Feedback Control of Dynamic Systems Global Edition 8th Edition

Insight into why the solution has certain features and how the system might be modified.

˙y + ky = u = δ(t), with initial condition y(0) = 0.

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.

The flow of water through two tanks.

The thermal capacity of the house, measured in BTU/°F.

If the input is delayed by τ seconds, then the output is also delayed by τ seconds.

To discuss fundamental mathematical tools needed for analysis in the s-plane, frequency response, and state space.

Transforms differential equations into an easier-to-manipulate algebraic form.

It simplifies the evaluation of the convolution integral.

∫₋∞^∞ f(τ)δ(t - τ) dτ = f(t).

It helps to see how well a trial design matches the desired performance.

Pump flow (in cubic-centimeters per minute) to h2.

90,000 BTU/h.

2 °F in 40 minutes.

The cold airflow from the air conditioner produces an equal amount of heat flow q out of each room.

The output response will also be scaled by the same factor.

Its poles and zeros, which indicate system characteristics including frequency response.

We need only find the response to a unit impulse.

Convolution as the response of a system to a series of short pulse (impulse) input signals.

200 g/min.

It causes a change in the output such that y(0+) - y(0-) = 1.

If the input is delayed or shifted in time, the output is unchanged except for being shifted by the same amount.

y(t) = ∫_{-∞}^{∞} h(τ) u(t - τ) dτ.

δ(t), where lim ε → 0 pε(t) = δ(t).

By superposition, the response is the sum of the individual outputs.

y(t) = Σ (from k=0 to ∞) u(kε) h( t - kε).

Because the response of an LTI system to an exponential input is also exponential.

The system output is the result of the convolution of the input signals with the system's impulse response.

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.

y(t) = α1 y1(t) + α2 y2(t).

H(s) is defined as the transfer function of the system.

The impulse and the exponential.

y(t) = ∫_{0}^{t} u(τ) h(t - τ) dτ.

Finding the differential equations relating the flow into the first tank to the flow out of the second tank.

The possibility of system instability.

Linear analysis and computer tools for solving the time response of linear systems.

t - τ = η.

Unit step 1(t) = { 0, t < 0; 1, t ≥ 0 }.

The values of C (thermal capacity) and R (thermal resistance).

H(s) = ∫(−∞ to ∞) h(τ) e^(−sτ) dτ.

A system where the output is not dependent on future inputs.

y(t) = ∫₋∞^∞ u(τ) h(t - τ) dτ.

They help in understanding the system's response through convolution.

C dT_h/dt = Ku - T_h - T_o/R.

It shows that superposition holds for the system modeled by this first-order linear differential equation.

Motion 'd' is proportional to the difference between T_act and T_amb due to thermal expansion.

It becomes more narrow and taller while holding a constant area.

Block diagrams.

Y(s) = ∫_{-∞}^{∞} y(t) e^{-st} dt.

The impulse response depends only on the difference between the time the impulse is applied and the time of observation.

Laplace transform (s-plane), frequency response, and state space analysis.

u(t) = α1 u1(t) + α2 u2(t).

2 °F.

The output is H(s) e^(st), differing from the input only in amplitude H(s).

All rooms are perfect squares, there is no heat flow through floors or ceilings, and the temperature is uniform throughout each room.

y(t) = ∫₀^∞ u(τ) h(t - τ) dτ.

They help manipulate system characteristics in a desired way.

Linear analysis techniques for quick approximations and numerical simulation of nonlinear equations for precise analysis.

Superposition applies if and only if the system is linear.

h(t) = e^(-kt) for t > 0.

If y1(t) is the output caused by u1(t), then y1(t - τ) will be the response to u1(t - τ).

The variable s may be complex, expressed as s = σ + jω, affecting both input and output.

By taking advantage of symmetry to reduce the number to three.

The convolution integral is replaced by a simple multiplication of the transforms, simplifying the analysis.

Routh’s test.

The representation of a general input signal as the sum of short pulses.

It represents the relationship between the input and the impulse response of a system.

If τ = 0 or if k(η + τ) = k = constant.

Decomposing the signal into a sum of elementary components and using superposition.

The Laplace transform.

As the net velocity change of the ball over a very short time period.

It describes the output of a system based on its impulse response and input.

The response is given by the convolution of the impulse response with the input.

It means that the system response starts before the input is applied, indicating a non-causal system.

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.

A rectangular pulse having unit area, defined as p(t) = {1, 0 ≤ t ≤ ε; 0, elsewhere}.

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.