P. A. COLLEGE OF ENGINEERING AND TECHNOLOGY POLLACHI - 642 002 EVEN SEMESTER 2015 – 16
QUESTION BANK IC 6601 – ADVANCED CONTROL SYSTEM
PREPARED PREPARED BY
S.ARUN
1
Un! – I STATE VARIABLE DESIGN PART - A
1. "#$! $%& !#& '%$()$*+, n !%$n,&% n*!/n /'& $n$,,3
Transfer function is defined under zero initial conditions. Transfer function is applicable to linear time invariant systems. Transfer function analysis is restricted to SISO systems.
It does not provides information regarding the internal state of the system. 2. "#$! , S!$!& $n' ,!$!& $%$)&3
The state is the condition of a system at any time instant, t. A set of variable which describes the state of the system at any time instant are called state variables. . "#$! , $ ,!$!& &*!/% 3
The state vector is a (n!"column matri (or vector"whose elements are state variables of the system,(where n is the order of the system".It is denoted by #(t".
4. "#$! $%& !#& $'$n!$&, / ,!$!& ,7$*& $n$,,3 89 The state space analysis is applicable to any type
of systems. They can be used for $odeling and analysis of linear and nonlinear systems, time invariant %time variant systems and $ultiple input %$ultiple output systems. (ii" The state space analysis can be performed with initial conditions. (iii" The variables used to represent the system can be any variables in the system. (iv"&sing this analysis the internal states of the system at any time instant can be predicted. !#
5. "%!& !#& ,!$!& /'& / n /%'&% ,,!&3
The state model of a system consists of state e'uation and output e'uation. The state model of a n th order system with minputs and poutputs are #(t" ) A#(t" * +&(t"
State e'uation
,(t" )-#(t" * .&(t"
Output e'uation
/here
#(t" ) state vector of order (n!" &(t" ) Input vector of order (m!" A ) system matri of order (nn" + ) Input matri of order (nm" (t" ) Output vector of order (p!" - ) Output matri of order (pn" a)transmission matri of order (pm" 2
6. T#& ,!$!& /'& / $ n&$% !& n$%$n! ,,!& , &n ) :8!9 ; A:8!9 < BU8!9= Y8!9 ;C:8!9 < DU8!9 . "%!& !#& &>7%&,,/n /% !%$n,&% n*!/n / !#& ,,!&.
(s"0 &(s")- (sI 1 A"
!
+&(s" *
?. "#$! , S!$!& '$%$3
The 2ictorial representation of the state model of the system is called state diagram. The state diagram of the system can be either in bloc3 diagram or signal flow graph form.
@. "%!& !#& 7%/7&%!&, / ,!$!& !%$n,!/n $!%>. i. 4(5" )e A 5 ) I (&nit matri " A t A t ! ! ii. 4(t" ) e )(e " )6 4(t"7 A ( t * t " A t! A t8 iii. 4(t !* t 8 " )e e ) 4(t!" ! 8 ) e . "%!& !#& ,/!/n / #//&n&/, ,!$!& &$!/n,.3 A t
The solution of homogeneous state e'uation is, #(t" ) e A t
4(t8" ) 4(t8" 4(t!"
5
/here , #(t" ) state vector at time , t e )State transition matri.
And #5 )Initial condition vector at t)5 10."%!& !#& ,/!/n / n/n- #//&n&/, ,!$!& &$!/n,.3
The solution of nonhomogeneous state e'uation is (tt5 " (t "
#(t" ) e A
#(t5 " * 9 e A
+.&( " d
11. "%!& $n !(/ 7%/7&%!&, / &&n$&,.
(i"A matri and its transpose have the same eigen values. (ii"The product of the eigen values (counting multiplicities"of the matri e'uals the determinant of the matri. 12."#$! , ,$%! !%$n,/%$!/n3
The 2rocess of transforming a s'uare matri A to another similar matri + by a transformation 2 ! A2 ) + is called similarity transformation. The matri 2 is called transformation matri.
1.D&n& */n!%/$)! $n' /),&%$)!.
A system is said to be completely state controllable if it is possible to transfer the system state from any initial state #( t 5 "at any other desired state #(t",in specified finite time by a control vector &( t ". A system is said to be completely state observable if every state #(t" can be completely identified by measurements of the output (t"over a finite time interval. 14."#$! , 7/& 7$*&&n! ) ,!$!& &&' )$*+3 3
The pole placement by state feedbac3 is a control system design techni'ue ,in which the state variables are used for feedbac3 to achieve the desired closed loop poles. 15."#$! , ,!$!& /),&%&%3
A device (or a computer program" that estimates or observes the state variables is called state observer
16."#$! , !#& n&&' /% ,!$!& /),&%&%3
In certain systems the state variables may not be available for measurement and feedbac3. In such situations we need to estimate the un measurable state variables from the 3nowledge of input and output. :ence a state observer is employed which estimates the state variables from the input and output of the system. The estimated state variable can be used for feedbac3 to design the system by pole placement.
1?."#$! , *$n/n*$ /% / ,!$!& /'&3
If the system matri, A is in the form of diagonal matri then the state model is called canonical form. 1@."#$! , &$n! ) '$/n$$!/n3
The process of converting the system matri A in to a diagonal matri by a similarity transformation using the modal matri $ is called diagonalization.
PART – B 1. erive the solution of homogeneous state e'uations. 2. (a" -onstruct a state model for a system characterized by the differential e'uation. d; y dt ;
d8 y
d y < = < !! < = y < > d 8 dt t
5
?ive the bloc3 diagram representation of the state model 8@9 And efine(i" @igen values (ii" @igen vectors (iii" state of a system 8@9 . @plain the pole placement design of continuous time system with a suitable eample. 4. @plain in detail about the design of state observer for continuous time systems 5. evelop the state model of a linear system and draw the bloc3 diagram of state model. !5 6. ?iven the transfer function
s
;
* ;s 8 . esign a feedbac3 controller so that the * 8s
eigen values of the closed loop system are at 8, !B! ?. -onstruct a state model for a system characterized by the differential e'uation C
C
y
* D y * E y * F y * u ) 5
8@9
4
Un! – II PHASE PLANE ANALYSIS PART - A
1. "#$! $%& n&$% $n' n/nn&$% ,,!&,3 G& &>$7&,.
The linear systems are systems which obeys the principle of super position. The systems which does not satisfy superposition principle are called nonlinear systems. @ample of linear system G y) a *b d 0 dt b @ample of nonlinear system G y) a8 * e
2. H/( n/nn&$%!&, $%& n!%/'*&' n !#& ,,!&,3
The nonlinearities are introduced in the system due to friction, inertia, stiffness, bac3lash, hysteresis, saturation and dead zone of the components used in the systems.
. "#$! $%& !#& &!#/', $$$)& /% !#& $n$,, / n/nn&$% ,,!&3
The two popular methods of analyzing nonlinear systems are 2hase plane method and describing function method.
4. "%!& $n !(/ P%/7&%!&, / n/nn&$% ,,!&,. 3
(i"The nonlinear systems may have Bump resonance in the fre'uency response. (ii"The output of a nonlinear system will have harmonics and sub harmonics when ecited by sinusoidal signals.
5. "#$! , $,n*#%/n/, &n*#n3
In a nonlinear system that ehibits a limit cycle of fre'uency Hl, it is possible to 'uench the limit cycle oscillation by forcing the system at a fre'uency w', where w' and Hl are not related each other. This phenomenon is called asynchronous 'uenching or signal stabilization. 6. "#$! , $!/n//, ,,!&3
A system which is both free (or unforced or zero input or constant input"and time invariant is called an autonomous system.
?. "#$! , 7#$,& 7$n&3
The coordinate plane with the state variables ! and 8 as two aes is called the 2hase plane (In phase plane #! is represented in is represented in ! ais,and 8 in yais." @. "%!& $n !(/ 7%/7&%!&, / n/n-n&$% ,,!&,.
The nonlinear systems may have Bump resonance in the fre'uency response. The output of a nonlinear system will have harmonics and subharmonics when ecited by sinusoidal signals.
. "#$! , Sn$% 7/n!3
A point in phaseplane at which the derivatives of all state variables are zero is called singular point. It is also called e'uilibrium point. 5
10.H/( !#& ,n$% 7/n!, $%& *$,,&'3
The singular points are classified as odal point, Saddle point, Jocus point and -entre or Korte point depending on the eigen values of the system matri. 11. D&n& !#& ,!$)! / $ n/n-n&$% ,,!& $! /%n.
The autonomous system defined by e'uation # ) J(#" is stable at the origin, if for every initial state #(t 5" which is sufficiently close to origin, #(t" remains near the origin for all t. 12."#$! , !#& '&%&n*& n ,!$)! $n$,, / n&$% $n' n/n-n&$% ,,!&,3
In linear system the stability of the system in the entire phaseplane can be Budged from the behaviour of the system at e'uilibrium state (i.e., at singular point", because the linear systems has only one e'uilibrium state. In nonlinear systems there may be multiple e'uilibrium states (singular points". The behaviour of nonlinear system about the e'uilibrium point may be different for small deviations and large deviations about the e'uilibrium point. :ence in nonlinear systems, stability is discussed relative to e'uilibrium state and the general stability of a system cannot be defined. 1."#$! , 7#$,& !%$&*!/%3
The locus of the state point ( !,8" in phase plane with time as running parameter is called phase traBectory. 14."#$! , 7#$,& 7/%!%$!3
A family of phase traBectories corresponding to various sets of initial conditions is called a phase portrait. 15. "#$! $%& !#& &!#/', $$$)& /% */n,!%*!n 7#$,& !%$&*!/%&,3
i"Analytical method, ii" Isocline method and iii" elta method.
PART – B !. @plain in detail about the behavior of nonlinear system and classifications of on linearites.
8. -onsider a system with an ideal relay as shown in fig. etermine the singular point. -onstruct phase traBectories corresponding to initial conditions. (i" c(5")8 L c (5" ! And,
(ii" c(5")8 L c (5"
! .E Ta3e r)8 volts % M)!.8 volts. 6
r
*
e
$
u
e
!
$
(
s
8
c
8169
;. A linear second order serve is described by the e'uation e < 8δw n e < w n8 e 5 where
N)5.!E, wn)! rad0sec. e(5")!.E and e(5" 5 . etermine the singular point. -onstruct the phase traBectory, using the method of isoclines. -hoose slope as8.5, 5.E,5,5.E % 8.5.(169
>. /hat is phase plane, phase traBectory and phase portrait. raw and eplain how to determine the stable and unstable limit cycles using phase portrait E. @plain the construction of phase traBectory using any two method
Un! – III DESCRIBING FUNCTION ANALYSIS PART - A 1. "#$! , 7 %&,/n$n*&3
In the fre'uency response of nonlinear systems, the amplitude of the response (output"may Bump from one point to another for increasing or decreasing values of H. This phenomenon is called is Bump resonance
2. "#$! $%& !#& n&$% $n' N/nn&$% S,!&,3 G& E>$7&,.
The linear systems are systems which obeys the 2rinciples of superposition. The system which does not satisfy the super positions are called nonlinear systems. eg of Pinear system G
y = ax + b
dx dt
bx eg of onlinear system G y = a x +e 2
. H/( n/nn&$%!&, $%& n!%/'*&' n !#& ,,!&3
The nonlinearities are introduced in the system due to friction, inertia, stiffness, bac3lash, hysteresis, saturation and deadzone of the components in the system.
4. H/( !#& n/nn&$%!&, $%& *$,,&'3 G& &>$7&,.
The nonlinearities are classified as incidental and intentional. The incidental nonlinearities are those which are inherently present in the system, -ommon eamples of incidental nonlinearities are saturation, deadzone, coulomb friction, stiction, bac3lash, etc..
7
The intentional onlinearities are those which are deliberately inserted in the system to modify the system characteristics. The most common eample of this type of nonlinearity is a relay. 5. "#$! $%& !#& &!#/', $$$)& /% !#& $n$,, / n/nn&$% ,,!&3
The two popular methods of analyzing nonlinear systems are phase plane method and describing function method.
6. "#$! , !#& '&%&n*& )&!(&&n 7#$,& 7$n& $n' '&,*%)n n*!/n &!#/', / $n$,,3 S.N /.
P#$,& P$n& M&!#/'
D&,*%)n Fn*!/n M&!#/'
!
&se complimentary approimations Jre'uency domain approach Qetains the nonlinearity as such and uses the second order approimations of a higher order linear part.
&se complimentary approimations Time domain approach
8 ;
Qetains the linear part and harmonically linearizes the nonlinearity.
?. S!$!& !#& !$!/n, / $n$n n/nn&$% ,,!&, ) '&,*%)n n*!/n $n' 7#$,& 7$n& &!#/',.
!. These methods are useful only for stability analysis and to study the behaviour of the system but cannot to used for system design 8. The phase plane method of analysis is useful only for second order systems with constant parameters and constant or Rero input. ;. In describing function analysis the accuracy of the information obtained is heavily dependent on the filtering property of the linear part of the system. >. This analysis does not give any useful information about transient response of the system.
@. "%!& $n! !(/ P%/7&%!&, / N/nn&$% ,,!&,.
!. The nonlinear systems may have Bump resonance in the fre'uency response. 8. The output of a nonlinear system will have harmonics and sub harmonics when ecited by sinusoidal.
. "%!& !#& $n'&% 7/, &$!/n /% n/nn&$% '$7n.
(
2
)
´ + B 1− x x ´ + Kx =0 M x
10."#$! $%& ! **&,3
The limit cycles are oscillations of the response of nonlinear systems with fied amplitude and fre'uency, If these oscillations or limit cycles eists when there is no input then they are called zero input limit cycles.
11. "#$! , ,$!%$!/n3 G& $n &>$7&. 8
In Saturation nonlinearity the output is proportional to input for a limited range of input signals, when the input eceeds this range, the output tends to become nearly constant. @g. Saturation in the o0p of electronic, rotating and flow amplifiers, speed and tor'ue saturation in electric and hydraulic motors. o0p Saturation
Approimated -haracteristics
i0p
Actual -haracteristics
Saturation
12."#$! , D&$'-/n&3
The deadzone is the region in which the output is zero for a given input, /hen the input is increased beyond this deadzone value, the output will be linear. o0p eadzone i0p
1."#$! $%& !#& !7&, / %*!/n3
The different type of friction are viscous friction, coulomb friction and stiction, the viscous friction is linear in nature, the coulomb friction and stiction are nonlinear frictions.
14. "#$! , #,!&%&,, $n' )$*+$,#3
The hysteresis is a phenomenon in which the output follows a different path for increasing and decreasing values of input. In fig(!", when the input is increased from a minimum value, the output follows path A+- and when the input is decreased from a maimum value, the output follows path -@A. )output )output @ #)input
#)input
The bac3lash nonlinearity is a type of hysteresis in mechanical gear trains and lin3ages, this is due to between the teeth of the drive gear and that of the driven gear. PART - B
9
1.
2.
Ex pl ai ni ndet ai l aboutt hebehav i orofnonl i nears ys t em andc l as s i fi cat i onsofNonl i near i t i es Cons i deras y s t em wi t ha ni deal r el a yass ho wni nfi g.De t er mi net hes i ngul ar poi nt .Cons t r uc tphas et r aj ec t or i es ,c or r es pondi ngt oi ni t i al c ondi t i ons .( i )c ( 0) =2; nd( i i )c ( 0) =2;c(0) = 1 . 5T ak er =2vol t s&µ=1. 2v ol t s c(0) = 1 A
r
+
e M
e
u
M
-
1 s
c
2
( 1 6 )
3.
Al i nears ec ondor ders er v ei sdes c r i bedbyt heequat i one + 2δwn
2
e+ wn e =
0
wher eδ=0. 15,wn=1r ad/ s ec .e ( 0) =1. 5ande(0) = 0 .Det er mi net hes i ngul arpoi nt . Cons t r uc tt hephas et r aj ec t or y , us i ngt heme t hodofi s oc l i nes .Choos es l opeas 2. 0, 0. 5, 0, 0. 5&2. 0.
4.Der i v et hedes cr i bi ngf unc t i onf oraBac kl as h nonl i near i t y( 1 6 ) 5. Wh ati sphas epl ane,phas et r aj ec t or yandphas epor t r ai t ?.Dr a w ande x pl ai nho w t odet er mi net hes t abl eanduns t abl el i mi tc y cl esus i ngphas epor t r ai t ? 6. T hei nput out putr el at i ons hi pofdeadz onenonl i near i t yi ss howni nt hefi gur e.The o/ pi sz er o,whent hei nputi sl es st hanD/ 2.Thei nput out putr el at i ons hi pi sl i near whent hei nputi sgr eat ert hanD/ 2.Ther es pons eoft henonl i near i t ywheni nputi s s i nus oi dofs i gnal ( x =Xs i nωt ) X -D/ 2
D/ 2
Sl o pek 7.
Thei nput out putr el at i ons hi pofs at ur at i onnonl i near i t yi sshowni nfi gur ebel ow. Thei / po/ pr el at i oni sl i nearf orx =0t os .whent hei nputx >s ,out putr eac hesa s at ur at edv al ueofk s.Ther es pons eoft henonl i near i t ywhent hei nputi s s i nus oi dal s i gnal ( x =Xs i nωt ) k S S
s l ope=k S
k S
( 16)
8.
Al i n ears ec on dor d ers er v oi sde sc r i bedb yt heequa t i on
10
y +
2δw
2
y
n
y
+ w n
= w
2
Whe r e
wn=1.y ( 0 ) = 2. 0 y(0) = 0. Det er mi net hesi ngul arpoi nt swhen( i ) δ= 0( i i )δ=0 . 6 Co ns t r u c t
phas et r aj ec t or yi neac hc as e.
Ex pl ai nt hec ons t r uc t i onofphas et r aj ec t or yus i ngan yt womet hod. 10. De r i v et hedes cr i bi ngf unc t i onf orr el aywi t hdeadz oneandhy s t er es i snon l i near i t y r i v et hedes cr i bi ngf unc t i onf ol l owi ngf unc t i onoft heel ementwhos ei nput 11. De out putc har ac t er i s t i c sar es howni nt hefi g. 9.
( 16) ( 16) ( 16)
o ut pu t s l op e=k D i nput D
x =xsi nωt
( 1 6 )
Un! – IV OPTIMAL ESTIMATION PART - A
1. "#$! $%& !#& ,!&7, n/&' n ,/!/n / /7!$ */n!%/ 7%/)& ,n !%$n,&% n*!/n $77%/$*#3
(i" (ii"
?iven a plant with transfer function ?(S", find the transfer function of the overall system which is optimal with respect to given performance criterion. -ompute the compensators for the system obtained from step (i"
2. "#$! $%& !#& ,!&7, n/&' n ,/!/n / /7!$ */n!%/ 7%/)& ,n ,!$!& $%$)& $77%/$*#3
(i"
(ii"
?iven a plant in the form of state e'uations. x t = x t + u t ; -ontinuous time system x + = x t + u ; iscrete time systems Qealize the control function obtained from step(i"
. "#$! , &$n! ) P/n!%$n P%n*7&3
This is based on the concept of calculus of variations also it is called as $inimum principle of pontryagin 4. D&n& B&$n P%n*7&3
This is based on the principle of invariant imbedding and leads to the principle of optimality that follows the basic laws of nature and does not need comple mathematical development to eplain its validity. 5. "#$! $%& !#& $77%/$*#&, /% n& !#& 7&%/%$n*& n'&>3
Once the performance inde for a system has been -hosen, the net tas3 is to 11
determine a control function that minimize the inde. Two approaches are available. (i" $inimum principle of pontryagin (ii" ynamic programming eveloped by +ellman. 6. "%!& '/(n !#& M$!%> R**$! E$!/n. P t
+
Q− P t B R
B P t + P t A + A P t =0
?. "%!& '/(n !#& R&'*&' M$!%> R**$! E$!/n A P + P A − P B R B P +Q =¿
12
