1. Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo Block Diagrams H(s) + - R(s) Y(s) E(s) B(s) Negative Feedback G(s)

2. Outline of Today’s Lecture Review A new way of representing systems Coordinate transformation effects hint: there are none! Development of the Transfer Function from an ODE Gain, Poles and Zeros The Block Diagram Components Block Algebra Loop Analysis Block Reductions Caveats

3. Alternative Method of Analysis Up to this point in the course, we have been concerned about the structure of the system and discribed that structure with a state space formulation Now we are going to analyze the system by an alternative method that focuses on the inputs, the outputs and the linkages between system components. The starting point are the system differential equations or difference equations. However this method will characterize the process of a system block by its gain, G(s), and the ratio of the block output to its input. Formally, the transfer function is defined as the ratio of the Laplace transforms of the Input to the Output:

4. Coordination Transformations Thus the Transfer function is invariant under coordinate transformation x1x1 x2x2 z2z2 z1z1

5. Linear System Transfer Functions General form of linear time invariant (LTI) system is expressed: For an input of u(t)=e st such that the output is y(t)=y(0)e st Note that the transfer function for a simple ODE can be written as the ratio of the coefficients between the left and right sides multiplied by powers of s The order of the system is the highest exponent of s in the denominator.

6. Simple Transfer Functions Differential Equation Transfer Function Name s Differentiator Integrator 2 nd order Integrator 1 st order system Damped Oscillator PID Controller

7. 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 value of G(s) is the zero frequency or steady state gain of the system

8. 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

9. 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

10. Block Algebra + - x y x-y + - y x + - x y + - z z-x+y - + x z + + z-x y z-x+y - + y x + z G x H xG xGH H x G xH xGH GH x xGH

11. 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

12. Block Algebra G x x Gx G x x + - x y x-y + - x y + - y + - x G H y + - y x H G

13. Closed Loop Systems H + + H + - A positive feedback system A negative feedback system r r y y

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

15. Loop Analysis 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)

16. Block Reduction Example + y x B A + - + - + - C D E F G + y x B A + - + - + - C D E F G First, uncross signals where possible + +

17. Block Reduction Example + - y x B A + - + - + - C D E F G + + - y x B A + - + - + - C D F G + Next: Reduce Feed Forward Loops where possible

18. Block Reduction Example Next: Reduce Feedback Loops starting with the inner most + - y x B A + - + - + - C D F G + y x B + - + - C F

19. Block Reduction Example + - x C x y x B + - + - C F y y

20. x y x y

21. 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

22. 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)

23. 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 + -

24. Summary The Block Diagram Components Block Algebra Loop Analysis Block Reductions Caveats Next: Bode Plots G(s) x xG(s) x + + x y x+y x x x H(s) + - R(s) Y(s) E(s) B(s) Negative Feedback G(s)