Modern Control Systems - Lecture 18: Linear Causal Time-Invariant ...

Modern Control Systems Matthew M. Peet Illinois Institute of Technology

Lecture 18: Linear Causal Time-Invariant Operators

Operators ˆ 2 space L2 and L

ˆ 2 are isomorphic, so are the sets of operators L(L2 ) Because L2 (−∞, ∞) and L ˆ and L(L2 ). ˆ 7→ φ−1 M ˆ φ. • Prove using the map M 7→ φM φ−1 and M How to parameterize L(L2 )?

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ˆ∞ L We now define the new space

Definition 1. ˆ ∞ (ıR) be the space of matrix-valued functions G ˆ : ıR → Cm×n such that Let L ˆ ˆ = kGk ˆ ∞ = ess sup σ ˆ kGk ¯ (G(ıω)) 0. • (Qu)(t − τ ) = Q(Sτ u)(t) • Initial time doesn’t matter. I

Identical signals applied at different times will produce the same output.

• Shifting the input shifts the output.

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Time-Invariant Systems Most Systems are Time-Invariant

Any state-space system is time-invariant x(t) ˙ = Ax(t) + Bu(t)

• Unless of course A varies with time (A(t))

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Time-Invariant Systems Multiplication Operators define Time-Invariant Operators

Lemma 5. ˆ φ−1 M ˆ φ is a time-invariant operator. For any G, G

Proof. Recall a system is time-invariant if Sτ G = GSτ . Examine the first term Sτ G = φ−1 MSˆ φφ−1 MGˆ φ = φ−1 MSˆ MGˆ φ = φ−1 MSˆGˆ φ = φ−1 MGˆ Sˆ φ = φ−1 MSˆ MSˆ φ = φ−1 MGˆ φφ−1 MSˆ φ = GSτ ˆ Sˆ = SˆG). ˆ This works because scalar multiplication commutes (G M. Peet

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Time-Invariant Systems Time-Invariant Operators define Multiplication Operators

More significantly, the converse is also true.

Theorem 6. An operator G : L2 (−∞, ∞) → L2 (−∞, ∞) is time-invariant if and only if ˆ∈L ˆ ∞ such that there exists some G G = φ−1 MGˆ φ ˆ∞. • All LTI systems can be represented using transfer functions in L • Note that not all transfer functions have a state-space representation. (e.g.

delay)

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Causal Systems The Truncation Operator

Linear, Causal, Time-Invariant Systems are those which are well-defined on L2 [0, ∞).

Definition 7. Define the Truncation Operator Pτ : L2 (−∞, ∞) → L2 (−∞, ∞) as ( u(t) t ≤ τ (Pτ u)(t) = 0 t>τ Truncation operator zeros out the signal after time τ .

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Causal Systems

An operator is causal if changes in the future input don’t create changes in past output. • If the output at time t, y(t) only depends on the input up to time t, u(s),

s ∈ (−∞, t].

Definition 8. An operator G ∈ L(L2 ) is Causal if Pτ GPτ = Pτ G for all τ ∈ R.

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Causal Systems Lemma 9. A linear time-invariant operator, G, is causal if and only if P0 GP0 = P0 G

Proof. First note that on L2 (−∞, ∞), Sτ is an invertible operator. Hence Pτ = Sτ P0 S−τ : we can shift truncation point to 0, truncate, then shift back. • This implies Pτ Sτ = Sτ P0 (truncation is not a time-invariant operator). • We have the following equivalence: G is causal if and only if

⇔ Pτ GPτ = Pτ G

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⇔ Pτ GPτ Sτ = Pτ GSτ

Sτ is invertible

⇔ Pτ GSτ P0 = Pτ Sτ G

G is LTI

⇔ Pτ Sτ GP0 = Sτ P0 G

G is LTI

⇔ Sτ P0 GP0 = Sτ P0 G

Pτ Sτ = Sτ P0

⇔ P0 GP0 = P0 G

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Causal Systems Corollary 10. If G ∈ L(L2 (−∞, ∞)) is LTI, then G is causal if and only if G : L2 [0, ∞) → L2 [0, ∞) Thus the subspace of Linear Causal Time-Invariant Operators is L(L2 [0, ∞))

Proof. We first show that 2) ⇒ 1). Suppose G : L2 [0, ∞) → L2 [0, ∞). • For any u ∈ L2 (−∞), P0 u ∈ L2 (−∞, 0] = L2 [0, ∞)⊥ . • Thus (I − P0 )u ∈ L2 [0, ∞). • Thus G(I − P0 )u ∈ L2 [0, ∞). • Thus P0 G(I − P0 )u = 0. • Thus P0 G = P0 GP0 . • Hence G is causal.

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Causal Systems Corollary 11. If G ∈ L(L2 (−∞, ∞)) is LTI, then G is causal if and only if G : L2 [0, ∞) → L2 [0, ∞)

Proof. Now we show that 1) implies 2). Suppose that P0 G = P0 GP0 . Then P0 G(I − P0 )u = 0. • Then (I − P0 )G(I − P0 ) = G(I − P0 ). • Note that for u ∈ L2 [0, ∞), we have (I − P0 )u = u. • Thus for u ∈ L2 [0, ∞),

Gu = G(I − P0 )u = (I − P0 )G(I − P0 )u ∈ L2 [0, ∞) since (I − P0 ) is the projection onto L2 [0, ∞) M. Peet

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Summary

An LTI operator is causal iff it maps L2 [0, ∞) → L2 [0, ∞) • Any LTI operator is defined by a multiplication operator (transfer function). • Which multiplication operators map H2 → H2 (causal operators) I Which operators define causal systems?

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