Empirical Differential Gramians forNonlinear Model Reduction

# Empirical Differential Gramians for Nonlinear Model Reduction

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###### Abstract

In this paper, we present an empirical balanced truncation method for nonlinear systems with constant input vector fields. First, we define differential reachability and observability operators by using the variational system. These two operators naturally induce two Gramians, which we call the differential reachability and observability Gramians, respectively. These two differential Gramians are matrix valued functions of the state trajectory (i.e. the initial state and input trajectory) of the original nonlinear system, and it is difficult to find them as functions of the state and input. The main result of this paper is to show that for a fixed state trajectory, it is possible to compute the values of these Gramians by using impulse and initial state responses of the variational system. Therefore, balanced truncation is doable along the fixed state trajectory without solving nonlinear partial differential equations, differently from conventional nonlinear balancing methods. We further develop an approximation method, which only requires trajectories of the original nonlinear systems. Our methods are demonstrated by a coupled van der Pol oscillators along a limit cycle and an RL network along a trajectory.

NL]Yu Kawano, NL]Jacquelien M.A. Scherpen

Jan C. Willems Center for Systems and Control, Engineering and Technology institute Groningen, Faculty of Science and Engineering, University of Groningen, Nijenborgh 4, 9747 AG Groningen, the Netherlands

Key words:  Model reduction; nonlinear systems; balanced truncation; proper orthogonal decomposition.

11footnotetext: A preliminary version of this paper is presented at the 20th IFAC World Congress, July 2017. Corresponding author Y. Kawano. Tel. +31 50 363 2048. Fax +31 50 363 3800.

## 1 Introdunction

Along with the development of new technologies, control systems are becoming more complex and large-scale. To capture systems’ components which are essential for controller design and analysis, model order reduction techniques have been established, see e.g. [1]. In systems and control, typical methods are balanced truncation and moment matching [1, 32], and both of them are extended to nonlinear systems [25, 5, 17, 3, 2, 10]. In contrast to successive theoretical developments, nonlinear model reduction methods still have computational challenges, since they require solutions to nonlinear partial differential equations (PDEs). There are few papers tackling this challenging problem such as [7, 24]. As a data driven model order reduction method, proper orthogonal decomposition (POD) [12] is often used in practice. However, POD is mainly proposed for noncontrolled systems.

For linear time-invariant (LTI) systems, POD and balancing are connected based on the fact that the controllability and observability Gramians can be computed by using impulse and initial state responses, respectively. That is, balanced truncation of LTI systems can be performed by using empirical data. This linear empirical method may also work for nonlinear systems as illustrated by several nonlinear examples [20, 9]. For nonlinear systems, recently, a connection between POD and nonlinear controllability functions is established by [13] in a stochastic setting. In a different problem setting, empirical nonlinear observability Gramians have also been proposed [19, 23]. However, there has been no uniform approach providing both empirical controllability and observability Gramians.

In this paper, we propose an empirical balancing method for nonlinear systems with constant input vector fields by utilizing its variational system. Since the variational system can be viewed as a linear time-varying system (LTV) along the trajectory of the original nonlinear system, one can extend the concept of the controllability and observability operators of the LTV system [18, 30]. These two extended operators naturally induce Gramians, which we call the differential reachability and observability Gramians, respectively. These differential Gramians depend on the state trajectory of the original nonlinear system. In general, it is not easy to obtain them as functions of the state trajectory. Nevertheless, we show that their values at each fixed trajectory can be computed on the basis of the impulse and initial state responses of the variational system along this fixed trajectory. These obtained trajectory-wise Gramians are constant matrices, and thus one can compute balanced coordinates and a reduced order model in a similar manner as the LTI case. A reduced order model may not give a good approximation for an arbitrary state trajectory but it does along the fixed state trajectory as illustrated by our examples.

The proposed empirical balanced truncation method requires the variational system model. For large-scale systems, computing it may be challenging. Therefore, we further develop approximation methods, which only require the original nonlinear system. Our approach is based on the fact that the variational system is a state space representation of the Fréchet derivative of an operator defined by the nonlinear system, and we use its discretization approximation. For the observability Gramian, similar methods to our approximation method are found in [19, 23]. However, there has been no corresponding method for the controllability Gramian, which has been a bottleneck for developing the corresponding balanced truncation method.

Similar nonlinear balanced realizations are found in flow balancing [28, 29] and in differential balancing [17] but they are not empirical methods and require solutions to nonlinear PDEs. Moreover, [17] does not give the concept of a Gramian. A preliminary version of our work is found in [16]. However, [16] does not give the concept of differential operators nor develop the discretization approximation methods. Moreover, we newly propose another differential balancing method for a class of nonlinear systems, which only requires the impulse response of the variational system.

The remainder of this paper is organized as follows. In Section 2, we define the differential reachability and observability Gramians. By using these Gramians, we define a differentially balanced realization along a trajectory of the system. In Section 3, we show that the values of the differential reachability and observability Gramians can be computed by using impulse and initial state responses of the variational system. Then, we develop approximation methods, which only require empirical data of the original nonlinear system. In Section 4, we give some remarks. First, we study positive definiteness of the differential reachability and observability Gramians related with nonlinear local strong accessibility and local observability in a specific case. Next, we propose another differential balancing method, which is further computationally oriented. In Section 5, examples demonstrate our methods for a coupled van der Pol oscillators and an RL network. Finally in Section 6, we conclude the paper.

## 2 Differential Reachability and Observability Gramians

In this paper, we present a nonlinear empirical balancing method for a system with a constant input vector field by using its variational system; the reason why considering the constant input vector field is elaborated in Remark 3.1 below. The proposed empirical balancing method is based on two Gramians, which we call differential reachability and observability Gramians. This section focuses on studying these two differential Gramians. First, we define differential reachability and observability operators. These operators naturally induce differential reachability and observability Gramians. Since the variational system can be viewed as a linear time-varying system along a trajectory of the original nonlinear system, our differential Gramians can be viewed as extensions of Gramians for linear time-varying (LTV) systems [18, 30].

### 2.1 Preliminaries

Consider the following nonlinear system with constant input vector field:

 Σ:{˙x(t)=f(x(t))+Bu(t),y(t)=h(x(t)),

where , , and ; and are class , and . Let denote the state trajectory of the system starting from for each choice of .

In our method, we use the prolonged system of the system , which consists of the original system and its variational system along ,

where , and . In the time interval such that exists and is class , exists for any bounded input because the variational system is a LTV system along .

First we consider to compute the solution of . For solution of the original system  starting from with the input , it follows from the chain rule that

 ddt∂φt−τ(xτ,u)∂xτ =∂∂xτdφt−τ(xτ,u)dt =∂f(φt−τ(xτ,u))∂xτ =∂f(φt−τ(xτ,u))∂φt−τ∂φt−τ(xτ,u)∂xτ =∂f(x(t))∂x∂φt−τ(xτ,u)∂xτ. (1)

That is, is the transition matrix of as an LTV system. Note that when , is the identity matrix. We also call it the transition matrix of the variational system along . By using this transition matrix, the solution to the variational system starting from with input along the state trajectory can be described as

 δx(t)= ∂φt−t0(x0,u)∂xδx0 +t∫t0∂φt−τ(x(τ),u)∂xBδu(τ)dτ. (2)

For the analysis, furthermore, we use a corresponding output when , namely

 δy(t)= ∂h(φt−t0(x0,u))∂x∂φt−t0(x0,u)∂xδx0. (3)

is used.

### 2.2 Differential Reachability and Observability Operators and Gramians

In this subsection, we define differential reachability and observability operators by using the nonlinear prolonged system. Then, we define differential reachability and observability Gramians. For nonlinear systems, the controllability and observability operators are already defined and used for analysis of nonlinear Hankel operators and their differentials [5, 6]. The proposed operators here are similar to the Fréchet derivatives of those controllability and observability operators but slightly different. The motivation for presenting two novel differential operators is to develop an empirical balancing method.

Inspired by LTV systems [18, 30], we define the differential reachability and observability operators and by using the transition matrix of the variational system as follows,

 dR(t0,tf,x0,u)∘δu:=tf∫t0∂φt−t0(x0,u)∂xBδu(t)dt, dO(t0,tf,x0,u)∘δx0 :=∂h(∂φt−t0(x0,u))∂x∂φt−t0(x0,u)∂xδx0.

Note that these operators are linear with respect to and , respectively. Therefore, one can compute their adjoints and as follows.

 dR∗(t0,tf,x0,u)∘δx0:=BT∂Tφt−t0(x0,u)∂xδx0, dO∗(t0,tf,x0,u)∘δua :=tf∫t0∂Tφt−t0(x0,u)∂x∂Th(∂φt−t0(x0,u))∂xδua(t)dt.

For LTV systems [18, 30], it is well known that the controllability (observability) Gramian is a composition of the controllability (observability) operator and its adjoint. Based on this fact, we define differential reachability and observability Gramians as follows.

###### Definition 2.1

The differential reachability Gramian is defined as

 GR(t0,tf,x0,u) :=tf∫t0∂φt−t0(x0,u)∂xBBT∂Tφt−t0(x0,u)∂xdt (4)

for and .

###### Definition 2.2

The differential observability Gramian is defined as

 GO(t0,tf,x0,u) :=tf∫t0∂Tφt−t0(x0,u)∂x∂Th(∂φt−t0(x0,u))∂φt−t0 ×∂h(∂φt−t0(x0,u))∂φt−t0∂φt−t0(x0,u)∂xdt (5)

for and .

Roughly speaking, and ; formally, one needs to consider operators defined by and , see e.g. [18]. These differential Gramians do not always exist for arbitrary and . However, if exists and is a function of the time in a finite time interval , these differential Gramians exist, since the trajectory of the variational system exists.

A similar observability Gramian is found in [19, 23]. However, they do not consider the reachability Gramian. Also, similarly both controllability and observability Gramians are found in flow balancing [28, 29]. For flow balancing, the reachability and observability Gramians are defined on different time intervals, and the input is fixed for any initial state. Thus, the Gramians for flow balancing are defined as functions of the initial states. In contrast, our differential Gramians also depend on the input trajectory in addition to the initial state. Moreover, the papers [28, 29] do not give an explicit expression for the transition matrix of the variational system . This explicit expression is essential to develop our empirical method.

###### Remark 2.3

For a linear time-invariant (LTI) system, the differential reachability and observability Gramians reduce to

 Gc=tf∫t0eA(t−t0)BBTeAT(t−t0)dt, Go=tf∫t0eAT(t−t0)CTCeA(t−t0)dt.

These are controllability and observability Gramians in a finite time interval [1].

###### Remark 2.4

Our differential Gramians are extensions of Gramians of LTV systems [18, 30]. By substituting , the differential reachability Gramian can be rearranged as

 GR(t0,tf,x0,u)=tf∫t0 ∂φtf−τ(xf,F−(u))∂xB ×BT∂Tφtf−τ(xf,F−(u)))∂xdτ,

where is the backward trajectory of system starting from with the input . This is an extension of the reachability Gramian for LTV systems in [30] to nonlinear prolonged systems. Similarly, the differential observability Gramian is an extended concept of the observability Gramian for LTV systems.

In [17], the following differential controllability and observability functions and are used for another type of differential balancing.

 LC(x0,u,δx0):=infδu∈Lm2(−∞,t0]12t0∫−∞∥δu(t)∥2dt, (6)

where , , and .

 LO(x0,δx0):=12∞∫t0∥δy(t)∥2dt,

where , , . The differential balancing in [17] is different from ours due to the difference between our differential reachability Gramian in (4) and the differential controllability function in (6). In contrast, the differential observability function and our differential Gramian are directly related, i.e.,

 LO(x0,u,δx0)=limtf→∞12δxT0GO(t0,tf,x0,u)δx0.

For the relationship between the differential reachability Gramian and differential controllability function, consider to find a differential Gramian corresponding to the differential controllability function. From (2), if the minimal norm solution to (6) exists, then it satisfies

 δx0=t0∫−∞∂φt0−τ(x(τ),u)∂xBδu(τ)dτ. (7)

This can be viewed as a definition of an operator from to , which we denote by , and call the differential controllability operator. It is well known that the minimum norm solution to (7) is given by , where is any solution to [21]. Define the differential controllability Gramian as , i.e.,

 GC(t0,x0,u) :=t0∫−∞∂φt0−τ(x(τ),u)∂xBBT∂Tφt0−τ(x(τ),u)∂xdτ.

If this differential controllability Gramian exists and is positive definite, the differential controllability function can be described as

 LC(x0,u,δxf)=12δxT0G−1C(t0,x0,u)δx0.

Note that the differential controllability Gramian is different from the differential reachability Gramian. The differential controllability Gramian is defined by using a backward trajectory of the original nonlinear system . In contrast, the differential reachability Gramian is defined by using a forward trajectory, which enables us to develop an empirical method.

### 2.3 Differentially Balanced Realization along a Fixed Trajectory

In a similar manner as a standard procedure, one can define a balanced realization between the differential reachability and observability Gramians. Since these differential Gramians are defined as functions of the trajectory , we define our balanced realization trajectory-wise as follows.

###### Definition 2.5

Let the differential reachability Gramian and differential observability Gramian at fixed be positive definite. A realization of the system is said to be a differentially balanced realization along if there exists a constant diagonal matrix

 Λ=diag{σ1,…,σn}, σ1≥⋯≥σn>0

such that .

In a similar manner as for the LTI case [1], it is possible to show that there always exists a differentially balanced realization along for any if positive definite differential reachability and observability Gramians exist. Positive definiteness of the differential Gramian will be discussed in Section 4.1 in a specific case when related with local strong accessibility and local observability of the original nonlinear system .

###### Theorem 2.6

Let the differential reachability Gramian and differential observability Gramian at fixed be positive definite. Then, there exists a non-singular constant matrix which achieves

 TGR(t0,tf,x0,u)TT = T−TGO(t0,tf,x0,u)T−1 = Λ:=diag{σ1,…,σn},

where . Then a differentially balanced realization along is obtained after a coordinate transformation .

If one can compute the differential reachability and observability Gramians along a fixed trajectory , then a differentially balanced realization and thus a reduced order model along this trajectory is computed by using a linear coordinate transformation. Clearly, this linear coordinate transformation changes for a different trajectory and time interval. However, in the LTI case, the reduced order model does not depend on the trajectory, since the Gramians do not depend on it; see Remark 2.3.

## 3 Empirical Differential Gramians

### 3.1 Empirical Differential Reachability and Observability Gramians

In the previous section, we defined a differentially balanced realization along a fixed trajectory . It is worth emphasizing that in order to obtain the differential Gramians as functions of , or equivalently and , one needs to solve nonlinear partial differential equations as for similar nonlinear balancing methods [17, 28, 29]. Therefore, we focus on computing the values of the differential Gramians trajectory-wise.

First, we show that the differential reachability Gramian along a fixed state trajectory can be computed by using an impulse response of . Define

 δxImp(t):=∂φt−t0(x0,u)∂xB.

Then, from the definition (4) of the differential reachability Gramian , we have

 GR(t0,tf,x0,u)=tf∫t0δxImp(t)δxTImp(t)dt. (8)

By using Dirac’s delta function , can be represented as

 δxImp(t)=t∫t0∂φt−τ(x(τ),u)∂xBδD(τ−t0)dτ. (9)

From (2), this is the impulse response of starting from the initial state along a fixed trajectory . Therefore, for each and , the constant matrix is obtained by using the impulse response of .

###### Remark 3.1

The equality (9) does not hold if is not constant. Indeed, for the system  and its trajectory , the differential reachability Gramian is

 ¯GR(t0,tf,x0,u) :=tf∫t0∂ψt−t0(x0,u)∂x∂f(ψt−t0(x0,u),u)∂u ×∂Tf(ψt−t0(x0,u),u)∂u∂Tψt−t0(x0,u)∂xdt.

However, the impulse response of the corresponding variational system is

 δ¯xImp(t) =t∫t0∂ψt−τ(x(τ),u)∂x∂f(ψτ−t0(x0,u),u)∂uδD(τ−t0)dτ =∂ψt−t0(x0,u)∂x∂f(x0,u(t0))∂u.

Therefore, the differential reachability Gramian and impulse response do not coincide with each other if is not constant.

Next, we show that differential observability Gramian can be computed by using initial state responses. Denote

 δyIsi(t)=∂h(φt−t0(x0,u))∂x∂φt−t0(x0,u)∂xei, i=1,…,n, (10)

where is the standard basis, i.e., whose th element is , and the other elements are the zeros. This is the initial output response of starting from the initial state with input along a fixed trajectory . From the definition (5) of the differential observability Gramian , we have

 GO(t0,tf,x0,u):=tf∫t0 [δyIs1(t)⋯δyIsn(t)]T ×[δyIs1(t)⋯δyIsn(t)]dt.

Therefore, the differential observability Gramian can be computed trajectory-wise by using initial output responses of starting from for each and .

In summary, by computing impulse and initial state responses of a variational system along a given state trajectory , one can compute trajectory-wise values of the differential reachability and observability Gramians. Therefore, trajectory-wise empirical differential balanced truncation is doable based on simulation data.

### 3.2 Discretization Approximation of the Fréchet Derivative

In the previous subsection, we have presented an empirical approach for trajectory-wise computation of the differential reachability and observability Gramians. The approach requires the variational system model in addition to the original system model. If the original nonlinear systems are large-scale, computing the variational system model may need an effort. Therefore, we present approximation methods, which do not require the variational system model.

In order to be self-contained, we first introduce the Fréchet derivative of a nonlinear operator. Consider a nonlinear operator defined by the system . A linear operator is said to be the Fréchet derivative if for each and , the following limit exists

 dΣ(x0,u)(δx0,δu) :=lims→0Σ(x0+sδx0,u+sδu)−Σ(x0,u)s

for all and . From its definition, the Fréchet derivative is given by the variational system .

Next, we consider to compute a discretization approximation of the Fréchet derivative. A simple approximation is

 dΣ(x0,u)(δx0,δu) ≈dΣapp(x0,u)(δx0,δu):=Σ(x0+sδx0,u+sδu)−Σ(x0,u)s.

Since the nonlinear operator is given by the system , a state space representation of the discretized approximation is obtained as follows.

 d Σapp(x0,u)(δx0,δu): Rn×Lm2[t0,tf]×Rn×Lm2[t0,tf]→Rn×Lp2[t0,tf], (x0,u,δx0,δu)↦(xvf,yv),

Therefore,  and  are approximately computed as the differences of pairs of state and output trajectories, respectively as  and , where  and coincide with the differences of pairs of the initial states  and inputs , respectively.

By using this state space representation of , one can compute approximations of impulse and initial state responses of a variational system . First, an approximation of the impulse response (9) is obtained as

 δxImp(t)≈x2(t)−x1(t)s, δx0=0,δu=δD(t−t0).

Therefore, we need two trajectories of the original nonlinear system to compute the differential reachability Gramian approximately.

Next, an approximation of the initial state response (10) is

 δyi(t)≈yv(t), δx0=ei,δu=0, i=1,…,n.

To obtain the differential observability Gramian approximately, we compute  for different initial states , . Note that in the approximations, is fixed, and depends on , . Therefore, we need trajectories of the original nonlinear system. Similar observability Gramians to our differential observability Gramians are proposed in [20, 9], which can be computed by using the trajectories of the original nonlinear systems. However, trajectories are required. Furthermore, our reachability Gramian is very different from [20, 9].

In summary, the differential reachability and observability Gramians can be approximately computed by using trajectories of the original nonlinear system, where  is same for the approximations of both differential reachability and observability Gramians. Note that even if one does not have an exact model of a real-life system, one only needs the impulse and initial state responses. Therefore, it may be possible to compute an approximation of a differentially balanced realization along by empirical data.

## 4 Further Analysis

In this section, we give some remarks for differential balancing proposed in this paper. First, we study positive definiteness of differential reachability and observability Gramians in terms of nonlinear local strong accessibility and local observability when . Next, we show that for a specific class of systems, one can achieve another empirical differential balancing only by using the impulse response of the variational system.

### 4.1 Positive Definiteness of Gramians along Autonomous System

The differentially balanced realization is defined for positive definite differential reachability and observability Gramians. In a specific case when , the positive definiteness implies local accessibility and observability of the original nonlinear system , and the converse is true for local observability; see [22] for the definitions of local strong accessibility and local observability.

###### Theorem 4.1

Let be in . Suppose that the solution to a system exists for any and in a finite time interval . Then, the system is locally strongly accessible if the differential reachability Gramian is positive definite for any and for any subinterval .

PROOF. For the sake of the simplicity of the discussion, we consider the single input case. Throughout the proof, we use the fact that the variational system along the trajectory is an LTV system. The differential reachability Gramian is nothing but the controllability Gramian [31] in the sense of LTV systems. For LTV systems, it has been shown in [26, 31] that the controllability Gramian is positive definite for any subinterval if and only if the LTV system satisfies the Kalman-like controllability rank condition [26]; the discussion until here holds for the multiple-input system. In the single input case, the Kalman-like controllability rank condition [26] for the variational system is as follows: there exists such that

where . In the multiple-input case, , are also needed to be taken into account. Condition (11) is a sufficient condition for local strong accessibility; see [22].

###### Remark 4.2

Theorem 4.1 gives a sufficient condition for local strong accessibility in terms of the differential reachability Gramian. To obtain a necessary and sufficient condition, one needs to compute multiple differential reachability Gramians by changing inputs. The gap between condition (11) and local strong accessibility is that the other Lie brackets that appear in the local strong accessibility rank condition [22], e.g., are missing; for more details see [22]. To cover such Lie brackets, one needs to compute multiple differential reachability Gramians by changing inputs.

Now, we provide the sketch of the idea of using multiple differential reachability Gramians in the single input case. Consider two differential reachability Gramians and , where . From the results on the controllability analysis of LTV systems [26, 31], one can confirm that if there exists a non-zero constant vector  such that

 GR(t1,t2,x0,0)v=0, GR(t1,t2,x0,u1)v=0 (12)

for any subinterval then

To cover all Lie brackets that appear in the local strong accessibility rank condition, one needs to compute a large amount of differential reachability Gramians corresponding to different inputs. This could even be an infinite number.

###### Theorem 4.3

Let and . Also let and be in . Suppose that the solution of a system exists for any in a time interval , and the observability codistribution [22] of has a constant dimension. Then, the system is locally observable if and only if the differential observability Gramian is positive definite for any and for any subintervals .

PROOF. When the observability codistribution has a constant dimension, a system with is locally observable if and only if the nonlinear observability rank condition holds for all initial states [22]. One can confirm that this nonlinear observability rank condition is nothing but the Kalman-like observability rank condition  [26] for the variational system along the trajectory as an LTV systems. That is, the system is locally observable if and only if its variational system is differentially observable [26, 31] under the assumption for the rank of the observability codistribution. Furthermore, the LTV system is differentially observable if and only if its observability Gramian is positive definite for any subinterval [31], which is nothing but .

The paper [23] gives a sufficient condition for local observability for non-zero . Theorem 4.3 implies that if one chooses , the necessity also holds. As well known for LTV systems, the differential reachability and observability Gramians along are positive definite if and only if the variational systems along is completely controllable and observable, respectively. The above theorems connect complete controllability [26] and observability [26] of the variational system and nonlinear local strong accessibility and observability, respectively. In addition, the theorems provide an empirical method for checking nonlinear local strong accessibility and local observability because one can compute the differential reachability and observability Gramians along by using impulse and initial state responses of the variational system, respectively. It is worth emphasizing that other empirical balancing methods in [20, 9] have not been connected with the system properties including accessibility and observability.

### 4.2 Another Differential Balancing Method for Variationally Symmetric Systems

For balancing methods including our differential balancing, two Gramians are needed in general. One is for controllability, and the other is for observability. However, for linear systems, there is a class of systems for which one Gramian is constructed from the other. Such systems are called symmetric [1, 27, 18]. Motivated by the results for symmetric systems, we develop another differential balancing method which requires only the impulse response of the variational system.

This symmetry concept is extended to nonlinear systems [11] and variational systems [15]. We further extend the latter symmetry concept.

###### Definition 4.4

The system is said to be variationally symmetric if there exists a class and non-singular such that

 n∑i=1∂S(x)∂xifi(x)+S(x)∂f(x)∂x=∂Tf(x)∂xS(x), (13) S(x)B=∂Th(x)∂x (14)

hold.

Variational symmetry implies that after a change of coordinates , the variational system becomes

 ⎧⎪⎨⎪⎩δ˙z(t)=∂Tf(x(t))∂xδx(t)+∂Th(x)∂xδu(t),δy(t)=BTδz(t). (15)

In the LTI case, the system (15) is called the dual system of the original system , and the variational symmetry property is called symmetry. For a symmetric LTI system, the controllability (observability) Gramian of the dual system corresponds to the observability (controllability) Gramian of the dual system [1, 27, 18]. Motivated by this fact, we consider to achieve model reduction based on the differential reachability Gramians of the original system and the system (15). For a variationally symmetric system, these two differential reachability Gramians are connected to each other. A similar relation holds for the differential observability Gramians. We leave to the reader this up.

###### Theorem 4.5

For the variationally symmetric system with respect to , the differential reachability Gramian of the system (15) satisfies

 G∗R(t0,tf,x0,u) =tf∫t0ST(φt(x0,u))∂φt(x0,u)∂xB ×BT∂Tφt−t0(x0,u)∂xS(φt(x0,u))dt.

for any and if it exists.

PROOF. By using (1) and (13), compute

 ddt (S(φt−τ(xτ,u))∂φt−τ(xτ,u)∂xτS−1(φt−τ(xτ,u))) = (n∑i=1∂S(φt−τ(xτ,u))∂xifi(φt−τ(xτ,u)) +S(φt−τ(xτ,u))∂f(φt−τ(xτ,u))∂x) ×∂φt−τ(xτ,u)∂xτS−1