudinal balance of the ship and the water flow of the engine, but not the lateral balance, i.e., we cannot regulate the first output y1 . Nevertheless, since the system is nonright-invertible, the feasibility of a reference does not imply that the tracking problem is solvable. We have two uncontrollable eigenvalues, so that Γ = {−0.0465, 0} and one of them is on the imaginary axis; therefore Λ = {0}. The dimension of the reachable subspace is dim R = 2, so we can assign two arbitrary closed-loop eigenvalues, e.g., π1 =

F. Padula, L. Ntogramatzidis and E. Garone / Systems & Control Letters 126 (2019) 8–20

19

Fig. 4. System outputs (left) and state (right).

−0.02, π2 = −0.03. First, we compute ⎡ 0.5178 ⎤ [ ] 0.5178 A B ⎦. T = ker = im ⎣ 00..4815 4815 C

D

0 0

Secondly, we check the solvability condition dim H = dim P (T + G ) + R

(

)

( = dim P T + ker[A + 0.0465 I

)

dim R

+

∑

B]

ker[A − πi I

B]

i=1

⎡ 0.5178 ⎤

⎛

0.5178 im 00..4815 4815 0 0 0 0 0 0 0.0207 −0.0207 0.1036 0.1036 −0.9944 −0.9944 0 0

= dim P ⎝

⎡ + im ⎣

⎡

0.7071 −0.7071 0 0 0 0

⎦ + im ⎣

⎣

0 0 0 0 0 1

0 0 −0.2946 0.3426 −0.8921 0

0 0 0 0 0 1

⎤ ⎦

⎤⎞

⎡ −ρ

⎦⎠ = 4.

Theorem 3.3 ensures that the tracking problem is solvable. We compute F , xss , uss as in the previous case obtaining F =

[ 3.8374 5.0906 0 −9.6 ] 0

0

0

0

,

[ xss =

2.5 2.5 1.325 1.325

]

,

uss =

[

0 10

]

and for the feasible reference r¯ = [0 −40000 10]⊤ we obtain the results in Fig. 4. In order to be able to laterally balance the ship one may try to open the valve r3 , with the idea of maintaining a constant flow from s3 to s4 , so that a steady difference between the levels of s3 and s4 can be maintained to laterally balance the ship. In this new configuration, A is the only matrix which changes:

⎡ −ρ

1

ρ1

⎤ 0

0

s4 r 2

s4 r2 r3

⎢ s1 r1 s1 r1 ⎥ ⎢ ⎥ ⎢ −ρ1 ρ1 ⎥ ⎢ ⎥ 0 0 ⎢ s2 r1 s2 r1 ⎥ ⎥. A = ⎢ ⎢ ⎥ −ρ ρ 2 2 ⎢ 0 ⎥ 0 ⎢ ⎥ s3 r 2 s3 r 2 ⎢ ⎥ ⎣ ρ2 −ρ2 (r2 + r3 ) ⎦ 0

0

At first glance this naive approach may seem promising because Yss = Y (the calculations are omitted for the sake of brevity). The fact that the space of feasible references coincides with the entire output space when the system is nonright-invertible should already suggest that the tracking problem is not solvable. Indeed, the condition of Theorem 3.3 is not satisfied. The fact that the problem is not solvable has a precise interpretation: the possibility to track every specific reference always depends upon the initial condition x0 , which implies that we are not able to track a feasible reference from every initial condition and ultimately to solve the tracking problem. In the specific example, if we achieve a given lateral balance y1 = r¯1 , the longitudinal balance y2 will always depend on the amount of crude oil contained in s1 and s2 , which is clearly uncontrollable and non-asymptotically stable. Let us consider a different solution. We close the valve s3 and we change the piping layout so that p3 pumps the cooling water of the engine from s3 into s4 (see Fig. 2). In this new configuration the matrices become 1

ρ1

⎤ 0

0

⎢ s1 r1 s1 r1 ⎢ ⎢ −ρ1 ρ1 ⎢ 0 ⎢ s2 r1 s2 r1 0 A=⎢ ⎢ −ρ2 ρ2 ⎢ 0 0 ⎢ s3 r2 s3 r2 ⎢ ⎣ ρ2 −ρ2 0

0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎦

⎡

0 0

0 0

⎤

⎢ 1 −1 ⎥ ⎢ ⎥ ⎥ B=⎢ ⎢ s3 s3 ⎥ , ⎣ 1 ⎦ 0

D=

[0 0] 00 01

s4

s4 r2 s4 r2

and the system is nonright-invertible. We have ℓ = 1, γ0 = 1, and ζ0 = 1 so that dim Yss = 2 and Yss = im

[ −0.0010 −1 0

1

−0.001 −0.0025

]

.

In this configuration the condition of Theorem 3.3 is satisfied. Let us choose the feasible reference Yss ∋ r¯ = [−4000 −32000 10]⊤ , which guarantees the correct flow rate of cooling water. When the nonright-invertibility prevents the possibility of tracking every reference signal in Y , this solution represents a good compromise, because it guarantees that the required flow of cooling water r¯3 = 10 is pumped through the engine, but also that the ship balance, r¯1 = −4000 and r¯2 = −32000, is close to the initially required values of −5000 and −40000. Following the aforementioned procedure, with π1 = −0.02, π2 = −0.03 (note

20

F. Padula, L. Ntogramatzidis and E. Garone / Systems & Control Letters 126 (2019) 8–20

Fig. 5. System outputs (left) and state (right).

that dim R = 2 so we do not need π3 ) we obtain F =

[

] −6.9632 −8.3897

7.7261 6.5520 −5.2518 10.5668 −6.3391 0.6241

,

[ xss =

2.5 2.5 1.025 2.025

]

,

uss =

[

0 10

]

and the results are shown in Fig. 5. Note that in this case the last row of F is nonzero and therefore the dynamics of the state influences also y3 despite the presence of a direct feedthrough. Therefore, the output y3 does not attain its final value instantaneously. Concluding remarks In this paper we provided necessary and sufficient constructive conditions for the solvability of the tracking problem for MIMO LTI systems that are possibly nonright-invertible, nonasymptotically stabilizable and possibly with invariant zeros at the origin. We only assumed the simple stabilizability, therefore addressing a wider class of systems. We first developed a stability framework which is appropriate to deal with nonasymptotically stabilizable systems. We proved that, under the simple stabilizability assumption, a feedback matrix that renders the closed loop BIBS stable also renders the closed loop simply stable. Then, we proposed a solution for the tracking problem which is articulated in a two-step procedure. Firstly, we provided a characterization of the feasible references. Finally, we found the conditions under which a stabilizing feedback exists such that every feasible reference can be tracked. Acknowledgments The authors are grateful to Prof. A. Ferrante and Prof. A. Visioli for their intelligent advices on the topic of this paper. Conflict of interest The authors declare that they have no conflict of interest. References [1] D. Limon, I. Alvarado, T. Alamo, E.F. Camacho, MPC for tracking piecewise constant references for constrained linear systems, Automatica 44 (9) (2008) 2382–2387.

[2] E. Garone, S. Di Cairano, I. Kolmanovsky, Reference and command governors for systems with constraints: A survey on theory and applications, Automatica 75 (2017) 306–328. [3] R. Schmid, L. Ntogramatzidis, A unified method for the design of nonovershooting linear multivariable state-feedback tracking controllers, Automatica 46 (2010) 312–321. [4] Y. He, B.M. Chen, C. Wu, Composite nonlinear control with state and measurement feedback for general multivariable systems with input saturation, Systems Control Lett. 54 (5) (2005) 455–469. [5] B.M. Mirkin, P.-O. Gutman, Output feedback model reference adaptive control for multi-input-multi- output plants with state delay, Systems Control Lett. 54 (10) (2005) 961–972. [6] R. Schmid, L. Ntogramatzidis, Nonovershooting and nonundershooting exact output regulation, Systems Control Lett. 70 (2014) 30–37. [7] A. Piazzi, A. Visioli, Using stable input–output inversion for minimum time feedforward constraint regulation of scalar systems, Automatica 41 (2) (2005) 305–313. [8] G.F. Franklin, J.D. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems, third ed., Addison-Wesley, Reading, MA, 1994. [9] L. Ntogramatzidis, J.-F. Trégouët, R. Schmid, A. Ferrante, Globally monotonic tracking control of multivariable systems, IEEE Trans. Automat. Control 61 (9) (2016) 2559–2564. [10] S. Galeani, A. Serrani, G. Varano, L. Zaccarian, On input allocationbased regulation for linear over-actuated systems, Automatica 52 (2015) 346–354. [11] H. Trentelman, A. Stoorvogel, M. Hautus, Control theory for linear systems, in: Ser. Communications and Control Engineering, Springer, Great Britain, 2001. [12] A.G.J. MacFarlane, N. Karcanias, Poles and zeros of linear multivariable systems: a survey of the algebraic, geometric and complex variable theory, Internat. J. Control 24 (1) (1976) 33–74. [13] H. Aling, J. Schumacher, A nine-fold canonical decomposition for linear systems, Internat. J. Control 39 (4) (1984) 779–805. [14] H.H. Rosenbrock, State-Space and Multivariable Theory, Wiley, New York, 1970. [15] W.M. Wonham, Linear Multivariable Control: A Geometric Approach, third ed., Springer-Verlag, 1985. [16] G. Basile, G. Marro, Controlled and Conditioned Invariants in Linear System Theory, Prentice Hall, Englewood Cliffs, New Jersey, 1992. [17] L. Ntogramatzidis, F. Padula, A general approach to the eigenstructure assignment for reachability and stabilizability subspaces, Systems Control Lett. 106 (2017) 58–67. [18] L. Ntogramatzidis, R. Schmid, Robust eigenstructure assignment in geometric control theory, SIAM J. Control Optim. 52 (2) (2014) 960–986. [19] R. Schmid, L. Ntogramatzidis, T. Nguyen, A.P. Pandey, A unified method for optimal arbitrary pole placement, Automatica 50 (8) (2014) 2150–2154. [20] G.P. Liu, R. Patton, Eigenstructure Assignment for Control System Design, in: Ser. Communications and Control Engineering, John Wiley & Sons, Inc., New York, NY, USA, 1998.