BASIC FORMULATION AND REDUCTION - AND AUTOMATED SIMULATION

AND AUTOMATED SIMULATION

5.3 BASIC FORMULATION AND REDUCTION

other bonds, but bond 3 has implications for bonds 8 and 9, as well as for bonds 10 and 11. Inspection of Figure 5.6d reveals that assigning causality to bond 5 (associated with the elementI5) determines the causality on bonds 6 and 7. With powers assigned as shown in parte, the bond graph is completely augmented.

Two further points may be made with respect to augmentation. The first is that, in assigning causality, the results do not depend on the order of bonds chosen except in special circumstances to be discussed later in Section 5.4. The second point is that assignment of causality and assignment of power directions are two entirely independent operations. Either one may be performed first. Typically, power directions will be first, but on occasion one may not bother to assign powers in studying aspects of system equation formulation.

FIGURE 5.7. Equation formulation. Example 1.

show p’s as the efforts on the˙ I-elements in integral causality and q’s as the˙ flows on C-elements in integral causality. In addition, it will prove useful to indicate the co-energy variables on the energy storage elements. These are the flow variables on the I-elements and the effort variables on theC-elements. We will discover that the co-energy variables will appear in the initial formulation and then will be eliminated during the reduction process. As a first example of basic formulation, consider the augmented bond graph in Figure 5.7a, which is repeated from Figure 5.4e. The causality has been assigned, and all energy storage elements are in integral causality, all bonds have a causal assignment, and there are no causal conflicts on the bonds of the constraint elements. The state variables are q2and p5. The formulation always starts by writing the rate of change of a state variable equals an effort or a flow. We then use causality to track the efforts and flow through the bond graph. Causality will show us the substitutions to make to eliminate unwanted variables while retaining appropriate ones, as the procedure leads us to explicit first-order equations. It makes no difference which state variable we start with. We will later learn that computer programs exist that can derive the equations following the same rules we use.

These programs number the bonds consecutively and derive equations starting with the state variable with the lowest bond number. We will do the same here.

Thus, starting withq2, we write,

q2=f2, (5.12)

and follow the causality of the model to direct us through the bond graph and lead us directly to the state equations. From Figure 5.7 we see thatf2is aninput to theC-element and anoutput from the 0-junction. This causal output is caused by the inputs,f1(t)andf3, and according to the power convention,

q2=f1(t)−f3. (5.13)

The flow f1(t) is the input flow from the flow source, and this variable is known and remains in the formulation. (The notation f1(t) may be used to remind us thatf1is aknown input function of time.) The flow,f3, is not wanted in the formulation, so we must look further.

We just usedf3as an input to the 0-junction, but it is also an output from the 1-junction. It is true that f3 equals bothf4and f5, but it is caused by f5 (look at the causality on the 1-junction). Thus, we write

q2=f1(t)−f5. (5.14)

The ﬂow,f5, is a co-energy variable and directly related to the state variable, p5, through the constitutive relationship for the element. If I5 is a nonlinear element, then

f5=⁻¹_I (p5), (5.15)

and, if I5is a linear element, as it is for this example, then f5= p5

. (5.16)

With this substitution we obtain the ﬁrst state equation as

q2=f1(t)− p5

. (5.17)

The second state equation starts with

p5=e5, (5.18)

and we need to follow the causal path and track down e5. From Figure 5.7 we see that e5 is aninput to the I-element and an output from the 1-junction. The 1-junction indicates thate5 is caused by e3 ande4, and according to the power convention,

e5=e3−e4; (5.19)

thus,

p5=e3−e4. (5.20)

Neither of these efforts is wanted in the ﬁnal formulation, so we must continue to use causality to determine what substitutions to make. The efforte3is an output from the 0-junction and is caused by e2. But e2 is a co-energy variable and is directly related to the state variable, q2, by the constitutive relationship for the C-element,

e2=⁻_C¹(q2), (5.21)

or, for the linear element used in this example, e2= q2

. (5.22)

This substitution will be made in (5.20), but ﬁrst let’s track down e4. The effort,e4, is an output from a resistance element where, in this causality,

e4=R(f4), (5.23)

or, for the linear assumption being used for this example,

e4=R4f4. (5.24)

Eq. (5.24) is used in (5.20) to eliminate e4, but we introduce, f4. This ﬂow is a causal output from the 1-junction where the causal ﬂow input is the co- energy variable,f5, which is directly related to the state variablep5, as indicated in (5.16). Using the substitutions from (5.22), (5.24), and (5.16) in (5.20), we obtain the second state equation as

p5=(1/C2)q2−(R4/I5)p5. (5.25) Equations (5.17) and (5.25) are the two state equations for this system. Since this example is linear, the ﬁnal step is to place these equations into the standard matrix form of (5.11),

d dt

0 −1/I5

1/C2 −R4/I5

1 0

f1(t). (5.26) This is the perfect starting point for analysis if the system is linear. One need only follow the causality, and the substitutions needed to eliminate unwanted variables while retaining state and source variables automatically occur. The reader may not believe this yet, so we present a second example.

Figure 5.8 shows an augmented bond graph that originally came from an electrical circuit. When causality was assigned, all energy storage elements ended up in integral causality, and all bonds had a causal assignment. This is always an indication that formulation of equations is straightforward. The state variables for this example are the momentum variable on bond 2, p2, and the displacement variable on bond 5, q5 and input variable is e_i(t). If the physical bond graph were nearby we would realize that the momentum variable is the ﬂux linkage for the inductor of L2 henrys and the displacement variable is the charge on the capacitor ofC5farads. The derivatives of the state variables are indicated on the energy storage elements, as are the associated co-energy variables, f2ande5.

The formulation starts by writing

p2=e2 (5.27)

e₁(t) 1

FIGURE 5.8. Equation formulation. Example 2.

and then pursuinge2 by using the causality of the bond graph. The effort,e2, is an output from the 1-junction and is caused by e1, e3, ande4, according to the power convention; thus,

p2=e1(t)−e3−e4. (5.28) The effort,e1(t), is a source variable and remains in the ﬁnal equations. The efforte4 is an output from the 0-junction and is caused by the causal input,e5. Bute5 is a co-energy variable directly related to a state variable by

e5= q5

. (5.29)

Before substituting into (5.28), notice that e3 is an output from a resistance with causality dictating

e3=R3f3, (5.30)

and f3 is caused by the co-energy variable, f2, and f2 is related directly to a state variable by

f2=(1/I2)p2. (5.31)

Using (5.29), (5.30), and (5.31) in (5.28) yields the state equation

p2=e1(t)−(R3/I2)p2−(1/C5)q5. (5.32) The second state equation starts with

q5=f5, (5.33)

followed by using causality to track the path forf5. The ﬂow,f5, is the input to C5and the output from the 0-junction. This output is caused by the causal inputs f4 andf6, according to the power convention; thus,

q5=f4−f6. (5.34)

The ﬂowf4is the output from the 1-junction, caused by the co-energy variable, f2, where

f2=(1/I2)p2. (5.35)

The ﬂow f6is the causal output from a resistance, where causality indicates that

f6=⁻¹_R (e6)=(1/R6)e6 (5.36) for this linear example. But e6is caused by the co-energy variable,e5, which is directly related to a state variable according to (5.29). Using (5.35) and (5.36) in (5.34) yields the ﬁnal state equation as

q5=(1/I2)p2−(1/R6C5)q5. (5.37) The ﬁnal step for linear systems is to put state equations (5.32) and (5.37) into the standard matrix format,

d dt

−R3/I2 −1/C5

1/I2 −1/R6C5

1 0

e1(t). (5.38) Perhaps the reader is starting to see the power of using causality and the equation formulation procedure presented in this section. Any approach to equation derivation always requires elimination of some variables in favor of others in order to end up with a computable set of equations. Bond graphs and causality simply make the substitutions obvious so that appropriate variables are eliminated while others are retained, without having to guess the appropriate substitutions to make. One more example should be sufﬁcient to demonstrate the consistency of our approach to the formulation of state equations.

Figure 5.9 is the augmented bond graph originally shown in Figure 5.6 for development of causality. The bond graph represents a physical system with hydraulic and mechanical parts. To keep track of what parts are hydraulic and what parts are mechanical, we really need to have the physical bond graph.

For equation formulation, we need only have the augmented bond graph since equation formulation follows the same procedure regardless of the physical system. This example has a transformer that couples the hydraulic and mechanical sides of the model. The modulus of the transformer is physically an area, A. In Figure 5.9, the modulus,m, is defined by noting the constitutive relationships for the transformer. We will derive the equations in terms of the generic definition of the transformer shown in the figure, but before we can actually use the equations to predict system response, we will have to determine how the modulus, m, relates to the area, A. (A physical variable bond graph would tell us whether m as defined is actually A or 1/A. How the transformer modulus is actually used depends on the causality that was determined for the transformer bonds.)

The causal assignment once again yielded all energy storage elements in integral causality and no bonds were left unassigned. The state variables areq3, the

FIGURE 5.9. Equation formulation. Example 3.

displacement variable on C3;q4, the displacement variable on C4; andp5, the momentum variable on I5. The input variables from the sources are e1ande2. Also noted on the bond graph are the associated co-energy variables. Equation derivation starts with

q3=f3, (5.39)

but

f3=f9−f10, and

f9=f8=(1/R8)e8=(1/R8)(e1−e9)

=(1/R8)(e1−e3)=(1/R8) (e1−(1/C3)q3) . (5.40) The flow f10 is the output flow from the transformer, caused by the input flow,f11, according to

f10=(1/m)f11=(1/m)f5=(1/mI5)p5. (5.41) Substituting (5.40) and (5.41) into (5.39) yields the ﬁrst state equation as

q3=(1/R8) (e1−(1/C3)q3)−(1/mI5)p5. (5.42) The second state equation is particularly simple,

q4=f4=f5=(1/I5)p5. (5.43)

And the ﬁnal state equation is

p5=e5=e2−e4−e6−e7+e11. (5.44) The effort e2 is a source variable and remains in the ﬁnal formulation. The efforte4 is directly related to a state variable,

e4=(1/C4)q4. (5.45)

The effortse6and e7 are outputs from resistances, where e6=R6f6=R6f5=(R6/I5)p5

e7=R7f7=R7f5=(R7/I5)p5 (5.46) and e11is the output effort from the transformer,

e11=(1/m)e10 =(1/m)e3=(1/m)q3/C3. (5.47) Substituting (5.45), (5.46), and (5.47) into (5.44) yields the ﬁnal state equation as

p5=e2−(1/C4)q4−[(R6+R7)/I5]p5+(1/mC3)q3. (5.48) Equations (5.42), (5.43), and (5.48) are a complete set of linear state equations and can be cast into the standard matrix format,

d dt

⎡

⎢⎣ q3

⎤

⎥⎦=

⎡

⎢⎣

−(1/R8C3) 0 −(1/mI5)

0 0 1/I5

1/mC3 −1/C4 −(R6+R7)/I5

⎤

⎥⎦

⎡

⎢⎣ q3

⎤

⎥⎦+

⎡

⎢⎣

1/R8 0

0 0

0 1

⎤

⎥⎦ e1

. (5.49) We hope that by now the reader is convinced that by choosing the state variables to be the p’s on I’s and theq’s on C’s in integral causality, and by following the input/output causal paths, the state equations are straightforwardly derived. If the system is linear, then the resulting equations can be put into a standard matrix format that is the perfect starting point for linear analysis, which is covered in the next chapter. If the system is nonlinear, equations are still derived easily by using bond graphs and causality, but there is no nice matrix representation from which to launch an analysis. Instead, computer simulation is required, which is covered later in this chapter and at an advanced level in Chapter 13.

Unfortunately, models do not always lend themselves to straightforward formulation. Sometimes modeling assumptions lead to perfectly good models that have mathematical formulation difﬁculties. Such problems are the topic of the next two sections.

5.4 EXTENDED FORMULATION METHODS—ALGEBRAIC LOOPS

Dalam dokumen Modeling, Simulation, and Control of Mechatronic Systems (Halaman 185-193)