Imaging system for processing a media

Abstract

An imaging system includes an imaging station, an actuator for driving a medium roller for controllably displacing the medium along a media transport path, and a controller assembly. The controller assembly includes a sensor device for generating a position signal and a processor for processing the position signal. The actuator is actuated in response to an actuation command from the controller assembly, which actuation command is derivable from a feedback component and a feedforward component. The feedback component is based on a position error. The processor is further arranged for determining a time dependent status parameter, which status parameter represents a status of the medium on the medium roller. The processor is further arranged for deriving an actuation command estimate based on the status parameter, and wherein the feedforward component includes the actuation command estimate.

Claims

1. An imaging system for processing a medium supplied from a medium roller, comprising: a media transport path; an imaging station arranged along said media transport path; an actuator configured to drive the medium roller for controllably displacing the medium in steps along the media transport path relative to said imaging station; and a controller assembly, wherein the controller assembly comprises: a sensor device configured to generate a position signal representing a position of the medium along the transport path; and a processor configured to process the position signal, wherein in operation, the actuator is actuated in response to an actuation command generated by the controller assembly, the actuation command being derivable from a feedback component for correcting incidental deviations in a behavior of the medium roller and a feedforward component, wherein the feedback component is based on a position error being the difference between a position setpoint and the position signal, wherein the feedforward component comprises an actuation command estimate derived by the processor from a time dependent status parameter, the time dependent status parameter representing an inertia of the medium on the medium roller and being derived by the processor from the position signal and the actuation command, and wherein during operation the processor is configured to continually fit the time dependent status parameter to the amount of medium on the medium roller.

2. The imaging system according to claim 1, wherein, when in use, the processor is configured to recursively determine the status parameter based on a previously determined value of the status parameter and a correction factor based on a difference between the sensed position of the medium and the desired position of the medium.

3. The imaging system according to claim 1, wherein the processor is further arranged for deriving the actuation command estimate from the status parameter and the position setpoint.

4. The imaging system according to claim 1, wherein the processor is further configured to determine the status parameter based on the position signal and a command error signal, the command error signal being the difference between the actuation command and the actuation command estimate.

5. The imaging system according to claim 1, wherein the controller assembly further comprises a memory arranged for storing a status parameter, and wherein the processor is configured to determine a subsequent status parameter from the status parameter stored on the memory.

6. The imaging system according to claim 1, wherein the processor configured to derive the actuation command estimate from a time derivative of the position setpoint.

7. The imaging system according to claim 6, wherein the processor configured to determine a setpoint velocity, a setpoint acceleration, and a setpoint jerk from the position setpoint, and to determine a velocity status parameter, an acceleration status parameter, and a jerk status parameter.

8. The imaging system according to claim 1, wherein the controller assembly and the actuator are arranged for stepwise driving the medium roller, and wherein the processor is configured to determine a displacement of the medium per step based on the position signal.

9. The imaging system according to claim 1, wherein the controller assembly further comprises a repetitive controller configured to model the eccentricity of the medium roller based on the position signal and to adjust the actuation command in correspondence to the eccentricity of the medium roller.

10. The imaging system according to claim 1, wherein the sensor device further comprises an observer configured to sense the angular position of the medium roll.

11. The imaging system according to claim 1, further comprising transport pinch rollers positioned upstream of the medium roller, and a passive buffer device positioned along the transport path between the medium roller and the transport pinch rollers for engaging the medium.

12. The imaging system according to claim 11, wherein the sensor device comprises a tension sensor configured to sense the position of the passive buffer.

13. The imaging system according to claim 1, wherein the processor is further configured to derive an actuation command estimate by means of an inverted model system of the imaging system, the inverted model system comprising the status parameter.

14. A method for actuating a medium roller in an imaging system, the imaging system comprising a media transport path and an imaging station arranged along said media transport path, the method comprising the steps of: inputting a position setpoint for positioning the medium roller at a predefined angular position; generating a first actuation command for actuating the medium roller based on the position setpoint; sensing an angular position of the medium roller; determining a feedback component based on a position error signal based on the difference between the position setpoint and the position signal for correcting incidental deviations in a behavior of the medium roller; deriving a feedforward component from the position setpoint; generating a second actuation command from the feedback component and the feedforward component; deriving a time dependent status parameter, the time dependent status parameter representing an inertia of the medium on the medium roller and being derived from the position signal and the first actuation command; and continually fitting the time dependent status parameter to the amount of medium on the medium roller during operation, wherein the step of determining the feedforward component further comprises the step of deriving an actuation command estimate from the time dependent status parameter and the position setpoint.

15. The method according to claim 14, further comprising the step of: deriving the feedforward component from the position signal sensed after actuating the medium roller by means of the first actuation command; and determining the actuation command estimate from said position signal by means of an adaptive feedforward algorithm, the adaptive feedforward algorithm comprising an inverted model system of the imaging system.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The present invention will become more fully understood from the detailed description given hereinbelow and the accompanying drawings which are given by way of illustration only, and thus are not limitative of the present invention, and wherein:

(2) FIG. 1 is a perspective schematic view of an imaging system according to the present invention;

(3) FIG. 2 is a block diagram representing an embodiment of a controller assembly of an imaging system as shown in FIG. 1;

(4) FIG. 3 is a detailed block diagram representing a further embodiment of a controller assembly of an imaging system as shown in FIG. 1;

(5) FIG. 4 is a block diagram representing the workings of the adaptive feedforward algorithm applied by the controller assembly in FIG. 3;

(6) FIG. 5 is a block diagram representing a further embodiment of the imaging system according to the present invention; and

(7) FIG. 6 is a block diagram representing the repetitive controller of the further embodiment of the imaging system in FIG. 5.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

(8) The present invention will now be described with reference to the accompanying drawings, wherein the same reference numerals have been used to identify the same or similar elements throughout the several views.

(9) A schematic drawing of the imaging system 1 according to the present invention is shown in FIG. 1. The medium roller 2 is located at the bottom of FIG. 1. The medium roller 2 holds the rolled up medium 3a. The medium roller 2 is rotatable by a means of an actuator 4 in the form of a motor. From the medium roller 2, the medium 3 extends towards a pinch roller 6. This pinch roller 6 is only used to lead the unspooled medium 3b to a transport pinch 8, when a leading edge of the medium 3 has not yet reached the transport pinch 8. After the pinch 6, the medium 3 passes the sensor device 5, comprising an encoder wheel 5. The encoder wheel 5 is a wheel that is rotatable over the medium 3 to measure the advancement of the transported medium 3. Upstream from the sensor device 5, a passive buffer device 7 is provided. In FIG. 1, the passive buffer 7 is a metal sheet 7, which pushes against the unspooled medium 3b by means of pulley-springs (not shown), thus decreasing the medium-tension to step-error ratio. The medium 3 then passes through the transport pinch 8. The transport pinch 8 is arranged to rotate and thereby advance the medium 3 in highly accurate steps, such that the medium 3 will be positioned correctly with respect to the print head of the imaging station 9.

(10) The medium 3 is advanced stepwise below the print head 9, which print head 9 swath-wise applies an image to the surface of the medium 3. Any inaccuracies originating from the step of medium roller 2 will cause the buffer 7 to be displaced with respect to the transport path P. This changes the tension in the medium 3, and affects the medium 3 positioning below the print head 9. This results in a decrease in print quality as the consecutively applied swaths are not properly aligned with respect to one another.

(11) The position of the buffer 7 with respect to the transport path is a measure for the tension in the unspooled medium 3b. The sensor device 5 may comprise a tension sensor 7a arranged for sensing the position of the passive buffer 7 and determining the tension in the medium 3 from said position. It is preferred that the controller assembly 10 is arranged for maintaining a substantially constant position of the buffer 7 to ensure accurate positioning of the unspooled medium 3b.

(12) The actuator 4 is provided for driving the medium roller 2. The actuator 4 is preferably an electric motor, especially an electric DC motor, arranged for stepwise rotating the medium roller over an angle based on an input signal or actuation command u. The input or actuation command u for the motor 4 is generally a voltage applied to the electric roll motor 4. A voltage is applied to a pinch motor (not shown) for driving the transport pinch roller 8. In an embodiment, the actuation command u is used for driving the pinch roller 8, such that the web 3 is advanced by means of the pinch roller 8. The actuator for driving the medium roller 2 is then formed by the motor for driving the pinch roller 8. When stepping (i.e. when advancing the medium 3 stepwise), the pinch roller 8 is driven such that there is a constant tension between the pinch roller 8 and the medium roller 2. As explained above, said tension may be determined by means of a tension sensor 7a. This gives a fixed relation between the input voltages for the motors for the transport pinch roller 8 and the medium roller 2. Thus, a further input voltage or further actuation command for the motor for the transport pinch 8 may be determined from the actuation command u or the input voltage for the medium roller motor 4 via said relation. The imaging system 1 in FIG. 1 comprises a controller assembly 10 for generating said input voltages or actuation commands u, which controller assembly 10 is schematically shown in FIG. 3.

(13) The controller assembly 10 comprises a sensor device 5, the output of which is applied for controlling the roll actuator 4. The sensor device in FIG. 1 comprises a buffer encoder 7a in the buffer 7 for determining the tension in the unspooled medium 3b, as discussed above. Preferably, there is a buffer encoder 7a on either side of the buffer 7 for accurately determining the tension in the unspooled medium 3b from the signals of both buffer encoders 7a. The sensor device 5 further comprises a position sensor, such as an encoder wheel, for determining the position and/or advancement of the medium 3 with respect to the transport path P. Further, the sensor device 5 may comprise an angular sensor, such as a roll-motor encoder 4a, for determining the rotational position of the medium roll 2 and/or the medium roll motor 4. A pinch encoder may further be provided for determining the angular position of the transport pinch roll 8 or it's motor (both not shown). The sensor device 5 is arranged for generating a position signal (y in FIG. 2) representing the position of the medium 3 along the transport path P based on measurements by the encoder wheel 5. The position signal y may further include data from the buffer encoder 7a and the roll-motor encoder 4a, which provide a signal representing the tension in the unspooled medium 3b and the angular position of the roll-motor 4 and/or medium roller 2.

(14) FIG. 2 illustrates a control diagram for a controller assembly 10 according to the present invention. Basically, a setpoint Sp is input to the controller assembly 10 for positioning a medium 3 at a desired position. From said setpoint, an actuation command u is derived for driving the roller motor 4 to move the medium 3 to the desired position. For accurate positioning, the actuation command u is composed of a feedback component u.sub.fb and a feedforward component u.sub.ff. The feedback component u.sub.fb is formed by inputting a position error e to a feedback filter C, while the feedforward component u.sub.ff is derived from a feedforward algorithm 11b. The feedforward algorithm 11b is adaptive to any changes in the medium 3a on the medium roller 2 by means of a status parameter determination algorithm 11a. The status parameter determination algorithm 11a derives a status parameter from among others the position signal y, such that the status parameter forms an accurate model system of the imaging system 1, specifically of the medium 3a on the medium roller 2 and preferably the medium roller 2.

(15) A position setpoint Sp is used as input on the left hand side of FIG. 2. This setpoint Sp corresponds to the desired position of the medium 3. The medium 3 is positioned by rotating the medium roller 2 of the imaging system 1 to an angular position. Said rotating is performed by inputting an actuation command u to the roller motor 4. To accurately control the positioning of the medium 3 and the medium roller 2, a position signal y is obtained. Said position signal y corresponds, for example to the medium 3 position or an angular position of the medium roller 2, as determined by the sensor device. On the bottom side of FIG. 2, the position signal y is used in a feedback filter or loop. The position signal y is compared to the setpoint Sp and their deviation is input to a feedback controller C as the position error e to determine the feedback component u.sub.fb. The feedback component u.sub.fb is combined with a feedforward component u.sub.ff to form actuation command u for driving the motor 4. The feedforward component u.sub.ff is determined by means of a feedforward filter 11b performed by the processor (11 in FIG. 3). The processor 11 applies the algorithm 11b to determine an actuation command estimate based on a previous actuation command u and the position signal y. From the position signal y, the processor 11 determines a status parameter , which is used as input for the algorithm 11b. The status parameter is continuously determined from the position signal y by means of an algorithm 11a, such that the value of the status parameter corresponds to the current status of the medium 3a on the medium roller 2. By using the status parameter as input, the feedforward algorithm 11b becomes adaptive to any changes in the inertia of the medium 3a on the medium roller 2. The status parameter forms an always up-to-date model system of the imaging system 1, thereby allowing for a highly accurate estimation of the desired actuation command u. This enables precise positioning of the medium 3 in the imaging system 1 according to the present invention.

(16) A controller assembly according to the present invention is depicted in detailed form in FIG. 3. The control scheme in FIG. 3 is similar to that in FIG. 2, but FIG. 3 comprises additional features and functionalities, which will be discussed below. Over the motor 4 a fixed feedback controller C is used that stabilizes the system 1 for all inertia's. Further, over the buffer 7 a fixed feedback controller C may be used. This controller C over the buffer 7 makes sure that at low frequencies the buffer 7 stays at a fixed position. The feedback controller C is creating a stable loop, but in practice may have a limited bandwidth. To obtain accurate stepping of the medium 3 advancement, a feedforward controller 11 is used, and because of the time variations in the system 1, and specifically in the medium 3a on the medium roller 2, an adaptive feedforward algorithm 11b is applied to estimate the actuation command required to maintain the following advancement step similar or even identical to the step before it.

(17) In FIG. 3, a block diagram for the controller assembly 10 according to the present invention is shown. A position setpoint Sp or position setpoint signal is input into the controller assembly 10 on the left side of FIG. 3. The position setpoint Sp comprises information representing the desired position of the medium 3 along the transport path P, such as position or step data. Further information representing the desired medium tension at e.g. the buffer 7 maybe provided in the position signal y. Preferably, the position signal y further comprises information representing the desired angular or rotational orientation of the medium roller 2 or its motor 4. The position setpoint Sp may comprise one or more input signals or voltages for the actuator 4 for rotating the medium roller 2 over a desired angle. The position setpoint Sp may be input prior to operation or during operation in a continuous manner.

(18) On the bottom right side of FIG. 3, the imaging system 1 is shown. By means of the sensor device 5, a position signal y is output. In the feedback controller C on the bottom side of FIG. 3, the position signal y is compared to the position setpoint Sp to determine a position error e. It will be appreciated that the angular position of the medium roller 2 may be determined from the position signal y by means of a Luenberger observer 5a. In practice, the direct measurement of the angular orientation of the medium roller 2 may be difficult to implement. The status of the medium roller 2 and/or its actuator 4 may then be determined from the position signal y by the observer 5a. For example, the angular position and rotational velocity of the actuator 4 and/or the medium roller 2 with the medium 3a may be derived from the position signal y, as well as from the current running through the motor 4. In an embodiment, the sensor device 5 provides a tension signal determined by the buffer encoder 7a and an angular signal determined by the angular sensor 4a, which signals are transmitted to the feedback controller C for determining the position error e.

(19) The position error e represents a deviation between the desired position of the medium 3 and the actual position of the medium 3 as determined by the sensor device 5. This position error is input to the feedback controller C for generating the feedback component u.sub.fb of the actuation command u. In an embodiment, the feedback filter C comprises a proportional component acting on the magnitude of the error signal and a derivative component acting on the rate of change of the error signal e. The resulting feedback component u.sub.fb will result in a fast correction of incidental disturbances, while the derivative component introduces enough damping to the controlled system to overcome problems due to overshoot. In imaging systems, it is undesired to oscillate a media during positioning thereof and the media should be in the correct position within a relatively small amount of time. Preferably, the feedback controller C comprises a P, PI, PID, ID, or PD controller. It will be appreciated that the feedback controller C in FIG. 3 may be implemented by means of the processor 11, i.e. as a software-based controller, or as a hardware-based feedback filter.

(20) The controller assembly 10 further comprises a processor 11, which has at least two main functions, namely determining the status parameters 1, 2, 3 and deriving the feedforward component u.sub.ff formed by the actuation command estimate . First, the processor 11 is arranged for determining the one or more time dependent status parameters 1, 2, 3, preferably by means of a status parameter determination algorithm 11a. The status parameters 1, 2, 3 are arranged to represent a status of the unspooled medium 3b on the medium roller 2. In FIG. 3, the status parameters 1, 2, 3 are derived from the position signal y as sensed by the sensor device 5, specifically the wheel encoder 5, and from the actuation command u generated by the controller assembly 10. In FIG. 3, the angular position of the medium roller 2 is determined from the position signal y by means of an observer 5a, such as a Luenberger observer 5a. Such an observer 5a may be applied when a direct measurement of the angular orientation of the medium roller 2 is not possible or is complicated. The angular orientation of the medium roller 2 is then derived by the observer 5a based on, e.g. signals representing the tension in medium 3b, the angular position of the actuator 4, and/or the position of the medium 3. The observer 5a outputs the roller orientation signal z, which signal z represents the angular orientation of the medium 3a on the medium roller 2 (or of the medium roller 2). The observer 5a increases the accuracy of the controller assembly by a precise determination of the medium roller's orientation. It will be appreciated that within the scope of the present invention, the position signal y may be used to determine the status parameters 1, 2, 3 without use of the observer 5a or the signal z. Alternatively, an encoder positioned at the circumference of the medium 3a on the medium roller 2 may be applied.

(21) The status parameters 1, 2, 3 are derived by means of the status parameter determination algorithm 11a. The algorithm 11a applies as inputs the position signal y and the actuation command u. The position signal y may in a preferred embodiment be converted into the roller orientation signal z by means of the observer 5a. Also, both signals y, z may be used. The actuation command u may in another embodiment be processed into a command error c, as will be discussed further on. The status parameter 1, 2, 3 represents the current status of the medium 3a on the medium roller 2. Since the parameters 1, 2, 3 are continuously adjusted and updated to reflect the present amount of medium 3a on the medium roller 2, the status parameter 1, 2, 3 may be considered to form an accurate model representation or system of the medium 3a and the medium roller 2. The status parameter determination algorithm 11a may for example be arranged to obtain the status parameters 1, 2, 3 from data formed by or based on one or more of the signals y, u, , , and/or z from sensors 4a, 5, 5a, 7a. The processor 11 is then arranged to analyze said data to determine the status parameters 1, 2, 3, for example by fitting the data to a model system or curve. Preferably, the processor 11 applies a recursive least squares algorithm 11a to recursively determine the status parameters 1, 2, 3. A recursive algorithm 11a has the advantage that computation time is reduced and the status parameters 1, 2, 3 may be determined with great accuracy within the time between two consecutive advancement steps. A further advantage of the recursive algorithm 11a is that it requires relatively little processor power, such that a cheap and/or simple processor 11 may be used. The status parameters 1, 2, 3 may be stored on the memory M for use in the algorithm 11a, for example by recursively determining a status parameters 1, 2, 3 from a previously determined status parameter 1, 2, 3 stored on the memory M. The memory M may further store information or data related to the position setpoint Sp, actuation command u, actuation command estimate , position signal y, and/or the roller orientation signal z. In a preferred embodiment, the algorithm 11a is a recursive least squares (RLS) algorithm, which advantageously provides for a rapid and efficient determination of the status parameters 1, 2, 3, as well as a fast convergence of said parameters 1, 2, 3 during the start-up phase of the printing process. Furthermore, the algorithm according to the present invention, specifically said RLS algorithm, is especially well suited for printing processing wherein step sizes are varied, as well as for processes wherein operational parameters such as the inertia of the medium roll 3a vary significantly. This a great advantage of the present invention over iterative learning control (ILC), since ILC is unable to cope properly with said varying step sizes and system parameters. Furthermore, ILC requires a number of ILC circuits specifically designed to a specific system, whereas the algorithm according to the present invention may be applied by means of a processor 11. Thereby, the present invention is easy and cheap to implement.

(22) The actuation command u, which may for example be an input voltage V for driving the electric medium roll motor 4, comprises a feedback and feedforward component u.sub.fb, u.sub.ff. The status parameters 1, 2, 3 are applied by the processor 11 for determining the feedforward component u.sub.ff. In FIG. 3, the position setpoint Sp, which for example is the desired angular position of the medium roller 2, is input to the processor 11 to derive an actuation command estimate u.sub.ff by means of a feedforward algorithm 11b. From the position setpoint Sp the processor 11 determines one or more setpoint parameters d/dt, d2/dt2, d3/dt3, such as the time derivatives d/dt, d2/dt2, d3/dt3. Any number or order of time derivatives may be applied. The processor 11 combines setpoint parameters d/dt, d2/dt2, d3/dt3 with the status parameters 1, 2, 3 to obtain an accurate estimate 0 of the actuation command u. By utilizing in the feedforward algorithm 11a the status parameters 1, 2, 3 determined by the algorithm 11a, the feedforward algorithm 11b uses a continuously up-to-date input, such that the estimate takes into account the decreasing inertia and outer radius of the medium roll 3a as it unspools. As such, the feedforward algorithm 11b provides an accurate estimate of the command estimate u, which allows for precise stepping and high quality printing.

Example 1

(23) The controller assembly 10 in FIG. 3 further comprises a feedforward controller 11b, which may be either a hardware or software-based controller. The position setpoint Sp is applied as input for determining the feedforward component u.sub.ff output by the feedforward controller 11b. From the position setpoint Sp, one or more time derivatives are derived as indicated by the blocks d/dt, d2/dt2, d3/dt3 in FIG. 3. Each block d/dt, d2/dt2, d3/dt3 corresponds to an order of the time derivative of the position setpoint Sp, i.e. d/dt is the first order time derivative of the position setpoint Sp, d2/dt2 is the second order time derivative of the position setpoint Sp, etc. By converting the position setpoint Sp into a plurality of time derivative signals d/dt, d2/dt2, d3/dt3, the dynamics of the system 1 may be identified. The time derivatives d/dt, d2/dt2, d3/dt3 are combined with the status parameters 1, 2, 3 to determine an actuation command estimate . Preferably, a status parameter 1, 2, 3 is determined by the parameter determination algorithm 11a for each of the time derivatives d/dt, d2/dt2, d3/dt3. The actuation command estimate , which correspond to the estimated driving voltage for medium roller motor 4, may then be expressed as:

(24) $\hat{u}{\left(\begin{array}{c}\frac{\mathrm{Ref}}{t}\\ \frac{{}^2\mathrm{Ref}}{t^2}\\ \frac{{}^3\mathrm{Ref}}{t^3}\end{array}\right)}^T\left(\begin{array}{c}{}_1\\ {}_2\\ {}_3\end{array}\right)$

(25) Thus, the adaptive feedforward algorithm 11b generates an actuation command estimate , which in FIG. 3 forms the feedforward component u.sub.ff. The feedfoward component u.sub.ff is combined with the feedback component u.sub.fb to form the actuation command u. This actuation command u is input into the medium roller motor 4 to advance the medium 3 by a step, such that the step distance applied during each advancement step of the medium 3 is constant or the medium is positioned at a desired and/or predefined position.

(26) The adaptive feedforward algorithm 11b is schematically illustrated in the block diagram in FIG. 4. In FIG. 4, the processor 11 with the adaptive feedforward algorithm 11b is placed in series with the imaging system 1. The medium 3 in the imaging system 1 is advanced based on the actuation command u and the advancement is sensed by means of the sensor device 5, which generates a position signal y, representing the position of the medium 5 along the transport path P. The sensed position signal y is the input to the adaptive feedforward algorithm 11b, which determines a number of time derivatives d/dt, d2/dt2, d3/dt3 from the position signal y. The time derivatives d/dt, d2/dt2, d3/dt3 of the position signal y are then each multiplied with a corresponding status parameter 1, 2, 3 and added together to form the actuation command estimate . The estimate is then subtracted from the actuation command u to yield the command error . From FIG. 4, it may be deduced that when the status parameters 1, 2, 3 form a perfect model of the imaging system 1 (or the medium roller 2), the command error would be zero. Basically, the status parameters 1, 2, 3 are arranged to form a model system, which is substantially the inverse of the imaging system 1. When inputting in the model system ha a detected position y of the medium 3 in the imaging system 1, which position y is the result of an actuation command u, the algorithm 11a would yield an estimate substantially similar or equal to the command u.

(27) To rapidly and accurately determine the status parameters 1, 2, 3, the controller assembly 10 according to the present invention applies a recursive least squares algorithm 11a. By minimizing the command error E between the actuation command u input to the actuator 4 and the actuation command estimate determined by the processor 11, the status parameters 1, 2, 3 are adjusted until the actuation command estimate accurately corrects the actuation command u to bring the medium 3 to a desired position. To this end, the algorithm 11a aims to minimize a cost-function, as defined by:

(28) $V\left(t\right)=\underset{i=0}{\overset{t}{.Math.}}\left({\left(i\right)}^2{}_1^{t-i}\right)$

(29) Wherein V(t) is the input voltage of the roller motor 4, and t may be the time or iteration number, corresponding, e.g. to the number of the current advancement step. , is a forgetting factor, generally smaller than 1, which allows the algorithm to weigh new measurements with regards to older ones. From FIG. 4, it can further be derived that the command error and the actuation command estimate may be defined as:
(t)=u(t){circumflex over (u)}(t)
{circumflex over (u)}(t)=(t).sup.T{circumflex over ()}(t)

(30) Wherein {circumflex over ()}(t) is a vector comprising the status parameters 1, 2, 3 at time t, while (t) is a vector comprising the measured signals y:

(31) $\left(t\right)=\left(\begin{array}{c}\stackrel{.}{y}\left(t\right)\\ \stackrel{\mathrm{.Math.}}{y}\left(t\right)\\ \stackrel{\mathrm{.Math.}}{y}\left(t\right)\end{array}\right)$

(32) The position signal y here preferably comprises a tension signal from the buffer encoder 7a, an angular signal from the roll-motor encoder 4a, and/or a position signal from the wheel encoder 5. Then, by setting the first derivative of V(t) equal to 0,

(33) $\frac{V}{\hat{}}\left(t\right)=0$

(34) the algorithm becomes:

(35) $\hat{}\left(t+1\right)=\hat{}\left(t\right)+F\left(t+1\right)\left(t+1\right)\left(t+1\right)$$F\left(t+1\right)=\frac{1}{{}_1}\left(F\left(t\right)-\frac{F\left(t\right)\left(t\right){}^T\left(t\right)F\left(t\right)}{{}_1+{}^T\left(t\right)F\left(t\right)\left(t\right)}\right)$

(36) Where, F determines the step-size and is also recursively updated. Note that if (t)=0, then F will be unstable, because .sub.1<1. Therefore an enable matrix Q(t) is used. This is a diagonal matrix with a one for the parameter to be updated and a zero for the parameter that needs to stay the same. For instance when only .sub.1 and .sub.3 need to be updated this matrix may be:

(37) $Q\left(t\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right)$

(38) A matrix W is used to replace 1/.sub.1 in the update of F. W is defined as:

(39) $W\left(t\right)=\left(I-Q\left(t\right)\right)+\frac{1}{{}_1}Q\left(t\right)$

(40) The algorithm then becomes:

(41) $\hat{}\left(t+1\right)=\hat{}\left(t\right)+Q\left(t\right)F\left(t+1\right)\left(t+1\right)\left(t+1\right)$$F\left(t+1\right)=W\left(t\right)\left(F\left(t\right)-Q\left(T\right)\frac{F\left(t\right)\left(t\right){}^T\left(t\right)F\left(t\right)}{{}_1+{}^T\left(t\right)F\left(t\right)\left(t\right)}\right)$

(42) To analyze the stability of the recursive least squares algorithm 11a passivity (hyperstability) or Lyapunov functions can be used. Tuning of the algorithm 11a can be done by choosing values for .sub.1 and F(0). .sub.1 is typically chosen between 0.85 and 1, and determines the weight on the older measurements. High values for .sub.1 averages zero mean noise better, while lower values will enhance convergence speed. For the imaging system 1, .sub.1=0.999 is preferably selected, as this reduces the effect of noise and the parameters follow the slowly time varying system well. F(0) is usually chosen as a diagonal matrix. The values on the diagonal reflect on the prior information about the optimal values for .

Example 2

(43) In FIG. 5, an imaging system 100 according to the present invention is illustrated as a block diagram. The imaging system 100 comprises a feedback controller assembly C and a feedforward controller with a feedforward algorithm 11b similar to those described with respect to FIGS. 1-3. The position signal y generated by the sensor device 5 is used as the basis for a feedback or a feedforward signal u.sub.fb, u.sub.ff. FIG. 5 illustrates the different components of the position signal y, namely the angular signal y.sub.1 representing the angular position of the actuator 4, the tension signal y.sub.2 representing the tension in medium 3b, and the advancement signal y.sub.3 corresponding to the position of the medium 3. The position signal y.sub.3 may be generated by the encoder wheel 5, the tension signal y.sub.2 by means of the tension encoder 7a in the buffer 7, while the roll-motor encoder 4a may be applied for obtaining the advancement signal y.sub.1.

(44) FIG. 5 shows that the observer 5a determines the medium roller orientation z from the position signal y comprising the signals y.sub.1, y.sub.2, and/or y.sub.3. Though the position signal y may be applied instead of the signal z, the observer 5a increases the accuracy of the controller assembly 10. The position error e is determined based on the angular signal y.sub.1 and the tension signal y.sub.2. The feedback controller assembly C comprises first and second feedback controllers C1, C2. The first feedback controller C1 converts the position error e to the feedback component u.sub.fb of the actuation command u, similar to the feedback controller C. Preferably, the first feedback controller C1 utilizes the advancement signal y.sub.1 as input and compares this to the position setpoint Sp. The second feedback controller C2 is arranged for adjusting the setpoint based upon the tension in the medium 3. The tension signal y.sub.2 is input into the second feedback controller C2 to correct the setpoint Sp, especially when the tension in the medium 3 deviates from a predefined value or reference. For example, when the buffer 7 moves, the tension signal y.sub.2 changes, and the second feedback controller C2 adjust the setpoint Sp in accordance with the recorded change in the tension signal y.sub.2. Thereby, any change in tension in the medium 3 is effectively corrected by adjusting the setpoint Sp. The feedforward component u.sub.ff comprises the actuation command estimate , derived by means of the adaptive feedforward algorithm 11b. The system parameters .sub.1, .sub.2, .sub.3 are recursively calculated from the medium roller orientation z and the actuation command u.

(45) The controller assembly 110 in FIG. 5 further comprises a repetitive controller 12, which is arranged to determine the eccentricity of the medium roll 3a from the position signal y, specifically from the medium roller orientation z and/or the advancement signal y.sub.3. The repetitive controller 12 may determine a cyclic disturbance in the position signal y by filtering the step error. Basically, the cyclic disturbance due to the eccentricity of the medium roll 3a is much lower in frequency than the step error in each advancement step. The repetitive controller 12 projects the position signal y onto one or more harmonic base functions or periodic functions, whose frequencies exceed the advancement step frequency. Thereby, the eccentricity may be filtered out from the position signal y. In an example, the cyclic disturbance d may be described by:

(46) $d=\underset{i=1}{\overset{N}{.Math.}}\left({}_i\sin \left(i{}_l+{}_i\right)\right)$

(47) and also by:

(48) 0$d={\left(\begin{array}{c}\sin \left({}_l\right)\\ \cos \left({}_l\right)\\ \sin \left(2{}_l\right)\\ \cos \left(2{}_l\right)\\ \mathrm{.Math.}\\ \sin \left(N{}_l\right)\\ \cos \left(N{}_l\right)\end{array}\right)}^T\left(\begin{array}{c}{}_1\\ {}_2\\ {}_3\\ {}_4\\ \mathrm{.Math.}\\ {}_{\left(2N-1\right)}\\ {}_{\left(2N\right)}\end{array}\right)={\tilde{H}}^T\left[k\right]$

(49) The algorithm may apply a plant model to filter the step error from the cyclic disturbance (FIG. 6). Then, the cyclic disturbance may be used to find an estimate for the disturbance using a linear combination of base functions. The linear combination of base functions may be found using a projection algorithm. Therein, this estimation s of the disturbance is subtracted from the position setpoint signal Sp to reject the cyclic disturbance at the output u. The input to the step-error filter is the estimated and real angular position {circumflex over ()}.sub.l, .sub.l of the medium-roll and the disturbance d. P is the actual imaging system 100 with the medium 3 and P is a model of this system 100. The output of this part can be described by:
u[k]={circumflex over (P)}{circumflex over ()}.sub.l[k]P(.sub.l[k]+d[k])
The model is correct for at least low frequencies, such that an actuation signal u[k] where the step-error is partially present, but will not drift away at low frequencies, and the cyclic disturbance is not affected by the step-error. The output u[k] can now be written as:
u[k]=H.sup.T+w[k]

(50) Where H.sup.T represents the cyclic disturbance caused by the roll 2, 3. H contains the base functions, and is for example the frequency of the roll 2:

(51) $H=\left(\begin{array}{c}\sin \left({}_l\left[k\right]\right)\\ \cos \left({}_l\left[k\right]\right)\end{array}\right)$

(52) is the linear combination of base functions. This determines the phase and amplitude of the disturbance. w[k] is a signal that includes the residual step-error, noise and higher harmonics not included in H. The buffer 7 or the encoder wheel 5 can be used for measuring the cyclic disturbance. The encoder wheel 5 may be used, to form a more accurate model for the encoder 5. The repetitive controller 12 further uses a projection algorithm to find the parameters. Here, H contains the base functions. The block g is a summation multiplied by a constant gain g. The position error e[k] may be written as:
e[k]=H.sup.T{circumflex over ()}[k]H.sup.T+w[k]=H.sup.T{circumflex over ()}[k]+w[k]

(53) Where {tilde over ()}[k] is an error between the parameter and {circumflex over ()}[k]. {circumflex over ()}[k+1] may then be written as:
{circumflex over ()}[k+1]={circumflex over ()}[k]gHe[k]

(54) Which may be rewritten to:
{tilde over ()}[k+1]=(IgHH.sup.T){tilde over ()}[k]gHw[k]

(55) The parameters will converge monotonic if:
(IgHH.sup.T)1

(56) Where (IgHH.sup.T) is the maximum singular value of (IgHH.sup.T). Monotonic convergence may advantageously be applied, because if the parameters move away from the determined optimum, then it could lead to amplification of the cyclic disturbance. Gain parameter g may be chosen, such that there is monotonic convergence. Choosing a large g lets the parameters converge fast. However, when w is large, {tilde over ()} will oscillate with the same period-time of .sub.l. Therefore {circumflex over ()} is preferably averaged over one period time of the medium-roll 3a rotation, which will decrease the oscillation. By fitting harmonic or periodic base functions with a period (greatly) exceeding the step frequency of the actuator 4, the eccentricity of the medium roll 2 may be derived from the position signal y. This eccentricity may then be applied to further improve the actuation command u by means of an eccentricity correction s, preferably as part of the feedback component u.sub.fb and/or the feedforward component u.sub.ff. This results in highly accurate stepwise advancement of the medium 3, and thereby in high quality printing, since consecutive printing swaths are precisely aligned with respect to one another. Advantageously, the repetitive controller 12 performs especially well in combination with the inverted model system according to the present invention, particularly in combination with the RLS algorithm. Basically, the model system or RLS algorithm filters the disturbances caused by the stepwise displacement of the medium 3 from the signal z, y.sub.3 used to determine the eccentricity, effectively providing a low-noise input signal z, y.sub.3 from which a highly accurate eccentricity may be determined.

(57) Although specific embodiments of the invention are illustrated and described herein, it will be appreciated by those of ordinary skill in the art that a variety of alternate and/or equivalent implementations exist. It should be appreciated that the exemplary embodiment or exemplary embodiments are examples only and are not intended to limit the scope, applicability, or configuration in any way. Rather, the foregoing summary and detailed description will provide those skilled in the art with a convenient road map for implementing at least one exemplary embodiment, it being understood that various changes may be made in the function and arrangement of elements described in an exemplary embodiment without departing from the scope as set forth in the appended claims and their legal equivalents. Generally, this application is intended to cover any adaptations or variations of the specific embodiments discussed herein.

(58) It will also be appreciated that in this document the terms comprise, comprising, include, including, contain, containing, have, having, and any variations thereof, are intended to be understood in an inclusive (i.e. non-exclusive) sense, such that the process, method, device, apparatus or system described herein is not limited to those features or parts or elements or steps recited but may include other elements, features, parts or steps not expressly listed or inherent to such process, method, article, or apparatus. Furthermore, the terms a and an used herein are intended to be understood as meaning one or more unless explicitly stated otherwise. Moreover, the terms first, second, third, etc. are used merely as labels, and are not intended to impose numerical requirements on or to establish a certain ranking of importance of their objects.

(59) The present invention being thus described, it will be obvious that the same may be varied in many ways. Such variations are not to be regarded as a departure from the spirit and scope of the present invention, and all such modifications as would be obvious to one skilled in the art are intended to be included within the scope of the following claims.