ME 233 Advanced Control II Lecture 15 Deterministic Input/Output Approach to SISO Discrete-Time Systems Repetitive Control
SISO ARMA model
Where all inputs and outputs are scalars: • u(k)   control input • d(k) is a periodic disturbance of period N • y(k)  output Repetitive control assumptions
Both the disturbance and the reference model output are periodic sequences,
where N is a known and large number 3
Also, the polynomials B(q 1 ) and (1-q -^) are co-prime Deterministic SISO ARMA models The zero polynomial:
has •  m v zeros that we do not want to cancel. •  m¸ zeros inside the unit circle (asymptotically stable) that we do want to cancel.
is anti-Schur has the zeros (in q) that we do not want to cancel Deterministic SISO ARMA models The zero polynomial:
Without loss of generality, we will assume that
6 Control strategy: We design the controller in two stages Minor-loop pole placement: Place minor-loop poles 1. (these will be cancelled later) Repetitive compensator: Reject periodic disturbance 2. Follow periodic reference 7
1.  Minor-loop Pole Placement: The poles of the minor-loop system are placed at specific locations in the complex plane. They will be cancelled later. . Minor-loop pole polynomial: Where: .   B s (q-1)   cancelable plant zeros .    A'c (q-') anti-Schur polynomial chosen by the designer
2.  Tracking: The output sequence  y(k)  must asymptotically follow a reference sequence  yd(k) which is periodic
• Error signal:
3.  Disturbance rejection: The closed loop system must reject a class of deterministic disturbances which satisfy
Diophantine equation: Obtain polynomials   R(g-1), S(g-1) that satisfy: Closed-loop poles      Plant poles We will factor out the  B³(q¯ polynomial next The disturbance annihilating polynomial has not been included
Diophantine equation: Obtain polynomials   R ′ (q − 1 ), S ′ (q − 1 ) which satisfy: Closed-loop Plant poles Unstable plant zeros poles R(q-1) = R'(q-1) Bs(q-1) Ac(q-1) = Bs(q-1) A'c(q-1) The disturbance annihilating polynomial has not been included Diophantine equation A'c(q-1) = A(q-1) R'(q-1) + q-dBw(q-1) S(q-1) Solution:
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Closed-loop dynamics
filtered repetitive disturbance –
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d f ( k ) Notice that is still a periodic  disturbance
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d f (k) where
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Repetitive compensator strategy: 1. Cancel stable poles and delay  A'c(q-'') 2. Zero-phase error compensation for   B ψ (q − 1 ) 3. Include annihilating polynomial in the   1 - denominator
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d f (k) Repetitive compensator :
so that CR is implementable 18
• Closed-loop dynamics d f (k)
19 Repetitive Controller Closed-loop dynamics: doing a bit of algebra, we obtain,
Where the closed-loop poles are the zeros of
since,
we obtain Acr(q)e(k) = 0 Where
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Theorem The tracking error  e(k) → 0    if the gains   kr, b are selected as follows: 1. b ≥ max | B (ejw) | 2 ωE|0,π 2. 0 < kr ＜ 2
Consider now the case when there are no unstable zeros,
23 Closed-loop poles for minimum phase zeros For the case when the are no unstable zeros, the closed-loop poles are given by the roots of
When 0 < k,< 2, we have N asymptotically stable closed-loop poles
24 Repetitive control example   (Bw( Assume that N = 4 Gc (q') y(k)
Choose    b = b2
d f (k) 25
d f (k) y(k) Open-loop TF Gc(q-1)CR(q-1)
4 Kr 1 Closed-loop poles: 1 1 kr 26
Now consider the general case, i.e. there are unstable zeros
The closed-loop poles are the roots of
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The closed-loop poles are the roots of
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Therefore  Acr(z) = 0  is equivalent to
By Nyquist’s theorem, the closed-loop system is  asymptotically stable if there are no encirclements around −1. This is guaranteed if the following hold for ω ∈ [0, π]
Closed-loop stability ↑
d f (k) Repetitive compensator:
The controller has N open-loop poles on the unit circle 33 Repetitive control example   (Bu(g-1) Assume that N = 4 Gc (q') y(k) d f (k)
Choose    b = b2
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d f (k) + y(k) 1 Open-loop TF Gc(q-1)CR(q-1)
4 Kr Closed-loop poles: 1 1 kr 35
d f (k)
N = 4 y,(k) + y(k) Gc (q') Plant:
therefore,
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d f (k) 0.8 kr (q = 0.2) (q 4 = 1) y(k) + Open loop TF Gc(q-1)CR(q-1) Closed-loop poles: 0.8 1 + kr (z − 0.2)(z4 − 1) 0
d f (k) + y(k) Repetitive control is not robust to unmodeled dynamics Closed-loop poles: 1 0.8 0 Kr
d f (k) y(k) CR(q
Q(q, q-1) moving average filter with zero-phase shift characteristics Controller’s N open-loop poles are no longer on the unit circle Add Q-filter
Q(q,q-1) moving average filter with zero-phase shift characteristics
Q(q,q-1)  has unit DC gain and gain decreases as frequency increases
Notice that the disturbance d(k) is no longer completely annihilated, since
However, with a proper choice of Q filter,
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y (k) +
Plant:
where,
d f (k) 42
d f (k) Closed-loop poles: 2.4 z 1 + kr (z − 0.2)(6 z 5 − z 2 − 4 z − Closed-loop system is asymptotically stable for a finite range of k, 0