Robotics process control book Part 10 docx

The vector mean value of the vector ξt is given as The expression Covξt1, t2 = Eξt1 − µt1ξt2 − µt2T 4.4.67 or is the corresponding auto-covariance matrix of the stochastic process vector

When replacing the arguments t1, t2 in Equations (4.4.58), (4.4.60) by t and τ then

and

If t = τ then

where Covξ(t, t) is equal to the variance of the random variable ξ The abbreviated form Covξ(t) = Covξ(t, t) is also often used

Consider now mutually dependent stochastic processes ξ1(t), ξ2(t), ξn(t) that are elements

of stochastic process vector ξ(t) In this case, the mean values and auto-covariance function are often sufficient characteristics of the process

The vector mean value of the vector ξ(t) is given as

The expression

Covξ(t1, t2) = E(ξ(t1) − µ(t1))(ξ(t2) − µ(t2))T

(4.4.67) or

is the corresponding auto-covariance matrix of the stochastic process vector ξ(t)

The auto-covariance matrix is symmetric, thus

If a stochastic process is normally distributed then the knowledge about its mean value and covariance is sufficient for obtaining any other process characteristics

For the investigation of stochastic processes, the following expression is often used

¯

µ = lim

T →∞

1 2T

Z T

−T

¯

µ is not time dependent and follows from observations of the stochastic process in a sufficiently large time interval and ξ(t) is any realisation of the stochastic process In general, the following expression is used

¯

µm= lim

T →∞

1 2T

Z T

−T

For m = 2 this expression gives ¯µ2

Stochastic processes are divided into stationary and non-stationary In the case of a stationary stochastic process, all probability densities f1, f2, fndo not depend on the start of observations and onedimensional probability density is not a function of time t Hence, the mean value (4.4.55) and the variance (4.4.56) are not time dependent as well

Many stationary processes are ergodic, i.e the following holds with probability equal to one

µ =

Z ∞

−∞

xf1(x)dx = ¯µ = lim

T →∞

1 2T

Z T

−T

The usual assumption in practice is that stochastic processes are stationary and ergodic

The properties (4.4.72) and (4.4.73) show that for the investigation of statistical properties of

a stationary and ergodic process, it is only necessary to observe its one realisation in a sufficiently large time interval

Stationary stochastic processes have a two-dimensional density function f2 independent of the time instants t1, t2, but dependent on τ = t2− t1that separates the two random variables ξ(t1), ξ(t2) As a result, the auto-correlation function (4.4.58) can be written as

Rξ(τ ) = E {ξ(t1)ξ(t2)} =

Z ∞

−∞

Z ∞

−∞

For a stationary and ergodic process hold the equations (4.4.72), (4.4.73) and the expression

E {ξ(t)ξ(t + τ)} can be written as

E {ξ(t)ξ(t + τ)} = ξ(t)ξ(t + τ)

= lim

T →∞

1 2T

Z T

−T

Hence, the auto-correlation function of a stationary ergodic process is in the form

Rξ(τ ) = lim

T →∞

1 2T

Z T

−T

Auto-correlation function of a process determines the influence of a random variable between the times t + τ and t If a stationary ergodic stochastic process is concerned, its auto-correlation function can be determined from any of its realisations

The auto-correlation function Rξ(τ ) is symmetric

For τ = 0 the auto-correlation function is determined by the expected value of the square of the random variable

For τ → ∞ the auto-correlation function is given as the square of the expected value This can easily be proved

Rξ(τ ) = ξ(t)ξ(t + τ ) =

Z ∞

−∞

Z ∞

−∞

For τ → ∞, ξ(t) and ξ(t + τ) are mutually independent Using (4.4.53) that can be applied to a stochastic process yields

Rξ(∞) =

Z ∞

−∞

x1f (x1)dx1

Z ∞

−∞

The value of the auto-correlation function for τ = 0 is in its maximum and holds

The cross-correlation function of two mutually ergodic stochastic processes ξ(t), η(t) can be given as

or

Rξη(τ ) =

Z ∞

−∞

Z ∞

−∞

x1y2f2(x1, y2, τ )dx1dy2

= lim

T →∞

1 2T

Z T

Consider now a stationary ergodic stochastic process with corresponding auto-correlation func-tion Rξ(τ ) This auto-correlation function provides information about the stochastic process in the time domain The same information can be obtained in the frequency domain by taking the Fourier transform of the auto-correlation function The Fourier transform Sξ(ω) of Rξ(τ ) is given as

Sξ(ω) =

Z ∞

−∞

Correspondingly, the auto-correlation function Rξ(τ ) can be obtained if Sξ(ω) is known using the inverse Fourier transform

Rξ(τ ) = 1

2π

Z ∞

−∞

Rξ(τ ) and Sξ(ω) are non-random characteristics of stochastic processes Sξ(ω) is called power spectral density of a stochastic process This function has large importance for investigation of transformations of stochastic signals entering linear dynamical systems

The power spectral density is an even function of ω:

For its determination, the following relations can be used

Sξ(ω) = 2

Z ∞ 0

Rξ(τ ) = 1

π

Z ∞ 0

The cross-power spectral density Sξη(ω) of two mutually ergodic stochastic processes ξ(t), η(t) with zero means is the Fourier transform of the associated cross-correlation function Rξη(τ ):

Sξη(ω) =

Z ∞

−∞

The inverse relation for the cross-correlation function Rξη(τ ) if Sξη(ω) is known, is given as

Rξη(τ ) = 1

2π

Z ∞

−∞

If we substitute in (4.4.75), (4.4.85) for τ = 0 then the following relations can be obtained

Eξ(t)2 = Rξ(0) = lim

T →∞

1 2T

Z T

−T

Eξ(t)2 = Rξ(0) = 1

2π

Z ∞

−∞

Sξ(ω)dω = 1

π

Z ∞ 0

The equation (4.4.91) describes energetical characteristics of a process The right hand side of the equation can be interpreted as the average power of the process The equation (4.4.92) determines the power as well but expressed in terms of power spectral density The average power is given by the area under the spectral density curve and Sξ(ω) characterises power distribution of the signal according to the frequency For Sξ(ω) holds

4.4.4 White Noise

Consider a stationary stochastic process with a constant power spectral density for all frequencies

This process has a “white” spectrum and it is called white noise Its power spectral density is shown in Fig.4.4.4a From (4.4.92) follows that the average power of white noise is indefinitely large, as

Eξ(t)2 = 1

πV

Z ∞ 0

Therefore such a process does not exit in real conditions

The auto-correlation function of white noise can be determined from (4.4.88)

Rξ(τ ) = 1

π

Z ∞ 0

where

δ(τ ) = 1

π

Z ∞ 0

because the Fourier transform of the delta function Fδ(jω) is equal to one and the inverse Fourier transform is of the form

δ(τ ) = 1

2π

Z ∞

−∞

Fδ(jω)ejωτdω

2π

Z ∞

−∞

ejωτdω

2π

Z ∞

−∞

cos ωτ dω + j 1

2π

Z ∞

−∞

sin ωτ dω

π

Z ∞ 0

The auto-correlation function of white noise (Fig.4.4.4b) is determined by the delta function and

is equal to zero for any non-zero values of τ White noise is an example of a stochastic process where ξ(t) and ξ(t + τ ) are independent

A physically realisable white noise can be introduced if its power spectral density is constrained Sξ(ω) = V for |ω| < ω1

The associated auto-correlation function can be given as

Rξ(τ ) = V

π

Z ω 1

0

cos ωτ dω = V

The following relation also holds

¯

µ2= D = V

2π

Z ω 1

−ω 1

dω = V ω1

Sometimes, the relation (4.4.94) is approximated by a continuous function Often, the following relation can be used

Sξ(ω) = 2aD

The associated auto-correlation function is of the form

Rξ(τ ) = 1

2π

Z ∞

−∞

2aD

The figure4.4.5depicts power spectral density and auto-correlation function of this process The equations (4.4.102), (4.4.103) describe many stochastic processes well For example, if a  1, the approximation is usually “very” good

Sξ Rξ

V

a)

b)

V δ(τ )

τ 0

Figure 4.4.4: Power spectral density and auto-correlation function of white noise

Sξ Rξ

a)

b)

τ 0

6

?

6

? 2D/a D

Figure 4.4.5: Power spectral density and auto-correlation function of the process given by (4.4.102)

and (4.4.103)

4.4.5 Response of a Linear System to Stochastic Input

Consider a continuous linear system with constant coefficients

dx(t)

where x(t) = [x1(t), x2(t), , xn(t)]T is the state vector, ξ(t) = [ξ1(t), ξ2(t), , ξm(t)]T is a stochastic process vector entering the system A, B are n×n, n×m constant matrices, respectively The initial condition ξ0is a vector of random variables

Suppose that the expectation E[ξ0] and the covariance matrix Cov(ξ0) are known and given as

Further, suppose that ξ(t) is independent on the initial condition vector ξ0 and that its mean value µ(t) and its auto-covariance function Covξ(t, τ ) are known and holds

E[(ξ(t) − µ(t))(ξ(τ) − µ(τ))T] = Covξ(t, τ ), for t ≥ 0, τ ≥ 0 (4.4.109)

As ξ0 is a vector of random variables and ξ(t) is a vector of stochastic processes then x(t)

is a vector of stochastic processes as well We would like to determine its mean value E[x(t)], covariance matrix Covx(t) = Covx(t, t), and auto-covariance matrix Covx(t, τ ) for given ξ0 and ξ(t)

Any stochastic state trajectory can be determined for given initial conditions and stochastic inputs as

x(t) = Φ(t)ξ0+

Z t

where Φ(t) = eAt is the system transition matrix

Denoting

then the following holds

¯

x(t) = Φ(t)x0+

Z t

This corresponds with the solution of the differential equation

d¯x(t)

with initial condition

¯

To find the covariance matrix and auto-correlation function, consider at first the deviation

x(t) − ¯x(t):

x(t) − ¯x(t) = Φ(t)[ξ0− ¯x0] +

Z t

It is obvious that x(t) − ¯x(t) is the solution of the following differential equation

dx(t)

dt −d¯x(t)dt = A[x(t) − ¯x(t)] + B[ξ(t) − µ(t)] (4.4.117) with initial condition

From the equation (4.4.116) for Covx(t) follows

Covx(t) = E[(x(t) − ¯x(t))(x(t) − ¯x(t))T]

= E



Φ(t)[ξ0− x0] +

Z t

0 Φ(t − α)B[ξ(α) − µ(α)]dα



×

×



Φ(t)[ξ0− x0] +

Z t

0 Φ(t − β)B[ξ(β) − µ(β)]dβ

T

(4.4.119) and after some manipulations,

Covx(t) = Φ(t)E[(ξ0− x0)(ξ0− x0)T]ΦT(t)

+

Z t

0

Φ(t)E[(ξ0− x0)(ξ(β) − µ(β))T]BTΦT(t − β)dβ

+

Z t

0 Φ(t − α)BE[(ξ(α) − µ(α))(ξ0− x0)T]ΦT(t)dα

+

Z t

0

Z t

0 Φ(t − α)BE[(ξ(α) − µ(α))(ξ(β) − µ(β))T]BTΦT(t − β)dβdα (4.4.120) Finally, using equations (4.4.107), (4.4.109), (4.4.110) yields

Covx(t) = Φ(t)Cov0ΦT(t)

+

Z t 0

Z t

0 Φ(t − α)BCovξ(α, β)BTΦT(t − β)dβdα (4.4.121) Analogously, for Covx(t, τ ) holds

Covx(t, τ ) = Φ(t)Cov0ΦT(t)

+

Z t 0

Z τ

0 Φ(t − α)BCovξ(α, β)BTΦT(τ − β)dβdα (4.4.122) Consider now a particular case when the system input is a white noise vector, characterised by E[(ξ(t) − µ(t))(ξ(τ) − µ(τ))T] = V (t)δ(t − τ)

for t ≥ 0, τ ≥ 0, V (t) = VT(t) ≥ 0 (4.4.123) The state covariance matrix Covx(t) can be determined if the auto-covariance matrix Covx(t, τ )

of the vector white noise ξ(t)

is used in the equation (4.4.121) that yields

Covx(t) = Φ(t)Cov0ΦT(t)

+

Z t 0

Z t

0 Φ(t − α)BV (α)δ(α − β)BTΦT(t − β)dβdα (4.4.125)

= Φ(t)Cov0ΦT(t) +

Z t

The covariance matrix Covx(t) of the state vector x(t) is the solution of the matrix differential equation

dCovx(t)

dt = ACovx(t) + Covx(t)A

with initial condition

The auto-covariance matrix Covx(t, τ ) of the state vector x(t) is given by applying (4.4.124)

to (4.4.122) After some manipulations follows

Covx(t, τ ) = Φ(t − τ)Covx(t) for t > τ

If a linear continuous system with constant coefficients is asymptotically stable and it is ob-served from time −∞ and if the system input is a stationary white noise vector, then x(t) is a stationary stochastic process

The mean value

is the solution of the equation

where µ is a vector of constant mean values of stationary white noises at the system input The covariance matrix

is a constant matrix and is given as the solution of

where V is a symmetric positive-definite constant matrix defined as

The auto-covariance matrix

E[(x(t1) − ¯x)(x(t2) − ¯x)T] = Covx(t1, t2) ≡ Covx(t1− t2, 0) (4.4.135)

is in the case of stationary processes dependent only on τ = t1− t2 and can be determined as Covx(τ, 0) = eAτCovx for τ > 0

Example 4.4.1: Analysis of a first order system

Consider the mixing process example from page 67given by the state equation

dx(t)

where x(t) is the output concentration, ξ(t) is a stochastic input concentration, a = −1/T1,

b = 1/T1, and T1= V /q is the time constant defined as the ratio of the constant tank volume

V and constant volumetric flow q

Suppose that

where ξ0 is a random variable

G(s)

Figure 4.4.6: Block-scheme of a system with transfer function G(s)

Further assume that the following probability characteristics are known

E[ξ0] = x0

E[(ξ0− x0)2] = Cov0

E[(ξ(t) − µ)(ξ(τ) − µ)] = V δ(t − τ) for t, τ ≥ 0

E[(ξ(t) − µ)(ξ0− x0)] ≡ 0 for t ≥ 0

The task is to determine the mean value E[x(t)], variance Covx(t), and auto-covariance function in the stationary case Covx(τ, 0)

The mean value E[x(t)] is given as

¯

x = eatx0− b

a 1 − eat
µ

As a < 0, the output concentration for t → ∞ is an asymptotically stationary stochastic process with the mean value

¯∞= −abµ

The output concentration variance is determined from (4.4.126) as

Covx(t) = e2atCov0− b

2 2a 1 − e2at
V Again, for t → ∞ the variance is given as

lim

t→∞Covx(t) = −b

2V 2a The auto-covariance function in the stationary case can be written as

Covx(τ, 0) = −ea|τ |b

2V 2a

4.4.6 Frequency Domain Analysis of a Linear System with Stochastic

Input

Consider a continuous linear system with constant coefficients (Fig 4.4.6) The system response

to a stochastic input signal is a stochastic process determined by its auto-correlation function and power spectral density The probability characteristics of the stochastic output signal can be found if the process input and system characteristics are known

Let u(t) be any realisation of a stationary stochastic process in the system input and y(t) be the associated system response

y(t) =

Z ∞

where g(t) is the impulse response The mean value of y(t) can be determined in the same way as E[y(t)] =

Z ∞

Analogously to (4.4.140) which determines the system output in time t, in another time t + τ holds y(t + τ ) =

Z ∞

−∞

The auto-correlation function of the output signal is thus given as

Ryy(τ ) = E[y(t)y(y + τ )]

= E

Z ∞

−∞g(τ1)u(t − τ1)dτ1

 Z ∞

−∞g(τ2)u(t + τ − τ2)dτ2



(4.4.143) or

Ryy(τ ) =

Z ∞

−∞

Z ∞

−∞

g(τ1)g(τ2)E[u(t − τ1)u(t + τ − τ2)]dτ1dτ2 (4.4.144)

As the following holds

E[u(t − τ1)u(t + τ − τ2)] = E [u(t − τ1)u{(t − τ1) + (τ + τ1− τ2)}] (4.4.145) then it follows, that

Ryy(τ ) =

Z ∞

−∞

Z ∞

−∞

where Ruu(τ + τ1− τ2) is the input auto-correlation function with the argument (τ + τ1− τ2). The mean value of the squared output signal is given as

y2(t) = Ryy(0) =

Z ∞

−∞

Z ∞

−∞

The output power spectral density is given as the Fourier transform of the associated auto-correlation function as

Syy(ω) =

Z ∞

−∞

Ryy(τ )e−jωτdτ

=

Z ∞

−∞

Z ∞

−∞

Z ∞

−∞

g(τ1)g(τ2)Ruu[τ + (τ1− τ2)]e−jωτdτ1dτ2dτ (4.4.148) Multiplying the subintegral term in the above equation by (ejωτ 1e−jωτ 2)(e−jωτ 1ejωτ 2) = 1 yields Syy(ω) =

Z ∞

−∞

g(τ1)ejωτ 1dτ1

Z ∞

−∞

g(τ2)e−jωτ 2dτ2

Z ∞

−∞

Ruu[τ + (τ1− τ2)]e−jω(τ +τ 1 −τ 2 )dτ (4.4.149) Now we introduce a new variable τ0= τ + τ1− τ2, yielding

Syy(ω) =

Z ∞

−∞

g(τ1)ejωτ1dτ1

 Z ∞

−∞

g(τ2)e−jωτ2dτ2

 Z ∞

−∞

Ruu(τ0)e−jωτ0dτ0

 (4.4.150) The last integral is the input power spectral density

Suu(ω) =

Z ∞

−∞

Ruu(τ0)e−jωτ 0

The second integral is the Fourier transform of the impulse function g(t), i.e it is the frequency transfer function of the system

G(jω) =

Z ∞

−∞

Finally, the following holds for the first integral

G(−jω) =

Z ∞