An entropy-based bound for the computational complexity of a switched system

TL;DR

This paper introduces an entropy-based bound for the computational complexity of switched systems, linking the joint spectral radius to system entropy and p-radius, and proposes a reduction method for low-rank matrices.

Contribution

It provides a new entropy-based guarantee for the sum of squares method's upper bound on the joint spectral radius of constrained switched systems.

Findings

The entropy and p-radius influence the accuracy of stability bounds.

A reduction method simplifies the computation of the joint spectral radius for low-rank matrices.

The approach enhances understanding of stability analysis in hybrid systems.

Abstract

The joint spectral radius (JSR) of a set of matrices characterizes the maximal asymptotic growth rate of an infinite product of matrices of the set. This quantity appears in a number of applications including the stability of switched and hybrid systems. A popular method used for the stability analysis of these systems searches for a Lyapunov function with convex optimization tools. We analyse the accuracy of this method for constrained switched systems, a class of systems that has attracted increasing attention recently. We provide a new guarantee for the upper bound provided by the sum of squares implementation of the method. This guarantee relies on the p-radius of the system and the entropy of the language of allowed switching sequences. We end this paper with a method to reduce the computation of the JSR of low rank matrices to the computation of the constrained JSR of matrices of small dimension.

Equations

$$x_k=A_{{\sigma }_k}{x}_{k-1},{\sigma }_k\in [m]$$

$$\rho (\mathcal{A})=\underset{k\to \infty }{\lim }\underset{\sigma \in [m{]}^k}{\max }\Vert A_{{\sigma }_k}\cdots A_{{\sigma }_2}{A}_{{\sigma }_1}{\Vert }^{1/k}.$$

$$x_k=A_{{\sigma }_k}{x}_{k-1},({\sigma }_1,\dots ,{\sigma }_k)\in G_k.$$

$$\rho (G,\mathcal{A})=\underset{k\to \infty }{\lim }{\hat{\rho }}_k(G,\mathcal{A},\Vert \cdot \Vert ),$$

$${\hat{\rho }}_k(G,\mathcal{A},\Vert \cdot \Vert )=\underset{s\in G_k}{\max }\Vert A_s{\Vert }^{1/k}.$$

$$A_1$$

$$A_3$$

$$A=\left(\begin{array}{cc}0.94& 0.56\\ 0.14& 0.46\end{array}\right)\text{ and }B=\left(\begin{array}{c}0\\ 1\end{array}\right).$$

$${\rho }_{\text{SOS-}2d}(\mathcal{A})$$

$$\underset{p_v(x)\in {\mathbb{R}}_{2d}[x],\overline{\gamma }\in \mathbb{R}}{\inf }\overline{\gamma }$$

$${\overline{\gamma }}^{2d}{p}_u(x)-p_v(A_{\sigma }x)$$

$$p_v(x)$$

$$\sum _{v\in V}{\int }_{{\mathbb{S}}^{n-1}}{p}_v(x)dx$$

$$\rho (G,\mathcal{A})\le {\rho }_{\text{SOS-}2d}(G,\mathcal{A}).$$

$$h(G)=\underset{k\to \infty }{\lim }\frac{1}{k}{\log }_2|G_k|.$$

$$h(E)=\underset{k\to \infty }{\lim }\frac{1}{k}{\log }_2|E_k|.$$

$${\rho }_p(G,\mathcal{A})=\underset{k\to \infty }{\lim }{\left[|E_k{|}^{-1}\sum _{s\in E_k}\Vert A_s{\Vert }^p\right]}^{\frac{1}{pk}}.$$

$$\underset{k\to \infty }{\lim }{\left[\sum _{s\in E_k}\Vert A_s{\Vert }^p\right]}^{\frac{1}{pk}}$$

$${\rho }_p(G,\mathcal{A})=2^{-h(E)/p}\underset{k\to \infty }{\lim }{\left[\sum _{s\in E_k}\Vert A_s{\Vert }^p\right]}^{\frac{1}{pk}}.$$

$$\begin{gathered} {\rho }_p(G,\mathcal{A})\le {\rho }_q(G,\mathcal{A})\le \rho (G,\mathcal{A}) \\ \le 2^{h(E)/q}{\rho }_q(G,\mathcal{A})\le 2^{h(E)/p}{\rho }_p(G,\mathcal{A}). \end{gathered}$$

$${\rho }_{\text{SOS-}2d}(G,\mathcal{A})\le 2^{h(E)/2d}{\rho }_{2d}(G,\mathcal{A})\le 2^{h(E)/2d}\rho (G,\mathcal{A}).$$

$${\rho }_{\text{SOS-}2d}(G,\mathcal{A})\le {\left(\genfrac{}{}{0ex}{}{n+d-1}{d}\right)}^{\frac{1}{2d}}\rho (G,\mathcal{A}).$$

$$\max\Big{\{}{n+d-1\choose d}^{-\frac{1}{2d}},2^{-h(E)/2d}\Big{\}}\rho_{\text{SOS-}2d}(G,\mathcal{A})\\ \leq\rho(G,\mathcal{A})\leq\rho_{\text{SOS-}2d}(G,\mathcal{A}).$$

$$p_{v,0}(x)$$

$$p_{v,k+1}(x)$$

$$p_{v,\infty }(x)=q_v(x)+\frac{1}{\tau }\sum _{(u,v,\sigma )\in E}{p}_{u,\infty }(A_{\sigma }x)$$

$$\tau p_{v,\infty }(x)-p_{u,\infty }(A_{\sigma }x)=\tau q_v(x)+\sum _{\begin{array}{c}(u^{\prime },v,{\sigma }^{\prime })\in E,\\ (u^{\prime },{\sigma }^{\prime })\ne (u,\sigma )\end{array}}{p}_{u^{\prime },\infty }(A_{{\sigma }^{\prime }}x)$$

$$p_{v,0}(x)$$

$$p_{v,k+1}(x)$$

$$p_{v,k}(x)=\frac{1}{{\tau }^k}\sum _{s\in E_k^{-}(v)}{q}_{s(1)}(A_{s}x)$$

$$p_{v,k}(x)$$

$$\le \frac{\beta }{{\tau }^k}\Vert x{\Vert }^{2d}\sum _{s\in E_k^{-}(v)}\Vert A_s{\Vert }^{2d}$$

$$\sum _{v\in V}{p}_{v,k}(x)$$

Full text

We analyse the accuracy of this method for constrained switched systems, a class of systems that has attracted increasing attention recently. We provide a new guarantee for the upper bound provided by the sum of squares implementation of the method. This guarantee relies on the $p$-radius of the system and the entropy of the language of allowed switching sequences.

We end this paper with a method to reduce the computation of the JSR of low rank matrices to the computation of the constrained JSR of matrices of small dimension.

I Introduction

In recent years, the study of the stability of hybrid systems has been the subject of extensive research using methods based on classical ideas from Lyapunov theory and modern mathematical optimization techniques. Even for switched linear systems, arguably the simplest class of hybrid systems, determining stability is undecidable and approximating the maximal asymptotic growth rate that a trajectory can have is NP-hard [1]. Despite these negative results, the vast range of applications has motivated a wealth of algorithms to approximate this maximal asymptotic growth rate.

A switched linear system is characterized by a finite set of matrices $\mathcal{A}\triangleq\{A_{1},A_{2},\ldots,A_{m}\}\subset\mathbb{R}^{n\times n}$ and the iteration

[TABLE]

where $[m]$ denotes the set $\{1,2,\ldots,m\}$.

The maximal asymptotic growth rate of this iteration is given by the joint spectral radius (JSR). The JSR $\rho(\mathcal{A})$ of a finite set of matrices $\mathcal{A}$ is defined as

[TABLE]

This definition is independent of the norm used.

The JSR was introduced by Rota and Strang [2] and has many applications such as co-simulation [3], wavelets, the capacity of some particular codes, zero-order stability of ordinary differential equations, congestion control in computer networks, curve design and networked and delayed control systems; see [4] for a survey on the JSR and its applications.

In some applications the values that $\sigma_{k}$ can take in (1) may depend on $\sigma_{k-1},\sigma_{k-2},\ldots$. These constraints are often conveniently represented using a finite automaton and the JSR under such constraints is called constrained joint spectral radius (CJSR) [5]; an example of constrained switched system is given by Example 1 and its automaton is illustrated by Figure 1. Constrained switched systems are used in a variety of applications including networked control [6, 7] and coordination of a network of autonomous agents [8]. Moreover, even if a switched system is unconstrained, studying an associated constrained system generated by path-complete methods enhance our ability to analyze the stability [9] or stabilize [10] the original unconstrained switched system.

The automaton representing the constraints can be represented by a strongly connected labelled directed graph $G(V,E)$, possibly with parallel edges. The labels are elements of the set $[m]$ and $E$ is a subset of $V\times V\times[m]$. We say that $(u,v,\sigma)\in E$ if there is an edge between node $u$ and node $v$ with label $\sigma$.

We use $E_{k}$ to denote the subset of $E^{k}$ (i.e. the $k$th cartesian power of $E$) that represents valid paths of length $k$. The $k$-tuple $(\sigma_{1},\sigma_{2},\ldots,\sigma_{k})$ is said to be $G$-admissible if $\sigma_{1},\ldots,\sigma_{k}$ are the respective labels of a path of length $k$. We denote the set of all $k$-tuples of $[m]^{k}$ that are $G$-admissible as $G_{k}$. The matrix product $A_{\sigma_{k}}\cdots A_{\sigma_{1}}$ is written $A_{s}$ when $s=(\sigma_{1},\ldots,\sigma_{k})$ or $s$ is a path with these respective labels.

The iteration (1) is rewritten as follows to take the automaton into account:

[TABLE]

The definition of the JSR is generalized as follows for constrained systems.

Definition 1** ([5]).**

The constrained joint spectral radius (CJSR) of a finite set of matrices $\mathcal{A}$ constrained by an automaton $G$, denoted as $\rho(G,\mathcal{A})$, is

[TABLE]

where

[TABLE]

The arbitrary switching case (1) can be seen as the particular case when the automaton has only one node and $m$ self-loops with labels $1,\ldots,m$.

Example 1** (Running example).**

We borrow the example of [11, Section 4]. It is based on a state-feedback control that might undergo dropouts in its state feedback. The set of matrices $\mathcal{A}$ is composed of the following four matrices

[TABLE]

where $k_{1}=-0.49$, $k_{2}=0.27$,

[TABLE]

The corresponding automaton is represented by Figure 1.

Approximating the CJSR usually consists in certifying upper bounds $\overline{\gamma}$ to the CJSR by exhibiting Lyapunov functions or invariant sets for the matrices $A_{i}/\overline{\gamma}$ (see Section II for precise definitions). The search for such Lyapunov functions can naturally be written as a convex optimization program using sum of squares (SOS) programming [12]. It turns out that these Lyapunov methods cannot produce an arbitrarily bad CJSR approximation: bounds are known on the accuracy of the estimate they deliver. Indeed, the following two bounds have been proved in the unconstrained case for the lowest upper bound $\overline{\gamma}$ that can be certified using sum of squares polynomials111A polynomial $p(x)$ is a sum of squares if there exists some natural number $k$ and $k$ polynomials $q_{i}(x)$ such that $p(x)=q_{1}^{2}(x)+\cdots+q_{k}^{2}(x)$. of degree $2d$, denoted $\rho_{\text{SOS-}2d}(\mathcal{A})$:

[TABLE]

The two guarantees are incomparable, as (3) depends on the dimension, and (4) depends on the number of matrices. However, only (3) has been generalized in the constrained case yet; see Theorem 3. Our main result is a generalization of the second guarantee: we relate the accuracy of the SOS-based approximation algorithm with the combinatorial complexity of the automaton. This complexity is measured by the entropy of the language of allowed switching signals. This new estimate of the accuracy of the SOS technique is always better than the previously existing one for sufficiently large sum of squares degree. According to the new estimate, the more constrained the system is, the smaller the entropy is and the better the accuracy of the method is. This shows that, in some sense, it is easier to analyse stability of constrained switched systems than unconstrained switched systems because the entropy of the language of allowed switching signals is smaller.

Constrained switched systems may also be useful to analyse abstraction techniques for complex control systems. Given a nonlinear system, an abstraction of the system can be constructed by a discretization of the state-space, such abstraction may enhance our ability to analyse the system [13]. The entropy of the language of allowed switching signals of the abstraction is related222The entropy of the abstraction with an $\varepsilon$-discretization measures the growth rate of the number of cells in which the state could be [14, Example 6.3.4] while the topological entropy is the limsup, with $\varepsilon\to\infty$, of the growth rate with $n$ of the cardinality of the largest $(n,\varepsilon)$-separated (or the smallest $(n,\varepsilon)$-spanning) set; see [15] for precise definitions. to the topological entropy of the nonlinear system [16, 15]. This suggests that the computational complexity of the abstraction is intrinsically related to the topological entropy of the nonlinear system and not to the specific choice of discretization, e.g. the value of $\varepsilon$. In [17], the authors use the Kullback-Leibler divergence of the uncertainty induced by a model to measure its fidelity. They measure the entropy of the uncertainty of the noise representing the part of the plant that is not accounted for in the model. This is similar to our work which measures the entropy of the uncertainty induced by an uncontrolled switching representing the loss of information due to the discretization. However, it is fundamentally different as we use this entropy to measure the computational complexity of the model and not the fidelity of the abstraction. Indeed, as we have seen, in our work this entropy is related to the topological entropy of the plant and not to the accuracy of the abstraction. Other appearances of the entropy in systems and control theory include [18, 19]; see [20] for an overview.

In [21], Ahmadi and Parrilo show how to reduce the computation of the JSR of matrices that are all of rank one to a combinatorial problem, which coincides with the CJSR of $1\times 1$ matrices (i.e. scalars). As a final contribution, we generalize this approach and give a reduction of the computation of the JSR (or CJSR) of matrices that are all of rank at most $r$ to the computation of the CJSR of $r\times r$ matrices.

The paper is organized as follows. In Section II, we give the SOS program searching for Lyapunov functions and we give our new estimate for its accuracy. The new bounds explicitly depend on the allowable transitions, through the graph $G(V,E)$. In Section III, we give the low rank reduction mentioned above.

Reproducibility

The code used to obtain the results is published on codeocean [22]. The algorithms are part of the SwitchOnSafety Julia [23] package [24] which computes invariant sets for hybrid sytems represented with the HybridSystems package [25]. The implementation relies on the SumOfSquares [26] and SetProg [27] extensions of JuMP [28]. The solver used is Mosek v8 [29].

II Stability and entropy

In this section, we give the SOS-based method to approximate the CJSR, we define the entropy of a constrained switching signal and the $p$-radius of a constrained switched system and we show how the performance guarantee of the method is related to the entropy of the switching signal and the $p$-radius of the switched system.

II-A Stability

As introduced in [12] and generalized in [11] for the constrained case, homogeneous333A homogeneous polynomial of degree $2d$ is a polynomial for which the degree of each monomial is $2d$. The polynomial is called homogeneous as for any real number $\lambda$, we have $p(\lambda x)=\lambda^{2d}p(x)$. polynomials of degree $2d$ can be used to certify upper bounds on the CJSR.

Proposition 1** ([30, Theorem 1]).**

Consider a finite set of matrices $\mathcal{A}$ constrained by an automaton $G(V,E)$. Suppose that there exist $|V|$ strictly positive homogeneous polynomials $p_{v}(x)$ of degree $2d$ such that $p_{v}(A_{\sigma}x)\leq\overline{\gamma}^{2d}p_{u}(x)$ holds for all edge $(u,v,\sigma)\in E$. Then $\rho(G,\mathcal{A})\leq\overline{\gamma}$.

We relax the positivity condition of Proposition 1 by the more tractable sum of squares (SOS) condition and define $\rho_{\text{SOS-}2d}(G,\mathcal{A})$ as the solution of the following sum of squares program.

Program 1** (Primal).**

[TABLE]

Remark 1**.**

In practice we can replace (6) and (7) by “$p_{v}(x)-\epsilon\|x\|_{2}^{2d}$ is SOS” for any $\epsilon>0$. This constrains $p_{v}(x)$ to be in the interior of the SOS cone, which is sufficient for $p_{v}(x)$ to be strictly positive. The bounds given in Section II-D are valid if $p_{v}(x)$ is in the interior of the SOS cone.

Remark 2**.**

The constraint (5) is equivalent to “$p_{u}(x)-p_{v}(A_{\sigma}x/\overline{\gamma})$ is SOS” hence the 1-sublevel sets of the polynomials $p_{v}$ provide invariant sets for the matrices $A_{\sigma}/\overline{\gamma}$ as claimed in the introduction.

By Proposition 1, a feasible solution of Program 1 gives an upper bound for $\rho(G,\mathcal{A})$, and thus, for any positive degree $2d$,

[TABLE]

Example 2**.**

Consider the unconstrained system [21, Example 2.1] with $m=3$: $\mathcal{A}=\{A_{1}=e_{1}e_{2}^{\top},A_{2}=e_{2}e_{3}^{\top},A_{3}=e_{3}e_{1}^{\top}\}$ where $e_{i}$ denotes the $i$th canonical basis vector. For any $d$, a solution to Program 1 is given by $(p(x),\gamma)=(x_{1}^{2d}+x_{2}^{2d}+x_{3}^{2d},1).$

Example 3**.**

Let us reconsider our running example; see Example 1. The optimal solution of Program 1 is represented by Figure 2 for $2d=2$, 4, 10 and 12.

II-B Entropy

The entropy of a regular language is defined as follows.

Definition 2** ([14, Definition 4.1.1]).**

Given a regular language recognized by an automaton $G$, we define the entropy of the language as

[TABLE]

The entropy of a language generated by an automaton is easily computable, as we now recall. The logarithm of the spectral radius of the adjacency matrix of an irreducible444An automaton is irreducible if for every pair of nodes $u,v$, there exists a path from $u$ to $v$ accepted by the automaton. automaton gives the entropy of its edge shift.

Definition 3** ([14, Definition 2.2.5]).**

The edge shift of an automaton $G=(V,E)$ is the language recognized by the automaton $G^{\prime}=(E,E^{\prime})$ with the transitions $((u,v,\sigma),(v,w,\sigma^{\prime}),(v,w,\sigma^{\prime}))\in E^{\prime}$ for each $(u,v,\sigma),(v,w,\sigma^{\prime})\in E$. We denote the entropy of the edge shift of $G$ as $h(E)=h(G^{\prime})$.

Particularizing equation (9) to the edge shift gives

[TABLE]

It turns out that the entropy of the edge shift is equal to the entropy of the language recognized by the automaton if the automaton is right-resolving [14, Proposition 4.1.13].

Definition 4** ([14, Definition 3.3.1]).**

An automaton $G$ is right-resolving if for every vertex $v$, the outgoing edges have different symbols.

Every regular language is recognized by a right-resolving automaton. Moreover, there are automated ways to obtain such an automaton from a starting representation of a language with an automaton that is not right-resolving [14, Section 3.3].

II-C Constrained $p$-radius

The constrained $p$-radius is defined as follows.

Definition 5**.**

The constrained $p$-radius of a finite set of matrices $\mathcal{A}$ constrained by an automaton $G(V,E)$, denoted as $\rho_{p}(G,\mathcal{A})$, is

[TABLE]

Thus, the CJSR can be defined as the constrained $p$-radius for $p=\infty$.

Theorem 1 shows a relation between entropy of the switching signals and the $p$-radius.

Lemma 1** ([31, Corollary B.5]).**

The limit

[TABLE]

converges.

Theorem 1**.**

Consider a finite set of matrices $\mathcal{A}$ constrained by an automaton $G$. The following relation holds

[TABLE]

Proof.

By Lemma 1, (11) converges and by (10), $\lim_{k\to\infty}|E_{k}|^{-\frac{1}{pk}}=2^{-h(E)/p}$. ∎

II-D Performance guarantees

In this section, we provide a new bound that relates the accuracy of Program 1 to the entropy of the switching signal and the $p$-radius of the switched system.

An important property of the $p$-radius is that it is increasing in $p$.

Lemma 2** ([31, Lemma 3.7]).**

Consider a finite set of matrices $\mathcal{A}$ constrained by an automaton $G$. For any integers $p\leq q$,

[TABLE]

This Lemma is already known in the unconstrained case where $2^{h(E)}=m$ [32].

Remark 3**.**

Lemma 2 shows that the $p$-radius provides an upper and lower bound on the CJSR. See [33, 12] for methods based on the veronese liftings computing the $2d$-radius either by computing a spectral radius or by solving a linear program (see [34] for computation algorithms when $p$ is not an even integer).

We show the following bound stating that the solution found by Program 1 is at least as good as the bound obtained by computing the $2d$-radius (see Lemma 2).

Theorem 2**.**

Consider a finite set of matrices $\mathcal{A}$ constrained by an automaton $G$. For any positive integer $d$, the approximation given by Program 1 using homogeneous polynomials of degree $2d$ satisfies:

[TABLE]

Note that the second inequality in (13) is simply (12). Theorem 2 is proven at the end of this section.

We can see with (13) that if $h(E)=0$, the approximation is exact. This corresponds to the case where every node of $G$ has indegree and outdegree 1. In that case, the graph forms a cycle of some length $k$ and the CJSR is simply the $k$th root of the spectral radius of the product of the matrices along this cycle.

For the unconstrained switching case, $2^{h(E)}$ is equal to the number of matrices $m$. Theorem 2 is therefore the generalization of (4) to the constrained case. A generalization of (3) to the constrained case was already known (note that the bound does not take into account the particular structure of the automaton):

Theorem 3** ([11, Theorem 3.6]).**

Consider a finite set of matrices $\mathcal{A}\subset\mathbb{R}^{n\times n}$ constrained by an automaton $G$ and a positive integer $d$. The approximation $\rho_{\text{SOS-}2d}(G,\mathcal{A})$ given by Program 1 using homogeneous polynomials of degree $2d$ satisfies:

[TABLE]

The results of Theorem 2, Theorem 3 and (8) are summarized by the following corollary.

Corollary 1**.**

Consider a finite set of matrices $\mathcal{A}\subset\mathbb{R}^{n\times n}$ constrained by an automaton $G$ and a positive integer $d$, the approximation given by Program 1 using homogeneous polynomials of degree $2d$ satisfies:

[TABLE]

We see that we can have arbitrary accuracy by increasing $d$.

Our proof technique for Theorem 2 relies on the analysis of an iteration in the vector space of polynomials of degree $2d$. When this iteration converges, it converges to a feasible solution of Program 1. By analysing this iteration as affine iterations in this vector space, we derive a sufficient condition for its convergence and thus an upper bound for $\rho_{\text{SOS-}2d}(G,\mathcal{A})$.

Consider the iteration

[TABLE]

for fixed homogeneous polynomials $q_{v}(x)$ of degree $2d$ in $n$ variables (not necessarily different) and a constant $\tau>0$.

When this iteration converges, it converges to a feasible solution of Program 1.

Lemma 3**.**

Consider a constant $\tau>0$. If there exist homogeneous polynomials $q_{v}(x)$ in the interior of the SOS cone such that iteration (14) converges then $\rho_{\text{SOS-}2d}(G,\mathcal{A})\leq\tau^{\frac{1}{2d}}.$

Proof.

Suppose the iteration converges to the polynomials $p_{v,\infty}(x)$. It is easy to show by induction that $p_{v,k}(x)$ is SOS for all $k$. It is trivial for $k=0$ and if it is true for $k$ then it is also true for $k+1$ by (14). Since the SOS cone is closed, $p_{v,\infty}$ is SOS. Now by (14), for each $v\in V$,

[TABLE]

so $p_{v,\infty}(x)$ is also in the interior of the SOS cone. For each edge $(u,v,\sigma)$, by manipulating the above equation, we have

[TABLE]

so $\tau p_{v,\infty}(x)-p_{u,\infty}(A_{\sigma}x)$ is SOS. Therefore $(\{\,p_{v,\infty}(x):v\in V\,\},\tau^{\frac{1}{2d}})$ is a feasible solution of Program 1. ∎

In view of Lemma 3, it is thus natural to analyse under which condition iteration (14) converges.

Proof of Theorem 2.

Iteration (14) is an affine map on the vector space of homogeneous polynomials of degree $2d$. It is well known that if the convergence is guaranteed when we only retain the linear part of the affine map then it is also guaranteed for the affine iteration.

Therefore we can analyse instead the following iteration

[TABLE]

We can see that

[TABLE]

where $s(1)$ denotes the first node of the path $s$.

Consider a norm $\|\cdot\|$ of $\mathbb{R}^{n}$ and its corresponding induced matrix norm of $\mathbb{R}^{n\times n}$. For each $v\in V$, we know by continuity of $q_{v}(x)$ that there exist $\beta_{v}>0$ such that $q_{v}(x)\leq\beta_{v}\|x\|^{2d}$ for all $x\in\mathbb{R}^{n}$. Let $\beta=\max_{v\in V}\beta_{v}$, then

[TABLE]

By Theorem 1, if $\tau>2^{h(E)}\rho_{2d}(G,\mathcal{A})^{2d}$, then $\lim_{k\to\infty}\sum_{v\in V}p_{v,k}(x)=0$ hence $\lim_{k\to\infty}p_{v,k}(x)=0$ $\forall v\in V$ since the polynomials $p_{v,k}$ belong to a proper cone. We obtain the result by Lemma 3. ∎

II-E Improving the automaton-dependent bounds

If strong duality holds for a convex problem, its feasibility is equivalent to the non-existence of an infeasibility certificate (see [35, Section 5.8]). An infeasibility certificate contains one entry per constraint and if this entry is zero for a given constraint then the infeasibility certificate remains valid if the constraint is removed from the problem. In this section, we show how this fact allows to improve the guarantee given by Theorem 2 using the sparsity of the infeasibility certificate.

We show in [31, Lemma A.1] that strong duality holds for Program 1 with a fixed $\overline{\gamma}$. This allows Program 1 to be solved by binary search on $\overline{\gamma}$: Given a fixed value $\gamma$, the problem is solved with $\overline{\gamma}=\gamma$; if a feasible solution is found, it means that ${\overline{\gamma}}^{\star}\leq\gamma$, otherwise, an infeasibility certificate is found showing that ${\overline{\gamma}}^{\star}\geq\gamma$. By Corollary 1, an infeasibility certificate for $\gamma$ provides the following lower bound certificate on the CJSR:

[TABLE]

In Theorem 4 we show a simple way to improve this lower bound certificate by inspecting the sparsity of the infeasibility certificate.

Definition 6**.**

Consider a finite set of matrices $\mathcal{A}$ constrained by an automaton $G(V,E)$. Given an infeasibility certificate $\widetilde{\mu}$ of Program 1, we denote by $E_{\widetilde{\mu}}$ the set of edges $e\in E$ such that the entry of $\widetilde{\mu}$ corresponding to constraint (5) with edge $e$ is nonzero.

Theorem 4**.**

Consider a finite set of matrices $\mathcal{A}$ constrained by an automaton $G(V,E)$. For any positive integer $d$, if there exists an infeasibility certificate $\widetilde{\mu}$ of Program 1 with $\overline{\gamma}=\gamma$ then

[TABLE]

Proof.

We consider the graph $G_{\widetilde{\mu}}(V,E_{\widetilde{\mu}})$. Since the infeasibility certificate $\widetilde{\mu}$ is zero for constraints (5) with edges $e\in E\setminus E_{\widetilde{\mu}}$, $\widetilde{\mu}$ remains a valid infeasibility certificate for Program 1 with input $(G_{\widetilde{\mu}},\mathcal{A})$ and $\overline{\gamma}=\gamma$, hence $\gamma\leq\rho_{\text{SOS-}2d}(G_{\widetilde{\mu}},\mathcal{A}).$ By Theorem 2, $2^{-h(E_{\widetilde{\mu}})/2d}\rho_{\text{SOS-}2d}(G_{\widetilde{\mu}},\mathcal{A})\leq\rho(G_{\widetilde{\mu}},\mathcal{A})$ and since $E_{\widetilde{\mu}}\subseteq E$, $\rho(G_{\widetilde{\mu}},\mathcal{A})\leq\rho(G,\mathcal{A})$. We obtain (15) by combining these three inequalities. ∎

Example 4**.**

Applying the result of this section to the running example gives the result of Figure 3. The “Kronecker lift” lower bound is the bound obtained by using the Kronecker lift to transform the constrained system with 9 edges into an unconstrained system with 9 matrices, one per edge. The upper bound obtained with both systems is the same [11, Proposition 3.9] hence we can use the guarantee for unconstrained systems (4) with $m^{\prime}=|E|=9$ for the constrained system.

The entropy of the switching signal $h(E)$ used in Theorem 2 is $\log_{2}(2.61803)$, while the value ${n+d-1\choose d}$ used in Theorem 3 is $d+1$ since $n=2$. Therefore, as we can see on the figure, the lower bound guaranteed by Theorem 3 is more accurate for $d=1$ only. The entropy $h(E_{\widetilde{\mu}})$ used in Theorem 4 is $\log_{2}(1.61803)$ for $d=1,2$ and $\log_{2}(1.83929)$ for $d=3,4,5,6$, it is more accurate than the three other lower bounds for every $d$.

The lower bound obtained by computing the $2d$-radius is the most accurate one among all lower bounds for the same $d$ for this example. In practice, better lower bounds can be obtained from the solution of Program 1 using the techniques of [30, 31].

III Low rank reduction

Suppose we want to compute the CJSR of a finite set of matrices $\mathcal{A}\triangleq\{A_{1},\ldots,A_{m}\}\subset\mathbb{R}^{n\times n}$ of rank at most $r$ constrained by an automaton $G(V,E)$. For $\sigma=1,\ldots,m$, since the matrix $A_{\sigma}$ has rank at most $r$, there exists $X_{\sigma},Y_{\sigma}\in\mathbb{R}^{n\times r}$ such that $A_{\sigma}=X_{\sigma}Y_{\sigma}^{\top}$. This can be used to build a new system with matrices of $\mathbb{R}^{r\times r}$ with the same CJSR. This new system can therefore be used to reduce the computation of the CJSR of a system of low rank matrices to a system of matrices of small size. Note that in the case $r=1$, it is known that the CJSR is computable in polynomial time [21].

Theorem 5** (Low Rank Reduction).**

Consider a finite set of matrices $\mathcal{A}\triangleq\{A_{1},\ldots,A_{m}\}\subset\mathbb{R}^{n\times n}$ of rank at most $r$ constrained by an automaton $G(V,E)$.

For a fixed decomposition $A_{\sigma}=X_{\sigma}Y_{\sigma}^{T}$ for $\sigma=1,\ldots,m$ where $X_{\sigma},Y_{\sigma}\in\mathbb{R}^{n\times r}$, denote the set of matrices $\mathcal{A}^{\prime}\triangleq\{A_{\sigma_{1}\sigma_{2}}^{\prime}\mid\sigma_{1},\sigma_{2}=1,\ldots,m\}\subset\mathbb{R}^{r\times r}$ where $A_{\sigma_{1}\sigma_{2}}^{\prime}=Y_{\sigma_{1}}^{T}X_{\sigma_{2}}$. Define the graph $G^{\prime}(V^{\prime},E^{\prime})$ with $V^{\prime}\triangleq E$ and

[TABLE]

Then the two CJSR are the same: $\rho(G,\mathcal{A})=\rho(G^{\prime},\mathcal{A}^{\prime}).$

Proof.

As the CJSR does not depend on the norm used, we choose a norm $\|\cdot\|$ that is submultiplicative, that is $\|AB\|\leq\|A\|\|B\|$ for all matrices $A,B$.

Let $\beta=\max_{\sigma=1}^{m}\max\{\|X_{\sigma}\|,\|Y_{\sigma}^{T}\|\}$. If $\beta=0$, then $\rho(G,\mathcal{A})=0=\rho(G^{\prime},\mathcal{A}^{\prime})$. Therefore we may assume that $\beta>0$. Consider a positive integer $k$. We first show that $[\hat{\rho}_{k}(G,\mathcal{A},\|\cdot\|)]^{k}\leq\beta^{2}[\hat{\rho}_{k-1}(G^{\prime},\mathcal{A}^{\prime},\|\cdot\|)]^{k-1}$ where $\hat{\rho}_{k}(G,\mathcal{A},\|\cdot\|)$ is defined in (2). For any $G$-admissible $(\sigma_{1},\sigma_{2},\ldots,\sigma_{k})$, we have

[TABLE]

using the submultiplicativity of the norm chosen, we have

[TABLE]

The same way, we now show that $[\hat{\rho}_{k-1}(G^{\prime},\mathcal{A}^{\prime},\|\cdot\|)]^{k-1}\leq\beta^{2}[\hat{\rho}_{k-2}(G,\mathcal{A},\|\cdot\|)]^{k-2}$. For any $G^{\prime}$-admissible $(\sigma_{2}\sigma_{1},\ldots,\sigma_{k}\sigma_{k-1})$, we have

[TABLE]

In summary, we have

[TABLE]

Taking the limit $k\to\infty$ we get $\rho(G,\mathcal{A})\leq\rho(G^{\prime},\mathcal{A}^{\prime})\leq\rho(G,\mathcal{A})$. ∎

Example 5**.**

Consider an unconstrained switched system with 2 rank $r$ matrices $A_{1},A_{2}$. This system is equivalent to the constrained switched system with automaton represented in Figure 4(a). Its low rank reduction is represented in Figure 4(b).

Remark 4**.**

The matrices $X_{\sigma},Y_{\sigma}$ of the factorization $A_{\sigma}=X_{\sigma}Y_{\sigma}^{T}$ are not unique. For any invertible matrix $S\in\mathbb{R}^{r\times r}$, $A_{\sigma}=(X_{\sigma}S)(S^{-1}Y_{\sigma}^{T})$ also gives a factorization. However, if $\rho(G^{\prime},\mathcal{A}^{\prime})$ is approximated using the sum of squares algorithm of Section II-A, any two factorizations will give the same approximation. The effect of using $X_{\sigma}S$ and $Y_{\sigma}S^{-T}$ instead of $X_{\sigma}$ and $Y_{\sigma}$ will simply be a linear change of variable of the polynomial $p_{\sigma}$; see Section II-A.

What is the impact of this reduction on the computational complexity and accuracy of the approximation ? The entropy of the language of allowed switching signals is the same for the initial system and the reduced system hence the guarantee in Theorem 2 is the same for both systems. However, the dimension of the matrices goes from the dimension of the matrices $n$ to their rank $r$ hence for low rank matrices the guarantee in Theorem 3 is improved.

In terms of computational complexity, there can be up to $m$ nodes and $m^{2}$ edges in the automaton of the reduced system. Therefore, even if the size of the matrices decreases from $n$ to $r$, the number of variables and constraints increases. This shows that the reduction only decreases the computational complexity if the rank of the matrices is sufficiently low.

IV Conclusion

This paper uncovers a first relation between the complexity of the discrete dynamic of a hybrid system and the computational performance of convex optimization methods analysing the stability of its continuous dynamic. The analysis is performed on discrete linear switched systems, a subclass of hybrid systems, but we believe that it should be extended to other classes of hybrid systems such as markovian switched systems where the entropy of the discrete dynamics is influenced by transition probabilities.