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lecture_5 - 2.160 System Identification Estimation and...

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2.160 System Identification, Estimation, and Learning Lecture Notes No. 5 February 22, 2006 4. Kalman Filtering 4.1 State Estimation Using Observers In discrete-time form a linear time-varying, deterministic, dynamical system is represented by x t + 1 x A t ) u B (1 + = t t t nx 1 rx 1 where x t R is a n -dimensional state vector, u k R is an input vector, and A , B are t t matrices with proper dimensions. Outputs of the system are functions of the state vector and are represented with a -dimensional vector y R x 1 : t y = x H t (2) t t xn where H t R is an observation matrix . ) , H t variables, one can simulate the system for predicting states and outputs in response to a time seq uence of inputs. S igure 4 his simulator may not workwell when ee F -1 below. T the model parameters are not ex actly k nown;actual outputs observed in the real system will differ from the predicted values. G iven those parameter matrices ( t , B A t and initial conditions of the state x t x + + u t B t A t x 0 H t y t t+1 unit time delay F igure 4 -1 Dynamic simulator of deterministic system 1
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A dynamic state observer is a real-time simulator with a feedbackmechanism for recursively correcting its estimated state based on the actual outputs measured from the real physical system. S igure 4 e a standard feedbackcontrol ee F -2 below. Note that, unlik system, the discrepancy between the predicted outputs y ı t and the actual outputs y t from the real system are fed back to the model rather than the real physical system. Using a nx feedbackgain matrix L R , the state observer is given by t x A ı t u B t L ( y x ı t + 1 + + = y ı ) t t t t t (3) y ı = H t x ı t t T o differentiate the estimated state from the actual state of the physical system, the estimated state residing in the real-time simulator is denoted x ı t . With this feedback the state of the simulator will follow the actual state of the real system, and thereby estimate the state accurately. If the system is observable , convergence of the estimated state to the actual state can be guaranteed with a proper feedbackgain. In other words, a stable observer can forget its initial conditions;regardless of an initial estimated state x ı 0 , the observer can produce the correct state as it converges. T tate his is Luenberger’s S Observer for deterministic systems. + + u t B t A t 0 ı x H t y t l i l G L t _ + + x t t y ı t x ı 1 ı + t x Model unit time delay Rea Phys ca System ain F -2 Luenberger’s state observer for igure 4 deterministic linear system ı A special case of the above state observer is estimation of constant parameters θ . ee eq n S uation (17) in Chapter 2. Replacing the state transition matrix A t by the n x identity matrixand setting inputs to zero leads to a recursive parameter estimation formula in (2-17): 2
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ı ( t ) = ı ( ı( θ θ ( t 1) + Κ ( t y ) t y ) ) (2-17) t T he difference from the previous parameter estimation problem is that in state estimation the state mak es “state transition” as designated by the state transition matrix A t and the input matrix B t
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