# Quasi-Steady-State Models of Three Timescale Systems: A Bond Graph Approach.

1. IntroductionThe singular perturbation theory provides a powerful tool for modelling and control design in dealing with the presence of parasitic parameters, for example, small time constants due to inductances, capacitances, and moment of inertia which increase the system order. The systems containing parasitic parameters are characterized by having slow and fast dynamics, and they are considered as timescale systems. In addition, the parasitic parameters can be neglected, and reduced order systems can be obtained [1, 2].

The mathematical model of systems represents a fundamental task in control theory. Frequently, these systems determine high-order state equations. However, many systems can contain different timescales due to the parasitic parameters. Thus, these systems can have different dynamics, and applying the appropriate procedures, these dynamics can be separated. The corresponding subsystems of these dynamics permit to have reduced models whose analysis and control design are direct and simpler [1, 2].

There are many papers and books dedicated to system analysis with singular perturbations. Also, many applications and control design for singularly perturbed systems have been proposed [1, 2]. For example, systems with high-gain feedback using singular perturbation methods have been studied in [3]. Nonlinear optimal control applying singular perturbations techniques is proposed in [4]. The nonlinear and robust composite control for a DC motor is presented in [5].

The control for a class of nonlinear singularly perturbed systems by exact design with manifold is introduced in [6]. The robust stability applied to singularly perturbed systems is described in [7]. The robustness of output feedback controllers for systems with singular perturbations is proposed in [8]. The observer problem by the need to obtain the states for a system with singular perturbations is solved in [9]. The composite control of singular perturbation systems using sidling modes is proposed in [10]. The feedback control design for systems with three timescales is introduced in [11]. A procedure to obtain reduced models of a system with multitimescales and applied to an electrical power system is presented in [12].

The nonlinear system analysis with multiple timescales is proposed in [13]. The problem for controlling linear systems with multiparameter singular perturbations is solved in [14]. The calculation of the average on cycles of three timescales systems is described in [15]. The asymptotic stability of a singularly perturbed nonlinear with three timescales is studied in [16]. The multiparameter asymptotic stability analysis for the three timescale singular perturbation problem of an autonomous helicopter is proposed in [17]. The analysis and control design of systems with two and three timescales and the application to fuel cells are presented in [18].

Bond graph theory offers a modelling and control platform for systems with different energy domains where important results in analysis and synthesis of systems have been found [19-21].

Some interesting papers using bond graphs for singular perturbations methods can be cited. The modelling and simplification of two timescale systems through bond graphs is described in [22]. A reciprocal system to obtain the fast dynamics is introduced in [23]. The quasi-steady-state model of a system in the physical domain is proposed in [24]. The analysis of a class of nonlinear systems with singular perturbations in bond graph is presented in [25]. Approximate bond graph models for two timescale systems are proposed in [26]. The analysis of a singular perturbed system with a feedback and observer in the physical domain is presented in [27].

A steel frame structure based on a bond graph model of distributed system using lumping technique is proposed in [28]. Also, this methodology can be used for stability and sensitivity analyses.

The slow and fast dynamics of a system represented by a bond graph determining the causal loop gains can be obtained [23, 29] where the bond graph methodology is a useful tool for system analysis.

In this paper, the modelling in bond graph of systems with three timescales is introduced. This bond graph is formed by three groups of storage elements in integral causality assignment (BGI). These storage elements represent the slow, medium, and fast dynamics of the system.

During the dynamic performance of a system with three timescales when the fast dynamics have converged, reduced systems can be obtained in a bond graph approach whose storage elements for the fast dynamics have to accept a derivative causality assignment and the storage elements for the medium and slow dynamics an integral causality is assigned. However, the subsystem for medium dynamics still has singular perturbation parameters. Hence, the most reduced system can be gotten by assigning a derivative causality to the storage elements of the fast and medium dynamics and an integral causality to the slow dynamics.

Therefore, the contribution of this paper is to obtain the quasi-steady-state models of three timescale systems modelled by bond graphs.

This paper is organized as follows: Section 2 gives the three timescale systems in the algebraic approach. The modelling in bond graph of systems with three timescales is described in Section 3. The reduced models obtaining the quasi-steady-state models in a bond graph approach is presented in Section 4. A case study of a Ward Leonard system applying the proposed methodology is developed in Section 5. Finally, in Section 6, the conclusions are given.

2. Three Timescale Systems

Consider a system with singular perturbations which is decomposed into slow, medium, and fast dynamics described by

[mathematical expression not reproducible], (1)

with the output

[mathematical expression not reproducible], (2)

where [x.sub.1](t) is an n x 1 vector, [x.sub.2](t) is an m x 1 vector, [x.sub.3](t) is an l x 1 vector, u(t) is an p x 1 vector input and y(t) is an q x 1 vector output.

It is assumed that system (1) has three different groups of eigenvalues: n eigenvalues are close to the origin and m and l eigenvalues are far and farther from the origin, respectively. The eigenspectrum e(A) of system (1) in the increasing order of absolute values is

[mathematical expression not reproducible]. (3)

System (1) is said to possess a three timescale property, if the largest absolute eigenvalue of the slow eigenspectrum e(As) is much smaller than the smallest absolute eigenvalue of the fast one eigenspectrum e([A.sub.m]) and if the largest absolute eigenvalue of the fast one eigenspectrum e([A.sub.m]) is much smaller than the smallest absolute eigenvalue of the fast two eigenspectrum e([A.sub.f]); that is [2],

[[epsilon].sub.1] = [parallel][p.sub.sn][parallel]/[parallel][p.sub.m1][parallel] [much less than] 1, (4)

[[epsilon].sub.2] = [parallel][p.sub.mm][parallel]/[parallel][p.sub.f1][parallel] [much less than] 1. (5)

By neglecting the fast dynamics of the complete system ([[epsilon].sub.2] = 0), the solution of the third line of (1) determines

[[??].sub.3](t) = -[A.sup.-1.sub.33][A.sub.31][[??].sub.1](t) - [A.sup.-1.sub.33][A.sub.32][[??].sub.2](t) - [A.sup.-1.sub.33][B.sub.3]u(t), (6)

and from the first and second lines of (1) and (6), the reduced models are expressed by

[mathematical expression not reproducible], (7)

where

[mathematical expression not reproducible]. (8)

The output of this reduced system is

[mathematical expression not reproducible], (9)

where

[mathematical expression not reproducible]. (10)

From the second line of (7) and doing [[epsilon].sub.1] = 0, the medium dynamics can also be removed by

[mathematical expression not reproducible], (11)

and substituting (11) into the first line of (7), the quasi-steady-state model is

[[??].sub.1](t) = [bar.[A.sub.11]][bar.[x.sub.1]](t) + [bar.[B.sub.1]]u(t), (12)

where

[mathematical expression not reproducible]. (13)

The output for the most reduced model is given by

[bar.y](t) = [bar.[C.sub.1]][bar.[x.sub.1]](t) + [bar.D]u(t), (14)

where

[mathematical expression not reproducible]. (15)

Bond graph models of singularly perturbed systems are described in the next section.

3. Modelling in Bond Graph of Systems with Singular Perturbations

A bond graph is a graphical representation of a system where the power interactions are described by lines called bonds. These bonds indicate the interactions between power variable pairs called effort e(t) and flow f(t), which is shown in Figure 1.

In order to obtain the sets of equations of a system modelled by bond graphs, the constitutive relations of the elements are required. These relationships can be dynamic or algebraic depending on the element and by the cause-effect assignment. In the bond graph, a bond with the causal stroke determines the causality assignment and the assignments of the half arrow and the causal stroke are independent as is shown in Figure 2.

The following physical elements can be used to build a dynamic system:

(i) The Active 1 Ports or Sources Denoted by ([MS.sub.e], [MS.sub.f]). These sources have only one causality, and this is shown in Figure 3.

(ii) The Passive 1 Ports. These elements are as follows:

(a) Resistance taking whatever causality shown in Figure 4

(b) Capacitance and inertance elements in an integral causality assignment where the input variable is integrated to produce the output variable which is shown in Figure 5

(c) Capacitance and inertance in a derivative causality assignment where the input and output have a derivative operation, and this is shown in Figure 6

(iii) The Ideal 2 Port Elements Denoted by (TF, GY) Representing Transformers and Gyrators. Figure 7 shows these elements.

(iv) The 3 Port Junctions Denoted by (1, 0) which Are Junctions That Determine the Different Connections between the Elements. These junctions are shown in Figure 8.

By using physical elements and junction structures, one can analyze systems containing complex multiport components applying bond graphs. Hence, a bond graph model with a preferred integral causality assignment (BGI) of a system with three timescales is shown in Figure 9. The block diagram of Figure 9 contains the following:

(i) Source field denoted by ([MS.sub.e], [MS.sub.f]) that determines the plant input u(t) [member of] [[Real part].sup.p].

(ii) Junction structure denoted by (0, 1, TF, GY) with 0 and 1 junctions, transformers TF, and gyrators GY.

(iii) Energy storage field denoted by (C, I) that defines energy variables q(t) and p(t) associated with C and I elements divided by the following:

(a) The states for the slow dynamics [x.sub.1](t) [member of] [[Real part].sup.n] and [mathematical expression not reproducible] associated with the storage elements in integral and derivative causality assignment, respectively

(b) The co-energy vectors for the slow dynamics [z.sub.1](t) [member of] [[Real part].sup.n] and [mathematical expression not reproducible] of the storage elements in integral and derivative causality assignment, respectively

(c) The states for the medium dynamics [x.sub.2](t) [member of] [[Real part].sup.m] and [mathematical expression not reproducible] associated with storage elements in integral and derivative causality assignment, respectively

(d) The co-energy vectors for the medium dynamics [z.sub.2](t) [member of] [[Real part].sup.m] and [mathematical expression not reproducible] of the storage elements in integral and derivative causality assignment, respectively

(e) The states for the fast dynamics [x.sub.3](t) [member of] [[Real part].sup.l] and [mathematical expression not reproducible] associated with the storage elements in integral and derivative causality assignment, respectively

(f) The co-energy vectors for the fast dynamics [z.sub.3](t) [member of] [[Real part].sup.l] and [mathematical expression not reproducible] of the storage elements in integral and derivative causality assignment, respectively

(iv) Energy dissipation field denoted by (R) that defines [D.sub.in](t) [member of] [[Real part].sup.r] and [D.sub.out](t) [member of] [[Real part].sup.r] as a mixture of power variables e(t) and f(t) indicating the energy exchanges between the dissipation field and the junction structure.

(v) Detector field denoted by ([D.sub.e]) that determines the plant output y(t) [member of] [[Real part].sup.q].

For systems of high order, the bond graph methodology permits to know the dynamics in an easy and direct way whose fast and slow dynamics of a bond graph model can be estimated by determination of causal loop gains [23, 29].

The constitutive relationships of the storage field for the slow dynamics are expressed by

[z.sub.1](t) = [F.sub.1][x.sub.1](t), (16)

[z.sup.d.sub.1](t) = [F.sup.d.sub.1] [x.sup.d.sub.1](t), (17)

for the medium dynamics,

[z.sub.2](t) = [F.sub.2][x.sub.2](t), (18)

[z.sup.d.sub.2](t) = [F.sup.d.sub.2][x.sub.2](t), (19)

for the fast dynamics,

[z.sub.3](t) = [F.sub.3][x.sub.3](t), (20)

[z.sup.d.sub.3](t) = [F.sup.d.sub.3][x.sub.3](t), (21)

and for the dissipation field,

[D.sub.out](t) = L[D.sub.in](t). (22)

The junction structure of the BGI is given by

[mathematical expression not reproducible]. (23)

Then, the matrices of the state variable representation are defined by

[mathematical expression not reproducible], (24)

[mathematical expression not reproducible], (25)

and the matrices of the output equation are described by

[mathematical expression not reproducible], (26)

where

[E.sub.1] = I - [S.sup.11.sub.14] [([F.sup.d.sub.1]).sup.-1] [S.sup.11.sub.41][F.sub.1], (27)

[E.sub.2] = I - [S.sup.22.sub.14] [([F.sup.d.sub.2]).sup.-1] [S.sup.22.sub.41][F.sub.2], (28)

[E.sub.3] = I - [S.sup.33.sub.14] [([F.sup.d.sub.3]).sup.-1] [S.sup.33.sub.41][F.sub.2], (29)

M = L[(I - [S.sub.22]L).sup.-1]. (30)

4. Reduced Models in the Physical Domain

An important advantage of the bond graph methodology is the capacity of deriving different models. Hence, the reduced models for the different dynamics of three timescale systems can be obtained. A direct and interesting result to get the quasi-steady-state model in a bond graph approach is proposed in [24, 26, 27]. However, these published results have been extended for three timescale systems in this paper.

A bond graph called singularly perturbed bond graph ([SPBG.sub.H]) is proposed where the derivative causality to the storage elements of the fast dynamics is assigned and the storage elements of the slow and medium dynamics maintain an integral causality assignment whose scheme is shown in Figure 10.

In order to obtain the [SPBG.sub.H], the key vectors [z.sub.3] and [[??].sub.3] of Figure 10 have been changed with respect to Figure 9. By the causality assignment of the storage elements, the dissipation elements may have different causality from the corresponding BGI and the new key vectors are defined by

[D.sup.h.sub.out](t) = [L.sub.h][D.sup.h.sub.in](t), (31)

where [L.sub.h] is the constitutive relationship of the dissipation elements in the bond graph [SPBG.sub.H].

The following lemma permits us to determine reduced systems with singular perturbations in the physical domain:

Lemma 1. Consider three timescale systems modelled by bond graphs where storage elements for the slow and medium dynamics have an integral causality assignment and the storage elements for the fast dynamics have a derivative causality which has been assigned whose junction structure is described by

[mathematical expression not reproducible], (32)

and then a state-space representation of linearly independent state variables of a reduced system is defined by

[mathematical expression not reproducible], (33)

where

[mathematical expression not reproducible], (34)

[mathematical expression not reproducible], (35)

with

[E.sub.h1] = I - [H.sup.11.sub.14] [([F.sup.d.sub.1]).sup.-1] [H.sup.11.sub.41][F.sub.1], (36)

[E.sub.h2] = I - [H.sup.22.sub.14] [([F.sup.d.sub.2]).sup.-1] [H.sup.22.sub.41][F.sub.2], (37)

[M.sub.h] = [L.sub.h] [(I - [H.sub.22][L.sub.h]).sup.-1], (38)

and the real roots of the algebraic equation is given by

[mathematical expression not reproducible], (39)

where

[[A.sup.h.sub.31] [A.sup.h.sub.32]] = [[F.sup.-1.sub.3] ([H.sup.31.sub.11] + [H.sup.31.sub.12][M.sub.h][H.sup.11.sub.21])[F.sub.1] ([H.sup.32.sub.11] + [H.sup.31.sub.12][M.sub.h][H.sup.12.sub.21])[F.sub.2]], (40)

[B.sub.h3] = [F.sup.-1.sub.3] ([H.sup.31.sub.13] + [H.sup.31.sub.12][M.sub.h][H.sub.23]]), (41)

with the output

[mathematical expression not reproducible], (42)

where

[[C.sup.h.sub.31] [C.sup.h.sub.32]] = [([H.sup.11.sub.31] + [H.sub.32][M.sub.h][H.sup.11.sub.21])[F.sub.1] ([H.sup.12.sub.31] + [H.sub.32][M.sub.h][H.sup.12.sub.21])[F.sub.2]], (43)

[D.sup.h] = [H.sub.33] + [H.sub.32][M.sub.h][H.sub.23]. (44)

The proof is presented in Appendix A.

A scheme to obtain the quasi-steady-state model of a three timescale system by removing medium and fast dynamics called [SPBG.sub.R] is shown in Figure 11.

The storage elements for the medium and fast dynamics have a derivative causality assignment and the storage elements for the slow dynamics an integral causality is assigned in Figure 11. The new key vectors denoted by [D.sup.r.sub.out](t) and [D.sup.r.sub.in](t) for the dissipation elements of the [SPBG.sub.R] are described by

[D.sup.r.sub.out](t) = [L.sub.r][D.sup.r.sub.in](t), (45)

where [L.sub.r] is the constitutive relationship.

The mathematical description of a system with singular perturbations related to Figure 11 in the following lemma is proposed.

Lemma 2. Consider a three timescale system modelled by bond graphs with storage elements for the slow dynamics having an integral causality assignment and a derivative causality to the storage elements for the medium and fast dynamics is assigned whose junction structure is defined by

[mathematical expression not reproducible], (46)

and then the quasi-steady-state model of the system is given by

[[??].sub.1](t) = [A.sup.r.sub.11][bar.[x.sub.1]](t) + [B.sup.r.sub.1]u(t), (47)

where

[A.sup.r.sub.11] = [E.sup.-1.sub.r] ([R.sup.11.sub.11] + [R.sup.11.sub.12][M.sub.r][R.sup.11.sub.21])[F.sub.1], (48)

[B.sup.r.sub.1] = [E.sup.-1.sub.r] ([R.sup.11.sub.13] + [R.sup.11.sub.12][M.sub.r][R.sub.23]), (49)

being

[E.sub.r] = I - [R.sup.11.sub.14] [([F.sup.d.sub.1]).sup.-1] [R.sup.11.sub.41][F.sub.1], (50)

[M.sub.r] = [L.sub.r] [(I - [R.sub.22][L.sub.r]).sup.-1], (51)

with the real roots of the algebraic equation for the medium dynamics described by

[bar.[x.sub.2]](t) = [A.sup.r.sub.21][bar.[x.sub.1]](t) + [B.sup.r.sub.2]u(t), (52)

where

[A.sup.r.sub.21] = [F.sup.-1.sub.2] ([R.sup.21.sub.11] + [R.sup.21.sub.12][M.sub.r][R.sup.11.sub.21])[F.sub.1], (53)

[B.sup.r.sub.2] = [F.sup.-1.sub.2] ([R.sup.21.sub.13] + [R.sup.21.sub.12][M.sub.r][R.sub.23]), (54)

and the output for the reduced model is

[bar.y](t) = [C.sup.r.sub.1][bar.[x.sub.1]](t) + [D.sup.r]u(t), (55)

where

[C.sup.r.sub.1] = [R.sup.11.sub.31] + [R.sub.32][M.sub.r][R.sup.11.sub.21])[F.sub.1], (56)

[D.sup.r] = [R.sub.33] + [R.sub.32][M.sub.r][R.sub.23]. (57)

The proof is presented in Appendix B.

5. Case Study

The DC motor has widely been used in variable drive systems.

Originally, the speed of DC motors was controlled by adjusting the current of the shunt field. This method of control was replaced by the Ward Leonard system [30]. This system is a motor-generator set to supply an adjustable voltage to the variable speed DC drive motor which is shown in Figure 12.

The Ward Leonard system is formed by an armature winding for the generator with resistance [R.sub.g] and inductance [L.sub.g], and the electromechanical conversion between velocity input w and the generated voltage is denoted by [n.sub.g]; the armature winding for the motor has the resistance [R.sub.m] and inductance [L.sub.m], the connection between the two armature windings of the generator and motor is done by a capacitor [C.sub.e] and a resistance [R.sub.e], and the motor drives a mechanical load from the electromechanical conversion [n.sub.m]. The mechanical load is the coupling of two inertia [J.sub.m] and [J.sub.c] by using a mechanical transformer a and a spring [K.sub.c] with the damping [b.sub.k].

The BGI of the Ward Leonard system is shown in Figure 13.

The key vectors and the constitutive relationships of the BGI for the slow dynamics are

[mathematical expression not reproducible], (58)

[mathematical expression not reproducible], (59)

for the medium dynamics,

[mathematical expression not reproducible], (60)

for the fast dynamics,

[x.sub.3] = [q.sub.7]; [[??].sub.3] = [f.sub.7]; [z.sub.3] = [e.sub.7]; [F.sub.3] = 1/[C.sub.e], (61)

for the dissipation elements,

[D.sub.in] = [[[f.sub.13] [e.sub.20] [f.sub.3] [f.sub.9] [e.sub.6]].sup.T]; [D.sub.out] = [[[e.sub.13] [f.sub.20] [e.sub.3] [e.sub.9] [f.sub.6]].sup.T]; L = diag {[b.sub.m], 1/[b.sub.k], [R.sub.g], [R.sub.m], 1/[R.sub.e]}, (62)

and the input u = [f.sub.1].

The system order is n = 2, [n.sub.d] = 1, m = 2, and l = 1; thus, n + m + l = 5. The junction structure of the BGI is described by

[mathematical expression not reproducible]. (63)

From (27), (58), (59), and (63),

[mathematical expression not reproducible]. (64)

From (24), (25), and (30) with (58) and (60)-(63), the state equation of the complete system is defined by

[mathematical expression not reproducible]. (65)

In order to remove the fast dynamics of the system, [SPBG.sub.H] is shown in Figure 14.

From the BGI in Figure 13, a derivative causality assignment to the storage element for the fast dynamics C : [C.sub.e] is applied which is shown in Figure 14. Hence, the new key vectors and the constitutive relationship for the dissipation elements are defined by

[D.sup.h.sub.in] = [[[f.sub.13] [e.sub.20] [f.sub.3] [f.sub.9] [f.sub.6]].sup.T]; [D.sup.h.sub.out] = [[[e.sub.13] [f.sub.20] [e.sub.3] [e.sub.9] [e.sub.6]].sup.T]; [L.sup.h] = diag {[b.sub.m], 1/[b.sub.k], [R.sub.g], [R.sub.m], [R.sub.e]}. (66)

The corresponding junction structure of the bond graph of Figure 14 is given by

[mathematical expression not reproducible]. (67)

From (39)-(41) with (58)-(61), (66), and (67), the matrices to get the algebraic roots for [x.sub.3] are

[A.sup.h.sub.31] = [0 0], (68)

[A.sup.h.sub.32] = [[C.sub.e][R.sub.e]/[L.sub.g] -[C.sub.e][R.sub.e]/[L.sub.m]], (69)

[B.sup.h.sub.3] = 0. (70)

From the second lines of (34) and (35) with (58), (60), (66), and (67), the matrices of the reduced system for the medium dynamics are

[mathematical expression not reproducible], (71)

[mathematical expression not reproducible], (72)

[mathematical expression not reproducible]. (73)

In this case, [E.sub.1h] = [E.sub.1], and from the first lines of (34) and (35) with (58), (60), (66), and (67), the matrices of the reduced system for the slow dynamics are

[mathematical expression not reproducible], (74)

[mathematical expression not reproducible], (75)

[mathematical expression not reproducible]. (76)

By substituting (68) to (76) into (33) and (39), the quasi-steady-state model is defined by

[mathematical expression not reproducible]. (77)

By assigning a derivative causality to the storage elements for the medium dynamics, I : [L.sub.g] and I : [L.sub.m], the [SPBG.sub.R] is shown in Figure 15.

With the [SPBG.sub.R] of Figure 15, the new key vectors and the constitutive relationship for the dissipation elements are

[D.sup.r.sub.in] = [[[f.sub.13] [e.sub.20] [e.sub.3] [e.sub.9] [f.sub.6]].sup.T]; [D.sup.h.sub.out] = [[[e.sub.13] [f.sub.20] [f.sub.3] [f.sub.9] [e.sub.6]].sup.T]; [L.sup.r] = diag {[b.sub.m], 1/[b.sub.k], 1/[R.sub.g], 1/[R.sub.m], [R.sub.e]}, (78)

and the junction structure is described by

[mathematical expression not reproducible]. (79)

In this case, [R.sub.22] [not equal to] 0, and from (51), (78), and (79),

[mathematical expression not reproducible], (80)

where [DELTA] = [R.sub.e][R.sub.g] + [R.sub.e][R.sub.m] + [R.sub.g][R.sub.m].

By substituting (58), (60), (79), and (80) into (53),

[mathematical expression not reproducible], (81)

from (54), (60), (79), and (80),

[mathematical expression not reproducible], (82)

and from (52), (81), and (82), the real roots of the medium dynamics are

[mathematical expression not reproducible]. (83)

In this case, [E.sub.r] = [E.sub.1] and substituting (58), (79), and (80) into (48), the state matrix for the reduced slow system is given by

[mathematical expression not reproducible], (84)

from (49), (79), and (80),

[mathematical expression not reproducible], (85)

and from (47), (84), and (85), the quasi-steady-state model of the system is defined by

[mathematical expression not reproducible]. (86)

In order to show the effectiveness of the proposed methodology, Table 1 gives the numerical parameters of the BGI illustrated in Figure 13.

The eigenvalues of the system defined in (65) are

e ([A.sub.s]) = {-0.10871 + j0.3086, -0.10871 - j0.3086}, e ([A.sub.m]) = {-9.9972, -178.3}; e ([A.sub.f]) = {-8922.6}, (87)

and to demonstrate that the dynamics of the storage elements I : [J.sub.m] and C : [K.sub.c] are slow, I : [L.sub.m] and I : [L.sub.g] are medium and C : [C.sub.e] is fast, and we apply the property of (4) and (5):

[mathematical expression not reproducible], (88)

[mathematical expression not reproducible], (89)

and from (88) and (89), we can conclude that the Ward Leonard system is a three timescale system.

Now, the simulation of this case study is obtained using 20-sim software. Figure 16 shows the state variables of the complete system.

The performance of the reduced system given in (77) is shown in Figure 17. Note that, the exact and reduced models are very close.

Figure 18 shows the most reduced system defined in (86). When the slow dynamics of a system are important for analysis or synthesis, the quasi-steady-state model instead of the original system can be used.

The use of sensitivity analysis can give information about the coupling of the different timescales. In general, systems with different timescales are by definition weakly coupled. However, in singular perturbation theory, this coupling is taken into account. At each step of reduction, the coupling of the neglected timescale is taken into account in the remaining equations. Sensitivity according to [31, 32] gives the augmented bond graph from which the sensitivity (92) is calculated:

L ([partial derivative]y/[partial derivative][[theta].sub.j] = [[C.sub.n] [(sI - [A.sub.n]).sup.-1] [C.sub.n] + [D.sub.n]] (LW)(s), (90)

where [C.sub.n] is selected as [P.sub.10] which is a medium dynamics, and it is coupled to the fast and slow dynamics.

The sensitivity of [p.sub.10] respect to [C.sub.e] is given by

[partial derivative][p.sub.10]/[partial derivative][C.sub.e] = [n.sub.g][S.sub.1]/[[C.sub.e][K.sub.c][L.sub.g][b.sub.k] ([J.sub.c][a.sup.2] + [J.sub.m])] * 1/[[DELTA].sup.2] * [partial derivative][DELTA]=[partial derivative][C.sub.e], (91)

where [DELTA] = det(sI - A) and

[S.sub.1] = ([b.sub.m] + [J.sub.m]s + [a.sup.2][b.sub.k] + [J.sub.c][a.sup.2]s + [J.sub.m][K.sub.c][b.sub.k][s.sup.2] + [K.sub.c][b.sub.k][b.sub.m]s + [J.sub.c][K.sub.c][a.sup.2][b.sub.k][s.sup.2]), (92)

The sensitivity of [p.sub.10] respect to [J.sub.m] is [partial derivative][p.sub.10]/[partial derivative][C.sub.e] = [n.sub.g][S.sub.1]/[C.sub.e][K.sub.c][L.sub.g][b.sub.k]([J.sub.c][a.sup.2] + [J.sub.m]) * 1/[[DELTA].sup.2] * [partial derivative][DELTA]/[partial derivative][J.sub.m]. (93)

In the case of singular perturbation, the sensitivity due to the changes in [C.sub.e] and [J.sub.m] can be better represented by the difference between the original state variable and the sensitivities in (91) and (93). Figures 19 and 20 present the sensitivity respect to [C.sub.e] and [J.sub.m].

6. Conclusion

A three timescale system modelled by bond graphs has been presented. The state-space representation of the full system including slow, medium, and fast dynamics based on a bond graph model with all storage elements in an integral causality assignment is determined. In order to obtain reduced models in the physical domain, two lemmas are proposed. Lemma 1 establishes a derivative causality assignment to the storage elements for the fast dynamics and an integral causality assignment to the storage elements for the slow and medium dynamics into the bond graph called [SPBG.sub.H] is maintained. From [SPBG.sub.H], the junction structure and the mathematical model of this system are gotten. Thus, the system described by using Lemma 1 determines a reduced model formed by slow and medium dynamics where the fast dynamics have converged.

The most reduced model is obtained by assigning a derivative causality to the storage elements for the fast and medium dynamics and an integral causality to the storage elements for the slow dynamics into the bond graph called [SPBG.sub.R] which is defined by using Lemma 2, and then the junction structure and the quasi-steady-state model through this lemma are determined.

Undoubtedly, the algebraic approach given in Section 2 to find reduced models of singularly perturbed systems can be used. However, the mathematical model of this system must be gotten. Also, the quasi-steady-state model removing the fast dynamics given by using (7) requires that the submatrix [A.sub.33] has to be nonsingular and an algebraic process. If [SPBG.sub.H] can be obtained, then A33 is nonsingular and Lemma 1 gives a direct way to find the previous results. The quasi-steady-state model for the slow dynamics by using the algebraic approach is given by [bar.[[??].sub.1]](t) = [[A.sub.11] - [A.sub.13][A.sup.-1.sub.33] [A.sub.31] - ([A.sub.12] - [A.sub.13][A.sup.-1.sub.33][A.sub.32])[([A.sub.22] - [A.sub.23][A.sup.-1.sub.33][A.sub.31]).sup.-1] ([A.sub.21] - [A.sub.23][A.sup.-1.sub.33] [A.sub.31])][bar.[x.sub.1]](t) + [[B.sub.1] - [A.sub.13][A.sup.-1.sub.33][B.sub.3] - ([A.sub.12] - [A.sub.13][A.sup.-1.sub.33][A.sub.32]) [([A.sub.22] - [A.sub.23] [A.sup.-1.sub.33][A.sub.31]).sup.-1] ([B.sub.2] - [A.sub.23][A.sup.-1.sub.33][B.sub.3])]u(t), and it is clear that the bond graph defined by [SPBG.sub.R] gives the same result in a direct and easier way.

Thus, reduced models based on the manipulation of the causality of the bond graphs have been proposed. A class of nonlinear systems and linear time varying (LTV) systems with three timescales represent the future works of this paper.

https://doi.org/10.1155/2019/9783740

Appendix

A. Proof of Lemma 1

From the seventh and eighth lines of (32) with (16) to (21) and applying the derivative with respect to the time,

[[??].sup.d.sub.1](t) = [([F.sup.d.sub.1]).sup.-1] [H.sup.11.sub.41][F.sub.1][[??].sub.1](t), (A.1)

[[??].sup.d.sub.2](t) = [([F.sup.d.sub.2]).sup.-1] [H.sup.22.sub.41][F.sub.2][[??].sub.2](t), (A.2)

and from the fifth line of (32) with (31)

[D.sup.h.sub.in](t) = [(I - [H.sub.22][L.sub.h]).sup.-1] ([H.sup.11.sub.21][z.sub.1](t) + [H.sup.12.sub.21][z.sub.2](t) + [H.sup.13.sub.21][[??].sub.3](t) + [H.sub.23]u(t). (A.3)

From the first to fourth lines of (32) with (31),

[mathematical expression not reproducible], (A.4)

and by substituting (A.1) to (A.3) into (A.4) with (38),

[mathematical expression not reproducible]. (A.5)

Considering the linearly independent state variables, (A.5) with (36) and (37) can be reduced to

[mathematical expression not reproducible], (A.6)

where

[mathematical expression not reproducible], (A.7)

and

[mathematical expression not reproducible], (A.8)

and from (A.6) to (A.8) with (34), (35), (40), and (41), the expressions (33) and (39) in terms of the submatrices of the [SPBG.sub.H] are proved.

From the sixth line of (32) with (31), (38), and (A.3), the output equation is

y = [H.sup.11.sub.31][z.sub.1] + [H.sup.12.sub.31][z.sub.2] + [H.sup.13.sub.31][[??].sub.3] + [H.sub.32][M.sub.h] ([H.sup.11.sub.21][z.sub.1] + [H.sup.12.sub.21][z.sub.2] + [H.sup.13.sub.21][[??].sub.3] + [H.sub.23]u, (A.9)

and by doing [[??].sub.3] = 0 with (43) and (44), (42) is proved.

From the third line of (A.6),

[[??].sub.3] = -[([A.sup.h.sub.33]).sup.-1] [A.sup.h.sub.31][x.sub.1] - [([A.sup.h.sub.33]).sup.-1] [A.sup.h.sub.32][x.sub.2] + [([A.sup.h.sub.33]).sup.-1] [x.sub.3] - [([A.sup.h.sub.33]).sup.-1] [B.sup.h.sub.3]u, (A.10)

and comparing (1) with (A.10), the relationships for the fast dynamics between BGI and [SPBG.sub.H] are

[A.sup.h.sub.33] = [A.sup.-1.sub.33] [[epsilon].sub.2], (A.11)

[A.sup.h.sub.31] = -[A.sup.-1.sub.33][A.sub.31], (A.12)

[A.sup.h.sub.32] = -[A.sup.-1.sub.33][A.sub.32], (A.13)

[B.sup.h.sub.3] = -[A.sup.-1.sub.33][B.sub.3], (A.14)

and by substituting (A.12), (A.13), and (A.14) into (6), the solution for the fast dynamics (39) is proved.

Now, substituting (A.10) into the second line of (A.6),

[mathematical expression not reproducible], (A.15)

and comparing the second line of (1) with (A.15), the relationships between BGI and [SPBG.sub.H] for the medium dynamics are

[A.sup.h.sub.23] = [A.sub.23][A.sup.-1.sub.33] [[epsilon].sub.2], (A.16)

[A.sup.h.sub.21] = [A.sub.21] - [A.sub.23][A.sup.-1.sub.33][A.sub.31], (A.17)

[A.sup.h.sub.22] = [A.sub.22] - [A.sub.23][A.sup.-1.sub.33][A.sub.32], (A.18)

[B.sup.h.sub.2] = [B.sub.2] - [A.sub.23][A.sup.-1.sub.33][B.sub.3]. (A.19)

From (A.10) and the first line of (A.6),

[mathematical expression not reproducible]. (A.20)

Comparing the first line of (1) with (A.20), the relationships between BGI and [SPBG.sub.H] for the slow dynamics are

[A.sup.h.sub.13] = [A.sub.13][A.sup.-1.sub.33] [[epsilon].sub.2], (A.21)

[A.sup.h.sub.11] = [A.sub.11] - [A.sub.13][A.sup.-1.sub.33][A.sub.31], (A.22)

[A.sup.h.sub.12] = [A.sub.12] - [A.sub.13][A.sup.-1.sub.33][A.sub.32], (A.23)

[B.sup.h.sub.1] = [B.sub.1] - [A.sub.13][A.sup.-1.sub.33][B.sub.3], (A.24)

from (A.17) to (A.19) with (A.22) to (A.24), the quasi-steady-state model given by (33) is proved.

By substituting (A.10) into (A.9),

[mathematical expression not reproducible]. (A.25)

Comparing (2) with (A.25), the relationships between BGI and [SPBG.sub.H] for the output are

[C.sup.h.sub.3] = [C.sub.3][A.sup.-1.sub.33][[epsilon].sub.2], (A.26)

[C.sup.h.sub.1] = [C.sub.1] - [C.sub.3][A.sup.-1.sub.33][A.sub.31], (A.27)

[C.sup.h.sub.2] = [C.sub.2] - [C.sub.3][A.sup.-1.sub.33][A.sub.32], (A.28)

[D.sup.h] = D - [C.sub.3][A.sup.-1.sub.33][B.sub.3], (A.29)

and from (9), (A.27), (A.28), and (A.29), (42) is proved.

B. Proof of Lemma 2

From the derivative with respect to time of the last line of (46) with (16) and (17),

[[??].sup.d.sub.1] = [([F.sup.d.sub.1]).sup.-1] [R.sup.11.sub.41][F.sub.1][x.sub.1], (B.1)

from the sixth line of (46) with (45),

[mathematical expression not reproducible], (B.2)

from the first line to fifth line of (46) with (45),

[mathematical expression not reproducible], (B.3)

and substituting (B.1) and (B.2) into (B.3) with (51),

[mathematical expression not reproducible]. (B.4)

From (50) and (B.4), the reduced state-space representation can be written by

[mathematical expression not reproducible], (B.5)

where

[mathematical expression not reproducible], (B.6)

and

[mathematical expression not reproducible], (B.7)

and from (B.5) to (B.7), the expressions (48), (49), (53) and (54) are proved.

The reduced system of (B.5) can be written by

[mathematical expression not reproducible], (B.8)

where [A.sup.r.sub.11] = [E.sup.-1.sub.r] [A.sup.r'.sub.11][F.sub.1], [A.sup.r.sub.12] = [E.sup.-1.sub.r] [A.sup.r'.sub.12], [A.sup.r.sub.21] = [F.sup.-1.sub.2] [A.sup.r'.sub.21][F.sub.1], [A.sup.r.sub.22] = [F.sup.-1.sub.2] [A.sup.r'.sub.22], and [B.sup.r.sub.1] = [E.sup.-1.sub.r] [B.sup.r'.sub.1].

From the second line of (B.8),

[[??].sub.2] = -[([A.sup.r.sub.22]).sup.-1] [A.sup.r.sub.21][x.sub.1] + [([A.sup.r.sub.22).sup.-1][x.sub.2] - [([A.sup.r.sub.22]).sup.-1] [B.sup.r.sub.2]u. (B.9)

Comparing (7) with (B.9) and doing [[??].sub.2] = 0, the relationships between BGI and [SPBG.sub.R] are

[mathematical expression not reproducible]], (B.10)

[mathematical expression not reproducible], (B.11)

[mathematical expression not reproducible], (B.12)

and by using (B.10) to (B.12), the algebraic equation (52) is proved.

By substituting (B.9) into the first line of (B.8), the relationships between BGI and [SPBG.sub.R] are

[mathematical expression not reproducible], (B.13)

[mathematical expression not reproducible], (B.14)

[mathematical expression not reproducible], (B.15)

and from (B.13) to (B.15) with (12), (47) is proved.

From the seventh line of (46) with (B.2) and (51),

[mathematical expression not reproducible], (B.16)

and by using (B.16), the expressions (56) and (57) are proved. Equation (B.16) is rewritten by

y = [C.sup.r.sub.1][x.sub.1] + [C.sup.r.sub.2][[??].sub.2] + [C.sup.r.sub.3][[??].sub.3] + [D.sup.r]u, (B.17)

where [C.sup.r.sub.1] = ([R.sup.11.sub.31] + [R.sub.32][M.sub.r][R.sup.11.sub.21])[F.sub.1], [C.sup.r.sub.2] = [R.sup.12.sub.31] + [R.sub.32][M.sub.r][R.sup.12.sub.21], [C.sup.r.sub.3] = [R.sup.13.sub.31] + [R.sub.32][M.sub.r][R.sup.13.sub.21], and [D.sup.r] = [R.sub.33] + [R.sub.32][M.sub.r][R.sup.23]. By substituting (B.9) into (B.17) with [[??].sub.3] = 0,

y = [[C.sup.r.sub.1] - [C.sup.r.sub.2][([A.sup.r.sub.22]).sup.-1] [A.sup.r.sub.21]][x.sub.1] + [C.sup.r.sub.2][([A.sup.r.sub.22]).sup.-1] [x.sub.2] - [C.sup.r.sub.2] [([A.sup.r.sub.22]).sup.-1] [B.sup.r.sub.2]u + [D.sup.r]u, (B.18)

and comparing (B.18) with y = [C.sub.1][x.sub.1] + [C.sub.2][x.sub.2] + Du, the relationships between BGI and [SPBG.sub.R] are

[C.sup.r.sub.2] = [C.sub.2] ([A.sup.r.sub.22]), (B.19)

[mathematical expression not reproducible], (B.20)

[mathematical expression not reproducible], (B.21)

and from (14), the output of this reduced system given by (55) is proved.

Data Availability

The typical and traditional data used to support the findings of this study are included within the article. In particular, we used the data for the Ward-Leonard system from the typical DC machines with mechanical loads.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Gilberto A. Gonzalez [ID], (1) Noe G. Barrera, (2) Gerardo Ayala, (3) J. Aaron Padilla, (4) and David Z. Alvarado (5)

(1) Faculty of Electrical Engineering, Graduate Studies Division of the Faculty of Mechanical Engineering, University of Michoacan, Morelia, Mexico

(2) Technological Institute of Morelia, Morelia, Mexico

(3) School of Sciences of Engineering and Technology, Autonomous University of Baja California, Baja California, Mexico

(4) Faculty of Electrical Engineering, University of Michoacan, Morelia, Mexico

(5) Graduate Studies Division of the Faculty of Mechanical Engineering, University of Michoacan, Morelia, Mexico

Correspondence should be addressed to Gilberto A. Gonzalez; gilmichga@yahoo.com.mx

Received 10 May 2019; Accepted 26 August 2019; Published 20 October 2019

Academic Editor: Ludovic Chamoin

Caption: Figure 1: A bond.

Caption: Figure 2: Causal bond.

Caption: Figure 3: Power sources.

Caption: Figure 4: Bond with resistance.

Caption: Figure 5: Bond in integral causality for capacitance and inertance.

Caption: Figure 6: Bond in derivative causality for capacitance and inertance.

Caption: Figure 7: Bonds for transformers and gyrators.

Caption: Figure 8: 1 and 0 junctions.

Caption: Figure 9: Bond graph with a predefined integral causality assignment.

Caption: Figure 10: Scheme of an [SPBG.sub.H].

Caption: Figure 11: Scheme of an [SPBG.sub.R].

Caption: Figure 12: Scheme of a Ward Leonard system with mechanical load.

Caption: Figure 13: BGI of the Ward Leonard system.

Caption: Figure 14: [SPBG.sub.H] to remove the fast dynamics.

Caption: Figure 15: [SPBG.sub.R] to remove the medium dynamics.

Caption: Figure 16: State variables of the complete system: (a) slow dynamics; (b) medium dynamics; (c) fast dynamics.

Caption: Figure 17: State variables of the reduced system: (a) slow dynamics [p.sub.14]; (b) slow dynamics [q.sub.19]; (c) medium dynamics [p.sub.4]; (d) medium dynamics [p.sub.10].

Caption: Figure 18: State variables for the slow dynamics: (a) [p.sub.14] and [bar.[p.sub.14]]; (b) [q.sub.19] and [bar.[q.sub.19]].

Caption: Figure 19: Sensitivity of [p.sub.10] respect to [C.sub.e].

Caption: Figure 20: Sensitivity of [p.sub.10] respect to [J.sub.m].

Table 1: Parameters of the case study. [R.sub.g] = 0.2[ohm] [C.sub.e] = 0.0001 F [K.sub.c] = 3.5 rad/N-m [R.sub.m] = 0.1[ohm] [b.sub.m] = 1.2N - m - s [n.sub.g] = 0.5 [R.sub.e] = 1.1[ohm] [b.sub.k] = 2.5N - m - s [n.sub.m] = 0.1 [L.sub.g] = 0.02H [J.sub.m] = 4N - m - a = 2 [s.sup.2] [L.sub.m] = 0.01H [J.sub.c] = 2N - m - w = 10 rad/s [s.sup.2]

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Title Annotation: | Research Article |
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Author: | Gonzalez, Gilberto A.; Barrera, Noe G.; Ayala, Gerardo; Padilla, J. Aaron; Alvarado, David Z. |

Publication: | Mathematical Problems in Engineering |

Geographic Code: | 1MEX |

Date: | Nov 1, 2019 |

Words: | 8371 |

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