The lq/lp Hankel norms of discrete-time positive systems across switching

ABSTRACT In this study, we focus on the Hankel norms of linear time-invariant (LTI) discrete-time positive systems across a single switching. The Hankel norms are defined as the induced norms from vector-valued past inputs to vector-valued future outputs across a system switching and a state transition at the time instant zero. A closed-form characterization of the Hankel norm in this switching setting for general LTI systems can readily be derived as the natural extension of the standard Hankel norm. Thanks to the strong positivity property, we show that we can successfully characterize the Hankel norms for the positive system switching case even in some combinations of p, q being . In particular, some of them are given in the form of linear programming (LP) and semidefinite programming (SDP). These LP- and SDP-based characterizations are particularly useful for the analysis of the Hankel norms where the systems of interest are affected by parametric uncertainties.


Introduction
This study is concerned with the analysis of the l q /l p Hankel norms of discrete-time positive systems across a single switching. Early studies on positive systems focused on controllability and reachability analysis [1,2], positive realization [3,4], and positive stabilization [5]. Then, the analysis and synthesis of positive systems using convex optimization have attracted great attention, and some fruitful results have been obtained to this date. Those results include the induced norm analysis using linear programming [6,7], Kalman-Yakubovich-Popov lemma with diagonal Lyapunov variables [8,9], positive system synthesis using geometric programming [10], and positive system analysis using copositive programming [11]. Surveys on recent studies of positive systems can be found at [12,13].
Analysis and synthesis of switched positive systems form a rather new and active research area in the theory of positive systems [14][15][16][17]. In [18], relatively new results on the induced norms of positive systems shown in [6,7,19,20] are also successfully extended to the switched case. On the other hand, the author of [21,22] considered the case where a general (i.e. nonpositive) continuous-time LTI system switches to another one at the time instant zero, and introduced the L 2 /L 2 Hankel norm as the induced norm from vector-valued L 2 past inputs to vector-valued L 2 future outputs. Here, the past input is injected to the system before switching, driving the initial state of the system after switching to some nonzero values along with the state transition at the time instant zero, and the future output corresponds to the initial response of the system after switching. By using this norm, we can quantitatively evaluate the magnitude of the "bumpy response" caused by switching.
The goal of this study is to derive the explicit characterizations of the l q /l p Hankel norms in the discretetime positive system switching setting. These norms are defined in exactly the same manner as [21,22], even though we evaluate the past inputs with l p norm and the future outputs with l q norm where p, q being 1, 2 or ∞. Namely, we deal with the l q /l p Hankel norms across a single switching, and these can be regarded (interpreted) as natural extensions of the standard (nonswitched) l q /l p Hankel norms. We focus only on single switching cases since in multiple switching cases such interpretation as Hankel norms is not possible. Similarly to [21,22], a closed-form characterization of the l 2 /l 2 Hankel norm in the general setting can readily be derived as the natural extension of the standard l 2 /l 2 Hankel norm of LTI systems. We then move on to the main issues and show that, thanks to the strong positivity property, we can successfully characterize the l q /l p Hankel norms for the positive system switching even in some combinations of p, q being 1, 2, ∞. In particular, some of them are given in the form of linear programming (LP) and semidefinite programming (SDP). These LP-and SDP-based characterizations are particularly useful for the analysis of the l q /l p Hankel norms where the systems of interest are affected by parametric uncertainties. We finally note that the continuous-time system counterpart of this study has been discussed in [23]. We use the following notation. The set of n × m real matrices is denoted by R n×m , and the set of n × m entrywise nonnegative (strictly positive) matrices is denoted by R n×m . For a matrix A, we also write A ≥ 0 (A > 0) to denote that A is entrywise nonnegative (strictly positive). We denote by 1 n ∈ R n the all-ones vector. The set of n × n real symmetric matrices is denoted by S n . For a matrix A ∈ S n , we write A 0 (A ≺ 0) to denote that A is positive (negative) definite. For a matrix A ∈ S n , we also denote by λ max (A) and d max (A) the maximum eigenvalue and the maximum diagonal entry of A, respectively. The maximum singular value of A ∈ R n×m is denoted by σ (A). Finally, for A ∈ R n×n , we denote by ρ(A) the spectral radius of A and we further define He{A} = A + A T .

Definition of the l q /l p Hankel norms across switching
Suppose two stable LTI systems p and f are given, which are the models of the system before and after switching at the time k = 0, respectively (see Figure 1). We assume that the state space realizations of p and f are given, respectively by f : Here, A p ∈ R n p ×n p , B p ∈ R n p ×n w , A f ∈ R n f ×n f , C f ∈ R n z ×n f with ρ(A p ) < 1 and ρ(A f ) < 1. We consider the case where the system p switches to the system f at k = 0 along with the state transition described by Here, S ∈ R n f ×n p is a given matrix. For the input signal w and the output signal z, we define For p, q = 1, 2, ∞ we also define Then, the l q /l p Hankel norm across switching from p to f with the state transition matrix S ∈ R n f ×n p is defined by (2), (3). (4) Note that x p (−∞) = 0 is tacitly assumed. In the following, we partition B p ∈ R n p ×n w and C f ∈ R n z ×n f as follows: ,

Basics of discrete-Time positive systems
Let us consider the LTI system G described by where A ∈ R n×n , B ∈ R n×n w , C ∈ R n z ×n . The impulse response g of the system G is given by The definition of the positivity of G and its characterization are given as follows.

Definition 3.1 ([24]):
The LTI system G given by (5) is called internally positive if its state x(k) and output z(k) are nonnegative for k ≥ 0 for any nonnegative input w(k) for k ≥ 0 and nonnegative initial state x(0).

Definition 3.3 ([24]):
The LTI system G given by (5) is called externally positive if its output z(k) is nonnegative for k ≥ 0 for any nonnegative input w(k) for k ≥ 0 and the zero initial state x(0) = 0.

Proposition 3.4 ([24]):
The system G given by (5) is In the following, we often identify p and f with p and f , respectively, where Then, we say that p and f are both internally positive iff p and f are both internally positive, i.e.
Similarly, we say that p and f are both externally positive iff p and f are both externally positive, i.e.

Existing results on the standard l q /l p Hankel norms positive systems
In the case where we can see that the l q /l p Hankel norm γ q/p defined by (4) reduces to the standard l q /l p Hankel norm of the system G which is denoted by G q/p . In the continuous-time system case, the L q /L p Hankel norms of general (i.e. nonpositive) LTI systems are studied by [25,26], and some of those results are made explicit for positive systems in [27]. Then, very recently, the counterpart results of [27] for discrete-time positive systems are derived in [28]. Some of them are given in the next proposition, where X ∈ S n and P ∈ S n stand for the controllability and observability Gramians of the system G given by (5), respectively. These are the unique solutions of the Lyapunov equations

Proposition 3.5 ([28]):
For the stable and externally positive system G given by (5) and (8), we have The characterizations (14), (15), (17), and (18) are valid even for general LTI systems and these are direct counterparts of the continuous-time results in [25]. The rest characterizations are direct counterparts of [27]. We finally note that explicit closed-form characterization of G ∞/1 is hardly available even for internally positive systems.

The l q /l p Hankel norms across switching: positive system results
We say that the system switching described by (1) In the following, we denote by X p ∈ S n p and P f ∈ S n f the controllability gramian of p and the observability Gramian of f , respectively. These are the unique solutions of the Lyapunov equations

Characterization of γ 2/2 for general system switching
We first note that γ 2/2 is given by This is the counterpart of the continuous-time system result by [22] and valid for general (i.e. nonpositive) switching. To provide a concise proof of (23), let us denote by the linear operator from w ∈ l p− to z ∈ l q+ in the system switching defined by (1), (2), (3). Namely, Then the result (23) readily follows from (24) as The worst-case input w ∈ l 2− that attains (23) can be given explicitly by where v ∈ R n p is the eigenvector corresponding to the It is very clear that (23) reduces to (17) in the standard Hankel norm setting (11).
In the case where (p, q) = (2, 2), on the other hand, explicit closed-form characterizations of γ q/p are hardly available for general switching. The difficulty in comparison with the standard l q /l p Hankel norms comes from the time-varying nature of the underlying system in the switched case. However, in the case of positive system switching defined by (10) and (22), we can still derive explicit characterizations for some combinations of (p, q). The key feature is that the linear operator given by (24) is positive under (10) and (22) in the sense that w ∈ l + p− leads to z ∈ l + q+ . This drastically facilitates the treatment of γ q/p particularly when p = ∞ and q = 1. The next lemma plays a key role in analysing the l q /l p Hankel norm γ q/p for the positive system switching. The proof of this lemma is given in the appendix section.  (22), suppose an input w ∈ l p− yields an output z ∈ l q+ where p, q being 1, 2, ∞. Define the input w ∈ l + p− associated with w ∈ l p− by w j (k) := |w j (k)| (k < 0, j = 1, . . . , n w ).
Then, the output z ∈ l q+ corresponding to the input w ∈ l + p− satisfies The next result follows from Lemma 4.1.  (22), suppose there exists an input w ∈ l p− with w p− = 1 such that the corresponding output z ∈ l q+ satisfies z q+ = γ for a given γ > 0. Then, there exists an input w ∈ l + p− with w p− = 1 such that the corresponding output z ∈ l + q+ satisfies z q+ ≥ γ .

Characterizations of γ q/∞ for positive system switching
In the case where we consider the l q /l p Hankel norms with p = ∞, we can readily see from Lemmas 4.1-4.3 that the next strong result holds.  (22), the l q /l ∞ Hankel norms with q being 1, 2, ∞ are attained by the input w ∈ l + ∞− given by w (k) = 1 n w (∀k < 0). This input leads to the initial state before switching and the initial state after switching From this lemma, we can readily obtain the next theorems.
Moreover, the following conditions are equivalent for a given γ > 0.
Furthermore, the next condition is also equivalent to (i) and (ii) if p and f are both internally positive, i.e. (9) holds.
The results (28), (31), and (33) readily follow by considering the initial response corresponding to the initial state (27). The proof for the equivalence of (i), (ii), and (iii) in Theorem 4.5 and (i) and (ii) in Theorem 4.6 are given in the appendix section. Important remarks on Theorems 4.5, 4.6, and 4.7 are as follows.

Remark 4.1: (i) It is clear that (28) and (31) reduce
to (19) and (20), respectively, in the case of (11). On the other hand, (33) looks much more complicated than (21), and we see that (21) can be obtained from (33) by assuming (11) and the maximum in (33) is attained at k f = 0. In the time-invariant case (11), it is allowed to consider the "shift" of the input signal w due to the time-invariant nature of the system and this intuitively explains the reason why the maximum is attained at k f = 0. In fact, if we assume (11) in (33), we have and hence the maximum is actually attained at k f = 0. However, in the switched case, the intrinsic time-varying nature of the system does not allow us to conclude in such a way and we have to take the maximum over k f ≥ 0 as in (33). (ii) The SDP-and LP-based characterizations (29) and (30)

Characterizations of γ 1/p for positive system switching
When considering the l q /l p Hankel norms for the positive system switching, we can confine ourselves to nonnegative input signals from Lemma 4.2. This leads to . Namely, we can characterize γ 1/p as follows: From this expression, we can see that γ 1/p is identical to the l ∞ /l p Hankel norm G ∞/p of the single-output, stable and externally positive LTI system G given by From this key observation and Proposition 3.5, we can obtain the next theorems.
(35) Moreover, the following conditions (i) and (ii) are equivalent for a given γ > 0.
We can verify (34) from γ 1/1 = G ∞/1 and (15). Similarly, we can verify (35) from γ 1/2 = G ∞/2 and (18). The equivalence of (i) and (ii) in Theorem 4.9 follows by exactly the same argument used in the proof of the equivalence of (i) and (ii) in Theorem 4.6. It should be noted that the expression of γ 1/∞ given by (28) can also be obtained from the fact that γ 1/∞ = G ∞/∞ and (21). Important remarks on Theorems 4.8 and 4.9 are as follows.

Remark 4.2:
(i) We can see that (35) reduces to (16) in the case of (11). On the other hand, we see that (34) can be reduced to (13) by assuming (11) and the maximum in (34) is attained at k p = 0. In fact, in the case where (11) holds in (34), we see that and hence the maximum is actually attained at k p = 0. (ii) The worst-case input w ∈ l 2− that attains (35) can be given explicitly by (iii) Obviously, the duality holds between γ 2/∞ given by (31) and γ 1/2 given by (35). Namely, we see that the l 2 /l ∞ Hankel norm for the positive system switching from p to f via S ∈ R n f ×n p + is equivalent to the l 1 /l 2 Hankel norm on the positive system switching from f to p via S T ∈ R and p :

The Hankel norms γ ∞/1 , γ ∞/2 and γ 2/1 for general switching
In this subsection, we provide explicit characterizations of γ ∞/1 , γ ∞/2 , and γ 2/1 . The results in this subsection can be derived without relying on the positivity and hence they are valid even for general (i.e. nonpositive) switching cases.
Theorem 4.10: For the system switching from p to f described by (1), (2), and (3), we have Remark 4.3: (i) We note that the expression of γ ∞/∞ given by (33) can also be obtained from the fact that and (21). (ii) We can see that (44) reduces to (15) in the case of (11). On the other hand, we see that (45) can be reduced to (18) by assuming (11) and the maximum in (45) is attained at k f = 0. In fact, in the case where (11) holds in (45), we see that and hence the maximum is actually attained at k f = 0.

Characterization of γ 2/1
We next consider the characterization of γ 2/1 . To this end, we define Then we can obtain the next lemma that is the discretetime system counterpart of the result in [29] dealing with continuous-time systems.

Lemma 4.11:
Let us consider the stable LTI system G given by (5) with x(0) = 0 and define its induced norm G ind (∞,2)/1 from w ∈ l 1+ to z ∈ l (∞,2)+ by Then, we have We now go back to the analysis of Hankel norm γ 2/1 . If we define C f := P f 1/2 S ∈ R n f ×n p , we can see from (24) that From this key observation and Lemma 4.11, the next theorem follows.

Numerical examples
Let us consider the case where the systems p , f , and the matrix S in (1), (2), and (3), respectively, are affected by polytopic-type uncertainty of the form ⎡ Here, we assume that the given matrices f , and S [l] (l = 1, . . . , N) that define the vertices of the polytope satisfy A [l] p ∈ R In the following, we denote by p,α , f,α , and S α the positive systems and the nonnegative matrix corresponding to the parameter α ∈ α. We assume that both p,α and f,α are stable for any α ∈ α. Under these assumptions, we denote by γ q/p (α) the l q /l p Hankel norm for the positive system switching from p,α to f,α via S α .
The problem we consider in this section is to compute the worst casel q /l p Hankel norm γ q/p defined by γ q/p := max α∈α γ q/p (α). Even though exact and efficient computation of γ q/p is hard, we can compute its upper bound efficiently by using the SDP and LP characterizations provided in the preceding section. For instance, from (29), we see that we can obtain an upper bound of γ 1/∞ by solving the following SDP: As a concrete example, let us consider the case where N = 2 and ⎡ ⎢ ⎣ C [1] f 0 0 A [1] f S [1] 0 0 A [1] p B [1] p ⎤ ⎥ ⎦ C [2] f 0 0 A [2] f S [2] 0 0 A [2] p B [2] p ⎤ ⎥ ⎦

Conclusion
In this paper, we analysed the l q /l p Hankel norms of positive systems across a single switching. We derived explicit representations of the l q /l p Hankel norms for p, q being 1, 2, ∞, where those new results for (q, p) = {(∞, 1), (∞, 2), (2, 1)} are valid even for general (nonpositive) switching cases. In particular, for

Disclosure statement
No potential conflict of interest was reported by the author.

Funding
This work was supported by JSPS KAKENHI grant number JP18K04200.