The technique for determining gain and phase margin of a system with adjustable parameters, as elaborated upon in the previous section, will now be applied to the control system of an aerodynamic maneuverable reentry vehicle. The block diagram corresponding to the control system is shown in Figure 3.
Figure 3: Basic Block Diagram of an Aerodynamic Reentry Vehicle
The transfer function associated with each block is defined below. The assumption is the system parameters, namely , , , and the frequency at which the system is run are all real numbers.
 (25) 
 (26) 
 (27) 
 (28) 
 (29) 
 (30) 
 (31) 
 (32) 
 (33) 
The transfer function representing GH1 is:
 (34) 
Similarly, the transfer function for GH2 is:
 (35) 
The transfer function for the simplified model is found by combining the transfer function equations for GH1 and GH2.
 (36) 
To determine the stability of the system, the closedloop system equation as defined in was modified to accommodate the inclusion of the gainphase margin tester as defined in equation (9). This results in an equation of the form:
where Denominator and Numerator refer to the denominator and numerator of , respectively. The numerator and denominator of can be extracted using the and commands.
 (37) 
 (38) 
Using equations (37) and (38) the closedloop system equation with the addition of a gainphase margin tester block can now be obtained.
 (39) 
The frequency response of the modified closedloop system is obtained by replacing the term s with .
 (40) 
To see the effects of parameter variations on the system, the equations , and defined respectively in (22), (23) and (24), will be calculated.
Recalling equations (20) and (21), respectively and , note that:
The real coefficients corresponding to are extracted using the following commands:
Similarly, the imaginary coefficients corresponding to are extracted using the following commands:
Using the sort command the values for , , , , , , and can be easily obtained.
 (41) 
 (42) 
 (43) 
 (44) 
 (45) 
 (46) 
 (47) 
Using the values obtained for and , the stability boundary plots for the system in the vs. plane can be obtained for different values of and . For the sake of efficiency, Maple procedures to calculate the values of and are generated from the equations for and as defined in equations (22) and (23)
The stability curves shown below assume the value of to be 30. The Maple code used to generated the plots is contained in the following code edit region.
Plot definition code
Stability Boundary Plots

Boundary Plots of Constant Phase Margins

Plot Command

Boundary Plots of Constant Gain Margins

Plot Command



Combining the boundary plots of constant phase margins and constant gain margins that lie within the stable region yields the plot shown below. Each region has a specified gain and phase margin. For instance, the region defined by will have a gain margin of and , and a phase margin of . While the region as defined by will have a gain margin of and , and a phase margin of .
The phase crossover frequency values can be obtained for any point along the constant gain margin boundary curves. Similarly, the gain crossover frequency values can be obtained for any point along the constant phase margin boundary curves. The phase crossover frequency values and the gain crossover frequency values for 6 data points are listed in the table below. The code used to obtain these values can be found in the following code edit region.
Crossover freq calc
Point

Value

Crossover Frequency

A

,

= (phase crossover frequency)

B

,

= (phase crossover frequency)

C

,

= (phase crossover frequency)

D

,

= (gain crossover frequency)

E

,

= (gain crossover frequency)

F

,

= (gain crossover frequency)


