Control Systems - Stability

Control systems are used to control the behavior of any dynamic system. It provides accurate information about the dynamic system so that it can work well. One of the important aspects of the control system is STABILITY. The stability of the system is important in order to get the desired output from the system. In this article, we will deal with how control system analysis helps in providing stability to the system. We will also study types of stability, applications, and many more.

What is Stability?

The stability of the system means when a controlled input is provided to any dynamic system, it must result in providing the controlled output. In other words, the system must be BIBO stable i.e., bounded input bounded output system. If the system is not in our control i.e., uncontrolled output is obtained on providing the bounded input then the system is said to be unstable.

The above image shows a Unit Step Signal which is an example of the bounded signal. When the value of time (t) on the x-axis increases, the output value remains 1. This shows that the above signal is stable.

The above image shows a Unit Ramp Signal which is an example of the un-bounded signal. When the value of time (t) on the x-axis increases, the output value increases continuously. This shows that the above signal is unstable.

Types of Stability

There are 3 types of stability which are as follows:

• Steady State Stability
• Transient Stability
• BIBO Stability

Steady State Stability

Steady-state stability means when a system is subjected to constant input for a long duration of time and the system results in a stable output, it is known as steady-state stability. When a dynamic system provides a stable output during any disturbance in the input, it is said to be a stable system.

Transient Stability

When a system changes its state, it is known as a transition. During the transition period, whether the system is stable or not when subjected to some disturbance is determined by the transient stability.

BIBO Stability

Bounded input and bounded output stability show a system is stable when the system returns the bounded output when the bounded input is given. When the output is controllable, the system is stable else it is unstable.

Types of System Based on Stability

There are 3 types of system based on stability:

• Completely stable system
• Marginally stable system
• Conditionally stable system
• Unstable System

Completely Stable System

As the name suggests, a completely stable system provides a stable output for all ranges of values. One way to identify a completely stable system is to check the poles of the transfer function. If the poles of the open and closed loop system lie in the left half of the s-plane, then the system is completely stable.

The graph given below shows the completely stable system.

Marginally Stable System

A marginally stable system is a system that is stable for the current or present value. Any disturbance in the input can make the output of the system unstable. The marginally stable system can be identified when the poles of the open loop and closed loop system lie on the imaginary axis of the s-plane. The graph given below is the example of marginally stable system.

Conditionally Stable System

When a system is stable for certain values, then it is known as a conditionally stable system. The system can become unstable during the transient response. In simple terms, a conditionally stable system is stable only when the loop gain of a system is in a particular range. The image given below shows a conditionally stable system.

Unstable System

A system is said to be unstable when it produces uncontrolled output. The unstable system can be identified when the open and closed loop poles are on the right half of the s-plane. The given graph shows the unstable system.

Methods to Analyze the Stability

The stability analysis in the control system is done using various methods. Some of the important methods are listed below:

• Routh-Hurwitz Stability Criterion
• Nyquist Stability Criterion
• Root Locus Method
• Bode Plot

Routh-Hurwitz Stability Criterion

It is a mathematical method that is used to determine the stability of the LTI system. It provides information about the roots in the right half of the s-plane by analyzing the coefficients of the characteristic equation of the system.

According to the Routh Hurwitz Criteria, the polynomial must satisfy the following 3 conditions:

• All coefficients of the polynomial must have same sign.
• All the terms in first column of the Routh's Array must have the same sign.
• All the power of 's' must be present in the characteristic equation.

If the above conditions are satisfied then the system is stable otherwise it is unstable.

Example: Examine the stability of given equation using Routh's method

s^{3}+4s^{2}+s+16=0

Solution:

Creating the Routh's Array:

There are 2 sign change when we do the transition from 4 to -3 and then -3 to 16. As there are 2 sign change, the system is unstable.

Nyquist Stability Criterion

A Nyquist plot is a graphical representation used in control engineering. It is used to analyze the stability and frequency response of a system. This criterion works on the principle of argument. According to the Nyquist Stability Criterion, the number of encirclements of the point (-1, 0) is equal to the P-Z times of the closed loop transfer function. If the number of encirclements is in the anticlockwise direction then the system is stable.

The equation for stability analysis is given below:

N = Z – P

Where,
P = open loop pole of the system on right hand side (RHP)
Z = close loop zero of the system on right hand side (RHP)
N = number of encirclement around (-1,0)

Note: ‘N’ is negative for anticlockwise encirclement around (-1,0) and positive for clockwise encirclement around (-1,0).

Example: Given below is the Nyquist Plot in terms of 'k'. Find the condition of 'k' for which the system is stable.

Solution

Case 1: If k< 240

The point -1+j0 is not encircled. This means that there are no poles on the right half of the plane. This means the system is stable for k less than 240.

Case 2: k>240

The point -1+j0 is encircled two times in the clockwise direction. This means that Z>P and hence the system is unstable.

Stability condition: 0 < K < 240

Root Locus Method

Root Locus Method plots the graph for the pole's movement. This helps in easy analysis of the dynamic system as it tells how the poles of the system move with the change in the input values. This helps in the identification at which point the system is stable or unstable.

• When the root locus plot is at the right hand side of the plane, the system is unstable.
• When the root locus plot is at the left hand side of the plane, the system is stable.

Example: Given below is the root locus plot for \frac{k}{(s+1)(s+2)(s+3)} . Comment on the stability of the system.

Solution:

From the graph, it is clear that for the low values of the gain 'k', the system is stable as the root locus plot is on the left-hand side of the plane. But when we go for a higher value of gain 'k', the plot moves towards the right-hand side of the plane and hence it becomes unstable.

Bode Plot

Bode plots describe linear time-invariant systems’ frequency response (change in magnitude and phase as a function of frequency). It helps in analyzing the stability of the control system. It applies to the minimum phase transfer function i.e. (poles and zeros should be in the left half of the s-plane).

Stability by bode plot:

 ωpc > ωgc ->System is stable

 ωpc < ωgc ->System is unstable

ωpc = ωgc ->System is marginally stable

Where 'w_pc' is gain cross over frequency and 'w_pc' is phase crossover frequency.

Gain crossover frequency: It is the frequency at which the magnitude of G(s) H(s) is unity.

|G(jω)H(jω)|_ω=ωgc = 1

Phase crossover frequency: It is the frequency where the phase angle of G(s) H(s) is -180 degrees.

∠G(jω)H(jω)∣_ω=ωpc= -180^∘

Example: Given below is the frequency response of the transfer function. By analyzing the graph, comment on the stability of the system.

Solution

The above figure shows the gain and phase plot. The gain cross over frequency (w_pc) and phase crossover frequency (w_pc) can be calculated using gain plot and phase plot respectively.

W_gc is the value at 0dB whereas W_pc is the value at -180^o.

Here ωpc < ωgc. This means the system is unstable

Applications of Control Systems - Stability

• The control system stability is important in the aerospace sector for ensuring the stability of the aircraft, and missiles which helps in maintaining the desired performance with accurate output and stability of the flight.

• In the automobile industry the stability of the control system is important in the stability of the electricity control (ESC), the anti-lock braking, and the high-accuracy active suspension system.

• It find its application in the sector of power system for maintaining the stability of the electric grids and blackout preventions.

Advantages and Disadvantages of Control Systems - Stability

The advantages and disadvantages of the stability are given below :

Advantages

• The open loop control system is very simple in its design which makes it economical. 
• Closed-loop systems are more precise and accurate as compared to open-loop due to their complex structure. They can also handle the non-linearity. 
• The control systems also remove errors in the signals which leads to a reduction in noise. 
• The closed-loop control systems are capable of controlling the external factors which makes them more stable and reliable. 
• The closed-loop systems are more resource-efficient. 

Disadvantages

• The open loop systems does not have feedback mechanism which makes them highly inaccurate and unreliable for output. 
• The open-loop systems are incapable of removing the disturbance which occurs due to external factors. 
• The control system requires precise integration and tuning which is a challenging task.
• In the closed loop control system, some oscillations may exist which leads to unstability.

Conclusion

In this article, we have studied stability in the control system. Stability is crucial for any dynamic system for proper functioning. There are various techniques from which we can determine the stability of the system which is discussed in the article. We have also studied its applications, advantages, and disadvantages for a better understanding of the concept. This sows that if the output is controlled then we can say that the system is stable or if in open loop transfer function, any two poles are present on the imaginary axis - then the system is said to be stable.