Patent Publication Number: US-11041473-B2

Title: Control device and control method for onboard internal combustion engine

Description:
BACKGROUND 
     1. Field 
     The following description relates to a control device and a control method for an onboard internal combustion engine. 
     2. Description of Related Art 
     Japanese Laid-Open Patent Publication No. 2012-036849 discloses a control device that controls an onboard internal combustion engine that executes an idling stop control. The idling stop control automatically stops and restarts the internal combustion engine to discontinue idling operation. In a vehicle where the idling stop control is executed, while the internal combustion engine is not operating, oxygen is absorbed by a catalyst device. Thus, immediately after restarting, oxygen is excessively absorbed by the catalyst device. This reduces the ability to purify exhaust gas. Accordingly, the control device described in Japanese Laid-Open Patent Publication No. 2012-036849 executes a rich reduction control to reduce oxygen absorbed in the catalyst device at the restarting time. In the rich reduction control, a fuel injection amount is increased such that the air-fuel ratio becomes richer than the stoichiometric air-fuel ratio. This introduces exhaust gas containing excessive fuel into the catalyst device. 
     To restore the ability to purify exhaust gas through the rich reduction control, it is desired that the reduction of oxygen be completed immediately to quickly restore the purification ability. 
     SUMMARY 
     It is an object of the present disclosure to provide a control device and a control method for an onboard internal combustion engine capable of immediately completing the reduction of excessive oxygen at a restarting time and quickly restoring the purification ability by expediting a reduction reaction in a catalyst device. 
     This Summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This Summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter. 
     To solve the above-described problem, according to a first aspect of the present disclosure, a control device for an onboard internal combustion engine is provided. The onboard internal combustion engine includes a fuel injection valve, an ignition device, a turbocharger equipped with a wastegate that controls a boost pressure by opening and closing a wastegate port, and a catalyst device arranged downstream of a turbine housing of the turbocharger in an exhaust passage, the catalyst device having an oxygen absorption ability and purifying exhaust gas. The control device includes an injection control unit that controls the fuel injection valve and performs a fuel cut-off control to stop supply of fuel to a combustion chamber during deceleration, an ignition control unit that controls the ignition device, an idling stop control unit that executes an idling stop control to discontinue idling operation by automatically stopping and restarting engine operation, and a boost control unit that controls opening and closing of the wastegate. The injection control unit is configured to execute a rich reduction control that makes an air-fuel ratio richer than a stoichiometric air-fuel ratio when the engine operation has been resumed by resuming the supply of the fuel to the combustion chamber. The boost control unit is configured to execute a valve-closing keeping control that keeps the wastegate closed until a condition for cancelling the valve-closing keeping control has been satisfied by the engine operation that was performed after closing the wastegate during execution of the fuel cut-off control. 
     To solve the above-described problem, according to a second aspect of the present disclosure, a control device for an onboard internal combustion engine is provided. The onboard internal combustion engine includes a fuel injection valve, an ignition device, a turbocharger equipped with a wastegate that controls a boost pressure by opening and closing a wastegate port, and a catalyst device arranged downstream of a turbine housing of the turbocharger in an exhaust passage, the catalyst device having an oxygen absorption ability and purifying exhaust gas. The control device includes circuitry that includes an injection control unit that controls the fuel injection valve and performs a fuel cut-off control to stop supply of fuel to a combustion chamber during deceleration, an ignition control unit that controls the ignition device, an idling stop control unit that executes an idling stop control to discontinue idling operation by automatically stopping and restarting engine operation, and a boost control unit that controls opening and closing of the wastegate. The injection control unit is configured to execute a rich reduction control that makes an air-fuel ratio richer than a stoichiometric air-fuel ratio when the engine operation has been resumed by resuming the supply of the fuel to the combustion chamber. The boost control unit is configured to execute a valve-closing keeping control that keeps the wastegate closed until a condition for cancelling the valve-closing keeping control has been satisfied by the engine operation that was performed after closing the wastegate during execution of the fuel cut-off control. 
     To solve the above-described problem, according to a third aspect of the present disclosure, a control device for an onboard internal combustion engine is provided. The onboard internal combustion engine includes a fuel injection valve, an ignition device, a turbocharger equipped with a wastegate that controls a boost pressure by opening and closing a wastegate port, and a catalyst device arranged downstream of a turbine housing of the turbocharger in an exhaust passage, the catalyst device having an oxygen absorption ability and purifying exhaust gas. The control device includes an injection control unit that controls the fuel injection valve and performs a fuel cut-off control to stop supply of fuel to a combustion chamber during deceleration, an ignition control unit that controls the ignition device, an idling stop control unit that executes an idling stop control to discontinue idling operation by automatically stopping and restarting engine operation, and a boost control unit that controls opening and closing of the wastegate. The injection control unit is configured to execute a rich reduction control that makes an air-fuel ratio richer than a stoichiometric air-fuel ratio when the engine operation has been resumed by resuming the supply of the fuel to the combustion chamber. The boost control unit is configured to execute a valve-closing keeping control that keeps the wastegate closed until a condition for cancelling the valve-closing keeping control has been satisfied by the engine operation that was performed after closing the wastegate prior to execution of the fuel cut-off control when a condition for executing the fuel cut-off control was satisfied. 
     To solve the above-described problem, according to a fourth aspect of the present disclosure, a control device for an onboard internal combustion engine is provided. The onboard internal combustion engine includes a fuel injection valve, an ignition device, a turbocharger equipped with a wastegate that controls a boost pressure by opening and closing a wastegate port, and a catalyst device arranged downstream of a turbine housing of the turbocharger in an exhaust passage, the catalyst device having an oxygen absorption ability and purifying exhaust gas. The control device includes circuitry that includes an injection control unit that controls the fuel injection valve and performs a fuel cut-off control to stop supply of fuel to a combustion chamber during deceleration, an ignition control unit that controls the ignition device, an idling stop control unit that executes an idling stop control to discontinue idling operation by automatically stopping and restarting engine operation, and a boost control unit that controls opening and closing of the wastegate. The injection control unit is configured to execute a rich reduction control that makes an air-fuel ratio richer than a stoichiometric air-fuel ratio when the engine operation has been resumed by resuming the supply of the fuel to the combustion chamber. The boost control unit is configured to execute a valve-closing keeping control that keeps the wastegate closed until a condition for cancelling the valve-closing keeping control has been satisfied by the engine operation that was performed after closing the wastegate prior to execution of the fuel cut-off control when a condition for executing the fuel cut-off control was satisfied. 
     To solve the above-described problem, according to a fifth aspect of the present disclosure, a control method for an onboard internal combustion engine is provided. The onboard internal combustion engine includes a fuel injection valve, an ignition device, a turbocharger equipped with a wastegate that controls a boost pressure by opening and closing a wastegate port, and a catalyst device arranged downstream of a turbine housing of the turbocharger in an exhaust passage, the catalyst device having an oxygen absorption ability and purifying exhaust gas. The control method includes controlling the fuel injection valve and performing a fuel cut-off control to stop supply of fuel to a combustion chamber during deceleration, controlling the ignition device, executing an idling stop control to discontinue idling operation by automatically stopping and restarting engine operation, controlling opening and closing of the wastegate, executing a rich reduction control that makes an air-fuel ratio richer than a stoichiometric air-fuel ratio when the engine operation has been resumed by resuming the supply of the fuel to the combustion chamber, and executing a valve-closing keeping control that keeps the wastegate closed until a condition for cancelling the valve-closing keeping control has been satisfied by the engine operation that was performed after closing the wastegate during execution of the fuel cut-off control. 
     To solve the above-described problem, according to a sixth aspect of the present disclosure, a control method for an onboard internal combustion engine is provided. The onboard internal combustion engine includes a fuel injection valve, an ignition device, a turbocharger equipped with a wastegate that controls a boost pressure by opening and closing a wastegate port, and a catalyst device arranged downstream of a turbine housing of the turbocharger in an exhaust passage, the catalyst device having an oxygen absorption ability and purifying exhaust gas. The control method includes controlling the fuel injection valve and performing a fuel cut-off control to stop supply of fuel to a combustion chamber during deceleration, controlling the ignition device, executing an idling stop control to discontinue idling operation by automatically stopping and restarting engine operation, controlling opening and closing of the wastegate, executing a rich reduction control that makes an air-fuel ratio richer than a stoichiometric air-fuel ratio when the engine operation has been resumed by resuming the supply of the fuel to the combustion chamber, and executing a valve-closing keeping control that keeps the wastegate closed until a condition for cancelling the valve-closing keeping control has been satisfied by the engine operation that was performed after closing the wastegate prior to execution of the fuel cut-off control when a condition for executing the fuel cut-off control was satisfied. 
     To solve the above-described problem, according to a seventh aspect of the present disclosure, a control device for an onboard internal combustion engine is provided. The onboard internal combustion engine includes a fuel injection valve, an ignition device, a turbocharger equipped with a wastegate that controls a boost pressure by opening and closing a wastegate port, and a catalyst device arranged downstream of a turbine housing of the turbocharger in an exhaust passage, the catalyst device having an oxygen absorption ability and purifying exhaust gas. The control device includes an injection control unit that controls the fuel injection valve, an ignition control unit that controls the ignition device, an idling stop control unit that executes an idling stop control to discontinue idling operation by automatically stopping and restarting engine operation, and a boost control unit that controls opening and closing of the wastegate. The injection control unit is configured to execute a rich reduction control that makes an air-fuel ratio richer than a stoichiometric air-fuel ratio when the engine operation has been resumed by resuming the supply of the fuel to the combustion chamber. The boost control unit is configured to execute a valve-closing keeping control that closes the wastegate when the idling stop control unit stops the supply of the fuel or before the idling stop control unit stops the supply of the fuel in a case in which a condition for executing the idling stop control is satisfied and keeps the wastegate closed until a condition for cancelling the valve-closing keeping control has been satisfied by the restarted engine operation. 
     To solve the above-described problem, according to an eighth aspect of the present disclosure, a control device for an onboard internal combustion engine is provided. The onboard internal combustion engine includes a fuel injection valve, an ignition device, a turbocharger equipped with a wastegate that controls a boost pressure by opening and closing a wastegate port, and a catalyst device arranged downstream of a turbine housing of the turbocharger in an exhaust passage, the catalyst device having an oxygen absorption ability and purifying exhaust gas. The control device includes an injection control unit that controls the fuel injection valve and performs a fuel cut-off control to stop supply of fuel to a combustion chamber during deceleration, an ignition control unit that controls the ignition device, and a boost control unit that controls opening and closing of the wastegate. The injection control unit is configured to execute a rich reduction control that makes an air-fuel ratio richer than a stoichiometric air-fuel ratio when the engine operation has been resumed by resuming the supply of the fuel to the combustion chamber. The boost control unit is configured to execute a valve-closing keeping control that closes the wastegate during execution of the fuel cut-off control or prior to the execution of the fuel cut-off control when a condition for executing the fuel cut-off control is satisfied and keeps the wastegate closed until a condition for cancelling the valve-closing keeping control has been satisfied after the fuel cut-off control was ended to resume the supply of the fuel. 
     Other features and aspects will be apparent from the following detailed description, the drawings, and the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a diagram showing the configurations of a control device and an onboard internal combustion engine that is subject to control according to an embodiment of the present disclosure. 
         FIG. 2  is a cross-sectional view showing the turbine housing in the turbocharger. 
         FIG. 3  is a flowchart illustrating the flow of processes in a routine for determining to start a rich reduction control. 
         FIG. 4  is a flowchart illustrating the flow of processes in a routine for determining to end the rich reduction control. 
         FIG. 5  is a flowchart illustrating the flow of processes in a routine for determining to start a valve-closing keeping control in a first embodiment. 
         FIG. 6  is a flowchart illustrating the flow of processes in a routine for determining to end the valve-closing keeping control. 
         FIG. 7  is a timing diagram illustrating the relationship between the timings of executing controls. 
         FIG. 8  is a flowchart illustrating the flow of processes in a routine for determining to start the valve-closing keeping control in a second embodiment. 
         FIG. 9  is a flowchart illustrating the flow of processes in a routine for determining to start a fuel cut-off control. 
         FIG. 10  is a timing diagram illustrating the relationship between the timing of executing the valve-closing keeping control and the timing of executing the fuel cut-off control. 
     
    
    
     Throughout the drawings and the detailed description, the same reference numerals refer to the same elements. The drawings may not be to scale, and the relative size, proportions, and depiction of elements in the drawings may be exaggerated for clarity, illustration, and convenience. 
     DETAILED DESCRIPTION 
     This description provides a comprehensive understanding of the methods, apparatuses, and/or systems described. Modifications and equivalents of the methods, apparatuses, and/or systems described are apparent to one of ordinary skill in the art. Sequences of operations are exemplary, and may be changed as apparent to one of ordinary skill in the art, with the exception of operations necessarily occurring in a certain order. Descriptions of functions and constructions that are well known to one of ordinary skill in the art may be omitted. 
     Exemplary embodiments may have different forms, and are not limited to the examples described. However, the examples described are thorough and complete, and convey the full scope of the disclosure to one of ordinary skill in the art. 
     First Embodiment 
     A control device  100  for an internal combustion engine  10 , which is an onboard internal combustion engine, according to a first embodiment will now be described with reference to  FIGS. 1 to 7 . 
     As shown in  FIG. 1 , the internal combustion engine  10 , which is an onboard internal combustion engine, is equipped with a turbocharger  50 , which includes a wastegate  60 . The turbocharger  50  includes a compressor housing  51  and a turbine housing  52 . The compressor housing  51  is arranged on an intake passage  12  of the internal combustion engine  10 . The turbine housing  52  is arranged on an exhaust passage  19  of the internal combustion engine  10 . The internal combustion engine  10  is controlled by the control device  100 . 
     An air flow meter  33  is arranged at a portion of the intake passage  12  located upstream of the compressor housing  51 . The air flow meter  33  detects an intake air amount and the temperature of intake air. An intercooler  70 , a throttle valve  31 , and an intake pressure sensor  36  are arranged in this order from the upstream side at portions of the intake passage  12  downstream of the compressor housing  51 . The intercooler  70  cools intake air through heat exchange with coolant. The throttle valve  31  is driven by a motor to adjust the intake air amount. 
     Further, the internal combustion engine  10  includes a port injection valve  14 , which is a fuel injection valve that injects fuel into intake air flowing through an intake port  13 . The port injection valve  14  is arranged on the intake port  13 , which is a portion that connects the intake passage  12  to a combustion chamber  11 . In addition, the combustion chamber  11  includes a direct injection valve  15  and an ignition device  16 . The direct injection valve  15  is a fuel injection valve that directly injects fuel into the combustion chamber  11 . The ignition device  16  performs spark discharge to ignite the air-fuel mixture of air and fuel introduced into the combustion chamber  11 . The combustion chamber  11  is connected to the exhaust passage  19  by an exhaust port  22 . 
     The internal combustion engine  10  is an inline four-cylinder internal combustion engine and includes four combustion chambers  11 .  FIG. 1  shows only one of the four combustion chambers  11 . When the air-fuel mixture burns in the combustion chamber  11 , a piston  17  reciprocates to rotate a crankshaft  18 , which is an output shaft of the internal combustion engine  10 . The exhaust gas subsequent to being burned is discharged from the combustion chamber  11  to the exhaust passage  19 . 
     The intake port  13  includes an intake valve  23 . The exhaust port  22  includes an exhaust valve  24 . The intake valve  23  is opened and closed by rotation of an intake camshaft  25 , to which rotation of the crankshaft  18  is transmitted. The exhaust valve  24  is opened and closed by rotation of an exhaust camshaft  26 , to which rotation of the crankshaft  18  is transmitted. 
     The intake camshaft  25  includes an intake-side variable valve timing mechanism  27 . The intake-side variable valve timing mechanism  27  varies the phase of the intake camshaft  25  relative to the crankshaft  18  to vary the timing of opening and closing the intake valve  23 . Further, the exhaust camshaft  26  includes an exhaust-side variable valve timing mechanism  28 . The exhaust-side variable valve timing mechanism  28  varies the phase of the exhaust camshaft  26  relative to the crankshaft  18  to vary the timing of opening and closing the exhaust valve  24 . 
     A timing chain  29  is wound around the intake-side variable valve timing mechanism  27 , the exhaust-side variable valve timing mechanism  28 , and the crankshaft  18 . Thus, when rotation of the crankshaft  18  is transmitted by the timing chain  29 , the intake camshaft  25  rotates together with the intake-side variable valve timing mechanism  27  and the exhaust camshaft  26  rotates together with the exhaust-side variable valve timing mechanism  28 . A catalyst device  80  is arranged at a portion of the exhaust passage  19  located downstream of the turbine housing  52 . The catalyst device  80  supports a three-way catalyst that reduces NOx and oxidizes CO and HC in exhaust gas at the same time. The catalyst device  80  has an oxygen absorption ability to absorb oxygen contained in the gas flowing through the exhaust passage  19 . 
     As shown in  FIG. 2 , an upstream exhaust pipe  20  and a downstream exhaust pipe  21 , which form the exhaust passage  19 , are connected to the turbine housing  52 . The turbine housing  52  accommodates a turbine wheel  54 . The compressor housing  51  accommodates a compressor wheel  53 . A bearing housing  56  accommodates a shaft  55 . The turbine wheel  54  is coupled to the compressor wheel  53  by the shaft  55 . The turbine wheel  54  is rotated by the stream of exhaust gas introduced into the turbine housing  52  through the upstream exhaust pipe  20 . This rotates the compressor wheel  53  to compress intake air and then deliver the intake air to the combustion chamber  11 . 
     The turbine housing  52  includes a wastegate port  57 . Exhaust gas passes through the wastegate port  57  to bypass the turbine wheel  54  and flow toward the downstream side of the turbine wheel  54 . The wastegate  60  opens and closes the outlet of the wastegate port  57  to control a boost pressure. That is, when the wastegate  60  is fully closed, the exhaust gas introduced into the turbine housing  52  through the upstream exhaust pipe  20  passes through the turbine wheel  54  and flows into the downstream exhaust pipe  21 . In this case, the turbine wheel  54  and the compressor wheel  53  rotate to increase the boost pressure. When the wastegate  60  is open, the exhaust gas introduced into the turbine housing  52  through the upstream exhaust pipe  20  bypasses the turbine wheel  54 , passes through the wastegate port  57 , and flows into the downstream exhaust pipe  21 . In this case, the boost pressure is low. The wastegate  60  is driven by an actuator  61 . The actuator  61  may be an electric motor or a device that is actuated using air pressure or hydraulic pressure. 
     As shown in  FIG. 1 , an upstream A/F sensor  34  is arranged on a portion of the exhaust passage  19  between the turbine housing  52  and the catalyst device  80 . The upstream A/F sensor  34  is a sensor that outputs a detection value corresponding to the oxygen concentration of gas flowing through the exhaust passage  19 , that is, an air-fuel ratio sensor that detects the air-fuel ratio of air-fuel mixture. Further, a downstream A/F sensor  35  is arranged on a portion of the exhaust passage  19  located downstream of the catalyst device  80 . The downstream A/F sensor  35  is an air-fuel ratio sensor in the same manner as the upstream A/F sensor  34 . 
     The control device  100  controls the internal combustion engine  10  by operating various devices subject to operation such as the throttle valve  31 , the port injection valve  14 , the direct injection valve  15 , the ignition device  16 , the intake-side variable valve timing mechanism  27 , the exhaust-side variable valve timing mechanism  28 , and the wastegate  60 . A detection signal of the operation amount of an accelerator of a driver is input to the control device  100  by an accelerator position sensor  30 . Further, a detection signal of a vehicle speed, which is a traveling speed of the vehicle, is input to the control device  100  by a vehicle speed, sensor  41 . 
     Furthermore, in addition to the air flow meter  33 , the upstream A/F sensor  34 , the downstream A/F sensor  35 , and the intake pressure sensor  36 , detection signals of various sensors are input to the control device  100 . For example, a throttle position sensor  32  detects an open degree of the throttle valve  31 . A crank position sensor  38  detects a rotation phase of the crankshaft  18 . A water temperature sensor  37  detects a coolant temperature, which is the temperature of coolant in the internal combustion engine  10 . From a detection signal of the crank position sensor  38 , the control device  100  calculates an engine rotation speed, which is a rotation speed of the crankshaft  18  of the internal combustion engine  10 . An intake-side cam position sensor  39  detects a rotation phase of the intake camshaft  25 . From a detection signal of the intake-side cam position sensor  39  and a detection signal of the crank position sensor  38 , the control device  100  calculates the phase of the intake camshaft  25  relative to the crankshaft  18 , which indicates the timing of opening and closing the intake valve  23 . An exhaust-side cam position sensor  40  detects a rotation phase of the exhaust camshaft  26 . From a detection signal of the exhaust-side cam position sensor  40  and a detection signal of the crank position sensor  38 , the control device  100  calculates the phase of the exhaust camshaft  26  relative to the crankshaft  18 , which indicates the timing of opening and closing the exhaust valve  24 . 
     The control device  100  receives output signals of various sensors and also performs various types of calculation based on these output signals. Further, the control device  100  executes various types of control for engine operation in accordance with the calculation results. The control device  100  includes, as control units that perform various types of control, an injection control unit  101 , an ignition control unit  102 , and a valve timing control unit  103 . The injection control unit  101  controls the port injection valve  14  and the direct injection valve  15 . The ignition control unit  102  controls the ignition device  16 . The valve timing control unit  103  controls the intake-side variable valve timing mechanism  27  and the exhaust-side variable valve timing mechanism  28 . Further, the control device  100  includes a boost control unit  104  and an idling stop control unit  105 . The boost control unit  104  controls the wastegate  60  by driving the actuator  61 . The idling stop control unit  105  executes an idling stop control, which discontinues idling operation, by automatically stopping and restarting the engine operation. 
     The injection control unit  101  calculates a target fuel injection amount, which is a control target value for the fuel injection amount, based on, for example, the operation amount of the accelerator, the vehicle speed, the intake air amount, the engine rotation speed, and an engine load factor. The engine load factor is the ratio of an inflow air amount per combustion cycle of a single cylinder to a reference inflow air amount. The reference inflow air amount is an inflow air amount per combustion cycle of a single cylinder when the open degree of the throttle valve  31  is the maximum. The reference inflow air amount is determined in accordance with the engine rotation speed. The injection control unit  101  basically calculates the target fuel injection amount such that the air-fuel ratio becomes the stoichiometric air-fuel ratio. Further, the injection control unit  101  calculates the control target values of the injection timings and fuel injection times of the port injection valve  14  and the direct injection valve  15 . The port injection valve  14  and the direct injection valve  15  are driven to open in accordance with these control target values. This causes an amount of fuel corresponding to the operating state of the internal combustion engine  10  to be injected and supplied to the combustion chamber  11 . In accordance with the operating state, the internal combustion engine  10  switches whether to inject fuel from the port injection valve  14  or the direct injection valve  15 . Thus, in the internal combustion engine  10 , in addition to injecting fuel both from the port injection valve  14  and the direct injection valve  15 , fuel may be injected only from the port injection valve  14  or only from the direct injection valve  15 . Additionally, the injection control unit  101  performs a fuel cut-off control in order to reduce a fuel consumption rate, for example, during deceleration in which the operation amount of the accelerator is zero. In the fuel cut-off control, the injection of fuel is stopped to hinder the supply of the fuel to the combustion chamber  11 . 
     The ignition control unit  102  calculates an ignition timing, which is the timing of spark discharge performed by the ignition device  16 , to operate the ignition device  16  and ignite the air-fuel mixture. The valve timing control unit  103  calculates the target value of the phase of the intake camshaft  25  relative to the crankshaft  18  and the target value of the phase of the exhaust camshaft  26  relative to the crankshaft  18  based on the engine rotation speed and the engine load factor to operate the intake-side variable valve timing mechanism  27  and the exhaust-side variable valve timing mechanism  28 . This causes the valve timing control unit  103  to control the timing of opening and closing the intake valve  23  and the timing of opening and closing the exhaust valve  24 . For example, the valve timing control unit  103  controls a valve overlap, which is a period during which the exhaust valve  24  and the intake valve  23  are both open. 
     The boost control unit  104  drives the actuator  61  to control the open degree of the wastegate  60  by calculating a target open degree of the wastegate  60 , for example, based on the vehicle speed and the accelerator operation amount or based on the engine rotation speed and the engine load factor. 
     The idling stop control unit  105  outputs commands to the injection control unit  101  and the ignition control unit  102 , to automatically stop the engine operation by stopping fuel supply and ignition while the vehicle is not operating and resume the engine operation by automatically resuming the fuel supply and ignition when the vehicle is started. That is, the idling stop control unit  105  executes the idling stop control, which discontinues idling operation, by automatically stopping and restarting the engine operation. 
     When the fuel cut-off control is executed to cause the vehicle to coast, air flows through the exhaust passage  19  into the catalyst device  80 . When the vehicle stops and the engine operation is stopped by the idling stop control or the like, the catalyst device  80  remains exposed to air. As a result, the catalyst device  80  absorbs a huge amount of oxygen. Thus, when the internal combustion engine  10  is restarted, the absorption amount of oxygen in the catalyst device  80  is excessively large. This may reduce the ability to purify exhaust gas. Thus, in the control device  100 , the injection control unit  101  executes a rich reduction control, which makes the air-fuel ratio richer than the stoichiometric air-fuel ratio, when the engine operation has been resumed by resuming the supply of fuel to the combustion chamber  11 . The execution of the rich reduction control causes excess fuel and exhaust gas to be introduced into the catalyst device  80 . Thus, when the oxygen absorbed by the catalyst device  80  reacts with fuel, the oxygen is reduced. 
     Next, a series of processes for the rich reduction control will be described with reference to  FIGS. 3 and 4 .  FIG. 3  illustrates the flow of the processes in a routine for determining to start the rich reduction control. This routine is repeatedly executed by the control device  100  while the control device  100  is running. 
     As shown in  FIG. 3 , when starting this routine, in the process of step S 100 , the control device  100  first determines whether the current time is the restarting time of the internal combustion engine  10  by the idling stop control. That is, the control device  100  determines whether the restarting is performed from a state in which the internal combustion engine  10  is automatically stopped by the idling stop control. 
     When determining that the current time is the restarting time by the idling stop control (step S 100 : YES), the control device  100  advances the process to step S 110 . In the process of step S 110 , the injection control unit  101  of the control device  100  starts the rich reduction control. In the rich reduction control, the injection control unit  101  makes the air-fuel ratio richer than when the rich reduction control is not executed, and injects fuel the amount of which is increased with respect to the target fuel injection amount such that the air-fuel ratio becomes richer than the stoichiometric air-fuel ratio. 
     Subsequently, in the process of step S 120 , the ignition control unit  102  of the control device  100  starts an ignition timing retardation control. In the ignition timing retardation control, the ignition control unit  102  corrects the ignition timing to be retarded than when the ignition timing retardation control is not executed, and performs spark discharge of the ignition device  16  at a timing that is more retarded than the ignition timing when the ignition timing retardation control is not executed. 
     In the process of step S 130  subsequent to step S 120 , the valve timing control unit  103  of the control device  100  starts a maximally-retarding exhaust control. In the maximally-retarding exhaust control, the valve timing control unit  103  uses the exhaust-side variable valve timing mechanism  28  to set the timing of opening and closing the exhaust valve  24  to be most retarded. With the timing of opening and closing the exhaust valve  24  set to be most retarded, the valve overlap is controlled by adjusting the timing of opening and closing the intake valve  23  using the intake-side variable valve timing mechanism  27 . That is, when executing the maximally-retarding exhaust control, the valve timing control unit  103  adjusts the timing of opening and closing the intake valve  23  with the timing of opening and closing the exhaust valve  24  set to be most retarded such that the same valve overlap can be achieved as when the maximally-retarding exhaust control is not executed. When starting the rich reduction control, the ignition timing retardation control, and the maximally-retarding exhaust control through the processes of step S 110  to step S 130 , the control device  100  ends this routine. 
     Further, as shown in  FIG. 3 , when determining in the process of step S 100  that the current time is not the restarting time by the idling stop control (step S 100 : NO), the control device  100  ends this routine without executing the processes of step S 110  to step S 130 . That is, when the current time is not the restarting time by the idling stop control, the control device  100  does not execute the rich reduction control, the ignition timing retardation control, and the maximally-retarding exhaust control. 
       FIG. 4  illustrates the flow of the processes in a routine for determining to end the rich reduction control. This routine is repeatedly executed by the control device  100  during the execution of the rich reduction control. 
     As shown in  FIG. 4 , when starting this routine, in the process of step S 200 , the control device  100  first determines whether a rear A/F value, which is a detection value of the downstream A/F sensor  35 , is less than or equal to a rich determination value. The rich determination value is a threshold value for determining that unburned fuel is contained in the exhaust gas flowing downstream of the catalyst device  80  based on the rear A/F value being less than or equal to the rich determination value. That is, the rich determination value is set to a value that is slightly smaller than a value indicating that the rear A/F value is the stoichiometric air-fuel ratio (i.e., a value indicating being rich). 
     When determining that the rear A/F value is less than or equal to the rich determination value (step S 200 : YES), the control device  100  advances the process to step S 210 . 
     The control device  100  ends the rich reduction control in the process of step S 210 . In the process of step S 210 , the injection control unit  101  of the control device  100  ends the rich reduction control. This causes the injection control unit  101  to stop increasing the fuel injection amount by the rich reduction control and execute fuel injection corresponding to the target fuel injection amount. 
     Subsequently, in the process of step S 220 , the ignition control unit  102  of the control device  100  ends the ignition timing retardation control. This causes the ignition control unit  102  to stop correcting the ignition timing by the ignition timing retardation control to be retarded and performs spark discharge of the ignition device  16  at an ignition timing at which correction with the ignition timing retardation control is not implemented. 
     In the process of step S 230  subsequent to step S 220 , the valve timing control unit  103  of the control device  100  ends the maximally-retarding exhaust control. This causes the valve timing control unit  103  to cancel the state in which the timing of opening and closing the exhaust valve  24  is set to be most retarded. Thus, the valve timing control unit  103  calculates the target value of the phase of the intake camshaft  25  relative to the crankshaft  18  and the target value of the phase of the exhaust camshaft  26  relative to the crankshaft  18  based on the engine rotation speed and the engine load factor to operate the intake-side variable valve timing mechanism  27  and the exhaust-side variable valve timing mechanism  28 . That is, the valve timing control unit  103  controls the valve overlap by operating both the timing of opening and closing the exhaust valve  24  and the timing of opening and closing the intake valve  23 . 
     When ending the rich reduction control, the ignition timing retardation control, and the maximally-retarding exhaust control through the processes of step S 210  to step S 230 , the control device  100  ends this routine. 
     Further, as shown in  FIG. 4 , when determining in the process of step S 200  that the rear A/F value is greater than the rich determination value (step S 200 : NO), the control device  100  ends this routine without executing the processes of step S 210  to step S 230 . 
     More specifically, when it can be estimated that the rear A/F value is greater than the rich determination value and unburned fuel is not contained in the exhaust gas flowing downstream of the catalyst device  80  although the rich reduction control is being executed, the control device  100  does not end the rich reduction control, the ignition timing retardation control, and the maximally-retarding exhaust control. In short, in the control device  100 , the injection control unit  101  continues the rich reduction control until the fuel passes through the catalyst device  80  and reaches the downstream A/F sensor  35  without being completely consumed through a reduction reaction in the catalyst device  80  as a result of reducing oxygen absorbed by the catalyst device  80  through the rich reduction control. 
     To restore the ability to purify exhaust gas through the rich reduction control, it is preferred that the reduction of oxygen be completed immediately in the catalyst device  80  to quickly restore the purification ability. Thus, the control device  100  executes a valve-closing keeping control, which keeps the wastegate  60  closed, in order to expedite the reduction of oxygen by the rich reduction control. 
     Next, the valve-closing keeping control will be described with reference to  FIGS. 5 and 6 .  FIG. 5  illustrates the flow of processes in a routine for determining to start the valve-closing keeping control. This routine is repeatedly executed by the control device  100  while the control device  100  is running. 
     As shown in  FIG. 5 , when starting this routine, in the process of step S 300 , the control device  100  first determines whether the fuel cut-off control is being implemented. When determining that the fuel cut-off control is being implemented (step S 300 : YES), the control device  100  advances the process to step S 310 . 
     In the process of step S 310 , the boost control unit  104  of the control device  100  starts the valve-closing keeping control. In the valve-closing keeping control, the boost control unit  104  closes the wastegate  60  and keeps the wastegate  60  closed. When determining that the fuel cut-off control is being executed, in a case where the valve-closing keeping control has already been implemented, the control device  100  continues the valve-closing keeping control without executing any process in the process of step S 310 . 
     When determining that the fuel cut-off control is not being executed (step S 300 : NO), the control device  100  ends this routine without executing the process of step S 310 . By repeatedly executing this routine while the internal combustion engine  10  is mining, the valve-closing keeping control is started from the point in time at which the fuel cut-off control is started. 
       FIG. 6  illustrates the flow of processes in a routine for determining to end the valve-closing keeping control. This routine is repeatedly executed by the control device  100  while the valve-closing keeping control is being executed. 
     As shown in  FIG. 6 , when starting this routine, in the process of step S 400 , the control device  100  first determines whether the rear A/F value is less than or equal to the rich determination value. When determining that the rear A/F value is less than or equal to the rich determination value (step S 400 : YES), the control device  100  advances the process to step S 410 . 
     The control device  100  ends the valve-closing keeping control in the process of step S 410 . In the process of step S 410 , the boost control unit  104  of the control device  100  ends the valve-closing keeping control. Thus, the boost control unit  104  calculates the target open degree of the wastegate  60 , for example, based on the vehicle speed and the accelerator operation amount or based on the engine rotation speed and the engine load factor to drive the actuator  61  and control the open degree of the wastegate  60 . 
     Further, as shown in  FIG. 6 , when determining that the rear A/F value is greater than the rich determination value (step S 400 : NO), the control device  100  ends this routine without executing the process of step S 410 . That is, the boost control unit  104  ends the valve-closing keeping control on the condition that the downstream A/F sensor  35  has detected that the air-fuel ratio is richer than the stoichiometric air-fuel ratio after the engine operation was resumed by resuming the supply of fuel to the combustion chamber  11 . 
     In this manner, when it can be estimated that the rear A/F value is greater than the rich determination value and unburned fuel is not contained in the exhaust gas flowing downstream of the catalyst device  80  although the valve-closing keeping control is being executed, the control device  100  does not end the valve-closing keeping control. In short, the condition for cancelling the valve-closing keeping control by the control device  100  is that the air-fuel ratio being richer than the stoichiometric air-fuel ratio has been detected by the downstream A/F sensor  35 . In the control device  100 , the valve-closing keeping control is continued until the fuel passes through the catalyst device  80  and reaches the downstream A/F sensor  35  without being completely consumed through a reduction reaction in the catalyst device  80  as a result of reducing oxygen absorbed by the catalyst device  80  through the rich reduction control. 
     Next, the operation of the first embodiment will be described with reference to  FIG. 7 .  FIG. 7  is a timing diagram illustrating a change in each control that occurs when the vehicle decelerates to stop and then restarts. 
     As shown in  FIG. 7 , when the vehicle starts to decelerate, at the point in time t 10 , the fuel cut-off control is started (step S 300 : YES) and the valve-closing keeping control is started (step S 310 : YES) to keep the wastegate  60  closed. The execution of the fuel cut-off control stops the supply of fuel, thereby causing air to pass through the combustion chamber  11  and flow into the exhaust passage  19 . Thus, a front A/F value, which is a detection value of the upstream A/F sensor  34 , and the rear A/F value, which is a detection value of the downstream A/F sensor  35 , both indicate that the front A/F value and the rear A/F value are lean. Air that does not contain fuel passes through the catalyst device  80 . Thus, the catalyst device  80  absorbs oxygen. 
     At the point in time t 11 , when a decrease in the vehicle speed stops the fuel cut-off control and shifts to idling operation, the supply of fuel is resumed. Thus, the front A/F value and the rear A/F value both change to be richer than the stoichiometric air-fuel ratio. At the point in time t 12 , when the vehicle is stopped and the idling stop control is performed to stop the operation of the internal combustion engine  10 , the supply of fuel stops. Then, the front A/F value and the rear A/F value both change to be approximate to the stoichiometric air-fuel ratio. While the internal combustion engine  10  is not operating in this manner, the catalyst device  80  is exposed to the air in the exhaust passage  19 . Thus, the catalyst device  80  absorbs oxygen. 
     At the point in time t 13 , when the stopping of the operation by the idling stop control is cancelled to restart the internal combustion engine  10  (step S 100 : YES), the rich reduction control, the ignition timing retardation control, and the maximally-retarding exhaust control are started (step S 110 , step S 120 , and step S 130 ). This causes fuel to be supplied with the air-fuel ratio increased to be richer than the stoichiometric air-fuel ratio, thereby introducing exhaust gas containing excess fuel into the catalyst device  80 . Thus, the front A/F value becomes richer. Immediately after the rich reduction control is started, the fuel contained in exhaust gas is consumed by the reduction of oxygen absorbed by the catalyst device  80  and thus does not reach the downstream A/F sensor  35 . Thus, the rear A/F value becomes approximate to the stoichiometric air-fuel ratio. When the rich reduction control continues, the reduction of oxygen progresses so that the absorption amount of oxygen in the catalyst device  80  decreases. Consequently, the fuel contained in the exhaust gas passes through the catalyst device  80  and reaches the downstream A/F sensor  35  without being completely consumed. 
     At the point in time t 14 , when the rear A/F value is less than or equal to the rich determination value (step S 200 : YES, step S 400 : YES), the rich reduction control ends (step S 210 ) and the valve-closing keeping control also ends (step S 410 ). At the same time, the ignition timing retardation control and the maximally-retarding exhaust control end (step S 220  and step S 230 ). 
     In the control device  100 , the valve-closing keeping control is started from the point in time at which the fuel cut-off control is started. Thus, when the rich reduction control is started, the wastegate  60  is already kept closed. During the execution of the rich reduction control, the valve-closing keeping control continues and the wastegate is kept closed. 
     Further, during the execution of the rich reduction control, the ignition timing retardation control is executed to perform the engine operation with the ignition timing retarded. In addition, during the execution of the rich reduction control, the maximally-retarding exhaust control is executed to control the overlap with the timing of opening and closing the exhaust valve  24  set to be most retarded. 
     The advantages provided by the control device  100  of the first embodiment will now be described. 
     (1) In a case in which the wastegate  60  is kept closed, the gas flowing through the exhaust passage  19  passes through the turbine wheel  54  of the turbocharger  50 . As the turbine wheel  54  rotates, the gas passing through the turbine wheel  54  and flowing toward the downstream side becomes a swirl flow and reaches the catalyst device  80 . Thus, as long as the valve-closing keeping control is executed, when the engine operation is resumed to execute the rich reduction control, exhaust gas containing excess fuel passes through the turbine wheel  54  and the exhaust gas, which is the swirl flow, is introduced into the catalyst device  80 . In this case, the exhaust gas is diffused in the exhaust passage  19  by a centrifugal force so that the exhaust gas containing fuel is uniformly introduced into the catalyst device  80  easily. Further, as compared to when exhaust gas flows straight toward the downstream side without swirling, the swirl flow can ensure the time for a catalyst and fuel to contact each other. Thus, the above-described configuration allows for efficient reduction of oxygen in the catalyst device  80  by the rich reduction control. 
     (2) The wastegate  60  is kept closed until the condition for cancelling the valve-closing keeping control has been satisfied by the engine operation that was performed after starting the valve-closing keeping control when the fuel cut-off control started and closing the wastegate  60  through the valve-closing keeping control. Thus, when the engine operation is resumed, the wastegate  60  is already closed. Accordingly, since exhaust gas passes through the turbine wheel  54  from when the rich reduction control is started, the operation resulting from the above-described swirl flow can be provided. Therefore, the above-described configuration expedites a reduction reaction in the catalyst device  80  with the swirl flow to immediately complete the reduction of an excessive amount of oxygen at the restarting time and quickly restore the purification ability. 
     (3) As long as the fuel cut-off control is executed, the output torque of the internal combustion engine  10  does not increase even if the wastegate  60  is closed. This allows the wastegate  60  to be kept closed in advance in preparation for the rich reduction control. In the above-described configuration, the valve-closing keeping control starts from the point in time at which the fuel cut-off control is started. Thus, the wastegate  60  can be kept closed in advance in preparation for the rich reduction control that is performed after the earliest point in time. 
     (4) When the fuel introduced together with exhaust gas by the rich reduction control is completely consumed through the reduction of oxygen absorbed by the catalyst device  80 , exhaust gas that does not contain fuel reaches the downstream A/F sensor  35 . When the reduction of oxygen progresses and the absorption amount of oxygen in the catalyst device  80  becomes small, fuel passes through the catalyst device  80  and reaches the downstream A/F sensor  35  without being completely consumed. The above-described configuration employs the configuration in which the valve-closing keeping control is ended on the condition that the air-fuel ratio being richer than the stoichiometric air-fuel ratio has been detected by the downstream A/F sensor  35 . Thus, it is possible to check that the reduction of oxygen progresses until fuel becomes unable to be completely consumed based on the detection result of the downstream A/F sensor  35 , thereby ending the valve-closing keeping control. 
     (5) Retarding the ignition timing limits the generation of NOx. In the above-described configuration, while the rich reduction control is incomplete, the ignition timing retardation control is executed to retard the ignition timing retardation control and limit the emission of NOx. This makes it possible to limit the emission of NOx until the purification ability of the catalyst device  80  restores. 
     (6) The emission of NOx and HC can be limited by causing exhaust gas to flow back into the combustion chamber  11  using the valve overlap. Like in the above-described configuration, when the maximally-retarding exhaust control to adjust the valve overlap is executed by adjusting the timing of opening the intake valve  23  with the timing of closing the exhaust valve  24  maximally retarded, the actual compression ratio can be reduced by maximally delaying the timing of closing the intake valve  23  while achieving the magnitude of a target valve overlap. Thus, the above-described configuration easily achieves the Atkinson cycle by delaying the timing of closing the intake valve  23  and achieves the target valve overlap. Thus, the pumping loss can be reduced using the Atkinson cycle to limit the consumption amount of fuel and limit the emission of NOx and HC. 
     The present embodiment may be modified as follows. 
     The valve-closing keeping control is started from the point in time at which the fuel cut-off control is started. Instead, the timing of starting the valve-closing keeping control does not have to be from the point in time at which the fuel cut-off control is started. The valve-closing keeping control simply needs to be started before the restarting is performed to start the rich reduction control. This allows for the advantage of uniformly introducing fuel into the catalyst device  80  using a swirl flow from the point in time at which the rich reduction control is started. 
     Second Embodiment 
     Subsequently, the control device  100  for the internal combustion engine  10 , which is an onboard internal combustion engine, according to a second embodiment will be described with reference to  FIGS. 8 to 10 . The same reference numerals are given to those components that are common to the first embodiment, and detailed explanations are omitted. In the first embodiment, the valve-closing keeping control is started from the point time at which the fuel cut-off control is started. In the control device  100  of the second embodiment, the valve-closing keeping control is started before the fuel cut-off control is started, and the wastegate  60  is closed prior to the execution of the fuel cut-off control. 
     In the control device  100  of the second embodiment, in the same manner as the control device  100  of the first embodiment, the rich reduction control is executed through the processes described with reference to  FIGS. 3 and 4 . In the control device  100  of the first embodiment, the valve-closing keeping control is started when the fuel cut-off control is started through the routine described with reference to  FIG. 5 . In the control device  100  of the second embodiment, instead of the routine illustrated in  FIG. 5 , a routine illustrated in FIG.  8  is executed. The routine illustrated in  FIG. 8  is repeatedly executed by the control device  100  while the control device  100  is running. 
     As shown in  FIG. 8 , when starting this routine, in the process of step S 500 , the control device  100  first determines whether a fuel cut-off execution condition has been satisfied. The fuel cut-off execution condition is a requirement for executing the fuel cut-off control. The fuel cut-off execution condition is the condition of the logical conjunction of the operation amount of the accelerator being zero and the engine rotation speed being greater than or equal to a fuel cut-off permission rotation speed. When determining that the fuel cut-off execution condition has been satisfied (step S 500 : YES), the control device  100  advances the process to step S 510 . 
     In the process of step S 510 , the boost control unit  104  of the control device  100  starts the valve-closing keeping control. In the valve-closing keeping control, the boost control unit  104  closes the wastegate  60  and keeps the wastegate  60  closed. In the process of step S 500 , in a case where the valve-closing keeping control has already been implemented when determining that the fuel cut-off execution condition has been satisfied, the control device  100  continues the valve-closing keeping control without executing any process in the process of step S 510 . 
     When determining that the fuel cut-off execution condition has not been satisfied (step S 500 : NO), the control device  100  ends this routine without executing the process of step S 510 . 
     By repeatedly executing this routine while the internal combustion engine  10  is running, the valve-closing keeping control is started from the point in time at which the fuel cut-off execution condition is satisfied. In the control device  100  of the second embodiment, the timing of ending the valve-closing keeping control is determined through the routine described with reference to  FIG. 6 . 
     Next, the determination of the timing of starting the fuel cut-off control in the control device  100  of the second embodiment will be described with reference to  FIG. 9 .  FIG. 9  illustrates the flow of processes in a routine for determining to start the fuel cut-off control in the control device  100  of the second embodiment. This routine is repeatedly executed by the control device  100  at predetermined cycles while the control device  100  is running. 
     As shown in  FIG. 9 , when starting this routine, in the process of step S 600 , in the same manner as the process of S 500 , the control device  100  first determines whether the fuel cut-off execution condition has been satisfied. When determining that the fuel cut-off execution condition has been satisfied (step S 600 : YES), the control device  100  advances the process to step S 610 . 
     The control device  100  increments a counter CNT in the process of step S 610 . The counter CNT is a counter for counting the time elapsed from when the fuel cut-off execution condition was satisfied. More specifically, the control device  100  increases the counter CNT one by one every time the control device  100  executes the process of step S 610 . Next, the control device  100  executes the process of step S 620 . In the process of step S 620 , the control device  100  determines whether the counter CNT is greater than or equal to a threshold value Cth. The threshold value Cth is set to a value that allows for determination based on the counter CNT having reached the threshold value Cth that the time from when the wastegate  60  started closing to when the wastegate  60  was completely closed has sufficiently elapsed after satisfying the fuel cut-off execution condition and starting the valve closing keeping control. That is, in step S 610 , based on the counter CNT being greater than or equal to the threshold value Cth, it is determined that the time for the wastegate  60  to be closed has sufficiently elapsed. 
     When determining that the counter CNT is greater than or equal to the threshold value Cth (step S 620 : YES), the control device  100  advances the process to step S 630 . In step S 630 , the injection control unit  101  of the control device  100  starts the fuel cut-off control. Then, the control device  100  resets the counter CNT to zero in the process of the subsequent step S 640  and temporarily ends this routine. When determining that the counter CNT is less than the threshold value Cth (step S 620 : NO), the control device  100  temporarily ends this routine without executing the process of step S 630  and the process of step S 640 . 
     When determining that the fuel cut-off execution condition has not been satisfied (step S 600 : NO), the control device  100  executes the process of step S 640  without executing the processes of step S 610  to step S 630  and then resets the counter CNT to zero to temporarily end this routine. 
     More specifically, the control device  100  performs this routine to start the fuel cut-off control after a certain delay time TD has elapsed since the fuel cut-off execution condition was satisfied. The period during which the counter CNT reaches the threshold value Cth corresponds to the delay time TD. The length of the delay time TD is set to time enough to close the wastegate  60  after starting closing the wastegate  60  since the fuel cut-off execution condition was satisfied. 
     Next, the operation of the second embodiment will be described with reference to  FIG. 10 .  FIG. 10  is a timing diagram showing a change in each control when the vehicle decelerates to stop. That is,  FIG. 10  illustrates a state on and before the point in time t 11  of  FIG. 7 . The change in each control subsequent to the point in time t 11  is the same as that of the first embodiment, which has been described with reference to  FIG. 7 . 
     As shown in  FIG. 10 , at the point in time t 7 , when the operation amount of the accelerator becomes zero, the fuel cut-off execution condition is satisfied. In  FIG. 10 , the accelerator is off when the operation amount of the accelerator is zero, and the accelerator is on when the accelerator is being operated. 
     When the fuel cut-off execution condition has been satisfied (step S 500 : YES, step S 600 : YES), at the point in time t 8 , the valve-closing keeping control is started (step S 510 ) to close the wastegate  60 . Further, while the fuel cut-off execution condition is satisfied, the counter CNT is repeatedly incremented (step S 610 ). 
     At the point in time t 9 , when the counter CNT is determined as being greater than or equal to the threshold value Cth (step S 620 : YES), the fuel cut-off control is started (step S 630 ). The execution of the fuel cut-off control stops the supply of fuel, thereby causing air to pass through the combustion chamber  11  and flow into the exhaust passage  19 . Thus, as has been described with reference to  FIG. 7 , the front A/F value, which is a detection value of the upstream A/F sensor  34 , and the rear A/F value, which is a detection value of the downstream A/F sensor  35 , both indicate that the front A/F value and the rear A/F value are lean. Since air that does not contain fuel passes through the catalyst device  80 , the catalyst device  80  absorbs oxygen. 
     At the point in time t 11 , when the engine rotation speed decreases to be less than the fuel cut-off permission rotation speed as the vehicle speed decreases, the fuel cut-off execution condition becomes unsatisfied. This stops the fuel cut-off control and shifts to idling operation. The shifting to the idling operation resumes the supply of fuel. Thus, the front A/F value and the rear A/F value both change to be richer than the stoichiometric air-fuel ratio. 
     The subsequent changes are the same as those in the first embodiment, which has been described with reference to  FIG. 7 . 
     More specifically, after the vehicle is stopped and the idling stop control is performed to stop the operation of the internal combustion engine  10 , the stopping of the operation by the idling stop control is cancelled to restart the internal combustion engine  10  (step S 100 : YES). As a result, the rich reduction control, the ignition timing retardation control, and the maximally-retarding exhaust control are started (step S 110 , step S 120 , and step S 130 ). This causes fuel to be supplied with the air-fuel ratio increased to be richer than the stoichiometric air-fuel ratio, thereby introducing exhaust gas containing excess fuel into the catalyst device  80 . Thus, the front A/F value becomes richer. When the rich reduction control continues, the reduction of oxygen progresses so that the absorption amount of oxygen in the catalyst device  80  decreases. Consequently, the fuel contained in the exhaust gas passes through the catalyst device  80  and reaches the downstream A/F sensor  35  without being completely consumed. 
     When the rear A/F value is less than or equal to the rich determination value (step S 200 : YES, step S 400 : YES), the rich reduction control ends (step S 210 ) and the valve-closing keeping control also ends (step S 410 ). At the same time, the ignition timing retardation control and the maximally-retarding exhaust control end (step S 220  and step S 230 ). In the control device  100  of the second embodiment, the wastegate  60  is kept closed until the condition for cancelling the valve-closing keeping control is satisfied by the engine operation that was performed after closing the wastegate  60  through the valve-closing keeping control. Thus, when the engine operation is resumed, the wastegate  60  is already closed. Accordingly, since exhaust gas passes through the turbine wheel  54  from when the rich reduction control is started, the operation resulting from a swirl flow can be provided in the same manner as the first embodiment. 
     Further, during the execution of the rich reduction control, the ignition timing retardation control is executed to perform the engine operation with the ignition timing retarded. In addition, during the execution of the rich reduction control, the maximally-retarding exhaust control is executed to control the overlap with the timing of opening and closing the exhaust valve  24  set to be most retarded. 
     During the execution of the fuel cut-off control, the supply of fuel to the combustion chamber  11  is not performed. Thus, although burning is not performed, intake and exhaust are performed with the intake air amount limited. Thus, the inside of the combustion chamber  11  is under negative pressure. Further, by closing the wastegate  60  during the execution of the fuel cut-off control, the open degree of the wastegate  60  decreases. When the wastegate  60  approaches a seat surface, the wastegate  60  is easily vibrated by the negative pressure in the combustion chamber  11  and the pulsation of exhaust gas. Thus, when the wastegate  60  strikes the seat surface while vibrating, noise is generated. Since burning is not performed during the execution of the fuel cut-off control, noise or vibration resulting from burning does not occur. Thus, the noise produced by the wastegate  60  striking the seat surface is noticeable. 
     In the control device  100  of the second embodiment, when the condition for executing the fuel cut-off control is satisfied, the boost control unit  104  starts the valve-closing keeping control to close the wastegate  60  at the point in time t 8  prior to the execution of the fuel cut-off control at the point in time t 9 . 
     In this configuration, prior to the execution of the fuel cut-off control, the wastegate  60  is closed to start the fuel cut-off control with the wastegate  60  closed. 
     The control device  100  of the second embodiment provides the following advantage in addition to the advantages that are the same as advantages (1), (2), and (4) to (6) of the first embodiment. 
     (7) In the second embodiment, the wastegate  60  is closed when burning is performed in the internal combustion engine  10  to limit the vibration of the wastegate  60  and make the noise produced by the wastegate  60  striking the seat surface unnoticeable. This makes it difficult for the occupant to hear the noise produced by the wastegate  60  striking the seat surface. 
     The second embodiment may be modified as follows. 
     In the above-described example, the elapse of the delay time TD is determined using the counter CNT. However, the fuel cut-off control does not have to be started by determining the elapse of the delay time TD. The fuel cut-off control may be started after checking with a different means that the wastegate  60  is closed. For example, the fuel cut-off control may be executed by determining that the wastegate  60  is closed based on the fact that the actuator  61  has stopped operating since the actuator  61  started closing the wastegate  60 . 
     The following are modifications commonly applicable to each of the above-described embodiments. The above-described embodiments, the above-described modifications, and the following modifications can be combined as long as the combined modifications remain technically consistent with each other. 
     In the above-described example, the A/F sensor in which the output value continuously changes in accordance with a change in the level of the oxygen concentration is employed as the air-fuel ratio sensor in the internal combustion engine. However, the air-fuel ratio sensor that detects the air-fuel ratio is not limited to the A/F sensor. For example, an O 2  sensor may be used. The O 2  sensor outputs an output value indicating that the air-fuel ratio is rich when the air-fuel ratio becomes rich when the output value greatly changes over the stoichiometric air-fuel ratio and outputs an output value indicating that the air-fuel ratio is lean when the air-fuel ratio becomes lean when the output value greatly changes over the stoichiometric air-fuel ratio. 
     The condition for ending the valve-closing keeping control is not limited to a condition in which the air-fuel ratio being richer than the stoichiometric air-fuel ratio has been detected by the air-fuel ratio sensor. Instead, for example, the condition for cancelling the valve-closing keeping control may be that the rich reduction control executed together with the valve-closing keeping control has continued for a certain period. 
     The timing of ending the rich reduction control, the ignition timing retardation control, the maximally-retarding exhaust control, and the valve-closing keeping control does not have to be the same as the condition for cancelling these controls. Instead, for example, the rich reduction control may be ended prior to the valve-closing keeping control. Alternatively, the valve-closing keeping control may be ended prior to the rich reduction control. If there is a period during which the rich reduction control is executed together with the valve-closing keeping control, fuel can be uniformly introduced into the catalyst device  80  using a swirl flow during that period. 
     In the above-described example, the rich reduction control is executed at the restarting time by the idling stop control. Instead, the rich reduction control may be executed when the fuel cut-off control is ended to resume the supply of fuel. Since oxygen is absorbed by the catalyst device  80  during the execution of the fuel cut-off control, the absorption amount of oxygen may become excessive. Also, when the fuel cut-off control is ended to resume the supply of fuel, fuel can be uniformly introduced into the catalyst device  80  using a swirl flow by executing the valve-closing keeping control in the same manner as the above-described embodiments. 
     The same configuration as that of the control device of each of the above-described embodiments may be applied to an internal combustion engine including two or more catalyst devices located in the exhaust passage  19 . When two catalyst devices are arranged, the rich reduction control may be continued until the reduction of oxygen has been completed in the downstream catalyst device. The operation resulting from a swirl flow generated by the valve-closing keeping control affects the catalyst device located on the most upstream side, which is most proximate to the turbine wheel  54 , but hardly affects the downstream catalyst device. Thus, in this case, the valve-closing keeping control may be ended at the point in time at which the reduction of oxygen has been completed in the upstream catalyst device. 
     The control device  100  is not limited to one that performs software processing on all processes executed by itself. For example, the control device  100  may include at least part of the processes executed by the software in the present embodiment as one that is executed by hardware circuits dedicated to execution of these processes (such as ASIC). That is, the control device  100  may be modified as long as it has any one of the following configurations (a) to (c): (a) a configuration including a processor that executes all of the above-described processes according to programs and a program storage device such as a ROM that stores the programs; (b) a configuration including a processor and a program storage device that execute part of the above-described processes according to the programs and a dedicated hardware circuit that executes the remaining processes; and (c) a configuration including a dedicated hardware circuit that executes all of the above-described processes. A plurality of software processing circuits each including a processor and a program storage device and a plurality of dedicated hardware circuits may be provided. That is, the above processes may be executed in any manner as long as the processes are executed by processing circuitry that includes at least one of a set of one or more software processing circuits and a set of one or more dedicated hardware circuits. 
     Various changes in form and details may be made to the examples above without departing from the spirit and scope of the claims and their equivalents. The examples are for the sake of description only, and not for purposes of limitation. Descriptions of features in each example are to be considered as being applicable to similar features or aspects in other examples. Suitable results may be achieved if sequences are performed in a different order, and/or if components in a described system, architecture, device, or circuit are combined differently, and/or replaced or supplemented by other components or their equivalents. The scope of the disclosure is not defined by the detailed description, but by the claims and their equivalents. All variations within the scope of the claims and their equivalents are included in the disclosure.