Frequency Domain Adjoint-Based Iterative Learning Control for MIMO Systems

Lee, Yu-Hsiu; Chou, Wei-Yi

doi:10.1115/1.4067535

Abstract

The adjoint of the system is a linear operator that can be interpreted as the transposition of the unit pulse response matrix with a Toeplitz structure. In the context of iterative learning control, the system adjoint can be used as the learning filter by performing dedicated experiments on the system, resulting in monotonically convergent algorithms. However, the required number of experiments equals the product of input and output channels, and the convergence speed is limited by the system peak gain. This work aims to reduce the number of dedicated experiments for multivariate systems and improve the convergence speed of the adjoint-based iterative learning control. The proposed algorithm constructs the system adjoint in frequency domain and therefore the required number of experiments equals the number of input channels. Exploiting the independence of frequency response of linear-time-invariant systems, the learning gain can be frequency-dependent, which further accelerates the convergence speed. Algorithm convergence under frequency domain uncertainties are ensured with learning scalar or diagonal gain design. It is shown that for a single-input-single-output system, the proposed approach becomes the inversion-based algorithms. Convergence condition is also developed for multivariable systems through complex analysis. The developed approach is validated through experiments on a multi-axis galvanometer.

1 Introduction

Iterative learning control (ILC) is used for high performance tracking, and it utilizes the repetition of the tasks and generates feedforward control commands in between iterations so as to overcome the limitations of conventional feedback control. Based on the knowledge of robot dynamics, Arimoto et al. [1] first developed P- and PD-type ILC. With the widespread applications, model-based approaches are proposed using unit pulse response representation [2,3], state-space representation [4,5], and transfer function representations [6,7].

Compared to model-based methods, data-driven approaches have the benefit of no model fitting loss [8,9]. For example, Janssens et al. used input and output signals to estimate a nonparametric convolution matrix of the system, followed by quadratic optimization to determine the optimal input signal [10]. Building on concepts from subspace identification algorithms, Ref. [11] further characterizes the system through input–output trajectories within the optimization framework. Chen and Tsao proposed a meta-learning approach wherein the learning of a nonparametric finite-impulse-response learning filter is formulated as an ILC problem [12]. Above methods often require the input signal to be persistently exciting to avoid algorithm singularity. In contrast, some algorithms solve for the optimal input specific to the tracked trajectory, making them less restrictive. Particularly, Sogo and Adachi [13] utilized time-reversed output error to construct control correction, which is equivalent to employing system adjoint as the learning function. The cascade of a system with its adjoint is zero phase, so the ILC design boils down to a more tractable scalar learning gain design. It is shown that this technique also has the interpretation of gradient descent when formulating as a quadratic optimization problem with the system adopting a unit pulse response Toeplitz matrix representation [14]. Recently, Bolder et al. [15] extend the idea to multivariable systems. Through channel manipulation and signal rerouting, the construction of the data-driven gradient requires $n_{i} \times n_{o}$ experiments per iteration for an $n_{i}$ -input– $n_{o}$ -output system. The scaling problem for massive multi-input–multi-output (MIMO) systems may be addressed by a stochastic approximation algorithm of the system adjoint [16].

In this letter, a different perspective of the system adjoint is adopted in the frequency domain. The Fourier transforms of the input and output signals are used to construct the frequency response function of the system adjoint, resulting in only $n_{i}$ experiments per ILC iteration in a deterministic setting. Furthermore, due to the independence of frequency response of linear time-invariant (LTI) systems, different learning gains can be assigned for different frequencies, accelerating the convergence speed of ILC. Scalar and diagonal learning gain design conditions are developed to ensure algorithm convergence under uncertainties, which can be analyzed using standard signal processing techniques.

The contributions of this letter are as follows:

A deterministic adjoint ILC algorithm for multivariable systems is developed in the frequency domain, requiring fewer experiments per iteration—from $n_{i} \times n_{o}$ to $n_{i}$ —and achieving faster convergence.
Convergence conditions are developed. Insights and connection to inversion-based algorithms are established.
The proposed approach is simulated and experimentally validated on a multi-axis galvanometer system.

The rest of the article is organized as follows. In Sec. 2, adjoint operation and conventional time domain adjoint-based algorithm are first introduced, followed by the proposed adjoint ILC in the frequency domain. Then, Sec. 3 considers a modification with diagonal learning gain design to further enhance the convergence speed. Monotonic convergence is ensured based on the Gershgorin circle theorem. It is shown that the proposed approach reduces to the model-free inversion-based iterative control for single-input–single-output (SISO) systems [17]. Section 4 provides simulation results, experimental validation, and a performance comparison of the algorithms in this work. Finally, conclusions and future directions are given in Sec. 5.

2 Data-Driven Adjoint Iterative Learning Control Algorithms

2.1 Conventional Time Domain Adjoint Iterative Learning Control.

Preliminaries of time domain adjoint ILC in Ref. [15] are briefly reviewed here. A SISO LTI system can be represented by the following infinite impulse-response (IIR) filter:

G (z) = \sum_{i = 0}^{\infty} g_{i} z^{- i}

(1)

where

z

is the

Z

-transform variable and

g_{i}

’s

\in R

are the system Markov parameters. Assuming the input and output signals are of finite length

N

⁠, the system output

y^{1}

under input

u^{1}

can then be expressed as

\underset{y^{1}}{\underset{⏟}{[\begin{matrix} y^{1} [0] \\ y^{1} [1] \\ ⋮ \\ y^{1} [N - 1] \end{matrix}]}} = \underset{G^{11}}{\underset{⏟}{[\begin{matrix} g_{0} & 0 & \dots & 0 \\ g_{1} & g_{0} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & 0 \\ g_{N - 1} & g_{N - 2} & \dots & g_{0} \end{matrix}]}} \underset{u^{1}}{\underset{⏟}{[\begin{matrix} u^{1} [0] \\ u^{1} [1] \\ ⋮ \\ u^{1} [N - 1] \end{matrix}]}}

(2)

with

u^{1} \in R^{N \times 1}

⁠,

y^{1} \in R^{N \times 1}

⁠, and

G^{11} \in R^{N \times N}

being the lifted representation of the system

G (z)

⁠. For a MIMO system of

n_{i}

input channels and

n_{o}

output channels, the input–output (I/O) relation can be expressed as follows:

\underset{y}{\underset{⏟}{[\begin{matrix} y^{1} \\ ⋮ \\ y^{n_{o}} \end{matrix}]}} = \underset{G}{\underset{⏟}{[\begin{matrix} G^{11} & \dots & G^{1 n_{i}} \\ ⋮ & ⋮ \\ G^{n_{o} 1} & \dots & G^{n_{o} n_{i}} \end{matrix}]}} \underset{u}{\underset{⏟}{[\begin{matrix} u^{1} \\ ⋮ \\ u^{n_{i}} \end{matrix}]}}

(3)

Consider the cost criterion

J (u) = \frac{1}{2} e^{⊺} e

(4)

to be minimized for ILC tracking. The tracking error

{e = y}^{des} - y

is defined as the difference between the desired output

y^{des} \in R^{n_{o} N \times 1}

and the system output

y \in R^{n_{o} N \times 1}

⁠. The optimal input for the cost

J

in can be solved iteratively using the gradient descent algorithm

u_{k + 1} = u_{k} - ρ \nabla J (u_{k})

(5)

where the subscript

k

denotes the iteration number,

ρ > 0

is the step size, and

\nabla J (u_{k}) = \partial J_{k} / \partial u_{k}

is the gradient. The gradient for the cost defined in is computed as

\nabla J (u_{k}) = {- G^{⊺} e}_{k}

(6)

The gradient descent algorithm uses the system adjoint $G^{⊺}$ for correction, and hence the name adjoint ILC.

For a SISO system

G^{11}

⁠, note the adjoint

(G^{11})^{⊺} = T G^{11} T

⁠, where the involutory permutation matrix

T = [\begin{matrix} 0 & \dots & 0 & 1 \\ ⋮ & 1 & 0 \\ 0 & ⋮ \\ 1 & 0 & \dots & 0 \end{matrix}] \in R^{N \times N}

(7)

can be interpreted as a time-reversal operator. Therefore, the adjoint of a MIMO system is similarly given by

\begin{aligned} G^{⊺} & = [\begin{matrix} (G^{11})^{⊺} & \dots & (G^{n_{o} 1})^{⊺} \\ ⋮ & ⋮ \\ (G^{n_{i} 1})^{⊺} & \dots & (G^{n_{o} n_{i}})^{⊺} \end{matrix}] \\ = \underset{T_{n_{i}}}{\underset{⏟}{[\begin{matrix} T & 0 \\ ⋱ \\ 0 & T \end{matrix}]}} \underset{\tilde{G}}{\underset{⏟}{[\begin{matrix} G^{11} & \dots & G^{n_{o} 1} \\ ⋮ & ⋮ \\ G^{1 n_{i}} & \dots & G^{n_{o} n_{i}} \end{matrix}]}} \underset{T_{n_{o}}}{\underset{⏟}{[\begin{matrix} T & 0 \\ ⋱ \\ 0 & T \end{matrix}]}} \end{aligned}

(8)

This representation, involving the involutory permutation matrix, shows that the adjoint-based correction ${G^{⊺} e}_{k}$ can be obtained through $n_{i} \times n_{o}$ experiments by manipulating the signal. For a two-by-two system, the procedure for constructing ${G^{⊺} e}_{k}$ is illustrated in Fig. 1.

Fig. 1

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Schematic of the experiment procedures of the time domain adjoint ILC algorithm

The maximum step size, namely the learning gain, can be determined by substituting into Eq.

\begin{aligned} u_{k + 1} & = u_{k} + ρ G^{⊺} e_{k} \\ = (I_{n_{i}} - ρ G^{⊺} G) u_{k} + ρ G^{⊺} y^{des} \end{aligned}

(9)

where

I_{n_{i}}

is an identity matrix of size

n_{i} \times n_{i}

⁠. The optimal learning gain is thus

ρ^{*} = \frac{1}{‖ G^{⊺} G ‖} = \frac{1}{‖ G ‖^{2}}

(10)

where

‖ \cdot ‖

denotes the matrix norm, and the optimal value can be estimated by the ratio of

{‖ e}_{k} ‖^{2}

over

{‖ G^{⊺} e}_{k} ‖^{2}

⁠.

According to Szego’s limit theorem, the Toeplitz matrix norm

‖ G ‖

approaches the infinity norm

‖ G (z) ‖_{\infty} = max_{ω} σ_{1} [G (e^{j ω})]

⁠, where

σ_{1}

denotes the maximum singular value, of an LTI system when the trajectory length

N

is sufficiently large [18]. Therefore, the learning gain in is equivalent to

ρ^{*} = \frac{1}{max_{ω} ‖ G (e^{j ω}) ‖^{2}} \approx \frac{‖ e_{k} ‖^{2}}{‖ {G^{⊺} e}_{k} ‖^{2}}

(11)

indicating the learning gain of time domain adjoint ILC is limited by the peak gain of the LTI system.

2.2 Proposed Frequency Domain Adjoint Iterative Learning Control.

The transposition of $G^{11}$ in Eq. (2) amounts to changing the variable from $z$ to $z^{- 1}$ of the IIR representation in (1). From the time domain definitions of the system and its adjoint in Eqs. (3) and (8), it is observed that the system adjoint in frequency domain should be the Hermitian transpose $G^{H} (e^{j ω}) \in C^{n_{i} \times n_{o}}$ of the frequency response matrix (FRM) $G (e^{j ω}) \in C^{n_{o} \times n_{i}}$ ⁠.

In this work, we propose estimating the adjoint

{\hat{G}}^{H}

in between ILC iterations, and the frequency domain adjoint-based update law is thus written as

U_{k + 1} (n) = U_{k} (n) + ρ_{k} (n) {\hat{G}}^{H} (e^{j ω_{n}}) E_{k} (n)

(12)

given that

\hat{G} (e^{j ω_{n}})

is the estimated FRM evaluated at

ω_{n} = 2 π n / N

(frequency index

n = 0, \dots, N - 1

⁠),

U_{k} (n) \in C^{n_{i} N \times 1}

and

E_{k} (n) \in C^{n_{o} N \times 1}

represent the discrete Fourier transform (DFT) of vector-valued signals of

u_{k} [t] \in R^{n_{i}}

and

e_{k} [t] \in R^{n_{o}}

with discrete time index

t

⁠,

ρ (n)

is a real-valued, frequency-dependent learning gain matrix.

The system FRM

\hat{G} (e^{j ω_{n}}) = [\begin{matrix} {\hat{G}}^{11} (e^{j ω_{n}}) & \dots & {\hat{G}}^{1 n_{i}} (e^{j ω_{n}}) \\ ⋮ & ⋮ \\ {\hat{G}}^{n_{o} 1} (e^{j ω_{n}}) & \dots & {\hat{G}}^{n_{o} n_{i}} (e^{j ω_{n}}) \end{matrix}]

(13)

is first constructed by injecting a signal into each input channel one at a time [19]. At iteration

k

⁠, there are a total of

n_{i}

dedicated experiments, and the channels of the trial input for the

j

th experiment (⁠

j = 1, \dots, n_{i}

⁠) are set as follows:

{\begin{matrix} {\bar{u}}_{k}^{j} [t] = u_{k}^{m} [t], & j = m \\ {\bar{u}}_{k}^{j} [t] = 0, & j \neq m \end{matrix}

(14)

The corresponding trial output signals are denoted by

{\bar{y}}_{k}^{i j}

(⁠

i = 1, \dots, n_{o}

⁠). Consequently, the

(i, j)

th entry of

\hat{G} (e^{j ω_{n}})

can be computed via

{\hat{G}}^{i j} (e^{j ω_{n}}) = \frac{{\bar{Y}}_{k}^{i j} (n)}{{\bar{U}}_{k}^{j} (n)} = \frac{F {{\bar{y}}_{k}^{i j} [t]}}{F {{\bar{u}}_{k}^{j} [t]}}

(15)

where

F {\cdot}

is the DFT operation. The FRM is estimated one column at a time, so only

n_{i}

experiments are required. For a two-by-two system, the procedure for constructing

\hat{G} (e^{j ω_{n}})

is illustrated in Fig. 2. Note that this is

n_{o}

times fewer experiments than the conventional time domain approach in Fig. 1 in a deterministic setting.

Fig. 2

View large Download slide

Schematic of the experiment procedures of the proposed frequency domain adjoint ILC algorithm

After obtaining

{\hat{G}}^{i j} (e^{j ω_{n}})

⁠, the estimated adjoint

{\hat{G}}^{H} (e^{j ω_{n}})

is readily available by taking the Hermitian transpose. Substituting

E_{k} (n) = Y^{des} (n) - Y (n)

into , the frequency domain recursion becomes

\begin{aligned} U_{k + 1} (n) & = (I_{n_{i}} - ρ_{k} (n) {\hat{G}}^{H} (e^{j ω_{n}}) G (e^{j ω_{n}})) U_{k} (n) \\ + ρ_{k} (n) {\hat{G}}^{H} (e^{j ω_{n}}) Y^{des} (n) \end{aligned}

(16)

The algorithm convergence is determined by the propagation factor

D (e^{j ω_{n}}) ≜ I_{n_{i}} - ρ_{k} (n) {\hat{G}}^{H} (e^{j ω_{n}}) G (e^{j ω_{n}})

(17)

Provided the estimate is exact, namely

{\hat{G}}^{H} = G^{H}

⁠, and assume the learning matrix

ρ_{k} (n) = ρ_{k} (n) I_{n_{i}}

is the product of a scalar and an identity matrix, then the optimal learning gain rendering

D

is

ρ_{k}^{*} (n) = \frac{1}{‖ G^{H} (e^{j ω_{n}}) G (e^{j ω_{n}}) ‖} = \frac{1}{‖ G (e^{j ω_{n}}) ‖^{2}}

(18)

Compared to Eq. (11), frequency domain adjoint ILC has the additional flexibility to assign frequency-dependent learning gain in accordance with the system magnitude, thus leading to improved convergence speed.

3 Learning Gain Design for Robustness and Convergence Speed

3.1 Motivating Example: Naive Learning Gain Design.

In the presence of measurement noise, ${\hat{G}}^{H}$ may not be exactly $G^{H}$ ⁠. In this case, the naive learning gain design in (18) may lead to algorithm divergence as the following example.

Example 1

Consider the learning of a two-by-two system. The estimated

{\hat{G}}^{H}

and

\hat{G}

at

ω_{n}

result in the propagation factor

D = \underset{I_{n_{i}}}{\underset{⏟}{[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]}} - ρ_{k} \underset{{\hat{G}}^{H}}{\underset{⏟}{{[\begin{matrix} g_{11} & 0 \\ 0 & g_{22} \end{matrix}]}^{H} [\begin{matrix} e^{j π} & 0 \\ 0 & e^{j 0} \end{matrix}]}} \underset{G}{\underset{⏟}{[\begin{matrix} g_{11} & 0 \\ 0 & g_{22} \end{matrix}]}}

The diagonal system only has phase uncertainty, and we assume $| g_{11} | > | g_{22} |$ without loss of generality. However, the choice of $ρ_{k} = 1 / ‖ G ‖^{2} = 1 / | g_{11} |^{2}$ will render the first eigenvalue of $D$ to be $1 + (| g_{11} |^{2} / | g_{11} |^{2}) = 2$ ⁠, which is outside the unit circle, thus leading to divergence.

3.2 Robust Learning Gain Design Through Complex Analysis.

To account for different uncertainties in estimating the columns of $\hat{G}$ ⁠, as well as to improve the convergence rate, herein a diagonal learning gain matrix $ρ (n) = diag (ρ_{1} (n), \dots, ρ_{n_{i}} (n))$ is employed.

The uncertainty is defined as the mismatch to the cascade of the system and its true adjoint:

\begin{aligned} Δ (e^{j ω_{n}}) & = {\hat{G}}^{H} (e^{j ω_{n}}) G (e^{j ω_{n}}) - G^{H} (e^{j ω_{n}}) G (e^{j ω_{n}}) \\ = ({\hat{G}}^{H} (e^{j ω_{n}}) - G^{H} (e^{j ω_{n}})) G (e^{j ω_{n}}) \\ = δ (e^{j ω_{n}}) G (e^{j ω_{n}}) \end{aligned}

(19)

where

δ (e^{j ω_{n}}) ≜ ({\hat{G}}^{H} (e^{j ω_{n}}) - G^{H} (e^{j ω_{n}}))

represents the difference to the true system adjoint. With the definition of

Δ

⁠, the propagation factor

D

in Eq. can be rewritten as

\begin{aligned} D & = I_{n_{i}} - {ρ \hat{G}}^{H} G \\ = I_{n_{i}} - ρ Δ - {ρ G}^{H} G \\ = I_{n_{i}} - ρ Δ - ρ H \end{aligned}

(20)

where

H ≜ G^{H} G

⁠. Note

H

is Hermitian and thus the diagonal elements

H_{i, i}

’s are positive real numbers. The goal is to design a proper

ρ

such that the eigenvalues of

D

lie within the unit circle, namely

| λ_{i} (D) | < 1

\forall i = 1, \dots, n_{i}

⁠.

Lemma 1

Let

Δ_{i, j}

and

H_{i, j}

be the

(i, j)

th entry of

Δ

and

H

⁠. The uncertainty is complex-valued and

Δ_{i, j} = M_{i, j} e^{j ϕ_{i, j}} = A_{i, j} + j B_{i, j}

⁠. Then the frequency domain MIMO adjoint ILC in Eq. converges at the

n

th frequency if

ρ_{i} (n) K_{i} (n) < 1

(21)

K_{i} (n) < H_{i, i} (n) + A_{i, i} (n)

(22)

ρ_{i} (n) < \frac{2 [H_{i, i} (n) + A_{i, i} (n) - K_{i} (n)]}{H_{i, i}^{2} (n) + M_{i, i}^{2} (n) + 2 H_{i, i} (n) A_{i, i} (n) - K_{i}^{2} (n)}

(23)

where

K_{i} ≜ \sum_{j \neq i} | Δ_{i, j} + H_{i, j} |

⁠.

Proof

From Gershgorin’s circle theorem [20], the eigenvalues of a given matrix

D \in R^{n_{i} \times n_{i}}

lie within the combined set of Gershgorin’s circles, each centered at

C_{i} = D_{i, i}

with a radius of

R_{i} = \sum_{j \neq i} | D_{i, j} |

⁠. The largest eigenvalue of the square matrix

D

can be bounded by the unit circle centered at the origin if

| C_{i} | + R_{i} < 1, i = 1, \dots, n_{i}

(24)

which can be expanded as

| 1 - ρ_{i} H_{i, i} - ρ_{i} Δ_{i, i} | < 1 - \sum_{j \neq i} | ρ_{i} Δ_{i, j} + ρ_{i} H_{i, j} |

(25)

The right-hand side of Eq. must be positive, which is true if

ρ_{i} K_{i} < 1

from Eq. . Squaring the above and substituting

Δ_{i, j} = A_{i, j} + j B_{i, j} = M_{i, j} e^{j ϕ_{i, j}}

gives

\begin{aligned} ρ_{i}^{2} (H_{i, i}^{2} + M_{i, i}^{2} - K_{i}^{2} + 2 H_{i, i} A_{i, i}) \\ < 2 ρ_{i} (H_{i, i} + A_{i, i} - K_{i}) \end{aligned}

(26)

Note that the right-hand side is positive if Eq. (22) is satisfied. The left-hand side is also positive because $K_{i}^{2} < (H_{i, i} + A_{i, i})^{2} < H_{i, i}^{2} + M_{i, i}^{2} + 2 H_{i, i} A_{i, i}$ using $A_{i, i} \leq M_{i, i}^{2}$ ⁠. This leads to the maximum learning gain in Eq. (23). □

In practice, these conditions can be evaluated by

ρ_{i} {\bar{K}}_{i} < 1

(27)

{\bar{K}}_{i} < {\hat{H}}_{i, i} - {\bar{Δ}}_{i, i}

(28)

{\bar{ρ}}_{i} = min_{p = \pm 1} \frac{2 [{\hat{H}}_{i, i} + p {\bar{Δ}}_{i, i} - {\bar{K}}_{i}]}{{\hat{H}}_{i, i}^{2} + 2 p {\bar{Δ}}_{i, i} {\hat{H}}_{i, i} + {\bar{Δ}}_{i, i}^{2} - {\bar{K}}_{i}}

(29)

The evaluation of the uncertainty

Δ

is achieved through the relationship

Δ = δ G

in Eq. , considering the worst-case scenario:

\begin{aligned} | Δ_{i, j} (e^{j ω_{n}}) | & \leq | {\bar{Δ}}_{i, j} (e^{j ω_{n}}) | \\ = \sum_{l = 1}^{n_{o}} | {\bar{δ}}_{i, l} (e^{j ω_{n}}) | \cdot | G_{l, j} (e^{j ω_{n}}) | \end{aligned}

(30)

where

{\bar{δ}}_{i, l}

is the estimated bound on the

(i, l)

th entry of matrix

δ

⁠, given by

{\bar{δ}}_{i, l} (e^{j ω_{n}}) = max_{k} | δ_{i, l} (e^{j ω_{n}}) |

(31)

and

| G_{l, j} (ω_{n}) |

is bounded by the maximum magnitude across iterations:

| G_{l, j} (e^{j ω_{n}}) | \leq max_{k} | G_{l, j} (e^{j ω_{n}}) |

(32)

Note that the subscript denotes the entry and the iteration index

k

is dropped for simplicity of notation. The worst-case

{\bar{K}}_{i}

is evaluated in a similar fashion. The diagonal elements

{\hat{H}}_{i, i}

’s of

G^{H} G

are obtained by averaging across ILC iterations when estimating

G

⁠. For optimal performance, the following learning gain is typically used together with conditions and .

\begin{aligned} ρ_{i}^{*} & = min_{p = \pm 1} \frac{{\hat{H}}_{i, i} + p {\bar{Δ}}_{i, i} - {\bar{K}}_{i}}{{\hat{H}}_{i, i}^{2} + 2 p {\bar{Δ}}_{i, i} {\hat{H}}_{i, i} + {\bar{Δ}}_{i, i}^{2} - {\bar{K}}_{i}} if {\bar{U}}_{k}^{j} (n) \neq 0 \\ = 0 if {\bar{U}}_{k}^{j} (n) = 0 \end{aligned}

(33)

The complete ILC algorithm is presented in Algorithm 1.

MIMO data-driven frequency domain adjoint ILC

Algorithm 1

Given a vector-valued reference signal $y^{des}$ ⁠, initialize the input signal as $u_{1} = y^{des}$ ⁠, and set the iteration index to $k = 1$ ⁠. Then, proceed with the following:

(1)
Apply input $u_{k} [t]$ to the system, then collect output $y_{k} [t]$ and compute $e_{k} [t] = y^{des} [t] - y_{k} [t]$ ⁠.
(2)
Perform $n_{i}$ experiments with ${\bar{u}}^{j}$ (⁠ $j = 1, \dots, n_{i}$ ⁠) as the nontrivial input at the $j$ th channel, and obtain the output ${\bar{y}}^{i j}$ at the $i$ th channel (⁠ $i = 1, \dots, n_{o}$ ⁠) for each experiment.
(3)
Perform DFT on time domain signals and obtain ${\bar{U}}^{j} (n)$ ⁠, ${\bar{Y}}^{i j} (n)$ ⁠. Estimate the system by evaluating each entry ${\hat{G}}^{i j} (e^{j ω_{n}}) = {\bar{Y}}^{i j} (n) / {\bar{U}}^{j} (n)$ ⁠.
(4)
Construct the estimated system $\hat{G} (e^{j ω_{n}})$ and take the Hermitian transpose to obtain the adjoint ${\hat{G}}^{H} (e^{j ω_{n}})$ ⁠.
(5)
Quantify the uncertainties ${\bar{Δ}}_{i, j}$ 's and ${\bar{K}}_{i}$ 's per Eq. (30), examine conditions (27) and (28), then design the learning gain $ρ_{i}$ by Eq. (33).
(6)
Update the control input via $U_{k + 1} (n) = U_{k} (n)$ $+ ρ_{k} (n) {\hat{G}}^{H} (e^{j ω_{n}}) E_{k} (n)$ ⁠. Then, perform inverse DFT to acquire the time domain input $u_{k + 1} [t]$ for the next iteration. Terminate iterations if the stopping criterion is met, otherwise increment the iteration index $k \leftarrow k + 1$ and go to step 1.

3.3 Connection to the Inversion-Based Algorithm for SISO Systems.

For SISO systems (⁠

{\bar{K}}_{i} = 0

⁠), when there is no uncertainty (⁠

{\bar{Δ}}_{i, i} = 0

⁠), Eq. becomes

ρ_{i}^{*} (n) = 1 / | G (e^{j ω_{n}}) |^{2}

⁠. The product of learning gain and the adjoint is

\frac{1}{G^{H} (e^{j ω_{n}}) G (e^{j ω_{n}})} \cdot G^{H} (e^{j ω_{n}}) = G^{- 1} (e^{j ω_{n}})

(34)

which is exactly the system inversion.

In practice, when

\hat{G}

is not exact, rewrite the propagation factor in for scalar

D

⁠:

\begin{aligned} D (e^{j ω_{n}}) & = 1 - ρ (n) {\hat{G}}^{H} (e^{j ω_{n}}) G (e^{j ω_{n}}) \\ = 1 - ρ (n) | \hat{G} (e^{j ω_{n}}) | \cdot | G (e^{j ω_{n}}) | e^{j (\hat{θ} (n) - θ (n))} \end{aligned}

(35)

where

G (e^{j ω_{n}}) = | G (e^{j ω_{n}}) | e^{j θ (n)}

is in phasor form.

The uncertainties, again under the worst-case scenario, can be evaluated by the maximum squared magnitude and the maximum difference in phase angle across ILC iterations:

{\begin{matrix} | \hat{G} (e^{j ω_{n}}) |_{max}^{2} = max_{k} | \hat{G} (e^{j ω_{n}}) |^{2} \\ Δ \hat{θ} (n) = max_{k} \hat{θ} (n) - min_{k} \hat{θ} (n) \end{matrix}

(36)

To ensure

| D | < 1

for algorithm convergence, the optimal learning gain

ρ

is designed as

ρ^{*} (n) = {\begin{matrix} \frac{\cos (Δ \hat{θ} (n))}{| \hat{G} (e^{j ω_{n}}) |_{max}^{2}} & if Δ \hat{θ} (n) \leq \frac{π}{2} \\ 0 & if Δ \hat{θ} (n) > \frac{π}{2} \end{matrix}

(37)

This design has a graphical interpretation in the complex plane as in Fig. 3: the squared inverse $1 / | G |^{2}$ limits the range of $D$ within a unit circle centered at $1 + j 0$ ⁠, and the factor $\cos Δ \hat{θ}$ produces the minimum distance to the origin when the phase condition is satisfied. When phase uncertainty $Δ \hat{θ} (n) > π / 2$ ⁠, the propagation factor $D$ will be outside of the unit circle centered at the origin for any $ρ (n) > 0$ ⁠.

Fig. 3

Complex plane visualization of the propagation factor D=1−ρG^G: (a) when Δθ^≤π/2, ρ*=cosΔθ^/|G^|max2 produces the fastest convergence and (b) when Δθ^>π/2, |D|>1 for any ρ>0

View large Download slide

Complex plane visualization of the propagation factor $D = 1 - ρ \hat{G} G$ ⁠: (⁠ $a$ ⁠) when $Δ \hat{θ} \leq π / 2$ ⁠, $ρ^{*} = \cos Δ \hat{θ} / | \hat{G} |_{max}^{2}$ produces the fastest convergence and (⁠ $b$ ⁠) when $Δ \hat{θ} > π / 2$ ⁠, $| D | > 1$ for any $ρ > 0$

Example 2

In this simulation study, we will consider the 200 Hz triangular waveform tracking (axis 1 in Fig. 6) of a galvanometer (GVSM002, Thorlabs) with a sampling frequency of 20 kHz. The measurement noise is white and zero-mean with a variance of

1.9 \times 10^{- 12}

⁠. This system has the following transfer function:

\begin{aligned} G (z) & = \frac{3.2657 \times 10^{- 5} (z^{2} - 6.333 z + 73.67)}{(z^{2} - 1.034 z + 0.5399) (z^{2} - 1.881 z + 0.898)} \\ \times \frac{(z^{2} + 0.1286 z - 0.8714) (z + 1) (z - 0.8714)}{(z^{2} - 1.267 z + 0.461) (z - 0.5753)} \end{aligned}

(38)

In Fig. 4, the root-mean-square (RMS) error of ILC tracking shows that the proposed frequency domain adjoint method outperforms the time domain adjoint algorithm due to the frequency-dependent learning gain, which is further illustrated in Fig. 5. The frequency domain adjoint ILC modulates the learning gain $ρ (n)$ at each frequency $ω_{n}$ according to the magnitude of the system, resulting in the inversion of the system and thus accelerating convergence. Meanwhile, the time domain method has a constant learning gain limited by the peak gain of the system.

Fig. 4

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ILC convergence of the frequency and time domain adjoint algorithms of the SISO system (38) from tracking a 200 Hz triangular waveform (reference trajectory of axis 1 in Fig. 6)

Fig. 5

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System magnitude $| G (e^{j ω_{n}}) |$ ⁠, the ideal learning gain $1 / | G (e^{j ω_{n}}) |^{2}$ of noise-free adjoint ILC, the frequency domain adjoint learning gain $ρ (n)$ from Eq. (37), and time domain adjoint learning gain from the ratio of $‖ e_{k} ‖^{2}$ over $‖ {G^{⊺} e}_{k} ‖^{2}$ ⁠. The patched areas are the regions where $ρ (n) = 0$ due to large phase uncertainty.

4 Experimental Results

4.1 Setup and Task Description.

The galvanometer in Example 2 has two orthogonal axes and will be used for experimental validation. Each axis has a pre-stabilized position servo control with a sampling frequency of 20 kHz, managed by a real-time target (PCIe-7841r, National Instruments). The motion of the two axes generates scanning patterns for applications in profilometry and additive manufacturing.

To showcase the proposed frequency domain algorithm against the conventional time domain approach for MIMO systems, cross-couplings are introduced: the input from axis 1 is multiplied by a weighting factor $w_{1} = 0.1$ and sent to the input of axis 2, and the input from axis 2 is multiplied by $w_{2} = 0.1$ and sent to the input of axis 1.

The tracking control objective is to follow a spatial raster scanning pattern. As shown in Fig. 6, the decomposed waveform of axis 1 is a 200 Hz triangular wave with a magnitude from 0 to $1 \deg$ ⁠, while the decomposed waveform of axis 2 is a stepping ramp with intermittent stops, also at 200 Hz. Both of which is filtered by a zero-phase low-pass filter $Q (z) = (z + 2 + z^{- 1}) / 4$ ⁠.

Fig. 6

View large Download slide

The desired reference trajectories decomposed from a raster scanning pattern for the two motion axes of the galvanometer

4.2 Results and Discussions.

In Fig. 7, the composite RMS error of frequency and time domain adjoint algorithms are presented. For this two-by-two system (⁠ $n_{i} = n_{o} = 2$ ⁠), the proposed algorithm reduces the number of dedicated experiments from $n_{i} \times n_{o} = 4$ to $n_{i} = 2$ per ILC iteration compared to the conventional time domain method. Additionally, due to the frequency-dependent learning gain, the proposed algorithm achieves faster convergence for each ILC update. The time domain error progression of each axis is supplemented in Fig. 8. The error is progressively flattened to zero as iterations advance. It should be noted that the Fourier transform assumes signal periodicity, making it more suitable for trajectories with repeated patterns, such as the sawtooth and staircase waveforms discussed. Furthermore, the proposed algorithm leverages the frequency independence inherent in LTI systems, which limits its applicability to strongly nonlinear or time-varying systems.

Fig. 7

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ILC convergence of the frequency and time domain adjoint algorithms of the MIMO system from tracking the trajectories presented in Fig. 6. Note that the horizontal axis denotes the number of experiments rather than ILC iterations to demonstrate the reduced interleaving trials.

Fig. 8

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Error progression of the frequency domain adjoint ILC for the MIMO galvanometer system

5 Conclusion and Future Works

In this work, a data-driven frequency domain adjoint-based ILC algorithm is developed for MIMO systems. First, fewer experiments for adjoint system identification are achieved by sequentially injecting signals to input channels. Second, the convergence speed is further improved with frequency-dependent learning gain, which is typically limited by the peak gain of the systems in conventional adjoint ILC. Third, by taking uncertainties into consideration when designing learning gain, algorithm robustness is validated through experiments. Future directions include less conservative learning gain design utilizing the knowledge of noise characteristics.

Funding Data

The National Science and Technology Council of Taiwan under Grant NSTC 112-2628-E-002-024 and NSTC 113-2628-E-002-031.
The National Taiwan University Excellence Research Program, Grant 113L893901.

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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Frequency Domain Adjoint-Based Iterative Learning Control for MIMO Systems

Abstract

1 Introduction

2 Data-Driven Adjoint Iterative Learning Control Algorithms

2.1 Conventional Time Domain Adjoint Iterative Learning Control.

2.2 Proposed Frequency Domain Adjoint Iterative Learning Control.

3 Learning Gain Design for Robustness and Convergence Speed

3.1 Motivating Example: Naive Learning Gain Design.

3.2 Robust Learning Gain Design Through Complex Analysis.

MIMO data-driven frequency domain adjoint ILC

3.3 Connection to the Inversion-Based Algorithm for SISO Systems.

4 Experimental Results

4.1 Setup and Task Description.

4.2 Results and Discussions.

5 Conclusion and Future Works

Funding Data

Conflict of Interest

Data Availability Statement

References

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Frequency Domain Adjoint-Based Iterative Learning Control for MIMO Systems

Abstract

1 Introduction

2 Data-Driven Adjoint Iterative Learning Control Algorithms

2.1 Conventional Time Domain Adjoint Iterative Learning Control.

2.2 Proposed Frequency Domain Adjoint Iterative Learning Control.

3 Learning Gain Design for Robustness and Convergence Speed

3.1 Motivating Example: Naive Learning Gain Design.

3.2 Robust Learning Gain Design Through Complex Analysis.

MIMO data-driven frequency domain adjoint ILC

3.3 Connection to the Inversion-Based Algorithm for SISO Systems.

4 Experimental Results

4.1 Setup and Task Description.

4.2 Results and Discussions.

5 Conclusion and Future Works

Funding Data

Conflict of Interest

Data Availability Statement

References

Get Email Alerts

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