Robust finite frequency range iterative learning control design and experimental verification

Control Engineering Practice 21 (2013) 1310–1320

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Robust finite frequency range iterative learning control design and experimental verification$ Wojciech Paszke a,n, Eric Rogers b, Krzysztof Gałkowski a, Zhonglun Cai c a

University of Zielona Góra, ul. Podgórna 50, 65-246 Zielona Góra, Poland Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, UK c Faculty of Engineering and the Environment, University of Southampton, Southampton SO17 1BJ, UK b

art ic l e i nf o

a b s t r a c t

Article history: Received 12 September 2012 Accepted 23 May 2013 Available online 5 July 2013

Iterative learning control is an application for two-dimensional control systems analysis where it is possible to simultaneously address error convergence and transient response specifications but there is a requirement to enforce frequency attenuation of the error between the output and reference over the complete spectrum. In common with other control algorithm design methods, this can be a very difficult specification to meet but often the control of physical/industrial systems is only required over a finite frequency range. This paper uses the generalized Kalman–Yakubovich–Popov lemma to develop a two-dimensional systems based iterative learning control law design algorithm where frequency attenuation is only imposed over a finite frequency range to be determined from knowledge of the application and its operation. An extension to robust control law design in the presence of norm-bounded uncertainty is also given and its applicability relative to alternative settings for design discussed. The resulting designs are experimentally tested on a gantry robot used for the same purpose with other iterative learning control algorithms. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Iterative learning control Controller design Kalman–Yakubovich–Popov lemma Linear matrix inequalities Robust control

1. Introduction Many industrial systems execute a task over a finite duration, reset to the starting location and then completes this task over and over again. Each execution is known as a trial or pass and the duration the trial length. Once a trial has been completed all data generated is available to update the control signal for the next trial and thereby improve performance from trial-to-trial. This area is known as Iterative Learning Control (ILC) and since the initial work, widely credited to Arimoto, Kawamura, and Miyazaki (1984), has been an established area of control systems research and application, where one starting point for the literature is the survey papers (Ahn, Chen, & Moore, 2007; Bristow, Tharayil, & Alleyne, 2006). Major application areas include robotics, with recent work in, for example (Barton & Alleyene, 2011), flexible valve actuation for non-throttled engine load control (Heinzen, Gillella, & Sun, 2011) and also a transfer from engineering to next generation healthcare for robotic-assisted upper limb stroke rehabilitation with supporting clinical trials (Freeman et al., 2009, Freeman, Rogers, Hughes, Burridge, & Meadmore, 2012). A significant part of the currently published ILC research starts from a linear time-invariant discrete model of the system ☆ This work is partially supported by Ministry of Science and Higher Education in Poland, Grant no. N N514 636540. n Corresponding author. Tel.: +48 683282611; fax: +48 683247295. E-mail addresses: [email protected] (W. Paszke), [email protected] (E. Rogers), [email protected] (K. Gałkowski), [email protected] (Z. Cai).

0967-0661/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conengprac.2013.05.011

dynamics in either state-space or shift operator/transfer-function form. In this case, one way to do control law design is, since the trial duration is finite, to define super-vectors for the variables. For example, let yk(p) be the scalar, for ease of presentation with a natural extension to the vector case, output on trial k, which is of length α o ∞. Then the super-vector, for linear dynamics and systems with a nonzero first Markov parameter, is Y k ¼ ½yk ð0Þ yk ð1Þ ⋯ yk ðα  1Þ⊤ ; and the ILC problem can hence be written as a system of linear difference equations with updating in k. The current trial error is the difference between the supplied reference signal and the trial output and, since the trial length is finite, trial-to-trial error convergence is independent of the state matrix. This, in turn, could lead to unacceptable along the trial dynamics. In the lifting approach, this problem can be addressed by applying a feedback control law to stabilize the system and/or improve transient performance and then design the ILC law for the controlled system. An alternative is to use a two-dimensional (2D) systems setting where ILC can be represented in this form with one direction of information propagation from trial-to-trial and the other along the trial. Given that the trial length is finite, ILC fits naturally into the repetitive process setting for analysis, where these processes (Rogers, Gałkowski, & Owens, 2007) have their origins in the mining and metal rolling industries and a substantial body of systems theory and control law design algorithms exists for them.

W. Paszke et al. / Control Engineering Practice 21 (2013) 1310–1320

Using the repetitive process setting, it is possible to simultaneously design a control law for trial-to-trial error convergence and along the trial performance. The method is to use a form of stability for repetitive processes that demands a bounded-input bounded-output property independent of the trial length. Control laws designed in this setting have been experimentally tested on a gantry robot replicating a robotic pick and place operation that often arises in industrial applications to which ILC is applicable (Hładowski et al., 2010, 2012). This previous ILC design in a repetitive process setting includes a stability condition that requires frequency gain attenuation over the complete frequency range and hence, by analogy with the standard linear systems case, is a very strict condition, especially as the reference signal many only have significant frequency content over a finite range. This paper develops a new control law design method where frequency attenuation is enforced over a finite range to be decided by frequency decomposition of the reference signal. The design makes use of the generalized Kalman–Yakubovich–Popov (KYP) lemma to establish the equivalence between frequency domain inequalities over finite and/or semi-finite frequency ranges for a transfer-function and a linear matrix inequality (LMI) defined in terms of its state-space realization (Iwasaki & Hara, 2005). The resulting design algorithm is experimentally applied to the gantry robot used in Hładowski et al. (2010, 2012), including the extension to robust design using a norm bounded uncertainty representation that offers advances not possible for the same problem in the lifted setting. The following notation is used throughout this paper. For a matrix X, X ⊤ and Xn denote its transpose and complex conjugate transpose respectively. The null and identity matrices with appropriate dimensions are denoted by 0 and I respectively. Moreover, the notation X≽Y (respectively X≻Y) means that the matrix X−Y is positive semidefinite (respectively, positive definite). Also symfXg is used to denote the symmetric matrix X þ X ⊤ and X ⊥ denotes the orthogonal complement of the matrix X, that is, a matrix whose columns form a basis of the nullspace of X. The symbol ð⋆Þ denotes block entries in symmetric matrices and ρðÞ denotes the spectral radius of its matrix argument, that is, if hi , 1 ≤i ≤h, is an eigenvalue of the h  h matrix H then ρðHÞ ¼ max1 ≤ i ≤ h jλi j.

The systems considered are linear and time-invariant and, after sampling if necessary, described in the ILC setting by the nominal state-space model xk ðp þ 1Þ ¼ Axk ðpÞ þ Buk ðpÞ; ð1Þ

where k≥0 is the trial number, α o ∞ is the number of samples along the trial, xk ðpÞ∈Rn is the state vector, yk ðpÞ∈Rm is the output vector and uk ðpÞ∈Rl is the control input vector. The uncertainty associated with the dynamics is modeled as additive perturbations ΔA, ΔB and ΔC to the matrices A, B and C, respectively, giving the model for design as xk ðp þ 1Þ ¼ ðA þ ΔAÞxk ðpÞ þ ðB þ ΔBÞuk ðpÞ; yk ðpÞ ¼ ðC þ ΔCÞxk ðpÞ:

ð2Þ

These perturbations are assumed to be of the norm-bounded form, that is, ΔA ¼ H 1 F ðpÞE1 ; ΔB ¼ H 1 F ðpÞE2 ; ΔC ¼ H 2 F ðpÞE1 ;

ð3Þ

where H1, H2, E1 and E2 are known real constant matrices of compatible dimensions and F ðpÞ is an uncertain perturbation satisfying F ðpÞF ⊤ ðpÞ⪯I:

Also F ðpÞ is assumed to be independent of the trial number k and hence the allowable uncertainties can vary along each trial but are assumed constant from trial-to-trial. Let rðpÞ∈Rm denote the reference vector and hence the error on trial k is ek ðpÞ ¼ rðpÞ−yk ðpÞ:

ð5Þ

A commonly used ILC strategy to construct the current trial input is as the sum of that used on the previous one plus a corrective term, that is, ukþ1 ðpÞ ¼ uk ðpÞ þ Δuk ðpÞ; k≥0;

ð6Þ

where Δuk ðpÞ denotes the modification term to the control input used on the previous trial. In the lifting approach (the survey papers Ahn et al., 2007; Bristow et al., 2006 are one starting point for the literature) the next step in design is to define the super-vector corresponding to yk of the previous section as Ek ¼ ½e⊤k ð0Þ e⊤k ð1Þ ⋯ e⊤k ðα−1Þ⊤ ; and proceed to write the controlled dynamics in the form Ekþ1 ¼ QEk , where Q is a block lower triangular matrix whose non-zero entries are formed from the Markov parameters of the system state-space model. This approach subsumes the along the trial dynamics and assumes that any requirements beyond trial-totrial error convergence arising in a particular application are, if required, met by first designing a feedback control loop for the system and then applying lifting to the resulting state-space model. This paper uses the repetitive process setting for ILC design where it is possible to simultaneously design a control law for trial-to-trial error convergence and performance along the trials and the extension to robust design is straightforward. A brief discussion of the advantages of robust ILC control law design in this setting as opposed to lifting designs is also given. To obtain a repetitive process description of the ILC dynamics introduce, for analysis purposes only, the vector ηkþ1 ðp þ 1Þ ¼ xkþ1 ðpÞ−xk ðpÞ: Suppose also that in the ILC law (6) Δukþ1 ðpÞ ¼ K 1 ηkþ1 ðp þ 1Þ þ K 2 ek ðp þ 1Þ;

ð7Þ

where K1 and K2 are matrices to be designed. Application of this control law to (2) gives the controlled dynamics state-space model

2. Preliminaries

yk ðpÞ ¼ Cxk ðpÞ; 0 ≤p ≤ α−1;

1311

ð4Þ

b b ηkþ1 ðp þ 1Þ ¼ Aη kþ1 ðpÞ þ B 0 ek ðpÞ; b b e ðpÞ ¼ C η ðpÞ þ D 0 e ðpÞ; kþ1

kþ1

k

ð8Þ

where b ¼ ðA þ ΔAÞ þ ðB þ ΔBÞK 1 ; D b 0 ¼ ðI−ðC þ ΔCÞ  ðB þ ΔBÞK 2 Þ; A b b B 0 ¼ ðB þ ΔBÞK 2 ; C ¼ −ðC þ ΔCÞ  ððA þ ΔAÞ þ ðB þ ΔBÞK 1 Þ:

ð9Þ

The state-space model (8) is that of a discrete linear repetitive process with pass output and state vectors ekþ1 and ηkþ1 , respectively, once the initial conditions are specified, that is, the pass state initial vector ηk ð0Þ, k≥1, and the initial pass profile e0 ðpÞ, 0 ≤p ≤α−1. As this paper uses the repetitive process setting for analysis and design, the word pass will be used instead of trial from this point onwards. Repetitive processes make a series of sweeps or passes through a set of dynamics defined over a finite interval or duration (Rogers et al., 2007). The pass profile is the name given to the output produced on each pass and once a pass is completed the process resets to the starting location and the next one commences. Moreover, the previous pass profile acts as a forcing function on, and hence contributes to, the dynamics of the next pass profile. This is the source of the unique control problem where the output sequence of pass profiles generated can contain oscillations that increase in amplitude in the pass-to-pass direction.

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The stability theory (Rogers et al., 2007) for repetitive processes imposes a bounded-input bounded-output (BIBO) property, in keeping with the unique control problem, on the sequence of pass profiles generated, that is, if the initial pass profile e0 is bounded then a bounded sequence of pass profile vectors fek gk must be generated, where the bounded property is defined in terms of the norm on the underlying function space. Asymptotic stability requires this BIBO property over the finite pass length whereas stability along the pass is stronger in requiring this property uniformly, that is, independent of the pass length. Consider the ILC system (8) in the absence of uncertainty. Then asymptotic stability holds in this case if and only if ρðI−CBK 2 Þ o 1. This is the error convergence result obtained in Kurek and Zaremba (1993) for the ILC law ukþ1 ðpÞ ¼ K 2 ek ðpÞ applied to the nominal model (1) using the stability theory of 2D discrete linear systems described by the Roesser state-space model. This condition imposes no constraint on the along the pass dynamics, due to the finite pass duration. One set of conditions for stability along the pass of (8) are given by the following result.

where Π is a given real symmetric matrix and Θ denotes the following frequency ranges Low frequency range Θ jθj ≤θl (ii) the LMI    A B ⊤ A Ξ I 0 I

B



0

Middle frequency range

High frequency range

θ1 ≤ θ ≤θ2

jθj≥θh

 þ

C 0

⊤  C Π 0 I

D

D I

 ≺0;

ð11Þ

where Q ≻0, P is a symmetric matrix and the matrix Ξ is partitioned as ð12Þ and specified as follows:  for the low frequency range

b B b 0 g is Lemma 1 (Rogers et al., 2007). Assuming that the pair fA; b observable, the ILC system described b ; Ag controllable and the pair fC by (8) is stable along the pass if and only if

ð13Þ

 for the middle frequency range b 0 Þ o1, (i) ρðD b o 1, (ii) ρðAÞ

ð14Þ

(iii) all eigenvalues of the transfer-function matrix GðzÞ ¼ b −1 B b ðzI−AÞ b0 þ D b 0 have modulus strictly less than unity for all C jzj ¼ 1.

 and for the high frequency range ð15Þ

Each of the conditions in this last result can be checked by applying standard discrete linear systems stability tests. The last condition (iii), in the single-input single-output (SISO) case for simplicity, requires that the frequency response of the transferfunction G(z) lies inside the unit circle in the complex plane. In control law design terms, this condition requires frequency gain attenuation over the complete frequency range and is a very strict condition, especially since many reference signals will have dominant frequency content only over a finite range of values. This paper develops new design algorithms where frequency attenuation is only imposed over a finite range of frequencies. The next section develops the stability tests in this setting. Throughout the paper, the following results are used, where the first is the generalized KYP lemma

Remark 1. As emphasized in Iwasaki and Hara (2007), the main use of the generalized KYP lemma is to compute a control law that allows the designer to specify the desired shape of the closed-loop frequency response in specified frequency ranges without the use of frequency gridding and weighting filters. In the ILC application it allows particular frequency ranges to be emphasized in the design. Since the bandwidth of the reference has most affect on the speed of convergence, the main focus in applications will be on the low frequency range. If some frequencies above the bandwidth are considered then better performance is a possibility. In the high frequency range, well above the reference signal bandwidth, noise and non-repeatable disturbances could be present and they cannot be effectively attenuated by the ILC law.

Lemma 2 (Iwasaki and Hara, 2007). For a discrete linear timeinvariant system with transfer-function matrix GðzÞ and frequency response matrix

Lemma 3 (Gahinet and Apkarian, 1994). Given a symmetric matrix Ψ ∈Rqq and two matrices Λ, Σ of column dimension q, there exists a matrix W such that the following LMI holds:

Gðejθ Þ ¼ Cðejθ I−AÞ−1 B þ D;

Ψ þ symfΛ⊤ WΣg≺0;

the following inequalities are equivalent:

if and only if the following two inequalities with respect to W are satisfied: ⊤

"

Gðejθ Þ I

#n " Π

# Gðejθ Þ ≺0 I

∀θ∈Θ;

⊤

Λ⊥ Ψ Λ⊥ ≺0; Σ ⊥ Ψ Σ ⊥ ≺0:

(i) the frequency domain inequality

ð10Þ

ð16Þ

ð17Þ

Lemma 4 (Khargonekar, Petersen, and Zhou, 1990). Let Σ 1 , Σ 2 be real constant matrices of appropriate dimensions then for any real matrix function F ðpÞ satisfying F ðpÞF ⊤ ðpÞ⪯I and a scalar ϵ 4 0 the

W. Paszke et al. / Control Engineering Practice 21 (2013) 1310–1320

following inequality holds: symfΣ 1 F ðpÞΣ 2 g⪯ϵ

−1

Σ 1 Σ ⊤1

þ

ϵΣ ⊤2 Σ 2 :

ð18Þ

3. Stability over a finite frequency range Let sðGðejθ ÞÞ denote the maximum singular value of the frequency response matrix Gðejθ Þ corresponding to G(z) in condition (iii) of Lemma 1. Then, on choosing the matrix Π as   I 0 Π¼ ; ð19Þ 0 −I (10) gives jθ n

jθ

Gðe Þ Gðe Þ o I

∀ θ∈Θ;

or

1313

and hence condition (i) of Lemma 1 is satisfied. Next, the Lyapunov interpretation of stability for standard discrete linear systems guarantees the feasibility of the LMI (23) and therefore condition (ii) of Lemma 1. To separate the matrix Ξ of (12) from the process model matrices and hence obtain (24), write (22) as 2 32 3 Ξ 12 0 2 ⊤ 3 Ξ 11 b B b0 A b ⊤ ⊤ 6 n 7 A I 0 b C b b D 7 b 0 76 4 ⊤ 56 Ξ 12 Ξ 22 þ C C ð26Þ 0 5≺0; 4 54 I b 0 I B ⊤ ⊤ 0 b b C b D b 0 −I 0 I 0 D D 0 0 which is of the form of the first inequality in (17) with 2 3 2 3 Ξ 12 0 Ξ 11 A B0 ⊤ ⊤ 6 n b C b b D 6 7 b0 7 7: Ξ Ξ 22 þ C C Λ⊥ ¼ 4 I 0 5; Ψ ¼ 6 4 12 5 ⊤ ⊤ b b b b 0 I 0 D0 C D 0 D 0 −I

ð27Þ

Also for Λ⊥ of (27)

sðGðejθ ÞÞ o 1

∀θ∈Θ:

ð20Þ

Also for all θ∈½0; 2π jθ

jθ

ρðGðe ÞÞ ≤sðGðe ÞÞ:

ð21Þ

Suppose that conditions (i) and (ii) of Lemma 1 hold, where these are standard discrete linear systems stability conditions (for b respectively). Consider Gðejθ Þ b 0 and A systems with state matrices D of (iii) of this lemma for a specified finite frequency range θ∈Θ over which stability is required, where the selection of Θ is application dependent and is discussed again in the experimental results section of this paper. Then a sufficient condition for the ILC scheme to be stable along the pass over the finite frequency range θ∈Θ is that the inequality (20) holds for this Gðejθ Þ. In the SISO case, equality holds in (21) and (20) is then a necessary and sufficient condition. Applying Lemma 2 gives that condition (iii) of Lemma 1 holds if #⊤ " # " #⊤ " # " b B b B b D b D b0 b0 b0 b0 C C A A Ξ Π ≺0; ð22Þ þ 0 I 0 I I 0 I 0 where Ξ (given by (12)) is the only matrix whose block entries depend on the chosen frequency range, that is, low, middle or high, as specified by (13), (14) and (15), respectively. The condition of (22) cannot, however, be directly applied to control law design since it will involve product terms in P, Q and the controlled process statespace model matrices for any frequency range. The following is the first new result in this paper and overcomes this difficulty.

b B b 0 : Λ ¼ ½−I A Next, to apply the result of Lemma 3 it is necessary to find a matrix Σ ⊥ that satisfies the second inequality in (17), where if 2 3 I 0 6 7 Σ ⊥ ¼ 4 0 0 5; 0 I then Σ ¼ ½0 I 0 and ⊤

Σ ⊥ Ψ Σ ⊥ ≺0: This last expression follows since in the case considered: " # Ξ 11 0 ⊤ ⊤ ≺0; Σ⊥ Ψ Σ⊥ ¼ b D b 0 D 0 0 −I

and hence the inequality holds if and only if the diagonal blocks in (28) satisfy Ξ 11 ≺0;

b ⊤D b D 0 0 −I≺0:

By Lemma 2, Ξ 11 ¼ −P for all considered frequency ranges and b⊤D b hence P≻0 is required. Moreover, D 0 0 −I≺0 is equivalent to (25) and hence to condition (i) of Lemma 1 for stability along the pass. Next, application of Lemma 3 gives that (26) is feasible if 2 3 Ξ 11 Ξ 12 0 ⊤ ⊤ 6 n b C b b D b0 7 6 Ξ 12 Ξ 22 þ C 7 þ symfΛ⊤ WΣg≺0: C ð29Þ 4 5 ⊤ ⊤ b b 0 −I b C b D 0 D D 0

Theorem 5. An ILC system of the form (8) is stable along the pass over the finite frequency range θ∈Θ, if there exist matrices S≻0, P≻0, Q ≻0 and W such that the following two LMIs are feasible: b ⊤ SA−S≺0; b A 2

Ξ 11 6 n 6 Ξ 12 −W ⊤ 6 6 6 0 4 0

ð23Þ Ξ 12 −W ⊤

0

b W þW A Ξ 22 þ A

W B

b⊤ W B b C

−I b D0

⊤b

⊤b

0

3

b 7 7 C 7 ⊤ 7≺0; b 7 D 0 5 −I ⊤

ð24Þ

where Ξ 11 , Ξ 12 and Ξ 22 are the block entries in a matrix Ξ with the structure (12), chosen according to the specific frequency range Θ of interest, that is, the low, middle or high frequency ranges given by (13), (14) and (15), respectively. Proof. If the LMI (24) is feasible then 2 3 b⊤ 4 −I D 0 5≺0; b 0 −I D

ð25Þ

ð28Þ

0

Finally, applying the Schur's complement formula to (29) gives (24) and the proof is complete. □ Remark 2. The last result also gives a direct method for improving the pass-to-pass error convergence by selecting the matrix Π in (19) as " # I 0 Π¼ ; 0 −γ 2 I and minimize γ under constraint 0 oγ ≤1. Using this minimization procedure, the eigenvalues of G(z) in Lemma 1 are assigned to locations inside the circle of radius γ in the complex plane. 4. Control law design Consider first the nominal model case. Then application of Theorem 5 yields matrix inequalities that are not convex. This difficulty is removed by the following result expressed in terms of LMIs that provides an algorithm for designing the control law, that is, stabilizing matrices K1 and K2 in (7) for the chosen frequency

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W. Paszke et al. / Control Engineering Practice 21 (2013) 1310–1320

range obtained by selecting the matrix Ξ to have the structure of (13), (14) or (15), respectively. Theorem 6. An ILC scheme described by the nominal model version of (8) (ΔA ¼ 0; ΔB ¼ 0; ΔC ¼ 0 in (9)) is stable along the pass over the b ≻0, c, b finite frequency range θ∈Θ if there exist matrices X1, X2, W S≻0, Q b ≻0 together with real scalars ρ and ρ such that the following LMIs P 1 2 are feasible: 2 3 bþρ W c −ρ BX 1 −ρ W c þρ W c⊤ c⊤ S −ρ2 AW 2 2 2 1 4 5≺0; ð30Þ c þ ρ BX 1 g ð⋆Þ −b S þ symfρ1 AW 1 2

b 11 Ξ

6 6 6 ð⋆Þ 6 6 6 ð⋆Þ 4

⊤

c b 12 −W Ξ n o c þ BX 1 b Ξ 22 þ sym AW

BX 2

ð⋆Þ

−I

ð⋆Þ

ð⋆Þ

ð⋆Þ

0

9 > > > > > ⊤ ⊤ ⊤ ⊤ ⊤ ⊤> c −W A C −X 1 B C = ≺0; > > > I−X ⊤2 B⊤ C ⊤ > > > ; −I 0

ð31Þ where the compatibly dimensioned matrices Ξ 11 , Ξ 12 , Ξ 22 form Ξ of (12) and ρ1 and ρ2 satisfy ρ21 −ρ22 o0:

ð32Þ

If the LMIs (30) and (31) are feasible, stabilizing control law matrices K1 and K2 in (7) are given by c K 1 ¼ X1W

−1

; K 2 ¼ X2:

Proof. By (31) " # −I I−X ⊤2 B⊤ C ⊤ I−CBX 2

−I

ð33Þ

≺0;

b 0 ¼ I−CBK 2 it is immediate that ρðD b 0 Þ o 1. and for X 2 ¼ K 2 and D Hence feasibility of (31) immediately ensures that the condition (i) of Lemma 1 holds. Next, by Lyapunov stability theory for standard discrete linear systems, condition (ii) of Lemma 1 is equivalent to " # h i S 0  A b ⊤ b b ⊤ SA−S≺0; b ð34Þ ¼A A I 0 −S I where S≻0 is a matrix variable. Introduce " # b A ⊥ Λ ¼ ; I and hence b Λ ¼ ½−I A: Then, for ρ1 and ρ2 satisfying (32), the LMI #  " ρ1 I S 0 ⊤ ¼ ðρ21 −ρ22 ÞS≺0; ½ρ1 I ρ2 I 0 −S ρ2 I holds for any S≻0. Next, introduce " # ρ1 I ; Σ⊥ ¼ ρ2 I and hence it is possible to set Σ ¼ ½−ρ2 I ρ1 I: Application of Lemma 3 now gives that (34) is feasible if and only if (" ) #   h i −I S 0 ρ1 I ≺0; ð35Þ þ sym ⊤ W −ρ2 I b 0 −S A

is solvable for W. If this is the case then S þ ρ2 W þ ρ2 W ⊤ ≺0 and if S≻0, W is invertible. Finally, post- and pre-multiplying (35) by " # 0 W −1 ; 0 W −1 c ¼ W −1 and and its transpose, respectively, and then setting W b c gives the LMI (30). c T SW S ¼W To establish the LMI (31), condition (iii) of Theorem 5 for the ILC system (8) in the nominal model case is 2

Ξ 11 6 Ξ ⊤ −W ⊤ 6 12 6 6 0 4 0

Ξ 12 −W ⊤

0 ⊤

Ξ 22 þ symfW ðA þ BK 1 Þg

W BK 2

K ⊤2 B⊤ W −CðA þ BK 1 Þ

−I I−CBK 2

3

0

−ðA þ BK 1 Þ C 7 7 7≺0: ðI−CBK 2 Þ⊤ 7 5 −I ⊤

⊤

ð36Þ Postand pre-multiplying this last inequality by diagfW −1 ; W −1 ; I; Ig and its transpose, respectively, and then introducing the change of variables b ¼W c; Q c⊤Q W c; X1 ¼ K 1W c; X2 ¼ K 2; b¼W c ⊤ PW P c ⊤ Ξ 11 W c ⊤ Ξ 12 W c ⊤ Ξ 22 W c; Ξ c; Ξ c; b 11 ¼ W b 12 ¼ W b 22 ¼ W Ξ gives immediately that (36) is equivalent to the LMI of (31) and the proof is complete. □ Remark 3. In Theorem 6 the scalars ρ1 and ρ2 must be selected prior to solving the LMIs (30)–(31). Selecting ρ1 and ρ2 to increase pass-to-pass error convergence is one option but the only available method is trial and error as part of the design and simulation of the controlled system prior to physical implementation. Consider the case when uncertainty is present in the model structure and hence the LMIs in the last theorem will include terms formed by multiplication of two matrices, for example, C þ ΔC and A þ ΔA representing the additive uncertainty. Then the result of Lemma 4 cannot be directly applied and the following analysis removes this obstacle to robust ILC law design. Introduce the notation 9 2 2 3 c⊤ b 11 b 12 −W Ξ 0 0 > Ξ > 0 > n o > 6 > 6 0 7 6 ð⋆Þ Ξ c þ BX 1 b 22 þ sym AW BX 2 0 = 6 7 6 ; J¼6 Φ¼6 7; > 4 0 5 6 ð⋆Þ > ð⋆Þ −I I > 4 > > ; −C ð⋆Þ ð⋆Þ ð⋆Þ −I 2

3 0 ⊤ ⊤ ⊤ ⊤ 6W 7 6 c A þ X1 B 7 N¼6 7: ⊤ ⊤ 4 5 X2 B 0

Then the LMI (31) for the nominal model can be written as  " # I Φ N ½I J ⊤ ≺0: J⊤ 0 N

ð37Þ

Also application of the result of Lemma 3 with " #   W1 Φ N ; Λ ¼ I; Σ ¼ ½J ⊤ −I; ; W ¼ Ψ¼ W2 N⊤ 0 gives that (37) holds for some W1 and W2 if and only if (" ) #   W1 ⊤ Φ N ½J −I ≺0: þ sym W2 N⊤ 0 Partitioning W1 as W 1 ¼ ½W ⊤11 W ⊤21 W ⊤31 W ⊤41 ⊤ ;

ð38Þ

W. Paszke et al. / Control Engineering Practice 21 (2013) 1310–1320

1315

enables (38) to be rewritten as 2

c⊤ b 12 −W Ξ

b Ξ 6 11 6 6 ð⋆Þ 6 6 ð⋆Þ 6 6 6 ð⋆Þ 4 ð⋆Þ

−W 11 C ⊤

0

c þ BX 1 g b 22 þ symfAW Ξ

−W 21 C ⊤

−BX 2

3

−W 11

7 7 W A þ X ⊤1 B⊤ −W 21 7 7 7≺0; ⊤ ⊤ X 2 B −W 31 7 7 ⊤ 7 −W 41 −CW 2 5 −W 2 −W ⊤2 ⊤ c ⊤

ð⋆Þ

−I

I−W 31 C ⊤

ð⋆Þ ð⋆Þ

ð⋆Þ ð⋆Þ

−I−symfW 41 C ⊤ g ð⋆Þ

ð39Þ where all terms involving the product of the nominal state-space model matrices are decoupled and W11, W21 and W31, W41 and W2 are new slack matrix variables of compatible dimensions. If uncertainty is present in the state-space model, (39) can be written as 2

b Ξ 6 11 6 6 ð⋆Þ 6 6 6 ð⋆Þ 6 6 6 ð⋆Þ 4 ð⋆Þ

c⊤ b 12 −W Ξ n o c þ BX 1 b 22 þ sym AW Ξ

"

E1 W ⊤11 0

−W 11

c⊤ ⊤

BX 2

−W 21 C ⊤

ð⋆Þ

−I

ð⋆Þ

ð⋆Þ

I−W 31 C ⊤   −I−sym W 41 C ⊤

ð⋆Þ

ð⋆Þ

ð⋆Þ

82 0 > > >6 > > > 6 <6 0 þsym 6 6 0 > > 6 > > >4 −H 2 > : 0

0

9 > > > > > > ⊤ ⊤ > W A þ X 1 B −W 21 > > = ⊤ ⊤ X 2 B −W 31 > > > > > −W 41 −CW ⊤2 > > > > ⊤ ; −W −W

−W 11 C ⊤

0

2

3

H1 7 7" 7 F ðpÞ 0 7 7 0 0 7 5 H1

E1 W ⊤21 c E1 W þ E2 X 1

#

0

4

E1 W ⊤31

E1 W ⊤41

E1 W ⊤2

(" þsym

−ρ2 H 1 ρ1 H 1

E2 X 2

0

0

≺0:

3

c þ E2 X 1  ≺0: F ðpÞ½0 E1 W

2

ð41Þ

c −ρ BX 1 −ρ W c ⊤ −ϵ1 ρ ρ H 1 H ⊤ −ρ2 AW 2 1 1 2 1 c þ ρ BX 1 g þ ϵ1 ρ2 H 1 H ⊤ −b S þ symfρ1 AW 1 1 1

ð⋆Þ

ð⋆Þ

ϵ2 4 0 such that the following LMIs are feasible: ⊤

ð⋆Þ

c b 12 −W Ξ c þ BX 1 g þ ϵ2 H1 H ⊤ b 22 þ symfAW Ξ 1

BX 2

−W 21 C ⊤

ð⋆Þ

ð⋆Þ

−I

I−W 31 C ⊤

ð⋆Þ

ð⋆Þ

ð⋆Þ

−I−symfW 41 C ⊤ g þ ϵ2 H 2 H⊤2

ð⋆Þ ð⋆Þ

ð⋆Þ ð⋆Þ

ð⋆Þ ð⋆Þ

ð⋆Þ ð⋆Þ

ð⋆Þ

ð⋆Þ

ð⋆Þ

ð⋆Þ

b 11 Ξ

W 41 E⊤1

ð⋆Þ

−ϵ2 I

ð⋆Þ

ð⋆Þ

0

W 31 E⊤1 W 2 E⊤1

3

7 c ⊤ E ⊤ þ X ⊤ E⊤ 7 −W 1 1 2 7 7 7 X ⊤2 E⊤2 7 7≺0; 0 7 7 7 0 7 7 0 5 −ϵ2 I

and ρ1 and ρ2 satisfy (32). Consequently, if the LMIs (42) and (43) are feasible, stabilizing control law matrices K1 and K2 in (7) are given by (33). In implementation terms, the control law (6) and (7) is

Theorem 7. An ILC scheme described by (8) with uncertainty structure modeled by (3) and (4) is stable along the pass over finite b b ≻0, P b ≻0, X1, X2, W c, frequency range Θ if there exist matrices S≻0, Q W2, W11, W21, W31, W41 together with real scalars ρ1 , ρ2 , ϵ1 4 0 and

6 6 6 6 6 6 6 6 6 6 6 6 4

−W 41 −CW ⊤2 −W 2 −W ⊤2 þ ϵ2 H 1 H ⊤1

0

ð43Þ

The following is the robust control version of the previous theorem and is proved by similar steps except finally the result of Lemma 4 is applied to (40) and (41) followed by application of the Schur's complement formula. Hence the details are omitted.

2

W 21 E⊤1

ð40Þ

)

#

⊤ 2 b c 6 S þ symfρ2 W g þ ϵ1 ρ2 H 1 H 1 6 6 ð⋆Þ 4

c ⊤ A⊤ þ X ⊤ B⊤ −W 21 þ ϵ2 H 1 H ⊤ W 1 1

#)

c −ρ BX 1 −ρ W c −ρ2 AW 2 1 5 b c −S þ symfρ1 AW þ ρ1 BX 1 g

ð⋆Þ

W 11 E⊤1

X ⊤2 B⊤ −W 31

⊤

b þ symfρ W cg S 2

−W 11 F ðpÞ

Moreover, there are no multiplications between the state-space model matrices in (30) and this LMI can be written as 2

Fig. 1. The gantry robot with the three axes marked.

2

−W 11 C ⊤

ukþ1 ðpÞ ¼ uk ðpÞ þ K 1 ðxkþ1 ðpÞ−xk ðpÞÞ þ K 2 ðrðp þ 1Þ−yk ðp þ 1ÞÞ;

ð44Þ

where the last term is phase-lead ILC. Such a term is allowed since once pass k is complete all data from it is available for use in computing the control to be applied on the next pass. The second term on the right-hand side of (44) involves the difference in the state vectors on successive trials and hence if all entries in the state vector are not available for measurement state estimation is required. In previous work (Hładowski et al., 2012), where stability along the pass is used, it has been shown that the state vector term in the control law can be replaced by pass profile information, that 3 0

7 7 c ⊤ E⊤ 7≺0; X ⊤1 E⊤2 þ W 1 5

ð42Þ

−ϵ1 I

is, directly measured data. This work can be extended in a natural manner to the finite frequency range case. Many alternative approaches to robust ILC control design for discrete linear systems have been reported and one option is to use the lifted model of the dynamics briefly discussed in Section 2 but this will encounter serious difficulties due to the presence of matrix products in the resulting model. For example, the ILC law ukþ1 ðpÞ ¼ uk ðpÞ þ Kek ðp þ 1Þ;

ð45Þ

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W. Paszke et al. / Control Engineering Practice 21 (2013) 1310–1320

the pick and place operation present in many industrial applications to which ILC is applicable. Each axis of the gantry robot is modeled based on frequency response tests where, since the axes are orthogonal, it is assumed that there is minimal interaction between them. Consider the X-axis, the one parallel to the conveyor in Fig. 1, for which frequency response tests, using the Bode gain and phase plots in Fig. 2, result in a 7th order differential linear system transfer-function as an adequate model of the dynamics to use for control law design. The transfer-function has been discretized with a sampling time of T s ¼ 0:01 s to develop a discrete linear state-space model of the form (1) with

results in error dynamics described by the lifted model Ekþ1 ¼ QEk ;

ð46Þ

where 2

3

I−KCB 6 −KCAB 6 Q¼6 4 ⋮

0

⋯

0

I−KCB

⋯

0

⋮

⋱

⋮

α−1

α−2

⋯

I−KCB

−KCA

B

−KCA

B

7 7 7: 5

ð47Þ

Hence when the state-space model matrices A, B, C have uncertainty associated with them that belong to a convex set, Q does not belong to a convex set and only a bound is possible that increases the conservativeness of the results. In the next section, the results of experimental implementation of the robust control law design in this section to a gantry robot undertaking a pick and place operation is given, including how to select the finite frequency range.

2

0:3879 6 −0:3898 6 6 6 0 6 6 0 A¼6 6 6 0 6 6 0 4 0

2

The multi-axis gantry robot (Ratcliffe, Lewin, Rogers, Hatonen, & Owens, 2006) shown in Fig. 1 with the axes marked replicates

0 0 0 0 0

0 0 0:2500

0:1041 0:0849 −0:2006

0 0 0

0 0 0 0

−0:3103 0 0 0

−0:1575 0 0 0

−0:0555 0:0353 −0:0164 0

0 0:5000 0:0353 0

0:0071

0

3 0:0832 0:0678 7 7 7 −0:1603 7 7 7 −0:0444 7; 7 0:2809 7 7 7 −0:2757 5 1:0000

3 7 7 7 7 7 7 7; 7 7 7 7 5 0:0146

0

 0:0057 :

The state matrix A has all eigenvalues inside the unit circle except for one of value unity on the real axis of the complex plane, and

−40

−60 RMS on each pass 10−1

−80 10−1

100

101

102

103

−90

10−2 RMS(e) [m]

Phase (degree)

0:2138 0:1744 −0:1575

0:0910  C ¼ 0:0391 0

−20 Magnitude (dB)

0

6 6 6 6 6 6 B¼6 6 6 6 6 4

5. Experimental verification

1:0000 0:3879 0

−135

10−3 −180 10−1

100

101

102

103 10−4

Frequency (rad/s)

0

Fig. 2. X-axis Bode gain and phase plots where the blue lines represent the measured data and the red lines those for the transfer-function used in design. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

50

100

150

Fig. 4. RMS values of the error for the X-axis over 200 passes.

Single−Sided Amplitude Spectrum

0.04

0.03

0.035 0.025 0.02

0.025 |Y(f)|

Displacement [m]

0.03

0.02

0.015

0.015 0.01 0.01 0.005

0.005 0

0 0

0.5

1 Time [sec]

1.5

200

the pass number

2

0

1

2

3

Frequency (Hz)

Fig. 3. (a) The reference trajectory for the X-axis. (b) Corresponding frequency spectrum.

4

5

0.16

0.16

0.14

0.14

0.12

0.12

1317

Single−Sided Amplitude Spectrum

0.1

0.1 |Y(f)|

Displacement [m]

W. Paszke et al. / Control Engineering Practice 21 (2013) 1310–1320

0.08

0.08

0.06

0.06

0.04

0.04

0.02

0.02 0

0 0

0.5

1

1.5

0

2

1

Time [sec]

2

3

4

5

Frequency (Hz)

Fig. 5. (a) The reference trajectory for the Y-axis. (b) Corresponding frequency spectrum.

Single−Sided Amplitude Spectrum

0.012

0.01 0.009

0.01

0.007

0.008

0.006 |Y(f)|

Displacement [m]

0.008

0.005

0.006

0.004 0.004

0.003 0.002

0.002

0.001 0

0 0

0.5

1

1.5

2

0

1

Time [sec]

2

3

4

5

Frequency (Hz)

Fig. 6. (a) The reference trajectory for the Z-axis. (b) Corresponding frequency spectrum.

RMS on each pass

10−1

RMS on each pass

10−2

10−3

RMS(e) [m]

RMS(e) [m]

10

−2

10−4

10−3 10−5

10−6

10−4 0

50

100

150

200

0

50

100

150

200

the pass number

the pass number

Fig. 7. RMS values of the error for the Y-axis over 200 passes.

Fig. 8. RMS values of the error for the Z-axis over 200 passes.

this model is unstable along the pass. Suppose also that the matrices defining the uncertainty model (3) are

chosen to demonstrate that the robust control design can actually be applied experimentally. Further development is required in this area but it is not possible to product explicit formulas for the terms in the uncertainty model used in this work or others. One area that could be addressed is to follow the tuning based ideas in Ratcliffe, Hatonen, Lewin, Rogers, and Owens (2008) for an adjoint optimal ILC algorithm, which started from an approximate axis model of an integrator and a gain.

H 1 ¼ ½0:3 0 0 0 0 0 0⊤ ; H 2 ¼ 0:01; E2 ¼ 0:01; E1 ¼ ½ 0:3

−0:1

0:1

−0:1

0:05

−0:05

0:1 :

Remark 4. As in all experimental implementations of robust control, modeling of the uncertainty is a critical aspect and is application specific. The uncertainty model in this work is

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W. Paszke et al. / Control Engineering Practice 21 (2013) 1310–1320

The reference signal of the robot is designed to simulate a pick and place operation of duration 2 s and results in the reference trajectory shown in Fig. 3(a) for the X-axis, with associated frequency spectrum in Fig. 3(b) obtained by applying the Fast Fourier Transform (FFT). Inspecting the amplitudes in the frequency spectrum, shows significant harmonics in the range from 0 to 5 Hz and this is taken as the low frequency range and hence θl ¼ 0:3142 and the matrix Ξ has the structure (13). Applying Theorem 7 with ρ1 ¼ 1 and ρ2 ¼ −2 gives the stabilizing robust

RMS(e) [mm]

X−axis

100

0

50

100

200

Y−axis

105 RMS(e) [mm]

150

100 0

50

100

150

200

RMS(e) [mm]

Z−xais

100

0

50

100

150

200

Pass number

Fig. 9. Experimentally measured MSE against pass number for each axis.

control law matrices K 1 ¼ ½−0:3256 −2:6079 −0:7878 −0:2023 −1:6026 0:3601 −10:5012; K 2 ¼ 38:6513:

As a preliminary to experimental verification, a simulation of the controlled system was performed over 200 passes and on completion of each one the Root Mean Square (RMS) value of the tracking error calculated using sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 N 2 ∑ e ðpÞ; RMSðek Þ ¼ αp¼1 k and plotted in Fig. 4 against pass number. This simulation confirms that the designed ILC control law is stabilizing and results in pass-to-pass error convergence. The design was also undertaken for the other two axes (Y and Z) where Ratcliffe et al. (2006) the state-space model matrices are 2 3 2 3 −0:1067 0:1250 0:0777 0 6 7 6 7 A ¼ 4 −0:0211 −0:1067 0:1016 5; B ¼ 4 0 5; 0 0 1:0000 0:0286   C ¼ 0:0360 0 0:0286 ; for the Y-axis model and 3 2 −0:0030 0:0625 0:0758 6 −0:0134 −0:0030 0:0637 7 A¼4 5; 0 0 1:0000   C ¼ 0:0232 0 0:0191 ;

2

3

0

6 B¼4

7 5;

0 0:0191

for the Z-axis. The reference trajectories for the Y and Z-axes are shown in Figs. 5 and 6, respectively, together with their frequency spectrums. In both cases 5 Hz can be chosen as the upper bound for the frequency range where the main harmonic content appears in both cases. For other work based on frequency decomposition of the reference signal, based on frequency sampling filters (Wang, 2009), see Bitmead and Anderson (1981) and the relevant cited references. The matrices defining the uncertainty model (3) for the

Input for pass 200

Input for pass 200

2

5

1 0 0 −5

−1 −2

0

20

40

60

80

100

120

140

160

180

200

−10 0

20

40

60

Output for pass 200

120

140

160

180

200

0.45

Reference Output

Reference Output

0.4

0.3

0.35

0.29

0.3 0

5

100

Output for pass 200

0.32 0.31

80

x 10

20

40

60

−4

80

100

120

140

160

180

200

0.25

Error for pass 200 2

0 x 10

20

40

60

−3

80

100

120

140

160

180

200

140

160

180

200

Error for pass 200

0

0

−2 −5

−4

−10

−6 0

20

40

60

80

100

120

140

Sample Number

Fig. 10. Experimental results for X-axis.

160

180

200

0

20

40

60

80

100

120

Sample Number

Fig. 11. Experimental results for Y-axis.

W. Paszke et al. / Control Engineering Practice 21 (2013) 1310–1320

Control input

Input for pass 200 0.8

1

0.6

0

0.4

−1

0.2 0

20

40

60

80

100

120

140

160

180

200

Output for pass 200 0.07 Reference

Output

control imput u

2

−2

1319

1st pass 10th pass 100th pass 200th pass

0 −0.2 −0.4 −0.6

0.06

−0.8 0.05 −1 0.04 0

2

x 10

20

40

60

−4

80

100

120

140

160

180

200

−1.2 0

0.5

1

1.5

2

time [s]

Error for pass 200

Fig. 13. Control inputs for the X-axis on four representative passes.

0 −2 −4 −6 0

20

40

60

80

100

120

140

160

180

200

Sample Number

Fig. 12. Experimental results for Z-axis.

Y and Z-axes are H 1 ¼ ½0:2 0 0⊤ ; H 2 ¼ 0:01; E2 ¼ 0:02; E1 ¼ ½0:5 −0:1 0:1; and H 1 ¼ ½0:3 0 0⊤ ; H 2 ¼ 0:01; E2 ¼ 0:01; E1 ¼ ½0:2 −0:1 0:1; respectively. Applying the design procedure of Theorem 7 for the low frequency range [0,5] Hz gives the corresponding control law matrices K 1 ¼ ½−4:5044 −0:0198 −27:8846;

K 2 ¼ 23:3258;

for Y-axis (when ρ1 , ρ2 are chosen as ρ1 ¼ 1; ρ2 ¼ −2) and K 1 ¼ ½−2:0851 −0:3364 −51:3834;

K 2 ¼ 102:8790;

for the Z-axis (when ρ1 ¼ 1; ρ2 ¼ −3). The simulated RMS data for the Y and Z-axes are shown in Figs. 7 and 8, respectively. This design has been applied experimentally to the gantry robot. The RMS for each axis plotted against pass number is shown in Fig. 9. Figs. 10, 11, and 12 show the input, output and error, respectively for each axis, against the pass number. These experimentally confirm the simulation predictions. Fig. 13 shows the control input on selected trials. These plots show that the control input is acceptable and converges as the trials proceed. Similar comments can be made for the other two axes.

allow norm-bounded uncertainty. Experimental verification of the control law design algorithm on a gantry robot has also been undertaken with very good agreement between predicted and measured data. In comparison to design using stability along the pass (Hładowski et al., 2010) no optimization is required, i.e., maximizing the value of K2 in the control law to increase the speed of passto-pass error convergence. Also in this previous design a low-pass filter is added to attenuate high frequencies but is not required in this new design. It is, however, possible to include a low-pass filter with pass-band edge above 5 Hz with the aim of increasing the pass-to-pass error convergence. The previous design was completed over the entire frequency range and then the pass-band limited by adding a low-pass filer, whereas the new design limits the frequency range within the design procedure. In the new design there exists possibility that the upper frequency can be maximized, that is, extend the bandwidth and hence the resulting control law will ensure better performance by trading-off between robustness and fast convergence (performance). Also in experimental verification of the design in Hładowski et al. (2010) it was necessary to zero-phase filter the error on each trial. The robust control design in this paper avoids an issue that arises in lifted model approaches by avoiding the product of matrices that define the uncertainty. These results require much further development and comparison with alternative, such as van de Wijdeven, Donkers, and Bosgra (2011) robust control design algorithms. Further work is also required to develop the option of directly maximizing the range of frequencies over which the learning is performed. This can be done by formulating the problem as the minimization of a linear objective function under LMI constraints.

6. Conclusions

References

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