Data Structures And Algorithms In C++ 2nd Edition Michael T. Goodrich, Roberto Tamassia, David M. Mount - Solutions

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Consider the helicopter altitude control problem shown in the following diagram,where Helicopter Altitude Control Problem J = helicopter inertia, Br(t) =reference helicopter attitude, B(t) =actual helicopter attitude, Ks = sensor gain, Kr =reference scaling gain, w(t) = disturbance torque.(a) Use
Consider the control system shown in the following diagmm._r_(t)_~®----1 D(s) I 1 y(t) . s(s+ 1)A Control System The controller D(s) is given by,() , s+a Ds=l\. 2 2 .s +w(a) Prove that the system is capable of tracking a sinusoidal reference input, r(t) = sin(wt), with a zero steady state
Design Project:The closed-loop control system of the DC motor can be derived and shown to be given by Pigure 4. 32. The terms S1( s), 0,. ( s) are the Laplace transforms of the actual angular speed w( t) and the desir-ed angular speed w,. ( t), respectively. E(s) and U(s) are the Laplace transforms
In the diagram for Problem 4.1, a PID controller is used in a unit feedback system such that Kr D(s) = Kp +- + KDs s1 G(s) = Js2 r(t) = 1 w(t) = 1.(a) What is the effect of the P ID on the stability of this specific systern?(b) Find the steady state errors , e~8 for the system.(c) What are the
Consider the following generic feedback control system.w(t)!_r_(t_) _+@--jD(s)j :-@---jG(s)l y(t).r 1 13 1 I A GeneTal Control System(a) Find the transfer function from the Teference r(t) to the output y(t)i.e. Tr(s).(b) Find the tmnsfeT function from the disturbance w(t) to the output y(t) i.e.
(a) Draw a state diagram for the following state equations:dx1(t)-;It = -2x1 (t) + 3x2(t)dx2(t)-;It= -Sx1(t)- Sx2(t) + 2r(t)(b) Find the characteristic equation of the system.(c) Find the transfer functions X1(s)jR(s) and X2(s)jR(s).Problem
The following diagram shows a contml system with conditional feedback. The transfer function G( s) denotes the controlled process, and D ( s) and H ( s) are the controller transfer functions.R(s) +0 l •01--Y(s)~~ L:cb--Conditional Feedback--Control System(a) Derive the transfer functions
Use block diaqrmn algebra or Mason's rule to determine the transfer function Y ( s) / R( s) joT the following block diagram.Problem
Derive the transfer function, the input-output differential equation and the state-variable matrix form for the following block diagram.~----~1 + Y(s)1-X--=3;--..-~Block Diagram Problem
Simplify the following block diagram and obtain the closedloop transfcT function Y ( 8) / R( 8).Problem
Simplify the following block diagram and obtain the closedloop transfer function Y(s)/R(s).~J;_ ~ 1--------r---Y_(_s)'-----{+~ L-~-Block Diagmm Problem
Simplify the following block diagram and obtain the closedloop transfer function Y ( s) / R( s).Problem
Show that the transfer functions obtained by using block diagram algebra and Mason's rule for the following block diagram are the same.Y(s)Block Diagram Problem
Simplify the following block diagram and obtain its overall transjcT function, Y(s)/ R(s).Problem
A dynamic system is described by the state-vaTiablc equations :X o= Ax and y = Cx, where and x(O) = [1 2r.(a) Obtain the state-transition matrix ¢(t).(b) Find the stat.e vaTiable responses x 1 (t) and x2(t).(c) Find the output Tesponse y( t).(d) For this system verify that:¢(0) =I q;-J (t) ¢(
(a) !lssurning zero initial conditions, find the transfer j1m.ction for a system. !.hot obeys the equation jj + 4y + 4y = u(t).(b) Fmm the transfer function obtain the unit step response of the system.(c) From the tmnsfer function obtain the impulse response of the system.(d) Differentiate the
Apply the initial and final value theorems to find y(o+) and y( oo) for- each of the following transforms. If a theorem is not applicable to a particular transform, explain why this is so.(a)(b)s3 + 2s + 4 Y(s) = s(s + 1) 2 (s + 2)( ) 4s2 + lOs + 10 y s = --;;--::-;;---:::--s3 + 2s2 + 5s(c)(d)Y(s)
Verify that the transfer .function for the circuit shown below is given by+vi (t)H(s) = V,(s) = s2 + 2s + 1_ Vi ( s) s 2 + 4s + 4 20+0.5F vc(b) Find the unit impulse response for the circuit.(c) Find the unit step response for the circuit.+10 Problem
The input-output differential equation of an electr·ical circuit whose input is a voltage vi(t) and the output is a voltage v 0 (t), is given by(a) Find the crpressions for the damping ratio C and the undamped natural frequency :»n.1 1 (b) Given the information: R1 = H1, R2 = 2n, C = 1F, L = 2H,
Consider the following diagram where the input is force f(t).~I m 1---f(t)Translational Mechanical System{a) Obtain a differential equation in x that describes the behavior of the system.{b) Use the following information: x(O) = 1, x(O) = 0, m = 1, b = 4, k = 3 and f(t) = 9, use Laplace transforms
Consider the RLC electTical cirw'it given below, where the input is cunmt ii ( t) and the outvut is voltage V0 ( t).(a) The input-output differential equation for the circuit is given by C .. 1 . 1 ·.( ) V0 + RV0 + LV0 = Z t .253 1 1 Oiven that R = 4o, L = 3H, C = 1F and assuming zero initial
(a) Derive the transfer function H(s) for the general secondorder system given by where y(t) is the output, u(t) is the input, and all initial conditions are assumed to be zero.(b) Find the natural 1·esponse of the second-order system.(c) When is this system stable?(d) Explain how the unit step
(a) Derive the transfer function H(s) joT the general fiTstorder system. given by y + O"Y = ku, where y(t) ·is the output, u(t) is the input, and all initial conditions are assumed to be zero.(b) Find the natural response of the first-or-der system.(c) When is this system-stable?(d) Obtain the
In the following mechanical system a velocity input, v(t) = 6 fort 2: 0, is applied as illustmted.(a) Wrile the system's differential equation in tenns of the velocity v1 (t).(b) What is the time constant T and the steady state velocity u1ss for the system?(c) Given the infor-mation: v 1 (0) = 10,
(a) Find the inverse Laplace tmnsform of the function 3s2 I 2s + 2 Y(s) = (s + 2)(s2 + 2s + 5)'by using a complete partial fmction e:~:pansion.251(b) Find the inverse Laplace lmnsform of the same function by using the method of completing the sq·uare.Problem :3.5 (a) Shov: that the following
Use partial fract'ion expansions to find the inverse Laplace transforms {the time system responses) of the following functions(a){b){c)5 yl ( s) = --,----,---,----,-s(s + 1)(s + 10)2s Y2 (s)= s2 +8s+16,3 + ') + 4 y; ( ) = s ~s 3 s s4- 16 Problem
Use the definition and properties of Laplace transforms to find the Laplace transform of each of the following functions:{a)Yl(t) =te-at+ e-at coswt{b)Y2(t) = te-2t cos :3t + t2 sin 2t(c)Problem
Find and sketch the response of a dynamic system described by the equation Gy + y = f(t), with the initial condition y(O) = 4 when the inp1d is:{a) f(t) = 10{b) j(t) = 5e-~{c) f(t) = 10 + 5e-!{d) f ( t) = sin t + cos t Problem
Determine the system response from experimental data Problein
In addition to using analytical methods, use numeric methods to obtain the system response of a linear or nonlinear model in numerical form
Find and analyze the impulse response, step response, and sinusoidal response
Use the Block diagram and Signal flow graph to determine the transfer function
Use the transfer function to determine system response
For a dynamic model use the Laplace transform toa) find the complete time responseb) determine the transfer function, its poles and :ocrosc) analy:oe stability, evaluate time constants, damping ratios, and undamped natural frequencies
For the first or second-order system model, solve the differential equations directly to determine the system response
The st'udent-teacher learning pmcess is inherently a feedback pmcess 'intended to re&uce the system erToT to a minirnum. The desired outp1tl is knowledge being studied, and the student can be considered the pmcess. ConstTvct a feedback contml system block diagram and identify the control blocks
An enginecTing organizational system is composed of major groups such as managernent, research and development, preliminary design, expeT'iments. pmd1tct design and drafting, fabrication and assembling, and testing. These gnmps are inteT"connected to make up the whole opemt'ion.The system can be
Many closed-loop and open-loop contml systems can be found in homes.(a) List six such examples (three open-loop and three closed-loop).(b) Construct feedback control system block diagrams for the six examples, and identify the control blocks (sensor, actuator, pmcess, controller, actual output, and
The following diagram depicts an automatic closed-loop system joT paper moisture level control.De~ired +(\______\Controller ~---1 mmsture~. . level c__ ___ _, Drier Moisture meter A Closed-Loop Control System for J'apeT Moisture(a) Explain how this co ntrol system works.(b) If the automatic
The following diagram depicts a closed-loop t.empemture control system.Desired temp.--- -------------' ': ~~Controller I ; ·I Compressor I , Heat flpw Room , "---./ - -- , dynanncs temp ! -~
n ·~~CtaJ;;~n~---~-!-------' ' I snnp . , ! Thermostat :I I_--------- ------------------ ------------ J Room Temperature Control Using a Thermostat(a) Explain how this control system works.(b) If the automatic control is replaced by manual control, explain how the new system will function.(c)
From the mathematical description of the system, construct a simplified version using idealized elements and define a suitable set of variables.
Use the appropriate element and interconnection laws to obtain a mathematical model generally consisting of ordinary differential equations.
If the model is nonlinear, determine the equilibrium conditions and, where appropriate, obtain a linearized model in terms of incremental variables.
Arrange the equations that make up the model in readily usable forms such as the state-variable form, input-output differential equation form, tmnsfer function form and the block diagmm.
Obtain system models from experimental data such as frequency response, transient response and stochastic steady state data.
Compare and contrast the different dynamic system model forms and be able to convert from one form to the other.
Identify or define the state variables, inputs, and outputs."' Establish the system equations by using dynamic system laws such as D'Alembert's law, KCL and KVL.
Obtain the desired form of the system model by manipulating the equations.
If the model is nonlinear, determine the equilibrium conditions and obtain a linearized model.
For the following translational mechanical system, the spTings are undefiected when ::r:1 X2 = 0.(a) Draw the free-body diagrams for the system.(b) Write down the dynamic equations of motion.(c) Choose the minimum set of state variables for the sysleTn and justify your choice.(d) Develop the system
For the system shown below, the springs are undeflected when x1 = x2 = 0, and the input is force f(t).~I f---- f(t)b2 /-'-;77---:H/L-.>..-Translational Mechanical System(a) Draw the free-body diagrams for the system.(b) Write down the dynamic equations of motion.(c) Choose the minimum set of state
For designing an automobile suspension, a two-mass system can be ·used for modeling as shown in the following diagram.This is called a quarter-car model because it comprises one of the four wheel suspensions. The car and wheel positions are denoted by y(t) and x(t) respectively. These
The input to the translational mechanical system shown in the following diagram is the displacement X3 ( t) of /,he Tight end of the spring k1 . The absolute displacements of m1 and m2 are x1 and x 2 , nspectively.The forces exeTted by the springs are zero when x 1 = x 2 = x 3 = 0.(a) Draw the
Consider the mechanical and the electrical systems shown in the following diagrams.T y(a) (b)(a) Obtain the transfer function of the mechanical system, where xi(t) is the input and X 0 ( t) is the output.(b) Obtain the transfer function of the electrical system, where Vi ( t) is the input and V 0
In the following mechanical system the input is a force f(t)and the output is the spting force in k2.(a) Draw the fret-body diagrams for the mass Tn an.d for /.he massless point A.(b) WTite down the eqv.ations of motion.(c) Choose the minimum number of state variables jirr the system and de'l'dop
In the following tmnslational mechanical system the input is f(t) and the output is /;he force in spring k 3 .~ 1 ~I j=}~:~.---m----, 4---- u bl Translational Mechanical System (a) Draw the free-body diagrams joT the mass m and the massless point A.(b) Write down the system eqnations of motion (c)
For a certain dynamic system the state-1wriable equations joT the position x1 and velocity v1 are given below:=VI 1 i11 = -[-(k1 + k2)x1- bv1 + bx2 + (t), where x 2 ( t) is the input and the output is x 1 . Use the s-opemtor to obtain the input-mdp1d differential equation which relates the output
Consider the following mechanical system.~-1~I f(t) ,-------,-~Is Translational Mechanical System The forces exerted by the springs an zero when x1 = :r:2 = X3 = 0. The input force is j(t) and the absolute displacements of m 1 , m2 and m3 arc:r1, xz and X3, respectively(a) Draw the free-body
The following data is provided for the tran.slationo.L mechanical .sysl.cm considered in Problem 2. 9.m1 = 5kg m2 = l5kg m3 = 50kg b1 = 500N/m/s b2 = 600N/m/s k1 = 5, OOON /m= lO,OOON/m Use MATLAB to find the following:(a) The output response (y) to an input force of 50N(b) The response to u.n
In some mechanical positioning systems the movement of a large object is contmlled by manipulating a much smaller object that is mechanically coupled with it. The following diagram depicts such a syste·m, where a force u(t) is applied to a small mass m in order to position a largeT mass M. The
In the following rotational mechanical system there is a driving torq11e T(t) exeTted on disk J 1 and a load torque TL(t) e:rerted on disk J2 . Both the applied torque T(t) and the load torque TL(t) aTe considered as inputs and the ou.tput is the angular velocity w3 ( t).(a) Draw the free-body
The diagram below shows a double pendulum system. Assume the displacement angles of the pendulums is small enough to ensure that the spring is always horizontal. The pendulum rods are taken to be massless, of length l, and the springs are attached 2/3 of the way down.m m Double Pendulum Derive two
Consider the RLC electrical circuit given below, where the input is current i 1 (t) and the output is voltage v 0 (t).5 In the figure below, determine the input-output differential equation of the circuit..--+-- +
Find the input-output differential equation relating V0 and Vi ( t) for the circuit shown below.
(a) For the following circuit find the state-variable matrix model (A, B, C, D) where v 0 is the output voltage and Vi is the input voltage.Lt Rt+ +vi (t) Rz Vo-Lz(b) Also find the input-output differential equation for the system.
Consider the following circuit where i; ( t) is the input current and i a ( t) is the output cuTTent.+ cl J-~ +i. (t) 1 c2 c(a) Obtain the state-variable matrix ·model (A, B, C, D) joT the circuit.(b) Obtain the input-oulput differential equation for the circuit.
for the following op-amp circuit derive the algebraic expression for the output voltage v 0 in terms of the two input voltages ·u 1 ( t)and v2 (t). Iwi'icate what mathematical operation the circuit performs.
(a) For the follow·ing op-amp circuit derive the input-output differential equation relating the output voltage v 0 and the input voltage Vi ( t),(b) Derive the circuit's state-variable matrix model (A, B, C, lJ), where the input is voltage v;( f,) and there are two outputs,· the volt:age output
Consider the following op-amp circnil where the ouLpnt is voltage V0 ( t) and the input is voltage Vi ( t).(a) Choose a set of state variables and justify your choice.(b) Obtain the state-variable matrix model (A, B, C, D) for the system.
Provide the details of an array implementation of the list ADT.
Implement a heap-based priority queue that supports the following additional operation in linear time:replace Comparator(c): Replace the current comparator with c. After changing the comparator, the heap will need to be restructured.
Give complete C++ code for a new class, ShrinkingVector, that extends the ArrayVector class shown in Code Fragment 6.2 and adds a function, shrinkToFit, which replaces the underlying array with an array whose capacity is exactly equal to the number of elements currently in the vector.Data from in
Provide the missing housekeeping functions (copy constructor, assignment operator, and destructor) for the class ArrayVector of Code Fragment 6.2.Data from in Code Fragment 6.2A vector implementation using an extendable array.The member data for class ArrayVector consists of the array storage A,
Consider the following fragment of C++ code, assuming that the constructor Sequence creates an empty sequence of integer objects. Recall that division between integers performs truncation (for example, 7/2 = 3).Sequence seq;for (int i = 0; i < n; i++)seq.insertAtRank(i/2, i);a. Assume that the
Suppose we want to extend the Sequence abstract data type with functions index Of Element(e) and position Of Element(e), which respectively return the index and the position of the (first occurrence of) element e in the sequence. Show how to implement these functions by expressing them in terms of
Describe the structure and pseudo-code for an array-based implementation of the vector ADT that achieves O(1) time for insertions and removals at index 0, as well as insertions and removals at the end of the vector. Your implementation should also provide for a constant time elemAtRank function.
Describe an efficient way of putting a vector representing a deck of n cards into random order. You may use a function, randomInteger(n), which returns a random number between 0 and n−1, inclusive. Your method should guarantee that every possible ordering is equally likely. What is the running
Design a circular node list ADT that abstracts a circularly linked list in the same way that the node list ADT abstracts a doubly linked list.
Show that only n−1 passes are needed in the execution of bubble-sort on a sequence with n elements.
Write a program that takes as input a rooted tree T and a node v of T and converts T to another tree with the same set of node adjacencies but now rooted at v.
Describe an algorithm for counting the number of left external nodes in a binary tree, using the Binary tree ADT.
What is the running time of algorithm height2(T,v) (Code Fragment 7.7) when called on a node v distinct from the root of T?Data from in Code Fragment 7.7A more efficient algorithm for computing the height of the subtree of tree T rooted at a node p. int height2(const Tree& T, const Position& p)
Extend the concept of an Euler tour to an ordered tree that is not necessarily a binary tree.
Let T be a tree with n nodes. Define the lowest common ancestor (LCA) between two nodes v and w as the lowest node in T that has both v and w as descendents (where we allow a node to be a descendent of itself). Given two nodes v and w, describe an efficient algorithm for finding the LCA of v and w.
Let T be a tree with n nodes, and, for any node v in T, let dv denote the depth of v in T. The distance between two nodes v and w in T is dv+dw−2du, where u is the LCA u of v and w (as defined in the previous exercise). The diameter of T is the maximum distance between two nodes in T. Describe an
Suppose each node v of a binary tree T is labeled with its value ∫ (v) in a level numbering of T. Design a fast method for determining ∫ (u) for the lowest common ancestor (LCA), u, of two nodes v and w in T, given ∫ (v) and ∫ (w). You do not need to find node u, just
Give a C++ implementation of a priority queue based on an unsorted list.
Develop a C++ implementation of a priority queue that is based on a heap and supports the locator-based functions.
Show that, given only the less-than operator (<) and the boolean operators and (&&), or (||), and not (!), it is possible to implement all of the other comparators: >, <=, >=, ==, !=.
Describe, in detail, an implementation of a priority queue based on a sorted array. Show that this implementation achieves O(1) time for operations min and removeMin and O(n) time for operation insert.
Write a program that can process a sequence of stock buy and sell orders as described in Exercise C-8.1.Data from in Exercise C-8.1An online computer system for trading stock needs to process orders of the form “buy 100 shares at $x each” or “sell 100 shares at $y each.” A buy order for $x
Implement the vector ADT by means of an extendable array used in a circular fashion, so that insertions and deletions at the beginning and end of the vector run in constant time.
Give a C++ code fragment for reversing an array.
Describe what changes need to be made to the extendable array implementation given in Code Fragment 6.2 in order to avoid unexpected termination due to an error. Specify the new types of exceptions you would add, and when and where they should be thrown.Data from in Code Fragment 6.2A vector
Describe what changes need to be made to the extendable array implementation given in Code Fragment 6.2 in order to shrink the size N of the array by half any time the number of elements in the vector goes below N/4.Data from in Code Fragment 6.2A vector implementation using an extendable
Implement the vector ADT using a doubly linked list. Show experimentally that this implementation is worse than the array-based approach.
Give a C++ code fragment for randomly permuting an array.
Give a C++ code fragment for circularly rotating an array by distance d.
Implement the sequence ADT by means of an extendable array used in a circular fashion, so that insertions and deletions at the beginning and end of the sequence run in constant time.

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Data Structures And Algorithms In C++ 2nd Edition Michael T. Goodrich, Roberto Tamassia, David M. Mount - Solutions

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