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NPTEL >> Mechanical Engineering >> Modeling and
Control of Dynamic electro-Mechanical System Module 4- Lecture
33
Optimal Controller Design Using Linear Quadratic
RegulatorOptimal Controller Design Using Linear Quadratic
Regulator
D Bi h kh Bh tt hDr. Bishakh Bhattacharya
Professor, Department of Mechanical Engineering
IIT Kanpur
Joint Initiative of IITs and IISc - Funded by MHRD
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NPTEL >> Mechanical Engineering >> Modeling and
Control of Dynamic electro-Mechanical System Module 4- Lecture
33
ThisLectureContains
Introduction to Optimal Control IntroductiontoOptimalControl
OptimalControllerDesign
AlgebraicRiccati Equation
Commentsonthechoiceofweightingmatrices
Joint Initiative of IITs and IISc - Funded by MHRD
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NPTEL >> Mechanical Engineering >> Modeling and
Control of Dynamic electro-Mechanical System Module 4- Lecture
33
Introduction t oduct o
So far, we have discussed about different techniques ofobtaining
the control gains to achieve desired closed loopobtaining the
control-gains to achieve desired closed-loopcharacteristics
irrespective of the magnitude of the gains.
It is to be understood though that higher gain implies largerg g
g p gpower amplification which may not be possible to realize
inpractice.
Hence there is a requirement to obtain reasonable closed loop
Hence, there is a requirement to obtain reasonable
closed-loopperformance using optimal control effort. A
quadraticperformance index may be developed in this
direction,minimization of which will lead to optimal
control-gain.
The process is elaborated farther in the following
discussion.
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NPTEL >> Mechanical Engineering >> Modeling and
Control of Dynamic electro-Mechanical System Module 4- Lecture
33
Optimal Control contd..Optimal Control contd..The dynamics of a
structure is represented in state space form as:
Where,A isthesystemmatrixderivedforthepassivesystemalongwith
mn Re,Re; uxuBAxxtheembeddedactuatorsandsensors,andB
representstheinfluencematrixcorrespondingtothedistributedcontroleffortu
forthem finitenumberofpatches.
ThederivationofA andB
arediscussedalreadyintheearliersection.Theconventionaltechniqueofminimizingaquadraticperformanceindex
for the design of feedback controller is as
followsindexforthedesignoffeedbackcontrollerisasfollows.
Theoutputormeasurementequationcanbewrittenas
CJointInitiativeofIITsandIISc Fundedby
MHRD 4
Cxy
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NPTEL >> Mechanical Engineering >> Modeling and
Control of Dynamic electro-Mechanical System Module 4- Lecture
33
Controller Design contd..Co t o e es g co td
Now, assume a static output feedback of the form
where,G isthegainmatrix.The objective is to design a controller
by choosing a proper controller gain
Gyu
TheobjectiveistodesignacontrollerbychoosingapropercontrollergainG,whichisalsooptimalinthesensethatitminimizesaperformanceindex:
Where,denotesexpectationwithrespecttotheinitialstate,Q andR are
dtEGJ x 00 )()( RuuQxx TTp p
weightingmatrices.Here,
00 XEx
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NPTEL >> Mechanical Engineering >> Modeling and
Control of Dynamic electro-Mechanical System Module 4- Lecture
33
Matrix Riccati Equation: at ccat quat o
Use of X0 eliminates the dependency of the feedback gain on the
i iti l diti i th ti l t t f db k t lnon-zero initial condition x0
in the optimal output feedback control.
The optimization results in a set of non-linear coupled matrix
equations which is given by:
0XMLML0RGCGCQKMMK
'
'''
Where,M denotestheclosedloopsystemas:M=A+BGC,Listhe
0LCKBRGCLC0XMLML
''1'0
p yLiapunov Matrix.
TheoptimalgainGmaybeobtainedbyiterativelysolvingtheeqn.set.
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NPTEL >> Mechanical Engineering >> Modeling and
Control of Dynamic electro-Mechanical System Module 4- Lecture
33
Algebraic Riccati EquationAlgebraic Riccati Equation However,
this method has certain difficulties in implementation. The
iterative algorithm suggested in this method requires initial t
bili i i hi h t b l il blstabilizing gains, which may not be always
available.
The available iterative schemes in the literature are
computationally intensive and the convergence is not always
ensured.
Hence, an alternate method is adopted for the controller design.
In , p gthis method, a nontrivial solution of the gain G is given
by:
1'''1 )(CLCCLKBRG )(CLCCLKBRG
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NPTEL >> Mechanical Engineering >> Modeling and
Control of Dynamic electro-Mechanical System Module 4- Lecture
33
Optimal ControllerOptimal ControllerHere, K and L are the
positive definite solutions of the following equations:
0KBRBKQKAKA '1' 0XKBRBALLKBBRA 0
'1'1 ')()(The first equation is known as the standard Riccati
equation The second one Is known as Lyapunov equation.
By tuning the weighting matrices Q and R, a sub-optimal
controller gain G can be achieved using this method. This method
gives acceptable solution in a single step (no iteration is
required).
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NPTEL >> Mechanical Engineering >> Modeling and
Control of Dynamic electro-Mechanical System Module 4- Lecture
33
Comments on Optimal Control:Comments on Optimal Control:The
principal drawback of this scheme is that it is not possible to
judge as to how far the sub-optimal solution is away from the
optimal (local and / or global) solution.
As a good engineering practice Q is tuned such that acceptedAs a
good engineering practice, Q is tuned such that accepted closed
loop response is obtained. Normally, Q is taken as the Modal matrix
of the system, while R is taken as an identity matrix.
The other drawback of this system is that the closed loop system
is not robust. This means that a slight variation of system
parameters may drastically affect the system performance.
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NPTEL >> Mechanical Engineering >> Modeling and
Control of Dynamic electro-Mechanical System Module 4- Lecture
33
Special References for this lectureSpecial References for this
lecture
Control System Design, Bernard Friedland, Dover
Control Systems Engineering Norman S Nise, John Wiley &
Sons
Design of Feedback Control Systems Stefani Shahian Savant
Hostetter Design of Feedback Control Systems Stefani, Shahian,
Savant, Hostetter
Oxford
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