Continuous-time system

Ingegneria dell ’Automazione - Sistemi in Tempo Reale

Selected topics on discrete-time and sampled-data systems

Luigi Palopoli

palopoli@sssup.it - Tel. 050/883444

Outline

Sample-rate selection Shannon theorem

smoothness of responses Noise rejection

Robustness

Multirate systems

Block-diagrams

Continuous-time system

Let the closed loop equation of the system be:

˙x = F x + G₁w where w is a zero mean white noise.

Let the power spectral sensity of w be Rw

The covariance of w is given by

E[w(t)w^T(t + τ )] = R_wδ(t)

Assume that w is incorrelated with x

Continuous-time system

Let the closed loop equation of the system be:

˙x = F x + G₁w where w is a zero mean white noise.

Let the power spectral sensity of w be Rw

The covariance of w is given by

E[w(t)w^T(t + τ )] = R_wδ(t)

Assume that w is incorrelated with x

Continuous-time system

Let the closed loop equation of the system be:

˙x = F x + G₁w where w is a zero mean white noise.

Let the power spectral sensity of w be Rw

The covariance of w is given by

E[w(t)w^T(t + τ )] = R_wδ(t) Assume that w is incorrelated with x

Continuous-time system

Let the closed loop equation of the system be:

˙x = F x + G₁w where w is a zero mean white noise.

Let the power spectral sensity of w be Rw

The covariance of w is given by

E[w(t)w^T(t + τ )] = R_wδ(t) Assume that w is incorrelated with x

CT system (Mean)

If the initial state has nonzero mean then

dm

dt = F m m(0) = m₀

where m(t) is the mean of x(t).

CT system (continued)

Introduce P = E{˜x˜x^T }, where ˜x = x − m Intuitive way:

dx˜

dt = F ˜x + G₁w

dx˜ = F ˜xdt + G₁wdt

Expressing variations of xx^T (from now on use x for ˜x):

d xx^T = (x + dx)(x + dx)^T = xdx^T + dxx^T + dxdx^T

CT system (continued)

Introduce P = E{˜x˜x^T }, where ˜x = x − m Intuitive way:

dx˜

dt = F ˜x + G₁w

dx˜ = F ˜xdt + G₁wdt Expressing variations of xx^T (from now on use x for ˜x):

d xx^T = (x + dx)(x + dx)^T = xdx^T + dxx^T + dxdx^T

CT system (continued)

Introduce P = E{˜x˜x^T }, where ˜x = x − m Intuitive way:

dx˜

dt = F ˜x + G₁w

dx˜ = F ˜xdt + G₁wdt

Expressing variations of xx^T (from now on use x for ˜x):

d xx^T = (x + dx)(x + dx)^T = xdx^T + dxx^T + dxdx^T

CT system (continued)

...from which

dxx^T = x(F x)^Tdt + x(G₁w)^Tdt + F xx^T + G₁wx^T + (F x)(F x)^Tdt² + (F x)(G^w₁ )^Tdt²+ +(G₁w)(F x)^Tdt² + (G₁w)(G₁w)^Tdt²

Taking expected values of both sides and considering that w is incorrelated with x, we get

dP

dt = P F^T + F P + G₁E(ww^T)G^T₁ dt

Taking the limit dt → 0 we get:

P˙ = P F^T + F P + G₁R_wG^T₁

CT system (continued)

...from which

dxx^T = x(F x)^Tdt + x(G₁w)^Tdt + F xx^T + G₁wx^T + (F x)(F x)^Tdt² + (F x)(G^w₁ )^Tdt²+ +(G₁w)(F x)^Tdt² + (G₁w)(G₁w)^Tdt²

Taking expected values of both sides and considering that w is incorrelated with x, we get

dP

dt = P F^T + F P + G₁E(ww^T)G^T₁ dt Taking the limit dt → 0 we get:

P˙ = P F^T + F P + G₁RwG^T₁

CT system (continued)

...from which

dxx^T = x(F x)^Tdt + x(G₁w)^Tdt + F xx^T + G₁wx^T + (F x)(F x)^Tdt² + (F x)(G^w₁ )^Tdt²+ +(G₁w)(F x)^Tdt² + (G₁w)(G₁w)^Tdt²

Taking expected values of both sides and considering that w is incorrelated with x, we get

dP

dt = P F^T + F P + G₁E(ww^T)G^T₁ dt Taking the limit dt → 0 we get:

P˙ = P F^T + F P + G₁RwG^T₁

Covariance

Definition r(s, t) = Cov{x(t), x(s)} = E{(x(t) − m(t))^T (x(s) − m(s))}

Let s ≥ t, the system evolution is given by:

x(s) = e^A(s−t)x(t) + R ^t

s e^{A(s−τ )}w(τ )dτ Computing the expectation value and

because x(t) is incorralated from w(t) we get r(s, t) = e^A(s−t)P (t), s ≥ t

Example

First order systems:

dx

dt = f x + g₁w, f < 0

var{x(t₀)} = r₀, mean(x(t₀)) = m(0) = m₀ Mean value:

dm

dt = f m → m(t) = m₀e^f(t−t0) Covariance function

Differential equation, let r1 = g₁²R_w,

P˙ = 2f P + g₁²R_w, P(t₀) = r₀ → P (t) = e^{2f (t−t0)}r₀ + r₁

2f (e^{2f (t−t0)} − 1) Assuming f < 0 and m0 = 0, we get: r(s, t) = _2f^r1e^f|t−s|

Sampled-data systems

Sampled-data equation:

x((k + 1)T ) = Φ(T )x(k) + e(kT ) Φ(t) = R T

0 e^Aτ)dτ e(kT ) = R _(k+1)T

kT eA((k+1)T −τ )G₁w(τ )dτ Mean value Evolution:

m(k + 1) = Φ(T )m(k), m(0) = m₀

Covariance evolution:

rxx(k, h) = cov(x(k), x(h)) = E{˜x(k)˜x(h)^T}, where ˜x = x − m

˜

x((k + 1)T ) = Φ(T )˜x(kT ) + e(kT )

Sampled-data systems

Sampled-data equation:

x((k + 1)T ) = Φ(T )x(k) + e(kT ) Φ(t) = R T

0 e^Aτ)dτ e(kT ) = R _(k+1)T

kT eA((k+1)T −τ )G₁w(τ )dτ Mean value Evolution:

m(k + 1) = Φ(T )m(k), m(0) = m₀ Covariance evolution:

rxx(k, h) = cov(x(k), x(h)) = E{˜x(k)˜x(h)^T}, where ˜x = x − m

˜

x((k + 1)T ) = Φ(T )˜x(kT ) + e(kT )

Sampled-data systems

Sampled-data equation:

x((k + 1)T ) = Φ(T )x(k) + e(kT ) Φ(t) = R T

0 e^Aτ)dτ e(kT ) = R _(k+1)T

kT eA((k+1)T −τ )G₁w(τ )dτ Mean value Evolution:

m(k + 1) = Φ(T )m(k), m(0) = m₀ Covariance evolution:

r_xx(k, h) = cov(x(k), x(h)) = E{˜x(k)˜x(h)^T}, where ˜x = x − m

˜

x((k + 1)T ) = Φ(T )˜x(kT ) + e(kT )

Sampled-data systems

Introduce P (k) = cov(x(k), x(k)) (T implied):

˜

x(k + 1)˜x^T(k + 1) = Φ˜x(k)˜x(k)^TΦ^T + Φ˜x(k)e(k)^T+ +e(k)(Φ˜x(k))^T + e(k)e(k)^T

Taking the expectation and considering that e(k) and x(k) are independent...

P(k + 1) = ΦP (k)Φ^T + E{e(k)e(k)^T} Considering that e(kT ) = R_kT^(k+1)T e^A((k+1)T −τ )G₁w(τ )dτ, we get:

C_d = E(e(kT )e(kT )^T) = Z _T

0

Φ(τ )G₁RwG^T₁ Φ(τ )dτ

To compute the covariance:

˜

x(k + 1)˜x(k)^T = (Φ˜xk + e(k))˜x(k)....

Because e(k) has zero mean and is independent from x(k)...

rxx(k + 1, k) = ΦP (k) from which r_xx(h, k) = Φ^h−kP(k), h ≥ k

Sampled-data systems

Introduce P (k) = cov(x(k), x(k)) (T implied):

˜

x(k + 1)˜x^T(k + 1) = Φ˜x(k)˜x(k)^TΦ^T + Φ˜x(k)e(k)^T+ +e(k)(Φ˜x(k))^T + e(k)e(k)^T

Taking the expectation and considering that e(k) and x(k) are independent...

P(k + 1) = ΦP (k)Φ^T + E{e(k)e(k)^T}

Considering that e(kT ) = R_kT^(k+1)T e^A((k+1)T −τ )G₁w(τ )dτ, we get:

C_d = E(e(kT )e(kT )^T) = Z _T

0

Φ(τ )G₁RwG^T₁ Φ(τ )dτ To compute the covariance:

˜

x(k + 1)˜x(k)^T = (Φ˜xk + e(k))˜x(k)....

Because e(k) has zero mean and is independent from x(k)...

rxx(k + 1, k) = ΦP (k) from which r_xx(h, k) = Φ^h−kP(k), h ≥ k

Sampled-data systems

Introduce P (k) = cov(x(k), x(k)) (T implied):

˜

x(k + 1)˜x^T(k + 1) = Φ˜x(k)˜x(k)^TΦ^T + Φ˜x(k)e(k)^T+ +e(k)(Φ˜x(k))^T + e(k)e(k)^T

Taking the expectation and considering that e(k) and x(k) are independent...

P(k + 1) = ΦP (k)Φ^T + E{e(k)e(k)^T}

Considering that e(kT ) = R_kT^(k+1)T e^A((k+1)T −τ )G₁w(τ )dτ, we get:

C_d = E(e(kT )e(kT )^T) = Z _T

0

Φ(τ )G₁RwG^T₁ Φ(τ )dτ

Example

First order systems:

dx

dt = f x + u + g₁w, f > 0

var{x(t₀)} = r₀, mean(x(t₀)) = m(0) = m₀

Design a discrete-time controller such that the equivalent CT system has a pole at s = s₀

First order systems (ZoH for outputs):

x(k + 1) = e^{f T}x(k) + ^{ef T}_f⁻¹u(k) + R _(k+1)T

kT g₁w(τ )dτ var{0)} = r₀, mean(x(0)) = m(0) = m₀

Choose u(k) = kx(k) such that e^{f T} + ^{(ef T}_f⁻¹⁾ = e^s0T Mean value:

m(k + 1) = e^s0Tm(k), m(0) = m₀ → m(k) = e^s0(k−k0)Tm₀

Example (continued)

Covariance function

Computation of P (k)

P(k + 1) = e^2s0TP(k) + Cd

C_d = R _T

0 e^{2f τ}G²₁R_ddτ = ^G21_2f^Rw(e^{2f T} − 1) = _2f^r1 (e^{2f T} − 1) P(k) = e^2s0(k−k0)Tr₀ + C_d^1−e2s0T (k−k0)

1−e^2s0T

Steady state (k → ∞):

m(k) → 0 P(k) → ^Cd

1−e^2s0T

rxx(k, h) = ^{Cdes0(k−h)T}

1−e^2s0T

Outline

Sample-rate selection Shannon theorem

smoothness of responses Noise rejection

Robustness

Multirate systems

Robustness

Consider a first order system: ˙x = ax + bu We consider a robustness problems

b is known with uncertainity b = ˜b + db

Sample with period T and design so that the closed loop poles are at e^s0T, s0 < 0 DT system:

x((k + 1)T ) = e^aTx(kT ) + e^aT − 1

a bu(kT ) Feedback: u(kT ) = γx(kT ) s.t.

e^aT + γ e^aT − 1

a b = e^s0T

Robustness with respect to

Stability |e^aT + γ ^eaT_a⁻¹(b + db)| ≤ 1 From which:

1 − e^s0T ≥ γ ^eaT_a⁻¹db ≥ −1 − e^s0T γ^eaT_a⁻¹ = ^es0T_b^−eaT

1−e^s0T

e^s0T−e^aT ≥ ^db_b ≥ −_es0T^1+e_−e^s0TaT

The measure of the maximum relative deviation:

µ_db =   

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