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State Feedback

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State Space Representation

  • Given a point mass on a line whose acceleration is directly controlled:

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  • We want to write this on a compact/general form

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  • On a state space form

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Block Diagram

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The Car Model

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  • If we care about/can measure the velocity:

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  • If we care about/can measure the position we have the same general equation with different matrices:

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Output Feedback

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Output Feedback

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Output Feedback in MATLAB

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State Feedback

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State Feedback

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State Feedback

  • Let's try
    • Asymptotically stable
    • Damped oscillations

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  • Let's do another attempt
    • Asymptotically stable
    • No oscillations

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State Feedback

  • Eigenvalues Matter
    • It is clear that some eigenvalues are better than others. Some cause oscillations, some make the system respond too slowly, and so forth ...
    • We will see how to select eigenvalues and how to pick control laws based on the output rather than the state.

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State Feedback in MATLAB

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State Feedback in MATLAB

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Pole Placement

  • Back to the point-mass, again

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Pole Placement

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Example

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Pole Placement in MATLAB

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Controllability

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Controllability

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Controllability

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Controllability in MATLAB

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Observer

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Observer

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Observer

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Observability

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Observability

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Observability

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Observability in MATLAB

  • obsv(A, C) is the MATLAB function to form a observability matrix

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Now, How Do We Put Everything Together ?

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Now, How Do We Put Everything Together ?

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The Separation Principle

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Everything in Block Diagram

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