Answers: 1. Guided missiles, automatic gain control in radio receivers, satellite tracking antenna 2. Yes - power gain, remote control, parameter conversion; No - Expense, complexity 3. Motor, low pass filter, inertia supported between two bearings 4. Closed-loop systems compensate for disturbances by measuring the response, comparing it to the input response (the desired output), and then correcting the output response. 5. Under the condition that the feedback element is other than unity 6. Actuating signal 7. Multiple subsystems can time share the controller. Any adjustments to the controller can be implemented with simply software changes 8. Stability, transient response, and steady-state error 9. Steady-state, transient 10. It follows a growing transient response until the steady-state response is no longer visible. The system will either destroy itself, reach an equilibrium state because of saturation in driving amplifiers, or hit limit stops. 11. Natural response 12. Determine the transient response performance of the system. 13. Determine system parameters to meet the transient response specifications for the system. 14. True 15. Transfer function, state-space, differential equations 16. Transfer function - the Laplace transform of the differential equation State-space - representation of an nth order differential equation as n simultaneous first-order differential equations Differential equation - Modeling a system with its differential equation

Chapter 3 1. Give two reasons for modeling systems in state space. 2. State an advantage of the transfer function approach over the state-space approach. 3. Define state variables. 4. Define state. 5. Define state vector. 6. Define state space. 7. What is required to represent a system in state space? 8. An eighth-order system would be represented in state space with how many state equations? 9. If the state equations are a system of first-order differential equations whose solution yields the state variables, then the output equation performs what function? 10. What is meant by linear independence'? 11. What factors influence the choice of state variables in any system? 12. What is a convenient choice of state variables for electrical networks? 13. If an electrical network has three energy-storage elements, is it possible to have a state-space representation with more than three state variables? Explain. 14. What is meant by the phase-variable form of the state-equation? 1. (1) Can model systems other than linear, constant coefficients; (2) Used for digital simulation 2. Yields qualitative insight 3. That smallest set of variables that completely describe the system 4. The value of the state variables 5. The vector whose components are the state variables 6. The n-dimensional space whose bases are the state variables 7. State equations, an output equation, and an initial state vector (initial conditions) 8. Eight 9. Forms linear combinations of the state variables and the input to form the desired output 10. No variable in the set can be written as a linear sum of the other variables in the set. 11. (1) They must be linearly independent; (2) The number of state variables must agree with the order of the differential equation describing the system; (3) The de gree of difficulty in obtaining the state equations for a given set of state variables. 12. The variables that are being differentiated in each of the linearly independent energy storage elements 13. Yes, depending upon the choice of circuit variables and technique used to write the system equations. For example, a three -loop problem with three energy storage elements could yield three simultaneous second-order differential equations which would then be described b y six, firstorder differential equations. This exact situation arose when we wrote the differential equations for mechanical systems and then proceeded to find the state equations. 14. The state variables are successive derivatives.

Chapter 4 1. Name the performance specification for first-order systems. 2. What does the performance specification for a first-order system tell us? 3. In a system with an input and an output, what poles generate the steady-state response? 4. In a system with an input and an output, what poles generate the transient response? 5. The imaginary part of a pole generates what part of a response? 6. The real part of a pole generates what part of a response? 7. What is the difference between the natural frequency and the damped frequency of oscillation? 8. If a pole is moved with a constant imaginary part, what will the responses have in common? 9. If a pole is moved with a constant real part, what will the responses have in common? 10. If a pole is moved along a radial line extending from the origin, what will the responses have in common? 11. List five specifications for a second-order underdamped system. 12. For Question 11 how many specifications completely determine the response? 13. What pole locations characterize (1) the underdamped system, (2) the overdamped system, and (3) the critically damped system? 14. Name two conditions under which the response generated by a pole can be neglected. 15. How can you justify pole-zero cancellation? 16. Does the solution of the state equation yield the output response of the system? Explain. 17. What is the relationship between (si — A), which appeared during the Laplace transformation solution of the state equations, and the state-transition matrix, which appeared during the classical solution of the state equation? 18. Name a major advantage of using time-domain techniques for the solution of the response. 19. Name a major advantage of using frequency-domain techniques for the solution of the response. 20. What three pieces of information must be given in order to solve for the output response of a system using statespace techniques? 21. How can the poles of a system be found from the state equations? 1.Time constant 2. The time for the step response to reach 63% of its final value 3. The input pole 4. The system poles 5. The radian frequency of a sinusoidal response 6. The time constant of an exponential response 7. Natural frequency is the frequency of the system with all damping r emoved; the damped frequency of oscillation is the frequency of oscillation with damping in the system. 8. Their damped frequency of oscillation will be the same. 9. They will all exist under the same exponential decay envelop. 10. They will all have the same percent overshoot and the same shape although differently scaled in time. 11. ζ, ωn, TP, %OS, Ts 12. Only two since a second-order system is completely defined by two component parameters 13. (1) Complex, (2) Real, (3) Multiple real 14. Pole's real part is large compared to the dominant poles, (2) Pole is near a zero 15. If the residue at that pole is much smaller than the residues at other poles 16. No; one must then use the output equation 17. The Laplace transform of the state transition matrix is (sI - A)-1 18. Computer simulation 19. Pole-zero concepts give one an intuitive feel for the problem. 20. State equations, output equations, and initial value for the state-vector 21. Det(sI-A) = 0

Chapter 5 1. Name the four components of a block diagram for a linear, time-invariant system. 2. Name three basic forms for interconnecting subsystems. 3. For each of the forms in Question 2, state (respectively) how the equivalent transfer function is found. 4. Besides knowing the basic forms as discussed in Questions 2 and 3, what other equivalents must you know in order to perform block diagram reduction? 5. For a simple, second-order feedback control system of the type shown in Figure 5.14, describe the effect that variations of forward-path gain, K, have on the transient response. 6. For a simple, second-order feedback control system of the type shown in Figure 5.14, describe the changes in damping ratio as the gain, K, is increased over the underdamped region. 7. Name the two components of a signal-flow graph. 8. How are summing junctions shown on a signal-flow graph? 9. If a forward path touched all closed loops, what would be the value of A^? Slate Space 10. Name five representations of systems in state space. 11. Which two forms of the state-space representation are found using the same method? 12. Which form of the state-space representation leads to a diagonal matrix? 13. When the system matrix is diagonal, what quantities lie along the diagonal? 14. What terms lie along the diagonal for a system represented in Jordan canonical form? 15. What is the advantage of having a system represented in a form that has a diagonal system matrix? 16. Give two reasons for wanting to represent a system by alternative forms. 17. For what kind of system would you use the observer canonical form? 18. Describe state-vector transformations from the perspective of different bases. 19. What is the definition of an eigenvector? 20. Based upon your definition of an eigenvector, what is an eigenvalue? 21. What is the significance of using eigenvectors as basis vectors for a system transformation? 1. Signals, systems, summing junctions, pickoff points 2. Cascade, parallel, feedback 3. Product of individual transfer functions, sum of individual transfer functions, forward gain divided by one plus the product of the forward gain times the feedback gain 4. Equivalent forms for moving blocks across summing junctions and pickoff points 5. As K is varied from 0 to ∞, the system goes from overdamped to critically damped to underdamped. When the system is underdamped, the settling time remains constant. 6. Since the real part remains constant and the imaginary part increases, the radial distance from the origin is increasing. Thus the angle θ is increasing. Since ζ= cos θ the damping ratio is decreasing. 7. Nodes (signals), branches (systems) 8. Signals flowing into a node are added together. Signals flowing out of a node are the sum of signals flowing into a node. 9. One 10. Phase-variable form, cascaded form, parallel form, Jordan canonical form, observer canonical form 11. The Jordan canonical form and the parallel form result from a partial fraction expansion. 12. Parallel form 13. The system poles, or eigenvalues 14. The system poles including all repetitions of the repeated roots 15. Solution of the state variables are achieved through decoupled equations. i.e. the equations are solvable individually and not simultaneously. 16. State variables can be identified with physical parameters; ease of solution of some representations 17. Systems with zeros 18. State-vector transformations are the transformation of the state vector from one basis system to another. i.e. the same vector represented in another basis. 19. A vector which under a matrix transformation is collinear with the original. In other words, the length of the vector has changed, but not its angle. 20. An eigenvalue is that multiple of the original vector that is the transformed vector. 21. Resulting system matrix is diagonal

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