Direct Synthesis Controller Tuning - Campus Tour

Colorado School of Mines CHEN403Direct Synthesis Controller TuningDirect Synthesis Controller TuningFirst Order Process . 2Second Order Process . 3Third Order & Higher Processes . 3Processes with Dead Time. 4FOPDT Example . 7Direct synthesis methods are based upon prescribing a desired form for the system’sresponse and then finding a controller strategy & parameters to give that response.L Ysp Gc-Ym P GvGLM Gp Y GmFor the feedback control loop above the overall transfer functions between the output Y and the set point & disturbance are:Gc GvGpY Ysp 1 Gc GvGpGmand:GLY . L 1 GcGvGpGmNote that for the set point transfer function, we can manipulate it to give:Gc Y Y . 1 G Y Y spGvGpmspOne implication is that if we pick a desired form for the response to a set point change,Y Ysp , then we have set out the desired form for the controller. For example, we mightthink that it would be great to have the output to immediately track the set point change,John Jechura (jjechura@mines.edu)April 23, 2017-1- Copyright 2017

Colorado School of Mines CHEN403Direct Synthesis Controller Tuningi.e., Y Ysp . However, a more practical response would be a first order decay into the finalvalue, or:Y 1. Ysp c s 1We can get this type of response with a controller strategy of the form: 1 c s 1 1 .Gc 1 GvGp c s 1 Gm GvGp 1 Gm c s 1 The direct synthesis equations are shown for Y Ysp but they really make more sense forlooking at the dynamic response of the measured variable, Ym Ysp . Then the transferfunction with respect to the set point and the implied controller strategy is:G GGG Ym Ysp Ym 1 m c v p Gc .Ysp 1 Gc GvGpGmGvGpGm 1 Ym Ysp GvGpGm c s Note that the product of the three transfer functions is simply the open-loop transferfunction, represented usually as GOL or some approximation of the actual transfer function,G: Ym Ysp 1 1 .GGYm c Gc Ysp 1 Gc GG 1 Ym Ysp G c sNotice that this shows there is an integral action to the Direct Synthesis controller strategy.First Order ProcessLet’s look at the Direct Synthesis controller strategy for a first order process:G Kp p s 1.Then the controller strategy is:John Jechura (jjechura@mines.edu)April 23, 2017-2- Copyright 2017

Colorado School of Mines CHEN403Gc 1Kp p s 1Direct Synthesis Controller Tuning 1 p s 11 p 1 . c s K p c s K p c p s Notice that this is simply PI control with the settings:Kc pK p cand I p .Second Order ProcessLet’s create a 2nd order process by assuming the process & actuator (valve) are both firstorder. Then:G KpKv p s 1 v s 1 2 p s 1 v s 1 1 p v s p v s 1 p vGc KpKv c sK pK v c sK pK v c 11 p v s . 1 p v s p v Notice that this is PID control with the settings:Kc p vK p K v c, I p v , and D p v p v.Further note that integral & derivatives times are only dictated by the process & not thedesired controller settling time.Third Order & Higher ProcessesProcesses that are 3rd order and higher will lead to controller strategies that are differentfrom PID controllers. However we can get PID controller settings by eliminating terms thatcorrespond to high order derivatives. For example:G Kan s an 1 s n 1 nJohn Jechura (jjechura@mines.edu)April 23, 2017 a0-3- Copyright 2017

Colorado School of Mines CHEN403an s n an 1 s n 1 KGc a0 Direct Synthesis Controller Tuning1 an s n an 1 s n 1 c sK c s a1K c a0 a0 1 a2a3 2 1 s s a1 a1 s a1 an n 1 s a1 So the appropriate PID control settings are:a1aa, I 1 , and D 2 .K ca0a1Kc Again note that integral & derivatives times are only dictated by process parameters & notthe desired controller settling time.Processes with Dead TimeLet’s consider a 1st order system with dead time (FOPDT):G K pe p s. p s 1If we keep the same desired form for the response to a set point change, then the controllerstrategy is:Gc p s 1K pe p s 11 p s p 1 e . c s K p c p s This does not correspond to a standard controller strategy (because of the exponentialterm). In fact, this exponential term shows a requirement for prediction of what is going tohappen, so this cannot be physically realized in a controller strategy.If we really wanted to do this though, we could approximate the exponential termwith a truncated Taylor series expansion:e p s 1 p sthen:John Jechura (jjechura@mines.edu)April 23, 2017-4- Copyright 2017

Colorado School of Mines CHEN403Direct Synthesis Controller Tuning p 1 1 1 p p p p s 1 1 p s K p c p s K p c s Gc p 11 p p p s 1 K p c p p s p p .Now we have PID control with the settings:Kc p pK p c, I p p , and D p p p p.However, we could change our desired form for the response to a set point change tosomething involving the same process dead time: sYm e p Ysp s 1then:Gc Y Y 1 eG 1 Y Y G s 1 em p sspm p scspand for this FOPDT process:Gc p s 1K pe p s e p s c s 1 e p s p s 1K p c s 1 e p s .Now we can make a truncated Taylor series approximation:e p s 1 p sto get:Gc p s 1K p c s 1 1 p s John Jechura (jjechura@mines.edu)April 23, 2017 p s 1K p c p s 1 1 p s K p c p p-5- Copyright 2017

Colorado School of Mines CHEN403Direct Synthesis Controller Tuningand we’re back to having PI control with the settings:Kc pand I p .K p c p We used a simple truncated Taylor series expansion for the dead time term here. Whathappens if we use a Padé approximation instead? Using a 1st order Pade approximation thecontroller strategy for a FOPDT is:Gc Gc p s 11 1 p s 2 K p c s 1 1 1 p s 2 s 1 1 12 s pp 11K p c s 1 1 p s 1 p s 22 s 1 sG K s 1 s 1 s 1 s G 12pcppc12c1212ppK p c p 12p2pp2cp1 s2 p 122 p p1s c p s 2 2 If we assume that the dead time is much less than the controller settling time ( p c )then we can neglect the quadratic term in the denominator. s 1 G pc12p1 s22 p pK p c p s1 p p p p 12Gc 1 s12 p p K p c p sp 2 p This is in the form of a PID controller where:Kc 12 p pK p c p 12, I p p , and D John Jechura (jjechura@mines.edu)April 23, 2017-6- p p2 p p. Copyright 2017

Colorado School of Mines CHEN403Direct Synthesis Controller TuningTo get the form reported by Smith & Corropio we have to make the further assumption thatthe dead time is also much less than the process time constant ( p p ):Gc 1 p s 1 K p c p p s 2 pand this is still in the form of a PID controller but now the settings are:Kc pK p c p 12, I p , and D p .FOPDT ExampleLet’s look at the response to a unit step change in the set point for a FOPDT process whereK p 0.25 , K L 0.75 , p L 10 , and p L 1 .Though it’s not needed for the Direct Synthesis method, let’s calculate the stability limit forP-control. Under P-control the characteristic equation is:1 Gc GvGpGm 0 1 K cK pe p s p s 1 0 p s 1 K c K pe p s 0.The stability limit can be determined by the direct substitution method: p u j 1 K cuK pe p u j 0 p u j 1 K cuK p cos p u j sin p u 0which leads to the two simultaneous equations:1 1 K cu K p cos p u 0 K cu K cos p u p p u sin p u K cu K p 0 p u tan p u 0The 2nd equation must be solved numerically. For this particular problem the equation &solution are:10 u tan u 0 u 1.632 Pu John Jechura (jjechura@mines.edu)April 23, 2017-7-2 3.85 and K cu 65.40 . u Copyright 2017

Colorado School of Mines CHEN403Direct Synthesis Controller TuningThe following table gives controller settings from various methods. The response time isconsidered that time at which the final response stays within 5% of the final value (0.95to 1.05). Three of the responses are depicted in the figure following this table.MethodKc I DDirect Synthesis - c 12010—Direct Synthesis c 0.526.710—Direct Synthesis - c 213.310—Original Zeigler-Nichols29.73.2—Original Zeigler-Nichols38.51.90.5ZN Some Overshoot21.61.91.3ZN No Overshoot14.41.91.3Rule of Thumb32.73.9John Jechura (jjechura@mines.edu)April 23, 2017-8-CommentsSlight overshoot – max value1.04. Response time is actuallybefore the first peak, about 3.4min.Even larger overshoot – maxvalue 1.2. Response time is afterthe first peak, about 5 min.No apparent overshoot –Response time about 6.4 min.Large overshoot – max valueabout 1.6. Response time after1st trough, about 8 min.Multiple harmonics at earlytimes. Again large overshoot –max value about 1.5. 1st troughjust outside range – responsetime about 8 min.Multiple harmonics over anextended period of time.Response time about 15 min.Long, broad cycling. Responsetime about 19 min. Iu 1.32 . Large overshoot –max value 1.6. Response time 9min. Copyright 2017

Colorado School of Mines CHEN403Direct Synthesis Controller Tuning1.6Original ZN PI1.4"Rule of Thum b" PI1.2Response (y)1.00.80.6Direct Synthesis, 10.40.20.0-0.20510152025Tim e (t)John Jechura (jjechura@mines.edu)April 23, 2017-9- Copyright 2017

and this is still in the form of a PID controller but now the settings are: 2Ip W p c W T p c p K K, W W, and W T 1 Dp. . Colorado School of Mines CHEN403 Direct Synthesis Controller Tuning Direct Synthesis - Direct Synthesis - Direct Synthesis - Colorado School of Mines CHEN403 Direct Synthesis Controller Tuning File Size: 822KB