Ingegneria dell Automazione - Sistemi in Tempo Reale p.1/28
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1 Ingegneria dell Automazione - Sistemi in Tempo Reale Selected topics on discrete-time and sampled-data systems Luigi Palopoli palopoli@sssup.it - Tel. 050/ Ingegneria dell Automazione - Sistemi in Tempo Reale p.1/28
2 Outline Representation of a sampled-data system Ingegneria dell Automazione - Sistemi in Tempo Reale p.2/28
3 DT models of sampled-data systems Using Z transform u(kt) y(t) y(kt) D/A (ZoH) G(s) A/D G(z) Supposed G(s) = H(s)e sλ, λ can be used to model computation delays The presence of a delay makes CT synthesis much more difficult (infinite dimension) Ingegneria dell Automazione - Sistemi in Tempo Reale p.3/28
4 Example Example: G(s) = e st a with a > 0 s+a Ingegneria dell Automazione - Sistemi in Tempo Reale p.4/28
5 Example Example: G(s) = e st a with a > 0 s+a Let λ = lt mt, with l N, m [0, 1) Ingegneria dell Automazione - Sistemi in Tempo Reale p.4/28
6 Example Example: G(s) = e st a with a > 0 s+a Let λ = lt mt, with l N, m [0, 1) G(z) = (1 z 1 )Z[L 1 [ G(s) s ]] Ingegneria dell Automazione - Sistemi in Tempo Reale p.4/28
7 Example Example: G(s) = e st a with a > 0 s+a Let λ = lt mt, with l N, m [0, 1) G(z) = (1 z 1 )Z[L 1 [ G(s) s ]] G(z) = (1 z 1 )Z[L 1 [e lst ( emst s (1 z 1 )z l Z[L 1 [ emst s emst s+a ] emst s+a )] = Ingegneria dell Automazione - Sistemi in Tempo Reale p.4/28
8 Example Z[ esmt s ] = z z 1 Z[ esmt s+a ] = ze amt z e at G(z) = (1 e amt z+α ), where α = e amt e at z l (z e at ) 1 e amt Ingegneria dell Automazione - Sistemi in Tempo Reale p.5/28
9 Matlab Control toolbox code 10cm Td = 1.5 a= 1 T = 1 sysc = tf(a, [1 a], td,td); sysd = c2d(sysc,t); Ingegneria dell Automazione - Sistemi in Tempo Reale p.6/28
10 Effects of delays Due to the delay l poles arose in the origin a zero arose at α α +0 when m 1 (small delay) α + when m 0 (large delay) And now let s review some concepts... Ingegneria dell Automazione - Sistemi in Tempo Reale p.7/28
11 Zeros and poles Poles are the zeros of the denonimator. A pole p i correspond to a free mode of the system associated with the time function z = p t i Zeros correspond to blocking properties of the the system: z = z i means that there exist initial conditions such that the system has response 0 to the signal z t i Ingegneria dell Automazione - Sistemi in Tempo Reale p.8/28
12 Zeros and poles - I Example: y(t + 1) ay(t) = u(t + 1) bu(t) Computing the Z transform we get: Y (z) = z z a (y 0 u 0 ) + z b z a U(z) u(t) = b t U(z) = Therefore z z b Y (z) = if y 0 = 1 then Y (z) = 0 z z a (y 0 1) Ingegneria dell Automazione - Sistemi in Tempo Reale p.9/28
13 Zeros and poles - II A more subtle example: y(t + 1) ay(t) = u(t + 1) au(t) The transfer function is z a ; is it equivalent to 1? z a Ingegneria dell Automazione - Sistemi in Tempo Reale p.10/28
14 Zeros and poles - II A more subtle example: y(t + 1) ay(t) = u(t + 1) au(t) The transfer function is z a ; is it equivalent to 1? z a NO! The evolution of the system is given by: y(t) = a t y 0 + u(t) Ingegneria dell Automazione - Sistemi in Tempo Reale p.10/28
15 Zeros and poles - II A more subtle example: y(t + 1) ay(t) = u(t + 1) au(t) The transfer function is z a ; is it equivalent to 1? z a NO! The evolution of the system is given by: y(t) = a t y 0 + u(t) Problems are limited only if a < 1 Ingegneria dell Automazione - Sistemi in Tempo Reale p.10/28
16 Non-minimum phase zeros Zeros outside of the unit circle correspond to performance limitations Ingegneria dell Automazione - Sistemi in Tempo Reale p.11/28
17 Non-minimum phase zeros Zeros outside of the unit circle correspond to performance limitations Example: Y (z) = z a U(z) z b with a > 1 Ingegneria dell Automazione - Sistemi in Tempo Reale p.11/28
18 Non-minimum phase zeros Zeros outside of the unit circle correspond to performance limitations Example: Y (z) = z a U(z) z b with a > 1 Suppose you want the signal c k with c < 1 Ingegneria dell Automazione - Sistemi in Tempo Reale p.11/28
19 Non-minimum phase zeros Zeros outside of the unit circle correspond to performance limitations Example: Y (z) = z a U(z) z b with a > 1 Suppose you want the signal c k with c < 1 You get U(z) = z(z b) (z a)(z c) Ingegneria dell Automazione - Sistemi in Tempo Reale p.11/28
20 Non-minimum phase zeros Zeros outside of the unit circle correspond to performance limitations Example: Y (z) = z a U(z) z b with a > 1 Suppose you want the signal c k with c < 1 You get U(z) = z(z b) (z a)(z c) U(z) can be expanded as: U(z) = 1 + a(a b) (a c)(z a) c(c b) (a c)(z c) Ingegneria dell Automazione - Sistemi in Tempo Reale p.11/28
21 Non-minimum phase zeros Zeros outside of the unit circle correspond to performance limitations Example: Y (z) = z a U(z) z b with a > 1 Suppose you want the signal c k with c < 1 You get U(z) = z(z b) (z a)(z c) U(z) can be expanded as: U(z) = 1 + a(a b) (a c)(z a) c(c b) (a c)(z c) u(t) grows unbounded: exact tracking with finite commands is impossible Ingegneria dell Automazione - Sistemi in Tempo Reale p.11/28
22 Root locus No delays Ingegneria dell Automazione - Sistemi in Tempo Reale p.12/28
23 Root locus No delays Ingegneria dell Automazione - Sistemi in Tempo Reale p.12/28
24 Root locus No delays h = 1, m = 0.4 Ingegneria dell Automazione - Sistemi in Tempo Reale p.12/28
25 Root locus No delays h = 1, m = Ingegneria dell Automazione - Sistemi in Tempo Reale p.12/28
26 Stabilising gains No delays: the system is stabilized for all positive gains Ingegneria dell Automazione - Sistemi in Tempo Reale p.13/28
27 Stabilising gains No delays: the system is stabilized for all positive gains In the presence of delays we have bounds on stabilising gains: are you able to find them? Ingegneria dell Automazione - Sistemi in Tempo Reale p.13/28
28 Stabilising gains No delays: the system is stabilized for all positive gains In the presence of delays we have bounds on stabilising gains: are you able to find them? Jury criterion! Ingegneria dell Automazione - Sistemi in Tempo Reale p.13/28
29 Jury stability criterion Consider the stability of A(z) = a 0 z n + a 1 z n a n Write the following table a 0 a 1... a n 1 a n a n a n 1... a 1 a 0 α n = a n a0 a n 1 0 a n a n 1 n 1 a n 1 n 1 a n 1 n 2... a n 1 0 α n 1 = an 1 n 1 a n a 0 0 where a k 1 i = a k i α ka k k i, α k = a k k /ak 0 Ingegneria dell Automazione - Sistemi in Tempo Reale p.14/28
30 The Jury-Criterion Theorem 0 (Jury 1961) If a 0 > 0 then the system has all roots inside the unit circle if and only if all a k 0 with k = 0, 1,..., n 1 are positive. If no a k 0 is zero, then the number of negative a k 0 is equal to the number of roots outside the unit circle. Ingegneria dell Automazione - Sistemi in Tempo Reale p.15/28
31 The two-dimensional case Consider the stability of A(z) = z 2 + a 1 z + a 2 Jurys table is 1 a 1 a 2 a 2 a 1 1 α 2 = a 2 1 a 2 2 a 1 (1 a 2 ) a 1 (1 a 2 ) 1 a 2 2 α 1 = a 1 a a 2 2 a2 1 1 a 2 1+a 2 Imposing conditions in the theorem one gets: a 2 < 1 a 2 > 1 + a 1 a 2 > 1 a 1 Ingegneria dell Automazione - Sistemi in Tempo Reale p.16/28
32 Going back to our example The equation is z 2 + (kω e at )z + kωα, where ω = 1 e amt, α = e amt e at ω applying the Jury criterion, the system is stable iff e at +1 2e amt e at 1 < k < < k < min{ e amt e at, e amt e at 1+e at 2e amt e at 1 } 0 < m < log e at 1 2 at otherwise The stabiliy margin decreases with the introduced dealy Ingegneria dell Automazione - Sistemi in Tempo Reale p.17/28
33 DT models of sampled-data systems Using state space representation Consider a system ẋ = F x + Gu, y = Hx The ZoH equivalent is given by: x((k + 1)T ) = Φx(k) + Γu(k) y = Cx(k) where Φ = e F T, Γ = T 0 ef τ dτg Ingegneria dell Automazione - Sistemi in Tempo Reale p.18/28
34 Example G(s) = 1/s 2 Ingegneria dell Automazione - Sistemi in Tempo Reale p.19/28
35 Example G(s) = 1/s 2 State space representations: ẋ = 0 1 x + 0 u, y = [1, 0]x Ingegneria dell Automazione - Sistemi in Tempo Reale p.19/28
36 Example G(s) = 1/s 2 State space representations: ẋ = 0 1 x + 0 u, y = [1, 0]x Φ = e F T = I + F T +... = 1 T 0 1 Ingegneria dell Automazione - Sistemi in Tempo Reale p.19/28
37 Example G(s) = 1/s 2 State space representations: ẋ = 0 1 x + 0 u, y = [1, 0]x Φ = e F T = I + F T +... = 1 T 0 1 Γ = [IT + F T 2 /2! + F 2 T 3 /3! +...]G = T 2 /2 T Ingegneria dell Automazione - Sistemi in Tempo Reale p.19/28
38 What about delay? Consider the system: ẋ = F x + Gu(t λ), y = Hx The general solution is given by x((k +1)T ) = e F T x(kt )+ (k+1)t kt e F ((k+1)t τ Gu(τ λ)dτ Let λ = lt mt, with l N, m [0, 1) mt Ingegneria dell Automazione - Sistemi in Tempo Reale p.20/28
39 Delays We come up with x((k+1)t ) = e F T x(kt )+Γ 1 u((k l)t )+Γ 2 u((k l+1)t ) where:γ 1 = kt +(1 m)t e F (k+1)t τ Gdτ, kt Γ 2 = (k+1)t kt +(1 m)t ef (k+1)t τ Gdτ with the change of variable: η = (k + 1)T τ we get: Γ 1 = T mt ef η Gdη Γ 2 = mt 0 e F η Gdη Ingegneria dell Automazione - Sistemi in Tempo Reale p.21/28
40 The case l = 1 Let s use x(k) for x(kt ) and u(k) for u(kt ) Ingegneria dell Automazione - Sistemi in Tempo Reale p.22/28
41 The case l = 1 Let s use x(k) for x(kt ) and u(k) for u(kt ) The DT system is described by x(k + 1) = Φx(k) + Γ 1 u(k 1) + Γ 2 u(k) Ingegneria dell Automazione - Sistemi in Tempo Reale p.22/28
42 The case l = 1 Let s use x(k) for x(kt ) and u(k) for u(kt ) The DT system is described by x(k + 1) = Φx(k) + Γ 1 u(k 1) + Γ 2 u(k) Introduce a new state variable z(k) = u(k 1) Ingegneria dell Automazione - Sistemi in Tempo Reale p.22/28
43 The case l = 1 Let s use x(k) for x(kt ) and u(k) for u(kt ) The DT system is described by x(k + 1) = Φx(k) + Γ 1 u(k 1) + Γ 2 u(k) Introduce a new state variable z(k) = u(k 1) The system dynamic equation becomes: x(k + 1) = Φ Γ 1 x(k) + Γ 2 u(k) z(k + 1) 0 0 z(k) 1 y(k) = [H, 0][ x(k) z(k) ] Ingegneria dell Automazione - Sistemi in Tempo Reale p.22/28
44 The case l > 1 The DT system is described by x(k + 1) = Φx(k) + Γ 1 u(k l) + Γ 2 u(k l + 1) Ingegneria dell Automazione - Sistemi in Tempo Reale p.23/28
45 The case l > 1 The DT system is described by x(k + 1) = Φx(k) + Γ 1 u(k l) + Γ 2 u(k l + 1) New state variables z 1 (k) = u(k l) z 2 (k) = u(k l + 1)... z l (k) = u(k 1) Ingegneria dell Automazione - Sistemi in Tempo Reale p.23/28
46 The case l > 1 - I The system dynamic equation becomes: Φ Γ 1 Γ x(k + 1) z 1 (k + 1) x(k) = z 1 (k) 0 + z l (k + 1) u(k) z l (k) x(k) z 1 (k) y(k) = [H, 0, 0..., 0]... z l (k) Ingegneria dell Automazione - Sistemi in Tempo Reale p.24/28
47 The case l = 0 Hey, this is prediction not delay? Does it make sense? Ingegneria dell Automazione - Sistemi in Tempo Reale p.25/28
48 The case l = 0 Hey, this is prediction not delay? Does it make sense? To start with, the system is causal: the predicted value will not manifest itself before the end of the sampling period. Ingegneria dell Automazione - Sistemi in Tempo Reale p.25/28
49 The case l = 0 Hey, this is prediction not delay? Does it make sense? To start with, the system is causal: the predicted value will not manifest itself before the end of the sampling period. The system with l = 0, m 0 will show the same response at t = 0 that the origal system would show at t = mt Ingegneria dell Automazione - Sistemi in Tempo Reale p.25/28
50 The case l = 0 Hey, this is prediction not delay? Does it make sense? To start with, the system is causal: the predicted value will not manifest itself before the end of the sampling period. The system with l = 0, m 0 will show the same response at t = 0 that the origal system would show at t = mt This is a tool to study the intersample behaviour (invented by Jury and originally called modified Z transform Ingegneria dell Automazione - Sistemi in Tempo Reale p.25/28
51 The case l = 0 The DT system is described by x(k + 1) = Φx(k) + Γ 1 u(k) + Γ 2 u(k + 1) Ingegneria dell Automazione - Sistemi in Tempo Reale p.26/28
52 The case l = 0 The DT system is described by x(k + 1) = Φx(k) + Γ 1 u(k) + Γ 2 u(k + 1) To eliminate u(k + 1) introduce the change of variable ξ(k) = x(k) Γ 2 u(k) Ingegneria dell Automazione - Sistemi in Tempo Reale p.26/28
53 The case l = 0 The DT system is described by x(k + 1) = Φx(k) + Γ 1 u(k) + Γ 2 u(k + 1) To eliminate u(k + 1) introduce the change of variable ξ(k) = x(k) Γ 2 u(k) After some computation we get: ξ(k + 1) = Φξ(k) + Γu(k) y(k) = Hξ(k) + Ju(k) where Γ = (ΦΓ 2 + Γ 1 ), J = HΓ 2 Ingegneria dell Automazione - Sistemi in Tempo Reale p.26/28
54 An example Consider a first order system ẋ = fx + gu Ingegneria dell Automazione - Sistemi in Tempo Reale p.27/28
55 An example Consider a first order system ẋ = fx + gu Γ 1 = R T mt efη gdη = eft e fmt f, Γ 2 = emft 1 f Ingegneria dell Automazione - Sistemi in Tempo Reale p.27/28
56 An example Consider a first order system ẋ = fx + gu Γ 1 = R T mt efη gdη = eft e fmt f Augmented discrete-time system:, Γ 2 = emft 1 f x(k + 1) 4 5 = 4 eft e ft e fmt f 5 4 x(k) emft 1 f 5 u(k) z(k + 1) 0 0 z(k) 1 Ingegneria dell Automazione - Sistemi in Tempo Reale p.27/28
57 An example Consider a first order system ẋ = fx + gu Γ 1 = R T mt efη gdη = eft e fmt f Augmented discrete-time system:, Γ 2 = emft 1 f x(k + 1) 4 5 = 4 eft e ft e fmt f 5 4 x(k) emft 1 f 5 u(k) z(k + 1) 0 0 z(k) 1 Control by static feedback u(k) = kx(x) yields the closed loop dynamic: 2 3 x(k + 1) 4 5 = z(k + 1) 2 4 eft + k emft 1 f e ft e fmt f k Ingegneria dell Automazione - Sistemi in Tempo Reale p.27/28
58 An example Consider a first order system ẋ = fx + gu Γ 1 = R T mt efη gdη = eft e fmt f Augmented discrete-time system:, Γ 2 = emft 1 f x(k + 1) 4 5 = 4 eft e ft e fmt f 5 4 x(k) emft 1 f 5 u(k) z(k + 1) 0 0 z(k) 1 Control by static feedback u(k) = kx(x) yields the closed loop dynamic: 2 3 x(k + 1) 4 5 = z(k + 1) 2 4 eft + k emft 1 f e ft e fmt f k Stability can be studied- via Jury criterion - on z 2 (e ft + k emft 1 )z k eft e fmt f f Ingegneria dell Automazione - Sistemi in Tempo Reale p.27/28
59 Summing up... Delays are relatively easy to model They extend the model with addtional state variables This effect has to be carefully accounted for in control design Ingegneria dell Automazione - Sistemi in Tempo Reale p.28/28
Ingegneria dell Automazione - Sistemi in Tempo Reale p.1/30
Ingegneria dell Automazione - Sistemi in Tempo Reale Selected topics on discrete-time and sampled-data systems Luigi Palopoli palopoli@sssup.it - Tel. 050/883444 Ingegneria dell Automazione - Sistemi in
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