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linear_systems_2009_11_02_01

linear_systems_2009_11_02_01 - 12 1 Linear Systems with...

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12 - 1 Linear Systems with Inputs and Outputs S. Lall, Stanford 2009.11.02.01 12. Linear Systems with Inputs and Outputs Systems with inputs and outputs Solution Impulse and step matrices Mass-spring example Discretization of continuous-time linear systems Example: forces applied to a mass Example: computing the state Solution of discrete-time LDS Convolution and impulse response Block Toeplitz matrices Example: controlling a hovercraft Comments on controlling a hovercraft
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12 - 2 Linear Systems with Inputs and Outputs S. Lall, Stanford 2009.11.02.01 The Key Points of This Section if we have a continuous-time linear dynamical system ˙ x ( t ) = Ax ( t ) + Bu ( t ) y ( t ) = Cx ( t ) + Du ( t ) we can discretize it to make a discrete-time LDS in the form x ( t + 1) = Ax ( t ) + Bu ( t ) y ( t ) = Cx ( t ) + Du ( t ) which gives the behavior at sample times discrete-time systems are just like continuous-time ones; they have a modal decomposition stability is determined by eigenvalues of A ; but we need | λ | < 1 not Re( λ ) < 0 they map u to y via convolution for control and estimation, we can form least-squares problems using the block Toeplitz matrix which maps u to y
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12 - 3 Linear Systems with Inputs and Outputs S. Lall, Stanford 2009.11.02.01 Inputs & outputs Continuous-time time-invariant LDS has form ˙ x = Ax + Bu y = Cx + Du Ax is called the drift term (of ˙ x ) Bu is called the input term (of ˙ x ) picture, with B R 2 × 1 : x ( t ) Ax ( t ) ˙ x ( t ) (with u ( t ) = 1 ) ˙ x ( t ) (with u ( t ) = - 1 . 5 ) B
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12 - 4 Linear Systems with Inputs and Outputs S. Lall, Stanford 2009.11.02.01 Interpretations ˙ x = Ax + Bu y = Cx + Du write ˙ x = Ax + b 1 u 1 + · · · + b m u m , where B = [ b 1 · · · b m ] state derivative is sum of autonomous term ( Ax ) and one term per input ( b i u i ) each input u i gives another degree of freedom for ˙ x (assuming columns of B inde- pendent) write ˙ x = Ax + Bu as ˙ x i = ˜ a T i x + ˜ b T i u , where ˜ a T i , ˜ b T i are the rows of A , B i th state derivative is linear function of state x and input u
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12 - 5 Linear Systems with Inputs and Outputs S. Lall, Stanford 2009.11.02.01 Solution The solution is x ( t ) = e tA x (0) + integraldisplay t 0 e ( t - τ ) A Bu ( τ ) y ( t ) = Ce tA x (0) + integraldisplay t 0 Ce ( t - τ ) A Bu ( τ ) + Du ( t ) check via differentiation at time t , current state x ( t ) and output y ( t ) depend on past input ( u ( τ ) for τ t ) that is, mapping from input to state and output is causal (with fixed initial state)
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12 - 6 Linear Systems with Inputs and Outputs S. Lall, Stanford 2009.11.02.01 Impulse matrix The impulse matrix is h ( t ) = Ce tA B + ( t ) with x (0) = 0 , y = h * u , i.e. , y i ( t ) = m summationdisplay j =1 integraldisplay t 0 h ij ( t - τ ) u j ( τ ) interpretations: h ij ( t ) is impulse response from j th input to i th output h ij ( t ) gives y i when u ( t ) = e j δ h ij ( τ ) shows how dependent output i is, on what input j was, τ seconds ago i indexes output; j indexes input; τ indexes time lag
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12 - 7 Linear Systems with Inputs and Outputs S. Lall, Stanford 2009.11.02.01 Step matrix The step matrix or step response matrix is given by s ( t ) = integraldisplay t 0 h ( τ ) interpretations: s ij ( t )
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