Repetitive Control
Professor Eric Rogers
Electronics and Computer Science
University of Southampton
Southampton, SO17 1BJ, UK
etar@ecs.soton.ac.uk
http://www.ecs.soton.ac.uk/
Professor Eric Rogers
ECS, University of Southampton, UK
Basics
• Many signals in engineering are periodic, or at least they
can be accurately approximated by a periodic signal over a
large time interval, e.g., engines, electrical motors and
generators, converters, or machines performing a
repetitive task.
• Control problem to try to track a periodic signal with the
output of the plant or reject a periodic disturbance acting
on a control system. In the former case the period, T, will
be known a priori by the nature of the task performed, in
the latter it can be identified using established methods.
• Repetitive control (RC) uses information from previous
periods or trials (also termed cycles in some literature) to
modify the control signal so that the overall system ‘learns’
to track perfectly a given T-periodic reference signal.
Professor Eric Rogers
ECS, University of Southampton, UK
Basics
• SISO plant with state-space model
ẋ(t) = Ax(t) + Bu(t), x(0) = x0
y(t) = Cx(t) + Du(t)
(1)
• Reference signal r(t) is given, and it is known that
r(t) = r(t + T), for a given T (in other words the actual
shape of r(t) is not necessarily known).
• Control design objective: find a feedback controller that
makes the system (1) track the reference signal as
accurately as possible (i.e., limt→∞ e(t) = 0,
e(t) := r(t) − y(t)), under the assumption that the reference
signal r(t) is T-periodic.
Professor Eric Rogers
ECS, University of Southampton, UK
Basics
• Solvability condition: the internal model principle must
hold, i.e., for controller
(2)
[Mu](t) = [Ne](t)
where M and N are suitable operators, a model of the
reference signal must be included in M.
• Since r(t) is T-periodic —its internal model is given by the
operator (1 − σT ), where [(1 − σT )v](t) = v(t) − v(t − T) for
an arbitrary v : R → R where R denotes the field of real
numbers.
• One simple RC law
u(t) = u(t − T) + e(t)
(3)
(M = (1 − σT )) and also [Nv](t) = v(t) for an arbitrary
v : R → R.
Professor Eric Rogers
ECS, University of Southampton, UK
Positive Realness
Definition
Let G(s) denote the transfer-function of the system (1), i.e.,
G(s) = C(sI − A)−1 B + D. Then (1) is positive-real if
1) Each entry in G(s) is analytic for Re s > 0
2) G(s) is real for real positive s
3) G(s) + G∗ (s) ≥ 0 for Re s ≥ 0
where the superscript ∗ denotes the complex conjugate and Re
denotes the real part.
Relaxed algorithm
u(t) = αu(t − T) + Ke(t)
(4)
where α ∈ (0, 1), is a relaxation parameter and K ∈ R, K > 0.
Professor Eric Rogers
ECS, University of Southampton, UK
Positive Realness
For x(0) = 0 and G(s) stable
U(s) =
αe−sT
K(s)r(s)
U(s) +
1 + G(s)K(s)
1 + G(s)K(s)
A sufficient condition (small gain theorem) for stability is
α
<1
sup ω≥0 1 + G(jω)K(jω)
(5)
(6)
For α = 1, this inequality is never met in when G(s) is strictly
proper, because if this holds then limω→∞ G(jω) = 0, and
1
therefore limω→∞ | 1+KG(jω)
| = 1.
Professor Eric Rogers
ECS, University of Southampton, UK
Positive Realness
• Condition (6) implies that the control law converges to a
T-periodic solution, and in the limit (4) becomes
u(t) =
K
e(t)
1−α
(7)
and y(t) converge to a T-periodic solution given by
Y(s) =
1
K
1−α G(s)
r(s)
K
+ 1−α
G(s)
(8)
• If α → 1 (giving the original law (3)), it is clear that an
infinite feedback gain is required. However, a positive-real
system in this case can tolerate such a gain. Therefore
stability holds for an arbitrary α ∈ [0, 1], if the original
system is positive-real.
Professor Eric Rogers
ECS, University of Southampton, UK
Positive Realness — SISO case
Nquist plot for G(s)
Nyquist Diagram of a PR system
0.2
0.15
Imaginary Axis
0.1
K
∞
0.05
0
−0.05
−0.1
−0.15
−0.2
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Real Axis
Professor Eric Rogers
ECS, University of Southampton, UK
Positive Realness — SISO case
• The control law (3) will result in convergent learning when
the original continuous-time system is positive-real.
• Problem: not possible to implement a delay block using
analogue components.
• Consider the SISO discrete-time system
x(t + 1) = Φx(t) + Γu(t),
y(t) = Cx(t) + Du(t)
x(0) = x0
(9)
and corresponding transfer-function
G(z) = C(zI − Φ)−1 Γ + D.
Professor Eric Rogers
ECS, University of Southampton, UK
Positive Realness — SISO case
Definition
Then (9) is said to be positive-real if
1) G(z) is analytic for |z| > 1
2) G(z) is real for real positive z
3) Re[G(ejθ )] ≥ 0 for all θ ∈ [0, 2π] (i.e., the Nyquist plot of G(z)
lies in the right-half plane)
Lemma
Suppose that the system (9) is positive-real and CB 6= 0 (the
assumption that CB 6= 0 is almost always true for a
continuous-time system sampled with zero-order hold) then
D 6= 0.
Professor Eric Rogers
ECS, University of Southampton, UK
Positive Realness — SISO case
• If the discrete control law u(t) = u(t − T) + Ke(t) gives
convergent learning, then infinite feedback gain results —
not implementable without instability.
• Hence the only option is the ‘relaxed’ version of the control
law (where α is typically replaced with a dynamic causal
filter), which does not give perfect tracking for an arbitrary
T-periodic reference.
• Next a computationally more complex algorithm for the
discrete-time case which results in perfect tracking is
developed.
Professor Eric Rogers
ECS, University of Southampton, UK
RC Design
• Sample the plant model every Ts seconds to give
(10)
A(z)y(z) = B(z)u(z)
where y(z) and u(z) denote the z transforms of the input
and output respectively and A(z) and B(z) are polynomials
in z−1 . Also it is assumed that (10) is both controllable and
observable.
• Control objective: track a T-periodic reference, i.e.,
r(t + T) = r(t).
Professor Eric Rogers
ECS, University of Southampton, UK
RC Design
Lemma
The tracking problem can be written equivalently in terms of the
original process model (10) as
Ã(z)e(z) = B(z)ũ(z)
Ã(z) = −(1 − z−T )A(z)
ũ(z) = (1 − z−T )u(z)
(11)
where e(z) = r(z) − y(z). Also a control law M(z)ũ(z) = N(z)e(z),
where M(z) and N(z) are polynomials in z−1 will drive the
tracking error to zero exponentially if it stabilizes the modified
system model (11).
Professor Eric Rogers
ECS, University of Southampton, UK
RC Design
• The design problem now is to force the output e(z) of (11)
to zero.
• Many possible solutions, e.g., robust or adaptive based
solutions.
• Here we use an optimal control based solution.
• Write the controlled system description as
e(z) =
−1 B(z)
ũ(z)
1 − z−T A(z)
(12)
which is a series connection of the polynomial systems S1
and S2 where
−1
S1 : up (z) = 1−z
−T ũ(z)
(13)
B(z)
S2 : e(z) = A(z) up (z)
Professor Eric Rogers
ECS, University of Southampton, UK
RC Design
A state-space model for S1 is
ξ(t + 1) = Ξξ(t) + Ωũ(t)
up (t) = −Ψξ(t) − ũ(t)
(14)
where




Ξ = 


0 0
1 0
0 1
.. . .
.
.
0
Professor Eric Rogers
0
0 ...
0 ...
0 ...
.. ..
.
.
0 ...

0 1
0 0 

0 0 

.. .. 
. . 
1 0
(15)
ECS, University of Southampton, UK
RC Design
Ω =
1 0 0 ... 0 0
T
Ψ =
0 0 0 ... 0 1
(16)
and for S2
xp (t + 1) = Ap xp (t) + Bp up (t)
(17)
e(t) = Cp x(t)
Combining these last two state-space models now yields
xm (t + 1) = Φm xm (t) + Γm ũ(t),
e(t) = Cm xm (t)
xm (0) = xm,o
(18)
where the dimension of xm (·) is n + T, and n is the order of the
original process model (10).
Professor Eric Rogers
ECS, University of Southampton, UK
RC Design
Ξ
0
−Bp Ψ Ap
0 Cp
=
Φm =
Cm
, Γm =
Ω
−Bp
(19)
Cost function
J 0 = min J(ũ, xm (0))
(20)
P
T
T
J(ũ, xm (0)) = ∞
i=1 e(i) Qe(i) + ũ (i)Rũ(i)
P∞
T QC x (i) + ũT (i)Rũ(i)
= i=1 xm (i)T Cm
m m
(21)
ũ∈l2
where
Q 0, R 0.
Professor Eric Rogers
ECS, University of Southampton, UK
RC Design
Solution:
u(t) = u(t − T) − Kxm (t)
(22)
K = (ΓTm SΓm + R)−1 ΓTm SΦm
(23)
where
and S is the solution of the algebraic Riccati equation
T
S = ΦTm [S − SΓm (ΓTm SΓm + R)−1 ΓTm S]Φm + Cm
QCm
(24)
or with the state estimator
x̂m (t + 1) = Φm x̂m (t) + Γm ũ(t) + L(e(t) − Cm x̂m (t))
(25)
where L is the observer gain, the control law is
u(t) = u(t − T) − Kx̂m (t)
Professor Eric Rogers
(26)
ECS, University of Southampton, UK
RC Design
Extension to include noise:
e
xm (t + 1) = Φm xm (t) + Γm ũ(t) + Hw(t)
e(t) = Cm xm (t) + v(n), xm (0) = xm,o
(27)
where w(t) and v(t) are zero mean Gaussian noise signals.
• Conceptually w(t) describes uncertainty in the state-space
model, whereas v(t) describes uncertainty in the
measurement process.
• If the covariance matrix Qn of v(t) and the covariance
matrix Rn of w(t) are known, it is possible to find an optimal
observer gain L that minimizes the variance of the
estimation error.
Professor Eric Rogers
ECS, University of Southampton, UK
RC Design
• Combining the optimal feedback controller and optimal
observer the resulting closed-loop system is stable, and
hence the expected value of e(t) will exponentially
converge to zero as t → ∞.
z
Professor Eric Rogers
ECS, University of Southampton, UK
Experimental Results
• The non-minimum phase electro-mechanical system.
Reference r(t)
Output u(t)
Input y(t)
12
Output (rads) / Input (V)
10
8
6
4
2
0
-2
0
5
10
Time (s)
15
20
20 UPM sinewave demand convergence.
Professor Eric Rogers
ECS, University of Southampton, UK
Reference r(t)
Output y(t)
Input(t)
Output (rads) / Input (V)
6
4
2
0
0
5
10
15
20
-2
-4
-6
Time (s)
20 UPM repeating sequence demand convergence.
Professor Eric Rogers
ECS, University of Southampton, UK
1
0.8
0.5
0.7
0.9
0.98
1
NE
0.6
0.4
0.2
0
0
50
100
150
200
250
300
350
400
Cycle No.
Trial error results for sinewave demand with relaxation.
Professor Eric Rogers
ECS, University of Southampton, UK
1
0.8
0.5
0.7
0.9
0.98
1
NE
0.6
0.4
0.2
0
0
50
100
150
200
250
300
350
400
Cycle No.
Trial error results for repeating sequence demand with
relaxation.
Professor Eric Rogers
ECS, University of Southampton, UK
1
Optimal based
Contraction mapping
Multiple lead
Phase-lead
0.8
NE
0.6
0.4
0.2
0
0
50
100
150
200
Cycle No.
250
300
350
400
Comparison of RC schemes for sinewave demand.
Professor Eric Rogers
ECS, University of Southampton, UK
1
Optimal based
Contraction mapping
Multiple lead
Phase-lead
0.8
NE
0.6
0.4
0.2
0
0
50
100
150
200
Cycle No.
250
300
350
400
Comparison of RC schemes for repeating sequence demand.
Professor Eric Rogers
ECS, University of Southampton, UK
Comparisons
• The previous two figures compare the best results
produced with this scheme with those of other RC laws
applied on the same plant, namely the phase-lead
algorithm, a multiple phase-lead algorithm and the
contraction mapping law. These methods also require a
plant model.
• It is clear that the optimality based algorithm has hugely
improved convergence when compared to these
alternatives, a similar final error bound and little sign of
greater instability.
• To match its convergence rate, the gain of these other
schemes can be increased but this always leads to the
rapid onset of instability.
Professor Eric Rogers
ECS, University of Southampton, UK
Comparisons
• Successive trial errors are linked through
e(z) =
γz−T
1−γ
e(z) +
r(z)
1 + G(z)H(z)
1 + G(z)H(z)
(28)
where
H(z) = K[zI − (Φm − LCm − Γm K)]−1 L
(29)
• A sufficient condition for convergence of the error
sequence is
γ 1 + GH < 1
(30)
for all |z| = 1.
Professor Eric Rogers
ECS, University of Southampton, UK
Comparisons
• Algorithm robustness can be started from that for (30) to
hold all roots of
1 − z−T + GH = 0
(31)
must have modulus less than unity.
• One way of proceeding from here is to plot the Nyquist
diagram of
point.
Professor Eric Rogers
GH
1−z−T
and check for encirclements of the −1
ECS, University of Southampton, UK
Comparisons
• A simpler alternative is to work with GH − z−T since
GH − z−T =
−z−T (Gn Hn + Gd Hd )
Gd Hd
(32)
where the subscripts n and d denote the numerator and
denominator polynomials of G and H.
Professor Eric Rogers
ECS, University of Southampton, UK
References
• The results in this section are from the reference below.
This is just one method from a well studied and still heavily
worked area.
• The references in this cited paper lead to some other
designs, but again nowhere near an exhaustive list.
C. T. Freeman, P. L. Lewin, E. Rogers and D. H. Owens and
J. J. Hätönen
An optimality-based repetitive control algorithm for
discrete-time systems
IEEE Transactions on Circutis and Systems-I: Regular
Papers, 55(1):412–423, 2008.
Professor Eric Rogers
ECS, University of Southampton, UK
Duality between ILC and RC
• It can be shown, under some assumptions, that the
structure of RC and ILC differ only in the location of an
internal model of the disturbance.
• The ability to treat RC and ILC in the same framework is
appealing since controllers found to operate well in one
area can be synthesized for application to the other.
• Firs work in the following paper.
D. de Roover, O. H. Bosgra and M. Steinbuch
Internal-model-based design of repetitive and iterative
learning controllers for linear multivariable systems
International Journal of Control, 73(10):914–929, 2000.
Professor Eric Rogers
ECS, University of Southampton, UK
Duality between ILC and RC
• More recent work and experimental verification using the
gantry robot can be found in the following paper.
C. T. Freeman, M. Ali Alsubaie, Z. Cai, E. Rogers and P. L.
Lewin
A common setting for the design of iterative learning and
repetitive controllers with experimental verification
International Journal of Adaptive Control and Signal
Processing, 2012, In press. Available online from
http://www.ecs.soton.ac.uk/
Professor Eric Rogers
ECS, University of Southampton, UK