lecture11_small

lecture11_small - State Feedback State Feedback State-space...

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State feedback State feedback control law: u ( t )= - Kx ( t ) C A 1 s K B P ( s ) ±² ³´ + - ±² ³´ + µ µ ± ± ± ± ± ² ³ ² ³ u ( s ) sx ( s ) x ( s ) x ( s ) y ( s ) Salient points: 1. There is no reference input (we will add one later). We are driving the state to zero. 2. We can measure all of the states which is often not realistic. We will remove this assumption later by designing an estimator . 3. The feedback gain, K , is static . This controller has no states. Roy Smith: ECE 147b 11 :2 State Feedback State Feedback State-space models allow us to use a very powerful design method: state feedback . To begin: assume that we can measure all of the state vector. We will address the case where this cannot be done later. State feedback control law: u ( t - ( t ) or, equivalently, u ( s - ( s ) C A 1 s K B P ( s ) ±² ³´ + - ±² ³´ + µ µ ± ± ± ± ± ² ³ ² ³ u ( s ) sx ( s ) x ( s ) x ( s ) y ( s ) Roy Smith: ECE 147b 11 :1

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State feedback Structure of the state feedback control gains: Examine the feedback gain matrix in more detail,. . . u ( k )= - Kx ( k ) = - ± K 1 K 2 ··· K n ² x 1 ( k ) x 2 ( k ) . . . x n ( k ) Note that K is a static nu × nx matrix. We have at least as many control gains in K as states in the system. We can often use state feedback to place the closed-loop poles anywhere we want. This may take a lot of gain. In other words, large values of K i . Roy Smith: ECE 147b 11 :4 State feedback Discrete-time state feedback Given a discrete-time state-space system, x ( k + 1) = Ax ( k )+ Bu ( k ) Recall that the poles of the system are the eigenvalues of A . State feedback law: u ( k - ( k ). This gives, x ( k + 1) = ( k ) - BK x ( k ) =( A - BK ) ³ ´µ x ( k ) closed-loop dynamics So state feedback changes the pole locations (which is what we expect feedback to do). The closed-loop poles are given by the eigenvalues of A - . Roy Smith: ECE 147b 11 :3
State feedback Controllable canonical form One particular state-space representation ( controllable canonical form ) makes this particularly easy. For example, P ( z )= b 1 z 2 + b 2 z + b 3 z 3 + a 1 z 2 + a 2 z + a 3 .

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lecture11_small - State Feedback State Feedback State-space...

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