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Selecting a proper initial input for Iterative Learning Control (ILC) algorithms has been shown to offer faster learning speed compared to the same theories if a system starts from blind. Iterative Learning Control is a control technique that uses previous successive projections to update the following execution/trial input such that a reference is followed to a high precision. In ILC, convergence of the error is generally highly dependent on the initial choice of input applied to the plant, thus a good choice of initial start would make learning faster and as a consequence the error tends to zero faster as well. Here in this paper, an upper limit to the initial choice construction for the input signal for trial 1 is set such that the system would not tend to respond aggressively due to the uncertainty that lies in high frequencies. The provided limit is found in term of singular values and simulation results obtained illustrate the theory behind.

Iterative Learning Control (ILC) [

Repetitive systems are those where the reference to follow, r ( t ) , has fixed time duration, 0 ≤ t ≤ T and has to be repeated large number of times which in turn tends to set the mathematical modelling expression outside the conventional control representation to a 2D system representation [

The error signal ( e k ( t ) ) is the forcing part to update the upcoming control input. After each trial, a system has to be reset to its initial position to start the next one. The resetting time, known as the stoppage time, is the time required for a system to do all the needed computations to update the next trial control input signal to start the next trial. One approach of ILC is of the form u k + 1 = u k + L e k , where u k is the trial k input; L is the learning gain and e k is the trial k error signal. One good starting point for rich information about ILC Theory and applications is [

Repetitive control (RC) [

Most of the literature suggest that the initial input can be an array of zeros such that the error for the first trial is the reference it self. This assures that the learning gain in the design theory is totally responsible for building up the control signal from the beginning, but if it could be possible to predict a better starting point that assures faster learning process it would be better in term of learning speed. This idea lead to several reported works in the literature such as those in [

This paper sets an upper limit condition in constructing the initial input using singular values based on system presence. It is required in input construction that it would not generate a high effort at the beginning to assure safe operation. This development is highly required in most industrial applications such as in robot arms and chemical patch processes. In parallel it speeds up learning process compared to the same method when the initial input construction omitted. In order to do so, the upper limit condition guarantees safe construction for such input to fulfill the operation requirements.

The following section reintroduces the work presented in [

In term of understanding the initial input construction reported in [

Let u = [ u ( 0 ) u ( 1 ) ⋯ u ( N − 1 ) ] T ∈ ℝ N be an array of N elements, then (under the necessary assumption relating to existence) the DFT of this array, denoted by u ^ , is defined as

u ^ i = ∑ n = 0 N − 1 u n e − j 2 π n i / N (1)

where u ^ ∈ ℂ N and i = { 0 , 1 , ⋯ , N − 1 } . As mentioned earlier, ILC trajectory reference has a fixed length of, T, and a sampling frequency must be chosen as

f s = N T for N ∈ 2 M , M ∈ ℤ . The Inverse Discrete Fourier Transform (IDFT) can be driven to be

u n = 1 N ∑ i = 0 N − 1 u ^ i e j 2 π n i / N (2)

Given g as the finite impulse response of an linear time-invariant (LTI) system, then the convolution between g and u produces the output sequence

y ( q ) = ∑ i = 0 q g ( q − i ) u ( q ) , q = 0 , 1 , ⋯ , N − 1 (3)

The DFT of y can then be calculated using

y ^ = g ^ ⊙ u ^ (4)

where ⊙ is the component-wise multiplication.

Now, given a reference trajectory y d of length N and it is required to construct an initial input vector u 0 * such that learning speed is improved when used with suitable ILC controller. In this paper an ILC law of the form

u k + 1 = u k + L e k (5)

will be considered where L is chosen to be the adjoint of the process matrix G, with u k = [ u k ( 0 ) u k ( 1 ) ⋯ u k ( N − 1 ) ] T , and e k = [ y d ( 0 ) − u k ( 0 ) y d ( 1 ) − u k ( 1 ) ⋯ y d ( N − 1 ) − u k ( N − 1 ) ] T . Let the associated LTI plant be given in state-space form of the following form

x ( q + 1 ) = A x ( q ) + B u (q)

y ( q ) = C x ( q ) , q = 0 , 1 , ⋯ , N − 1 (6)

where the sample time has been set at unity for notational simplicity, x ( ⋅ ) ∈ ℝ n , x ( 0 ) = 0 , and the operators A, B and C are of appropriate dimensions. Then using the plant model y k = G u k [

G = [ D 0 0 ⋯ 0 C B D 0 ⋯ 0 C A B C B D ⋯ 0 ⋮ ⋮ ⋮ ⋱ ⋮ C A N − 2 B C A N − 3 B ⋯ ⋯ D ] (7)

with y k = [ y k ( 0 ) y k ( 1 ) ⋯ y k ( N − 1 ) ] T , is the lift form, then the error evolution equation can be driven easily to be

e k + 1 = ( I − G L ) e k (8)

Taking the DFT of both sides of (5) now gives

u ^ k + 1 = u ^ k + l ^ ⊙ e ^ k (9)

and likewise for (8)

e ^ k + 1 = ( I ^ − g ^ ⊙ l ^ ) ⊙ e ^ k (10)

where I ^ ∈ ℝ N and has each entry equal to unity. Repeated application leads to

e ^ k + 1 = ( I ^ − g ^ ⊙ l ^ ) k ⊙ e ^ 0 (11)

where the power operation is applied in component-wise fashion. Now consider the error progression starting from an arbitrary initial input, u 0 ,

‖ e k + 1 ‖ 2 = ∑ i = 0 N − 1 | e ^ k + 1 , i | 2 = ∑ i = 0 N − 1 | ( 1 − g ^ i l ^ i ) k e ^ 0 , i | 2 = ∑ i = 0 N − 1 | 1 − g ^ i l ^ i | 2 k | e ^ 0 , i | 2 = ∑ i = 0 N − 1 | 1 − g ^ i l ^ i | 2 k | y ^ d , i − g ^ i u ^ 0 , i | 2 (12)

This is minimized with respect to the initial input by setting u ^ 0 equal to

u ^ 0 , i * = y ^ d , i g ^ i , i = 0 , 1 , ⋯ , N − 1 (13)

This effectively generates a steady-state inverse (over the duration of the trial). The above derivation is found in [

This section represents the novelty of this paper where a new condition is found that sets the maximum number of frequency components to include in constructing the initial input for trial 1 in any selected ILC method based on the presence of the plant model. This construction is as pointed out earlier includes system model presence. The key start was to consider the ILC design given in [

σ ¯ ( d k ) < σ ¯ ( ∑ i = 0 k ( Ψ i ) − ∑ j = 0 k − 1 ( d j ) − G u 0 ) − σ _ ( ∑ h = 0 k − 1 ( Ψ h ) − ∑ v = 0 k − 1 ( d v ) − G u 0 ) (14)

where d is the load disturbance, Ψ is the system output with the presence of load disturbances given by [

Ψ k ( t + δ ) = G ( q ) u k ( t ) + d k ( t ) , (15)

y k ( t ) = Ψ k ( t ) + n k ( t ) , t = 0 , 1 , ⋯ , n − 1. (16)

G is the process matrix and u 0 is the initial input for the first trial. Here in this argument, two cases are to consider where the first is to assume that disturbances influence is still acting on the system while starting the operation; and this is a very small possibility in term of quality production. The second case is to consider system operation after the disturbances influence is vanished. Thus, in this case we can assume d k = d k − 1 = d k − 2 = d 0 . Now, we go through the following to find the upper limit to the initial input for the first trial

0 < σ ¯ ( ∑ i = 0 k ( Ψ i ) − 0 − G u 0 ) − σ _ ( ∑ h = 0 k − 1 ( Ψ h ) − G u 0 ) (17)

σ _ ( ∑ h = 0 k − 1 ( Ψ h ) − G u 0 ) < σ ¯ ( ∑ i = 0 k ( Ψ i ) − G u 0 ) (18)

Breaking up the last equation leads to

σ ¯ ( G u 0 ) − σ _ ( G u 0 ) < σ ¯ ( ∑ i = 0 k ( Ψ i ) ) − σ _ ( ∑ h = 0 k − 1 ( Ψ h ) ) (19)

σ ¯ ( G u 0 ) < σ ¯ ( ∑ i = 0 k ( Ψ i ) ) (20)

where σ _ ( G u 0 ) , and σ _ ( ∑ h = 0 k − 1 ( Ψ h ) ) are assumed to be zero for simplicity. Thus, the result obtained in (20) says that for a linear repetitive system to start its operation with better performance, a designer can use the above inequality as a guide or reference when constructing the initial input such that the condition above is met. This condition says if a designer has the performance of first three trials in any previous operation, those information can be used to set the initial input according to (20).

Simulation results presented in this section are obtained for a gantry robot Z-axis. The gantry robot shown in ^{rd} order

representation and given as in [

G X ( z ) = 15.8869 ( s + 850.3 ) s ( s + 353.81 + 461.03 j ) ( s + 353.81 − 431.03 j ) (21)

For enhancing system performance, stability and disturbance rejection, the gantry system is fitted in a feedback loop with a PID controller whose parameters can be found in [

One common ILC method is the adjoint method addressed in; for example, [

In this paper, the step size is chosen to be 0.5 and the system is operated for 10 trials; to show the advantage of using input prediction over the unpredicted case. The input is constructed using (13) for the first 5 frequency components and it showed an advantage of using an input with such construction method. In this example, the model given is treated as an exact model due to the fact that those results are obtained in simulation, but for the case of experimental implementation, it would be very sensitive in constructing the input and (20) should be considered since all models are an approximation to the exact behaviour.

Applying (20) as an upper limit to the construction of u 0 leads us to next table where the first column represents the frequency component while the second represents the result of applying (20). It can be seen that for the first 5 frequency component and the sum of the first 5 frequency components, the norm of u 0 is still less than 1, thus there is no harm to use the sum of the first 5 frequency components to construct u 0 according to (20).

error norm here is for an array of 1500 points representing the reference length; T / T s , where T is the length of the reference in seconds (3 s) and T s is the sampling period which is 1/500 s.

Overall the initial input construction provides better error start which in turn speeds up learning process depending on the ILC method chosen and enhances the repetitive system performance. Notice the uncertainty effect where it clearly shows an increase in the control effort as the number of frequency components increases in constructing u 0 . Thus the condition given in (20) minimizes the effect of the uncertainty depending on limiting the first trial control effort by not exceeding the limit in (20).

This paper uses the design model used to set the upper limit of load disturbances acting on a system [

In future, verifying the proposed work experimentally over the gantry is under consideration. Checking the validity of the proposed condition over different types of systems such as minimum-phase plants will be investigated experimentally in the future.

Alajmi, N., Alobaidly, A., Alhajri, M., Salamah, S. and Alsubaie, M. (2017) An Upper Limit for Iterative Learning Control Initial Input Construction Using Singular Values. Intelligent Control and Automation, 8, 154-163. http://dx.doi.org/10.4236/ica.2017.83012