Transfer Function Gain and Relative Stability

Key Takeaways

• Transfer function gain=Yssr(t), where Yss represents output  y(t) at steady-state and r(t) is the input.

• The transfer function gain is the magnitude of the transfer function, putting s=0. Otherwise, it is also called the DC gain of the system, as s=0 when the input is constant DC.

• If Ka is the given transfer function gain and Kc is the gain at which the system becomes marginally stable, then GM=KcKa

Linear system

Transfer function, steady-state, and stability are some terms that instantly pop up when we think about a control system. The steady-state and stability can be defined using the transfer function of the system. The transfer function gain is a parameter that connects the steady-state conditions and stability with the transfer function. It is the ratio of what you receive from the system as output to what you input to the system, under steady-state condition. 

Let’s define the transfer function gain with an example: consider kicking a ball with your palm and bouncing it back and forth. The height to which the ball bounces up and the hand kick we apply share a certain relation. Mathematically, the relationship can be defined as the ratio of height to which the ball bounces to the force of the kick given and symbolizes the transfer function gain at the steady-state condition. 

Defining Transfer Function Gain

Consider a linear system with input r(t) and output y(t). The output settles to a steady state after transients. Let R(s) and Y(s) be the Laplace transform of the input and output, respectively. Let G(s) be the open-loop transfer function of the system. Provided the initial conditions are zero, the equation is:  

The transfer function gain can be defined as the ratio of y(t) at steady-state, represented by 

Y_ss to the input r(t):

We assume that the steady-state output is attained as time, t, tends to infinity. The steady-state output can be defined as: 

The output y(t) is bounded for bounded input r(t). Now we will find the steady-state output Y_ss(s) using the final value theorem:

Obtain Y(s) from equation (1), and we get: 

Substituting equation (5) in (4):

Let’s say R(s) is a step input equal to . Substituting in equation (6), it is reduced to:

Putting equation (7) in equation (2):

From equation (8), we can conclude that the transfer function gain is the magnitude of the transfer function, putting s=0. Otherwise, it is called the DC gain of the system, as s=0 when the input is constant DC. With this knowledge, just consider a first-order system, with transfer function: 

The transfer function gain is obtained as K, substituting s=0.

So the transfer function is given in the form: 

where N(s) and D(s) are the numerator and denominator polynomials respectively. K represents the transfer function gain, irrespective of the order of the function. 

Relative Stability in Terms of  Transfer Function Gain

Every control system designer aims for a stable system, since stability is an important factor for a system to behave as expected. For high efficiency and cost-effectiveness, in some scenarios, we need to pull the system operation up to saturation regions or non-linearities. In such systems, the relative stability is a significant parameter. The relative stability is the measure of how close the system is to instability. It is usually defined using gain margin and phase margin. 

Consider that a unity feedback system is defined by the open-loop transfer function shown in equation (10). The factor by which the transfer function, gain K, can be multiplied to make the system marginally stable is called gain margin, GM. If K_a is the given transfer function gain and K_c is the gain at which the system becomes marginally stable, then:  

The gain margin is usually expressed in decibels:

The Routh Hurwitz criteria and Bode plots can be utilized for finding the gain margin of a stable system. 

When you are designing a control system, stability is of prime importance. If you know the transfer function gain of the system, calculating the gain margin is an easy way to check for relative stability. The Bode plots and Routh Hurwitz are some methods to find out the gain margin. The Cadence Design tools can be utilized for plotting frequency response from which the stability can be analyzed. PSpice offers the best simulation tools available to plot the Bode and frequency curves. 

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