Lecture 3 Intellectual themes Design of complex systems Modeling and controlling physical systems Augmenting physical systems with computation Building systems that are robust to uncertainty Intellectual themes are developed in context of a mobile robot The goal is to convey a distinct perspective about engineering Models Signals and systems Circuits Probability Planning Approach focus on key concepts and pursue in depth Software Engineering We will use programming throughout 6 01 as an essential tool for engineers and scientists to facilitate learning The Signals and Systems Abstraction Programming is intrinsically interesting exemplifying the two most important themes in 6 01 Abstraction and modularity Represent a system physical mathematical or computational by the way it transforms input signal into an output signal Difference equation y n x n x n 1 Block diagram Signal Representation From Samples to Signals Lumping all of the samples possibly in nite into a single object the signal simpli es its manipulation This lumping is analogous to Representing coordinates in three space as points Representing lists of numbers as vectors in linear algebra Creating an object in Python Operators manipulate signals rather than individual samples Nodes represent whole signals i e X and Y The boxes operate on those signals Delay shift whole signal to the right one time step Add sum two signals Gain multiply by 1 Signals are primitives Operators are the means of combination Operator Notation Symbols can compactly represent diagrams Let R represent the right shift operator Systems are concisely represented with operators The block diagram is the difference machine Equivalent representation with R Y X RX 1 R X Check yourself Operator Notation Correct answer 2 Operator Representation of a Cascade System System operations have simgple operator representations Cascade systems multiply operator expressions Using operator notation Let Y RX Given that equation we know that y n 1 x n for all n is true Y 1 R X Y 1 R Y Substituting for Y Y 1 R 1 R X Operator Algebra Expressions involving R obey many familiar laws of algebra such as commutativity R 1 R X 1 R RX This is easily proved by the de nition of R and it implies that cascaded systems commute assuming initial rest Multiplication distributes over addition Y RX where X represents the whole input signal x n for all n and Y represents the whole output signal y n for all n Operator The associative property similarly holds for operator expressions Equivalent operator expression 2 R R 1 R 2 R R 1 R Check Yourself Operator Algebra Correct answer 3 All three of these systems are equivalent Operator Approach Feedback Y 1 R X Y X RY Example Accumulator To nd Y try step by step analysis Start at rest Applies your existing expertise with polynomials to understand block diagrams and thereby understand systems Feedback complicates relation between input and output Without feedback output signal is linear combination of shifted versions of input signal Feedback introduces a similar constraint but now the input signal is a linear combination of shifted versions of the output signal Check yourself Feedback Correct answer 2 The difference equation that relates X and Y is y n x n 1 y n 2 The response of the accumulator systems could also be generated by a system with in nitely many paths from input to output each with one unit of delay more than the previous The system functional for the accumulator is the reciprocal of a polynomial R The product 1 R x 1 R R R are reciprocals thus we can write Geometric Growth The signal can increase or decrease with time if the loop gain is not 1 Geometric growth gives us the equation This geometric growth arises from cyclic signal ow paths Cyclic signal ow paths persistent responses to transient inputs These system responses can be characterized by a single number the pole which is the base of the geometric sequence Check yourself Geometric Growth Correct answer 2 What value of p represents the signal p 0 5 The value of p determines the rate of growth p 1 magnitude diverges monotonically 0 p 1 magnitude converges monotonically 1 p 0 magnitude converges alternating sign p 1 magnitude diverges alternating sign Second Order Systems The unit sample responses of more complicated cyclic systems are more complicated This response is not geometric It grows then decays Second Order Systems Equivalent forms Factor the operator expression to break the system into two similar systems Y X 1 6RY 0 63R Y 1 1 6R 0 63R Y X 1 0 9R 1 0 7R Y X Higher Order Systems Poles Systems that can be represented by linear difference equations with constant coef cients have operator representations that are the ratio of polynomials in R Poles can be identi ed by expanding the system functional in partial fractions The poles are p for 0 i n where n is the order of the n denominator Poles can be identi ed by replacing each R in the system functional with 1 z Then the poles are the roots of the denominator polynomial in z Complex Poles Powers of complex numbers are easy to compute using polar forms Express the pole at z a jb as re where r a b and tan b a Then the mode is re r e geometric growth of magnitude linear growth of angle Complex Pole Example Consider a complex pole at re where r 0 98 and 0 2 The nth sample of the corresponding mode is re r e Complex Roots Difference equations that represent physical systems have real valued coef cients Difference equations with real valued coef cients generate real valued outputs from real valued inputs
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