Patent Publication Number: US-2021178786-A1

Title: Sheet conveyor and image forming system

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     The present application claims priority from Japanese Patent Applications No. 2019-226414 filed on Dec. 16, 2019 and No. 2020-039178 filed on Mar. 6, 2020, the disclosures of which are incorporated herein by reference in its entirety. 
     BACKGROUND 
     Field of the Invention 
     The present disclosure relates to a sheet conveyor and an image forming system. 
     Description of the Related Art 
     There is conventionally known a printing apparatus that pulls out a sheet from a sheet roll (rolled sheet) and prints an image on the pulled-out sheet (see, for example, Japanese Patent Application Laid-open No. 2014-076669). The sheet is pulled out toward a recording head by use of a conveyance roller. 
     In the printing apparatus, a diameter of the sheet roll is determined. The diameter of the sheet roll determined is used for controlling conveyance of the sheet. The diameter of the sheet roll is measured, for example, by a distance sensor. Or, the diameter of the sheet roll is estimated based on a conveyance amount of the sheet under a condition that the sheet is conveyed by the conveyance roller and a rotation amount of the sheet roll. 
     SUMMARY 
     A sheet conveyor that pulls out a sheet from a sheet roll and conveys the sheet may include a tensioner positioned between the sheet roll and a conveyance roller and configured to apply tension to the sheet. 
     The tensioner includes a movable part. The movable part comes into contact with the sheet to apply tension to the sheet. The movable part is displaced upon receiving urging force from an elastic material such as a spring to apply appropriate tension to the sheet. 
     In the conveyor including the above-described tensioner, a posture of the sheet pulled out from the sheet roll changes by the displacement of the tensioner. It is thus not possible to accurately estimate the diameter of the sheet roll in the above-described conventional method in which the diameter of the sheet roll is estimated based on the conveyance amount of the sheet by the conveyance roller and the rotation amount of the sheet roll. 
     An object of the present disclosure is to provide a sheet conveyor including a tensioner in which a diameter of a sheet roll can be accurately estimated based on a conveyance amount of a sheet or the like without actually measuring the diameter of the sheet roll by a sensor. 
     According to the first aspect of the present disclosure, there is provided a sheet conveyor, including: a holder; a rotation measuring device; a conveyance roller; a tensioner; a position detector; a motor; and controller. The holder is configured to detachably hold a sheet roll. The rotation measuring device is configured to measure a rotation amount of the sheet roll. 
     The conveyance roller is configured to convey the sheet pulled out from the sheet roll held by the holder. The tensioner is provided between the holder and the conveyance roller. The tensioner is movable and configured to contact with the sheet pulled out from the sheet roll toward the conveyance roller. The position detector is configured to detect a position of the tensioner. 
     The motor is configured to generate driving force for rotating the conveyance roller. The controller is configured to control the motor. The controller is configured to estimate a diameter of the sheet roll based on a conveyance amount of the sheet caused by rotation of the conveyance roller driven by the motor, the rotation amount of the sheet roll measured by the rotation measuring device, and the position of the tensioner detected by the position detector. 
     This sheet conveyor can estimate the diameter of the sheet roll with high accuracy based on the conveyance amount of the sheet by the conveyance roller and the rotation amount of the sheet roll, even under a condition that a posture of the sheet pulled out from the sheet roll changes depending on the position of the tensioner in a section ranging from the sheet roll to the conveyance roller. It is thus possible to estimate the diameter of the sheet roll with high accuracy without actually measuring the diameter of the sheet roll by a sensor. 
     According to the second aspect of the present disclosure, there is provided an image forming system including: the sheet conveyor as defined in the first aspect and a process unit. The process unit is provided downstream of the conveyance roller in a conveyance direction of the sheet and configured to form an image on the sheet. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  depicts a configuration of an image forming system according to an aspect of the present disclosure. 
         FIG. 2  illustrates functions of a tensioner. 
         FIG. 3  is a block diagram of an electrical configuration of the image forming system. 
         FIG. 4  is a flowchart of a process executed by a main controller. 
         FIG. 5  is a block diagram of a velocity controller. 
         FIG. 6  is a block diagram of a configuration of a tension controller. 
         FIG. 7  is a flowchart of a process executed by a roll diameter estimator. 
         FIG. 8  is a graph indicating relationships between tensioner positions and route lengths. 
         FIG. 9  illustrates a method of estimating a roll diameter. 
         FIG. 10  is a graph indicating a change in an estimated value of the roll diameter. 
         FIG. 11  is a flowchart indicating part of an initial process. 
         FIG. 12  is a flowchart of an origin setting process executed by the main controller in the second embodiment. 
         FIG. 13  is a flowchart of a process executed by the roll diameter estimator in the third embodiment. 
         FIG. 14  is a flowchart of a switching process of state quantity noise executed by the roll diameter estimator in the third embodiment. 
         FIG. 15  is a block diagram of a configuration of the tension controller in the third embodiment. 
     
    
    
     DESCRIPTION OF THE EMBODIMENTS 
     First the Embodiment 
     A first embodiment of the present disclosure will be described with reference to the accompanying drawings. An image forming system  1  of this embodiment depicted in  FIG. 1  is configured so as to form an image on a sheet Q pulled out from a sheet roll Q 0  (rolled sheet). The image forming system  1  functions as a sheet conveying device for conveying the sheet Q toward the lower side of the recording head  40 . 
     The image forming system  1  includes a holder  10  (see  FIG. 3 ), a tensioner  15 , a conveying roller  20 , a nip roller  25 , a belt mechanism  30 , and a recording head  40 . In  FIG. 1 , configuration downstream of the belt-mechanism  30  in a conveyance direction of the seat Q is omitted. Downstream of the belt mechanism  30 , for example, a cutter for cutting the sheet Q is provided. 
     The holder  10  removably holds the sheet roll Q 0 . The sheet roll Q 0  is formed by the sheet Q wound around a hollow core material. The holder  10  includes a rotation shaft  10 A inserted through the core material of the sheet roll Q 0 , and a holder main body (not depicted in the drawing(s)) that rotatably holds the rotation shaft  10 A. 
     The holder main body is fixed in a casing (not depicted in the drawing(s)) of the image forming system  1 . The core material of the sheet roll Q 0  is fixed so as not to slide with respect to the rotation shaft  10 A of the holder  10 . As a result, the sheet roll Q 0  rotates together with the rotation shaft  10 A of the holder  10 . 
     After the sheet roll Q 0  is mounted on the rotating shaft  10 A, the sheet Q is manually pulled out from the sheet roll Q 0  via the tensioner  15  to a position where the sheet Q has passed through the nip point P 2  between the conveying roller  20  and the nip roller  25 . Point P 1  in  FIG. 1  indicates the point at which the sheet Q is pulled out of the sheet roll Q 0 . 
     The conveying roller  20  conveys the sheet Q nipped between the conveying roller  20  and the nip roller  25  in the conveyance direction indicated by the thick arrow. As depicted in  FIG. 2 , the tensioner  15  is connected to the spring member  16  to be displaceable in a front-rear direction. The front side corresponds to the downstream side of the conveyance direction and the rear side corresponds to the upstream side of the conveyance direction. 
     The tensioner  15  is disposed at the rear side of the spring material  16 . The tensioner  15  applies tension to the sheet Q by backward urging force from the spring material  16 . 
     The tensioner  15  is displaced frontward by receiving pressing force from the sheet Q under a condition that the sheet Q pulled out from the sheet roll Q 0  is conveyed by the conveyance roller  20 . The pressing force from the sheet Q corresponds to tension of the sheet Q.  FIG. 2  depicts a state where the tensioner  15  and the sheet Q are displaced from a position depicted by solid lines to a position depicted by dotted lines due to the pressing force from the sheet Q. 
     A position in the front-rear direction of the tensioner  15  is stabilized at a position where the urging force of the spring material  16  and the tension of the sheet Q are balanced. The image forming system  1  is configured to estimate the tension of the sheet Q and control the tension by using the position of the tensioner  15  as an index (details are described below). 
     The belt mechanism  30  is disposed downstream of the conveyance roller  20  in the conveyance direction. The belt mechanism  30  conveys the sheet Q conveyed from the conveyance roller  20  further downstream. As depicted in  FIG. 1 , the belt mechanism  30  includes a driving roller  31 , a driven roller  32 , and a belt  33  stretched between the driving roller  31  the driven roller  32 . The driving roller  31  rotates the belt  33  by rotary motion synchronized with the conveyance roller  20 . 
     The belt mechanism  30  further includes the first facing roller  35  and the second facing roller  36 . The first facing roller  35  faces the driving roller  31  with the belt  33  interposed therebetween. The second facing roller  36  faces the driven roller  32  with the belt  33  interposed therebetween. 
     The sheet Q conveyed from the conveyance roller  20  passes between the first facing roller  35  and the belt  33  due to the rotation of the belt  33 , so that the sheet Q is conveyed downstream. The sheet Q is conveyed further downstream by passing between the second facing roller  36  and the belt  33 . A conveyance velocity of the sheet Q by the rotation of the belt  33  is the same as a conveyance velocity of the sheet Q by use of the conveyance roller  20 . 
     For example, the belt mechanism  30  can have an air absorbing function. That is, the belt  33  may have fine holes through which air passes. A suction device (not depicted) that suctions air may be provided below the belt  33 . The sheet Q may be conveyed while being sucked or absorbed to a surface of the belt  33  by the air suction performed by the suction device. 
     The recording head  40  is provided above the belt mechanism  30  to form an image on the sheet Q passing below the recording head  40 . The recording head  40 , which is a line head, simultaneously forms images for an entirety in a line direction of the sheet Q passing below the recording head  40 . The line direction is a direction along the surface of the sheet Q. The line direction is the conveyance direction of the sheet Q, in other words, a direction orthogonal to a longitudinal direction of the sheet Q. 
     The recording head  40  may be, for example, an ink-jet head that forms an image on the sheet Q in accordance with an ink-jet system. The recoding head  40  may be a thermal head that forms an image on the sheet Q in accordance with a thermosensitive system or a thermal transfer system. 
     Under a condition that the recording head  40  forms an image in accordance with the ink-jet system, a fixer  45  for drying and fixing ink may be provided downstream of the recoding head  40  in the conveyance direction. As depicted by the dotted line in  FIG. 1 , the fixer  45  may be provided above the belt mechanism  30  such that the fixer  45  is adjacent to the recording head  40 . 
     Subsequently, an electrical configuration of the image forming system  1  is explained in detail. As depicted in  FIG. 3 , the image forming system  1  includes the controller  50  that controls an entire system. The image forming system  1  further includes a supply motor  61 , a motor driver  63 , a rotary encoder  65 , and a measurement circuit  67  to control rotation of the sheet roll Q 0  installed in the holder  10 . 
     The supply motor  61  is connected to the rotation shaft  10 A of the holder  10  via a gear (not depicted). The supply motor  61  applies power to the rotation shaft  10 A. The rotation shaft  10 A of the holder  10  rotates under a condition that the rotation shaft  10 A receives power from the supply motor  61 . The sheet roll Q 0  rotates along with the rotation shaft  10 A. 
     The supply motor  61  may be a direct-current motor (DC motor). The supply motor  61  drives the rotation shaft  10 A by generating rotation torque depending on a drive current input from the motor driver  63 . The motor driver  63  inputs, to the supply motor  61 , a drive current depending on a control input U SR  input from the controller  50 . 
     The rotary encoder  65  is provided in the rotation shaft  10 A of the holder  10  or a rotation shaft of the supply motor  61 . The rotary encoder  65  outputs an encoder signal depending on rotation. The measurement circuit  67  measures a rotation position and a rotation velocity (i.e., rotation angle and angular velocity) of the sheet roll Q 0  as a physical quantity related to the rotary motion of the sheet roll Q 0 . The measurement circuit  67  inputs, to the controller  50 , the rotation position and the rotation velocity measured. The rotation position and the rotation velocity of the rotation shaft  10 A correspond to the rotation position and the rotation velocity of the sheet roll Q 0 . 
     The image forming system  1  further includes a conveyance motor  71 , a motor driver  73 , a rotary encoder  75 , and a measurement circuit  77  as the configuration for controlling the rotation of the conveyance roller  20 . 
     The conveyance motor  71  may be a direct-current motor (DC motor). The conveyance motor  71  is connected to the conveyance roller  20  via a gear. The conveyance motor  71  rotates and drives the conveyance roller  20  by generating rotation torque depending on a drive current input from the motor driver  73 . The motor driver  73  inputs, to the conveyance motor  71 , a drive current depending on a control input U PF  input from the controller  50 . 
     The rotary encoder  75  is provided in a rotation shaft of the conveyance roller  20  or a rotation shaft of the conveyance motor  71 . The rotary encoder  75  outputs an encoder signal depending on rotation. The measurement circuit  77  measures a rotation position and a rotation velocity (i.e., rotation angle and angular velocity) of the conveyance roller  20  as a physical quantity related to the rotary motion of the conveyance roller  20 . The measurement circuit  77  inputs, to the controller  50 , the rotation position and the rotation velocity measured. 
     The image forming system  1  further includes a position detector  80  that detects a position of the tensioner  15 . The position detector  80  detects a position X in the front-rear direction of the tensioner  15  with reference to a predefined origin position. The position detector  80  inputs the detected position X to the controller  50 . The position detector  80  may be configured, for example, by a linear encoder. 
     The image forming system  1  further includes a head driver  90 , a registration sensor  91 , a user interface  95 , and a communication interface  97 . The head driver  90  is configured to drive the recording head  40  in accordance with a control signal from the controller  50 . The registration sensor  91  is provided upstream of the belt mechanism  30 . The registration sensor  91  is configured to detect a leading edge of the sheet Q passing therethrough, and to input the detected signal to the controller  50 . 
     The user interface  95  includes a display section for displaying a variety of information for a user and an input section for receiving instructions from the user. The display section is, for example, a liquid crystal display. The input section is, for example, a touch panel on the liquid crystal display. 
     The communication interface  97  is configured to communicate with an information device in the wired or wireless communication. The communication interface  97  may be a USB interface or a wired/wireless LAN interface. The information device may be a personal computer or a tablet terminal owned by the user. 
     The controller  50  includes a main controller  51 , a printing controller  53 , a velocity controller  55 , and a tension controller  57 . The main controller  51  includes a processor  51 A and a memory  51 B. The memory  51 B includes a Random Access Memory (RAM) and a flush memory. 
     The processor  51 A executes a variety of processes in accordance with computer programs stored in the memory  51 B. In the following, it can be understood that processes executed by the main controller  51  are executed by the processor  51 A in accordance with the computer program(s). 
     The printing controller  53 , the velocity controller  55 , and the tension controller  57  are configured, for example, by an ASIC. Image data of a printing object is input from the main controller  51  to the printing controller  53 . The printing controller  53  inputs, to the head driver  90 , a control signal for causing the recording head  40  to print an image based on the image data of the printing object. 
     The velocity controller  55  determines the control input U PF  for the controller motor  71  so that the conveyance roller  20  rotates at a target rotation velocity in accordance with an instruction from the main controller  51 . The velocity controller  55  inputs, to the motor driver  73 , the control input U PF  determined. In this embodiment, the control velocity of the sheet Q is controlled by controlling the rotation velocity of the conveyance roller  20 . 
     The tension controller  57  determines the control input U SR  for the supply motor  61  so that the sheet Q is conveyed while having target tension in accordance with an instruction from the main controller  51 . The tension controller  57  inputs, to the motor driver  63 , the control input U SR  determined. 
     Under a condition that the main controller  51  receives a printing instruction and image data of a printing object from an information device through the communication interface  97 , the main controller  51  executes processes indicated in  FIG. 4 . The main controller  51  controls respective sections of the image forming system  1  in cooperation with the printing controller  53 , the velocity controller  55 , and the tension controller  57  so that an image based on the image data of the printing object received is formed on the sheet Q. 
     Under a condition that the processes indicated in  FIG. 4  are started, the main controller  51  executes an initial process (S 110 ). The initial process includes a process for arranging the leading edge of the sheet Q at a predefined starting point. The starting point may be a position where the leading edge of the sheet Q is detected by the registration sensor  91 , or a position shifted downstream in the conveyance direction by a predefined distance from the position where the leading edge of the sheet Q is detected by the registration sensor  91 . 
     The starting point may be a position where the leading edge of the sheet Q enters the belt mechanism  30 . Alternatively, the starting point may be a point that is upstream in the conveyance direction from a position, where image formation is performed on the sheet Q by the recording head  40 , by a distance required for acceleration of the sheet Q. 
     After arranging the leading edge of the sheet Q at the starting point, the main controller  51  starts a conveyance process of the sheet Q (S 120 ). In the conveyance process, the main controller  51  inputs a velocity profile to the velocity controller  55 . The velocity controller  55  controls the rotation velocity of the conveyance roller  20  in accordance with the velocity profile. The velocity profile indicates a target rotation velocity of the conveyance roller  20  until the sheet Q is stopped at a target stop position. 
     Specifically, the velocity profile indicates a target rotation velocity in an acceleration section, a target rotation velocity in a constant velocity section, and a target rotation velocity in a deceleration section. The sheet Q accelerates until the sheet Q reaches a predefined velocity by controlling the rotation velocity of the conveyance roller  20  in accordance with the velocity profile. After reaching the predefined velocity, the sheet Q moves at a constant velocity, and then decelerates. 
     In the conveyance process, the main controller  51  further inputs a tension profile to the tension controller  57 . The tension controller  57  executes tension control of the sheet Q in accordance with the tension profile. The tension profile indicates target tension until the sheet Q is stopped at the target stop position. 
     After starting the conveyance process, the main controller  51  waits until the sheet Q reaches the predefined velocity (S 130 ). The main controller  51  starts the printing process under a condition that the sheet Q starts constant velocity movement (S 140 ). In the printing process, the main controller  51  causes the printing controller  53  to execute drive control of the recording head  40  for forming the image based on the image data of the printing object on the sheet Q. The recording head  40  repeatedly executes the image forming operation in the line direction in synchronization with movement in the conveyance direction of the sheet Q. 
     The main controller  51  executes an ending process (S 160 ) under a condition that the printing process and the conveyance process are completed (S 150 : Yes). The ending process includes a process in which the user is informed of the completion of printing through the user interface  95 . Then, the main controller  51  ends the processes indicated in  FIG. 4 . 
     Referring to  FIG. 5 , a detailed configuration of the velocity controller  55  is explained below. The velocity controller  55  calculates the control input U PF  for the conveyance motor  71  based on a deviation between the rotation velocity of the conveyance roller  20  measured by the measurement circuit  77  and the target rotation velocity. The velocity controller  55  is configured to execute feedback control for the conveyance roller  20  based on the control input U PF  calculated. In the following, the actual rotation velocity means a measured value of the rotation velocity. 
     The velocity controller  55  includes a velocity instruction device  101 , a deviation calculator  103 , a PID controller  105 , a static friction compensation device  107 , and an adder  109 . The velocity instruction device  101  outputs a target rotation velocity ω r  at each point of time from the start of control in accordance with the velocity profile input from the main controller  51 . 
     The deviation calculator  103  calculates a deviation E V =(ω r −ω PF ) between the target rotation velocity ω r  output from the velocity instruction device  101  and an actual rotation velocity ω PF  input from the measurement circuit  77 . The PID controller  105  calculates a control input U V  for the conveyance motor  71  based on the deviation E V  input from the deviation calculator  103 . 
     The PID controller  105  includes: a proportional element that amplifies the deviation E V  with a gain G p  and outputs it; an integral element that amplifies an integral value INT(E V ) of the deviation E V  with a gain G i  and outputs it; and a differential element that amplifies an integral value DIF(E V ) of the deviation E V  with a gain G d  and outputs it. The PID controller  105  calculates a total of the output from the proportional element, the integral element, and the differential element as the control input U V  for the conveyance motor  71 . 
     The static friction compensation device  107  outputs a compensation amount C for compensating for the shortage of control input U V  caused by static friction. The compensation amount C is a fixed value under a condition that the actual rotation velocity ω PF  is zero, that is, in a static state. The compensation amount C is zero under a condition that the actual rotation velocity ω PF  is not zero, that is, in a non-static state. 
     The adder  109  corrects the control input U V  output from the PID controller  105  by the compensation amount C, and inputs, to the motor driver  73 , a control input U PF =U V +C after correction. The motor driver  73  inputs, to the conveyance motor  71 , a drive current corresponding to the control input U PF  input from the velocity controller  55 , and drives the conveyance motor  71  so that rotation torque corresponding to the control input U PF  is generated. The rotation velocity of the conveyance roller  20  and the conveyance velocity of the sheet Q corresponding to the rotation velocity of the conveyance roller  20  are subjected to the feedback control by the velocity controller  55 . 
     The tension control is executed by the tension controller  57  depicted in  FIG. 6 . The tension controller  57  calculates the control input U SR  for the supply motor  61  based on a deviation between tension of the sheet Q estimated from the position X of the tensioner  15  detected by the position detector  80  (hereinafter referred to as estimated tension) and the target tension. The tension controller  57  thus executes the feedback control for the tension of the sheet Q based on the control input U SR  calculated. 
     As depicted in  FIG. 6 , the tension controller  57  includes a tension instruction device  110 , a tension estimator  120 , a deviation calculator  130 , a PID controller  140 , a roll diameter estimator  150 , a feedforward controller  160 , an adder  170 , and a gain setter (gain setting device)  180 . 
     The tension instruction device  110  outputs target tension T r  at each point of time from the start of control in accordance with the tension profile input from the main controller  51 . The tension estimator  120  estimates tension T acting on the sheet Q based on the position X of the tensioner  15  input from the position detector  80 . Specifically, the tension estimator  120  can calculate, as estimated tension T, a value k·X obtained by multiplying the position X of the tensioner  15  by a certain proportional efficient k. 
     The deviation calculator  130  calculates a deviation E T =T r −T between the target tension T r  and the estimated tension T. The PID controller  140  calculates a feed back control input U B  for the supply motor  61  based on the deviation E T  input from the deviation calculator  130 . 
     As depicted in  FIG. 6 , the PID controller  140  includes a proportional gain amplifier  141 , an integral gain amplifier  142 , a differential gain amplifier  143 , an integrator  145 , a differentiator  146 , and an adder  148 . The deviation E T  calculated by the deviation calculator  130  is input to the proportional gain amplifier  141 , the integrator  145 , and the differentiator  146 . The proportional gain amplifier  141  amplifies the deviation E T  input from the deviation calculator  130  with the gain K p  and outputs it. 
     The integrator  145  executes integral calculation for the deviation E T  and inputs an integral value INT(E T ) of the deviation E T  to the integral gain amplifier  142 . The integral gain amplifier  142  amplifies the integral value INT(E T ) of the deviation E T  input from the integrator  145  with a gain K i  and outputs it. 
     The differentiator  146  executes differential calculation for the deviation E T , and inputs a differential value DIF(E T ) of the deviation E T  to the differential gain amplifier  143 . The differential gain amplifier  143  amplifies the differential value DIF(E T ) of the deviation E T  input from the differentiator  146  with a gain K d  and outputs it. 
     The adder  148  adds K p ·E T  output from the proportional gain amplifier  141 , K i ·INT(E T ) output from the integral gain amplifier  142 , and K d ·DIF(E T ) output from the differential gain amplifier  143 . The adder  148  outputs an addition value K p ·E T +K i ·INT(E T )+K d ·DIF(E T ) as the feedback control input U B  for the supply motor  61 . 
     The adder  170  outputs an addition value U B +U F  obtained by adding the feedback control input U B  input from the PID controller  140  and a feedforward control input U F  input from the feedforward controller  160  as the control input U SR  for the supply motor  61 . 
     The feedforward controller  160  includes a differentiator  161 , an acceleration torque estimator  163 , and an FF gain amplifier  165 . The differentiator  161  differentiates the rotation velocity ω PF  of the conveyance roller  20  input from the measurement circuit  77  to calculate rotation acceleration α of the conveyance roller  20 . The rotation acceleration corresponds to angle acceleration. In the following, the rotation acceleration α calculated from the differentiator  161  is expressed as an actual rotation acceleration α. 
     The acceleration torque estimator  163  estimates acceleration torque τ of the supply motor  61  required for acceleration of the conveyance roller  20  based on the actual rotation acceleration α, in other words, required for rotation of the sheet roll Q 0  depending on the acceleration of the sheet Q. Specifically, the acceleration torque τ is calculated in accordance with an equation τ=J(R)·(R P /R)·α based on the actual rotation acceleration α, the roll diameter R that is a radius of the sheet roll Q 0 , a radius R P  of the conveyance roller  20 , and an inertia J(R) of the sheet Q 0  estimated from the roll diameter R. 
     Under a condition that the rotation acceleration of the conveyance roller  20  is α, the acceleration of the sheet Q conveyed by the rotation of the conveyance roller  20  is R P ·α. The sheet roll Q 0  is required to rotate at rotation acceleration (R P /R)·α to pull out the sheet Q from the sheet roll Q 0  at the same acceleration. Under a condition that the inertia is J, the acceleration torque required for achieving this rotation is J·(R P /R)·α. 
     A function J(R) for calculating the inertia J(R) of the sheet roll Q 0  with the roll diameter being R is prepared in advance. The radius R P  of the conveyance roller  20  is a fixed value of the image forming system  1 . The roller diameter R of the sheet roll Q 0  is estimated by the roll diameter estimator  150  (details thereof are described below). 
     The acceleration torque estimator  163  calculates the acceleration torque τ based on the above equation and information of the roll diameter R input from the roll diameter estimator  150 . The FF gain amplifier  165  adjusts the acceleration torque τ calculated so that the acceleration torque τ calculated is amplified by a gain K FF , and outputs an acceleration torque K FF ·τ after adjustment as the feedforward control input U F . The gain K FF  is normally a value 1, and the gain K FF  may be finely adjusted from the value 1 depending on machine characteristics of a rotation system. 
     The motor driver  63  inputs a drive current corresponding to the control input U SR =U F +U B  to the supply motor  61 , and drives the supply motor  61  so that rotation torque corresponding to the control input U SR  is generated. The tension of the sheet Q is controlled to the target tension by executing the feedforward control and the feedback control for the supply motor  61 . 
     In this embodiment, the gain setter  180  is configured to adjust the gains K p , K i , and K d  in the PID controller  140  based on the roll diameter R estimated by the roll diameter estimator  150 . The gains K p , K i , and K d  are set to K p =K p (R), K i =K i (R), K d =K d (R) in accordance with the functions K p (R), K i (R), and K d (R) of which variables are the roll diameter R. The functions K p (R), K i (R), and K d (R) are determined in advance through an examination. 
     Subsequently, an estimation operation of the roll diameter R by the roll diameter estimator  150  is explained. The roll diameter estimator  150  estimates the roll diameter R for each sampling period t s  by executing the processes in  FIG. 7  (S 260 ). The processes in  FIG. 7  are started under a condition that the conveyance of the sheet Q is started, and the processes in  FIG. 7  are ended under a condition that the conveyance of the sheet Q is ended. 
     Under the condition that the processes in  FIG. 7  are started, the roll diameter estimator  150  sets a current time t as a reference time T 0  (S 210 ). Until conditions for starting the estimation of the roll diameter R are satisfied, the roll diameter estimator  150  accumulates and stores a rotation position θ PF (t) of the conveyance roller  20 , a rotation position θ SR (t) of the sheet roll Q 0 , and a position X(t) of the tensioner  15  for each sampling period t s , as data required for the estimation of the roll diameter R (S 210 ). 
     The period until the estimation starting conditions are satisfied may be a period from the reference time t 0  until a predefined time t 1  described below elapses, or a period shorter than said period. The rotation position θ PF (t) indicates a rotation position θ PF  of the conveyance roller  20  measured by the measurement circuit  77  at the time t. The rotation position θ SR (t) indicates a rotation position θ SR  of the sheet roll Q 0  measured by the measurement circuit  67  at the time t. The position X(t) indicates a position X of the tensioner  15  detected by the position detector  80  at the time t. 
     After completing the process of S 210 , the roll diameter estimator  150  repeatedly executes the processes of S 220  to S 290  for each sampling period t s . In S 220 , the roll diameter estimator  150  stores the rotation position θ PF (t) of the conveyance roller  20 , the rotation position θ SR (t) of the sheet roll Q 0 , and the position X(t) of the tensioner  15  at the current time t. 
     After that, the roller diameter estimator  150  calculates a sheet conveyance amount L(t) from the reference time t 0  to the current time t based on the radius R p  of the conveyance roller  20 , a rotation position θ PF (t 0 ) of the conveyance roller  20  at the reference time t 0 , and the rotation position θ PF (t) of the conveyance roller  20  at the current time t (S 230 ). Specifically, the roll diameter estimator  150  calculates the sheet conveyance amount L(t) in accordance with an equation L(t)=R p ·(θ PF (t)−θ PF (t 0 )). 
     The roll diameter estimator  150  calculates a rotation amount δθ(t) of the sheet roll Q 0  from the referent time T 0  to the current time t (S 240 ). In the following, a rotation amount of the sheet roll Q 0  is also referred to as a roll rotation amount. 
     In S 240 , the roll diameter estimator  150  calculates the roll rotation amount δθ(t)(=θ SR (t)−θ SR (t 0 )) based on the rotation position θ SR (t 0 ) of the sheet roll Q 0  at the reference time to and the rotation position θ SR (t) of the sheet roll Q 0  at the current time t. The roll rotation amount δθ(t) is calculated by a dimension of an angle that does not require information of the roll diameter R. 
     The roll diameter estimator  150  calculates a change amount δD(t) of a route length D from the reference time t 0  to the current time t (S 250 ). Here, the route length D means a length of the conveyance route from the point P 1  to the point P 2  depicted in  FIG. 1 . In other words, the route length D is coincident with a length of the sheet Q from the point P 1  to the point P 2 . 
     The tensioner  15  has flexibility. Change in the position X of the tensioner  15  changes the length of the conveyance route from the point P 1  to the point P 2  as depicted in  FIG. 2 . As depicted in  FIG. 8 , the route length D is shorter as the position X of the tensioner  15  increases, that is, the route length D is shorter as the tensioner  15  moves forward. 
     The roll diameter estimator  150  calculates a route length D(X(t)) corresponding to the position X(t) of the tensioner  15  at the current time t. Further, the roll diameter estimator  150  calculates a route length D(X(t 0 )) corresponding to a position X(t 0 ) of the tensioner  15  at the reference time T 0 . 
     The roll diameter estimator  150  calculates the route lengths D(X(t 0 )), D(X(t)) in accordance with a table or a function D(X) indicating correspondence relationships between the positions X and the route lengths D stored in advance. 
     The function D(X) may be theoretically derived based on a geometrical relationship between the position X of the tensioner  15  and the points P 1 , P 2 . Regarding the position X and the route length D, route lengths D 1 , D 2 , D 3  . . . in the respective positions X 1 , X 2 , X 3  . . . are measured, and the function D(X) may be calculated by fitting the function to the measured data. 
     The roll diameter estimator  150  may include a table that stores the route lengths D 1 , D 2 , D 3  . . . in the respective positions X 1 , X 2 , X 3  . . . . The roll diameter estimator  150  may refer to the table and may calculate the route lengths D(X(t 0 )), D(X(t)) corresponding to the positions X(t 0 ), X(t) through liner interpolation. 
     In S 250 , the roll diameter estimator  150  calculates a change amount of the route length D δD(t) (=D(X(t))−D(X(t 0 )) based on the calculated route lengths D(X(t 0 )), D(X(t)). 
     After that, the roll diameter estimator  150  calculates the roll diameter R of the sheet roll Q 0  (=(L(t)+δD(t))/δD(t)) as an estimated value based on the sheet conveyance amount L(t), the rotation amount δD(t) of the sheet roll Q 0 , and the change amount δD(t) of the route length D (S 260 ). 
     (L(t)+δD(t)) corresponds to the length of the sheet Q that is pulled out from the sheet roll Q 0  from the reference time t 0  to the time t. δθ(t) corresponds to the rotation amount of the sheet roll Q 0  under a condition that the sheet Q is pulled out by a length (L(t)+δD(t)). 
     It is thus possible to calculate the radius R of the sheet roll Q 0  under a condition that the sheet Q is conveyed by the length (L(t)+δD(t)) in accordance with the equation R=(L(t)+δD(t))/δD(t). According to this equation, the radius R can be calculated accurately under a condition that the radius R is approximately constant from the time t 0  to the time t. 
     After calculating the roll diameter R in S 260 , the roll diameter estimator  150  determines whether end conditions are satisfied in S 270 . Under a condition that the roll diameter estimator  150  has determined that the end conditions are satisfied (S 270 : Yes), the roll diameter estimator  150  ends the processes in  FIG. 7 . Before ending the processes in  FIG. 7 , the roll diameter estimator  150  causes the main controller  51  to save, in the memory  51 B, the roll diameter R calculated in S 260  as the latest estimated value of the roll diameter R (S 275 ). 
     The roll diameter estimator  150  determines that the end conditions are satisfied (S 270 : Yes) under a condition that the sheet conveyance is normally ended and under a condition that the sheet conveyance is abnormally ended. Then, the roll diameter estimator  150  ends the processes in  FIG. 7 . 
     Under a condition that the roll diameter estimator  150  has determined that the end conditions are not satisfied (S 270 : No), the roll diameter estimator  150  determines whether a time 2t 1  that is twice as long as the predefined time t 1  has elapsed from the reference time t 0  (S 280 ). Under a condition that the roll diameter estimator  150  has determined that the time 2t 1  has not elapsed (S 280 : No), the roll diameter estimator  150  executes the process of S 220 . 
     Under a condition that the roller diameter estimator  150  has determined in S 280  that the time 2t 1  has elapsed (S 280 : Yes), the roll diameter estimator  150  updates the reference time t 0  to a time (t 0 +t 1 ) elapsed from the reference time t 0  by the time t 1  (S 290 ). Then, the roll diameter estimator  150  executes the process of S 220 . As described above, the roll diameter estimator  150  executes the processes of S 220  to S 290  for each sampling period t s . In S 260 , the roll diameter estimator  150  estimates the roll diameter R. 
     In  FIG. 9 , each broken line indicates a state where the reference time t 0  is shifted every time t 1 . In  FIG. 9 , the time t 1  is 0.5 seconds. The roll diameter estimator  150  calculates the roll diameter R for each sampling period t s  during each period from the time (t 0 +t 1 ) to a time (t 0 +2t 1 ) hatched in  FIG. 9 . In this case, the sampling period t s  may be one millisecond. 
     In  FIG. 10 , change in the roll diameter R calculated by the above method is indicated by a solid line. In  FIG. 10 , a correct roll diameter R is 69 mm. It can be evaluated that the roll diameter R is calculated accurately under a condition that the roll diameter R calculated is in a range of 67 to 71 mm. 
     In this embodiment, the roll diameter R is calculated by including not only the conveyance amount L(t) of the sheet Q by the conveyance roller  20  but also the change amount δD(t) of the route length D. It is thus possible to estimate the roll diameter R quickly and accurately. 
     In  FIG. 10 , a broken line indicates a roll diameter R′ estimated in accordance with an equation R′=L(t)/δθ(t) without including the change amount δD(t) of the route length D. As understood from a comparison between the roll diameter R indicated by the solid line and the roll diameter R′ indicated by the broken line, it is possible to estimate the roll diameter R more quickly and accurately by the technique of the present disclosure in which the change amount δD(t) of the route length D is included, than a case where the change amount δD(t) is not included. A dot-dash chain line in  FIG. 10  indicates a difference of the route length D from a specific route length D 0 . 
     Referring to  FIG. 11 , details of the initial process (S 110  in  FIG. 4 ) executed by the main controller  51  are explained. Under a condition that the initial process is started (S 110 ), the main controller  51  determines whether the sheet conveyance executed last time (executed previously or most recently) is ended normally (S 310 ). 
     Under a condition that the main controller  51  has determined that the initial process is ended normally (S 310 : Yes), the main controller  51  executes a process of S 330 . Under a condition that the main controller  51  has determined that the initial process is ended abnormally (S 310 : No), the main controller  51  determines whether the previous abnormal end (last abnormal end) is caused by a jam (S 320 ). 
     Under a condition that the main controller  51  has determined that the previous abnormal end is caused by the jam (S 320 : Yes), the main controller  51  executes the process of S 330 . Under a condition that the main controller  51  has determined that the previous abnormal end is caused by any other cause than the jam (S 320 : No), the main controller  51  executes a process of S 360 . 
     The abnormal end due to any other cause than the jam may be an abnormal end caused under a condition that the sheet Q of the sheet roll Q 0  is used up or finished up. Under the condition that the sheet roll Q is used up, a new sheet roll Q 0  is installed in the holder  10 . Under a condition that the previous sheet conveyance (last sheet conveyance) is not executed, the main controller  51  formally determines as “No” in S 310  and S 320 , and executes the process of S 360 . 
     In S 330 , the main controller  51  sets a latest estimated value saved in the memory  51 B as the roll diameter R of the sheet roll Q 0  to be used for the tension control performed in a period until the estimation of the roll diameter R is started. 
     Specifically, the main controller  51  sets the latest estimated value read from the memory  51 B, as a current roll diameter R, for the feedforward controller  160  and the gain setter  180  included in the tension controller  57 . Under a condition that the roll diameter estimator  150  newly estimates the roll diameter R, the roll diameter R set in the feedforward controller  160  and the gain setter  180  is updated to the newly estimated value. 
     After executing the process of S 330 , the main controller  51  activates the feedforward control of the tension controller  57  by permitting operation of the feedforward controller  160  (S 340 ). 
     Further, the main controller  51  inputs a velocity profile for the high velocity conveyance, as a velocity profile that defines the target rotation velocity of the conveyance roller  20 , to the velocity controller  55 . Accordingly, the conveyance mode of the sheet Q is set to a “high velocity” mode (S 350 ). 
     After that, the main controller  51  executes a process of S 390 . In S 390 , the main controller  51  causes the velocity controller  55  and the tension controller  77  to execute the rotation velocity control and the tension control so that the sheet Q is conveyed with high velocity to the starting point in accordance with the velocity profile input in S 350 . Then, the main controller  51  ends the processes in  FIG. 11 . 
     In S 360 , the main controller  51  discards the latest estimated value of the roll diameter R saved in the memory  51 B. Then, the main controller  51  invalidates the feedforward control by prohibiting operation of the feedforward controller  160  (S 370 ). 
     In S 380 , the main controller  51  inputs a velocity profile for low velocity conveyance to the velocity controller  55 , and sets the conveyance mode of the sheet Q to a “low velocity” mode. In the velocity profile for the lower velocity mode, the target rotation velocity in the constant velocity section after the sheet Q reaches a maximum velocity is lower than that in the high velocity mode. 
     After that, the main controller  51  causes the velocity controller  55  and the tension controller  77  to execute the rotation velocity control and the tension control so that the sheet Q is conveyed with low velocity to the starting point in accordance with the velocity profile input in S 380 . Then, the main controller  51  ends the processes in  FIG. 11 . 
     The operation prohibition state of the feedforward controller  160  during the low velocity conveyance is maintained until the estimation of the roll diameter R by the roll diameter estimator  150  is newly started. In other words, the feedforward controller  160  can calculate the feedforward control input U F  based on the roll diameter R under a condition that information of the roll diameter R that is newly estimated by the roll diameter estimator  150  is obtained. 
     In addition, under a condition that the latest estimated value of the roll diameter R is discarded in S 360 , the gain setter  180  sets, to the PID controller  140 , gains K p =K d (R s ), K i =K i (R s ), and K d =K d (R s ) in accordance with a standard roll diameter R s  given from the main controller  51  until the roll diameter R is newly estimated by the roll diameter estimator  150 . 
     In the image forming system  1  of this embodiment described above, the roll diameter estimator  150  of the controller  50  estimates the roll diameter R, which is a radius of the sheet roll Q 0 , based on the sheet conveyance amount L(t) measured, the rotation amount δθ(t) of the sheet roll Q 0  measured, and the position X of the tensioner  15  detected. 
     Specifically, the roll diameter estimator  150  estimates the change amount δD(t) of the conveyance route length D(t) of the sheet Q from the first point P 1  where the sheet Q is pulled out from the sheet roll Q 0  to the second point P 2  where the sheet Q is nipped between the conveyance roller  20  and the nip roller  25 , based on the position X(t) of the tensioner  15 . 
     The roll diameter estimator  150  estimates the roll diameter R based on the roll rotation amount δθ(t) and a pulling-out amount of the sheet Q (L(t)+δD(t)) determined from the sheet conveyance amount L(t) caused by the rotation of the conveyance roller  20  and the change amount δD(t) of the route length D(t). 
     Specifically, the roll diameter estimator  150  estimates the roll diameter R (=(L(t)+δD(t))/δD(t)) based on the sheet conveyance amount L(t), the roll rotation amount δθ(t) of the sheet roll Q 0 , and the change amount δD(t) of the route length D during a period t 0  to t from the reference time T 0  to an estimated time t. 
     Thus, in the image forming system  1  of this embodiment, the roll diameter R can be estimated with high accuracy without providing a dedicated sensor such as an optical distance sensor even under a condition that the change in the position X of the tensioner  15  changes the posture and the route length D of the sheet Q. In other words, it is possible to estimate the roll diameter R with high accuracy without actually measuring the roll diameter R by the dedicated sensor. 
     In this embodiment, as depicted in  FIG. 9 , the roll diameter estimator  150  updates the reference time t 0  to a time elapsed from the reference time t 0  by the time t 1  every time the time 2 t1  elapses from the reference time t 0 . The roll diameter estimator  150  estimates, for each sampling period t s , the roll diameter R based on the sheet conveyance amount L(t), the rotation amount δθ(t) of the sheet roll Q 0 , and the change amount δD(t) of the route length D during a period from the reference time t 0  to the time t. 
     In this embodiment, the roll diameter R at the time t is estimated based on an observation value related to the sheet conveyance during the observation period t 0  to t. The observation period is updated to partially overlap with the last observation period (previous observation period), thus continuously changing the observation period. Accordingly, in this embodiment, it is possible to inhibit a high-frequency error component of the roll diameter R estimated and to estimate the roll diameter R stably compared to a case where the observation period is updated not to overlap with the last observation period. 
     Further, in this embodiment, the roll diameter R estimated by the roll diameter estimator  150  is stored in the memory  51 B under a condition that the sheet conveyance ends (S 275 ). In the next sheet conveyance, the feedforward control input U F  is calculated based on the roll diameter R stored last time (stored most recently or previously) until the estimation of the roll diameter R is started by the roll dimeter estimator  150 . The supply motor  61  is controlled by the control input U SR  based on the feedforward control input U F  and the feedback control input U B  calculated. 
     However, under a condition that the last sheet conveyance is ended abnormally and this abnormal end is caused by any other cause than the jam, the roll diameter R saved last time (saved most recently or previously) is discarded and the feedforward control is inactivated in view of the possibility that sheet roll Q 0  is being replaced. That is, the tension controller  57  controls the rotation of the supply motor  61  only by the feedback control, in other words, only by the feedback control input U B . 
     However, under the condition that only the feedback control is used to control the rotation of the supply motor  61 , a conveyance error of the sheet Q until the sheet Q is arranged at the starting point may be larger than a case where the sheet Q is conveyed using the feedforward control and the feedback control. Thus, in this embodiment, under the condition that the feedforward control is inactivated, the velocity profile for low velocity conveyance is set in the velocity controller  55  and the sheet Q is conveyed at low velocity. 
     That is, under the condition that the supply motor  61  is controlled by the feedforward control and the feedback control, the sheet Q is conveyed at high velocity through the control of the controller  20  based on the velocity profile for high velocity conveyance. Under the condition that the supply motor  61  is controlled only by the feedback control, the sheet Q is conveyed at low velocity through the control of the conveyance roller  20  based on the velocity profile for low velocity conveyance. 
     Thus, in this embodiment, it is possible to inhibit deterioration in control accuracy by executing the conveyance control so that the sheet Q is conveyed at low velocity even under a condition that the control with high accuracy can not be executed due to the shortage of information of the roll diameter R. If the sheet Q is arranged at the starting point with low control accuracy, the sheet Q is liable to be excessively conveyed due to the control error and the sheet Q may come into contact with the recording head  40 . In this embodiment, the conveyance control of the sheet Q at low velocity can inhibit such a problem. 
     As described above, in this embodiment, under the condition that the abnormal end is caused by any other cause than the jam, the feedforward control that needs the information of the roll diameter R is inactivated. The reason thereof is that there is possibility that the sheet roll Q 0  is being replaced and a degree of the accuracy in which the roll diameter R is the last estimated value is low. 
     As understood from this context, the processes of S 310  and S 320  correspond to determination whether or not the roll diameter R saved in the memory  51 B is accurate. Thus, the inactivation of the feedforward control may be executed by any other method than the processes of S 310  and S 320  through which it is determined whether or not the roll diameter R saved in the memory  51 B is accurate. 
     Under a condition that the controller  50  has determined that the accuracy of the roll diameter R saved in the memory  51 B is high, the feedforward control can be executed based on the latest estimated value of the roll diameter R saved last time (saved most recently or previously) in the memory  51 B until the estimation of the roll diameter R by the roll diameter estimator  150  is started. Under a condition that the controller  50  has determined that the accuracy is low, the feedforward control can be inactivated. 
     Subsequently, image forming systems  1  according to the second embodiment and the third embodiment are explained below. The image forming systems  1  according to the second and third embodiments are modified examples of the image forming system  1  according to the first embodiment. 
     In the following, configurations and processes of the image forming systems  1  according to the second and third embodiments that are different from those of the first embodiment are selectively explained. The configurations and processes similar to those of the first embodiment are designated by the same reference numerals and step numbers as the first embodiment, and explanation therefor is omitted. 
     Second Embodiment 
     The image forming system  1  according to the second embodiment is configured so that the main controller  51  executes an origin setting process indicated in  FIG. 12 . 
     In the origin setting process, the position X of the tensioner  15  detected by the position detector  80  is set to zero at the origin position of the tensioner  15 . The origin setting process is thus executed immediately after the image forming system  1  starts up before the conveyance process of the sheet Q starts. The origin setting process may be executed before or after the leading edge of the sheet Q is arranged at the starting point defined in advance in the initial process (S 110 ). 
     The origin position of the tensioner  15  is a position where the tensioner  15  is static under a condition that the spring material  16  has a natural length and no elastic force from the spring material  16  is generated. Under a condition that the tensioner  15  is in the origin position, the tension of the sheet Q acting on the tensioner  15  is zero. 
     Under a condition that the origin setting process starts, the main controller  51  sends or feeds the sheet Q by rotating the supply motor  61  in a normal rotation direction by a predefined amount in a state where the conveyance roller  20  is stopped (S 410 ). The normal rotation direction of the supply motor  61  is a rotation direction in which the sheet Q moves in the conveyance direction. 
     The predefined amount is previously set to a feeding amount of the sheet Q to result in the following situation. That is, even under a condition that the tensioner  15  is in a limit position corresponding to maximum tension before the process of S 410 , the sheet Q being fed or sent out flexes or bends to lose the tension of the sheet Q, and the tensioner  15  moves to the origin position and is static at the origin position. 
     Thus, under a condition that the process of S 410  is completed, the tensioner  15  typically moves to the origin position. The main controller  51  determines that the tensioner  15  is in the origin position and resets the detected position X of the tensioner  15  detected by the position detector  80  to zero (S 420 ). The initial setting of the position detector  80  is executed as described above so that the position X of the tensioner  15  is detected as a relative position from the origin position. 
     After that, the main controller  51  rewinds the sheet Q by reversely rotating the supply motor  61  in a state where the conveyance roller  20  is stopped, and ends the origin setting process indicated in  FIG. 12  (S 430 ). 
     In S 430 , the main controller  51  controls, based on the position X of the tensioner  15  detected by the position detector  80 , the supply motor  61  through the tension controller  57 . In this configuration, the rewinding of the sheet Q is completed under a condition that the tension of the sheet Q reaches predefined constant tension determined in advance. The constant tension may be the target tension under a condition that the sheet Q is conveyed thereafter. 
     As described above, the position detector  80  may be the linear encoder. The linear encoder may be an encoder that has a fixed mechanical origin position on an encoder scale to detect a position with reference to the origin position, an encoder that can detect a position as a relative position from a position reset by software, or the like. 
     According to the second embodiment, the tension of the sheet Q can be accurately estimated based on the detected position X by the position detector  80  also under a condition that the latter encoder is used as the position detector  80 . 
     Third Embodiment 
     Subsequently, the image forming system  1  of the third embodiment is explained. The image forming system  1  of the third embodiment is configured to estimate the roll diameter r by estimating a state quantity related to the rotation system of the sheet roll Q 0  by use of an extended Kalman filter. Although the roll diameter is explained using the variable R in the first embodiment, the roll diameter is explained using a variable r in the third embodiment. 
     The Kalman filter is a technique in which the state quantity is estimated with high accuracy by feeding back an error between an actual observation value and an observation value obtained from an observation model set in advance and by correcting the state quantity. In the extended Kalman filter, a state equation and an observation equation are Taylor-expanded and linearly approximated for application to nonlinear systems. 
     In the third embodiment, the extended Kalman filter is set as follows. Under a condition that the sheet Q has a thickness of μ and that the sheet roll Q 0  rotates 2π, the roll diameter r changes by the thickness μ. A relationship between the roll diameter r and a rotation amount θ of the sheet roll Q 0  is expressed by the following equation. 
     
       
         
           
             
               
                 d 
                  
                 r 
               
               
                 d 
                  
                 
                     
                 
                  
                 θ 
               
             
             = 
             
               - 
               
                 μ 
                 
                   2 
                    
                   π 
                 
               
             
           
         
       
     
     The rotation amount θ of the sheet roll Q 0  can be observed through the rotary encoder  65  and the measurement circuit  67 . A time differential dr/dt of the roll diameter r can be expressed by the following equation by using an angular velocity ω of the sheet roll Q 0 . 
     
       
         
           
             
               
                 d 
                 
                   d 
                    
                   t 
                 
               
                
               r 
             
             = 
             
               
                 r 
                 . 
               
               = 
               
                 
                   
                     
                       d 
                        
                       r 
                     
                     
                       d 
                        
                       θ 
                     
                   
                   · 
                   
                     
                       d 
                        
                       θ 
                     
                     
                       d 
                        
                       t 
                     
                   
                 
                 = 
                 
                   
                     
                       
                         d 
                          
                         r 
                       
                       
                         d 
                          
                         θ 
                       
                     
                      
                     ω 
                   
                   = 
                   
                     
                       - 
                       
                         μ 
                         
                           2 
                            
                           π 
                         
                       
                     
                      
                     ω 
                   
                 
               
             
           
         
       
     
     A state quantity x related to the rotation system of the sheet roll Q 0  is defined as follows by using the roll diameter r, the rotation amount θ of the sheet roll Q 0 , and a rotation velocity ω of the sheet roll Q 0  (i.e., angular velocity ω). A superscript T is a transposition symbol. 
         x =[ r θω] T  
 
     On the assumption that the roll diameter r is estimated by using the extended Kalman filter during conveyance of the sheet Q at constant velocity, a time evolution model (time development model) of the state quantity x (i.e., state equation) is defined by the following equations. 
     
       
         
           
             
               x 
               . 
             
             = 
             
               f 
                
               
                 ( 
                 x 
                 ) 
               
             
           
         
       
       
         
           
             
               
                 r 
                 . 
               
               = 
               
                 
                   - 
                   
                     μ 
                     
                       2 
                        
                       π 
                     
                   
                 
                  
                 ω 
               
             
             , 
             
               
                 θ 
                 . 
               
               = 
               ω 
             
             , 
             
               
                 ω 
                 . 
               
               = 
               0 
             
           
         
       
     
     A Jacobian matrix A of the multivariable vector-valued function f is defined by the following equation. 
     
       
         
           
             
               A 
               = 
               
                 
                   
                     ∂ 
                     f 
                   
                   
                     ∂ 
                     x 
                   
                 
                 = 
               
             
              
             
               [ 
               
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                       - 
                       
                         μ 
                         
                           2 
                            
                           π 
                         
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     1 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
               
               ] 
             
           
         
       
     
     A relationship between the conveyance amount L of the sheet Q and the rotation amount θ of the sheet roll Q 0  and a relationship between a conveyance velocity V of the sheet Q and the rotation velocity ω of the sheet roll Q 0  are expressed as follows. 
     
       
      
       L=rθ−M  
      
     
     
       
      
       V=rω−ζ 
      
     
     The above relational expression related to the conveyance amount L of the sheet Q is an approximate expression because the roll diameter r changes with time. A parameter M corresponds to the change amount δD of the conveyance route length D of the sheet Q from the point P 1  to the point P 2  depicted in  FIG. 1 . A parameter ζ corresponds to a time differential of the change amount δD as well as a difference between the conveyance velocity V of the sheet Q and a circumferential velocity r·ω of the sheet roll Q 0  caused by the change in the conveyance route length D. Thus, as explained in the first embodiment, the parameter M and the parameter ζ can be specified from the position X of the tensioner  15  detected by the position detector  80 . 
     Here, an observation amount y is defined by using the conveyance amount L of the sheet Q, the conveyance velocity V of the sheet Q, and the rotation amount θ of the sheet roll Q 0 . 
         y =[ LV θ] T  
 
     An observation model related to the observation amount y (i.e., observation equation) is defined by the following equation. In the following equation, carets or hats mean estimated values. 
         y=h ( x ) 
         {circumflex over (L)}=rθ−M,{circumflex over (V)}=r ω−ζ,{circumflex over (θ)}=θ
 
     A Jacobian matrix C of the multivariable vector-valued function h is defined by the following equation. 
     
       
         
           
             C 
             = 
             
               [ 
               
                 
                   
                     θ 
                   
                   
                     r 
                   
                   
                     0 
                   
                 
                 
                   
                     ω 
                   
                   
                     0 
                   
                   
                     r 
                   
                 
                 
                   
                     0 
                   
                   
                     1 
                   
                   
                     0 
                   
                 
               
               ] 
             
           
         
       
     
     In this embodiment, the roll diameter r is estimated by estimating the state quantity x by use of the extended Kalman filter based on the above state equation and the observation equation. In the estimation based on the extended Kalman filter, the state quantity x is estimated by repeating a prediction step and a filtering step. 
     In the prediction step, a prior estimated value x t  (caret or hat) and a prior error covariance matrix P t  of the state quantity x are calculated by the following equations based on a posterior estimated value x (caret or hat) certain time before, an error covariance matrix P certain time before, and a state quantity noise Q given in advance. The following equation includes the Jacobian matrix A. 
       {dot over ( {circumflex over (x)} )} t   =A{umlaut over (x)}   
     
       
      
       P 
       t 
       =APA 
       T 
       +Q  
      
     
     In the filtering step, an estimated value x t+1  (caret or hat) and an error covariance matrix P t+1  of the state quantity x are calculated based on the prior estimated value x t  (caret or hat) and the prior error covariance matrix P t  of the state quantity x calculated in the prediction step, an observation amount noise R given in advance, the actual observation amount y, and a prior estimated value y (caret or hat) of the observation amount y. The following equation includes a Jacobian matrix C. 
         {circumflex over (x)}   t+1   ={circumflex over (x)}   t   +K ( y−ŷ ) 
         P   t+1 =(1− KC ) P   t  
 
         K=P   t   C   T ( R+CP   t   C   T ) −1    
     The prior estimated value y (caret or hat) of the observation amount y is calculated by the observation equation and the prior estimated value x t  (caret or hat) of the state quantity x. The parameter M and the parameter ζ are calculated based on the change amount in the position X of the tensioner  15  observed. 
     The roll dimeter r estimated by the roll diameter estimator  150  in this embodiment is an estimated value of the roll diameter r included in an estimated value of the state quantity x calculated here. The conveyance velocity V of the sheet Q included in the observation amount y may be a conveyance velocity R P ·ω PF  measured based on the rotation velocity ω PF  of the conveyance roller  20  measured by the measurement circuit  77  and the radius R P  of the conveyance roller  20 . Or, the conveyance velocity V of the sheet Q included in the observation amount y may be a target conveyance velocity R P ·ω r  of the sheet Q based on the target rotation velocity ω r  from the velocity instruction device  101 . The conveyance amount L of the sheet Q may be a measured value or a value calculated as a time differential of a conveyance velocity V=R P ·ω r  based on the target rotation velocity ω r . 
     The state quantity x may be estimated by defining an observation equation h(x) without including the parameter ζ, that is, on the assumption that the parameter ζ is zero. The state quantity x may be estimated by defining the observation equation h(x) on the assumption that the parameter M is zero and the parameter ζ is zero. 
     The roll diameter estimator  150  executes the processes indicated in  FIG. 13 . The roll diameter estimator  150  can estimate the roll estimator r by executing the prediction step and the filtering step for each sampling period t s  and updating the state quantity x. 
     An equation L=rθ−M related to the conveyance amount L of the sheet Q holds only approximately because the roll diameter r varies. The roll diameter estimator  150  sectionally applies the extended Kalman filter to such an equation that holds approximately, and estimates the roll diameter r. 
     Under a condition that the processes in  FIG. 13  are started, the roll diameter estimator  150  initializes the state quantity x and the error covariance matrix P (S 510 ). After that, the roll diameter estimator  150  sets the rotation amount θ of the sheet roll Q 0  that is an element of the estate quantity x to zero. Further, the roll diameter estimator  150  sets the rotation velocity ω of the sheet roll Q 0  that is an element of the state quantity x to a rotation velocity ω=(R p /r)·ω r  corresponding to the target rotation velocity ω r  of the conveyance roller  20  based on the radius R p  of the conveyance roller  20  and the estimated value of the roll diameter r. Further, the roll diameter estimator  150  sets, as current observation values, the conveyance amount L of the sheet Q and the rotation amount θ of the sheet roll Q 0  that are elements of the observation amount y (S 520 ). 
     After completion of the process of S 520 , the roll diameter estimator  150  executes the prediction step and the filtering step based on the extended Kalman filter, and calculates the estimated value of the state quantity x (S 530 ). In the filtering step, the roll diameter estimator  150  obtains the current observation amount y measured. After that, the roll diameter estimator  150  outputs the estimated value of the roll diameter r in the state quantity x (S 540 ). 
     The roll diameter estimator  150  executes the processes of S 530  to S 540 , updates the state quantity x, and outputs the estimated value of the roll diameter r for each sampling period t s  until a reset timing of the state quantity x arrives (S 550 : YES), or until the end conditions for the processes in  FIG. 13  are satisfied (S 560 : Yes). 
     Under a condition that the reset timing has arrived (S 550 : Yes), the roll diameter estimator  150  executes the process of S 520  to once reset a calculation result based on the extended Kalman filter before the reset timing. After executing the process of S 520 , the roll diameter estimator  150  executes, in S 530 , the prediction step and the filtering step based on the value updated in S 520 , and calculates the estimated value of the state quantity x. 
     The roll diameter estimator  150  sectionally estimates the state quantity x based on the extended Kalman filter by repeatedly executing the above processes. Under a condition that the roll diameter estimator  150  has determined that the end conditions are satisfied (S 560 : Yes), the roll diameter estimator  150  ends the processes in  FIG. 13 . 
     Specifically, the reset timing may be a timing at which the conveyance length L of the sheet Q corresponding to the rotation amount of the conveyance roller  20  exceeds an initial value by a predefined amount. The roll diameter estimator  150  thus calculates the estimated value of the state quantity x including the roll diameter r for each conveyance of the sheet Q by the predefined amount by use of the extended Kalman filter sectionalized. 
     The roll diameter estimator  150  can estimate the state quantity x based on the extended Kalman filter by switching a setting value of the state quantity noise Q in S 530  as indicated in  FIG. 14 . 
     In  FIG. 14 , the roll diameter estimator  150  sets the state quantity noise Q to the first state quantity noise Q 1  (S 620 ) at the beginning of the estimation of the roll diameter r (S 610 : Yes). In any other period than the above (S 610 : No), the roll diameter estimator  150  sets the state quantity noise Q to the second state quantity noise Q 2  (S 630 ). 
     The state quantity noise Q contains an element expressing variation in the state quantity x (in particular, distribution). At the beginning of the estimation of the roll diameter r, the likelihood (the degree of certainty) of the roll diameter r is low. The first state quantity noise Q 1  is thus defined in advance to express larger distribution than the second state quantity noise Q 2 . 
     Accordingly, the roll diameter estimator  150  appropriately estimates the roll diameter r by switching the setting value of the state amount noise Q between the first state quantity noise Q 1  and the second state quantity noise Q 2  depending on the degree of certainty of the roll diameter r estimated. An initial period of the estimation of the roll diameter r, in other words, the first period from the start of estimation of the roll diameter r may be a period during which the estimated value of the roll diameter r is stabilized after the sheet roll Q 0  is installed in the holder  10 . 
     Further, the roll diameter estimator  150  may configure the extended Kalman filter by including the rotation torque τ acting on the sheet roll Q 0  from the supply motor  61  and the tension T of the sheet Q. For example, a state equation may be defined as follows. 
     
       
         
           
             
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     The roll diameter estimator  150  may estimate the state quantity x from the extended Kalman filter based on this state equation and the observation equation described above to estimate the roll diameter r. 
     The estimation of the roll diameter r of the sheet Q based on the extended Kalman filter described above is especially significant under a condition that the image forming system  1  includes a tension controller  200  having a configuration depicted in  FIG. 15  instead of the tension controller  57 . 
     As depicted in  FIG. 15 , the tension controller  200  that replaces the tension controller  57  includes the tension instruction device  110 , the tension estimator  120 , the deviation calculator  130 , the PID controller  140 , the roll diameter estimator  150 , the gain setter (gain setting device)  180 , a primary delay filter  210 , a target velocity generator  220 , a deviation calculator  230 , an adder  240 , a supply velocity controller  250 , a feedforward controller  260 , and an adder  270 . 
     The deviation calculator  130  calculates a deviation E T =T r −T between the target tension T r  from the tension instruction device  110  and the estimated tension T from the tension estimator  120  similar to the first embodiment. The PID controller  140  calculates a tension control input U T =K p ·E T +K i ·INT(E T )+K d ·DIF(E T ) based on the deviation E T  input from the deviation calculator  130 . 
     The tension control input U T  corresponds to the feedback control input U B  of the first embodiment. The gain setter  180  is configured to adjust the gains K p , K i , and K d  in the PID controller  140  based on the roll diameter r estimated by the roll diameter estimator  150 . 
     The target velocity generator  220  calculates a target rotation velocity ω sr  of the sheet roll Q 0  based on the target rotation velocity ω r  of the conveyance roller  20  input from the velocity instruction device  101  of the velocity controller  55  via the primary delay filter  210  and the roll diameter r estimated by the roll diameter estimator  150 . 
     The target velocity generator  220  calculates the target rotation velocity ω sr  in accordance with an equation ω sr =(R p /r)·ω r  so that the conveyance roller  20  and the sheet roll Q 0  rotate at the same circumferential velocity. R p  is a radius of the conveyance roller  20  as described above. The target rotation velocity ω sr  of the sheet roll Q 0  corresponds to the target rotation velocity ω sr  of the rotation shaft  10 A. 
     The deviation calculator  230  calculates a deviation E W =(ω sr −ω) between the target rotation velocity ω sr  from the target velocity generator  220  and the rotation velocity ω of the sheet roll Q 0  measured by the measurement circuit  67 . 
     The adder  240  calculates a control input U C =(U T +E W ) by adding the deviation E W  to the tension control input U T  from the PID controller  140 . The supply velocity controller  250  is configured as the PID controller to calculate a feedback control input U B * by adding a velocity control component to the control input U C  from the adder  240 . Instead of the feedback control input U B  in the first embodiment, the feedback control input U B * is input to the adder  270  in this embodiment. 
     Specifically, the supply velocity controller  250  outputs a value U B *=K wp ·U C +K wi ·INT(U C )+K wd ·DIF(U C ) as the feedback control input U B *. DIF(U C ) is a differential value of the control input U C . The value INT(U C ) is an integral value of the control input U C . K wp  is a proportional gain. K wi  is an integral gain. K wd  is a differential gain. 
     Instead of the actual rotation velocity ω PF  of the conveyance roller  20 , the feedforward controller  260  calculates the acceleration torque τ similar to the first embodiment by use of the target rotation velocity ω r  of the conveyance roller  20  input from the velocity instruction device  101  of the velocity controller  55  via the primary delay filter  210 . 
     The feedforward controller  260  inputs the feedforward control input U F  corresponding to the calculated acceleration torque τ to the adder  270  similar to the first embodiment. Or, the feedforward controller  260  inputs, to the adder  270 , a feedforward input U F * obtained by adding a compensation amount for a viscous friction torque and a dynamic friction torque to the feedforward input U F . 
     The adder  270  outputs an addition value U B *+U F  obtained by adding the feedback control input U B * input from the supply velocity controller  250  and the feedforward control input U F  input from the feedforward controller  260  or an addition value U B *+U F * obtained by adding the feedback control input U B * input from the supply velocity controller  250  and the feedforward control input U F *, as the control input U SR  for the supply motor  61 . 
     The reason why the tension controller  200  does not use the actual rotation velocity ω of the conveyance roller  20  but uses the target rotation velocity ω r  is as follows. A power transmission system such as a gear is provided between the supply motor  61  and the rotation shaft  10 A of the holder  10 . Thus, there is a time lag until driving of the supply motor  61  is reflected in the rotary motion of the sheet roll Q 0 . The time lag may cause a control error if the feedforward control input U F  is calculated based on the actual rotation velocity ω PF  of the conveyance roller  20  to control the supply motor  61 . 
     In the third embodiment, the feedforward control input U F  or the feedforward control input U F * is calculated based on the target rotation velocity ω r  of the control roller  20 . In this case, the rotary motion of the sheet roll Q 0  can be controlled by controlling the supply motor  61  while inhibiting the effect of the time lug. 
     Under the condition that the target rotation velocity ω r  is used, a tensioning process, in which the tension of the sheet Q is set to the target tension T r  by reversely rotating the supply motor  61  in a state where conveyance roller  20  is stopped, may be executed similarly to the process of S 430  in order to make the tension of the sheet Q the target tension T r  before starting the conveyance process of the sheet Q. 
     In the third embodiment, since the roll diameter r is estimated by using the extended Kalman filter, it is possible to estimate the roll diameter with high accuracy while inhibiting effects of an observation error, a signal noise, and the like. Under the condition that the velocity control is performed based on the estimated value of the roll diameter r as depicted in  FIG. 15 , the velocity control is affected by an estimation error of the roll diameter r. Thus, estimating the roll diameter r with high accuracy improves the conveyance control of the sheet Q. 
     It is needless to say that the present disclosure is not limited to the above exemplary embodiments and can take various aspects. In the first embodiment, the acceleration torque τ may be estimated based on the target rotation velocity ω r  of the conveyance roller  20  and the feedforward control input U F  may be calculated similarly to the third embodiment. The conveyance amount L of the sheet Q may be calculated based on an integral value of the target rotation velocity ω r . 
     The holding structure of the sheet roll Q 0  by the holder  10  and the driving system of the sheet roll Q 0  are not limited to the above embodiments. In each of the image forming systems  1  described above, the sheet roll Q 0  is rotatably supported by the rotation shaft  10 A of the holder  10  by inserting the core material of the sheet roll Q 0  into the rotation shaft  10 A of the holder  10 . 
     However, the holder may be formed from a hollow cylindrical material of which inside has an accommodation space for the sheet roll Q 0 . The holder may be configured so that an inner surface defining the accommodation space for the sheet roll Q 0  rotates. The sheet roll Q 0  may rotate depending on the rotation of the inner surface of the holder in a state of being accommodated in the holder. Further, a roller brought into contact with an outer circumferential surface of the sheet roll Q 0  may be provided. The sheet roll Q 0  may rotate by rotation of this roller. 
     The holder  10  may have no shaft. In this configuration, a roller may be provided between the tensioner  15  and the sheet roll Q 0 . The rotation amount of the sheet roll Q may be measured from a rotation amount of the roller. 
     The technique of the present disclosure may be applied to an image forming system not including the belt mechanism  30 . In this case, the image forming system may include a platen for supporting the sheet Q, instead of the belt mechanism  30 . The technique of the present disclosure may be applied to an image forming system in which a recording head of a serial driving system is provided as the recording head  40  instead of the line head. In this case, the recording head forms an image on the sheet Q by reciprocatingly moving in the line direction. The technique of the present disclosure may be applied to an image forming system of an electrophotographic system. 
     The technique of the present disclosure may be applied to a system for forming an image on a surface of the sheet Q that faces the outside in a radial direction of the sheet roll Q 0 . Or, the technique of the present disclosure may be applied to a system for forming an image on a back surface of the sheet Q that faces the inside in the radial direction of the sheet roll Q 0 . The technique of the present disclosure may be applied to a system for forming an image on both surfaces of the sheet Q. The position of the conveyance roller  20  and the position of the nip roller  25  may be exchangeable or replaceable. 
     The technique of the present disclosure can be applied not only to the system for forming an image on the sheet Q by use of a color material but also to a variety of systems. For example, the technique of the present disclosure may be applied to a system for making a mark in the sheet Q through perforation or to a system for irradiating a surface of the sheet Q with light to sterilize the surface. The technique of the present disclosure may be applied to a system for forming a trace pattern on a sheet-like substrate. The sheet roll Q and the sheet Q may be paper, vinyl, or a flexible print substrate (FPC). 
     The configuration of the tensioner  15  is not limited to the above embodiment. The tensioner may be configured as an arm in which the first end is supported pivotally and the second end has a roller, like a pendulum arm. Further, the technique of the present disclosure does not limit the configuration of the rotary encoders  65  and  75 . The rotary encoders  65  and  75  may be magnetic rotary encoders instead of optical rotary encoders. 
     The printing controller  53 , the velocity controller  55 , and the tension controller  57  may be configured by combining the CPU and the ASIC. In each of the controller  50 , the main controller  51 , the printing controller  53 , the velocity controller  55 , and the tension controller  57 , the number of the CPU(s) and the ASIC(s) and whether or not the CPU and the ASIC is provided therein is not limited to the above specific examples. 
     The function provided in one component in each of the above exemplary embodiments may be distributed in components. The function provided in components may be integrated in one component. Part of the configuration according to each of the above exemplary embodiments may be omitted. The embodiments of the present disclosure include various embodiments or aspects that are included in the technical ideas specified by the following claims. 
     There is a correspondence relationship between the words and terms as follows. The process of S 240  executed by the rotary encoder  65 , the measurement circuit  67 , and the roll diameter estimator  150  corresponds to an exemplary process achieved by a rotation measuring device. The process of S 230  executed by the rotary encoder  75 , the measurement circuit  77 , and the roll diameter estimator  150  corresponds to an exemplary process achieved by a conveyance measuring device. The conveyance motor  71  corresponds to an exemplary first motor. The supply motor  61  corresponds to an exemplary second motor. 
     The deviation calculator  103  and the PID controller  105  of the velocity controller  55  correspond to an exemplary first feedback control element. The deviation calculator  130  and the PID controller  140  of the tension controller  57  correspond to an exemplary second feedback control element. The feedforward controller  160  corresponds to an exemplary feedforward control element.