Department of AUTOMATIC CONTROL Multivariable Control Exam Exam 4 8 Grading All answers must include a clear motivation and a well-formulated answer. AnswersmaybegiveninEnglishorSwedish.Thetotalnumberofpointsis5.The maximum number of points is specified for each subproblem. Accepted aid The textbook Glad& Ljung, standard mathematical tables like TEFYMA, an authorized Formelsamling i Reglerteknik / Collection of Formulas and a pocket calculator. Handouts of lecture notes and lecture slides are also allowed. Results The results will be reported via LADOK.
. Consider the feedback system in Figure. v n r e u y Σ 4 Σ Σ s Figure ThesysteminProblem. a. Giveastatespacerealizationofthesystemwithinputsr,vandnand outputse,uandy. (p) b. Determine the entries of the transfer matrix P er (s) P ev (s) P en (s) P ur (s) P uv (s) P un (s) P yr (s) P yv (s) P yn (s) mappingtheinputsr,vandntotheoutputse,uandy. (p) c. After permutation (re-ordering) of the inputs and outputs, the step responsesofthesystemhavebeenplottedinfigure.determinethenew order of inputs and outputs. ( p).5 4 4.5.5.5 Time (sec) Figure Stepresponse plotsfor Problem.Theinputsandoutputshavebeen reordered.
d. Suppose that the load disturbance v and the measurement disturbance n arebothconstant andzero,whilethereference signal r isazero-mean stochastic process withspectral density Φ r (ω) =.Determine the ω spectral density of e. ( p) a. Thefirstorderprocesscanbewritten Combining this with gives the state realization b. The transfer matrix is P er (s) P ev (s) P en (s) ẋ= xuv u=4e=4(r y)=4[r (xn)=4r 4x 4n r ẋ= }{{} 6x 4 4 v A }{{} n B e r u = 4 x 4 4 v y n }{{}}{{} C D P ur (s) P uv (s) P un (s) =C(sI A) BD P yr (s) P yv (s) P yn (s) = 4 4 () 4 4 4 4 4 = 4 6 4 4 6 4 4 c. ThestartingpointsofthestepresponsesshouldbegivenbytheD-matrix, sotheinputorderis(n,v,r)andtheoutputorderis(e,y,u). d. Thespectrumofeisgivenbytheformula Φ e (ω)=p er (iω)φ r (ω)p er (iω) ( = 4 ) ( iω6 ω 4 ) iω6 = (ω 4) (ω 36)(ω ) 3
. Yourbosshasheardthatyouaregreatatautomaticcontrolandwants somehelpfindingagoodcontrollerfortheprocessp(s)= s e.5s. a. Hewantsafastsystem;thebandwidth ω b shouldbeatleastrad/s.ifit is possible, find a stabilizing controller that meets the specification. If it is not possible, explain why. ( p) b. Yourbossgetsalittleimpatientandtriestofindacontrollerhimself.He claimsthathewillgetafastenoughsystemwiththecontroller C(s)= 5s e.5s. s HeshowsyouthemarginplotoftheopenloopsystemPCwhereeverything looksnice,seefigure3.youdohoweverseeaverybigproblemwiththis design.whatistheproblemthatyouneedtoexplaintoyourboss? (p) 5 Bode Diagram Gm = Inf, Pm = 7.7 deg (at.9 rad/s) Magnitude (abs) 9 Phase (deg) 5 (rad/s) Frequency Figure3 Marginplotfortheopenloopsysteminproblemb. a. It is not possible to fulfill the specification due to the fundamental limitation onthebandwidthofatime-delayedsystem.asystemontheformg(s)= G (s)e sl,whereg isarationalfunction,willhavethelimitedbandwidth ω B </Lwhichinthiscasemeansthatyoucangetabandwidthofat most rad/s. b. You can not realize acontroller withthefactor e.5s init, since sucha controller would be non-causal. 3. Consider the double integrator [ ẋ= x z=[ x. [ u, 4
a. Find the LQR controller that minimizes ( ) z T (t)z(t)u T (t)u(t) dt. (.5 p) b. Determinethestationaryvalueu (r)oftheoptimalinputuwhenthereferencevalueforzisr. (.5p) c. Find the LQR controller that minimizes ( (z(t) r) (u(t) u (r)) ) dt. (p) a. The fastidious would start by ascertaining controllability of the system. Sincewearenot,weskipthatstep.Let [ s s S= s s 3 be the unique positive definite solution to the Riccati equation M T Q MA T SSA (SBQ, )Q (SBQ,) T =, where [ A=, B= [, M=[, Q =I, Q =I, Q, =. This yields [ T [ [ [ [ T s s s s [ s s 3 s s 3 [ [ [ s s T [ s s = s s 3 s s 3 [ [ [ [ s s [s s 3 = 4 s s s s 3 [ s s s s 3 s s s 3 4s s =. 3 s =, s =, s =s s 3, s 3 = s = 6, 4s s 3 = 6. 5
This gives the optimal state feedback gain L=Q (SBQ,) T = Thus, the sought controller is [ T [ 6 =[ 6. 6 u(x)= [ 6x. b. In stationarity we have ż=mẋ=ẋ ẋ = ẋ = ẋ. () Instationaritywealsohave u=,whichtogetherwiththesecondstate equation yields = u=ẋ. () Combining the two state equations with equations() and() yields in stationarity u=x =ẋ = ẋ =. The stationary input value u (r)= isthusnecessaryinorderforz=rinstationarity,foranyr. c. The integral is minimized with the LQR controller u= Lxl r r, where Lwasdeterminedinproblema.,andl r ischosentogetunitary stationary gain in the closed-loop system, that is ( ([ l r =(M(BL A) B) = [ [ [ ) [ ) 6 ( [ [ ) ( [ [ ) 6 = [ = [ 6 ( [ ) = [ =. Thus, the sought controller is u(x,r)= [ 6xr. Notethatthesolutionisentirelyindependentofu,whichitwouldhave beenevenifu werenon-zero. 4. 6
a. Consider control of the process If a proportional controller P(s)= s. C(s)=K isused,forwhatvaluesof K willtheclosed-loopsysteminfigure4be stable? (p) r Σ e C u P y Figure 4 Block diagram of the closed-loop system in Problem 4. b. Consider the situation where C instead is an unknown time-varying functionk(t),withanupperandlowerlimitof±α.thatis,foralltimes α K(t) α. For what positive values of α can stability of the closed-loop system be guaranteed? Hint: Use the small-gain theorem! (3 p) a. The closed-loop transfer function is given by s K s G(s)= P(s)C(s) K P(s)C(s) = = K sk. The closed-loop system is thus(asymptotically) stable if and only K>. b. Werewritetheblockdiagramintoaformcompatiblewiththesmallgain theorem, which yields the block diagram in Figure 5. r Σ e C u y P Figure 5 Block diagram of the closed-loop system in Problem 4. 7
Since both C and P are stable systems, we can apply the small gain theorem. Since P is linear, time-invariant and SISO, its system gain is P =sup G(iω) =sup ω ω iω =sup ω ω =. ForCwehave C =sup e= = α. K e e =sup e= (K(t) e(t)) dt (e(t)) dt The small gain theorem thus gives stability if C P < C < P α<. sup e= α (e(t)) dt (e(t)) dt 5. ConsiderasystemS describedby [ [ ẋ= x u, y=[ x. a. Compute the Hankel singular values. ( p) b. Balancedtruncationisperformedtoyieldafirst-orderapproximationS of theoriginalsystems.provideanupperboundon S S.Notethatyou donotneedtocalculates. (p) a. Let [ [ s s o o S=, O= s s 3 o o 3 be the controllability and observability Gramians, respectively. The controllability Gramian is found as the unique symmetric solution of the Lyapunov equation ASSA T BB T = [ [ [ [ s s s s s s 3 s s 3 [ [ s s s s s s s s s 3 s 4=, s s =, s s 3 = s s s 3 s =, s =, s 3 =. [ [ = [ 4 = 8
The observability Gramian is found as the unique symmetric solution of the Lyapunov equation A T OOAC T C= [ [ [ [ o o o o o o 3 o o 3 [ [ o o o o 3 o o o o o 3 o o =, o o 3 =, o 3 4= o o 3 o 3 o =, o =, o 3 =. [ [ = [ = 4 The Gramians are thus [ S=, O= TheHankelsingularvalues σ aregivenby [ =det(so σ I)= 3 σ 4 3 σ =(3 σ ) 8= σ 4 6σ σ =3± 8. Since the Hankel singular values are positive, we get b. Wehave σ=. 3± 8= 3± = (± ) = ±. S S S (u) S (u) =sup u= u supσ r =σ r, u= where σ r isthehankelsingularvalueofthetruncated state. Thus,by choosing σ r =,weobtain S S. 6. Consider a system described by the following transfer function matrix G(s)= [ s s s s s s Suggest a suitable transformation for doing decoupled control and provide the overall structure of the controller.(you do not have to design any specific controller for the decoupled systems) ( p) 9
Wenoticethatthefirstoutputisdrivenbythedifferenceofthecontrolsignal,thesecondoutputisdrivenbythesum,wecanthusmakeacoordinate change in the output of our controllers exploiting this information. 7. u = Fû, Û = C Y, Û = C Y [.5.5 F =.5.5 [ s = Y = GFÛ= Û s s WethusseethatthischoiceofFcompletelydecouplesoursystem. a. Design an internal model controller for the system P(s)= s s. s4 Ifyouhavethepossibilitytodecidethecrossoverfrequency ω c fortheopen loop of any of the subsystems, choose your design parameters such that ω c =rad/s. (3p) b. Internal model controllers can also be designed by solving a convex optimization problem of the form min Q T T Q. HereT andt canbechosensothat ( WS S T T Q= W T T wheres=(ipc) isthesensitivityfunctionandt=pc(ipc) is thecomplementarysensitivityfunctionandw S aswellasw T areweighting functions. After finding the optimal Q, the controller can be obtained as C=(I QP) Q.FindexpressionsforT andt intermsofageneral processpaswellasgeneralweightingfunctionsw S andw T. Hint:UseST=I! ), (p)
a. By following the recommendations in the book for how to handle nonminimumphasezerosandhowtomakeqproperweget Q(s)= s s The corresponding controller is given by C(s)=(I Q(s)P(s)) Q(s)= = s λs λs s s. s4 λs s (s ) s s4 λs = λs λs s s4 λs. Q(s)= The open-loop transfer function is P(s)C(s) = s s s4 s s4 λs = s. λs Ifwechoosethedesignparameter λ=.5,thesubsystem(,)willgetthe crossoverfrequency ω c =rad/s. b. WefirstsolvethecontrollerequationforQ,thatis Thus C=(I QP) Q (I QP)C=Q C=QQPC C(IPC) =Q. ( ) ( ) ( ( WS S WS (I T) ) WS I PC(IPC) ) T T Q= = = ( W T T W T T W ) T PC(IPC) ( ) ( ) ( ) WS (I PQ) WS WS P = = Q. W T PQ W }{{} T P }{{} T T