NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-mechanical System Module 4- Lecture 3 Reduced Order Observer Design Dr Bishakh Bhattacharya h Professor, Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc - Funded by MHRD
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-mechanical System Module 4- Lecture 3 This Lecture Contains Introduction to Reduced OrderObserverObserver Governing Equation Estimator Design Reduced Order Observer Block Diagram Joint Initiative of IITs and IISc - Funded by MHRD
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-mechanical System Module 4- Lecture 3 Introduction to Reduced Order Observer Design In the last lecture, we have introduced the concept of controlling a dynamic system where none of the states are available for direct measurement and hence the states are to be estimated through the design of an observer nother trivial way of solving this problem could be by the use of system output, provided the number of outputs available equal to the order of the system Following the output eqn: Y CC X X C Y However, there are many cases, where, number of outputs t available are less (say r numbers which is less than n orderof the system) In such a case you need a reduced order observer to estimate the n-r number of states This is also known as Luenberger Observer
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-mechanical System Module 4- Lecture 3 Governing Equation Let us consider, the governing equation for the dynamic system (Plant) in state-space space form as: X X Bu, Y C X Consider, Y to be of size r, and partition the state vector into two parts such that X = [X X ] T, where X is of size r and X is of size n-r The governing equation could be subdivided similarly such that: X X X BU X X X BU Joint Initiative of IITs and IISc Funded by MHRD 4
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-mechanical System Module 4- Lecture 3 Estimator Equation The states X could be estimated based on directly the measured output Y such that: X C Y The states X, however, has to be estimated following a similar strategy as had been done for the full order observer Xˆ C ˆ Y X B Consider, the above equation in terms of a new state vector Z, such that u Z Xˆ LY, Xˆ Z LY Joint Initiative of IITs and IISc Funded by MHRD 5
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-mechanical System Module 4- Lecture 3 Observer e Design based on Reduced States The reduced order system may be epressed in terms of states Z as: z Q z R y S u Now, defining the error for the reduced order system, we can obtain the error dynamics as: e ˆ e M B u B u LC M Q z R y S u
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-mechanical System Module 4- Lecture 3 Observer e design contd The error dynamics could be further simplified as: e Q e ( ( LC LC Q) R C ( B QLC ) LC B S) u To obtain an error dynamics which will be independent of,, and u, the following conditions must be satisfied: Q R S B LC C LCB LC C QL By selecting the observer gain L, one can obtain Q, R and S
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-mechanical System Module 4- Lecture 3 Reduced Order Observer e in a block-diagram daga y C ˆ G = ( L C )C u G S /s L Q
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-mechanical System Module 4- Lecture 3 ssignment ssignment Consider a networked first order hydraulic system with the following state ti equation u 5 u 0 0 0 0 5 3 3 Design a reduced order observer when only the first state is directly measured Joint Initiative of IITs and IISc Funded by MHRD 9
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-mechanical System Module 4- Lecture 3 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 Oford Joint Initiative of IITs and IISc Funded by MHRD 0