Given a differential equation with an excitation signal, the steps involved in obtaining the solution are the same. 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. For a system with a staedy-state gain of unity, we can outline the method of solution for the ramp input, as shown below.
For unit step input, the particular solution is equal to 1, if the system has a steady-state gain of unity. The ramp function is an integral of the unit step function. Hence the particular solution due to the ramp function is the integral of the particular solution due to the unit step function. The integral of 1 is time t, and a constant arising out of integration. The slope of the particular solution is the same as that of the input ramp function. If the slope of the ramp function is one, then the slope of the particular solution is also one. If the slope of the ramp function is 2, then the slope of the particular solution is also 2. Remember that integration gives rise to a constant, known as the constant of integration. This constant, called as B here, is to be evaluated by substituting the particular solution and its derivative into the given differential equation.
Since the ramp input is the integral of unit-step input, it is possible to get the response due to the ramp input by integrating the unit-step response. This technique is also illustrated, but it may prove to be more difficult to obtain the solution in some cases.
The first order differential equation with the ramp input is shown by equation (6.1). The slope of the ramp function is only one. The coefficient of y on the left hand side of equation is k, and the coefficient of the ramp function on the right hand side of equation is also k. In such a case, the slope of the ramp function is only one. If the coefficient of the ramp function on the right hand side of equation is k times m, instead of just k, then the slope of the ramp function is m. This aspect will become clearer, when we form differential equation for a system. Here our task is to obtain the solution given a differential equation. For the system defined by equation (6.1), the neper frequency of the system is k, and the slope of the ramp input is 1. Since we are seeking the ramp response, the initial condition is assumed to be zero. The response to the ramp input is obtained as follows.
Here is repetition of what has been stated earlier. For the unit step input, the particular solution is equal to 1, as shown by equation (6.2). The ramp function is an integral of the unit step function, as shown by equation (6.3). Hence the particular solution due to the ramp function is an integral of the particular solution due to the unit step function. The integral of one is time t, plus a constant arising out of integration, as shown by equation (6.4). The slope of the particular solution is the same as that of the input ramp function. Remember that integration gives rise to a constant, known as the constant of integration. This constant, called as B here, is to be evaluated by substituting the particular solution and its derivative into the given differential equation.
Since the particular solution is known, its derivative can be obtained. It is equal to one. The particular solution satisfies the differential equation. Replace y and its derivative by the particular solution and its derivative, as shown by equation (6.5). The value of the unknown constant B, is obtained as shown next.
From equation (6.6), we get that (1 + B.k) is zero. Equation (6.6) shows the value of B. With this value of B, the particular solution can be expressed, as shown by equation (6.7).
Given the differential equation, if the slope of the particular solution is not known, the particular solution can be assumed to be of the form, expressed by equation (6.8). Both A and B can be determined as shown below.
Since the particular solution is known, its derivative can be obtained. It is equal to A. The particular solution satisfies the differential equation. Replace y and its derivative by the particular solution and its derivative, as shown by equation (6.9). The values of the unknown constants, A and B , are obtained as shown below.
We can form two equations, based on equation (6.11). The terms with time t, on either side of equal sign can be equated with each other. It turns out that the value of A is 1. It is seen that (A + B.k) is zero, since there is no constant term on the right hand side of equation (6.10). The value of B is obtained, as shown by equation (6.10). With these values of A and B, the particular solution can be expressed, as shown by equation (6.11). It is the same expression obtained earlier.
The complementary solution is the response obtained from the homogeneous equation. The complementary solution is of the form, as shown by equation (6.12).
The response to ramp input is obtained as the sum of the complementary integral and the particular integral, as shown by equation (6.13). Since the response to the ramp signal is to be obtained, we can ignore the response that may arise due to the initial condition of the system. This implies that the initial condition should be taken to be zero. When the initial condition is zero, the value of C, is equal to the reciprocal of k, the neper frequency. The reciprocal of k is the time constant of the system, and hence the value of C is equal to the time constant of the system. The ramp response is expressed by equation (6.14). The response to ramp can also be called the zero-state response, since the initial condition is set to be of zero value.
The ramp response can be obtained by integrating the response to unit-step signal, since the ramp signal is the integral of the unit-step signal, and the system under consideration is a linear system. Let the unit step response be called a(t). We have obtained the unit step response of first order system earlier, and it is presented by equation (6.15). The ramp response is obtained by integrating the unit step response, as shown by equation (6.16). We get a constant, as a result of integration. This constant, called B, is evaluated from the initial value of the ramp response. The initial value of the ramp response is assumed to be zero, and hence the value of B, can be determined. The ramp response is expressed by equation (6.17). It is the same expression obtained earlier.
When the initial condition is not zero, the response obtained is called as the total response. The total response of the system can be obtained as shown now. The total response is obtained as the sum of the complementary integral and the particular integral, as shown by equation (6.18). We know the particular solution, as it has been obtained earlier. The complementary integral is assumed to be an exponential function. The value of A is obtained from the initial value of total response, as shown by equation (6.19). The total solution is expressed by equation (6.20).
A numerical example is presented now. Equation (6.21) presents a differential equation for a first order system with ramp input. The initial condition is zero. The assumed complementary solution is expressed by equation (6.22). The assumed particular solution is presented by equation (6.23). Since the assumed particular solution satisfies the differential equation, we get equation (6.24).
From equation (6.24), the value of B is obtained to - 0.5. The ramp response is obtained as the sum of the complementary integral and the particular integral, as shown by equation (6.25). Based on zero initial value of the ramp response, we get the value of C, as 0.5. Equation (6.27) presents the ramp response.
We can obtain the solution, starting from the unit-step response. For the same system, the unit-step response can be obtained and it is expressed by equation (6.28). Integrate the unit-step response, as shown by equation (6.29). Since the initial condition is of zero value, the value of C can be evaluated to be - 0.5, and the ramp response is expressed by equation (6.30).
When the initial condition is not zero, the response obtained is called as the total response. The total response of the system can be obtained as shown now. The total response is obtained as the sum of the complementary integral and the particular integral, as shown by equation (6.31). We know the particular solution, since it has been obtained earlier. The complementary integral is assumed to be an exponential function. The value of C is obtained from the initial value of total response, as shown by equation (6.32). The total solution is expressed by equation (6.33).
Fig. 1: Response of a first-order System with Ramp Input
The plots of the total response, the zero input response, and the ramp response are shown now. The total response is the sum of the ramp response and the zero input response. It follows closely the zero input response at start, whereas it becomes the same as the ramp response for relatively large values of time. Once the transients due to the poles of the system have decayed, both the ramp response and the total response vary linearly with time. The total response or the ramp response keeps varying with time, and the system does not reach a steady state in this case. In this case, the input is a ramp function, and the ramp response is also increasing linearly with time, after a time lapse equaling five times the time constant. The question can be asked how well the response tracks the input. This question is answered next.
The ramp response is displayed by equation (6.34). After a time lapse greater than five times the time constant, the ramp response is expressed by equation (6.35). In such a case, the error, the difference between the ramp input and the ramp response is expressed by equation (6.36). The reciprocal of k st is the time constant of the first order system presented here. The sketch of the ramp input and the ramp response is shown in Fig. 13. The steady state error is the vertical offset, between the ramp input and the ramp response, after the exponential part of the response has decayed to nearly zero value. It is equal to the value of time constant t of the system. Larger the time constant is, larger the error is. The time lag is the horizontal offset, between the ramp input and the ramp response, and it equals the time constant. A first order system tracks the ramp input with an error equaling the time constant. In other words, the response of first order system lags the ramp input by a period equaling the time constant.
When the time varying input has a slope of n, the response of the first order system is presented by equation (6.37). After the exponential part of the response has decayed to nearly zero value, the response is expressed by equation (6.38). In such a case the error is n times the time constant, as stated by equation (6/39). In this case also, the response of first order system lags the input by a period equaling the time constant, even though the error at a given instant is n times the time constant.
The differential equation of a second order over damped system with ramp input is shown by equation (6.40). The slope of the ramp function is only one. The coefficient of y on the left hand side of equation is (k.m), and the coefficient of ramp function on the right hand side of equation is also (k.m). In such a case, the slope of the ramp input function is only one. Since we are seeking the ramp response, the initial conditions are assumed to be of zero value. As in the case of the first order system, the ramp response can be obtained as the sum of the complementary solution and the particular solution. The nature of particular solution is dependent on the input, and the system parameters do not influence it. Hence the nature of the particular solution is the same irrespective of the order of the system. We know that the particular solution for unit step input is one, as stated by equation (6.41). Since the ramp function is the integral of the unit step function, the particular solution for the ramp input is integral of the particular solution for the unit step input, as shown by equation (6.42). The particular solution due to ramp input function is time t, plus a constant, arising due to integration. This approach is valid, since we are dealing with linear systems.
The assumed particular solution is shown by equation (6.43). Its derivative can be obtained, and it is equal to one. The second derivative of the particular solution is zero, as shown by equation (6.44). The given differential equation is presented by equation (6.45). Since the assumed particular solution satisfies the differential equation, replace y and its derivatives by the particular solution and its derivatives. Then equation (6.45) appears as shown below.
After substituting the particular solution, and its derivatives into the given differential equation, we get equation (6.46). The value of the unknown constant C, is obtained as shown by equation (6.47). Equation (6.48) displays the particular solution. The value of the constant in equation (6.47) depends on the values of the poles of the over damped system. The poles of the over-damped system are at - k, and at - m.
The ramp response is the sum of the complementary solution and the particular solution. Equation (6.49) presents the sum of the assumed complementary solution and the particular solution obtained earlier. The system is over-damped and it has two distinct poles. Hence the assumed complementary solution is presented as the sum of two exponential functions, corresponding to the two poles. Since the initial value of ramp response and the initial value of its derivative are of both zero value, we get equation (6.50). From the two simultaneous equations, we get the values of A and B, as shown by equation (6.51). The ramp response of the over damped system is shown by equation (6.52). Dealing with symbolic expressions involving k and m, instead of real values, is a bit daunting, but it is necessary to get used to symbolic expressions.
The ramp response can be obtained by integrating the response to unit-step signal, since the ramp signal is the integral of the unit-step signal and the system we are dealing with is a linear system. Let the unit step response be called a(t). We have obtained the unit step response of second order over-damped system earlier, and it is presented by equation (6.53). The ramp response is obtained by integrating the unit step response, as shown by equation (6.54). We get a constant, as a result of integration. This constant is evaluated from the initial value of the ramp response. The initial value of the ramp response is assumed to be zero and hence the value of the constant can be determined. The ramp response is expressed by equation (6.54). It is the same expression obtained earlier.
A numerical example is presented now. Equation (6.55) presents a differential equation for a second order over-damped system with a ramp input. The initial conditions have zero values. The assumed complementary solution is expressed by equation (6.56), and the assumed particular solution is presented by equation (6.57). Since the assumed particular solution satisfies the differential equation, we get the equations, shown next.
The assumed particular solution is presented by equation (6.58) and the derivatives of the assumed particular solution are presented by equation (6.59). Since the assumed particular solution satisfies the differential equation, we get equation (6.60) after substituting the particular solution and its derivatives into the given differential equation. From equation (6.61), the value of C is obtained as equal to - (4/3). Equation (6.62) presents the particular solution.
The ramp response is obtained as the sum of the complementary integral and the particular integral, as shown by equation (6.63). The initial value of the ramp response and the initial value of its derivative are both of zero value. Based on the initial values of the ramp response and its derivative, we get equation (6.64). The values of A and B are obtained from the two simultaneous equations, derived from the initial conditions. Equation (6.65) presents the ramp response.
We can obtain the ramp response from the unit-step response. For the system described the equation (6.55), the unit-step response obtained is displayed by equation (6.65). On integrating the unit-step response, we obtain the ramp response, as shown by equation (6.66). The value of C is evaluated from the value of y(0). Since y(0)= 0, we get the value of C, as shown by equation (6.67).
Equation (6.67) presents the homogeneous differential equation of a second order over-damped system, and the initial conditions are also specified by equation (6.67). The assumed complementary solution is presented by equation (6.68). The assumed zero-input response is the sum of two exponential functions, corresponding to the two distinct roots, at - 1 and at - 3. Based on the initial values of the ramp response and its derivative, we get equation (6.69). The values of A and B are obtained from the two simultaneous equations, derived from the initial conditions. Equation (6.70) presents the zero-input response.
When the initial conditions are not of zero value, the response obtained is called as the total response. The total response of the system can be obtained as shown now. The total response is obtained as the sum of the zero-state response, and the zero-input response. Equation (6.71) presents the differential equation of a second order over-damped system with ramp input, along with the non zero initial conditions. In this case, the zero-state response is the same as the ramp response. The ramp response obtained earlier for this system is presented by equation (6.72). The zero-input response with these initial conditions has been obtained in the previous sub-section, and is expressed by equation (6.73). Hence the total solution can be expressed, as shown by equation (6.74).
Fig. 2: Response of a II order Over damped System with Ramp Input
The plots of the ramp input, and the ramp response are shown in Fig. 2. The ramp response increases slowly at start, and then increases linearly with time, after some time lapse equaling about six seconds. We see next how well the second order over damped system tracks the ramp input.
We find out the steady state error, and the time lag of the system. Equation (6.75) presents the ramp response of the over-damped system. After the exponential functions have decayed to zero value, the ramp response obtained is as presented by equation (6.76). The error, the difference between the input and the response, is obtained as shown by equation (6.77). We can express the error in a different way, as shown next.
A second order over-damped system has two poles, and there is a time constant associated with each pole. The time constants are expressed as t1 and t2 , and they can be expressed as the reciprocals of k and m, respectively. Then the error is seen to be sum of the time constants, as expressed by equation (6.78). The time lag of the over-damped system is equal to the sum of the two time constants.
The differential equation of a second order critically-damped system with a ramp input is shown by equation (6.79). The slope of the ramp function is only one. The coefficient of y on the left hand side of equation is k2, and the coefficient of ramp function on the right hand side of equation is also k2. In such a case, the slope of the ramp input function is only one. Since we are seeking the ramp response, the initial conditions are assumed to be of zero value. As before, the ramp response of a second order critically-damped system can be obtained as the sum of the complementary solution and the particular solution.
We know that the particular solution for unit step input is one if the system has a steady-state gain of unity.. Since the ramp function is the integral of the unit step function, the particular solution for the ramp input is integral of the particular solution for the unit step input. We get the assumed particular solution, as the sum of time t and a constant, as shown by equation (6.80). Its derivative can be obtained, and it is equal to one. The second derivative of the particular solution is zero, as shown by equation (6.81). Since the assumed particular solution satisfies the differential equation, replace y and its derivatives in the given differential equation by the particular solution and its derivatives, and this leads to equation (6.81). By matching terms on either side, we get the value of constant C, as shown by equation (6.82). Then the particular solution is obtained, as shown by equation (6.83).
Equation (6.84) presents the assumed complementary solution. The ramp response is the sum of the complementary solution and the particular solution. The particular solution has been obtained earlier. The system is critically-damped and has two poles at the same location. Hence the assumed complementary solution is presented as the product of an exponential function and a first order polynomial in time t. The exponential function corresponds to the pole, and the first order polynomial is due to multiplicity of pole being 2. The ramp response is expressed by equation (6.85). We get the values of A and B from the initial conditions, which are both of zero value.
Since the initial value of ramp response is of zero value, we get the value of B as 2/k. Since the initial value of its derivative is of zero value, we get the value of A as 1. The values of A and B are shown by equation (6.86). The ramp response of the critically-damped system is shown by equation (6.87).
The ramp response can also be obtained by integrating the unit-step response. It is somewhat difficult to obtain the solution, because we need to integrate by parts here. The unit step response of a critically-damped second order system is presented by equation (6.88). We have obtained this expression earlier. The ramp response is the integral of the unit step response, and it is displayed by equation (6.89). It is shown next, how we can carry out integration by parts.
Equation (6.90) shows how integration by parts is to be carried out. The parts are defined by equation (6.91), and the result is shown by equation (6.92). Using this result, we get equation (6.93). All the steps are not shown, but it is not difficult to get equation (6.93). When the initial conditions are not zero, the zero-input response can be obtained separately, and then the total response can be obtained as the sum of ramp response and zero-input response. A numerical example is presented below.
Equation (6.94) presents a differential equation for a second order critically-damped system with a ramp input. The initial conditions have zero value. The assumed particular solution is presented by equation (6.95). The derivatives of the assumed particular solution are shown by equation (6.96). Since the assumed particular solution satisfies the differential equation, we get equations, shown below.
Replace y and its derivatives in the given differential equation by the assumed particular solution and its derivatives. Then we get equation (6.97). From equation (6.97), the value of C is obtained as equal to - 1. Equation (6.98) presents the particular solution.
The ramp response is the sum of the complementary solution and the particular solution. The particular solution has been obtained earlier. Equation (6.99) presents the assumed complementary solution. The system is critically-damped and has two poles at the same location. Hence the assumed complementary solution is presented as the product of an exponential function, and a first order polynomial in time t. The exponential function corresponds to the pole and the first order polynomial is due to multiplicity of the pole being 2. The ramp response is expressed by equation (6.100). The initial value of the ramp response and the initial value of its derivative are both of zero value. Based on the initial values of the ramp response and its derivative, we get equation (6.101). Since the initial value of the ramp response is zero, we get that B = 1. Since the initial value of the derivative of the ramp response is zero, the value of A is 1, as stated by equation (6.101). Equation (6.102) presents the ramp response.
The ramp response can also be obtained by integrating the unit-step response. The differential equation with the ramp input is stated by equation (6.94). The unit step response of this critically-damped second order system is presented by equation (6.103). In order to get the ramp response, we need to integrate by parts, as shown next.
Equation (6.104) shows that the ramp response is the integral of the unit step response. Equation (6.105) shows the next step, where the last term in equation (6.105), containing both time t and the exponential function, is integrated by parts. Equation (6.106) shows the next step, and equation (6.107) shows the simplified result. The same result has been obtained earlier, using a different technique. When the initial conditions are not zero, the zero-input response can be obtained separately, as shown next.
When the system has non-zero initial conditions, the zero input response can be obtained as shown now. Equation (6.108) presents the homogeneous differential equation of a critically-damped system, with non zero initial conditions. Equation (6.109) presents the assumed complementary solution. A first order polynomial in time t is part of the assumed solution, because of the multiplicity of the pole at - 2 is two. From the initial conditions, the values of A and B are obtained to be 26 and 10, respectively. The complementary solution can then be represented by equation (6.111).
When the initial conditions are not of zero value, the response obtained is called as the total response. The total response of the system can be obtained as the sum of the zero-state response, and the zero-input response. Equation (6.112) presents the differential equation of a second order critically-damped system with a ramp input, along with the non zero initial conditions. In this case, the zero-state response is the same as the ramp response. The ramp response obtained earlier for this system is presented by equation (6.113). The zero-input response with these initial conditions has been obtained in the previous sub-section, and is expressed by equation (6.114). Hence the total response can be expressed, as shown by equation (6.115).
We find out the steady-state error and the time lag of the system. Equation (6.116) presents the ramp response of the critically-damped system. After the exponential functions have decayed to zero value, the ramp response obtained is as presented by equation (6.117). The error, the difference between the input and the response, is obtained as shown by equation (6.118). It is seen that the error is two times the time constant, where the time constant is the reciprocal of the neper frequency, where the neper frequency is equal to k.
Fig. 3: Response of a II order Critically damped System with Ramp Input
The plots of the ramp input and the ramp response are shown in Fig. 3. The ramp response increases slowly at start, and then increases linearly with time after some time lapse equaling about four seconds. The ramp response of the critically damped system looks very similar to that of the over- damped system.
The differential equation of an under damped system with ramp input is presented by equation (6.119). Equation (6.120) shows how the undamped natural frequency wn, the damped natural frequency wd, the damping coefficient x, and the neper frequency k, are inter-related. For an under-damped system, the damping coefficient lies between zero and 1. For equation (6.119), the slope of the ramp input function is 1. We get the ramp response as follows.
The ramp response of a second order under-damped system can be obtained as the sum of the complementary solution and the particular solution. When the system is under-damped, it has a complex conjugate pair of roots. In such a case, the complementary solution is assumed to be of the form, shown by equation (6.121). The real part of the pole is reflected by the exponential function. The imaginary part of the root defines the oscillating frequency of the sinusoidal terms.
We know that the particular solution for unit step input is one. Since the ramp function is the integral of the unit step function, the particular solution for the ramp input is integral of the particular solution for the unit step input. We get the assumed particular solution, as the sum of time t and a constant, as shown by equation (6.123). It can be noticed that the particular solution is dependent on the input signal, and not on the system. Hence the particular solution has the same form, irrespective of the order of the differential equation.
From the assumed particular solution, the first derivative can be obtained, and it is equal to one. The second derivative of the particular solution is zero. The assumed particular solution satisfies the differential equation. Replace y and its derivatives in the given differential equation by the particular solution, and its derivatives, and this leads to equation (6.124). By matching terms on either side, we get the value of constant C, as shown by equation (6.124). Then the ramp response is obtained, as shown by equation (6.125), as the sum of the complementary solution and the particular solution. In order to determine the values of constants A and B, forming part of the assumed complementary integral, we make use of the initial conditions, which are assumed to be of zero value. We can find the initial value of ramp response at time t = 0, from equation (6.125). The first derivative of the ramp response is expressed by equation (6.126). It is a bit long. The initial value of the derivative of the ramp response can be obtained from equation (6.126).
The values of constants A and B, can be obtained from the initial conditions, and their values are expressed by equation (6.127). By using these constants, the ramp response can be expressed, as shown by equation (6.128). This equation is somewhat long. With real values, the expression for the ramp response is not forbidding.
It is too messy to obtain the ramp response from the unit-step response and hence it is not shown. The response of the under-damped system is fast and it more or less tracks the input with only a small-error when the damping factor is low.
A numerical example is presented now. Equation (6.129) presents a differential equation for a second order under-damped system with a ramp input. The initial conditions have zero value. The assumed particular solution and its derivatives are presented by equation (6.130). Since the assumed particular solution satisfies the differential equation, we get equation (6.131), and the particular solution obtained is shown by equation (6.132).
The ramp response is the sum of the complementary solution, and the particular solution. The particular solution has been obtained earlier. Equation (6.133) presents the auxiliary equation, and equation (6.134) presents the assumed complementary solution. The system is under-damped, and has a complex conjugate pair of poles. Hence the envelope of assumed complementary solution is an exponential function, with its neper frequency being equal to the absolute value of real part of poles. The oscillating frequency in radians per second is the imaginary part of the complex pair of poles. The ramp response is expressed by equation (6.135). The initial value of the ramp response, and the initial value of its derivative, are both of zero value. Based on the initial values of the ramp response and its derivative, we get equation (6.136), and the values of A and B are obtained, based on the initial values. Equation (6.136) specifies the obtained values of A and B. Equation (6.137) presents the ramp response.
The same solution can be obtained, based on the unit-step response. Equation (6.138) presents the same differential equation with a step input. Equation (6.139) presents the unit-step response. The value of x1 is obtained as shown above, and its value is expressed by equation (6.140a).
The value of x2 is obtained as shown above, and its value is expressed by equation (6.140b).
Equations (6.141) and (6.142) show the remaining steps for obtaining the ramp response from the unit-step response. In the case of an under-damped system, obtaining the ramp response from the unit-step response is not recommended. The solution is shown here for the sake of completeness. It can be done but it is complicated.
The plot of the response is shown next.
Fig. 4: Response of a II order Under-damped System with Ramp Input
The plot of the ramp response of the under damped system is shown in Fig. 4. It is seen that this system tracks the ramp input function fairly fast, and the error is much less. In general, under-damped systems have a faster response, and hence many of the measurement systems tend to be under-damped.
The differential equation of a second order system with zero damping is displayed by equation (6.143). This system has a ramp input, and it is the aim to get the ramp response. We get the particular solution first. Since the assumed particular solution is as expressed by equation (6.144), we can get its first derivative to be 1, and its second derivative is zero. It can be verified that the value of constant C is zero and the particular solution varies linearly with time.
The complementary solution is assumed to be of the form, expressed by equation (409). The ramp response is expressed by equation (6.145). The values of constants, A and B, can be determined from the initial conditions, as shown next.
Fig. 5: Response of a II order Undamped System with Ramp Input
The plot of the ramp response is shown in Fig. 5, with the value of w being 2. The oscillations in response do not die down.
When a second-order system has a ramp input and a step input, the differential equation appears as shown by equation (6.150). The ramp response has already been obtained and it is presented by equation (6.151). The step response can be obtained, as shown by equation (6.152). The total response is presented by equation (6.153).
This page has described how the ramp response of a first and a second order system can be obtained. The next page describes how the response of a first and a second order system can be obtained, if the input is an exponential signal.