Compositional abstraction of interconnected control systems under dynamic interconnection topology

In this work, we derive conditions under which compositional abstractions of networks of control systems, interconnected via some dynamic interconnection topology, can be constructed using the dynamic interconnection and joint dissipativity-type properties of subsystems and their abstractions. In the proposed framework, the abstraction, itself a system (possibly with a lower dimension), can be used as a substitute of the original system in the controller design process. Moreover, we derive conditions for the construction of abstractions for a class of control systems involving nonlineari-ties satisfying an incremental quadratic inequality. We provide an example to illustrate the effectiveness of the proposed dissipativity-type compositional reasoning by reducing a 150-dimensional nonlinear system to a 3-dimensional one.


I. INTRODUCTION
For large interconnected control systems, controller design to achieve some complex specifications in a reliable and cost effective way is a challenging task. One direction which has been explored to overcome this challenge is to use a simpler (e.g. lower dimension) (in)finite approximation of the given system as a replacement in the controller design process. This allows for a design of a controller for the approximation, which can be refined to the one for the original complex system. The error between the output behaviour of the concrete system and the one of its approximation can be quantified in this detour controller synthesis approach.
Rather than treating the interconnected system in a monolithic manner, an approach which severely restricts the capability of existing techniques to deal with many number of subsystems, compositional schemes provide network-level certifications from main structural properties of the subsystems and their interconnections. Recently, there have been several results on the compositional construction of (in)finite abstractions of deterministic control systems including [1], [2], [3], [4], and probabilistic control systems [5]. These results use a small-gain type condition to enable the compositional construction of abstractions. However, as shown in [6], this type of condition is a function of the size of the network and can be violated as the number of subsystems grows.
Recently in [7], a compositional framework for the construction of abstractions of networks of control systems has been proposed using dissipativity theory. In this result a notion of storage function is proposed which describes joint dissipativity properties of control systems and their abstractions. This notion is used to derive compositional conditions under which a network of abstractions approximate a network of the concrete subsystems. Those conditions can be independent of the number of the subsystems under some properties on the interconnection topologies.
In this work, we extend the results in [7] to networks of control systems in which the interconnection topology is dynamic [8]. In such interconnected systems, the additional dynamics introduced due to the interconnection/interaction system has to be taken into account in the compositional reasoning. We derive conditions under which we can form compositional abstractions of a given interconnected control system under dynamic interconnection.
In addition, for a class of control systems which takes into account a more general class of nonlinearities than the one considered in [7], we derive a set of linear matrix (in)equalities facilitating the construction of their abstractions together with the corresponding storage functions. We illustrate the effectiveness of the proposed results by deriving a compositional abstraction for a network of control systems of this class in which compositionality conditions are satisfied independent of the number or gains of the subsystems.

II. CONTROL SYSTEMS A. Notation
The sets of non-negative integer and real numbers are denoted by N and R, respectively. Those symbols are footnoted with subscripts to restrict them in the usual way, e.g. R >0 denotes the positive real numbers. The symbol R n×m denotes the vector space of real matrices with n rows and m columns. The symbols 1 n , 0 n , I n , 0 n×m denote the vector in R n with all its elements to be one, the zero vector, identity, and zero matrices in Similarly, we use X = [X 1 ; . . . ; X N ] to denote the matrix in R n×m with n = N i=1 n i , given N ∈ N ≥1 , matrices X i ∈ R ni×m , n i ∈ N ≥1 , and i ∈ [1; N ]. Given a vector x ∈ R n , we denote by x the Euclidean norm of x. Given a matrix M = {m ij } ∈ R n×m , we denote by M the Euclidean norm of M . Given matrices M 1 , . . . , M n , the notation diag(M 1 , . . . , M n ) represents a block diagonal matrix with diagonal matrix entries M 1 , . . . , M n . Given a symmetric matrix A, λ min (A) and λ max (A) denote the minimum and maximum eigenvalues of A, respectively. Given a function f : R ≥0 → R n , the (essential) supremum of f is denoted by f ∞ := (ess)sup{ f (t) , t ≥ 0}. A continuous function γ : R ≥0 → R ≥0 , is said to belong to class K if it is strictly increasing and γ(0) = 0; γ is said to belong to K ∞ if γ ∈ K and γ(r) → ∞ as r → ∞. A continuous function β : R ≥0 × R ≥0 → R ≥0 is said to belong to class KL if, for each fixed t, the map β(r, t) belongs to class K with respect to r, and for each fixed nonzero r, the map β(r, t) is decreasing with respect to t and β(r, t) → 0 as t → ∞.

B. Control Systems
Here, we define the class of control systems being investigated in this paper.
Definition 2.1: The class of control systems studied in this paper is a tuple , and R q2 are the state, external input, internal input, external output, and internal output spaces, respectively; • U and W are subsets of sets of all measurable functions of time taking values in R m and R p , respectively; • f : R n ×R m ×R p → R n is a continuous map satisfying the following Lipshitz assumption: for every compact set D ⊂ R n , there exists a constant Z ∈ R >0 such that f (x, u, w) − f (y, u, w) ≤ Z x − y for all x, y ∈ D, all u ∈ R m , and all w ∈ R p ; • h 1 : R n → R q1 is the external output map; for any υ ∈ U and any ω ∈ W, where a locally absolutely continous curve ξ : R ≥0 → R n is called a state trajectory of Σ, ζ 1 : R ≥0 → R q1 is called an external output trajectory of Σ, and ζ 2 : R ≥0 → R q2 is called an internal output trajectory of Σ. We call the tuple (ξ, ζ 1 , ζ 2 , υ, ω) a trajectory of Σ, consisting of a state trajectory ξ, output trajectories ζ 1 and ζ 2 , and input trajectories υ and ω, satisfying (II.1). We also write ξ aυω (t) to denote the value of the state trajectory at time t ∈ R ≥0 under the input trajectories υ and ω from initial condition ξ aυω (0) = a, where a ∈ R n . We denote by ζ 1aυω and ζ 2aυω the external and internal output trajectories corresponding to the state trajectory ξ aυω .
Remark 2.2: If the control system Σ does not have internal and external feedforward maps, the description of the system defined in Definition 2.1 reduces to the tuple Correspondingly, equation (II.1) describing the evolution of state and output trajectories reduces to:

(II.2)
We use the notion of control system in (II.2) later to refer to control subsystems in an interconnected system. Remark 2.3: If the control system Σ does not have internal inputs and outputs, the description of the control system in Definition 2.1 reduces to the tuple Correspondingly, the equation (II.1) describing the evolution of state and output trajectories reduces to: We use the notion of control system in (II.3) later to refer to a dynamical interconnection topology in an interconnected system. Remark 2.4: If the control system Σ does not have internal inputs and outputs, and external feedforward map, the definition of the control system in Definition 2.1 reduces to the tuple Correspondingly, the equation (II.1) describing the state and output trajectories reduces to: We use the notion of control system in (II.4) later to refer to an overall interconnected control system.

III. STORAGE FUNCTION
In this section, we recall the notion of so-called storage function introduced in [7] with some modifications to accommodate for dynamic interconnection topology.
Definition 3.1: Let be two control subsystems with the same external output space dimension. A continuously differentiable function V : , some matrices W,Ŵ , and H of appropriate dimensions, and some symmetric matrix X of appropriate dimension with conformal block partitions X ij , i, j ∈ [1; 2], such that for any x ∈ R n and anyx ∈ Rn, one has and ∀x ∈ R n , ∀x ∈ Rn, ∀û ∈ Rm, ∃u ∈ R m , such that ∀ŵ ∈ Rp ∀w ∈ R p , one obtains .
We now recall the notion of simulation functions introduced in [9] with some modification.
and ∀x ∈ R n , ∀x ∈ Rn, ∀û ∈ Rm, ∃u ∈ R m such that The next theorem, borrowed from [7], shows the importance of the existence of a simulation function by quantifying the error between the output behaviours of Σ and the ones of its abstractionΣ.
. Suppose V is a simulation function fromΣ to Σ. Then, there exists a KL function ϑ such that for any x ∈ R n ,x ∈ Rn,υ ∈Û , there exists υ ∈ U such that the following inequality holds for any t ∈ R ≥0 :

IV. INTERCONNECTED SYSTEMS
Here, we define interconnected control systems under dynamic interconnection topology.
where the inputs to Σ o andΣ o are the internal outputs of the subsystems Σ i andΣ i , respectively, i.e.  2 We interpret inequality s ≥ 0 component-wise.
η ∈ K ∞ , and ψ ext ∈ K ∞ ∪ {0} are defined as Hence we conclude that V is a simulation function fromΣ to Σ.

Remark 4.3:
Note that the case of static interconnection and its associated conditions presented in [7]  In the following section, we consider a specific class of control subsystems Σ, and a specific candidate storage function V . We derive conditions under which candidate V is a storage function from an abstractionΣ to Σ. Those conditions facilitate the construction ofΣ.

V. A CLASS OF NONLINEAR CONTROL SYSTEMS
We consider a specific class of control subsystems given byξ where A ∈ R n×n , B ∈ R n×m , D ∈ R n×p , E ∈ R n×l k , F ∈ R l k ×n , C 1 ∈ R q1×n , and C 2 ∈ R q2×n . The time-varying non-linearity is the one considered in [10] satisfying an incremental quadratic inequality. Similar to [10], we assume that ϕ(t, F x) ∈ R l k is defined by an implicit relation: where M is the set of symmetric matrices referred to as incremental multiplier matrices, the following incremental quadratic constraint holds for all t ∈ R ≥0 , and all k 1 , k 2 ∈ R l k : We recall the assumption given in [10] which will be used in the rest of the paper. Assumption 1: There is a continuous function ς such that, for all t ∈ R ≥0 and z k = F x, H1ĥ21(x1) . . . H1ĥ21(x1) . . . H1ĥ21(x1) . . . H1ĥ21(x1) . . .
To facilitate subsequent analysis, we write the incremental multiplier matrixM in the following conformal partitioned formM We use the tuple to refer to the class of system of the form (V.1). We now consider a candidate function and derive conditions under which it is a storage function from aΣ to Σ.

A. Storage function
Here, we consider a candidate storage function of the form where P and M 0, are matrices of appropriate dimensions. In order to show that V (x,x) in (V.2) is a storage function from an abstractionΣ to the concrete system Σ, we require the following assumptions on the concrete system Σ. Assumption 2: Let Σ = (A, B, C 1 , C 2 , D, E, F, ϕ). There exist matrices M 0, K, W, X 11 , X 12 , X 21 , X 22 , L n , L 1 , and Z of appropriate dimensions, and positive constantκ, such that the following (in)equalities hold: Then function V defined in (V.2) is a storage function from Σ to Σ, where X in Definition 2.1 is given by: . Proof: Note that from (V.6b), and for all x ∈ R n , and x ∈ Rn, we have implying that inequality (III.1) holds with α(r) = λmin( M ) λmax(C T 1 C1) r 2 for any r ∈ R ≥0 , which is a K ∞ function. We proceed to prove inequality (III.2). By the definition of V , one has For any x ∈ R n ,x ∈ Rn, one obtains: Given any x ∈ R n ,x ∈ Rn, andû ∈ Rm, we use the following interface function providing u ∈ R m : andR is a matrix of appropriate dimension. Using the interface function in (V.7), and the conditions (V.6a), (V.6d), (V.6e), (V.6f), and (V.6g), one obtains: Using Young's inequality [11], Cauchy-Schwarz inequality, (V.3), (V.4), (V.5), and (V.6c), one obtains the upper bound forV (x,x) in (V.8) for any positive π <κ. Remark that we used the following inequality [10] for any x ∈ R n and x ∈ Rn to show inequality (V.8): x − Px δϕ Using the upper bound in (V.8), the inequality (III.2) is readily satisfied with η(s) = (κ − π)s, and ψ ext (s) = √ M (BR−PB) 2 π s 2 , ∀s ∈ R ≥0 , and matrix X = X 11 X 12 X 21 X 22 . In the next section, we provide a practical example for compositional abstraction of an interconnected system with dynamic interconnection topology.

VI. EXAMPLE
Consider n first order resistance-capacitance (R-C) circuits, interconnected via resistance-inductance (R-L) series branches. The i-th R-C circuit has the dynamics given by: where i ∈ [1; n], v si ∈ R represents the input source voltage (external input), v ci ∈ R the voltage across capacitor, C i the capacitance, R i the resistance, andw i ∈ R the total current inflow from other R-L branches connected to the R-C circuit.
Assuming identical values of the resistance and inductance in all R-L branches, one can write the dynamics of the total current inflow for the i-th R-C circuit as: . . . ; e ini ], e ij ∈ R 1×n is a row vector whose k-th element is defined as . . . ;w n ], and ϕ : R → R is defined as The dynamic of the interconnection topology Σ o is given by (a o1 , . . . , a on ), . . . ; b on ], C o = I n , and D o = 0 n . We aggregate each Σ i into a scalar abstractionΣ i = (Â i ,B i ,Ĉ 1i , 1, 1, 1, 1, ϕ) given by the following dynamicŝ and with α i (r) = 1 λmax(C T 1i C1i) r 2 , η i (r) = 2r, ψ iext (r) = 0, ∀r ∈ R ≥0 . The interface function in (V.7) is given here by By selecting μ 1 = · · · = μ N = 1, the function ( 1 n1 , . . . , 1 n N ), is a simulation function fromΣ to Σ, whereΣ is the interconnection of the abstract subsystems with the dynamic interconnection topologyΣ o satisfying conditions (IV.2) and (IV.3). For this example, we choose C 1i = 1 0 . . . 0 , and the dynamic interconnection system as follows

VII. CONCLUSION
In this paper, we derived conditions under which compositional abstractions of networks of control systems under dynamic interconnection topologies can be constructed using abstractions of components. The approximation errors are quantified using the notion of simulation function. For a class of nonlinear control systems, we derived a set of linear matrix (in)equalities facilitating the construction of their abstractions. Finally, we showed the effectiveness of u y Abstractionŷ u Fig. 3: 150-dimensional interconnected system is reduced to a 3-dimensional interconnected system. the results on a 150-dimensional network of R-C circuits by reducing it to a 3-dimensional abstraction.