Given a differential equation with an excitation signal, the steps involved are the same.
The response obtained with zero initial conditions is known as the zero-state response. In the case of step input, the response obtained with zero initial conditions is known as the step response. The step response is due to the step input only, and hence the initial conditions are of zero value. We can also use terms such as the forced response and the natural response. They are interpreted differently in text books and hence these terms are not used often in this book. We deal with the zero-state response and the zero-input response, since there is no ambiguity associated with these responses. If a system has an excitation signal and non-zero initial conditions, then the response obtained is called the total response, which is the sum of the zero-state response and the zero-input response,
It is necessary to remember that the particular solution is dependent on the excitation signal, whereas the complementary solution is based on the homogeneous equation of the system. Excitation signal dictates what the particular solution should be. The order of the system does not affect the type of response due to the excitation signal. For a step input, the input remains constant for t > 0, and hence the rate of change of input is zero, during t > 0. Hence the particular solution should also be a constant, with the rate of change in particular solution being zero. This approach simplifies the solution. This approach is logical too, since the excitation signal is driving the system, and the system response, which is the particular solution, should resemble the excitation signal.
The first order differential equation with a step input is shown by equation (4.1). The neper frequency of the system is k, and the magnitude of step input is m. The initial condition is assumed to be of zero value. On the right hand side of equation (4.1), we have the product of k and m. Here k is the neper frequency, and m is the magnitude of the step input. When you form the differential equation for a system, this is how the differential equation will get formed if the system has steady-state unity gain. What is stated is explained later. The point being made here is that the magnitude of step input is only m, and it is not equal to the product of k and m.
Equation (4.2) presents the same equation with a first-order polynomial in D and an exponential input function that does not decay. The response is obtained as follows.
Since the input signal is a step input signal, it remains constant for t > 0, and hence the rate of change in input is zero during t > 0. Hence the particular solution is also a constant, with the rate of change in particular solution being zero. The particular solution satisfies equation (4.3). Replace y and its derivative by the particular solution and its derivative, as shown by equation (4.4). Since the derivative of particular solution is zero, the particular solution equals m, which is the magnitude of step input.
We can also use the technique that is going to be employed repeatedly for exponential excitation functions. A step input function can be viewed as an exponential function that does not decay. Let us consider equation (4.5). Since the input function is an exponential function, the particular solution is assumed to be an exponential function and it is evaluated, as shown by equation (4.6). Since the step function has a constant value for t > 0, the value of s in equation (4.5) is zero, the particular solution is expressed by equation (4.7). When the magnitude of step function, defined here as m, is unity, the particular solution is also equal to unity. Then the system can be said to have a steady-state gain equal to unity. This way of getting the particular solution may appear to be tortuous, but the idea is to show how the same technique can be used for a variety of input functions.
The next step is to get the complementary solution. To get the complementary solution, we need to form the homogeneous equation. The homogeneous equation is displayed now, and equation (4.8) presents the assumed complementary solution. We have already learnt how to get the complementary solution of first order system. The value of constant A, in equation (4.8) is obtained from the initial condition.
The total solution is the sum of the particular solution and the complementary solution. In this case, we are seeking the zero-state response or the step response. Equation (4.9) presents the sum of the particular solution and the complementary solution. The value of total solution, at t = 0, is equal to the sum of m and A. In other words, A equals the difference of the value of total solution at t = 0 and m, the magnitude of the step input. With this value of A, we present the total solution by equation (4.10). It is more useful to present equation (4.10) in a different way, as displayed by equation (4.11). The merit of equation (4.11) lies in the fact, that it presents the total solution as the sum of the response due to the input signal, and the response due to the initial condition. To get the step response, we can set the initial condition to be zero.
Equation (4.12) presents the unit step response of a system that has a steady-state unity gain. The steady-state gain is the gain the system after the transient, expressed by the decaying exponential function, dies down or becomes nearly zero. For unit step, the magnitude of input is 1. Set m equal to 1, and we get the response. The initial condition is assumed to be zero, in order to get the unit step response.
When the initial condition is not zero, the total solution is the sum of the step response and the zero input response. Equation (4.13) presents the step response due to the step input with a magnitude of m.
The zero input response is presented by the equation below equation (4.14). Then the total solution is obtained, as shown by equation (4.15). Given an input signal, and non-zero initial conditions, the response obtained is known as the total response. It may also be called as the complete response. In this text, we call response as the total response. We can state that the total response is the sum of zero-state response and the zero-input response. This statement is valid for any input. When the input function is a step function, the zero-state response and the step response are the same.
Let us take up some numerical examples now.
For example 1, unit step response is to be obtained, and the differential equation is shown by equation (4.16).
First the particular solution is obtained. It is a constant, and hence its derivative is equal to zero, as shown by equation (4.17). Equation (4.18) shows the differential equation in terms of the particular solution and its derivative. As shown by equation (4.19), the particular solution is a constant, and its value is one. Another way of obtaining the particular solution is illustrated by equation (4.20).
Next we obtain the complementary solution. It is assumed to be an exponential function, as shown by equation (4.21). Note that the homogeneous equation states what the neper frequency is, and the exponential function in the assumed solution corresponds to this neper frequency. The unit step response is the sum of the particular solution and the complementary solution. Equation (4.22) presents the unit step response, wherein the value of constant A is yet to be determined. The value of constant A is obtained from the initial condition which is set equal to zero. The value of A is equal to - 1, and equation (4.23) presents the unit step response.
Another example is considered. Example 2 is presented now by equation (4.24), where the magnitude of step input is 3. We obtain 3 by dividing 15 by 5.
First the particular solution is obtained. It is a constant, and hence its derivative is equal to zero. Equation (4.25) shows the differential equation in terms of the particular solution and its derivative.
Since the magnitude of step input is 3, the particular solution equals 3, as shown by equation (4.26). You should notice by now that the magnitude of the particular solution equals the magnitude of the step input, when the system has a steady-state unity gain.
Next we obtain the complementary solution. It is assumed to be an exponential function, as shown by equation (4.27). Note that the homogeneous equation states what the neper frequency is and the exponential function in the assumed solution corresponds to this neper frequency. The step response is the sum of the particular solution and the complementary solution. Equation (4.28) presents the sum of the step response and the complementary solution , wherein the value of constant A is yet to be determined. The value of constant A is obtained from the initial condition. It is equal to - 3, and equation (4.29) presents the step response.
Let us take up example 3. A first-order differential equation, with non zero initial condition is presented by equation (4.30). This system has non-zero initial condition.
The total response is the sum of the particular solution and the complementary solution. Since the magnitude of step input is 3, the particular solution equals 3, as shown by equation (4.31). Equation (4.31) presents the sum of the step response and the complementary solution, wherein the value of constant A is yet to be determined. The value of constant A, is obtained from the initial condition. It is equal to 7, and equation (4.32a) presents the total response. The total response is split into two parts, and it is the sum of the step response and the zero input response, as shown by equation (4.32b).
Now three cases are considered. The first case is about obtaining the unit step response, described by equations (4.33) and (4.34). The second case is about obtaining the total response given negative initial condition. The third case relates to obtaining the total response for the same the differential equation, but with a positive initial condition. In the end, the plots of three responses are shown. Here the unit step response has been obtained for the given differential equation.
For the first-order system, the initial condition is set to be - 0.5. The total response obtained is shown above, by equations (4.35) and (4.36).
For the same system, the initial condition is set to be 0.5. The total response obtained is shown by equations (4.37) and (4.38).
Fig. 1: Response of a first-order System with Unit Step Input
The three responses, corresponding to three cases, are shown above. It is seen that irrespective of the initial value, the final value of each of the responses is the same, which is unity. The system reaches the final value in 5 seconds. For the case considered, the time constant is 1 second, and hence it can be stated that the system reaches the final value after a time lapse that equals 5 times the time constant. This period is known as the settling time.
Given a differential equation with an excitation signal, the steps involved in obtaining the solution are the same, whether it is a first-order or a second-order system. The first step is to obtain the particular solution. The second step is to get the assumed complementary solution. The third step is to write the total solution as the sum of the particular solution and the complementary solution. The fourth step is to determine the unknown constants in the complementary solution from the initial conditions.
The differential equation of an over-damped system with unit step input is shown by equation (4.39). The aim is to get the unit step response, and hence the initial conditions are assumed to be of zero value. The magnitude of the step input is only one, since the coefficient of y(t), on the left hand side of equation is equal to the coefficient of the unit step function on the right hand side. The particular solution is also a constant, with the rate of change in the particular solution being zero. The particular solution satisfies equation (4.39). The particular solution and its derivatives are shown by equation (4.40). Replace y(t) and its derivatives by the particular solution and its derivatives. Then the particular solution has a value of one, which is the magnitude of unit-step input.
Equation (4.41) is the characteristic equation of the system. The roots are at - k and - m. The assumed complementary solution in equation (4.42) contains two exponential functions corresponding to the two roots. The values of coefficients, A and B, are evaluated from the initial conditions. To evaluate two coefficients, we need two equations. We form these equations based on the two initial conditions of a second order system.
The assumed unit step response is the sum of the particular solution and the complementary solution, as shown by equation (4.43). Since the value of unit step response at t = 0, is zero, we get equation (4.44). The first derivative of unit step response, evaluated at t = 0, is also zero. To get this value, we first differentiate equation (4.43), and equation (4.45) shows the result.
Since the first derivative of unit step response, evaluated at t = 0, is also zero, we get equation (4.46). The values of A and B are determined, and they are expressed by equation (4.47). The unit step response is displayed by equation (4.48). Next we take up a numerical example.
Example 5 is presented here. Equation (4.49) presents the differential equation of a second order over damped system with unit step input. The initial conditions are not specified, and they are assumed to be of zero value. We first obtain the particular solution. Given a unit step input, the particular solution is a constant. The derivatives of the particular solution are of zero value, as shown by equation (4.50). The particular solution satisfies the differential equation, and hence the particular solution has a value of one. Equation (4.51) presents the auxiliary equation, and the roots are distinct and real. They are located at - 2 and - 3. Hence the assumed complementary solution in equation (4.52) contains two exponential functions corresponding to the two roots. The values of coefficients, A and B, are evaluated from the initial conditions. A second order system has two initial conditions, which are both of zero value in this instance.
The assumed unit step response is the sum of the complementary solution and the particular solution, as shown by equation (4.53). Since the value of unit step response at t = 0 is zero, we get equation (4.54). The first derivative of unit step response, evaluated at t = 0, is also zero. To get this value, we first differentiate equation (4.53) and then set t = 0. Equation (4.55) shows the result. On solving the two simultaneous equations (4.54) and (4.55), we obtain the values of A and B, as shown by equation (4.56). The unit step response is then obtained, as shown by equation (4.57). Next we consider the case when initial conditions are not zero. The particular solution and the assumed complementary solution remain the same. But the values of A and B are different.
Next we obtain the total response given non-zero initial conditions. Equation (4.58) states the two initial conditions. Based on the initial conditions, we obtain equations (4.59) and (4.60). The values of A and B are stated by equation (4.61). The total response is expressed by equation (4.62). We can express the total response as the sum of the particular solution and the complementary solution. Alternatively, it is possible to express the total response as the sum of the unit step response and the zero input response. We have already obtained the unit step response. Hence we get the zero input response due to the initial conditions next.
The zero-input response is due to initial conditions only, with the input assumed to be zero. Hence we need to deal with the assumed complementary solution only. Equation (4.63) presents the assumed zero-input response and it is called yN. Based on the initial conditions, we obtain equations (4.64) and (4.65). The values of A and B are stated by equation (4.66). The zero input response is expressed by equation (4.67).
We can express the total response as the sum of zero-state response and the zero-input response. In this case, the zero-state response and the unit step response are one and the same. Equation (4.68) presents the unit-step response. Equation (4.69) presents the zero-input response. The sum of these two responses is the total response, presented by equation (4.70). We had obtained the same result earlier for the total response, presented by equation (4.62).
Fig. 2: Response of a second-order over damped System with Unit Step Input
The plots of unit step response, the zero-input response and the total response are shown in Fig. 2. The total response equals the zero-input response at start, whereas it becomes the same as the unit-step response for relatively large values of time. Once the transients due to the poles, or the neper frequencies of the system have decayed, both the unit step response and the total response settle down to a constant value of one. When the response equals unity, the response has reached a steady state, and we call this part of response as the steady-state response. Now the question can be asked when the system reaches the steady state. The system has two poles. The response due to the pole nearest to the origin decays more slowly. In this example, the pole nearest to the origin is located at - 2. The time constant corresponding to this pole is half a second. The response due to this pole is less than 1 percent, after a time lapse equaling five times the time constant. Hence after two and a half seconds, the system reaches the steady state. In some cases, we may be satisfied when the system has reached 98 percent of the final value. In such a case, the settling time is taken to be four times the time constant.
It is necessary to know what we mean by the forced response and the natural response. The forced response is due to the applied signal. Hence it can be taken to represent the zero-state response. But it is preferable to state that the forced response is the same as the particular solution. The other terms of the response containing the neper frequencies of the system contain the characteristic or the natural modes of the system and these terms combine to form the natural response. Part of the natural response stems from the non-zero initial conditions and the other part is due to zero initial conditions and the applied signal. In the absence of any input, the natural response is the same as the zero-input response.
In this sub-section, we determine the response of a second order critically-damped system to unit step input. The differential equation of a critically-damped system with unit step input is shown as equation (4.73). The aim is to get the unit step response, and hence the initial conditions are assumed to be of zero value.
The particular solution is also a constant, with the rate of change in particular solution being zero. The particular solution satisfies the given differential equation, presented by equation (4.73). The particular solution, and its derivatives are as shown by equation (4.74). Replace y(t) and its derivatives by the particular solution and its derivatives. Then the particular solution has a value of one, which is the magnitude of unit step input.
Equation (4.75) is the characteristic equation of the system. Both the roots are at - k. The assumed complementary solution presented by equation (4.75), contains an exponential function and it is multiplied by a first order polynomial in t, since the multiplicity of roots at - k is 2. The values of coefficients, A and B, are evaluated from the initial conditions. As we know, a second order system has two initial conditions.
The assumed unit step response is the sum of the complementary solution and the particular solution, as shown by equation (4.77). Since the value of unit step response and the value of its derivative at t = 0 are both zero, we get equation (4.78). On solving the two simultaneous equations, we obtain the values of A and B, as shown by equation (4.79). The unit step response is then obtained, as shown by equation (4.80) . Next we take up a numerical example.
Example 6 is presented now. Equation (4.81) presents the differential equation of a second order critically-damped system with unit step input. The initial conditions are not specified, and they are assumed to be of zero value. Equation (4.82) presents the auxiliary equation, and both the roots are at - 3 . Hence the assumed complementary solution presented by equation (4.83) contains an exponential function and it is multiplied by a first order polynomial in time t, since the multiplicity of roots is 2. The values of coefficients, A and B, are evaluated from the initial conditions. A second order system has two initial conditions, which are both of zero value in this instance.
We obtain next the particular solution. Given a unit step input, obtain the particular solution, as a constant. The derivatives of the particular solution are of zero value, and the particular solution satisfies the differential equation. Hence the particular solution has a value of one, as shown by equation (4.84).
The assumed unit step response is the sum of the complementary solution and the particular solution, as shown by equation (4.85). Since the value of unit step response and the first derivative of unit step response, evaluated at t = 0, are both zero, we get equation (4.86). On solving the two simultaneous equations, we obtain the values of A and B, as shown by equation (4.87). The unit step response is then obtained, as shown by equation (4.88).
We can express the unit-step response as the sum of the forced response and the natural response. The forced response is the same as the particular solution, as expressed by equation (4.89). The other terms of the unit-step response containing the neper frequencies of the system contain the characteristic or the natural modes of the system and these terms combine to form the natural response. The natural response is expressed by equation (4.90). Next we consider the case when initial conditions are not zero. The particular solution and the assumed complementary solution remain the same. But the values of A and B are different.
Next we obtain the total response given non-zero initial conditions. The two initial conditions are stated to be - 1 and + 1. The assumed total response is expressed by equation (4.91). Based on the initial conditions, we obtain equation (4.91). The values of A and B are obtained as - 5 and - 2, as expressed by equation (4.92). The total response is expressed by equation (4.93). Alternatively, it is possible to express the total response as the sum of unit step response and the zero input response. We already have the unit step response. Hence we get the zero-input response next. Then we can find out the total response, and then verify whether we get the same result, as shown by equation (4.93).
The zero input response is due to initial conditions only, with the input assumed to be zero. Hence we need to deal with the assumed complementary solution only. The initial conditions have been given, and they are equal to - 1, and + 1. Equation (4.94) presents the assumed zero state response, and it is called yN. Based on the initial conditions, we obtain equation (4.95). The values of A and B are obtained as - 2, and - 1 respectively. The zero-input response is expressed by equation (4.96).
We can express the total response as the sum of the zero-state response and the zero-input response. Equation (4.97) presents the zero-state response, and equation (194) presents the zero-input response. The sum of these two responses is the total response, presented by equation (4.99). We had obtained the same result earlier for the total response.
Fig. 3: Response of a second-order critically damped System with Unit Step Input
The plots of the unit step response, the zero input response, and the total response are shown in Fig. 3. The total response equals the zero input response at start, whereas it becomes the same as the unit step response for relatively large values of time.
Once the transients due to the poles of the system have decayed, we find that both the unit step response and the total response settle down to a constant value of one. When the response equals unity, the response has reached the steady state, and we call this part of response as the steady-state response. Now the question can be asked about when the system reaches the steady state. The system has two poles, at - 3, and it contains a term increasing linearly with time. In such a case, it is difficult to find when the total response will reach 99 percent of final value. The time constant in this case is one third of a second, and after a time lapse of 2.4 seconds, the total response is within 1 percent of final value. This period corresponds to 7.2 times the time constant. From the plots, it is difficult to make out the difference between the response of a critically-damped system, and that of an over-damped system. The response of an under-damped system is different. It exhibits some oscillations, as we will see next.
In this sub-section, we determine the response of an under-damped second order system to unit step input. The differential equation of an under damped system with unit step input is shown as equation (4.100). The aim is to get the unit step response, and hence the initial conditions are assumed to be of zero value. Equation (4.101) presents the assumed complementary solution. An under-damped system has a complex, conjugate pair of poles. The exponential function corresponds to the real part of the poles, and the oscillating frequency wd, corresponds to the imaginary part of the poles.
The particular solution is a constant, with the rate of change in particular solution being zero. The particular solution satisfies the given differential equation, presented by equation (4.100). Since the particular solution is a constant, its derivatives are of zero value. Replace y(t) and its derivatives, by the particular solution and its derivatives. Then the particular solution has a value of one, which is the magnitude of unit step input, as shown by equation (4.102). The assumed unit-step response is the sum of the complementary solution and the particular solution, as shown by equation (4.103). The derivative of unit-step response is shown by equation (4.104). To get the derivative, you need to know how to differentiate an expression by parts.
Since the value of unit step response is zero at t = 0, we get equation (4.105). We obtain the value of B as - 1. The first derivative of unit-step response, evaluated at t = 0, is also zero, and we get equation (4.106). We obtain the value of A from this condition. . The unit-step response is then obtained, as shown by equation (4.107). It is possible to present the unit step response in a form that is more meaningful. For this purpose, we need to know how the undamped natural frequency can be represented in terms of its real and imaginary part, as shown by the diagram shown next.
The relationship between the real part, the imaginary part and the magnitude of the complex frequency, that can be associated with an under-damped second order system, can be explained with the help of a right angled triangle. The real part of the pole corresponds to the neper frequency, and it is the part that determines how fast the envelope of the response decays exponentially. The frequency of oscillation of response is the damped frequency wd, and this is the imaginary part of the complex frequency. The magnitude is wn, which is the undamped natural frequency of the system, and it is represented by the hypotenuse. The cosine of the angle between the hypotenuse and the horizontal side is the damping coefficient. Given a fixed magnitude, the real part shrinks and the imaginary part grows, as the damping coefficient diminishes in value. For an under-damped system, the range of damping coefficient is from zero to one.
Based on this diagram, we get equation (4.108). The unit-step response, obtained earlier, is presented again, as equation (4.109). From the right angle triangle, we get values for k and wd, expressed by equation (4.110). Replace k and wd in equation (4.109) by the expressions in (4.110). Then the unit-step response can be presented by equation (4.111). The expression for the unit-step response can be simplified and we get the simpler expression for the unit-step response, as shown by equation (4.112). The trigonometric identity that has been used for simplification is shown as equation (4.113).
The identities that can be derived from the right angle triangle are shown by equation (4.114). Using the relationship that expresses the ratio of damped oscillating frequency wd to the undamped natural frequency wn as a function of damping coefficient, we can express the unit-step response, presented by equation (4.112), slightly differently, as shown by equation (4.115). It is seen that the damping coefficient figures prominently in equation (4.115). Since the unit step response is oscillatory, it is possible to find when the response peaks and express the peak value, only as a function of damping coefficient. Equation (4.115) is used to find out what the peak value of unit-step response is.
If a second order system is under-damped, the peak value of unit step response is higher than unity, and the peak overshoot and the instant at which it occurs can be determined as follows. The unit-step response is differentiated with respect to time and the derivative is set to zero to determine the instant at which the derivative is zero. From this value of time, the peak overshoot can be obtained. Differentiate the unit step response expressed by equation (4.115) and equate the derivative to zero, as shown by equation (4.116). Since the exponential function cannot be zero, the expression within the square brackets is zero, as shown by equation (4.117). We can move part of the expression from the left-hand side to the right-hand side , as shown by equation (4.118). From equation (4.118), we get equation (4.119). From the triangle relating the components of the complex frequency, we get the simplified equation, shown in equation (4.119).
The first peak of the response occurs at an instant equal to p over wd , as shown by equation (4.120). Using this relationship, we can simplify equation (4.112). The steps used to get equation (4.120) are shown by equation (4.121). Substitute the value of instant of the first peak in equation (4.120) into equation (4.112) to get the peak value of unit step response. The peak value of response and the peak overshoot are obtained as shown by equation (4.122). Peak overshoot is the amount by which the response shoots past the steady-state value of one. It can be seen that the peak value of response and the peak overshoot can be expressed as a function of damping coefficient alone.
In case, where the initial conditions are not zero, the total response is the sum of the particular solution and the complementary solution, as shown by equation (4.123). The constants in the complementary response can be obtained from the initial conditions, as shown by equations (4.124a) and (4.124b). Then the total response is expressed, as shown by equation (220). Next we take up a numerical example.
Example 7 is presented now. Equation (4.126) presents the differential equation of a second order under-damped system with unit step input. The initial conditions are not specified, and they are assumed to be of zero value. Equation (4.127) presents the auxiliary equation and the system has a complex, conjugate pair of roots. Hence the assumed complementary solution in equation (4.128) contains an exponential function, corresponding to the real part of the roots, and the oscillating frequency of the sinusoidal terms corresponds to the imaginary part of the poles. The values of coefficients, A and B, are evaluated from the initial conditions. A second-order system has two initial conditions, which are of zero value in this instance.
We obtain next the particular solution. Given a unit step input, obtain the particular solution is a constant. The derivatives of the particular solution are of zero value, and the particular solution satisfies the differential equation. Hence we get the particular solution to have a value of one, as expressed by equation (4.129).
The assumed unit step response is the sum of the complementary solution and the particular solution, as shown by equation (4.130). Since the value of unit step response at t = 0 is zero, we get equation (4.131), and we get that B equals - 1. The first derivative of unit step response, evaluated at t = 0, is also zero. To get this value, we first differentiate equation (4.130), and then set t = 0. Equation (4.132) shows the result, and we get that A equals - 0.5. The unit step response is then obtained, as shown by equation (4.133). Next we consider the case when initial conditions are not zero. The particular solution and the assumed complementary solution remain the same. But the values of A and B are different.
The zero-input response is due to initial conditions only, with the input assumed to be zero. Hence we need to deal with the assumed complementary solution only, and equation (4.134) presents the assumed complementary solution. The initial conditions have been given in equations (4.135) and (4.136). From the value of the initial condition at t = 0, we get the value of B. From the value of derivative of the complementary solution at t = 0, we get the value of A. Equation (136) presents the zero state response.
Given non-zero initial conditions, the total response is the sum of the unit-step response and the zero- input response. Equation (4.137) presents the differential equation with initial conditions. Equation (4.138) presents the total response.
Fig. 4: Response of a second-order under damped System with Unit Step Input
The plots of the unit step response, the zero input response, and the total response are shown in Fig. 4. The total response equals the zero input response at start, whereas it becomes the same as the unit step response for relatively large values of time.
Once the transients due to the poles of the system have decayed, we get that both the unit step response, and the total response settle down to a constant value of one. When the response equals unity, we can state that the response has reached a steady value, and we call this part of response as the steady-state response.
In this sub-section, we obtain the unit step response of a second-order system with no damping. The differential equation is presented by equation (4.139), and the initial conditions are of zero value. The assumed complementary solution is presented by equation (4.140).
The particular solution is equal to one, as shown by equation (4.141). The unit step response is presented by equation (4.142), as the sum of the particular solution and the complementary solution. The derivative of the unit step response is presented by equation (4.145). The values of A and B are obtained from the initial conditions.
Since initial value of the response is zero, B equals - 1, as shown by equation (4.146). Since the derivative has zero value at t = 0, the value of A is zero, as shown by equation (4.147). Then the unit step response of the system with no damping can be obtained and it is displayed by equation (4.148).
Fig. 5: Response of a second-order undamped System with Unit Step Input
The plot of the unit step response of a second order system with zero damping is shown in Fig. 5. The undamped natural frequency of the system has been set to be 8 radians per second. It can be seen that the response is oscillatory and it will last forever, since there is no damping.
Three more problems are presented here. The difference is that the system has two inputs, the unit-step input and its derivative. We can use the principle of superposition to get the solution.
The differential equation of a second-order over-damped system is presented by equation (4.149). We can express the response as the sum of two responses as shown by equation (4.150), the first due to the step input and the second due to its derivative. The differential equation with the step input is shown in equation (4.151).
Equation (4.152) presents the unit-step response. Refer to example 5 for getting this response. Since we are dealing with a linear system, the response to derivative of the unit-step input is the derivative of the unit-step response. Hence we get equation (153). Equation (4.154) presents the total response.
The differential equation of a second-order critically-damped system is presented by equation (4.155). We can express the response as the sum of two responses as shown by equation (4.156), the first due to the step input and the second due to its derivative. The differential equation with the step input is shown in equation (4.157).
Equation (4.158) presents the unit-step response. Refer to example 6 for getting this response. Since we are dealing with a linear system, the response to derivative of the unit-step input is the derivative of the unit-step response. Hence we get equation (159). Equation (4.160) presents the total response.
The differential equation of a second-order under-damped system is presented by equation (4.161). We can express the response as the sum of two responses as shown by equation (4.162), the first due to the step input and the second due to its derivative. The differential equation with the step input is shown in equation (4.163).
Equation (4.164) presents the unit-step response. Refer to example 7 for getting this response. Since we are dealing with a linear system, the response to derivative of the unit-step input is the derivative of the unit-step response. Hence we get equation (165). Equation (4.166) presents the total response.
This page has described how the step response of a first and a second order system can be obtained. The next page describes how the impulse response of a first and a second order system can be obtained.