Stability and robustness are fundamental design requirements of any control system. Consequently, stability analysis is a vital stage in the design and development process of a control system. In addition to providing information about the inherent stability of a system, stability analysis techniques are often employed to gain insight into the degree of stability of a system.

The most common methods used for determining the stability margins of a system, namely gain margin and phase margin, are based on frequency domain approaches such as the Nyquist, Bode, and Nichols method. These methods are limited to systems with less than two adjustable parameters. Systems that have more than two adjustable parameters require more complex control strategies such as the Vishnegradskii diagram, the parameter plane method or the stability equation method. These methods are used to plot the stability boundary of the system and to determine the effects of parameter variation on system stability but give no information about the stability margins (gain margins or phase margins) of the system.

The method described by Chang and Han [1] in their paper has been shown to be an effective method in determining the gain margin and phase margin of a system with adjustable parameters, such as a space shuttle. Their method combines commonly used frequency domain approaches with the parameter plane stability method to obtain boundary plots of constant gain margin and constant phase margin.

[1]  Chang, C-H., Han K-W. Gain Margins and Phase Margins for Control Systems with Adjustable Parameters. Journal of Guidance, Control, and Dynamics, (1989): 13(3), 404-408

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The following section presents a brief overview of the approach taken by Chang and Han in determining the gain and phase margin of a system with adjustable parameters.

Consider the block diagram of a basic closed-loop control system shown in Figure 1.

Figure 1: Basic Block Diagram of a Closed-Loop System

The open loop transfer function for this system is defined as:

Substituting $s\=j\cdot \mathrm{\ω}$ into equation (1) yields:

Equation (2) can then be rewritten as:

Expressing equation (3) in terms of its magnitude and phase yields the following equation:

where

$∣G\left(\mathrm{j\ω}\right)∣\= \sqrt{{\left(\mathrm{\ℜG}\left(\mathrm{j\ω}\right)\right)}^2\+{\left(\mathrm{\ℑG}\left(\mathrm{j\ω}\right)\right)}^2}$

and

$\mathrm{\Φ}\mathit{ }\mathit{\=}\mathit{ }\angle G\left(\mathrm{j\ω}\right)\= \arctan \left\{\frac{\mathrm{\ℑG}\left(\mathrm{j\ω}\right)}{\mathrm{\ℜG}\left(\mathrm{j\ω}\right)}\right\}$

Combining equations (2) and (4) gives:

Dividing both sides of equation (5) by $∣G\left(\mathrm{j\ω}\right)∣{\ⅇ}^{j\cdot \mathrm{\Φ}}$ results in:  

Let's define:

$A\=\frac{1}{∣G\left(\mathrm{j\ω}\right)∣}$

and

$\mathrm{\Theta }\= \mathrm{\Phi } \+ 180$

where A is the gain margin of the system at $\mathrm{\Θ} \= 0$, and $\mathrm{\Theta }$ is the phase margin of the system when $A\=1$.

Then equation (6) can be rewritten as:

or as

The term $A{\ⅇ}^{-j\cdot \mathrm{\Θ}}$ is commonly referred to as the gain-phase margin tester. The gain-phase margin tester can be used to determine the gain and phase margin of a system with adjustable parameters by incorporating it as an additional block in the closed loop system, such as the one shown in Figure 2.

Figure 2: Basic Block Diagram of a Closed-Loop System with Gain-Phase Margin Tester

The gain-phase margin tester can also be expressed as: $$

where X and Y are:

Substituting X and Y into equation (8) yields:

which can be expressed in terms of the real and imaginary parts, namely:

Assuming that X and Y are parameters, we obtain the following expressions:

where $\mathrm{B\_\_1}$, $\mathrm{C\_\_1}$, $\mathrm{D\_\_1}$, $\mathrm{B\_\_2}$, $\mathrm{C\_\_2}$, and $\mathrm{D\_\_2}$ are functions of $\mathrm{\ω}$.

Solving equations (14) and (15) simultaneously yields:

and

where

If the system has adjustable parameters then equation (8) can be written as: