Robust Iterative Learning Control with Quadratic Performance Index

ARTICLE pubs.acs.org/IECR

Robust Iterative Learning Control with Quadratic Performance Index Zuhua Xu,†,‡ Jun Zhao,† Yi Yang,†,* and Zhijiang Shao† †

National Laboratory of Industrial Control Technology, Department of Control Science and Engineering, Zhejiang University, Hangzhou 310027, People's Republic of China ‡ Key Laboratory of Advanced Control and Optimization for Chemical Processes, Shanghai 200237, People's Republic of China

Furong Gao Department of Chemical & Biomolecular Engineering, Hong Kong University of Science & Technology, Clear Water Bay, Kowloon, Hong Kong ABSTRACT: In this paper, a robust iterative learning control (ILC) designed through a linear matrix inequality (LMI) approach is proposed first, based on the worst-case performance index with ellipsoidal uncertainty and polytopic uncertainty, respectively. Since the design based on worst-case performance index is too conservative, a novel ILC design based on nominal performance index is further proposed, and its robust convergence properties are proven. The latter can give better performance when the nominal model is close to the true process. Simulations have demonstrated the effectiveness and excellent performance of the proposed methods.

μ analysis, van de Wijdeven et al.15 proposed a robust monotonic convergence analysis approach for uncertain systems and applied to linear quadratic norm optimal ILC. In contrast to analysis of the performances of given ILC algorithms, another important issue is to systematically consider the model uncertainty from the ILC design aspect. Lee et al.9 presented a constrained robust Q-ILC control based on the worst-case performance index to account for model uncertainty explicitly. However, uncertain representation of the process model and the solution for this method have not been clearly presented. Nguyen et al.16 solved the robust Q-ILC problem for linear systems by employing an upper bound of the worst-case performance index. However, using upper bound of the worst-case performance index leads to a more conservative control, and only one single uncertainty was considered in this method. In this paper, a robust ILC designed through a linear matrix inequality (LMI) approach, based on the worst-case performance index with ellipsoidal uncertainty and polytopic uncertainty, is proposed first. Considering the conservative nature of the worstcase performance index, a novel robust ILC design based on a nominal performance index is further proposed, with its robust convergence properties proven and its superior performance demonstrated, especially when the nominal model becomes close to the true process. The rest part of this paper is organized as follows. In section 2, robust ILC problem is formulated, including model uncertainty and process constraint. The robust ILC design based on the worst-case performance index is presented in section 3, while the robust ILC based on nominal performance index is introduced in section 4. In section 5, robust convergence properties are proven. Numerical examples are illustrated in section 6, and, finally, the conclusions are drawn in section 7.

1. INTRODUCTION Iterative learning control (ILC) is a control technique originally proposed for systems that repetitively perform the same task. It improves the set point tracking accuracy progressively through repetitive learning.1,2 After its initial application for industrial robots, ILC has been adopted as an effective control methodology in many industrial processes with repetitive natures. The robust design and robust analysis of ILC have always been a hot research topic. By establishing the connection between the ILC convergence condition and the well-known robust performance condition, Tayebi et al.3 proposed a robust ILC design method for an uncertain linear time-invariant system. Shi et al.4 developed a robust feedback feedforward ILC design method with uncertain 2D Roesser system representation, which can allow the designer to choose balanced optimization objectives to meet the performance requirements. Applying the supervector approach to ILC, Ahn et al.5 presented a stability analysis of the ILC problem for discrete-time systems when the plant Markov parameters were subject to interval uncertainty. Based on the internal model control (IMC) structure, Liu et al.6 proposed a robust ILC method for batch processes with uncertain time delay. In van de Wijdeven et al.,7 a robust ILC control strategy based on H∞ optimization was presented for systems with additive model uncertainty, and the resulting controller was not restricted to be casual and inherently operated on a finite time interval. Recently, the ILC algorithm based on a quadratic performance index has been widely studied.811 Gao et al.12 analyzed the robustness and convergence issues of norm optimal ILC algorithm for processes with uncertain initializations and disturbances, and established a sufficient and necessary condition to ensure robust BIBO stability. Gorinevsky13 presented the analysis and design method for LQ/LTR ILC via a loop-shaping method, including robust stability, nominal performance, and actuator move magnitude. Ghosh et al.14 discussed the robustness of LQ norm-optimal ILC based on a frequency domain representation. Based on the finite time interval representation and r 2011 American Chemical Society

Received: August 31, 2011 Accepted: December 4, 2011 Revised: November 14, 2011 Published: December 05, 2011 872

dx.doi.org/10.1021/ie201962z | Ind. Eng. Chem. Res. 2012, 51, 872–881

Industrial & Engineering Chemistry Research

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where θ = [θ1 θ2 3 3 3 θp]T and W = WT g 0. In other words, G is an affine function of uncertain parameter θ, which lies in an ellipsoid set with its center at θ̅ . Matrix W defines the size and orientation of the ellipse. 2.2. Polytopic Uncertainty. Polytopic uncertain systems are widely accepted in robust analysis and robust control. For polytopic uncertain systems, the uncertainty set Ω is defined as

2. PROBLEM DESCRIPTION Without any loss of generality, consider a discrete-time linear system: y k ¼ Guk þ d where yk ¼ and uk ¼

h

h

yk ð1ÞT

uk ð0ÞT

ð1Þ

yk ð2ÞT

uk ð1ÞT

333

333

yk ðNÞT

iT

uk ðN 1ÞT

Ω ¼ CoðG1 , 3 3 3 , Gp Þ

where Co refers to the convex hull defined by vertices Gi(i = 1, 3 3 3 , p). In other words, if G∈Ω then, for some non-negative number θ1, 3 3 3 , θp summing to 1, we have G = θ1G1 + 3 3 3 + θpGp. There are some constraints that must be set on the control input in industrial application, to ensure safe and smooth operations, as listed below: (1) For the raw values of inputs: u1 e uk e uh (2) For the rate of input changes, with respect to the time index: δu1 e δuk e δuh (3) For the rate of input changes, with respect to the batch index: Δu1 e Δuk e Δuh All the above constraint equations can be rearranged as follows:

iT

In the above equation, yk(t) and uk(t) represent the output and input of the system at the tth time instant of the kth run, d is the disturbance vector, and G is a low-triangular matrix consisting of system pulse response coefficients. Let ek =r yk represents the output error, where r is the desired output trajectory. The following transition model for tracking error trajectory then can be derived: ek ¼ ek1 GΔuk

ð2Þ

ΓΔuk g Λk

where Δuk = uk uk1 is the difference of the control input between iterations. The ILC algorithm design problem can now be formulated to find a control law such that the system output has the asymptotic convergence property, i.e., ekf0 as k f∞. There are different choices of designs to solve this ILC problem. The quadratic (norm) optimal formulation is adopted here, because of its superior performance and the capability of dealing with multivariate constrained problems.811 At each iteration, the following quadratic performance index is minimized to obtain the input vector: min kek k2Q þ kΔuk k2R Δuk

where

Δuk

G∈Ω

ð7Þ

3 I 6 6 I 7 7 7 Γ¼6 6 J7 4 5 J

2

2

3 maxðul -uk1 , Δul Þ 6 6 minðuh uk1 , Δuh Þ 7 7 7 Λk ¼ 6 6 7 δul Juk1 4 5 δuh þ Juk1

ð3Þ

where Q and R are positive definite matrices. Any model can have model mismatch problems. Therefore, it is important to consider the model uncertainty during ILC design. In ref 9, Lee et al. have used the following worst-case performance index to develop a robust ILC algorithm: min max kek k2Q þ kΔuk k2R

ð6Þ

2

I 6 6 I 6 J ¼6 6 0 6 l 4 0

ð4Þ

where Ω is a uncertain set of process models. However, the uncertainty of process model is not rigorously defined in their work, and the solution for this problem has not been presented. In this paper, a robust ILC design, through the LMI approach, is proposed first, based on the worst-case performance index with ellipsoidal and polytopic uncertainties. Since the min-max paradigm of the optimization performance for the “worst-case” are way too conservative for real applications and may yield poor performance, a novel robust ILC algorithm based on nominal performance index is further proposed with some additional constraints, to ensure robust convergence. In this paper, the following uncertain representations of process model are considered in the design of a robust ILC algorithm: 2.1. Ellipsoidal Uncertainty. For ellipsoidal uncertain systems, the uncertainty set Ω is defined as

0 I I ⋱ 0

333 333 ⋱ ⋱

0 0 l ⋱ I

3 0 7 07 7 07 7 l7 5 I

Recently, the LMI technique has become a useful tool for solving a wide variety of optimization and control problems. In this paper, the LMI technique is adopted to design the robust ILC. Detailed information on the LMI technique can be found in Boyd et al.17 Moreover, in order to design robust ILC through LMI, the following lemma is introduced. Lemma (S-Procedure)17. Let F0, 3 3 3 , Fp be the quadratic functions of the variable ξ∈Rp: Fi ðξÞ } ξT Ti ξ þ 2uTi ξ þ vi where Ti =

i ¼ 0; :::; p

TTi .

The following condition on F0, 3 3 3 , Fp: F0 ðξÞ g 0 for all ξ

ð8Þ

such that

Ω ¼ fG ¼ G0 þ θ1 G1 þ 3 3 3 þ θp Gp jðθ θÞT Wðθ θÞ e Fg

Fi ðξÞ g 0

ð5Þ 873

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holds if there exists t1 g 0; :::; tp g 0

With the above notation, the min-max problem (eq 10) can be cast into the following formulation:

ð9Þ

min λ

such that, for all ξ, p

∑ ti Fi ðξÞ g 0

subject to

i¼1

When p = 1, the converse holds, provided that there is some ξ0 such that F1(ξ0) > 0.

kMðΔuk Þθ þ f ðΔuk Þk22 þ khðΔuk Þk22 e λ 2 " θ ∈ fθjθ θW e FgΓΔuk g Λk

3. ROBUST ILC ALGORITHM BASED ON WORST-CASE PERFORMANCE INDEX In order to incorporate model uncertainty information explicitly into the ILC algorithm, the following robust ILC algorithm, based on the worst-case performance index, is formulated: min max kek k2Q þ kΔuk k2R ð10Þ Δuk

)

Note that condition θ θ̅ " #T 2 T 1 4 F θ Wθ θ Wθ

G∈Ω

2 W

)

F0 ðξÞ

ð11Þ

λ, Δuk

e F can be transformed to

3" # T θ W5 1 g0 θ W

ð12Þ

kek k2Q þ kΔuk k2R ¼ Q 1=2 ðG0 þ

" #T 2 λ khðΔuk Þk22 kf ðΔuk Þk22 1 4 θ MðΔuk ÞT f ðΔuk Þ

¼ kGΔuk ek1 k2Q þ kΔuk k2R p

∑

i¼1

MðΔuk Þ ¼ Q

½ G1 Δuk

3" # f ðΔuk ÞT MðΔuk Þ 1 5 g0 θ MðΔuk ÞT MðΔuk Þ

kMðΔuk Þθ þ f ðΔuk Þk22 þ khðΔuk Þk22 e λ

Gp Δuk

333

e λ can be

Using the S-Procedure, we have

2

G2 Δuk

+ h(Δuk)

ð13Þ

2 θi Gi ÞΔuk Q 1=2 ek1 þ jjR 1=2 Δuk jj22

defined as 1=2

2 2

)

Condition M(Δuk)θ + f(Δuk) rearranged as

In this section, the LMI technique is adopted to solve the minmax optimization problem 10 for linear systems with ellipsoidal uncertainty and polytopic uncertainty. 3.1. Ellipsoidal Uncertainty

2 2

)

)

ΓΔuk g Lk

)

subject to

2 for every θ, θ θW e F

f ðΔuk Þ ¼ Q ðG0 Δukk ek1 Þ hðΔuk Þ ¼ R 1=2 Δuk 1=2

if and only if there exists a scalar t g 0 such that, for every θ∈Rp,

" #T 2 T λ kf ðΔuk Þk22 khðΔuk Þk22 tðF θ WθÞ 1 4 θ MðΔuk ÞT f ðΔuk Þ tWθ

3" # T f ðΔuk ÞT MðΔuk Þ tθ W 1 5 g0 T θ MðΔuk Þ MðΔuk Þ þ tW

ð14Þ

Note that eq 14 can be written as 2

T

λ kf ðΔuk Þk22 khðΔuk Þk22 tðF θ WθÞ 4 MðΔuk ÞT f ðΔuk Þ tWθ

3 T f ðΔuk ÞT MðΔuk Þ tθ W 5g0 MðΔuk ÞT MðΔuk Þ þ tW

Using the Schur complements, it can be written as 2 3 T T λ tðF θ WθÞ tθ W f ðΔuk ÞT hðΔuk ÞT 6 7 6 7 tWθ tW MðΔuk ÞT 0 6 7g0 6 7 Þ MðΔu Þ I 0 f ðΔu k k 4 5 hðΔuk Þ 0 0 I

ð15Þ

Therefore, the robust ILC algorithm, based on worst-case performance, can be solved by the following optimization problem over LMIs: min

λ, t, Δuk

λ

ð18Þ

subject to 2 T λ tðF θ WθÞ 6 6 tWθ 6 6 f ðΔu Þ k 4 hðΔuk Þ

ð16Þ Note that t g 0 is implied by tW g 0 in eq 16. The linear constraint ΓΔuk g Λk can be expressed as an LMI with diagonal matrices: diagðΓΔuk Lk Þ g 0 ð17Þ

T

tθ W tW MðΔuk Þ 0

f ðΔuk ÞT MðΔuk ÞT I 0

3 hðΔuk ÞT 7 7 0 7g0 7 0 5 I

diagðΓΔuk Λk Þ > 0 874

dx.doi.org/10.1021/ie201962z |Ind. Eng. Chem. Res. 2012, 51, 872–881


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4. ROBUST ILC ALGORITHM BASED ON NOMINAL PERFORMANCE INDEX As remarked by Ding et al.18 and Badgwell,19 for model predictive control, the min-max paradigm of optimizing the worst-case performance index represents extreme cases in real application and may yield conservative control and poor performance. A more sensible approach is to minimize the nominal performance index with additional constraints to ensure robust convergence. Hence, the following robust ILC algorithm is proposed based on a nominal performance index:

3.2. Polytopic Uncertainty. The min-max problem described by eq 10 is equivalent to the following formulation:

min λ

ð19Þ

λ, Δuk

subject to max kGΔuk ek1 k2Q þ kΔuk k2R e λ

G∈Ω

ΓΔuk g Λk Using the triangle inequality, the first term in the first constraint can be expressed as kGΔuk ek1 k2Q

2 2 min kGΔu ̅ k ek1 kQ þ kΔuk kR

2 3 3 3 þ θp ðGp Δuk ek1 Þ Q 2 2 eθ1 kG1 Δuk ek1 kQ þ 3 3 3 þ θp Gp Δuk ek1 Q

subject to kGΔuk ek1 k2Q þ kΔuk k2R e kek1 k2Q ΓΔuk g Lk

eðθ1 þ 3 3 3 þ θp Þ max kGi Δuk ek1 k2Q i ¼ max kGi Δuk ek1 k2Q i

This equality holds if and only if

kGΔuk ek1 k2Q þ kΔuk k2R e kek1 k2Q

1 for i ¼ j 0 for i 6¼ j GiΔukek1 2Q. Therefore,

max kGΔuk ek1 k2Q þ kΔuk k2R

G∈Ω

¼ max kGi Δuk ek1 k2Q þ kΔuk k2R

ð21Þ

i

Hence, the optimization problem described by eq 19 can be transformed to the following formulation: min λ

ð22Þ

t, Δuk

min λ

subject to ΓΔuk g Lk

þ kΔuk k2R

eλ

i ¼ 1; :::; p

subject to 2 2 kGΔu ̅ k ek1 kQ þ kΔuk kR e λ

kGΔuk ek1 k2Q þ kΔuk k2R e kek1 k2Q ΓΔuk g Λk

Using the Schur complements, it can be further written as the following optimization problem over LMIs: min λ

ð23Þ

λ, Δuk

subject to 2

λ 6 6 G Δu e k1 4 i k Δuk

ðGi Δuk ek1 ÞT Q 1 0

4

3 ΔuTk 7 0 7 5 g 0 i ¼ 1; :::; p R 1

Using the S-Procedure, the second constraint in problem 26 is equivalent to

T

kek1 k2Q kf ðΔuk Þk22 khðΔuk Þk22 tðF θ WθÞ MðΔuk ÞT f ðΔuk Þ tWθ

"G∈Ω

The first constraint in problem 26 can be converted to an LMI by Schur complements: 2 3 T λ ðGΔu ΔuTk ̅ k ek1 Þ 6 7 6 GΔu Q 1 0 7 ð27Þ k ek1 4̅ 5g0 0 R 1 Δuk

diagðΓΔuk Lk Þ > 0 2

ð25Þ

ð26Þ

λ, Δuk

kGi Δuk ek1 k2Q

"G∈Ω

This implies that the quadratic performance index for each model in the uncertainty set is nonincreasing, which guarantees the robust convergence of ILC algorithm (see section 5). The advantage of this approach is that control performance can be greatly improved when the nominal model is close to the actual process. Another advantage of this approach is that no additional tuning parameters are needed to ensure robust convergence of the ILC. Moreover, the robust optimization problem 24 is always feasible at each run, because Δuk = 0 is a feasible solution in any case. In this section, the LMI technique is applied to the robust constrained ILC design, based on the nominal performance index. 4.1. Ellipsoidal Uncertainty. Optimization problem 24 can be cast into the following formulation:

)

2 Q = maxi

)

)

where j meets GjΔukek1

)

θ¼

"G∈Ω

where G∈Ω is the nominal model. Consider the first constraint in problem 24:

ð20Þ

(

ð24Þ

Δuk

¼ θ1 ðG1 Δuk ek1 Þ þ

T

f ðΔuk ÞT MðΔuk Þ tθ W MðΔuk ÞT MðΔuk Þ þ tW 875

3 5g0



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It can be further converted into the following LMI by Schur complements: T

kek1 k2Q tðF θ WθÞ

T

tθ W

f ðΔuk ÞT

hðΔuk ÞT

T

tWθ f ðΔuk Þ hðΔuk Þ

tW MðΔuk Þ MðΔuk Þ I 0 0

0 0 I

7 7 7g0 7 5

Therefore, the robust ILC algorithm based on the nominal performance index can be solved by min λ

Therefore, the robust ILC algorithm based on the nominal performance index can be solved via λ, t, Δuk

subject to

λ

2

3 T λ ðGΔu ΔuTk ̅ k ek1 Þ 6 7 6 GΔu Q 1 0 7 k ek1 4̅ 5g0 1 0 R Δuk 3 2 kek1 k2Q ðGi Δuk ek1 ÞT ΔuTk 7 6 6 Gi Δuk ek1 Q 1 0 7 5 g 0 i ¼ 1; :::; p 4 1 0 R Δuk

ð29Þ

3 λ ð ̅GΔuk ek1 ÞT ΔuTk 7 6 6 GΔu Q 1 0 7 k ek1 5g0 4̅ 1 Δuk 0 R 2 T T kek1 k2Q tðF θ WθÞ tθ W f ðΔuk ÞT 6 6 tWθ tW MðΔuk ÞT 6 6 f ðΔu Þ MðΔuk Þ I k 4 hðΔuk Þ 0 0 2

hðΔuk ÞT 0 0 I

3

diagðΓΔuk Λk Þ > 0

7 7 7g0 7 5

5. CONVERGENCE ANALYSIS AND COMPUTATIONAL ISSUES

diagðΓΔuk Ωk Þ > 0

5.1. Convergence Analysis. Convergence will be proved in the absence of process constraint ΓΔuk g Λk. Extending the proof to the constrained algorithm may involve some technical hurdle, as detailed in ref 9. The following assumptions are made to prove the convergence properties of the robust ILC algorithm: (A1) G has a full row-rank for every G ∈Ω (A2) Q and R are positive definite The above assumption A1 is needed to ensure zero tracking error of the robust ILC algorithm. When G does not have a full row-rank, system (2) has uncontrollable modes at (1,0), which means that it is impossible to achieve zero tracking error at general. Assumption A2 is reasonable. Theorem 1. Under assumptions A1-A2, system 2 converges to the origin under the robust ILC algorithm, based on the worstcase performance index, i.e., Δukf0 and ekf0 as kf∞. Remark 1. The above-mentioned theorem has been established by Lee et al.9 and also Nguyen et al.16 Theorem 2. Under assumptions A1 and A2, system 2 converges to the origin under the robust ILC algorithm based on nominal performance index, i.e., Δukf0 and ekf0 as kf∞. Proof. Let Vk = maxG∈ΩΦk(G,uk), where Φk(G,uk) = ek (G,Δuk) 2Q + Δuk 2R According to robust convergence constraint 25,

4.2. Polytopic Uncertainty. Optimization problem 24 can be transformed to the following formulation:

min λ

ð30Þ

λ, Δuk

subject to 2 2 kGΔu ̅ k ek1 kQ þ kΔuk kR e λ

kGΔuk ek1 k2Q þ kΔuk k2R e kek1 k2Q ΓΔuk g Λk

"G∈Ω

)

)

The first constraint in problem 30 can be converted into an LMI by Schur complements: 2 3 T ΔuTk λ ðGΔu ̅ k ek1 Þ 6 7 6 GΔu Q 1 0 7 ð31Þ k ek1 4̅ 5g0 1 Δuk 0 R

Vk

Using the triangle inequality, one can get G∈Ω

kGΔuk ek1 k2Q

þ kΔuk k2R

i

Hence, the second constraint in problem 30 is equivalent to

i

kGi Δuk ek1 k2Q

þ kΔuk k2R

e

kek1 k2Q

¼ e

kek1 k2Q þ kΔuk1 k2R kΔuk1 k2R h i max kek1 ðG, Δuk1 Þk2Q þ kΔuk1 k2R kΔuk1 k2R

¼ e

V k1 ΔuTk 1 RΔuk1 V k1

G∈Ω

¼ max kGi Δuk ek1 k2Q þ kΔuk k2R

max

ð34Þ

λ, Δuk

subject to

max

3 ΔuTk 7 0 7 5 g 0 i ¼ 1; :::; p 1 R

ð33Þ

ð28Þ

min

ðGi Δuk ek1 ÞT Q 1 0

kek1 k2Q 6 6 G Δu e k1 4 i k Δuk

)

6 6 6 6 4

2

3

)

2

which can be further converted to LMIs via Schur complements:

e

kek1 k2Q

ð35Þ

where the last inequality results from the positive definiteness of R. Therefore, sequence {Vk } is nonincreasing. It is also bounded from below by zero (0). Consequently, sequence {Vk}

ð32Þ 876



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Figure 1. Output response of robust ILC based on the worst-case performance index for θ = 0.1.

Figure 3. Output response of robust ILC based on the nominal performance index for θ = 0.1.

Figure 2. Input response of robust ILC based on the worst-case performance index for θ = 0.1.

Figure 4. Input response of robust ILC based on the nominal performance index for θ = 0.1.

converges. From eq 35, we get ΔuTk 1 RΔuk1 e V k V k1 w lim

ΔuTk 1 RΔuk1 e

w lim

ΔuTk 1 RΔuk1 ¼ 0

ks f∞ ks f∞

lim ðV k V k1 Þ ¼ 0

ks f∞

This implies that Δukf0 as kf∞. In addition, limkf∞(ek1 ek) = limkf∞GΔuk = 0, which implies that ekfe∞. Next, the convergence of ek to zero can be shown by contraction. Assume that e∞ 6¼ 0. The gradient of Φ at Δu∞ = 0 then is ∇Φ∞ ðG, Δu∞ ÞjΔu∞ ¼ 0 ¼ GQ e∞ Since the ellipsoidal uncertainty set (5) and polytopic uncertainty set (6) are compact and convex sets, Π = {GQe∞: G∈Ω} forms a convex compact set. In addition, Π does not include the origin, as Q is positive definite, G has a full row-rank, and e∞ 6¼ 0 by assumption. This means that there exists a

Figure 5. Q-norm of error of robust ILC system versus the number of run for θ = 0.1. 877



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Figure 6. Q-norm of error of robust ILC system versus the run number for θ = 0.16.


Figure 9. Output response of robust ILC based on the worst-case performance index for α = 0.3.


0

nonzero direction p that satisfies: eT∞ Q GT p < 0

5.2. Computational Issue. As shown in section , robust ILC

"G∈Ω

algorithm based on the worst-case and nominal performance indices can be rearranged as optimization of objective functions over a set of LMIs. It is well-known that LMI-based optimization is convex and is therefore a fundamentally tractable optimization problem. Hence, LMI-based optimization has low computational complexity and can be solved in polynomial time. For most ILC systems, since the computation is performed between runs with sufficient time, it is conceivable to carry out LMI-based optimization. However, these types of algorithms would not be suitable for fast batch processes, such as mechanical systems, because the time between runs is limited.

By Taylor’s theorem, we have Φ∞ ðG, tpÞ ¼ Φ∞ ðG, Δu∞ ÞjΔu∞ ¼ 0 þ ∇ΦT∞ ðG, Δu∞ ÞjΔu∞ ¼ 0 3 tp þ oðtÞ where t is a positive scalar. For sufficiently small t, the remainder term is eventually dominated by the first-order term; that is, Φ∞ ðG, tpÞ ≈ eT∞ Q e∞ teT∞ Q GT p < eT∞ Q e∞

"G∈Ω

Since the nominal modelG∈Ω, we have Φ∞(G,tp) < eT∞Qe∞ = Φ∞(G,0). Hence, p is a feasible direction leading away from Δu∞ = 0, along which the nominal performance index decreases while satisfying robust convergence constraint (25). This contradicts the result that Δu∞ = 0 is the optimal solution, hence proves ekf0 as k f∞.

6. NUMERICAL ILLUSTRATIONS Performance of the proposed robust ILC designs is illustrated through two numerical examples. 878



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Figure 10. Input response of robust ILC based on the worst-case performance index for α = 0.3.

Figure 12. Input response of robust ILC based on the nominal performance index for α = 0.3.

The constraints of control input for system are specified by ul ¼ 10,

uh ¼ 10, Δul ¼ 5, Δuh ¼ 5, δul ¼ 5, δuh ¼ 5

In this example, ellipsoidal uncertainty is used to describe uncertain systems. Hence, we have θ0 = 0.345, F = 0.255, and W = I. To illustrate the time responses of the ILC control system, the sampling period is chosen to be ts = 1 and the number of samples is N = 40. The design parameters of the robust ILC are chosen as follows: Q =I, R = 0.02I. The nominal model, 1 0:8es 0:345 GðsÞ ¼ ̅ ð5s þ 1Þð3s þ 1Þ 5s þ 1 The parameter θ for the real process is assumed to be 0.1 and kept constant for the entire iteration. In this work, software CVX20 is used to solve the LMI problem. Figures 1 and 2 plot the output and input of the robust ILC based on the worst-case performance index, while Figures 3 and 4 show the output and input of the robust ILC based on the nominal performance index. The dotted lines represent the output and input of the first run, the dashed lines denote the results of the 5th run and the solid lines plot the results of the 10th run. The gray lines in Figures 1 and 3 represent the reference trajectory. It is clearly shown that the both algorithms can lead to a converged control; however, the robust ILC based on nominal performance index converges much faster than the one based on the worst-case performance index. Figure 5 compares ek Q of these two algorithms under ellipsoidal uncertainty. It confirms that robust ILC based on the nominal performance index gives better performance when the nominal model is close to the true process. The total computation time required for 10 runs of the worstcase performance index and the nominal performance index were ∼8.9 and ∼13.4 s, respectively (i.e., 0.89 and 1.34 s per run), accomplished on a PC with Intel DuoCore 2.65 GHz CPU and 2G memory using Matlab. As mentioned above, the robust ILC algorithm based on the nominal performance index gives better performance when the nominal model is close to the true process. It is necessary to

Figure 11. Output response of robust ILC based on the nominal performance index for α = 0.3.

Example 1. In the first example, it is assumed that the process is described by 1 0:8es þθ GðsÞ ¼ ð5s þ 1Þð3s þ 1Þ 5s þ 1

) )

where 0:85 e θ e 0:16: This linear time-delay system has been widely used by related references, e.g., Lee et al.9 and Nguyen et al.13 The desired reference trajectory is 8 > >0 > < 0:1ðt 5Þ rðtÞ ¼ 0 > > > : 1 0:1ðt 25Þ

t t t t

∈ ∈ ∈ ∈

½0, 5 ∪ ½36, 40 ½6, 15 ½16, 25 ½26, 35 879



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Figure 14. Q-norm of error of the robust ILC versus the run number for α = 0.9.


explore their performances when there is a big model mismatch between the nominal model and the true process. Figures 6, 7, and 8 compare the performance of these two robust ILC algorithms for the real process with parameter θ = 0.16, 0.6, and 0.8, respectively. As illustrated in the figures, the robust ILC algorithm based on the nominal performance index gives excellent performance in most of the uncertainy set (0.6 e θ e 0.16). Only for the extreme cases where 0.85 e θ e 0.6, the robust ILC algorithm based on the worst-case performance index gives a better performance. Example 2. In the second example, the process is described by k where k ¼ 1, 0:1 e α e 0:9 GðsÞ ¼ s þ α The desired reference trajectory is 8 > 0 t∈ > > < 0:1ðt 5Þ t∈ rðtÞ ¼ 0 t∈ > > > : 1 0:1ðt 25Þ t ∈

½0, 5 ∪ ½36, 40 ½6, 15 ½16, 25 ½26, 35


The design parameters of the robust ILC algorithm are chosen as Q =I, R = 0.02I. The nominal model is

The constraints of control input for the system are specified by ul ¼ 8,

uh ¼ 8,

Δul ¼ 3,

GðsÞ ¼ ̅

Δuh ¼ 3,

δul ¼ 3, δuh ¼ 3

The parameter α for the true process is assumed to be 0.3 and kept constant for the entire iteration. Figures 9 and 10 respectively show the output and input of the robust ILC based on the worst-case performance index, while Figures 11 and 12 respectively illustrate the output and input of the robust ILC based on the nominal performance index. The dotted lines represent the output and input of the first run, the dashed lines denote the results of the third run, and the solid lines plot the results of the fifth run. Again, the gray lines represent the reference trajectory. Both algorithms can provide good control performance and convergence to the desired trajectory as the run number increases. Figure 13 compares ek Q of these two robust ILC algorithms under polytopic uncertainty. It is obvious that the robust ILC based on the nominal performance index gives better

The sampling period is again chosen to be ts = 1 and the number of samples is N = 40. In this example, polytopic uncertainty is used to describe uncertain systems. Since 0.1 e α e 0.9, we conclude that G(s)∈Ω = Co(G1(s),G2(s)), where G1 ðsÞ ¼

1 s þ 0:1

G2 ðsÞ ¼

1 s þ 0:9

1 s þ 0:5

) )

and

880



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’ REFERENCES (1) Ahn, H. -S.; Chen, Y. Q.; Moore, K. L. Iterative learning control: brief survey and survey and categorization. IEEE Trans. Syst., Man, Cybern., Part C 2007, 37, 1099. (2) Wang, Y. Q.; Gao, F. R.; Doyle, F. J., III. Survey on iterative learning control, repetitive control, and run-to-run control. J. Process Control 2009, 19, 1589. (3) Tayebi, A.; Zaremba, M. B. Robust iterative learning control design is straightforward for uncertain LTI systems satisfying the robust performance condition. IEEE Trans. Autom. Control 2003, 48, 101. (4) Shi, J.; Gao, F. R.; Wu, T. J. Robust design of integrated feedback and iterative learning control of a batch process based on a 2D Roesser system. J. Process Control 2005, 15, 907. (5) Ahn, H.-S.; Moore, K. L.; Chen, Y. Q. Stability analysis of discrete-time iterative learning control systems with interval uncertainty. Automatica 2007, 43, 892. (6) Liu, T.; Gao, F. R.; Wang, Y. Q. IMC-based iterative learning control for batch processes with uncertain time delay. J. Process Control 2010, 2, 173. (7) van de Wijdeven, J. J. M.; Donkers, M. C. F.; Bosgra, O. H. Iterative learning control for uncertain systems: Noncausal finite time interval robust design. Int. J. Robust Nonlinear Control 2011, 21, 1645. (8) Amann, N.; Owens, D. H.; Rogers, E. Iterative learning control using optimal feedback and feedforward actions. Int. J. Control 1996, 65, 277. (9) Lee, J. H.; Lee, K. S.; Kim, W. C. Model-based iterative learning control with a quadratic criterion for time-varying linear systems. Automatica 2000, 36, 641. (10) Chin, I.; Qin, S. J.; Lee, K. S.; Cho, M. A two-stage iterative learning control technique combined with real-time feedback for independent disturbance rejection. Automatica 2004, 40, 1913. (11) Shi, J.; Gao, F. R.; Wu, T. J. Single-cycle and multi-cycle generalized 2D model predictive iterative learning control (2D-GPILC) schemes for batch processes. J. Process Control 2007, 17, 715. (12) Gao, F. R.; Yang, Y.; Shao, C. Robust iterative learning control with application to injection molding process. Chem. Eng. Sci. 2001, 56, 7025. (13) Gorinevsky, D. Loop shaping for iterative control of batch processes. IEEE Control Syst. Mag. 2002, 6, 55. (14) Ghosh, J.; Paden, B. Pseudo-inverse based Iterative Learning Control for linear nonminimum phase plants with unmodeled dynamics. J. Dyn. Syst., Meas., Control 2004, 126, 661. (15) Van de Wijdeven, J.; Donkers, T.; Bosgra, O. Iterative learning control for uncertain systems: robust monotonic convergence analysis. Automatica 2009, 45, 2383. (16) Nguyen, D. H.; Banjerdpongchai, D. An LMI approach for robust iterative learning control with quadratic performance criterion. J. Process Control 2009, 19, 1054. (17) Boyd, S.; Ghaoui, L. E.; Feron, E.; Balakrishnan, V. Linear Matrix Inequalities in System and Control Theory; SIAM: Philadelphia, PA, 1994, (18) Ding, B. C.; Xi, Y. G.; Cychowski, M. T.; O’Mahony, T. Improving off-line approach to robust MPC based-on nominal performance cost. Automatica 2007, 43, 158. (19) Badgwell, T. A. Robust model predictive control of stable linear systems. Int. J. Control 1997, 68, 797. (20) Grant, M.; Boyd, S.; Ye, Y. CVX: Matlab Software for Disciplined Convex Programming; Stanford University. Available via the Internet at http://www.stanford.edu/∼boyd/cvx, 2008.


) )

performance than robust ILC based on the worst-case performance, in terms of convergence rate and final control error. For the computations, the total time required to compute 10 runs for the worst-case performance index and the nominal performance index were ∼13.5 and ∼18.4 s, respectively (i.e., 1.35 and 1.84 s per run), on a PC with Intel DuoCore 2.65 GHz CPU and 2G memory using Matlab. In Figures 14, 15, and 16, the ek Q values of these two algorithms are compared for the real process parameters (α = 0.9, 0.2, and 0.15, respectively). As illustrated in thefigures, the robust ILC algorithm based on the nominal performance index gives excellent performance in most of the uncertainy set (0.2 e α e 0.9).

7. CONCLUSION In this paper, a novel linear matrix inequality (LMI) approach for robust iterative learning control (ILC) based on the worstcase performance with ellipsoidal uncertainty and polytopic uncertainty is proposed first. Because the min-max optimization results in conservative control and poor performance, another robust ILC algorithm based on the nominal performance index is developed further, with additional constraints to ensure robust convergence. The latter gives better performance when the nominal model is close to the true process. Illustrative examples are performed to demonstrate the effectiveness and merits of the proposed control scheme. ’ AUTHOR INFORMATION Corresponding Author

*Tel.: 86-571-87951011. E-mail: [email protected].

’ ACKNOWLEDGMENT This study is financially supported by National Science Foundation of China (Nos. 60934007 and 60804023), 973 Program of China (Nos. 2012CB720503, 2009CB320603), and Foundation of Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education. We acknowledge their support. 881


Robust Iterative Learning Control with Quadratic Performance Index

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