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Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo Root Locus Diagrams

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Outline of Todays Lecture Review The Block Diagram Components Block Algebra Loop Analysis Block Reductions Caveats Poles and Zeros Plotting Functions with Complex Numbers Root Locus Plotting the Transfer Function Effects of Pole Placement Root Locus Factor Responses

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Block Diagrams Throughout this course, we have used block diagrams to show different properties Here, we will formalize the meaning of block diagrams Sense Compute Actuate Controlle r Plant Sensor Dc1c1 c2c2 c n-1 cncn a1a1 a2a2 a n-1 anan … … … u y z1z1 z2z2 z n-1 znzn Disturbance Controller Plant/Process Input r Output y x -K krkr State Feedback Prefilter State Controller u

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Components The paths represent variable values which are passed within the system Blocks represent System components which are represented by transfer functions and multiply their input signal to produce an output Addition and subtraction of signals are represented by a summer block with the operation indicated on the arrow G(s) x xG(s) x + + x y x+y x x x Branch points occur when a value is placed on two lines: no modification is made to the signal

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Block Algebra G x H Hx + - Gx (G-H)x G-H (G-H)x x G x + - Gx Gx-z z G + - x z G G(x-z) + - x z G + - x z G Gx Gz G(x-z) G x Gx G x G

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Loop Analysis (Very important slide!) H(s) + + R(s) Y(s) E(s) B(s) Positive Feedback H(s) + - R(s) Y(s) E(s) B(s) Negative Feedback G(s)

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Loop Nomenclature Reference Input R(s) + - Output y(s) Error signal E(s) Open Loop Signal B(s) Plant G(s) Sensor H(s) Prefilter F(s) Controller C(s) + - Disturbance/Noise The plant is that which is to be controlled with transfer function G(s) The prefilter and the controller define the control laws of the system. The open loop signal is the signal that results from the actions of the prefilter, the controller, the plant and the sensor and has the transfer function F(s)C(s)G(s)H(s) The closed loop signal is the output of the system and has the transfer function

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Caveats: Pole Zero Cancellations Assume there were two systems that were connected as such An astute student might note that and then want to cancel the (s+1) term This would be problematic: if the (s+1) represents a true system dynamic, the dynamic would be lost as a result of the cancellation. It would also cause problems for controllability and observability. In actual practice, cancelling a pole with a zero usually leads to problems as small deviations in pole or zero location lead to unpredictable dynamics under the cancellation. R(s) Y(s)

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Caveats: Algebraic Loops The system of block diagrams is based on the presence of differential equation and difference equation A system built such the output is directly connected to the input of a loop without intervening differential or time difference terms leads to improper block interpretations and an inability to simulate the model. When this occurs, it is called an Algebraic Loop. Such loops are often meaningless and errors in logic. 2 + -

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Gain, Poles and Zeros The roots of the polynomial in the denominator, a(s), are called the poles of the system The poles are associated with the modes of the system and these are the eigenvalues of the dynamics matrix in a state space representation The roots of the polynomial in the numerator, b(s) are called the zeros of the system The zeros counteract the effect of a pole at a location The variable s is a complex number: The value of G(0) is the zero frequency or steady state gain of the system

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Plotting functions on the Complex Plane Plotting functions on the complex plane is more complicated than the real plane because of unexpected forms that occur Consider an equation such as If z is limited to real numbers, z must be 1 for any n BUT, this is not the case if z is allowed to be a complex number if n = 3, then If n = 4, then Consider a function such as If z were real, a hyperbola results BUT, if z is a complex number, a totally different result occurs Both a and b vary with results in surface rather than a curve The result of the function could be either real or complex Therefore, visualization is difficult

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Root Locus The root locus plot for a system is based on solving the system characteristic equation The transfer function of a linear, time invariant, system can be factored as a fraction of two polynomials When the system is placed in a negative feedback loop the transfer function of the closed loop system is of the form The characteristic equation is The root locus is a plot of this solution for positive real values of K Because the solutions are the system modes, this is a powerful design tool While we focus here on the gain, K, we can plot any parameter this way

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Plotting a Transfer Function Root Locus The path is determined from the open loop transfer function by varying the gain s as used in a transfer function is a complex number Poles will be marked with X Zeros with be marked with an O Each path represents a branch of the transfer function in the complex plane All paths start at poles and end at zeros mirror across the real axis There must be a zero for each pole Those that are not shown on the plot are at infinity Matlab command rlocus(sys)

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Paths of the Transfer Function K=1 K=0.1 K=3 K=10

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Paths of the Transfer Function The real values of the gain move the poles along the root loci Notice that the placement of the gain moved poles dictates the output response of the system Poles in the right half plane are unstable responses K=1 K=0.1 K=3 K=10

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The effect of placement on the root locus Imaginary axis Real Axis j j d n n sin -1 ( ) The magnitude of the vector to pole location is the natural frequency of the response, n The vertical component (the imaginary part) is the damped frequency, d The angle away from the vertical is the inverse sine of the damping ratio,

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Root Locus Factor Responses Real Axis j A complete system will sum all of these effects that are present in the systems response The dominating effects will be from the poles closest to the origin

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Example A radar tracking antenna (Nise, 1995) has the position control transfer function of The antenna must have a 5% settling time of less than 2 seconds with an over damped response.

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Example

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Current system can not meet either requirement with gain alone: By adding a zero at -1.34, a pole at -11 and a gain of 271, we get Is this the best controller?

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Summary Poles and Zeros Plotting Functions with Complex Numbers Root Locus Plotting the Transfer Function Effects of Pole Placement Root Locus Factor Responses Next: Bode Plots Imaginary axis Real Axis j j d n n sin -1 ( )

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