p.3
Dynamic Response and System Analysis
What does linear analysis provide in terms of system response?
Insight into why the solution has certain features and how the system might be modified.
p.9
Differential Equations for Room Temperature Control
What is the differential equation describing the system in Example 3.3?
˙y + ky = u = δ(t), with initial condition y(0) = 0.
p.11
Transfer Functions and Frequency Response
What is the relationship between Y(s), U(s), and H(s) in Laplace transforms?
Y(s) = U(s) H(s), where U(s) is the Laplace transform of the input and H(s) is the Laplace transform of the impulse response.
p.2
Fluid Flow Dynamics in Two-Tank Systems
What is the focus of the laboratory experiment described in Chapter 2?
The flow of water through two tanks.
p.2
Heating Dynamics in Residential Buildings
What does C represent in the heating equation?
The thermal capacity of the house, measured in BTU/°F.
p.6
Dynamic Response and System Analysis
What does time invariance imply about the system's response?
If the input is delayed by τ seconds, then the output is also delayed by τ seconds.
p.4
Dynamic Response and System Analysis
What is the purpose of Chapter 3?
To discuss fundamental mathematical tools needed for analysis in the s-plane, frequency response, and state space.
p.4
Dynamic Response and System Analysis
What does the Laplace transform do?
Transforms differential equations into an easier-to-manipulate algebraic form.
p.10
Transfer Functions and Frequency Response
What is the purpose of the Laplace transform in system analysis?
It simplifies the evaluation of the convolution integral.
p.8
Impulse Response and Convolution
What is the sifting property of the impulse?
∫₋∞^∞ f(τ)δ(t - τ) dτ = f(t).
p.3
Dynamic Response and System Analysis
What is the importance of obtaining the dynamic model of a system in control system design?
It helps to see how well a trial design matches the desired performance.
p.2
Fluid Flow Dynamics in Two-Tank Systems
What parameters are used to compute a linearized model and transfer function in the two-tank system?
Pump flow (in cubic-centimeters per minute) to h2.
p.1
Air Conditioning Systems in High-Rise Buildings
What is the governing factor for the temperature in each room on the fourth floor of the high-rise building?
The cold airflow from the air conditioner produces an equal amount of heat flow q out of each room.
p.6
Dynamic Response and System Analysis
What happens to the output response when the input signal is scaled?
The output response will also be scaled by the same factor.
p.4
Dynamic Response and System Analysis
What can be identified from the transfer function?
Its poles and zeros, which indicate system characteristics including frequency response.
p.8
Impulse Response and Convolution
What does the principle of superposition tell us about system response?
We need only find the response to a unit impulse.
p.7
Convolution and Impulse Response
What does the illustration in Figure 3.1 represent?
Convolution as the response of a system to a series of short pulse (impulse) input signals.
p.9
Dynamic Response and System Analysis
What is the effect of the impulse δ(t) on the system at t = 0?
It causes a change in the output such that y(0+) - y(0-) = 1.
p.5
Linear Time-Invariant Systems (LTIs)
What does it mean for a system to be time invariant?
If the input is delayed or shifted in time, the output is unchanged except for being shifted by the same amount.
p.10
Dynamic Response and System Analysis
What is the integral equation for the output of a dynamic response system?
y(t) = ∫_{-∞}^{∞} h(τ) u(t - τ) dτ.
p.8
Impulse Response and Convolution
What is the mathematical representation of an impulse signal?
δ(t), where lim ε → 0 pε(t) = δ(t).
p.6
Dynamic Response and System Analysis
How is the response to multiple short pulses determined?
By superposition, the response is the sum of the individual outputs.
p.6
Dynamic Response and System Analysis
What is the total response to a series of short pulses at time t?
y(t) = Σ (from k=0 to ∞) u(kε) h( t - kε).
p.4
Linear Time-Invariant Systems (LTIs)
Why are Fourier and Laplace transforms useful in studying LTI systems?
Because the response of an LTI system to an exponential input is also exponential.
p.7
Convolution and Impulse Response
What is the relationship between input signals and system output in convolution?
The system output is the result of the convolution of the input signals with the system's impulse response.
p.5
Linear Time-Invariant Systems (LTIs)
What does the principle of superposition state?
If the system has an input expressed as a sum of signals, the response can be expressed as the sum of the individual responses to those signals.
p.5
Linear Time-Invariant Systems (LTIs)
How is the output expressed when superposition holds?
y(t) = α1 y1(t) + α2 y2(t).
p.11
Transfer Functions and Frequency Response
What does H(s) represent in the context of Laplace transforms?
H(s) is defined as the transfer function of the system.
p.6
Dynamic Response and System Analysis
What are the most common candidates for elementary signals in linear systems?
The impulse and the exponential.
p.10
Dynamic Response and System Analysis
What is the integral expression for causal systems when t = 0?
y(t) = ∫_{0}^{t} u(τ) h(t - τ) dτ.
p.1
Fluid Flow Dynamics in Two-Tank Systems
What is the focus of Problem 2.27 regarding the two-tank fluid-flow system?
Finding the differential equations relating the flow into the first tank to the flow out of the second tank.
p.4
Dynamic Response and System Analysis
What is introduced when feedback is applied to a system?
The possibility of system instability.
p.3
Dynamic Response and System Analysis
What is the focus of the chapter mentioned in the text?
Linear analysis and computer tools for solving the time response of linear systems.
p.9
Linear Time-Invariant Systems (LTIs)
What is the unit-step function defined as?
Unit step 1(t) = { 0, t < 0; 1, t ≥ 0 }.
p.2
Heating Dynamics in Residential Buildings
What are the unknown values to be determined for the house?
The values of C (thermal capacity) and R (thermal resistance).
p.11
Transfer Functions and Frequency Response
What is the integral expression for H(s)?
H(s) = ∫(−∞ to ∞) h(τ) e^(−sτ) dτ.
p.10
Dynamic Response and System Analysis
What is a causal system?
A system where the output is not dependent on future inputs.
p.8
Impulse Response and Convolution
What is the output for a general input in a time-invariant system?
y(t) = ∫₋∞^∞ u(τ) h(t - τ) dτ.
p.7
Convolution and Impulse Response
What is the significance of short pulse inputs in system analysis?
They help in understanding the system's response through convolution.
p.2
Heating Dynamics in Residential Buildings
What is the equation for heating a house as described in the text?
C dT_h/dt = Ku - T_h - T_o/R.
p.5
Linear Time-Invariant Systems (LTIs)
What is the significance of the equation ˙y + ky = u in relation to superposition?
It shows that superposition holds for the system modeled by this first-order linear differential equation.
p.1
Differential Equations for Room Temperature Control
What is the relationship between motion 'd' and the difference between T_act and T_amb?
Motion 'd' is proportional to the difference between T_act and T_amb due to thermal expansion.
p.8
Impulse Response and Convolution
What happens to the basic pulse as ε approaches 0?
It becomes more narrow and taller while holding a constant area.
p.10
Transfer Functions and Frequency Response
What is the Laplace transform of the output y(t)?
Y(s) = ∫_{-∞}^{∞} y(t) e^{-st} dt.
p.8
Impulse Response and Convolution
What characterizes a linear time-invariant (LTI) system?
The impulse response depends only on the difference between the time the impulse is applied and the time of observation.
p.3
Dynamic Response and System Analysis
What are the three domains to study dynamic response?
Laplace transform (s-plane), frequency response, and state space analysis.
p.5
Linear Time-Invariant Systems (LTIs)
What is the mathematical expression for a composite input in the context of superposition?
u(t) = α1 u1(t) + α2 u2(t).
p.11
Transfer Functions and Frequency Response
What happens to the output when the input is of the form e^(st)?
The output is H(s) e^(st), differing from the input only in amplitude H(s).
p.1
Air Conditioning Systems in High-Rise Buildings
What assumptions are made about the rooms in the air conditioning problem?
All rooms are perfect squares, there is no heat flow through floors or ceilings, and the temperature is uniform throughout each room.
p.8
Impulse Response and Convolution
What does the convolution integral represent?
y(t) = ∫₀^∞ u(τ) h(t - τ) dτ.
p.4
Dynamic Response and System Analysis
What is the significance of poles and zeros in system analysis?
They help manipulate system characteristics in a desired way.
p.3
Dynamic Response and System Analysis
What are the two approaches to solving dynamic equations?
Linear analysis techniques for quick approximations and numerical simulation of nonlinear equations for precise analysis.
p.5
Linear Time-Invariant Systems (LTIs)
Under what condition does superposition apply?
Superposition applies if and only if the system is linear.
p.9
Convolution and Impulse Response
What is the solution for the impulse response of the system described?
h(t) = e^(-kt) for t > 0.
p.5
Linear Time-Invariant Systems (LTIs)
How is time invariance mathematically expressed?
If y1(t) is the output caused by u1(t), then y1(t - τ) will be the response to u1(t - τ).
p.11
Transfer Functions and Frequency Response
What is the significance of the variable s in Laplace transforms?
The variable s may be complex, expressed as s = σ + jω, affecting both input and output.
p.1
Differential Equations for Room Temperature Control
How can the number of differential equations for the temperature in each room be reduced?
By taking advantage of symmetry to reduce the number to three.
p.11
Transfer Functions and Frequency Response
What does the convolution integral represent in the context of Laplace transforms?
The convolution integral is replaced by a simple multiplication of the transforms, simplifying the analysis.
p.7
Convolution and Impulse Response
What does Figure 3.2 illustrate?
The representation of a general input signal as the sum of short pulses.
p.9
Convolution and Impulse Response
What is the convolution integral used for in the context of Laplace transforms?
It represents the relationship between the input and the impulse response of a system.
p.6
Dynamic Response and System Analysis
Under what conditions does Eq. (3.3) satisfy Eq. (3.2)?
If τ = 0 or if k(η + τ) = k = constant.
p.6
Dynamic Response and System Analysis
What is the principle used to solve for the response of a linear system to a general signal?
Decomposing the signal into a sum of elementary components and using superposition.
p.8
Impulse Response and Convolution
How can the effect of a collision between a bat and a baseball be summarized?
As the net velocity change of the ball over a very short time period.
p.10
Dynamic Response and System Analysis
What is the significance of the convolution integral in system analysis?
It describes the output of a system based on its impulse response and input.
p.9
Convolution and Impulse Response
How does the impulse response relate to the system's response to a general input?
The response is given by the convolution of the impulse response with the input.
p.10
Dynamic Response and System Analysis
What does it mean if h has values for negative time?
It means that the system response starts before the input is applied, indicating a non-causal system.
p.11
Transfer Functions and Frequency Response
Why must care be taken when using the results of Laplace transforms?
The integrals of the transforms usually do not converge for all values of s and are only defined for a finite region in the s-plane.
p.6
Dynamic Response and System Analysis
What defines a short pulse p(t) in the context of the system?
A rectangular pulse having unit area, defined as p(t) = {1, 0 ≤ t ≤ ε; 0, elsewhere}.
p.4
Linear Time-Invariant Systems (LTIs)
What are the two attributes of linear time-invariant systems (LTIs)?
1. A linear system response obeys the principle of superposition. 2. The response can be expressed as the convolution of the input with the unit impulse response.
