Create a Spark Motivational Contest

Professor Walter W. OlsonDepartment of Mechanical, Industrial and Manufacturing EngineeringUniversity of ToledoBlock DiagramsH(s)+-R(s)Y(s)E(s)B(s)Negative FeedbackG(s)1Outline of Todays LectureReviewA new way of representing systemsCoordinate transformation effectshint: there are none!Development of the Transfer Function from an ODEGain, Poles and ZerosThe Block DiagramComponentsBlock AlgebraLoop AnalysisBlock ReductionsCaveats

Alternative Method of AnalysisUp to this point in the course, we have been concerned about the structure of the system and discribed that structure with a state space formulationNow 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:

Coordination TransformationsThus the Transfer function is invariant under coordinate transformationx1x2z2z1

Linear System Transfer Functions

General form of linear time invariant (LTI) system is expressed:

For an input of u(t)=est such that the output is y(t)=y(0)est

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.Simple Transfer FunctionsDifferential EquationTransfer FunctionNamesDifferentiatorIntegrator2nd order Integrator1st order systemDamped OscillatorPID Controller

Gain, Poles and ZerosThe roots of the polynomial in the denominator, a(s), are called the poles of the systemThe poles are associated with the modes of the system and these are the eigenvalues of the dynamics matrix in a state space representationThe roots of the polynomial in the numerator, b(s) are called the zeros of the systemThe zeros counteract the effect of a pole at a locationThe value of G(s) is the zero frequency or steady state gain of the system

Block DiagramsThroughout this course, we have used block diagrams to show different propertiesHere, we will formalize the meaning of block diagramsSenseComputeActuate

ControllerPlantSensorDc1c2cn-1cn-1a1a2an-1anSSSSSSSS

uyz1z2zn-1znS

DisturbanceControllerPlant/ProcessInputrOutputyxS-KkrState FeedbackPrefilterState ControlleruComponentsThe paths represent variable values whichare passed within the systemBlocks represent System components whichare represented by transfer functions and multiplytheir input signal to produce an outputAddition and subtraction of signals are representedby a summer block with the operation indicatedon the arrowG(s)xxG(s)x++xyx+yxxxBranch points occur when a value is placed on two lines: no modification is made to the signalBlock Algebra+-xyx-y+-yxx-y+-xy+-x-yzz-x+y-+xz++z-xyz-x+y-+yxz-x+y+zGxHxGxGHHxGxHxGHGHxxGHBlock AlgebraGxHHx+-Gx(G-H)xG-H(G-H)xxGx+-GxGx-zzGGx-z+-x

z

GG(x-z)+-xzG+-xzGGxGzG(x-z)GxGxGxGxGGxGxBlock AlgebraGxxGxGxGxx

+-xyx-yx-y+-xyx-yx-y+-y+-xGHy+-yxHG

Closed Loop SystemsH++H+-A positive feedback systemA negative feedback systemrryyLoop Analysis(Very important slide!)

H(s)+-R(s)Y(s)E(s)B(s)Negative FeedbackG(s)

Loop AnalysisH(s)++R(s)Y(s)E(s)B(s)

Positive FeedbackH(s)+-R(s)Y(s)E(s)B(s)Negative FeedbackG(s)Block Reduction Example+yxBA+-+-+-CDEFG+yxBA+-+-+-CDEFGFirst, uncross signals where possible++Block Reduction Example+-yxBA+-+-+-CDEFG++-yxBA+-+-+-CDFG+Next: Reduce Feed Forward Loops where possible

Block Reduction ExampleNext: Reduce Feedback Loops starting with the inner most+-yxBA+-+-+-CDFG+

yxB+-+-CF

Block Reduction Example+-xC

x

yxB+-+-CF

y

y

Block Reduction Examplex

y

xy

Loop NomenclatureReferenceInputR(s)+-Outputy(s)ErrorsignalE(s)Open LoopSignalB(s)PlantG(s)SensorH(s)PrefilterF(s)ControllerC(s)+-Disturbance/NoiseThe 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

Caveats: Pole Zero CancellationsAssume there were two systems that were connected as such

An astute student might note thatand then want to cancel the (s+1) termThis 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)

Caveats: Algebraic LoopsThe 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+-SummaryThe Block DiagramComponentsBlock AlgebraLoop AnalysisBlock ReductionsCaveatsNext: Bode PlotsG(s)xxG(s)x++xyx+yxxxH(s)+-R(s)Y(s)E(s)B(s)Negative FeedbackG(s)