Time Response Analysis of Continuous Time Systems

Sources

1. W. Bolton, Elsevier, “System Parameters” - 2002
   • In Transient Response
2. Bernard Friedland, Courier Corporation, “Dynamics of linear systems” - 2005
   • In Transient Response and State Space Representation
3. Arie Nakhmani, McGrawHill, “Introduction to Control Systems” - 2020
   • In State Variables
4. Arie Nakhmani, McGrawHill, “State-Space Representations” - 2020
   • In Stability of Continuous-Time Systems
5. Roland Burns, Elsevier, “Time Domain Analysis” - 2001
   • In Step Response
6. Sohrab Rohani, Yuanyi Wu, Elsevier, “Development of Linear State-Space Models and Transfer Functions for Chemical Processes” - 2017
   • In Unit Impulse Function
7. Taan S. ElAli, Tyler & Francis, “Signal Representation” - 2020
   • In Impulse Response
8. Pierre Muret, Wiley, “Continuous-time Systems: General Properties, Feedback, Stability, Oscillators” - 2018
   • Convergence and Divergence of the Impulse Response

Transient Response

• Transient response is the response of the system before settling down to steady state value, and after being subjected to an input.
• Investigating the transient response is important because it can allow us to identify the factors that affect the response speed and overshooting amount.
• A stable system necessitates that the transients plummet to a steady state value after a period of time.
• An unstable system is one that oscillates with an increasing amplitude, preventing it from reaching a steady state value.

Second-Order Systems

The transient-response characteristics of a second-order control system are the rise time, peak time, overshoot, decay ratio, settling time, and number of oscillations to settling time.

Rise time

The rise time refers to the period where the response of the system rises from 10% to 90% of the steady state value. We can find it using a formula consisting of the undamped natural angular frequency and the damping factor:

$$t_r=\frac{\pi }{2w_n\sqrt{1-{\zeta }^2}}$$

Peak time

The peak time describes the time that starts at 0 and ends where the first peak value occurs. It is given by the formula

$$t_p=\frac{\pi }{w_n\sqrt{1-{\zeta }^2}}$$

In a critically damped system (i.e., $\zeta =1$), the peak time becomes infinite, and, therefore, the response never reaches the steady state value despite coming near it.

Overshoot

The overshoot measures the highest amount reached by the response after surpassing the steady state value. In this regard, it is also equivalent to the first peak’s amplitude. It is typically expressed as a percentage of the steady state value:

$$\%OS=e^{-(\zeta \pi /\sqrt{1-{\zeta }^2})}\times 100\%$$

Subsidence Ratio

The subsidence ratio specifies how rapidly the oscillations decay. It is obtained by dividing the second overshoot’s amplitude by the first overshoot’s amplitude, thereby:

$$\text{subsidence ratio}=\frac{\text{second overshoot}}{\text{first overshoot}}=e^{-(2\zeta \pi /\sqrt{1-{\zeta }^2})}$$

Settling time

The settling time refers to the time where the magnitude of the response drops and stays within a specific percentage of the steady-state value, indicating how long it takes for the oscillations to dissipate. For a 2% steady-state value, the settling time is mathematically expressed as

$$t_s=\frac{-\ln (0.02\sqrt{1-{\zeta }^2})}{\zeta w_n}$$

which, when $0<\zeta <0.9$, can be approximated as

$$t_s=\frac{4}{\zeta w_n}$$

Number of Oscillations to Settling time

The number of oscillations to settling time is equals to the settling time divided by the periodic time. Hence, after some derivations, a settling time defined for 2% of the steady-state value will have a number of oscillations mathematically expressed as

$$\text{number of oscillations}=\frac{2}{\pi }\sqrt{\frac{1}{{\zeta }^2}-1}$$

State-Space Representation

Transient Response and State Space Representation

• The state-space representation is useful for simulating the transient response: the transfer function is typically converted into the state-space form because it is much easier to integrate the resulting differential equation from the state-space than solving the inverse Laplace transform through numerical methods.
• Besides this, the state-space representation is also useful for applying state-space methods on subsystems without needing to turn it into a set of first-order differential equations.

State Variables

• State space analysis can also be applied to systems with multiple input and multiple output (MIMO).
• They have state variables that indicate the whole state of the system at any given time through its set of internal signals. To put it in another way, it allows for state prediction given specific input signals
• All possible values of these state variables are found in the state space.

Stability of Continuous-time Systems

• For continuous-time systems, we can use the state-space representation to determine the system stability by using the eigenvalues of A vector.

Responses in Analysis

Step Response

• In the state-space approach, the step function is represented by $x_i(t)=B$ and $X_i(s)=\frac{B}{s}$ when time is greater than 0. For unit step one, however, it is particularly $x_i(t)=1$ and $X_i(s)=\frac{1}{s}$.
• Given an underdamped second-order system and a step response input, we can find the damping ratio either by using the percentage overshoot or the logarithmic decrement. The latter method requires more than one overshoot and the damp to be light. Moreover, the obtained time constant formula can be used to compute for the undamped natural frequency.
• The three step response performance specification used to specify the performance in the time domain:
  • Rise time
  • Overshoot
  • Settling time

Impulse Response

Unit Impulse Function

The unit impulse function $\delta (t)$ is the limit of a rectangular pulse function with area = 1 because it is equivalent to 1 at ${\int }_{-\infty }^{\infty }\delta (t)dt$ when $t=0$. Furthermore, it also has an area of unity, a width of zero, and an infinite height at $t=0$. For this reason, a unit impulse input is characterized by having a height of infinity at $t=0$, thereby also having a unity area at that time where height is infinity, and a Laplace transform of 1.

Impulse Response

The impulse response has an infinite magnitude at $t=0$ since it is the derivative of the step signal—a signal that is discontinuous at $t=0$.

Convergence and Divergence of the Impulse Response

A stable system is one that has an output that approaches 0 when the transient excitation $x(t)$ on the input returns to zero. Otherwise, it is unstable.

The inverse Laplace transform of the gain of the feedback path $H(s)$ is the impulse response. Accordingly, the convergence of the impulse response output signal toward 0 indicates that the system is stable, while its divergence indicates that it is unstable.