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Sang–Chul Lee DCASL 2009.03.11 POLE PLACEMENT STATE OBSERVER EFFECTS OF THE ADDITION OF THE OBSERVER ON A CLOSED-LOOP SYSTEM INTRODUCTION OF THE DISTURBANCE OBSERVER Open-loop system closed-loop control system with u = -Kx (Regulator system) What is the difference between two figures ?? x Ax Bu y Cx Where choose the Control signal -> (1) x = state vector (n-vector) u = control vector (sclar) A = n x n constant matrix B = n x 1 constant matrix C = 1 x n constant matrix u Kx (2) Substituting Eq (2) into Eq (1) gives x(t ) ( A BK ) x(t ) x(t ) e( A BK )t x(0) If the eigenvalues of matrix (A-BK) are asymptotically stable, x(t) approaches 0 as t approaches infinity The problem of placing the closed-loop poles at the desired location is called a pole-placement problem So, What is the pole-placement technique ?? Define T(transformation matrix) by M is the controllability matrix T MW M B AB A n1B And Refer to the notes please. Page. 1 The ai ' s are coefficients of the characteristic polynomial. Define a new state vector x̂ by Eq (1) can be modified to (Controllable canonical form) Where , Refer to the notes please. Page.1 Let us choose a set of the desired eigenvalues as 1 , 2 , then the desired characteristic ( s 1 )( s 2 ) , n . Eq. becomes ( s n ) s n 1s n 1 n 1s n 0 (3) Let us write when is used to control system The characteristic Eq. is (4) This characteristic Eq. is the same as the desired characteristic Eq. for the system, when ‘u = -Kx’ is used as the control signal. We can rewrite the Eq. (4) as s n (a1 1 ) s n 1 (an 1 n 1 ) s (an n ) 0 This is the characteristic Eq. for the system with state feedback. Therefore, it must be equal to Eq.(3) So we get And then, we obtain What is the STATE OBSERVER ?? - A state observer estimates the state variables based on the measurements of the output and control variables. Why we use the STATE OBSERVER ?? - Not all state variables are available for feedback! x Ax Bu (5) y Cx x Ax Bu K e ( y Cx) ( A K eC) x Bu K e y (6) Model of the state observer Block diagram of the system and full-order state observer Subtract Eq.(6) from Eq.(5). So we can obtain the observer error Eq. x x Ax Ax K e (Cx Cx) ( A K eC)( x x) Let’s Define Then Eq.(7)becomes (7) e xx e (A K eC)e The dynamic behavior of the error vector is determined by the eigenvalues of matrix (A K eC) . thus the problem here becomes the same as the pole-placement problem. It’s very similar to the pole-placement technique Refer to the notes please. Consider the system defined by x Ax Bu y Cx u Kx And, the observer Eq. is The laplace transform of u is x ( A K eC) x Bu K e y U (s) KX(s) (8) The laplace transform of observer Eq. is sX( s ) ( A K eC) X ( s) BU ( s) K eY ( s ) If x(0) = 0, then we can obtain X( s) (Is A K eC BK ) 1 K eY ( s) By substituting this Eq. into (8) , we get U ( s) K (Is A K eC BK ) 1 K eY ( s) Block diagram representation of system U ( s) K (Is A K eC BK ) 1 K eY ( s) Note that the transfer function acts as a controller for the system. The application of disturbance observer to practice J : inertia Kt : torque coefficient of electric motor Tl : load torque When the disturbance ‘Tl’ is generated ?? How can we use the estimated value ‘Tdis’ ?? Disturbance observer in motion control (designed by Gopinath’s method) Thank you!