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Unformatted text preview: State estimation Propagating the model: C A z 1 B a14a13 a15a12 + a27 a27 a27 a27 a118 a45 a54 y ( k ) x ( k ) Model: P ( z ) C A z 1 B a14a13 a15a12 + a27 a27 a27 a27 a118 a45 a54 y ( k ) x ( k ) Plant: P ( z ) a27 a118 a27 u ( k ) To calculate an estimated state, x ( k ), we must choose an initial estimated state, x (0), and run it through out model. Note that our controller will generate u ( k ) so we know what this is for times, k = 0 , . . . , k 1. Roy Smith: ECE 147b 12 : 2 State Estimation State Estimation State feedback design assumes that we can measure the complete state. What do we do if we cannot? Estimate it . Approach: create a model of the system and use its state instead of the measured state. C A z 1 B a14a13 a15a12 + a27 a27 a27 a27 a118 a45 a54 y ( k ) x ( k ) Model: P ( z ) C A z 1 B a14a13 a15a12 + a27 a27 a27 a27 a118 a45 a54 y ( k ) x ( k ) Plant: P ( z ) a27 a118 a27 u ( k ) Roy Smith: ECE 147b 12 : 1 State estimation Error properties: Look at the solution to the state equations: x ( k ) = A k x (0) + k summationdisplay j =0 A ( k j ) B u ( j ) x ( k ) = A k x (0) + k summationdisplay j =0 A ( k j ) B u ( j ) Subtracting these gives, x ( k ) = A k x (0) the estimation error depends only on the initial error. Again, its easy to see that if A is stable the transient caused by the initial estimation error will decay to zero. Can we do better? Make use of the measurement, y ( k ). Roy Smith: ECE 147b 12 : 4 State estimation Error properties: Define the state estimation error: x ( k ) := x ( k ) x ( k ) . Applying the state equation for both the model and the plant gives, x ( k + 1) = x ( k + 1) x ( k + 1) = A x ( k ) A x ( k ) = A ( x ( k ) x ( k )) = A x ( k ) . So the dynamics of the error, x ( k ), are the same as the openloop dynamics of the plant. If the plant is openloop unstable, the state estimation error, x ( k ), will blow up. Roy Smith: ECE 147b 12 : 3 State estimator Error properties The estimated state update equation is now, x ( k + 1) = A x ( k ) + B u ( k ) + L ( y ( k ) y ( k )) = A x ( k ) + B u ( k ) + LC ( x ( k ) x ( k )) Now subtract this from the true state update equation to get the error equation, x ( k + 1) = x ( k + 1) x ( k + 1) = A x ( k ) + B u ( k )...
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This note was uploaded on 04/06/2010 for the course ECE 145 taught by Professor Rodwell during the Spring '07 term at UCSB.
 Spring '07
 RODWELL

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