Bi kh Bh tt ac arya Professor Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc Funded by MHRD brPage 2br NPTEL Mechanical Engineering Modeling and Control of Dynamic electroMechanical System Module 4 Lecture 33 Jo ID: 30114 Download Pdf

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Bi kh Bh tt ac arya Professor Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc Funded by MHRD brPage 2br NPTEL Mechanical Engineering Modeling and Control of Dynamic electroMechanical System Module 4 Lecture 33 Jo

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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 Regulator Optimal Controller Design Using Linear Quadratic Regulator Bi h kh Bh tt h r. Bi kh Bh tt ac arya 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 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 In oduct n t oduct o So far, we have discussed about different techniques of obtaining the control gains to achieve desired closed loop obtaining the control gains to achieve desired closed loop characteristics irrespective of the magnitude of the gains. It is to be understood thou hthathi her ain im lies lar er power amplification which may not be possible to realize in practice. Hence there is requirement to obtain reasonable closed loop Hence there is requirement to obtain

reasonable closed loop performance using optimal control effort. A quadratic performance 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 r epresented in state space form as: Re Re Ax

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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic

electro-Mechanical System Module 4- Lecture 33 Co ll r D es n co td .. Co t o e es g co td Now, assume a static output feedback of the form Gy dt Ru Qx

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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 4- Lecture 33 at rix Ri ccat iE quat n: at ccat quat o Use of X eliminates the dependency of the feedback gain on the iitil diti ith ti l ttfdbk tl non-zero iti con diti on x n th e op ti ma ou pu ee db ac con ro . The optimization results in a set of non-linear coupled matrix equations which is given by: ML RGC LC GCLC ML

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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 4- Lecture 33 Algebraic Riccati Equation Algebraic Riccati Equation • However, this method has certain di fficulties in implementation. The iterative algorithm suggested in this method requires initial t bili i i hi h t b l il bl bili ng ga ns, w hi may no e a ways ava il bl e. • The available iterative schemes in the literature are computationally intensive and the convergence is not always ensured. • Hence an alternate method is ado ted for the controller desi n. In ,pg this

method, a nontrivial soluti on of the gain G is given by: (CLC (CLC

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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 4- Lecture 33 Optimal Controller Optimal Controller Here, K and L are the positive definite solutions of the following equations: KA BR 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 accepted As good engineering practice , 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 lecture Special 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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