Lecutre 32

[Music] [Music] in this lecture we'll be discussing how the Clos of control system can be analyzed in the frequency domain in the presence of disturbance and noise so let me write the title feedback control problem in the frequency domain we will revisit the close loop control system what we have done before for the airc attitud control problem so let me draw the close loop diagram says summing point where we are having this reference input to be track and we have have controller it's giving the desired control input to the system denoted by GC office and we are having another summing point where control input and disturbance coming into picture let's do the disturbance signal MDS this is U office with is given by the controller and the sum control input and disturbance which is denoted by [Music] U going to the plant to be controlled GP offs and this the output from the plant y office and this is the feedback signal here also we are assuming noise coming into the system which is dened by WS plus and we are having feedback here the feedback and the output from this summing point let's by y Ms and the error in the presence of measurement noise we can denote as so here uh let me write EMS this is basically measurement noise and YMS is the measure output and U is is basically us Plus MDS which is nothing but effective control input to the plant effective control input the plant and this is if you say if you have actuator we can say actuator input plus disturbance now the measurement error we can write on the block diagram let me write this is figure one close slope control system from the figure one we can write the EMS measurement error EMS we can write RS minus y Ms or we can write RS minus ysus ws and also you can write es s minus WS where ES basically RS minus Ys which is basically tracking error now we'll find us us from the Block D from figure one us from figure one us we can write equal Al to GC of s RS minus ysus WS or okay uh also can write y of we can write GP of s into H2 control input on we can write in U or we can write GPS us plus MDS if you expand this expression we can further write GP of s GC ofs you can write RS - Ys - WS plus GP of s and mdms again you can write y of s 1 + z GP of s GC of s equal to GP of s GC of s into RS minus WS plus GP of s MDS so here actually we are going to find the overall error uh how we can find the error due to the reference signal and due to the disturbance and noise we'll combine them so this is how we this is already have done but here actually we are going to find y ofs equal to can write GP offs GC offs 1 plus GP of s GC of s s into rsus WS plus uh GP of s 1 plus GP of s GC of s into [Music] MDS now we can find the tracking error tracking error can find e of s equal to R of s minus y of equal [Music] to we can substitute the expression of Y is in this expression this let me equation number one if you substitute equation number one in this expression we'll have 1 minus GP of s GC of s 1 + GP of s GC of s office plus GP office GC ofs 1+ GP ofs GC of s into ws and minus GP ofs 1 plus GP ofs gcfs into MDS so this is how we can write the overall error in the system with respect to reference signal noise ws and disturbance MDS and if you denote if you denote G of s for the unity feedback system we can write GP of s CC of this is basically open loop transer function this you already have expl explained and in terms of G of s we can write equation number two we can write equation two can be written as right e of s = to 1 by 1 + G of s RS plus G is 1 + G of s ws and minus GP of s 1 + G of s MDS so this expression already you have found while we are discussing on the how we can neglect the state are in the system but here the combined equation with respect to reference signal Omega s and MDS and this expression we can write in simplified form e RS plus e ws and E MD s so this is the first term this is how the error and reference signal are connected or related the second term how the error and measurement noise are related and the third term how the disturbance and error is connected so from this let me note this is equation number equation number three here we can write where you r s equal to can write 1 + G of s RS so this is how the reference signal and error are related and ews we can write G of s 1 + G Nots WS this is how the eror and measurement NOS related and the third term e MDS equal to we can write minus GP of is 1 + G of s now if youd like to write the measurement error equation we can write now EMS basically EMS in the control system the blog dagram we have considered here only EMS is there this the EMS so EMS we can write now our main aim is how we can neglect EMS how we can make es or EMS to zero As Time tends to Infinity so once we achieve this condition our control system can track the reference signal in the presence of disturbance and noise in the system so EMS we can write es minus WS or if you substitute es what you have found here if you sub equation number three here we can write 1 by 1 + C of is into RS into RS plus G of s 1 + G of s -1 w s and plus can write minus GP of s 1 + G of this into this let me denote this equation number four now we can find the overall control or the control input uh given by the controller we can write the control input the control input by them controller we can write us equal to gcfs and M of this this easily can find from the block diagram so here we can write U of S equal to GC of s into EMS so can write so if you substitute the expression EMS in this equation we have the control input given by the controller if you Subs number four in this equation have GC of s 1 + G of s into RS plus minus GC of s 1 + G of s into w s and plus minus G of s 1 + G of s into MDS uh if you look carefully this is this expression the first expression we can write U RS and second term we can write U WS and third term we can write U MDS so this is basically the first term Ur is basically the relation between the control input and the reference signal and second term relates the control input and measurement noise and the third term relates between the disturbance and control input umds so now we where we can write URS equal to can write c c of is 1 + C not of s r s and WS minus GC of this 1 + C not of this and U MDS can write minus 3 of this 1 + this and let's denote this equation number five so this is how we can find the control input and measurement error equation and our motivation is how we can neglect or mitigate the disturbance and measurement noise with designing suitable control algorithm in frequency domain before we proceed to that part I'd like to highlight few important points which are very useful for Designing controller in frequency domain as a control engineer ideally we'd like to have our objective is objective is e RS should be zero e DS should be zero and E WS instead of EDS we can write MD ews should be zero once you achieve this condition we can track Ys to RS so but it is not possible to have this condition it is quite difficult to achieve this this objective for this if you would like to go to certain range or certain value close to this condition it should be noted that first one is uh the disturbance it should be this is very important Point while designing the control algorithm in frequency domain noted that the disturbance should be rejected or you can WR disturbance should be rejected Haring relatively slowly with time so it means the disturbance signal coming to the system varies very slowly with time so or in other words you can write the disturbance signal signal [Music] have low frequency content this is very while designing the control system we need to consider this point disturbance signal slowly time and second point the measurement noise [Music] typically varies rapidly with time the MDS has high frequency content this is very very important to design the control system how we can neglect the measurement noise with suitable design of control algorithm or also we have another point in addition to this we need we need to consider e RS and E MDS over low frequencies to ensure put reference signal tracking because our min how can track the reference signal and disturbance rection it means uh our reference signal also varies slowly with time so that's why uh ER is and emds over low frequencies to ensure good reference signal tracking and disturbance and while ews only needs to we consider over high frequencies to Ure good measurement noise so this is important Point while desiging the cont system how we can design the suitable control algorithms such that over low frequencies it can reject the error tracking error and the error due to the disturbance now um let's stop it here uh we'll continue from the next lecture how we can choose the control uh transer function or controller in such way that this condition can be [Music] achieved [Music] [Music] [Music]