Modern Control Systems Matthew M. Peet Illinois Institute of Technology Lecture 18: Linear Causal Time-Invariant Operators
Operators L 2 and ˆL 2 space Because L 2 (, ) and ˆL 2 are isomorphic, so are the sets of operators L(L 2 ) and L(ˆL 2 ). Prove using the map M φmφ 1 and ˆM φ 1 ˆMφ. How to parameterize L(L 2 )? M. Peet Lecture 18: 2 / 16
ˆL We now define the new space Definition 1. Let ˆL (ır) be the space of matrix-valued functions Ĝ : ır Cm n such that Ĝ ˆL = Ĝ = ess sup σ(ĝ(ıω)) < ω R Every element of ˆL defines a multiplication operator. Definition 2. Given Ĝ ˆL (ır), define (MĜû)(ıω) = Ĝ(ıω)û(ıω) M. Peet Lecture 18: 3 / 16
ˆL Every multiplication operator defined by ˆL is a bounded linear operator. Proposition 1. For any Ĝ ˆL (ır), MĜ ˆL. Furthermore MĜ ˆL = Ĝ ˆL Proof. Sufficiency is easy. MĜû 2ˆL2 = = = sup ω (MĜû)(ıω) (MĜû)(ıω)dω û(ıω) Ĝ(ıω) Ĝ(ıω)û(ıω)dω sup Ĝ(ıω) 2 û(ıω) 2 dω ω Ĝ(ıω) 2 û(ıω) 2 dω = Ĝ 2ˆL û 2ˆL2 M. Peet Lecture 18: 4 / 16
ˆL Because the Fourier Transform is unitary, MĜ also defined an operator in L with equivalent norm. If G = φ 1 MĜφ, then G L(L2) = MĜ L( ˆL2) = Ĝ ˆL. Question: For every G L does there exist some Ĝ L such that G = φ 1 MĜφ M. Peet Lecture 18: 5 / 16
Time-Invariant Systems To answer the previous question affirmatively, we must consider a subspace of linear operators. Definition 3. Define the shift operator S τ : L 2 (, ) L 2 (, ) by (S τ u)(t) = u(t τ) Also called the delay operator Well defined on both L 2 (, ) and L 2 [0, ). Invertible on L 2 (, ) but not on L 2 [0 ). The shift operator can be defined by a multiplication operator S τ = φ 1 MŜφ where Ŝ(ıω) = e ıωt M. Peet Lecture 18: 6 / 16
Time-Invariant Systems Definition 4. An operator Q is Time-Invariant if S τ Q = QS τ for all τ > 0. (Qu)(t τ) = Q(S τ u)(t) Initial time doesn t matter. Identical signals applied at different times will produce the same output. Shifting the input shifts the output. M. Peet Lecture 18: 7 / 16
Time-Invariant Systems Most Systems are Time-Invariant Any state-space system is time-invariant ẋ(t) = Ax(t) + Bu(t) Unless of course A varies with time (A(t)) M. Peet Lecture 18: 8 / 16
Time-Invariant Systems Multiplication Operators define Time-Invariant Operators Lemma 5. For any Ĝ, φ 1 MĜφ is a time-invariant operator. Proof. Recall a system is time-invariant if S τ G = GS τ. Examine the first term S τ G = φ 1 MŜφφ 1 MĜφ = φ 1 MŜMĜφ = φ 1 MŜĜ φ = φ 1 MĜŜ φ = φ 1 MŜMŜφ = φ 1 MĜφφ 1 MŜφ = GS τ This works because scalar multiplication commutes (ĜŜ = ŜĜ). M. Peet Lecture 18: 9 / 16
Time-Invariant Systems Time-Invariant Operators define Multiplication Operators More significantly, the converse is also true. Theorem 6. An operator G : L 2 (, ) L 2 (, ) is time-invariant if and only if there exists some Ĝ ˆL such that G = φ 1 MĜφ All LTI systems can be represented using transfer functions in ˆL. Note that not all transfer functions have a state-space representation. (e.g. delay) M. Peet Lecture 18: 10 / 16
Causal Systems The Truncation Operator Linear, Causal, Time-Invariant Systems are those which are well-defined on L 2 [0, ). Definition 7. Define the Truncation Operator P τ : L 2 (, ) L 2 (, ) as { u(t) t τ (P τ u)(t) = 0 t > τ Truncation operator zeros out the signal after time τ. M. Peet Lecture 18: 11 / 16
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(L 2 ) is Causal if for all τ R. P τ GP τ = P τ G M. Peet Lecture 18: 12 / 16
Causal Systems Lemma 9. A linear time-invariant operator, G, is causal if and only if Proof. P 0 GP 0 = P 0 G First note that on L 2 (, ), S τ is an invertible operator. Hence P τ = S τ P 0 S τ : we can shift truncation point to 0, truncate, then shift back. This implies P τ S τ = S τ P 0 (truncation is not a time-invariant operator). We have the following equivalence: G is causal if and only if P τ GP τ = P τ G P τ GP τ S τ = P τ GS τ P τ GS τ P 0 = P τ S τ G P τ S τ GP 0 = S τ P 0 G S τ is invertible G is LTI G is LTI S τ P 0 GP 0 = S τ P 0 G P τ S τ = S τ P 0 P 0 GP 0 = P 0 G S τ is invertible M. Peet Lecture 18: 13 / 16
Causal Systems Corollary 10. If G L(L 2 (, )) is LTI, then G is causal if and only if G : L 2 [0, ) L 2 [0, ) Thus the subspace of Linear Causal Time-Invariant Operators is L(L 2 [0, )) Proof. We first show that 2) 1). Suppose G : L 2 [0, ) L 2 [0, ). For any u L 2 ( ), P 0 u L 2 (, 0] = L 2 [0, ). Thus (I P 0 )u L 2 [0, ). Thus G(I P 0 )u L 2 [0, ). Thus P 0 G(I P 0 )u = 0. Thus P 0 G = P 0 GP 0. Hence G is causal. M. Peet Lecture 18: 14 / 16
Causal Systems Corollary 11. If G L(L 2 (, )) is LTI, then G is causal if and only if G : L 2 [0, ) L 2 [0, ) Proof. Now we show that 1) implies 2). Suppose that P 0 G = P 0 GP 0. Then P 0 G(I P 0 )u = 0. Then (I P 0 )G(I P 0 ) = G(I P 0 ). Note that for u L 2 [0, ), we have (I P 0 )u = u. Thus for u L 2 [0, ), Gu = G(I P 0 )u = (I P 0 )G(I P 0 )u L 2 [0, ) since (I P 0 ) is the projection onto L 2 [0, ) M. Peet Lecture 18: 15 / 16
Summary An LTI operator is causal iff it maps L 2 [0, ) L 2 [0, ) Any LTI operator is defined by a multiplication operator (transfer function). Which multiplication operators map H 2 H 2 (causal operators) Which operators define causal systems? M. Peet Lecture 18: 16 / 16