Synthesis SP Skogestad And Postlethwaite(1996 .
Topic #2416.31 Feedback Control Systems H SynthesisSP Skogestad and Postlethwaite(1996) Multivariable Feedback Control Wiley.JB Burl (2000). Linear Optimal Control Addison-Wesley.ZDG Zhou, Doyle, and Glover (1996). Robust and Optimal Control Prentice Hall.MAC Maciejowski (1989) Multivariable Feedback Design Addison Wesley.Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare(http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Fall 200716.31 24–1Synthesis For the synthesis problem, typically define a generalized version ofthe system dynamics (SISO or MIMO)wz u Pzw (s) Pzu(s)Pyw (s) Pyu(s)Gc y Signals:Generalized plant: Pzw (s) Pzu(s)P (s) Pyw (s) Pyu(s)z – Performance outputw – Disturbance/ref inputscontains plant G(s) and all performance/uncertainty weightsy – Sensor outputsu – Actuator inputs With the loop closed (u Gcy), can show that z Pzw PzuGc(I PyuGc) 1Pyw w Fl (P, Gc)w– Called a (lower) Linear Fractional Transformation (LFT). Define the H norm for a TFM P (s) as P (s) sup σ[P (jω)]ω– Basically finds the peak amplification possible for the TFM overall frequencies (works for MIMO).December 9, 2007Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare(http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Fall 200716.31 24–2 Design Objective: Find Gc(s) to stabilize the closed-loop system and minimize Fl (P, Gc) . Hard problem to solve, so typically consider suboptimal problem:– Find Gc(s) to satisfy Fl (P, Gc) γ– Then use bisection (called a γ iteration) to find the smallestvalue (γopt) for which Fl (P, Gc) γopt hopefully get that Gc approaches Goptc Consider the suboptimal H synthesis problem:1Find Gc(s) to satisfy Fl (P, Gc) γ A Bw BuPzw (s) Pzu(s): Cz 0 Dzu P (s) Pyw (s) Pyu(s)Cy Dyw 0where it is assumed that:1. (A, Bu, Cy ) is stabilizable/detectable (essential)2. (A, Bw , Cz) is stabilizable/detectable (essential)T3. Dzu[ Cz Dzu ] [ 0 I ] (simplify/essential) 0BwTDyw (simplify/essential)4.IDyw1 SP367December 9, 2007Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare(http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Fall 200716.31 24–3 There exists a stabilizing Gc(s) such that Fl (P, Gc) γ iff(1) X 0 that solves the AREAT X XA CzT Cz X(γ 2Bw BwT BuBuT )X 0 and Rλi A (γ 2Bw BwT BuBuT )X 0 i(2) Y 0 that solves the AREAY Y AT BwT Bw Y (γ 2CzT Cz CyT Cy )Y 0 and Rλi A Y (γ 2CzT Cz CyT Cy ) 0 i(3) ρ(XY ) γ 2ρ is the spectral radius (ρ(A) maxi λi(A) ). Given these solutions, the central H controller is given by A (γ 2Bw BwT BuBuT )X ZY CyT Cy ZY CyTGc(s) : BuT X0where Z (I γ 2Y X) 1– Has as many states as the generalized plant. Note that this design does not decouple as well as the regulator/estimator for LQGDecember 9, 2007Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare(http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Fall 200716.31 24–4Simple Design Example r–e z1Ws Gc(s)whereG uWuz2 ỹG(s) 200(0.05s 1)2(10s 1) Note there is 1 input (r) and two performance outputs - one thatpenalizes the sensitivity S(s) of the system, and the other thatpenalizes the control effort used. To achieve good low frequency tracking and a crossover frequencyof about 10 rad/sec, pickWs s/1.5 10s (10) · (0.0001)Wu 1– Typically treat Wu as a constant (similar role as ρ as the LQRcontrol control cost)December 9, 2007Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare(http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Fall 200716.31 24–5 Generalized system in this case:Pwwsruz1zz2wuG eyGcFigure by MIT OpenCourseWare.Figure 1: Rearrangement of original picture in the generalized plant format. Easy to show that the closed-loop is: z1Ws S rWu Gc Sz2– Input r acts as “disturbance input” w to generalized system. Ws(s) Ws(s)G(s)z1 Ws(s)(r Gu) Wu(s)P (s) 0z 2 Wu u1 G(s) e r GuPzw (s) Pzu(s) u GcePyw (s) Pyu(s)PCL Fl (P, Gc) Ws WsGGc(I GGc) 11 Wu0 Ws WsGGcSWsS Wu G c SWu Gc SDecember 9, 2007Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare(http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Fall 200716.31 24–6 In state space form, let A BG(s) : C 0ẋẋwz1z2e Ws(s) : Aw BwCw Dw Wu 1 Ax BuAw xw Bw e Aw xw Bw r Bw CxCw xw Dw e Cw xw Dw r Dw CxWu ur Cx A0 0 B B C A B0 www P (s) : Dw C Cw Dw 0 00 0 Wu C0 1 0 Now use the mu-tools code to solve for the controller. (Could alsohave used the robust control toolbox code).A [Ag zeros(n1,n2);-Bsw*Cg Asw];Bw [zeros(n1,1);Bsw];Bu [Bg;zeros(n2,1)];Cz [-Dsw*Cg Csw;zeros(1,n1 n2)];Cy [-Cg zeros(1,n2)];Dzw [Dsw;0];Dzu [0;1];Dyw [1];Dyu 0;P pck(A,[Bw Bu],[Cz;Cy],[Dzw Dzu;Dyw Dyu]);% call hinf to find Gc (mu toolbox)[Gc,G,gamma] hinfsyn(P,1,1,0.1,20,.001);December 9, 2007Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare(http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Fall 200716.31 24–7 Results from the γ-iteration showing whether the various X, Y ,ρ(XY ) tests are passed/failed during the bisection searching overγ, starting at the initial bound of 20.Resetting value of Gamma min based on D 11, D 12, D 21 termsTest bounds: 461.3841.3651.3751.3701.3681.3661.3660.6667 hamx eig9.6e 0009.6e 0009.5e 0009.5e 0009.4e 0009.1e 0009.3e 0009.2e 0009.2e 0009.2e 0009.2e 0009.2e 0009.2e 0009.2e 0009.2e 0009.2e 000Gamma value achieved:gamma xinf eig hamy eig6.2e-008 1.0e-0036.3e-008 1.0e-0036.3e-008 1.0e-0036.5e-008 1.0e-0036.9e-008 1.0e-003-1.2e 004# 1.0e-0037.3e-008 1.0e-0037.6e-008 1.0e-003-6.4e 004# 1.0e-0037.7e-008 1.0e-003-1.9e 006# 1.0e-0037.7e-008 1.0e-0037.7e-008 1.0e-0037.7e-008 1.0e-0037.7e-008 1.0e-003-1.3e 007# 1.0e-00320.0000yinf eignrho xyp/f0.0e 0000.0000p0.0e 0000.0000p0.0e 0000.0000p0.0e 0000.0000p0.0e 0000.0000p-4.5e-0100.0000f0.0e 0000.0000p0.0e 0000.0000p0.0e 0000.0000f0.0e 0000.0000p0.0e 0000.0000f-4.5e-0100.0000p0.0e 0000.0000p0.0e 0000.0000p0.0e 0000.0000p0.0e 0000.0000f1.3664 Since γmin 1.3664, this indicates that the controller does notmeet the desired goal of S 1/ Ws (can only say that S 1.3664/ Ws ).– Confirmed by the plot, which that the blue line passes abovethe magenta But note that, even though this design fails the sensitivity weight- the controller still gets pretty good performance– For performance problems, can think of the objective of gettingγmin 1 as a “design goal” it is “not crucial”– Use Wu to tune the control designDecember 9, 2007Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare(http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Fall 200716.31 24–8Figure 2: Visualization of the weighted sensitivity tests.Figure 3: Time response of controller that yields γmin 1.3664.December 9, 2007Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare(http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Fall 200716.31 24–9Controller Interpretations Given these solutions, the central H controller is given by A (γ 2Bw BwT BuBuT )X ZY CyT Cy ZY CyTGc(s) : BuT X0where Z (I γ 2Y X) 1 Can develop a further interpretation of this controller if rewrite thedynamics as:x̂ Ax̂ γ 2Bw BwT X x̂ BuBuT X x̂ ZY CyT Cy x̂ ZY CyT yu BuT X x̂ x̂ Ax̂ Bw γ 2BwT X x̂ Bu BuT X x̂ ZY CyT [y Cy x̂] x̂ Ax̂ Bw γ 2BwT X x̂ Buu L [y Cy x̂]looks very similar to Kalman Filter developed for LQG controller. The difference is that there is an additional input ŵworst γ 2BwT X x̂that enters through Bw .– wworst is an estimate of worst-case disturbance to the system. Finally, note that a separation rule does exist for the H controller.But it is much more complicated than for LQG.December 9, 2007Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare(http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Fall 200716.31 24–10Code: H Synthesis12345678% Hinf example% 16.31 MIT Fall 2007% Jon tSize’,20)91011121314151617clear allif exist(’yprev’)yprev [1 1]’;tprev [0 1]’;Sensprev [1 1];fprev [.1 100];Wu 1;end181920212223242526272829303132333435363738% define plant[Ag,Bg,Cg,Dg] tf2ss(200,conv(conv([0.05 1],[0.05 1]),[10 1]));Gol ss(Ag,Bg,Cg,Dg);% define sensitivity weightM 1.5;wB 10;A 1e-4;[Asw,Bsw,Csw,Dsw] tf2ss([1/M wB],[1 wB*A]);Ws ss(Asw,Bsw,Csw,Dsw);% form augmented P dynamicsn1 size(Ag,1);n2 size(Asw,1);A [Ag zeros(n1,n2);-Bsw*Cg Asw];Bw [zeros(n1,1);Bsw];Bu [Bg;zeros(n2,1)];Cz [-Dsw*Cg Csw;zeros(1,n1 n2)];Cy [-Cg zeros(1,n2)];Dzw [Dsw;0];Dzu [0;Wu];Dyw [1];Dyu 0;P pck(A,[Bw Bu],[Cz;Cy],[Dzw Dzu;Dyw Dyu]);3940414243% call hinf to find Gc (mu toolbox)diary hinf1 diary[Gc,G,gamma] hinfsyn(P,1,1,0.1,20,.001);diary off444546[ac,bc,cc,dc] unpck(Gc);ev max(real(eig(ac)/2/pi))474849505152535455PP ss(A,[Bw Bu],[Cz;Cy],[Dzw Dzu;Dyw Dyu]);GGc ss(ac,bc,cc,dc);CLsys feedback(PP,GGc,[2],[3],1);[acl,bcl,ccl,dcl] ssdata(CLsys);% reduce closed-loop system so that it only has% 1 input and 2 outputsbcl bcl(:,1);ccl ccl([1 2],:);dcl dcl([1 2],1);CLsys ss(acl,bcl,ccl,dcl);56575859606162636465666768f logspace(-1,2,400);Pcl freqresp(CLsys,f);CLWS squeeze(Pcl(1,1,:)); % closed loop weighted sensWS freqresp(Ws,f); % sens weightSensW squeeze(WS(1,1,:));Sens CLWS./SensW; % divide out weight to get closed-loop neWidth’,2)hold loglog(fprev,abs(Sensprev),’r.’)December 9, 2007Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare(http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
Fall 2007697071727316.31 24–11legend(’S’,’1/W s’,’W sS’,’Location’,’SouthEast’)hold offxlabel(’Freq 9na size(Ag,1);nac size(ac,1);Acl [Ag Bg*cc;-bc*Cg ac];Bcl [zeros(na,1);bc];Ccl [Cg zeros(1,nac)];Dcl 0;Gcl ss(Acl,Bcl,Ccl,Dcl);[y,t] �LineWidth’,2.5)axis([0 1 0 1.1])hold old offxlabel(’Time sec’)ylabel(’Step response’)8788899091%yprev y;%tprev t;%Sensprev Sens;%fprev f;9293return949596print -dpng -r300 hinf1.pngprint -dpng -r300 hinf12.png97December 9, 2007Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare(http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
16.31 Feedback Control Systems H Synthesis SP Skogestad and Postlethwaite(1996) Multivariable Feedback Control Wiley. JB Burl (2000). Linear Optimal Control Addison-Wesley. ZDG Zhou, Doyle, and Glover (1996). Robust and Optimal Control Prentice Hall. MAC Maciejowski (1989) Multivariable Feedback Design Addison Wesley.
A: SISO Feedback Control. A.2 Sensitivity and Feedback Performance. Reference: A.3 Loop Shaping [SP05] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control; Analysis and Design ,
Skogestad and Postlethwaite, Multivariable Feedback Control, 2nd ed. Supporting text: Zhou, Doyle and Glover, Robust and Optimal Control 8 homeworks, compulsory download from homepage after each lecture, hand in within one week require Matlab with Robust Control toolbox 1-day take home open book exam, within 6 weeks after last lecture
SP Skogestad and Postlethwaite(1996) Multivariable Feedback Control Wiley. JB Burl (2000). Linear Optimal Control Addison-Wesley. ZDG Zhou, Doyle, and Glover (1996). Robust and Optimal Control Prentice Hall. MAC Maciejowski (1989) Multivariable Feedback Design Addison Wesley.
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
Milli-Q Synthesis/Synthesis A10 1 Chapter 1 INTRODUCTION 1-1 USING THIS MANUAL MATCHING THIS MANUAL WITH YOUR MILLI-Q This manual is intended for use with a Millipore Milli-Q Synthesis or Milli-Q Synthesis A10 Water Purification System. This Owner s Manual is a guide for use during the in
Organic Synthesis What are the Essentials in Synthesis? 5 Since organic synthesis is applied organic chemistry, to stand a realistic chance of succeeding in any synthesis, the student ought to have a good knowledge-base of organic chemistry in the following areas: Protecting group chemistry Asymmetric synthesis
2 Materials and methods 14 2.1 Materials 15 2.2 Synthesis of ITO NPs 16 2.3 Synthesis of TiO 2 NPs 17 2.4 Synthesis of PES support layer 18 2.5 Synthesis of PES-ITO nanocomposite membrane 18 2.6 Synthesis of TFC membranes 19 2.7 Characterization of synthesized NPs and membranes 20
Grouted pile connections shall be designed to satisfactorily transfer the design loads from the pile sleeve to the pile as shown in . Figure K.5-1. The grout packer may be placed above or below the lower yoke plate as indicated in Figure K.5-2. The connection may be analysed by using a load model as shown in Figure K.5-3. The following failure modes of grouted pile to sleeve connections need .
