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Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo State Feedback Disturbance Controller Plant/Process Output y x -K krkr State Feedback Prefilter State Controller u

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Outline of Todays Lecture Review Reachability Testing for Reachability Control System Objective Design Structure for State Feedback State Feedback 2 nd Order Response State Feedback using the Reachable Canonical Form

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Reachability We define reachability (often times called controllability) by the following: A state in a system is reachable if for any valid states of the system, say, initial state at time t=0, x 0, and a state x f, there exists a solution for t>0 such that x(0) = x 0 and x(t)=x f. There are systems which we can not control the states are not reachable with our input. There in designing control systems, it is important to know if the system is controllable. This is closely linked with the concept of ergodicity of the system in which we ask the question whether or not it is possible to with some measure of our system to measure every possible state of the system.

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Reachability For the system,, all of the states of the system are reachable if and only if W r is invertible where W r is given by

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Canonical Forms The word canonical means prescribed In Control Theory there a number transformations that can be made to put a system into a certain canonical form where the structure of the system is readily recognized One such form is the Controllable or Reachable Canonical form.

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Reachable Canonical Form A system is in the reachable canonical form if it has the structure Such a structure can be represented by blocks as Dc1c1 c2c2 c n-1 cncn a1a1 a2a2 a n-1 anan … … … u y z1z1 z2z2 z n-1 znzn

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Control System Objective Given a system with the dynamics and the output Design a linear controller with a single input which is stable at an equilibrium point that we define as

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Our Design Structure Input r Disturbance Controller Plant/Process Output y x -K krkr State Feedback Prefilter State Controller u

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Our Design Structure Disturbance Controller Plant/Process Input r Output y x -K krkr State Feedback Prefilter State Controller u

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Restated Control System Objective (Eigenvalue Assignment Problem) Given a system with the dynamics and the output Design a linear controller with a single input which is stable at an equilibrium point that we define as with a state feedback controller such that Note that k r does not affect stability, is a scalar, and can be chosen as for y e =r

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Example Design a controller that will control the angular position to a given angle, 0

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Example Design a controller that will control the angular position to a given angle, 0

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2 nd Order Response As the example showed, the characteristic equation for which the roots are the eigenvalues allow us to design the reachable system dynamics When we determined the natural frequency and the damping ration by the equation we actually changed the system modes by changing the eigenvalues of the system through state feedback 1 Re( ) Im( ) x x x x x x x x x x n =1 n 1 Re( ) Im( ) x x x x x x n =1 n =2 n =4

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State Feedback Design with the Reachable Canonical Equation Since the reachable canonical form has the coefficients of the characteristic polynomial explicitly stated, it may be used for design purposes:

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Example: Inverted Pendulum Design a controller that will stabilize the Segway forward velocity at a given position, r 0

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Example

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Summary Control System Objective Design Structure for State Feedback State Feedback 2 nd Order Response State Feedback using the Reachable Canonical Form Next: State Observers Disturbance Controller Plant/Process Input r Output y x -K krkr State Feedback Prefilter State Controller u

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