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Chapter 5. Equations

5. Equations

You can define a set of differential equations, algebraic equations, and formulas to describe continuous changes of variables over time. You may associate equations (hereinafter we use the term equations for both equations and formulas) with:

•An active object;

•A particular state of a statechart.

At any given moment of time there exists the global active set of equations in a model. This set is constructed of all currently active equations of all objects and all statechart states. Equations associated with an active object are active during the whole object lifetime. Equations associated with a statechart state are active while the statechart is at that state. A statechart contributes to the global set not only the equations of the current simple state, but also of all its container states.

AnyLogic supports matrices and hyper-arrays as well as primitive types in equations, see Chapter 6, “Matrices and hyper-arrays”.

5.1 Equation types

An equation can be of three types:

•Differential equation (see section 5.1.1.1, “Differential equations”)

•Algebraic equation (see section 5.1.1.2, “Algebraic equations”)

•Formula (see section 5.1.1.3, “Formulas”)

5.1.1.1 Differential equations

Differential equations are defined in the form

d(x)/dt=F(x,y,t,…)

AnyLogic V User’s Manual

where x is an output or state variable of type double or of matrix type, and F(x,y,t,…) is a well-formed arithmetic expression which may contain any variables of the active object and the special symbol t denoting time. The expression F(x,y,t,…) can contain any arithmetic operations and method calls, such as, e.g. sin(), cos(), sqrt(), etc., or calls to your own methods.

F(x,y,t,…) should not call those methods that indirectly use variables of the active object; the methods should take variables as parameters. The best way to ensure this is to use static methods only.

F(x,y,t,…) should never call those methods that change variables of the active object.

F(x,y,t,…) should not contain conditional operators e.g. b>0 ? c:d.

You can define any number of differential equations. It is an error if a variable occurs more than once in the left-hand side in the set of equations. An input variable is not allowed to appear in the left-hand side.

Example

In the example shown in Figure 44, the behavior of the variable x is defined by the differential equation d(x)/dt=t.

Figure 44. Differential equations (I)

If you solve this equation analytically, you get x(t)=t^2/2 + C, which, if you take into account the initial value x(0)=1, gives x(t)=t^2/2+1. This is a parabola. Figure 45 shows this parabola calculated and drawn by AnyLogic.

AnyLogic V User’s Manual

Figure 47. Coordinate and velocity of the mathematical pendulum

5.1.1.2 Algebraic equations

Algebraic equations are defined in the form

x=F(x,y,t,…)

where x is an output or state variable of type double or of a matrix type, and F(x,y,t,…) is a well-formed arithmetic expression. For the rules that apply to F(x,y,t,…) see section 5.1.1.1, “Differential equations”.

Also, you specify the set of unknown variables, assuring that the number of unknown variables equals the number of algebraic equations:

find(x, y, …)

You can define any number of algebraic equations. You cannot specify an input variable as an unknown one.

AnyLogic V User’s Manual

5.2 Defining an equation

You specify an equation as a string. For convenience you can specify equations in the Equations code section of an active object class or in the Equation section of a variable. Those two places are self-synchronized; that is, if you write or remove an equation in one place, the equation is added or removed in the other place automatically. In the Equations code section of an active object class, you can specify all kinds of equations. However, in variable’s properties, you can define only differential equations and formulas for this particular variable.

►To define an equation in active object class code

1.In the Project window, right-click the Code item of the active object class and choose Open Code from the popup menu, or

Double-click the Code item of the active object class.

The Code window of the active object is displayed, see Figure 49.

2.Type the equation in the Equations section.

Figure 49. Code window of the active object class

►To define a differential equation for a variable

1.Select the variable on the structure diagram.

AnyLogic V User’s Manual

Figure 50. Dependencies displayed on the structure diagram

You may associate equations with a particular state of a statechart (composite as well as simple). You define equations for the statechart state into the property Equations of the state.

►To define equations for a statechart state

1.Click the state on the statechart diagram.

2.In the Properties window, type equations in the Equations section.

5.3 Functions

AnyLogic enables using custom functions in equations, namely:

•AnyLogic provides a set of predefined functions – the most frequently used mathematic functions.

•AnyLogic supports a special type of function – a lookup table. A lookup table is a continuous function defined in the table form. You may need it to define a complex non-linear relationship which cannot be described as a composition of standard functions, or to bring data defined as a table function to a continuous mode.

•AnyLogic enables you to define custom functions. This is frequently needed when there is a standard composition of functions used in multiple equations. In this case