Solutions of system differential equations-the total response of the system

2-2 Solutions of system differential equations-total response of the system

1. Proof of linearity of linear differential equations

The linear system must satisfy both homogeneity and superposition. Therefore, to prove whether the differential equation of a linear system is linear, we must prove whether it satisfies both homogeneity and superposition.

The general form of the differential equation of a linear system is

(2-5)

Assuming that the solution of this equation to the input f1 (t) is y1 (t), then

(2-6)

If the solution of the equation to the input f2 (t) is y2 (t), then

(2-7)

Multiply both ends of the equal sign of formula (2-6) by an arbitrary constant A1, and multiply both ends of equal sign of formula (2-7) by an arbitrary constant A2, then have

Add these two forms and you have

This means that if f1 (t) y1 (t), f2 (t) y2 (t) then A1f1 (t) + A2f2 (t) A1y1 (t) + A2y2 (t), that is, the system described by equation (2-5) is linear.

Second, the solution of the system differential equation-the total response of the system

Seeking a solution to the system's differential equations is actually seeking the system's full response y (t). The solution of the system differential equation is the total response y (t) of the system. The full response y (t) of a linear system can be decomposed into the superposition of zero input response yx (t) and zero state response yf (t), ie . The following proves this conclusion.

In Figure 2-2, if the excitation f (t) = 0, but the initial condition of the system is not equal to zero, then the system response is the zero input response yx (t), as shown in Figure 2-4 (a). According to equation (2-5), the differential equation of the system at this time is written as:

(2-8)

In Figure 2-2, if the incentive However, the initial condition of the system is equal to zero (that is, a zero-state system), and the response of the system at this time is the zero-state response yf (t), as shown in Figure 2-4 (b). According to equation (2-5), the differential equation of the system at this time can be written as

(2-9)

Add formula (2-8) and formula (2-9)

which is

In the formula

visible It is indeed the solution of the system differential equation (2-5). This conclusion provides a way and method to find the system's full response y (t), that is, first to obtain the zero input response yx (t) and the zero state response yf (t), and then the yx (t) and yf (t) Superposition, that is, the system's full response y (t), namely . This method is called the zero input zero state method.