A block diagram showing plant dynamics in normal mode coordinates, a linear feedback controller with command limits, and actuator dynamics with rate and displacement limits is given. The objective is to examine the effects of joint rate and displacement saturation limits on the stability regions. An unstable short period dynamic model is transformed into normal mode coordinates and is augmented with actuator dynamics. A linear feedback controller is used to provide closed-loop stability. The stability with constrained actuator rate limits under varying bandwidth, displacement and command limits is examined

Shrivastava, P. C.

English

NASA Technical Reports Server

N81-21606 
EFFECTS OF JOINT RATE AND DISPLACEMENT 
CONSTRAINTS ON STABILITY REGIONS 
P. C. Shrivastava 
Department o f  Mechanical and Aerospace Engineering 
P r i  nceton Un ive rs i t y  
Princeton, New Jersey 
Th is  block diagram shows the  p lan t  dynamics i n  normal mode coordinates, a 
1 i near feedback con t ro l  1 e r  w i t h  command 1 i m i  t s ,  and the  ac tua tor  dynamics w i t h  
r a t e  and displacement l i m i t s .  
j o i n t  r a t e  and d i  spl acement sa tu ra t i on  1 i r n i  t s  on the  s t a b i  1 i ty regions. 
The ob jec t i ve  i s  t o  examine the  e f f e c t s  o f  
SCHEMATI C BLOC K -  D I AGRAM 
Actuator 
Rate-limit l im i t  I I 
0 
-.i b2 
Comnand 
I im iter 
q1 
PRE~WNG PAGE BLANK NOT F I L W  
87 
https://ntrs.nasa.gov/search.jsp?R=19870018173 2020-03-20T10:13:51+00:00Z
The u n s t a b l e  s h o r t - p e r i o d  dynamic model i s  t r ans fo rmed  i n t o  normal mode 
c o o r d i n a t e s ,  and i t  i s  augmented w i t h  a c t u a t o r  dynamics. 
sidered a s  the  unstable mode. A l inear feedback c o n t r o l l e r  i s  used t o  p r o v i d e  
c l o s e d - l o o p  s t a b i l i t y .  We examine the s t a b i l i t y  boundar i e s  w i t h  c o n s t r a i n e d  
a c t u a t o r  r a t e  limits under v a r y i n g  bandwidth,  d i sp l acemen t  and command limits. 
The s t a b i l i t y  boundar i e s  a r e  u n s t a b l e  limit cycles i n  a phase-plane p l o t  whose 
axes a r e  the u n s t a b l e  mode and c o n t r o l  d e f l e c t i o n .  
The 9 mode i s  con- 
PROBLEM FORMULATION 
0 SHORT-PERIOD DYNAMICS 
0 ACTUATOR DYNAMICS 
0 
d u = -  au + au 
0 AUGMENTED SYSTEM DYNAMICS 
(NORMAL MODE) 
'0' 
0 
a 
d U 
88 
u = -clql - c2q2 - c3u d 
The closed-loop stability by feedback of only the unstable mode requires 
the root  magnitude t o  be greater t h a n  t h a t  of the unstable eigenvalues. In 
other words, the control bandwidth  must be h igh  enough t o  stabilize the 
system. This requirement i s  relaxed when the unstable mode and the 
control deflection are fed back because the effective actuator bandwidth  i s  
increased. Furthermore, feedback of control deflection a1 so permits reduction 
i n  the feedback gain  of the unstable mode. 
CLOSED-LOOP DYNAMICS 
FEEDBACK OF UNSTABLE MODE ( 4 1  - MODE) [CONTROL LAW 1 1  
- CHARACTERISTIC  EQUATION: 
S2 + ( a  - A11 S + $ b l C 1  -aX1) 
- S T A B I L I T Y  CONDITIONS: 
C1> Xl/bi  
FEEDBACK OF IlNSTARLE MODE AND CONTROL DEFLECTION fa: AND u)  [CONTROL LAW 21 
- CHARACTERISTIC  EQUATION: 
(S - X2) S2 + ( a  +aC3 - X1) S t a (blC1 - c 
- S T A B I L I T Y  CONDIT IONS 
blCl>X1(l + C3)+ blCl>Xi/a S I G N  OF C j ?  
89 
The increase i n  the sizes of stability regions i s  not  proport ional  t o  the 
bandwidth .  A t  h i g h  bandwidths, the sizes of stability regions become almost 
independent of the bandwidth  because the ac tua to r  i s  almost always rate- 
saturated. Feedback o f  control deflection permits increased region of rate- 
unsaturated operation due t o  reduced feedback gain  requirement on the unstable 
mode. T h i s  helps increase the size o f  stability region size. 
EFFECTS OF BANDWIDTH VARIATIONS (MCE GAIN) 
CONTROL L A W  1 
c *
-1 
0: ;i/ 
CONTROL L A W  2 
RATE LIMIT = 4 0  rad/s 
While increase i n  t h e  system feedback increases the  c l  osed-1 oop 
s t a b i l i t y ,  i t  causes e a r l y  r a t e  saturation thereby reducing t h e  s izes  o f  
stab i  1 i t y  regions. 
EFFECTS OF BANDWIDTH VARIATIONS (TWICE MCE GAIN) 
CONTROL LAW 1 CONTROL LAW 2 
0 h
-i 
I # 
-0.m -0 .20  -ai c.10 0 . 2 0  0.3s 
(11 -nooE 
e 
RATE L I M I T  = 4 0  rad/s 
e 
? 
2 
c . 
91 
Increasing rate  limits implies relaxing the constraints,  and the rate  
saturation occurs a t  relatively larger values o f  the control and  s t a t e  
vari ab1 es. Therefore, the stabi 1 i ty  reg1 on s izes  increase w i t h  ra te  1 i m i  ts.  
EFFECTS OF RATE-LIMIT VARIATIONS (MCE GAIN) 
CONTROL MW 1 CONTROL L A W  2 
L 
100 
= 10 rad/s 
Upon displacement saturation, the system i s  essentially open-1 oop and 
rate limit has no significance. The displacement limits restrain the area of 
closed-1 oop operation i n  the phase-pl ane, which  causes drastic reduction i n  
the sizes of stability regions. For this reason, the sizes o f  s t a b i l i t y  
regions do n o t  show s i g n i f i c a n t  size increases despite large increases i n  the 
rate limits. 
EFFECTS OF IMPOSING DISPLACEMENT LIMIT (MCE GAIN) 
CONTROL L A W  1 CONTROL L A W  2 
- 
h 
-I 
D I,
-1 
BANDWIDTH = 10 rad/s 
Marginal increases i n  the sizes of s t a b i l i t y  reqions obtained for larqe 
increases i n  rate l imits  under displacement constraints suqaest the 
possibi l i ty  of invariance of s t a b i l i t y  regions w i t h  respect t o  the ra te  
l imits.  
rate-unsaturated region which i s  essentially sustained by the displacement 
l imits .  
the s tab i l i ty  boundary w i t h i n  the rate-unsaturated region. 
there exists an  upper bound on the rate limits beyond which the s t a b i l i t y  
boundary i s  no longer influenced by the rate  limits, as illustrated. 
Notice that  a large p a r t  of the s t a b i l i t y  boundary l i e s  w i t h i n  the 
Increasing rate limits effectively b r i n g  i n  an even larger portion of 
Clearly, therefore 
INVARIANCE OF STABILITY REGIONS WITH RATE LlMllS 
(MCE G A I N  CASE) 
CONTROL L A W  1 CONTROL L A W  2 
P = 250 rad/s \ ' 
94 
DISPLACEMENT LIMIT = 20° 
UANDWI DT1I 
6 = 3 7 5  rad/s 
a- c -  
The large command limits do n o t  affect the s t ab i l i t y  reqions because the 
1 i near control 1 e r  meets the demands required for cl osed-1 oop s t a b i  1 i ty .  
Reduction i n  these 1 i m i  t s  deteriorates the control 1 e r  capahi 1 i t y  . 
system divergence, due  t o  the i n a b i l i t y  of the controller t o  meet the demanded 
control, causes a l i t t l e  expansion i n  the Stabi l i ty  reqion; t h i s  diverqence i s  
overcome because of the reduced demand on the controller by a laruer p a r t  of 
the s t a b i l i t y  boundary. As the command limit i s  reduced further,  the system 
becomes unstable for relatively lonser time placing even larqer demands on the 
control 1 e r .  
sa t i s f ied  by the controller. 
A smal 1 
The s t a b i l  i ty region vanishes because these demands cannot be 
EFFECTS OF COMMAND-LIMIT VARIATIOHS (MCE GAIN) 
CONTROL L A W  1 
WITH AND WITHOUT 
7 -  
ludl = 22 deg. 
WITH  LIMIT:^ 
t' 
ludl = 21 deg. 
?' 
ludl = 20.5 deg. 
BANDWIDTH = 1 0  rad/s 
RATE L I M I T  = 40 rad/s 
95 
96 
The 1 imi t cycle, representing the s tabi l  i t y  boundary i n  the phase-pl ane 
of unstable mode and control deflection, implies t h a t  a hypercylinder 
represents the overall s t a b i  1 i t.y reoion i n  the normal mode coordi nates. With  
the s t a t e  space coordinates located i n  the plane of normal modes, the 
s t ab i l i t y  regions clearly become functions of control deflection limits under 
rate constraints. Imposition of displacement limits implies t h a t  a chopped 
hypercyl inder represents the s t ab i l i t y  reqion. Therefore, f i n i t e  s t ab i l i t y  
regions can ex is t  i n  the s t a t e  space coordinates only f o r  those controls t h a t  
sa t i s fy  the displacement saturation constraints. 
INTERPRETATION OF RESULTS IN PHYSICAL COORDINATES 
Under ra te  constraints, a hypercylinder i s  shown t o  represent the 
s t ab i l i t y  region i n  the normal mode coordinates when a saddle-point type 
system i s  controlled by feedback of the unstable mode and control 
deflection. I ts  intersection w i t h  the plane of unstable mode and control 
deflection i s  an unstable limit cycle. An increase i n  ra te  l imits  resul ts  i n  
increased s ize  of the s t a b i l i t y  regions, b u t  imposinq d i s p l a  ement constraints 
causes drast ic  reduction i n  the i r  sizes. That is ,  the stabi i t y  reaions are 
largely dependent upon the displacement limits, and they can even be made 
independent of the rate  limits. Feedback of control deflect  on effectively 
increases the region of rate-unsaturated operation result ing i n  increased 
s izes  of s t a b i l i t y  regions. However, the s izes  of s t ab i l i t y  regions are 
reduced w i t h  increased feedback gain due t o  early ra te  saturation. 
case of j o i n t  ra te  and displacement constraints where the existence o f  a 
s t ab i l i t y  region is always guaranteed, i n  the case of j o i n t  ra te  and conand 
constraints,  the s t a b i l i t y  region can disappear when the command limit i s  
reduced below a certain limit. 
Unlike the 
CONCLUSIONS 
0 LIMIT CYCLE REPRESENTS STABILITY BOUNDARY 
0 MARGINAL EFFECT OF ACTUATOR BANDWIDTH 
0 INCREASE IN STABILITY REGION SIZE WlTH RATE LIMITS 
0 SIGNIFICANT SIZE REDUCTION UNDER CONSTRAINED DEFLECTION 
-0 LARGER REGION WITH FEEDBACK OF CONTROL DEFLECTION 
0 REDUCED SIZES WITH INCREASED GAIN 
0 INTOLERANCE OF STABILITY REGIONS TO COMMAND LIMITS 
97 


Effects of joint rate and displacement constraints on stability regions

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