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Professor Walter W. OlsonDepartment of Mechanical, Industrial and Manufacturing EngineeringUniversity of ToledoState FeedbackS

DisturbanceControllerPlant/ProcessOutputyxS-KkrState FeedbackPrefilterState Controlleru1Outline of Todays LectureReviewReachabilityTesting for ReachabilityControl System ObjectiveDesign Structure for State FeedbackState Feedback2nd Order ResponseState Feedback using the Reachable Canonical Form

ReachabilityWe 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, x0 , and a state xf , there exists a solution for t>0 such that x(0) = x0 and x(t)=xf.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.ReachabilityFor the system, ,

all of the states of the system are reachable if and only if Wr is invertible where Wr is given by

Canonical FormsThe word canonical means prescribedIn 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 recognizedOne such form is the Controllable or Reachable Canonical form.Reachable Canonical FormA system is in the reachable canonical form if it has the structure

Such a structure can be represented by blocks as

Dc1c2cn-1cn-1a1a2an-1anSSSSSSSS

uyz1z2zn-1znControl System ObjectiveGiven a system with the dynamics and the output

Design a linear controller with a single input which isstable at an equilibrium point that we define as

Our Design StructureInputrS

DisturbanceControllerPlant/ProcessOutputyxS-KkrState FeedbackPrefilterState Controlleru8Our Design StructureS

DisturbanceControllerPlant/ProcessInputrOutputyxS-KkrState FeedbackPrefilterState Controlleru

9Restated Control System Objective(Eigenvalue Assignment Problem)Given a system with the dynamics and the output

Design a linear controller with a single input which isstable at an equilibrium point that we define aswith a state feedback controller such that

Note that kr does not affect stability, is a scalar, and can be chosen as

for ye=r

Example

Design a controller that will control the angular position to a given angle, q0

Example

Design a controller that will control the angular position to a given angle, q0

2nd Order ResponseAs the example showed, the characteristic equation for which the roots are the eigenvalues allow us to design the reachable system dynamicsWhen 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

z-1-11Re(l)Im(l)xxxxxxxxxxz=1z=0.6z=0.4z=0.1z=0z=0z=0.1z=0.4z=0.6z=1wn=1

wn-1-11Re(l)Im(l)xxz=0.6xxxxwn=1wn=1wn=2wn=2wn=4wn=4State Feedback Design with the Reachable Canonical EquationSince the reachable canonical form has the coefficients of the characteristic polynomial explicitly stated, it may be used for design purposes:

Example: Inverted PendulumDesign a controller that will stabilize the Segway forward velocity at a given position, r0

Example

SummaryControl System ObjectiveDesign Structure for State FeedbackState Feedback

2nd Order ResponseState Feedback using the Reachable Canonical FormNext: State ObserversS

DisturbanceControllerPlant/ProcessInputrOutputyxS-KkrState FeedbackPrefilterState Controlleru