Vulnerabilidad de las zonas urbanas en el contexto del cambio climático

Having studied the intrinsic system property of stability, we now reintroduce the input term in the first equation (1.52) and seek its formal solution, which is similar to Equation 1.47 with the difference that in place of the summation, we now have a convolution integral representing the input contribution

x

( )

t ⁼unforced responsee^Atx^{0 +}

∫

0^te^{A(t− )}bu d

input forcing

 ^σ ( )σ σ



 (1.63)

As with discrete-time LTI system, linearity allows for the superposition of the effects, so that the first contribution, controlled by the initial conditions x0 and the matrix A, is summed to the input contribution, represented by the convolution integral. Equation 1.63, like its discrete-time version Equation 1.47, is conceptually important to investigate the system properties, but is hardly ever used to determine the system response in practice because the input u may not have a simple mathemati-cal form, and therefore, the integral may not have an analytimathemati-cal solution. As will be fully treated in the following, in most cases numerical methods will be preferred to simulate the system behaviour.

However, in very simple cases (e.g. a constant input), Equation 1.63 can actually be solved. Let us consider a single-input system with a constant value u₀. In this case, the convolution integral in Equation 1.63 can be handled via Taylor series expansion of the transition matrix, whereby each term can be separately integrated

e u d e e u d e e u d

Now, consider the even simpler case in which the system matrix A reduces to a scalar, that is, A = −a. Thus, the system dynamics reduces to the scalar differential equation

k = 1 Exponentially stable Exponentially unstable Simply unstable

t t t

if σ < 0 if σ > 0 if σ = 0

FIGURE 1.29 Response produced by an eigenvalue of multiplicity k, depending on its real part.

x= − ⋅ + ⋅a x b u0 (1.65) Assuming that the initial condition is zero, that is, x0= , the system (1.65) can be thought to be 0 driven abruptly from a zero steady state

(

x =0 to another steady state determined by the constant

)

input u0 applied at t = +0 . This situation is often referred to as step response, meaning that the input abruptly changes from 0 to u0 in a step-wise mode. The first term in Equation 1.63 vanishes, and the convolution integral becomes

The response of this system is graphically presented in Figure 1.30, showing that the steady-state value depends not only on the input value u₀ but also on the system parameters a and b, whereas the speed with which the asymptotic value is reached depends only on a, the system time constant, as shown by Equation 1.66.

Now consider again the system (1.65), but this time the input is represented by a Dirac δ

( )

t impulse, which can be defined as follows:

( ( )

As shown in Figure 1.31, the Dirac impulse can be viewed as the limit of a distribution of rectan-gular pulses of duration a and amplitude 1/a. When a tends to zero, the amplitude tends to infinity but the strength of the impulse (height times duration) is still unit.

Basically, the Dirac impulse is non-zero only at t = 0 and has unit area. Now suppose again that the system starts from zero initial condition, so that its evolution is entirely determined by the convolution integral, which can be solved by using the above properties (1.67) of the Dirac pulse

FIGURE 1.30 Response of the system (1.65) to a constant input u0 starting with zero initial condition.

Therefore, the state and output of the system are

By inspection of Equation 1.69, it appears that the application of a Dirac impulse is equivalent to altering the initial system condition from 0 to b, the input matrix. From then on, the subsequent evolution is that of an autonomous system starting from the new initial condition b.

1.8.3.1 An Application of an Impulse-Drive Linear Model: The Nash Reservoir

A simple yet effective linear model is due to the famous mathematician John Nash, who inspired the successful movie A Beautiful Mind starring Russell Crowe in the title role and who was killed in a car crash in May 2015. Nash developed a simple model of a reservoir by cascading a certain number of elementary reservoirs (Rinaldi et al., 1979), as illustrated in Figure 1.32. The system states are the volumes of each tank and the outflow is proportional to the last volume, that is, q k x= ⋅ . The n

dynamics of each reservoir is given by the difference between the incoming and outgoing flows, similar to the example of Section 1.3.1

x k xi= ⋅ i−1− ⋅k xi (1.70)

The system representation of the ensemble of n tanks in series is therefore



FIGURE 1.31 The Dirac impulse can be viewed as a distribution of pulses of constant unit area, but decreasing duration.

To obtain the impulse response of the system (1.71), we can use the result of Equation 1.69 and ini-tialize the system with the initial condition x

( )

0 = appearing in Equation 1.71. Considering as an b example, the cascade of five reservoirs, the state impulse response is shown in Figure 1.32, whereas the output variations for several values of the reservoir constant k are shown in Figure 1.33. The analysis of the system can be carried a step further. Consider again the output impulse response in the form of the second Equation 1.69 and notice that the matrix A of the system (1.71) is lower trian-gular; therefore, it has n eigenvalues, all equal to −k. Thus, we can express the impulse response as

y t e

T n t

T e T

k

t

n t

( )

^{= =}

(

₋

)

^  T

 =

− −

c ^A b 1 1

1 1

! where (1.72)

The time of the maximum flow can be computed by setting the output derivative to zero and solving for t to yield

tmax=

(

n−^{1 (1.73)}

)

T

This result will be considered again in Chapter 6, when dealing with the impulse response of real reactors. As a concluding remark of this section, analysing the system response to apparently

0 50 100 150 200 250 300 350 400

0 10 20 30 40 50 60 70 80 90 100

Time (h)

Volume (m3)

x1

x₂ x₃ x4

x5

u(t) x1

x₂ x₃

x₄

x₅ q= kx₅

FIGURE 1.32 Schematic representation of the Nash reservoir model, consisting of a number of cascaded tanks. The volume variations in response to a flow pulse are shown in the left-hand diagram.

0 100 200 300 400 500

k = 0.04 k = 0.03 k = 0.02 k = 0.01

600 700 800 900 1000

0 0.02 0.04 0.06 0.08

Time (h)

Flow (m

3 s−1)

FIGURE 1.33 Impulse response of the Nash reservoir model for several values of the flow constant k.

unrealistic inputs might appear to be an arid academic exercise. The contrary is in fact true because step and impulse inputs (or their real-world approximations) will be often encountered in practice, and the notions learned here will undoubtedly become useful in due course.

In document Reflexiones a propósito de la revisión del Plan General de Madrid (página 187-192)