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lecture12_small

lecture12_small - State Estimation State Estimation State...

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State estimation Propagating the model: C A z 1 B a14a13 a15a12 + a27 a27 a27 a27 a118 a45 a54 ˆ y ( k ) ˆ x ( k ) “Model:” ˆ 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, ˆ x ( k ), we must choose an initial estimated state, ˆ 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 ˆ y ( k ) ˆ x ( k ) “Model:” ˆ 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
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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 ) ˆ x ( k ) = A k ˆ 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, it’s 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 ) ˆ x ( k ) . Applying the state equation for both the model and the plant gives, ˜ x ( k + 1) = x ( k + 1) ˆ x ( k + 1) = A x ( k ) A ˆ x ( k ) = A ( x ( k ) ˆ x ( k )) = A ˜ x ( k ) . So the dynamics of the error, ˜ x ( k ), are the same as the open-loop dynamics of the plant. If the plant is open-loop unstable, the state estimation error, ˜ x ( k ), will blow up. Roy Smith: ECE 147b 12 : 3
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State estimator Error properties The estimated state update equation is now, ˆ x ( k + 1) = A ˆ x ( k ) + B u ( k ) + L ( y ( k ) ˆ y ( k )) = A ˆ x ( k ) + B u ( k ) + LC ( x ( k ) ˆ x ( k )) Now subtract this from the true state update equation to get the error equation, ˜ x ( k + 1) = x ( k + 1) ˆ x ( k + 1) = A x ( k ) + B u ( k ) [ A ˆ x ( k ) + B u ( k ) + LC ( x ( k ) ˆ x ( k )) ] = A ( x ( k ) ˆ x ( k )) LC ( x ( k ) ˆ x ( k )) = A ˜ x ( k ) LC ˜ x ( k )) = ( A LC ) ˜ x ( k ) .
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