Till 5th Jan - Include 6th Asgn Part_2

7/28/2019 Till 5th Jan - Include 6th Asgn Part_2

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Poles can be placed as per own requirement Example


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Controllability is too strong a condition A weaker notion than controllability is

sufficient that is named as stabilizability It refers to the ability to move only the

unstable modes of the system System is stabilizable if the unstable modes

are controllable or if the uncontrollablemodes are stable


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Example Neither of these systems are controllable

X= [2 0 ; 0 -1] x + [1 0] u (where X and Z are xdot and zdot)Stable mode at -1 is not controllable where as unstablemode at 2 is controllable , so system is stabilizable

Z= [2 0 ; 0 1] z + [1 0] uunstable mode at 1 is not controllable, so it is notstabilizable as well

Ackermanns Formula with example


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Can relocate all system closed loop poles whereas in classical control design, a designer can onlyhope to achieve a pair of complex conjugate

poles that are dominant Moving a poles around is costly A faster system may require more accurate

sensors along with larger and stronger actuators Guide line for moving the poles is

RH Poles be reflected as RH poles One should avoid to shift the poles far into left half

plane, as system bandwidth increases and systembecome more sensitive to noise


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State feedback does not affect the systemzeros

As steady state tracking properties dependsupon poles and zeros so tracking propertiescannot be helped by a sate feed back alone


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State feedback is not usually practical

Leads to PD compensators, which have infinite

bandwidth Not possible and practical to sense all the states

and feedback them all

For this output feedback is used


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StefaniDrill Problems D 9.1(a,b) ----- (page No 635)

Part of Assignment Six


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An issue of steady state error is not addresstill this point

Input is given as


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Preceding technique places a gain outside thefeedback loop

When elements of the control system are notwith in the feed back loop, the overall systemwill be quite sensitive to the elements outsidethe loop

An alternate method for achieving zerosteady state error to the step input is knownas integral control


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Stefani

Drill Problems D 9.3 (a,b) ----- (page No 639)Part of Assignment Six


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Typically some of the states are measured bysensors and the rest will be observed by

another device Observer is the device which simulates the

original system


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Observer error will go to zero asymptotically if and only if thematrix (A-LC) is a stable matrix L is an observer gain Controllability allows plant pole placement where as observability

allows observer pole placement IfMc is transpose it will equal to Mo (observability matrix)


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In first system mode at -1 is unobservable, butdetectable In second system mode at 1 is unobservable and

undetectable Example


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Example

Observer poles at


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Stefani

Drill Problems D 9.5 (a,b) ----- (page No 642)Part of Assignment Six


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Can observe states be able to stabilize thesystem


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Control gain calculated from previousexample


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State transformation is given as

Eigen values of the controller and the observerwill be at the diagonal

This property of observation through this block

triangle structure is known as separation property

Till 5th Jan - Include 6th Asgn Part_2

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