Data-Driven Reachability Analysis from Noisy Data

Data-Driven Reachability Analysis from Noisy Data

Amr Alanwar¹, Anne Koch², Frank Allgöwer², and Karl Henrik Johansson¹

1KTH Royal Institute of Technology Email: {alanwar,kallej}@kth.se

2Institute for Systems Theory and Automatic Control, University of Stuttgart Email: {anne.koch,frank.allgower}@ist.uni-stuttgart.de

Abstract—We consider the problem of computing reach- able sets directly from noisy data without a given sys- tem model. Several reachability algorithms are presented, and their accuracy is shown to depend on the underlying system generating the data. First, an algorithm for com- puting over-approximated reachable sets based on matrix zonotopes is proposed for linear systems. Constrained matrix zonotopes are introduced to provide less conser- vative reachable sets at the cost of increased computa- tional expenses and utilized to incorporate prior knowledge about the unknown system model. Then we extend the approach to polynomial systems and under the assumption of Lipschitz continuity to nonlinear systems. Theoretical guarantees are given for these algorithms in that they give a proper over-approximative reachable set containing the true reachable set. Multiple numerical examples show the applicability of the introduced algorithms, and accuracy comparisons are made between algorithms.

I. INTRODUCTION

Reachability analysis computes the union of all possible trajectories that a system can reach within a finite or infinite time when starting from a bounded set of initial states [1]. The most popular approaches for computing reachable sets are set- propagation and simulation-based techniques. Set-propagation techniques propagate reachable sets for consecutive time steps.

The efficiency depends on the set representation and the used technique. For instance, Taylor series methods propa- gate enclosures over discrete time by constructing a Taylor expansion of the states with respect to time and bounding the coefficients [2]. The resulting enclosure is then inflated by a bound on the truncation error. Other set representations are polyhedra [3], zonotopes [4], (sparse) polynomial zonotopes [5], ellipsoids [6], and support functions [7]. Zonotopes have favorable properties as they can be represented compactly, and they are closed under Minkowski sum and linear mapping.

The simulation-based approach in [8] over-approximates the reachable set by a collection of tubes around trajectories such that the union of these tubes provides an over-approximation of the reachable set. Sampling-based reachability analysis utilizes random set theory in [9]. Other simulation-based techniques are proposed in [10]–[14].

While there is a considerable amount of literature on com- puting reachable sets for different model classes, these meth- ods assume an a priori given model. Obtaining a model that

Fig. 1: The reachable sets consistent with noisy input-state data are computed while making use of side information if available.

adequately describes the system at hand from first principles or noisy data is usually a challenging and time-consuming task. Simultaneously, system data in the form of measured trajectories is often readily available in many applications.

Therefore, we are interested in reachability analysis directly from noisy data of an unknown system model. One recent contribution in this direction can be found in [15], where the authors introduce two data-driven methods for computing the reachable sets with probabilistic guarantees. The first method represents the reachability problem as a binary classification problem, using a Gaussian process classifier. The second method makes use of a Monte Carlo sampling approach to compute the reachable set. A probabilistic reachability analysis is proposed for general nonlinear systems using level sets of Christoffel functions in [16] where they provide a guarantee that the output of the algorithm is an accurate reachable set approximation in a probabilistic sense. Over- approximating reachable sets from data are considered in [17]

based on interval Taylor-based methods applied to systems with dynamics described as differential inclusions; however, the proposed approach only works under the assumption of noise-free data. Another interesting method is introduced in [18], where the model is assumed to be partially known and data is used to learn an additional Lipschitz continuous state- dependent uncertainty, where the unknown part is assumed to be bounded by a known set. We believe that computing

arXiv:2105.07229v1 [eess.SY] 15 May 2021

guaranteed reachable sets from noisy data is still an open problem.

The main idea underlying the introduced data-driven reach- ability framework is visualized in Fig. 1 where we compute data-driven reachable sets based on matrix zonotope recursion.

In order to guarantee that the reachable set encloses all system trajectories from finite noisy data, we compute a matrix zonotope that encloses all models that are consistent with the noisy data. We then propagate the initial set forward, utilizing this matrix zonotope to compute the reachable set.

We also provide an improved reachability analysis algorithm that provides a less conservative over-approximation of the reachable set at the cost of additional computational expenses.

This improved analysis scheme is enabled by introducing a new set representation, namely constrained matrix zonotopes.

We utilize this novel set representation and the corresponding operations additionally to provide a systematic approach on how supplementary prior knowledge about the unknown model like states decoupling, partial model knowledge, or given bounds on certain entries in the system matrices can be incorporated into the reachability analysis. We then extend our approach to two classes of nonlinear systems: polynomial systems and Lipschitz systems. All used codes to recreate our findings are publicly available¹.

We will specifically build upon ideas used in [19]–[21]

and [22]–[25] for data-driven analysis and data-driven con- troller design, respectively. In these works, data is used to characterize all models that are consistent with the data. This characterization enables a computational approach for direct systems analysis and design without explicitly identifying a model.

The main contributions of this paper are as follows:

• An algorithm is proposed to compute the reachable sets of an unknown control system from noise-corrupted input- state measurements using matrix zonotopes (Algorithm 1). The resultant reachable sets are shown to over- approximate the model-based reachable sets for LTI sys- tems (Theorem 1).

• A new set representation named constrained matrix zono- tope (Definition 5) and its essential operations are pro- posed.

• The constrained matrix zonotope is exploited in Algo- rithm 2 by providing a less conservative reachable sets us- ing the exact noise description. The computed reachable sets over-approximate the model-based reachable sets for LTI systems (Theorem 2).

• A general framework is introduced for incorporating side information like states decoupling, partial model knowledge, or given bounds on certain entries in the system matrices about the unknown model (Algorithm 3) into the reachability analysis, which further decreases the conservatism of the resulting reachable sets. The re- sultant reachable sets over-approximate the model-based reachable set for LTI systems (Theorem 3).

• An algorithm is proposed for reachability analysis in the existence of process and measurement noise for LTI

1https://github.com/aalanwar/Data-Driven-Reachability-Analysis

systems (Algorithm 4).

• An algorithm (Algorithm 5) is proposed for computing the reachable sets that are guaranteed to over-approximate the exact reachable set for polynomial systems (Theorem 4). A variant of the LTI side information framework can be used for the polynomial systems.

• An algorithm is proposed for computing the reachable set (Algorithm 6), which results in reachable sets that are guaranteed to over-approximate the exact reachable sets of nonlinear systems under a Lipschitz continuity assumption (Theorem 5).

• A comparison between the proposed algorithms and the alternative direction of system identification and model- based reachability analysis is provided.

As discussed in more detail before, there have been a few approaches to infer the reachable sets directly from data, mostly without providing guarantees in the case of noisy data. An alternative approach is to apply well-known system identification approaches and consecutively do model-based reachability analysis. Thus, we include a comparison to a standard system identification method in Section VI while con- sidering 2σ uncertainty bound in the model-based reachability analysis. Interesting recent results on system identification with probabilistic guarantees from finite noisy data include concentration bounds [26]. Another very related approach is set membership estimation (see, e.g., [27]), where the trade-off between conservatism and computational expenses is usually of central importance, which we will also encounter in the course of this paper.

This paper is an extension to our previous work in [28]

where we introduced the basic idea of computing the reachable set by using matrix zonotopes over-approximating the set of models consistent with the data. In this work, we significantly extend and improve the ideas in [28] by introducing con- strained matrix zonotopes and its essential set of operations which allow the incorporation of side information about the unknown model and provide less conservative reachable sets.

We also enhance the proposed nonlinear Lipschitz reachability analysis method in [28]. Furthermore, we propose a new approach for computing the reachable sets of polynomial systems, and we consider measurement noise for linear sys- tems. Our initial ideas in [28] have been utilized in different applications like data-driven predictive control [29] and set- based estimation [30].

The rest of the paper is organized as follows: the problem statement and preliminaries are defined in Section II. The new set representation (constrained matrix zonotope) is proposed in Section III. Data-driven reachability analysis for LTI systems is proposed in Section IV including a framework to include prior knowledge into the analysis approach. Then, we extend the proposed approach to nonlinear systems in Section V.

The introduced approaches are applied to multiple numerical examples in Section VI, and Section VII concludes the paper.

II. PROBLEMSTATEMENT ANDPRELIMINARIES

We start by defining some set representations that are used in the reachability analysis.

A. Set Representations We define the following sets:

Definition 1: (Zonotope [31]) Given a center c_Z ∈ Rⁿ and γ_Z ∈ N generator vectors in a generator matrix GZ = h

g⁽¹⁾_Z . . . g^(γ_Z^Z)

i∈ R^n×γZ, a zonotope is defined as

Z =n x ∈ Rⁿ

x = c_Z+

γ_Z

X

i=1

β⁽ⁱ⁾g_Z⁽ⁱ⁾, −1 ≤ β⁽ⁱ⁾≤ 1o . (1) We use the shorthand notation Z = hc_Z, G_Zi for zonotopes.

A linear map L is defined as LZ = hLcZ, LGZi. Given two zonotopes Z1 = hcZ₁, GZ₁i and Z2 = hcZ₂, GZ₂i, the Minkowski sum is

Z1+ Z2=D

c_Z1+ c_Z2, [G_Z1, G_Z2]E

. (2)

For simplicity, we use the notation + instead of ⊕ to denote the Minkowski sum as the type can be determined from the context. Similarly, we use Z1− Z2to denote Z1+ −1Z2. We define the Cartesian product of two zonotopes Z1 and Z2 by

Z1× Z2= z₁ z₂

    

z₁∈ Z1, z₂∈ Z2



=DcZ₁

cZ₂



,GZ₁ 0 0 GZ₂

E

. (3)

Definition 2: (Matrix Zonotope [32, p.52]) Given a center matrix CM ∈ R^n×p and γM ∈ N generator matrices G˜_M=h

G⁽¹⁾_M . . . G^(γ_M^M)

i∈ R^n×(pγM), a matrix zonotope is defined as

M=n

X ∈ R^n×p  

X=C_M+

γM

X

i=1

β⁽ⁱ⁾G⁽ⁱ⁾_M, −1 ≤ β⁽ⁱ⁾≤ 1o . (4) We use the shorthand notation M = hC_M, ˜G_Mi for matrix zonotopes.

Zonotopes have been extended in [33] to represent arbitrary convex shapes by applying constraints on the factors multi- plied with the generators.

Definition 3: (Constrained Zonotope [33, Prop. 1]) An n- dimensional constrained zonotope is defined by

C = {x ∈ Rⁿ| x = cC+ GCβ, ACβ = bC, kβk∞≤ 1} , (5) where cC ∈ Rⁿ is the center, GC ∈ R^n×ng is the generator matrix and AC ∈ R^nc×ng and bC∈ R^ncdenote the constraints.

In short, we use the shorthand notation C = hcC, G_C, A_C, b_Ci for constrained zonotopes.

B. Problem Statement

We consider a discrete-time system

x(k + 1) = f (x(k), u(k)) + w(k),

y(k) = x(k) + v(k). (6)

where w(k) ∈ Zw⊂ Rⁿdenotes the noise bounded by a noise zonotope Zw, u(k) ∈ Uk ⊂ R^mthe input bounded by an input zonotope Uk, y(k) ∈ Rⁿthe measured state that is additionally corrupted by measurement noise v(k) ∈ Zv ⊂ Rⁿ bounded by measurement noise zonotope Z_v, and x(0) ∈ X₀⊂ Rⁿ the initial state of the system bounded by the initial set X₀.

Reachability analysis computes the set of states x(k) which can be reached given a set of uncertain initial states X0⊂ Rⁿ

and a set of uncertain inputs Uk. More formally, it can be defined as follows:

Definition 4: (Exact Reachable Set) The exact reachable set RN after N time steps subject to inputs u(k) ∈ Uk ⊂ R^m∀k = {0, . . . , N − 1}, noise w(·) ∈ Zw, is the set of all states trajectories starting from initial set X0 ⊂ Rⁿ after N steps:

RN =x(N ) ∈ Rⁿ

x(k+1) = f (x(k), u(k)) + w(k), x(0) ∈ X0, u(k) ∈ Uk, w(k) ∈ Zw:

∀k ∈ {0, ..., N −1} . (7) We aim to compute an over-approximation of the exact reachable sets when the model of the system in (6) is unknown, but input and noisy state trajectories are available. More specifically, we aim to compute data-driven reachable sets in the following cases:

1) LTI systems in Subsection IV-A:

x(k + 1) = A_trx(k) + B_tru(k) + w(k).

2) LTI systems given additional side information about the unknown model in Subsection IV-B.

3) LTI systems with measurement noise in Subsection IV-C:

x(k + 1) = Atrx(k) + Btru(k) + w(k), y(k) = x(k) + v(k).

4) Polynomial systems in Subsection V-A.

5) Lipschitz nonlinear systems in Subsection V-B.

Instead of having access to a mathematical model of the sys- tem, we consider K input-state trajectories of different lengths Ti, i = 1, . . . , K denoted by {u⁽ⁱ⁾(k)}^T_k=0ⁱ⁻¹, {x⁽ⁱ⁾(k)}^T_k=0ⁱ , i = 1, . . . , K. We collect the set of all data sequences in the following matrices

X =x⁽¹⁾(0) . . . x⁽¹⁾(T₁) . . . x^(K)(0) . . . x^(K)(T_K) , U₋=u⁽¹⁾(0) . . . u⁽¹⁾(T1−1) . . . u^(K)(0) . . . u^(K)(TK−1) . Let us further denote

X₊=x⁽¹⁾(1) . . . x⁽¹⁾(T₁) . . . x^(K)(1) . . . x^(K)(T_K) , X₋=x⁽¹⁾(0) . . . x⁽¹⁾(T1−1) . . . x^(K)(0) . . . x^(K)(TK−1) . The total amount of data points from all available trajectories is denoted by T = PK

i=1Ti, and we denote the set of all available data by D = {U−, X}. Note that when dealing with measurement noise, we will consider the trajectories {y⁽ⁱ⁾(k)}^T_k=0ⁱ instead of {x⁽ⁱ⁾(k)}^T_k=0ⁱ .

C. Preliminaries

We denote the unknown process noise in state trajectory i by ˆw⁽ⁱ⁾(·). It follows directly that the stacked matrix of the noise ˆw⁽ⁱ⁾(k) in the collected data:

Wˆ₋= ˆw⁽¹⁾(0) . . . ˆw⁽¹⁾(T₁−1) . . . ˆw^(K)(0) . . . ˆw^(K)(T_K−1) is an element of the set Mw where Mw = hC_Mw, ˜G_Mwi, with

G˜_Mw =h G⁽¹⁾_M

w . . . G^(γ_M^Mw)

w

i

. (8)

Note that Mw is the matrix zonotope resulting from the concatenation of multiple noise zonotopes Zw = hc_Zw,h

g⁽¹⁾_Z

w . . . g_Z^(γZw)

w

ii as follows:

C_Mw=c_Zw . . . cZ_w , (9)

TABLE I: Notations summary for the main used sets.

Notation Description

Uk Input zonotope at time k.

X0 Initial reachable set.

Zw Zonotope bounding the process noise.

Mw Matrix zonotope bounding the process noise matrix ˆW₋.

Nw Constrained matrix zonotope as a subset of Mw given additional constraints from data for LTI system.

MΣ Matrix zonotope consistent with the data and noise bound for LTI system.

NΣ Constrained matrix zonotope as a subset of MΣ given additional constraints from data for LTI system.

Rk Exact reachable set.

Rˆ_k Reachable set computed using zonotopes for LTI system.

R¯k Reachable set computed given exact noise description using constrained zonotopes for LTI system.

R¯^s_k Reachable set computed given side information using constrained zonotopes for LTI system.

Rˆ^m_k Reachable set computed using zonotopes in case of measurement noise for LTI system.

R¯^m_k Reachable set computed using constrained zonotopes in case of measurement noise for LTI system.

R˜^m_k Reachable set computed using a data-based approximation in case of measurement noise for LTI system.

Rˆ^p_k Reachable set computed using zonotopes for polynomial systems.

R¯^p_k Reachable set computed using constrained zonotopes for polynomial systems.

R¯^s,p_k Reachable set computed given side information using constrained zonotopes for polynomial systems.

R⁰_k Reachable set for Lipschitz nonlinear systems.

G(1+(i−1)T )

Mw =h

g⁽ⁱ⁾_Z

w 0_{n×(T −1)} i

, G(j+(i−1)T )

Mw =h

0_n×(j−1) g⁽ⁱ⁾_Z

w 0_{n×(T −j)} i

, G(T +(i−1)T )

Mw =h

0_{n×(T −1)} g⁽ⁱ⁾_Z

w

i .

(10)

∀i = {1, . . . , γ_Zw}, j = {2, . . . , T − 1}. We denote the Kronecker product by ⊗. We denote the element at row i and column j of matrix A by (A)i,j and column j of A by (A).,j. For vectors, we denote the element i of vector a by (a)i. For a given matrix A, A⁰_(i,j) denotes a matrix of same size as A with zero entries everywhere except for the value (A)i,j at row i and column j. The vectorization of a matrix A is defined by vec(A). The element-wise multiplication of two matrices is denoted by . The right inverse is denoted by ^†. We denote the over-approximation of a reachable set Rk by an interval by int(Rk). We define also for N time steps

F = ∪^N_k=0(Rk× Uk). (11) Finally, we denote all system matrices A B that are consistent with the data D = (U−, X) by NΣ:

NΣ= {A B | X+= AX−+ BU−+ W−, W−∈ Mw}.

By assumption, A_tr B_tr ∈ NΣ. Throughout the paper, we keep record of the used set descriptions in Table I.

III. CONSTRAINEDMATRIXZONOTOPE

Inspired by constrained zonotopes, we propose to extend the notion of matrix zonotopes to constrained matrix zonotopes as follows:

Definition 5: (Constrained Matrix Zonotope) Given a center matrix CN ∈ R^n×p and a number γN ∈ N of generator matrices ˜G_N = [G⁽¹⁾_N . . . G^(γ_N^N)] ∈ R^n×(pγN), as well as matrices ˜A_N = [A⁽¹⁾_N . . . A^(γ_N^N)] ∈ R^nc×(naγN) and B_N ∈ R^nc×na constraining the factors β^(1:γN), a constrained

matrix zonotope is defined by N =n

X ∈ R^n×p  

X = C_N +

γ_N

X

i=1

β⁽ⁱ⁾G⁽ⁱ⁾_N,

γ_N

X

i=1

β⁽ⁱ⁾A⁽ⁱ⁾_N = B_N, −1 ≤ β⁽ⁱ⁾≤ 1o . Furthermore, we define the shorthand notation N = hCN, ˜G_N, ˜A_N, B_Ni for the constrained matrix zonotope.

The constrained matrix zonotopes are closed under Minkowski sum and multiplication by a scalar which can be done as follows:

Proposition 1: For every N1 = hC_N1, ˜G_N1, ˜A_N1, B_N1i ⊂ R^n×p, N2= hC_N2, ˜G_N2, ˜A_N2, B_N2i ⊂ R^n×p, and R ∈ R^k×n the following identities hold

RN₁= hRC_N1, R ˜G_N1, ˜A_N1, B_N1i, (12) N1+ N2=D

C_N1+ C_N2, [ ˜G_N1, ˜G_N2], ˜A_N12,B_N1 0 0 BN2

E , (13) where

A˜_N12=

"

A⁽¹⁾_N

1 0

0 0

 . . .

"

A^(γ_N^N1)

1 0

0 0

#0 0 0 A⁽¹⁾_N

2

 . . .

"

0 0

0 A^(γ_N^N2)

2

##

. A proof of Proposition 1 is provided in the Appendix. Dur- ing the propagation of the reachable sets, we additionally need to multiply the constrained matrix zonotope by a zonotope or constrained zonotope. The result of both operations can be over-approximated by a constrained zonotope. We provide these operations in the next two propositions.

Proposition 2: For every N = hCN, ˜G_N, ˜A_N, B_Ni ⊂ R^p×n, and Z = hcZ, G_Zi ⊂ Rⁿ the following identity holds

N Z ⊂D

C_Nc_Z,G˜_Nc_Z C_NG_Z G_f ,

A_{N Z} 0 0 , vec(BN)E

, (14)

where A_{N Z} =h

vec(A⁽¹⁾_N ) . . . vec(A^(γ_N^N)) i

, G_f=h

g_f⁽¹⁾ . . . g_f^(γZγN)i ,

g^(k)_f = f⁽ⁱ⁾G⁽ⁱ⁾_Ng^(j)_Z , ∃ k ∀i = {1, . . . , γ_N}, and

j = {1, . . . , γ_Z} such that k = {1, . . . , γ_Zγ_N}, (15) f⁽ⁱ⁾= max(|β_L,N⁽ⁱ⁾ |, |β_U,N⁽ⁱ⁾ |), (16) β_L,N⁽ⁱ⁾ = min

β⁽ⁱ⁾

β⁽ⁱ⁾:

γN

X

j=1

β^(j)A^(j)_N = BN, kβk∞≤ 1, (17)

β_U,N⁽ⁱ⁾ = max

β⁽ⁱ⁾

β⁽ⁱ⁾:

γ_N

X

j=1

β^(j)A^(j)_N = BN, kβk∞≤ 1. (18) A proof of Proposition 2 is provided in the Appendix.

Proposition 3: For every N = hCN, ˜GN, ˜AN, BNi ⊂ R^p×n, and C = hcC, G_C, A_C, b_Ci ⊂ Rⁿ the following identity holds

N C ⊂D

C_Nc_C,G˜_Nc_C C_NG_C G_f ,

A_{N C} 0 0

0 A_C 0



,vec(BN) b_C

E

, (19)

where A_{N C}=h

vec(A⁽¹⁾_N ) . . . vec(A^(γ_N^N)) i

, G_f =h

g⁽¹⁾_f . . . g^(γ_f^CγN) i

,

g^(k)_f = f^(k)G⁽ⁱ⁾_Ng_C^(j), ∃ k ∀i = {1, . . . , γ_N}, and

j = {1, . . . , γC} such that k = {1, . . . , γCγN}, (20) f^(k)= max(|β_L,N⁽ⁱ⁾ β_L,C^(j)|, |β_L,N⁽ⁱ⁾ β^(j)_U,C|, |β_U,N⁽ⁱ⁾ β_L,C^(j)|, |β_U,N⁽ⁱ⁾ β_U,C^(j)|),

(21) β_L,N⁽ⁱ⁾ = min

β⁽ⁱ⁾

β⁽ⁱ⁾:

γN

X

j=1

β^(j)A^(j)_N = B_N, kβk_∞≤ 1, (22)

β_U,N⁽ⁱ⁾ = max

β⁽ⁱ⁾

β⁽ⁱ⁾:

γN

X

j=1

β^(j)A^(j)_N = B_N, kβk_∞≤ 1, (23) β_L,C^(j) = min

β^(j)

β^(j): ACβ = bC, kβk∞≤ 1, (24) β_U,C^(j) = max

β^(j)

β^(j): A_Cβ = b_C, kβk_∞≤ 1. (25) A proof of Proposition 3 is provided in the Appendix.

IV. DATA-DRIVEN REACHABILITY FORLINEAR SYSTEMS

We consider in this section LTI systems given (i) data corrupted by process noise, (ii) data corrupted by process noise while having additional prior information on the system matri- ces, and (iii) data corrupted by process noise and measurement noise.

A. Linear Systems with Process Noise Consider a discrete-time linear system

x(k + 1) = A_trx(k) + B_tru(k) + w(k), (26) where Atr ∈ R^n×n, and Btr ∈ R^n×m. Due to the presence of noise, there generally exist multiple matrices A B that are consistent with the data. To provide reachability analysis

guarantees, we need to consider all models that are consistent with the data. Therefore, we are interested in computing a set MΣthat contains all possibleA B that are consistent with the input-state measurements and the given noise bound. We build upon ideas from [20] to our zonotopic noise descriptions, which yields a matrix zonotope MΣ⊇ N_Σpaving the way to a computationally simple reachability analysis.

Lemma 1: Given input-state trajectories D = (U−, X) of the system (26) and a matrix H such that

X−

U−



H = I, (27)

then the matrix zonotope

MΣ= (X+− Mw)H (28)

contains all matricesA B that are consistent with the data D = (U−, X) and the noise bound, i.e., NΣ⊆ MΣ.

Proof: For anyA B ∈ NΣ, we know that there exists a W−∈ Mw such that

AX₋+ BU₋= X+− W₋. (29) Every W− ∈ Mw can be represented by a specific choice βˆ_M⁽ⁱ⁾

w, −1 ≤ ˆβ_M⁽ⁱ⁾

w ≤ 1, i = 1, . . . , γM_w, that results in a matrix inside the matrix zonotope Mw:

W−= CM_w+

γ_Mw

X

i=1

βˆ_M⁽ⁱ⁾

wG⁽ⁱ⁾_M

w.

Multiplying H from the right to both sides in (29) yields

A B = X+− CMw+

γ_Mw

X

i=1

βˆ_M⁽ⁱ⁾

wG⁽ⁱ⁾_M

w

!

H. (30) Hence, for all A B ∈ NΣ, there exists βˆ⁽ⁱ⁾_M

w,

−1 ≤ βˆ_M⁽ⁱ⁾

w ≤ 1, i = 1, . . . , γMw, such that (30) holds. Therefore, for all A B ∈ NΣ, it also holds that

A B ∈ MΣas defined in (28), which concludes the proof.

Remark 1: The condition (27) in Lemma 1 requires that there exists a right-inverse of the matrix X−

U−



. This is equivalent to requiring this matrix to have full row rank, i.e.

rankX₋ U₋



= n + m. This condition can be easily checked given the data. Note that for noise-free measurements this rank condition can also be enforced by choosing the input persistently exciting of order n + 1 (compare to [34, Cor. 2]).

To guarantee an over-approximation of the reachable sets for the unknown system, we need to consider the union of reachable sets of all A B that are consistent with the data. We apply the results of Lemma 1 and do reachability analysis to all systems in the set M_Σ. Let ˆR_k denotes the reachable set computed based on the noisy data using matrix zonotopes. We propose Algorithm 1 to compute ˆRk as an over-approximation of the exact reachable set Rk. The set of models that is consistent with data is computed in line 2 which is then utilized in the recursion of computing the reachable set Rˆk+1in line 4. The following theorem proves that ˆRk⊇ Rk. Theorem 1: Given input-state trajectories D = (U₋, X) of the system in (26) and a matrix H as defined in (27), then the reachable set computed in Algorithm 1 over-approximates the exact reachable set, i.e., ˆRk ⊇ Rk.

Algorithm 1 LTI-Reachability

Input: input-state trajectories D = (U−, X), initial set X0, process noise zonotope Z_w and matrix zonotope M_w, input zonotope Uk, ∀k = 0, . . . , N − 1

Output: reachable sets ˆRk, ∀k = 1, . . . , N

1: Rˆ0= X0

2: MΣ= (X₊− Mw)X₋ U₋

† 3: for k = 0 : N − 1 do

4: Rˆk+1= MΣ( ˆRk× Uk) + Zw 5: end for

Proof: The reachable set computed based on the model can be found using

Rk+1=Atr Btr (Rk× Uk) + Zw. (31) Since A_tr B_tr ∈ MΣ according to Lemma 1 and both Rk

and ˆRkstart from the same initial set X0, it holds that Rk+1⊆ Rˆk+1.

Lemma 1 provides a matrix zonotope MΣwhich comprises all A B that are consistent with the data and the noise bound. However, not all elements of the matrix zonotope MΣ correspond to a system in (26) that can explain the data given the noise bound, i.e., MΣ is in fact a superset of all

A B that are consistent with the data (NΣ ⊆ MΣ). As discussed in [20], [23], X+−W− might not be explainable by AX− + BU− for all possible W− ∈ Mw. More precisely, there might not exists a solution A B to the system of linear equations

A BX₋ U−



= X+− W₋

for all W₋ ∈ Mw. An exact description for all systems consistent with the data and the noise bound would therefore be the set

NΣ= (X+− Nw)H (32)

with

Nw= {W₋∈ Mw| (X+− W−)X₋ U₋

⊥

= 0}, (33)

where X₋ U₋

⊥

denotes a matrix containing a basis of the kernel of X₋

U₋



. Representing Nw and NΣ is not possible using state of the art set representations. Therefore, we propose the constrained matrix zonotope introduced in Section III as a new set representation that can represent the sets Nw and thereby N_Σ to compute a less conservative reachable set ¯R_k at the cost of increasing the computational complexity in Algorithm 2. Due to adding constraints, ¯Rk is a constrained zonotope different from ˆRk which is a zonotope. We first compute the exact noise description Nw in line 2 to line 5.

Then, we compute the set of models NΣthat is consistent with the exact noise description in line 6 which is further utilized in the recursion of computing the reachable set ¯Rk+1 in line 8. The following theorem proves that Rk ⊆ ¯Rk.

Theorem 2: Given input-state trajectories D = (U₋, X) of the system in (26) and a matrix H as defined in (27), then the

Algorithm 2 LTI-Constrained-Reachability

Input: input-state trajectories D = (U−, X), initial set X0, process noise zonotope Z_w and matrix zonotope M_w, input zonotope Uk, ∀k = 0, . . . , N − 1

Output: reachable sets ¯Rk, ∀k = 1, . . . , N

1: R¯0= X0 2: A⁽ⁱ⁾_N

w = G⁽ⁱ⁾_M

w

X₋ U₋

⊥

, ∀i = {1, . . . , γZwT }

3: A˜_Nw =h A⁽¹⁾_N

w . . . A^(γ_N^{ZwT )}

w

i

4: B_Nw = (X+− C_Mw)X₋ U−

⊥ 5: Nw= hCM_w, ˜GM_w, ˜AN_w, BN_wi

6: NΣ= (X+− Nw)X₋ U−

† 7: for k = 0 : N − 1 do

8: R¯_k+1= N_Σ( ¯R_k× U_k) + Z_w

9: end for

reachable set computed in Algorithm 2 over-approximates the exact reachable set, i.e., ¯Rk ⊇ Rk.

Proof: As pointed out in [23], the condition for the existence of a solution F[A B]to the system of linear equations

F_{[A B]}X₋ U₋



= X+− W₋, or equivalently

X−

U−

^>

F_{[A B]}^> = (X+− W₋)^> (34) can be reformulated via the Fredholm alternative as

X₋ U₋



˜

z = 0 ⇒ (X+− W₋)˜z = 0,

which means that any vector ˜z ∈ R^T in the kernel of X₋ U₋



must also lie in the kernel of X+−W−. SinceX₋ U−

⊥

contains a basis of the kernel of X₋

U₋



, another equivalent condition for the existence of a solution F_{[A B]} in (34) is hence

(X+− W₋)X₋ U₋

⊥

= 0. (35)

Considering the constraint (35) together with the bounding matrix zonotope Mw= hC_Mw, ˜G_Mwi, we find:

(X+− C_Mw−

γ_ZwT

X

i=1

β⁽ⁱ⁾G⁽ⁱ⁾_M

w)X₋ U₋

⊥

= 0. (36) Rearranging (36) results in

(X₊− CMw)X₋ U₋

⊥

| {z }

B_Nw

=

γ_ZwT

X

i=1

β⁽ⁱ⁾G⁽ⁱ⁾_M

w

X₋ U₋

⊥

| {z }

A⁽ⁱ⁾_Nw

.

Remark 2: Algorithm 2 provides a less conservative de- scription of the data-driven reachable set compared to Algo- rithm 1 by utilizing a less conservative description of the set of systems consistent with the data. To be more precise, NΣ

in (32) with (33) is an equivalent description of all systems consistent with the data and the noise bound (compare [20, Lemma 8]). However, applying the reachability analysis in line 8 of Algorithm 2 requires multiplying constrained matrix zonotopes by zonotopes and constrained zonotopes. For this multiplication, we introduced a guaranteed over-approximation in Proposition 2 and Proposition 3, respectively, which hence introduces conservatism into the proposed reachability analysis approach.

Note that initial zonotope X0 captures all the uncertainty in the initial state. Next, we provide a general framework for incorporating side information about the unknown model.

B. Linear Systems with Side information

Consider a scenario in which we have prior side information about the unknown model from the physics of the problem or any other source. It would be beneficial to make use of this side information to have less conservative reachable sets.

In the following, we propose a framework to incorporate side information about the unknown model like decoupled dynamics, partial model knowledge, or prior bounds on entries in the system matrices. More specifically, we consider any side information that can be formulated as

| ¯QA_tr Btr − ¯Y | ≤ ¯R, (37) where ¯Q ∈ R^ns×n, ¯Y ∈ R^ns×(n+m), and ¯R ∈ R^ns×(n+m) are matrices defining the side information which is known to hold for the true system matrices A_tr Btr. Here, the absolute |.| and ≤ are element-wise operators. To incorporate such side information into the reachability analysis, we utilize once again the newly introduced concept of constrained matrix zonotopes. We introduce a reachability analysis in Algorithm 3 on the basis of the set of system matricesA B that are consistent with the data (including the less conservative noise handling in Nw) as well as the a priori known side information in (37). We denote the reachable set computed based on the side information by ¯R^s_k. Algorithm 3 summarizes the required computation to incorporate the side information. After setting R¯^s₀ = X0 in line 1, we compute the exact noise description Nwand exact set of models NΣconsistent with the noisy data in lines 2:6 similar to Algorithm 2. Next, we compute the set of models Ns consistent with the side of the information in line 7 to line 14. Finally, we compute the recursion of the reachable sets in line 16. The following theorem proves that R¯^s_k ⊇ Rk.

Theorem 3: Given input-state trajectories D = (U−, X) of the system in (26), a matrix H as defined in (27), and side information in form of (37). Let T > 2(n + m). Then, the reachable set computed in Algorithm 3 over-approximates the exact reachable set, i.e., ¯R^s_k⊇ R_k.

Proof: For all matrices A_s B_s that satisfy the side information (37), there exists a matrix ¯D ∈ R^ns×(n+m) with ( ¯D)i,j∈ [−1, 1] such that

Q¯As Bs − ¯Y =

n

X

i=1 m

X

j=1

R¯⁰_(i,j) ¯D. (38) Additionally, we know that all system matrices consistent with the dataA B ∈ NΣare bounded by the constrained matrix

Algorithm 3 LTI-Side-Info-Reachability

Input: input-state trajectories D = (U−, X), initial set X0, process noise zonotope Z_w and matrix zonotope M_w, side information in terms of ¯Q, ¯Y , ¯R. input zonotope Uk, ∀k = 0, . . . , N − 1

Output: reachable sets ¯Rk, ∀k = 1, . . . , N

1: R¯^s₀= X0

Equivalent to lines 2:6 of Algorithm 2

7: G⁽ⁱ⁾_N

s = G⁽ⁱ⁾_N

Σ, ∀i = {1, . . . , γZ_wT }

8: G⁽ⁱ⁾_N

s = 0, ∀i = {γ_ZwT + 1, . . . , γ_ZwT + nm}

9: G˜_Ns=h G⁽¹⁾_N

s . . . G^(γ_N^{ZwT +nm))}

s

i

10: A⁽ⁱ⁾_N

s =

"

A⁽ⁱ⁾_N

Σ

QG⁽ⁱ⁾_N

Σ 0

#

, ∀i = {1, . . . , γ_ZwT }

11: A^(γ_N^{ZwT +k)}

s =

 0

− ¯R⁰_(i,j) 0



, ∃ k ∀ i = {1, . . . , n}, j = {1, . . . , m}, such that k = {1, . . . , nm}

12: A˜_Ns =h A⁽¹⁾_N

s . . . A^(γ_N^{ZwT +nm)}

s

i

13: B_Ns =

 B_NΣ Y − ¯¯ QC_NΣ 0



14: N_s= hC_NΣ, ˜G_Ns, ˜A_Ns, B_Nsi

15: for k = 0 : N − 1 do

16: R¯^s_k+1= N_s( ¯R^s_k× U_k) + Z_w

17: end for

zonotope NΣ, i.e.

A B = CNΣ+

γ_ZwT

X

i=1

β_N⁽ⁱ⁾

ΣG⁽ⁱ⁾_N

Σ, (39)

with

γ_ZwT

X

i=1

β⁽ⁱ⁾_N

ΣA⁽ⁱ⁾_N

Σ = B_NΣ. (40)

Inserting (39) in (38) results in Y − ¯¯ QC_NΣ = ¯Q

γ_ZwT

X

i=1

β_N⁽ⁱ⁾

ΣG⁽ⁱ⁾_N

Σ−

n

X

i=1 m

X

j=1

R¯⁰_(i,j) ¯D. (41) With ( ¯D)i,j∈ [−1, 1], we can concatenate ( ¯D)i,jto β_NΣ con- stituting β_Ns. Then, combining (40) with the new constraints in (41) yields ˜AN_s and BN_s. We add zero generators to maintain the correct number of generators.

Remark 3: Note that for most applications, T > 2(n+m) is automatically satisfied. Therefore, this assumption was taken in the above theorem to ease the notation. In the case of T ≤ 2(n + m), special attention must be given in the setup of the matrices in ˜AN_s and BN_s. The main idea, however, can nevertheless be straightforwardly applied.

Remark 4: Note that the reachable sets computed in Al- gorithm 3 using side information are less conservative than the ones computed in Algorithm 2 using constrained matrix zonotopes which in turn is less conservative than the ones computed in Algorithm 1 using the matrix zonotope, i.e., Rk⊆ ¯R^s_k ⊆ ¯Rk ⊆ ˆRk, as additional information is included in form of additional constraints.

Next, we consider dealing with measurement noise in com- bination with process noise.

C. Linear Systems with Measurement Noise

In the following, we consider measurement noise in addition to process noise, i.e.,

x(k + 1) = A_trx(k) + B_tru(k) + w(k),

y(k) = x(k) + v(k). (42)

Besides the input data matrix U₋, we collect the noisy state measurements Y in the matrices

Y₊=y⁽¹⁾(1) . . . y⁽¹⁾(T1) . . . y^(K)(1) . . . y^(K)(TK) , Y₋=y⁽¹⁾(0) . . . y⁽¹⁾(T1−1) . . . y^(K)(0) . . . y^(K)(TK−1) . Additionally, let ˆO = ˆV+− A ˆV− with

Vˆ+=ˆv⁽¹⁾(1) . . . ˆv⁽¹⁾(T₁) . . . ˆv^(K)(1) . . . ˆv^(K)(T_K) , Vˆ₋=ˆv⁽¹⁾(0) . . . ˆv⁽¹⁾(T1−1) . . . ˆv^(K)(0) . . . ˆv^(K)(TK−1) , where ˆv⁽ⁱ⁾(k), k = 0, 1, . . . , Ti, denotes again the actual measurement noise sequence on trajectory i that led to the measured input-state trajectories. If we assume knowledge of a bound on ˆO, the same approach as presented before can be pursued.

Assumption 1: The matrix ˆO is bounded by a matrix zono- tope ˆO ∈ M_o which is known.

Proposition 4: Given input-state trajectories (U−, Y ) of the system in (26). If there exists a matrix ˜H such that

 Y₋ U−



H = I,˜ (43)

then

Rˆ^m_k+1= MΣ˜( ˆR^m_k × Uk) + Zw, Rˆ^m₀ = X0, (44) with

MΣ˜ = (Y+− Mo− Mw) ˜H (45) over-approximates the exact reachable set, i.e., Rk⊆ ˆR^m_k.

Proof: With

Y₊− (V+− AtrV₋) − W₋= A_trY₋+ B_trU₋, the proof follows the proofs of Lemma 1 and Theorem 1 given Assumption 1.

Next, we utilize the introduced constrained matrix zonotope in Section III to find a less conservative set given Assump- tion 1.

Proposition 5: Given input-state trajectories (U−, Y ) of the system in (26) and a matrix ˜H as defined in (43), then

R¯^m_k+1= NΣ˜( ¯R^m_k × Uk) + Zw, R¯^m₀ = X0, (46) with

NΣ˜ = hC_M˜

Σ, ˜G_M˜

Σ, ˜A_N˜

Σ, B_N˜

Σi, (47)

A˜_N˜

Σ =h A⁽¹⁾_N

Σ˜ . . . A^(γ_N^Mo+γMw)

Σ˜

i , A⁽ⁱ⁾_N

Σ˜ = G⁽ⁱ⁾_M

w

 Y₋ U₋

⊥

, i = {1, . . . , γMw},

A⁽ⁱ⁾_N

Σ˜ = G⁽ⁱ⁾_M

o

 Y₋ U₋

⊥

, i = {γ_Mw+ 1, . . . , γ_Mo+ γ_Mw},

B_N˜

Σ = (Y₊− CMw− CMo) Y₋ U₋

⊥

, where C_M˜

Σ and ˜G_M˜

Σ are defined in (45).

Proof: Similar to Theorem 2, we have (Y₊− ˆW₋− ˆO) Y₋

U₋

⊥

= 0.

We do not know ˆW₋ and ˆO but we can bound them by Wˆ₋ ∈ M_w= hC_Mw, ˜G_Mwi and ˆO ∈ M_o = hC_Mo, ˜G_Moi.

Therefore, we have:



Y+− C_Mw− C_Mo−

γ_Mw

X

i=1

β_M⁽ⁱ⁾

wG⁽ⁱ⁾_M

w

−

γ_Mo

X

i=1

β_M⁽ⁱ⁾

oG⁽ⁱ⁾_M

o

 Y₋ U₋

⊥

= 0. (48)

Let βNΣ˜ =β_Mw β_Mo. Thus we rewrite (48) as



Y+− CM_w− CM_o

 Y₋ U−

⊥

| {z }

B_{N ˜}

Σ

=

γ_Mw

X

i=1

β_N⁽ⁱ⁾

Σ˜G⁽ⁱ⁾_M

w+

γ_Mo

X

i=1

β_N^(γMw+i)

Σ˜ G⁽ⁱ⁾_M

o

 Y₋ U₋

⊥

, (49) which yields B_N˜

Σ and A_N˜

Σ.

Note that a similar assumption to Assumption 1 has been taken in [22, Asm. 2]. However, it might be difficult in practice to find a suitable set Mo even with a given bound on v(k), k = 0, 1, . . . , T , since A is assumed to be unknown.

Therefore, we introduce a data-based approximation for the reachable set under the influence of the measurement noise from data. Instead of Assumption 1, we now only consider a bound on v(k) ∈ Zv. Similar to the matrix zonotope Mw of the modeling noise, we have Mv = hC_Mv, ˜G_Mvi where ˆV₊, ˆV₋ ∈ Mv. Algorithm 4 summarizes the proposed approach to deal with measurement noise. The general idea can be described as follows:

1) Obtain an approximate model ˜M .

2) Obtain a zonotope that gives an over-approximation of the model mismatch between the true model and the approximate model ˜M , and the term AtrV₋ from data.

Algorithm 4 LTI-Meas-Reachability

Input: input-state trajectories D = (U−, Y ), initial set X0, process noise zonotope Zw and matrix zonotope Mw, mea- surement noise zonotope Zv and matrix zonotope Mv, input zonotope Uk, ∀k = 0, . . . , N − 1

Output: reachable sets ˆRk, ∀k = 1, . . . , N

1: R˜^m₀ = X0

2: M = (Y˜ +− CM_v− CM_w) Y₋ U₋

†

3: AV = maxj (Y+)_.,j− ˜M (Y₋)_.,j (U₋)_.,j

!

4: AV = minj (Y+)_.,j− ˜M (Y−)_.,j (U−)_.,j

!

5: ZAV = zonotope(AV , AV ) − Zw− Zv 6: for k = 0 : N − 1 do

7: R˜^m_k+1= ˜M



( ˜R^m_k + Zv) × U



+ ZAV + Zw 8: end for