Lesson 10: I/O representation (Deterministic and Stochastic)

Systems and Control Theory Lecture Notes

Laura Giarr´ e

Lesson 10: I/O representation (Deterministic and Stochastic)

 I/O representation

 CT case

 DT case

 AR(n)

 MA(m)

 ARMA(n,m)

 From I/O to state representation

I/O representation (SISO p = m = 1)

 Let us define an operator: D  _dt ^d (TC) and D  Forward shift operator (TD).

 Then, D ^k y(t) = ^d

_dt ^y(t)

and D ⁰ y(t) = y(t) in the CT case, or Dy(t) = y(t + 1), D ^k y(t) = y(t + k) in the DT case.

 Then, the transfer function H(s) (CT) or H(z) (DT) can be rewritten substituting s or z with the operator D or (z ⁻¹ with D ⁻¹ ):

H(D) = B(D)

A(D) = b ₀ D ^m + b 1 D ^m−1 + . . . + b m−1 D + b m

D ⁿ + a ₁ D ⁿ⁻¹ + . . . + a _n−1 D + a _n

I/O representation

 Correspondingly, the relation between input and output (zero i.c. or steady state) is:

A(D)y(t) = B(D)u(t)

I/O representation

 (CT) A differential equation:

(D ⁿ + a 1 D ⁿ⁻¹ + . . . + a n−1 D + a n )y(t) (b ₀ D ^m + b ₁ D ^m−1 + . . . + b _m−1 D + b _m )u(t)

 Such that:

d ⁿ y(t)

dt ⁿ + a ₁ d ⁿ⁻¹ y(t)

dt ⁿ⁻¹ + . . . + a _n y(t) = b ₀ d ^m u(t)

dt ^m + b ₁ d ^m−1 u(t)

dt ^m−1 + . . . + b _m u(t)

I/O representation (DT)

 (DT) A differential equation:

y(t + n) + a ₁ y(t + n − 1) + . . . + a _n y(t) = b ₀ u(t + m) + b ₁ u(t + m − 1) + . . . + b _m u(t)

 If we consider the Backward operator, D ⁻¹ then:

y(t) + a ₁ y(t − 1) + . . . + a _n y(t − n) =

b ₀ u(t) + b ₁ u(t − 1) + . . . + b _m u(t − m)

From I/O to I/S/0: Realization _

case 1: deg B = n − 1 < deg A = n ( Strictly proper)

 A(D) = D ⁿ + a ₁ D ⁿ⁻¹ + . . . + a _n−1 D + a _n , B(D) = b ₁ D ^m−1 + . . . + b n−1 D + b n

 where A(D)y(t) = B(D)u(t),

 We define a intermediate state as ξ(t) = _A(D) ^u(t) and then y(t) = B(D)ξ(t)

 This leads to a realization in state space called the Reachability Canonical Form:

A =

⎡

⎢ ⎢

⎣

0 1 0 . . . 0

0 0 1 . . . 0

0 0 . . . . . . 0

−a n −a n−1 −a n−2 . . . −a 1

⎤

⎥ ⎥

⎦ B =

⎡

⎢ ⎢

⎢ ⎢

⎣ 0

.. . .. . 1

⎤

⎥ ⎥

⎥ ⎥

⎦

C = [b _n b _n−1 . . . b ₁ ], D = 0

From I/O to I/S/0: Realization

 case 2: deg B = deg A = n ( proper)

 Step 1: rewrite the transfer function as H(D) = _A(D) ^¯B(D) + β ₀

 Then the realization is the same for the matrices A and B, but

 C = [β _n β _n−1 . . . β ₁ ], D = β ₀

 Being β _i the coefficients of polynomial ¯ B(D)

AR(n) processes

 Let e(k) be a stochastic process

 Assume that H(D) = _A(D ¹

)

 where A(D ⁻¹ ) = 1 +  _n

i=1 a _i D ⁻ⁱ that is to say A(z ⁻¹ ) = 1 +  _n

i=1 a _i z ⁻ⁱ is a polynomial with zeros such that

|z _i | < 1 and H(z) = _A(z ¹

)

 The process y(k) = H(z)e(k) satisfies the recursive equation Y (k) + a ₁ Y (k − 1) + . . . + a _n Y (k − n) = e(k)

 Y (k) is called an Auto Regressive process of order n, AR(n).

 The covariance function R _y (l) satisfies the recursion:

R _y (l) + a 1 R _y (l − 1) + ... + a n R _y (l − n) = 0, forl > n

MA(m) processes

 Assume that H(z) = C (z ⁻¹ )

 with C (z ⁻¹ ) = 1 +  _n

i=1 c _i z ⁻ⁱ

 The process y(k) = H(z)e(k) satisfy the recursive equation Y (k) = e(k) + c ₁ e(k − 1) + . . . + c _m e(k − m)

 Y (k) is called a Moving Average process of order m, MA(m).

 The covariance function R _y (l) is given by:

R _y (l) =

c _l + c ₁ c _l+1 + . . . c _m−l c _m , for l ≤ m

0 for l > m

ARMA(n, m) processes

 We prefer AR processes because 1)more parsimonious, 2) easier to evaluate the covariance function.

 For a complete description of processes with rational spectrum we need to introduce the general description:

H(z) = C (z ⁻¹ ) A(z ⁻¹ )

 Y (k) satisfies the recursive equation:

Y (k) + a ₁ Y (k − 1) + . . . + a _n Y (k − n) = e(k) + c ₁ e(k − 1) + . . . + c _m e(k − m)

 Y (k) is called an Auto Regressive Moving Average process of

order n, m, ARMA(n, m)

Thanks