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Repetitive Control (RC) designed with state feedback that includes past error feedforward and current error feedback schemes for linear time-invariant systems is reintroduced. Periodic disturbances are common within repetitive systems and can be represented with a time-delay model. The proposed design focuses on isolating the disturbance model and finding the overall transfer function around the delay model. The use of the small gain theorem around the delay model assures disturbance accommodation if stability conditions are achieved. This paper reintroduces the designed RC controller within the state feedback in the presence of both past error and current error structures. Robustness conditions are investigated and set to enhance system performance in the presence of modelling mismatch, which represents the novel contribution in this paper. Simulations demonstrate the advantages of the robust conditions obtained while improving system performance for dynamic perturbations.

Systems of a repetitive [

RC was first mentioned with the reported work of [

RC has a direct impact on industry, which utilizes RC applications, such as systems with rotary movement such as disc drives [

The idea behind RC is to use previous periods/trials to modify the control signal such that the overall system learns to follow a periodic reference trajectory with period T to a high precision. Most previously designed frameworks report their work in a continuous time-domain due to the nature of the repetitive system and use time instants t to form the forcing function in the update law for a certain upcoming time instant denoted by t + T . In literature, [

Repetitive control and iterative learning control (ILC), which is another technique used to accommodate periodic disturbances and to enhance the performance of repetitive systems, are not similar even though they use the same updating technique. The main difference is that in RC there is no resetting between trials; the reference that is followed is continuous with r ( t ) = r ( t + N ) , where N is the number of samples. That is, the system initial states for trial k are those of the final states of trial k − 1 . In comparison, for ILC, the system resets to the home position after each trial to start the next trial. A comparison between RC and ILC that clarifies similarities and differences can be found in [

Based on the above statements, a lifted form that maps the problem structure from the property of being expressed in time and trial indexes to a uniform structure that depends on the trial index alone can be generally considered. Thus the design starts with first defining the periodic signal and then setting the required steps to design the RC controller in the lifted model with the presence of the delay model in the feedback loop. Any periodic signal can be generated by an autonomous system containing a delay model along the forward path with a positive feedback loop [

This paper reintroduces the RC design within the proposed framework in a state feedback structure. As well it presents new robust conditions that set limitations towards the design that are different than those introduced in [

The following section briefly discusses RC design in a general case under the proposed framework in [

Starting with a linear time-invariant system with m outputs, p inputs and n states having a discrete overall transfer function in the state space form given by P ( z ) = C ( z I n − A ) − 1 B + D . The matrices A, B, C and D are of proper dimensions. Also let the system output be y ( z ) and the input u ( z ) , then the process output equation is y ( z ) = P ( z ) u ( z ) .

The general platform to describe the RC controller or the ILC controller is the same due to the similarities they both hold, thus it is no harm to initially describe the system in the lifted form as a start. Consider a “single trial” with a finite time duration with N samples, where the model of the system dynamics at trial k can be expressed as

x k ( i + 1 ) = A x k ( i ) + B u k ( i ) , x k ( 0 ) = x k − 1 ( N − 1 ) y k ( i ) = C x k ( i ) + D u k ( i ) (1)

where 0 ≤ i ≤ N − 1 . In the above equation, the RC controller does not reset to the initial state after each trial as done by ILC; x k ( 0 ) = x 0 . Now, introduce the input and output vectors as

u k = [ u k ( 0 ) , u k ( 1 ) , ⋯ , u k ( N − 1 ) ] T

y k = [ y k ( 0 ) , y k ( 1 ) , ⋯ , y k ( N − 1 ) ] T

Then the dynamics for each trial can be written in the form of

y k = P u k (2)

where

P = [ 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 C A N − 4 B ⋯ D ]

where its elements are the Markov parameters. Defining the reference; in discrete form, to hold the vector elements of

r = [ r ( 0 ) , r ( 1 ) , ⋯ , r ( N − 1 ) ] T

As discussed, the RC problem can be illustrated with the structure presented in

A periodic signal that is considered as an autonomous system consisting of a positive feedback control loop with a pure time delay in the forward path with appropriate initial conditions can generate a periodic signal with appropriate boundary conditions by modelling with a signal of length N in discrete-time as

x w ( t k + 1 ) = A w x ( t k ) , x w ( t 0 ) = x w 0

w ( t k ) = C w x ( t k ) (3)

where the N × N matrix A w is given by

A w = [ 0 1 0 ⋯ 0 ⋮ ⋮ ⋮ ⋱ ⋮ 0 0 0 ⋯ 1 1 0 0 ⋯ 0 ]

and the 1 × N row vector C w as

C w = [ 1 0 0 ⋯ 0 ]

A robust controller K ( z ) (where z denotes the discrete-time delay operator) is required for the robust periodic control problem, which is defined as:

Given a m × l transfer-function matrix P ( z ) with an input vector that consists of the plant input and a disturbance input; u = u p + u w , the output signal as defined in (2) and a reference signal r ( t k ) = r ( t k + N ) , t k = 0 , Δ T , 2 Δ T , ⋯ with N sampling time. The design of K ( z ) requires that the overall closed loop system is asymptotically stable, and the tracking error; e k = r − y k , tends to zero along the trial domain and the previous two conditions are robust.

The solution considered in [

This paper considers the RC design scheme in the state feedback reported in [

For a single channel, consider the system in (3), and also introduce the following N × 1 vector

B w = [ 0 ⋯ 0 0 1 ] T

and

D w = ( 0 ifpasterrorfeedforwardcase 1 ifcurrenterrorfeedbackcase

for a multi-input multi-output (MIMO) case define A r to be a diagonal matrix consisting of A w along its diagonal.

A r = diag { A w }

and the same is true for B r , C r and D r where each diagonal block is repeated m times (acting on the system output). Thus, if considering the periodic problem proposed in

C r ( z I N m − A r ) − 1 B r + D r = ( ( z N I m − I m ) − 1 if D w = 0 ( I m − z − N I m ) − 1 if D w = 1

Now, the design considers whether the state feedback can be found with more details in [

[ x r ( i + 1 ) x ( i + 1 ) ] = [ A r − B r C 0 A ] [ x r ( i ) x ( i ) ] + [ − B r D B ] u k ( i ) + [ B r 0 ] r ( i ) (4)

where stabilising this system guarantees periodic disturbance accommodation since the output of the combined system is the plant output and its input is the control input signal, where x r is the internal model system state. Manipulate the combined system by choosing the control input of the combined system as

u ( i ) = − K r [ x ^ r ( i ) x ^ ( i ) ]

with an observer to estimate the states. This in turn will end up with the overall system of the form [

[ x ^ r ( i + 1 ) x ^ ( i + 1 ) ] = [ A r − B r C 0 A ] [ x ^ r ( i ) x ^ ( i ) ] − [ − B r D B ] K r [ x ^ r ( i ) x ^ ( i ) ] + L r ( v ( i ) − ( [ C r − D r C ] + D r D K r ) [ x ^ r ( i ) x ^ ( i ) ] ) (5)

The overall structure description can be found in [

‖ H ( z ) ‖ < 1 (6)

where the overall transfer function around the delay operator, H ( z ) , differs depending on the error case considered, either past error feedforward or current error feedback, for a Past error feedforward case

H ( z ) = ( G ( z ) + P ( z ) ) G ( z ) − 1 (7)

and for a Current error feedback case

H ( z ) = G ( z ) ( G ( z ) + P ( z ) ) − 1 (8)

where G ( z ) in both cases is governed by the following

G ( z ) = [ C r − D r C ] ( z I − [ A r − B r C 0 A ] + [ B r D − B ] K r ) − 1 [ − B r D B ] − D r D

The solution required depends on solving the linear quadratic regulator to find K via the Riccati equation, such that the model considers the difference between the combined system around the plant and the delay model and the estimator structure to minimize the required cost function [

In this section the robustness property of the two designed RC controllers in [

For past error feedforward in state feedback design the starting point is the stability condition given in (6) where the induced norm has to be less than 1 to guarantee system stability. A more conservative restriction is to consider singular values instead, thus the condition will be

σ ( H ( z ) ) < 1

which clearly indicates that all the eigenvalues of H ( z ) are inside the unit circle once the maximum singular value is considered. Verifying this condition in the maximum case assures reference tracking and periodic disturbance accommodation. Now, consider the case where unmodelled system dynamics or system uncertainty defined as ( Δ ) act on the system in operation. To examine this case define [ P = P o + P o Δ W ] where P o , Δ , W are the nominal plant, the uncertainty, and the uncertainty weight, respectively. Each of the defined variables is stable, causal and linear time invariant for simplicity. Now, in combination with the definition of H ( z ) given in (7), we can write the following derivation:

σ ( G + P o + P o Δ W ) < σ ( G ) ≤ 1

σ ( G + P o ) + σ ( P o Δ W ) < σ ( G ) ≤ 1

taking the uncertainty part on one side and the other parts from the right side yields

σ ( P o Δ W ) < σ ( G ) − σ ( G ) − σ ( P o ) (9)

maximizing the left-hand side will give the possible variation in system dynamics; meanwhile, the right-hand side, sets the upper bound for the system so as to not have unwanted performance through the operation. This can be found if the right-hand side was of the form σ ¯ ( G ) − σ _ ( G ) − σ ¯ ( P ) . To extend the previous property and set a weight for the uncertainty that gives a better upper bound and permits the system to deal with unmodelled dynamics through the operation can be found if we manipulate Equation (9) to be of the form

σ ( Δ ) < σ ( G ) − σ ( G ) − σ ( P o ) σ ( W ) σ ( P o ) (10)

maximizing the left-hand side of Equation (10), such that the right-hand side is kept at a minimum, can be seen as solving the following

σ ¯ ( Δ ) < min max = σ ¯ ( G ) − σ _ ( G ) − σ _ ( P o ) σ ( W ) σ ¯ ( P o ) (11)

Now, σ ¯ ( G ) ≠ σ _ ( G ) unless G is a scalar multiplied by the identity, which is not true in our design. Thus, returning back to Equation (11), to suppress the uncertainty effect to a higher level, further investigation toward weight (W) is taken into account and can be expressed by σ ¯ ( Δ ) < 1 , in the following

1 < σ ¯ ( G ) − σ _ ( G ) − σ _ ( P o ) σ ( W ) σ ¯ (Po)

σ ¯ ( W ) < σ ¯ ( G ) − σ _ ( G ) − σ _ ( P o ) σ ¯ ( P o ) < 1 (12)

The condition given in Equation (12) will set the upper limit to the weighting factor such that uncertainty is extended and avoids a high level of unmodelled dynamics compared to a case where there is no consideration of a weighting factor.

For current error feedback in the state feedback design the starting point again is the stability condition given in (6) where the induced norm has to be less than 1 to guarantee system stability. Again consider system uncertainty as ( Δ ) acting on the system in operation. To examine this case again, define [ P = P o + P o Δ W ] with the definitions given and properties considered in the past-error case. Following the same steps, in combination with the definition of H ( z ) given in (8), we can write the following derivation in term of the singular values as

σ ( G G − P o − P o Δ W ) < 1

which leads to writing the above after manipulation as

σ ¯ ( Δ ) > σ ( G ) − σ ( G ) − σ ( P o ) σ ( W ) σ ( P o ) (13)

Since the uncertainty is assumed to be stable ( σ ¯ ( Δ ) < 1 ), then equation (13) can be written as

σ ( G ) − σ ( G ) − σ ( P o ) σ ( W ) σ ( P o ) < 1 (14)

Equation (14) will give the proper condition for the weighting factor (W) such that the left-hand side is minimized, which will be

1 > σ ¯ ( W ) > σ ¯ ( G ) − σ _ ( G ) − σ _ ( P o ) σ ¯ ( P o ) (15)

Condition (15) is the same as that in (12) to a limit where in (15) it sets the lower limit to the weight selection while (12) sets the upper limit to the uncertainty weight, which has a wider and better range than that of (15). This result supports the experimental results obtained for the past error feedforward case in ILC, instead of the current error feedback in [

The next section presents simulation results obtained on a non-minimum phase plant where the results show a performance improvement against system uncertainty and modelling mismatch when considering the robust design to previously reported design frameworks for the RC case.

This section presents simulation results obtained for a non-minimum phase plant (NMP) to verify that the proposed design suppresses the dynamic matrix changes of a plant with a difficult mathematical structure, such as with NMP.

In this example, a NMP plant was tested against dynamic matrix changes with both the absence and the presence of a weighting factor. The physical system was constructed to implement both ILC and Repetitive controller (RC) schemes, which led to the verity of reported works, for example [

P ( s ) = 1.202 ( 4 − s ) s ( s + 9 ) ( s 2 + 12 s + 56.25 ) (16)

This system has been tested in two different cases, the first is where the weighting factor is ignored, while the second case considers the presence of the

weighting factor. A reference signal of 2 seconds was applied with a selected sampling frequency of 100 Hz generating 200 sample points that were recorded in each trial. The system can be operated for large number of trials, but only the first 10 cycles are presented since they include all the needed information to clarify the weighting factor effect.

In this paper conditions were set to extend the range of linear system uncertainty based on the singular value principle for the RC design presented in [

The authors declare no conflicts of interest regarding the publication of this paper.

Alsubaie, M., Alhajri, M. and Altowaim, T. (2018) Repetitive Control Uncertainty Conditions in State Feedback Solution. Intelligent Control and Automation, 9, 95-106. https://doi.org/10.4236/ica.2018.94008