Terminal Iterative Learning Control with Multiple Intermediate Pass Points Tong Duy Son and Hyo-Sung Ahn Distributed Control & Autonomous Systems Laboratory School of Mechatronics, Gwangju Institute of Science and Technology (GIST), Korea
July 1, 2011
Tong Duy Son , DCAS Lab. (GIST) - 2011 American Control Conference, San Francisco, California, USA, 2011
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Terminal Iterative Learning Control with Multiple Intermediate Pass Points
Outline
Introduction Preliminaries: Research motivation Single terminal point Multiple intermediate pass points Simulation Conclusion
Tong Duy Son , DCAS Lab. (GIST) - 2011 American Control Conference, San Francisco, California, USA, 2011
2/19
Terminal Iterative Learning Control with Multiple Intermediate Pass Points Introduction
Multiple points tracking control MIMO plant model: xk (t + 1) = Axk (t) + Buk (t) yk (t) = Cxk (t) k is the iteration index; t = 0, 1, 2 . . . , N − 1 is the time index. Given desired outputs at given time instants t1 , t2 , . . . , tM : yd (t1 ),
yd (t2 ),
...,
yd (tM )
where 1 ≤ t1 < t2 < ... < tM ≤ N . Constructs a control law that drives the output goes through given terminal points.
Tong Duy Son , DCAS Lab. (GIST) - 2011 American Control Conference, San Francisco, California, USA, 2011
3/19
Terminal Iterative Learning Control with Multiple Intermediate Pass Points Introduction
Applications Satellite antenna tracking control
Figure: Satellite antenna control.
Given desired azimuths and elevations from tracking profile, control the antenna to point toward the desired ground within the sampling interval. Other examples: pick and place robots, temperature profile control of rapid thermal processing systems. Tong Duy Son , DCAS Lab. (GIST) - 2011 American Control Conference, San Francisco, California, USA, 2011
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Terminal Iterative Learning Control with Multiple Intermediate Pass Points Preliminaries
Conventional approach Dividing into trajectory planning and trajectory tracking control. Trajectory planning: Interpolation splines, control theoretic interpolating splines. 6
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Figure: Interpolation splines.
Trajectory ILC tracking control: For the linear plant yk = P uk , uk+1 = Tu uk + Te (yd − yk ) Converged error: e∗ = I − P (I − Tu + Te P )−1 Te yd Tong Duy Son , DCAS Lab. (GIST) - 2011 American Control Conference, San Francisco, California, USA, 2011
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Terminal Iterative Learning Control with Multiple Intermediate Pass Points Preliminaries
Limitations
Computational analysis and memory requirements in trajectory planning stage. The reference trajectory might not be optimal in the ILC controller. The existence of errors in both stages. When there is a change in the tracking profile, the system needs to start from beginning.
Tong Duy Son , DCAS Lab. (GIST) - 2011 American Control Conference, San Francisco, California, USA, 2011
6/19
Terminal Iterative Learning Control with Multiple Intermediate Pass Points Preliminaries
Research objective
Investigate learning knowledge through trials to incorporate trajectory planning and ILC tracking control. Decrease computational and memory cost, increase system performance. Reference trajectory free.
Tong Duy Son , DCAS Lab. (GIST) - 2011 American Control Conference, San Francisco, California, USA, 2011
7/19
Terminal Iterative Learning Control with Multiple Intermediate Pass Points Main results Single terminal point
Initial iterative learning for single terminal point The input signal is constant at all sampling times in the iteration Output at the terminal point y(N ): yk (N ) = CAN xk (0) + C
N −1 X
AN −j−1 Buk (j)
j=0
= PN u k where PN = C
NP −1
AN −j−1 B, and PN is full rank.
j=0
Cost function J = kek+1 (N )kQ + kuk+1 − uk kR + kuk+1 kS Q, R and S are real symmetric positive definite matrices. Tong Duy Son , DCAS Lab. (GIST) - 2011 American Control Conference, San Francisco, California, USA, 2011
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Terminal Iterative Learning Control with Multiple Intermediate Pass Points Main results Single terminal point
Controller ILC Controller: uk+1 = Lu uk + Le ek (N )
(1)
where Lu = (PNT QPN + R + S)−1 (PNT QPN + R) Le = (PNT QPN + R + S)−1 PNT Q Theorem The ILC learning algorithm (1) is asymptotically stable for all symmetric positive definite matrices Q, R, and S. Proof sketch Consider L = (Lu − Le PN )−1 = I + M . λ(I + M ) = λ(M ) + 1 > 1. ρ(Lu ) = max |λi (Lu )| < 1. Tong Duy Son , DCAS Lab. (GIST) - 2011 American Control Conference, San Francisco, California, USA, 2011
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Terminal Iterative Learning Control with Multiple Intermediate Pass Points Main results Single terminal point
Monotonic convergence Lemma The ILC learning algorithm (1) guarantees monotonic convergence if Q, R, S are chosen as diagonal positive definite matrices: Q = qI, R = rI, and S = sI. Proof sketch Applying the contraction mapping theorem with uk+1 = f (uk ), kf (u1 ) − f (u2 )k = kL(u1 − u2 )k ≤σ ¯ (L) ku1 − u2 k where σ ¯ (L) =
p
ρ(LT L).
L is symmetric positive definite, LT = L. p p ρ(LT L) = ρ(L)2 = ρ(L).
Tong Duy Son , DCAS Lab. (GIST) - 2011 American Control Conference, San Francisco, California, USA, 2011
10/19
Terminal Iterative Learning Control with Multiple Intermediate Pass Points Main results Multiple terminal point
Problem settings Multiple points: yd (t1 ),
yd (t2 ),
...,
Terminal output at ti : yk (ti ) = C
yd (tM ).
tP i −1
Ati −j−1 Buk (j).
j=0
Define pi (t) as pi (t) =
CAti −t−1 B 0
if t < ti . if t ≥ ti
Supervectors: uk =
uTk (0)
pi =
pi (0)
uTk (1) pi (1)
Terminal output at ti : yk (ti ) =
... ...
NP −1 t=0
T uTk (N − 1) T pi (N − 1) .
pi (t)uk (t) = pTi uk .
Tong Duy Son , DCAS Lab. (GIST) - 2011 American Control Conference, San Francisco, California, USA, 2011
11/19
Terminal Iterative Learning Control with Multiple Intermediate Pass Points Main results Multiple terminal point
Controller Cost function: J=
M X
eTk (ti )Qi ek (ti ) + uTk+1 Suk+1 + uk+1 − uk
T
R uk+1 − uk
i=1
where ek (ti ) = yd (ti ) − pTi uk+1 Supervectors: T yd = ydT (t1 ) ydT (t2 ) . . . ydT (tM ) T . P = pT1 pT2 . . . pTM Errors at terminal points: M X
T eTk (ti )Qi ek (ti ) = yd − Puk+1 Q yd − Puk+1
i=1
where Q = diag(Q1 , Q2 , . . . , QM ). Tong Duy Son , DCAS Lab. (GIST) - 2011 American Control Conference, San Francisco, California, USA, 2011
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Terminal Iterative Learning Control with Multiple Intermediate Pass Points Main results Multiple terminal point
Controller Theorem The ILC learning algorithm uk+1 = Lu uk + Le ek
(2)
where −1 T Lu = PT QP + R + S P QP + R −1 T Le = PT QP + R + S P Q. is asymptotically stable for all symmetric positive definite matrices Q, R and S. Proof sketch Resulting from the previous theorem Tong Duy Son , DCAS Lab. (GIST) - 2011 American Control Conference, San Francisco, California, USA, 2011
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Terminal Iterative Learning Control with Multiple Intermediate Pass Points Main results Multiple terminal point
Monotonic convergence
Lemma The ILC learning algorithm for multiple points (2) guarantees monotonic convergence if Q, R and S are chosen as diagonal positive definite matrices. Proof sketch Resulting from the previous lemma Performances: h −1 T i e∞ = I − P PT QP + S P Q yd .
Tong Duy Son , DCAS Lab. (GIST) - 2011 American Control Conference, San Francisco, California, USA, 2011
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Terminal Iterative Learning Control with Multiple Intermediate Pass Points Main results Multiple terminal point
Comments
Learning through iterations, the ILC algorithm produces the control signal iteratively, while the output curves track the data profile. The converged input is achieved, which its output is the optimal reference trajectory. Decrease order of P, Q and yd .
Tong Duy Son , DCAS Lab. (GIST) - 2011 American Control Conference, San Francisco, California, USA, 2011
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Terminal Iterative Learning Control with Multiple Intermediate Pass Points Simulation results
Single terminal point 2 1.8 1.6 1.4
Errors
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2nd iteration 20th iteration Desired Points
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Output Curves
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Figure: Convergence of error and output curves. Tong Duy Son , DCAS Lab. (GIST) - 2011 American Control Conference, San Francisco, California, USA, 2011
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Terminal Iterative Learning Control with Multiple Intermediate Pass Points Simulation results
Multiple terminal points 3.5
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2nd iteration 20th iteration Reference trajectory Desired Points
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(a) Multiple points tracking
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(b) Reference trajectory tracking
Figure: Comparison of multiple point and reference trajectory tracking (which is made from interpolation splines and conventional ILC). Tong Duy Son , DCAS Lab. (GIST) - 2011 American Control Conference, San Francisco, California, USA, 2011
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Terminal Iterative Learning Control with Multiple Intermediate Pass Points Simulation results
Compare control energy in both cases Tracking multiple points: Z 1 20 2 u (t) dt = 19.72 2 0 Tracking reference trajectory: Z 1 20 2 u (t) dt = 21.05 2 0 A reference trajectory is made from the interpolation splines technique, and then an ILC controller is applied to track that trajectory. Similar performance and less energy without trajectory planning step.
Tong Duy Son , DCAS Lab. (GIST) - 2011 American Control Conference, San Francisco, California, USA, 2011
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Terminal Iterative Learning Control with Multiple Intermediate Pass Points Conclusion
Conclusion We have proposed and analyzed the ILC problem in single and multiple points tracking control for MIMO systems. By learning from trials, it is possible to incorporate trajectory planning and ILC tracking control. Advantages in reference trajectory free, reducing order, thus decreasing the complexity and computational effort. Future works: robustness, constraints and smoothing. An Interpolation Method for Multiple Terminal Iterative Learning Control, 2011 IEEE MSCC. Iterative Learning Control for Optimal Multiple-Point Tracking, submitted to 2011 IEEE CDC.
Tong Duy Son , DCAS Lab. (GIST) - 2011 American Control Conference, San Francisco, California, USA, 2011
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