控制理论课程设计及Matlab代码

The gain margin and phase margin of the closed loop system is positive infinite and can also be found in the following two aspects from Bode plot andNyquistplot respectively.
On one hand, according to the Bode plot of the resultant system, since the Magnitude plot does not cross the 0 (dB),we can’t determine the phase margin. Besides, we can’tfind the gain margin either because the Phrase plotdoes not cross the -180(deg). As a result, for this resultant system, we can’t identify both thegain margin and phase margin.
,
Such that
,
whichcan be simplified as
Step 3:Define
Such that
Left and right multiply it by P gives
whichcan be simplified by applying as
Step 4:If wechoose , then the above equation is equivalent to
Determine the controllability of the given system.
The controllability matrix of the given open-loop system
,
The given system is controllable since the determinant of the controllability matrix
,
Where W, , are constants, , , are the functions ofu,w,q, , , and , and
and .
Linearize the longitudinal equations, we have
,
Around the trim condition
and ,
Assume all the states are measurable, design a state feedback controller ,where , such that the resultant system is asymptotically stable
andat the same time, meet the following requirements on rise time , the settling time and the overshoot OS%;
The overall system has the state-space form
(a)Given the state vectorandthe control input vector as
and ,
Then, the longitudinal equations of motion for the aircraft can be written in the following form:
Compare the system with state feedback controller with the original system.
On one hand, from the step response plot, we could tell that although bothsystemare asymptotically stable, the performance of the system with state feedback controller is betterfor itsshorterrise time and settling time, as well as lower overshoot. On the other hand, according to the Bode plot ofthetwo system, the magnitude of the peak drops and the phase changes more smoothly between and .
where
so is Hurwitz.
Step 2: Since is Hurwitz,
isasymptotically stable. ApplyLyapunovtest ofobservabilitytoit, there exists a positive symmetric matrix
,
with
The system is observable since the determinant of theobservabilitymatrix
Assuming that only the output is measurable, use the same state feedback gainKas in (b) to design anobserver-based feedback controller such that the eigenvalues of the state estimatormatr-ix (A-LC) lie to the left of -5 on the left-hand plane.
(c) Continue with (b) and perform the followinganalysis and designing:
Determinetheobservabilityof the system define by (6).
Theobservabilitymatrix of thegiven open-loop system is
Where A and B are
,
,
(b)Now consider an autopilot designing problem. The linearized equae-spaceform as
where
, , ,
Plot the unit step response(Figure 1),Bodeplot(Figure 2), andNyquistplot(Figure 3)of the given open-loop system in MATLAB.
Thereal partsofeigenvalues of the resultant system dynamic matrix are-17.8064 and -2.2394,both liein the lefthand planeof the complex plane, so the closed loop systemis asymptotically stable. In addition,the corresponding step response information isthe following:
Where
Step 5:Now we can write the observer-based feedback controlleras
where , , , .
Socontroller is
Explain how you can make sure the overall feedback system is asymptotically stable.
On the other hand, these margins have simple geometric interpretations on theNyquistdiagram ofthe transfer function.The gain margin is given by the inverse ofthe distance to the nearest point between −1 and 0 where the loop transfer functioncrosses the negative real axis. The phase margin is given by the smallest angle onthe unit circle between −1 and the loop transfer function. From figure 6, however, we can easily tell that the plot lies on the right hand plane of the complex plane, where noloop transfer function cross the interval -1 and origin, again, we can’t determine the gain margin and phase margin.
The closed-loop system need to satisfy fourconditions above, we canusetrailand error method to find two poles ofthe closed-loop system, then use‘place’MATLABcommand find the correspondinggain , thenwrite a MATLAB function that take gainKas inputandoutput theeigenvaluesof the matrix, step response information and check whether the resultant systemindeedsatisfy all the conditions.