A system can be represented by a delay or gain or the combination of the two systems using a Cascade or FeedforwardAdd or FeedbackAdd Lecture 4 Multiple Representations of Systems Operator representations are compact and preserve ow information Combine best features of difference equations and block diagrams Provide a convenient way to reason out a system Example Y 1 R X Operator Expressions Obey Rules of Polynomial Algebra Operator expressions are commutative and associative and multiplication distributes over addition Equivalent polynomials equivalent systems when started at rest System Functionals We can represent an entire system by its operator representation which we call the system function Representing a system by a system functional Example one Y HX Example two Y HX Functionals for systems without feedback are polynomials in R General Framework for Thinking about Systems Primitives Delays Gains Composition Cascade FeedforwardAdd FeedbackAdd The rules of composition are recursive Composing Complicated Systems Any of the rules of composition can be added to any two systems to generate a new type of system Cascade Feedforward Feedback Check yourself System Functionals Correct answer 1 Determine the system functional H Black s Equation Black s equation has two common forms Difference between the two the change of sign G The second form is useful in most control applications where the goal is to make Y and X converse Check yourself Representing systems Correct answer 2 Assume that F and G can be represented by polynomials in R How many of the following systems can be represented by a polynomial in R Cascade F G F x G is a polynomial FeedforwardAdd F G F G is a polynomial FeedbackAdd F Gain 1 FeedbackAdd Gain 1 G F 1 FG FeedbackAdd F G F 1 F is generally not a polynomial Check yourself Representing systems Correct answer 5 Assume that each of F and G can be represented by a ratio of two polynomials in R How many of the following systems can be represented by a ratio of two polynomials Cascade F G FeedforwardAdd F G FeedbackAdd F Gain 1 FeedbackAdd Gain 1 G FeedbackAdd F G System Functionals as Abstraction Systems that can be represented by a Delay or a gain or by the combination of two systems using Cascade FeedforwardAdd or FeedbackAdd can also be represented by a ratio of polynomials in R R 1 K and K 1 R Cascade Feedforward Feedback System Functionals can be used interchangeably with primitives PCAP Framework for Managing Complexity A system can be represented by a delay or a gain or the combination of two systems using Cascade FeedforwardAdd or FeedbackAdd Primitives Delays Gains Composition Cascade FeedforwardAdd FeedbackAdd Abstraction System Functional Poles Complex Poles Example Ways to nd the poles of a system Write the system functional in special form Write the operator expression in special form Substitute R 1 z and nd the roots of the denominator All the methods give the same result What if a pole has a non zero imaginary part The poles are at z corresponding modes are Powers of complex numbers are easy to compute using polar forms Express the pole at z as re where r b and tan b a Then the mode is re r e geometric growth of magnitude linear growth of angle Complex Roots An isolated complex root can result only from a difference equation with complex valued coef cients Example Corresponding difference equation 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 But systems with real valued coef cients can have complex valued poles The sum of the modes associate with complex conjugates is real Check yourself Complex Poles Correct answer 2 Output of a system with poles at z What are the values for r and 0 5 r 1 and 0 5 Designing a Control System We have already built several feedback systems to control the robot wallFinder move forward or backward to position robot a desired distance from the wall in front of it wallFollower move the robot parallel to a wall maintaining a desired distance from the wall We can use the Signals and Systems Abstraction to gain insight into the general problem of designing a feedback controller Controlling Accumulation Both wallFinder and wallFollower contained accumulators to model integrative processes wallFinder command forward velocity to set forward distance wallFollower command rotational velocity to set lateral distance Accumulator model d n 1 d n Tv n Integrator Finding the Optimal Gain Consider a proportional controller for the accumulator system Replace accumulator with equivalent system functional This gives us an equivalent system with a single block Unit Sample Response The system functional contains a single pole at The numerator is just a gain and a delay Step Response We are often interested in the step response of a control system Start with the output d n 0 while the input di n is held constant where d n is the sonarDistance and di n is the desiredDistance Calculating unit step response Unit Step response s n is the response of H to the unit step signal u n which is constructed by accumulation of the unit sample n Commute and relabel signals The unit step response s n is equal to the accumulated responses to the unit sample response h n Root Locus The poles of the system functional provide insight for choosing K Check yourself Root Locus Correct answer 2 Find KT for fastest convergence of unit sample response KT 1 Destabilizing Effect of Delay The optimum gain allows the proportional controller system to converge in a single step Adding a single delay causes stability Check yourself System Functional Correct answer 4 Find the system functional Feedback and Control Poles If KT is small the poles at z KT and z 1 KT z Pole near 0 generates fast response Pole near 1 generates slow response The slow mode dominates the response Check yourself Poles Correct answer 5 The period of oscillation is 6 Check yourself Poles Correct answer 2 The fastest response is when KT 1 4 Destabilizing Effect of Delay Adding delay tends to destabilize control systems Check yourself Possible Test Question Correct answer How many of the following statements are true This system has three poles Unit sample response is the sum of three geometric sequences Unit sample response is y n 0 0 0 1 0 0 1 0 0 1 0 0 1 Unit sample response is y n 1 0 0 1 0 0 1 0 0 1 0 0 1 One of the poles is at z 1 Designing Control Systems
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