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Lecture: Pole placement by dynamic output feedback Automatic Control 1 Pole placement by dynamic output feedback Prof. Alberto Bemporad University of Trento Academic year 2010-2011 Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 1 18

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Lecture: Pole placement by dynamic output feedback Introduction Output feedback control We know how to arbitrarily place the closed-loop poles by state feedback However, we may not want to directly measure the entire state vector Can we still place the closed-loop poles arbitrarily even if we only measure the output y Open-loop model: ) = Ax )+ Bu ) = Cx {z state-spacemodel )= {z transferfunction Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 2 18

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Lecture: Pole placement by dynamic output feedback Root locus Static output feedback (and “root locus”) Simple static feedback law: )= Ky Closed-loop poles can be only placed on the root locus by changing the gain Examples: MATLAB rlocus(sys) Root locus of a system with two asymptotically stable open-loop poles. The system is closed-loop asymptotically stable Root locus of a system with two asymptotically stable poles and an unstable open-loop pole. The system is closed-loop unstable (Walter R. Evans, “Graphical analysis of control systems”, 1948) (1920-1999) Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 3 18

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Lecture: Pole placement by dynamic output feedback State feedback control State feedback control (review) Assume the system is completely reachable State feedback control law )= Kx )+ Closed-loop system ) = ( BK )+ Bv ) = Cx )= where zI BK Adj zI BK det zI BK We can assign the roots of arbitrarily in the complex plane by properly choosing the state gain (complex poles must have their conjugate) Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 4 18

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Lecture: Pole placement by dynamic output feedback State feedback control State-feedback control (review) Assume in canonical reachability form . . . Let . . . The closed-loop matrix BK . . . is also in canonical form, so by choosing we can decide its eigenvalues arbitrarily Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 5 18

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Lecture: Pole placement by dynamic output feedback State feedback control Zeros of closed-loop system Fact Linear state feedback does not change the zeros of the system: )= Example for Change the coordinates to canonical reachability form 0 1 0 0 0 1 Compute Adj zI Adj zI does not depend on the coefﬁcients Then Adj zI BK also does not depends on Hence )= Adj zI Adj zI BK Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 6 18

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Lecture: Pole placement by dynamic output feedback State feedback control Potential issues in state feedback control Measuring the entire state vector may be too expensive (many sensors) even impossible (high temperature, high pressure, inaccessible environment) Can we use the estimate instead of to close the loop ? Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 7 18

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Lecture: Pole placement by dynamic output feedback Dynamic compensator Dynamic compensator Assume the open-loop system is completely observable (besides being reachable) Construct the linear state observer )= )+ Bu )+ )) Set )= )+ The dynamics of the error estimate )= is )= Ax )+ Bu Bu )+ Cx ))=( LC The error estimate does not depend on the feedback gain Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 8 18

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Lecture: Pole placement by dynamic output feedback Dynamic compensator Closed-loop dynamics Let’s combine the dynamics of the system, observer, and feedback gain ) = Ax )+ Bu ) = )+ Bu )+ )) ) = )+ ) = Cx Take as state components of the closed-loop system (itisindeedachangeofcoordinates) The closed-loop dynamics is BK BK LC ) = Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 9 18

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Lecture: Pole placement by dynamic output feedback Dynamic compensator Closed-loop dynamics The transfer function from to is ) = zI BK BK zI LC zI BK zI LC zI BK Even if we substituted with , the input-output behavior of the closed-loop system didn’t change ! The closed-loop poles can be assigned arbitrarily using dynamic output feedback, as in the state feedback case The closed-loop transfer function does not depend on the observer gain Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 10 18

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Lecture: Pole placement by dynamic output feedback Dynamic compensator Separation principle Separation principle The design of the control gain and of the observer gain can be done independently Watch out ! )= zI BK only represents the I input output) behavior of the closed-loop system The complete set of poles of the closed-loop system are given by det zI BK BK LC )= det zI BK det zI LC )= A zero pole cancellation of the observer poles has occurred: )= zI BK BK LC Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 11 18

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Lecture: Pole placement by dynamic output feedback Dynamic compensator Transient effects of the estimator gain The estimator gain seems irrelevant ... However, consider the effect of a nonzero initial condition for ) = Cx ) = —” BK BK LC —” —” BK BK LC BK CBK ) = —” BK BK LC —” BK CBK BK BK BK CBK The observer gain has an effect on the transient ! Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 12 18

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Lecture: Pole placement by dynamic output feedback Dynamic compensator Choosing the estimator gain Intuitively, if is a poor estimate of then the control action will also be poor Rule of thumb : place the observer poles 10 times faster than the controller poles Optimal methods exist to choose the observer poles (Kalman ﬁlter) Fact: The choice of is very important for determining the sensitivity of the closed-loop system with respect to input and output noise Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 13 18

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Lecture: Pole placement by dynamic output feedback Dynamic compensator Zero pole cancellations We have zero pole cancellations, the system has uncontrollable and or unobservable modes Intuitively: does not depend on is not controllable depends on during transient observable The reachability matrix is BK BK LC ··· BK BK LC BK ··· BK 0 0 ··· Since is reachable, rank )= uncontrollable modes The observability matrix doesn’t have a similar structure Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 14 18

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Lecture: Pole placement by dynamic output feedback Dynamic compensator Dynamic compensator The state-space equations of the dynamic compensator are ) = ( BK LC )+ Bv )+ Ly ) = )+ Equivalently, its transfer function is given by (superposition of effects) )=( zI BK LC )+ zI BK LC {z dynamicoutputfeedback MATLAB con=-reg(sys,K,L) Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 15 18

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Lecture: Pole placement by dynamic output feedback Dynamic compensator Example: Control of a DC Motor )= MATLAB K=1; beta=.3; alpha=1; G=tf(K,[1 beta alpha 0]); ts=0.5; % sampling time Gd=c2d(G,ts); sysd=ss(Gd); [A,B,C,D]=ssdata(sysd); % Controller polesK=[-1,-0.5+0.6 j,-0.5-0.6 j]; polesKd=exp(ts polesK); K=-place(A,B,polesKd); % Observer polesL=[-10, -9, -8]; polesLd=exp(ts polesL); L=place(A’,C’,polesLd)’; MATLAB % Closed-loop system, state=[x;xhat] bigA=[A,B K;L C,A+B K-L C]; bigB=[B;B]; bigC=[C,zeros(1,3)]; bigD=0; clsys=ss(bigA,bigB,bigC,bigD,ts); x0=[1 1 1]’; % Initial state xhat0=[0 0 0]’; % Initial estimate T=20; initial(clsys, [x0;xhat0],T); pause t=(0:ts:T)’; v=ones(size(t)); lsim(clsys,v); Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 16 18

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Lecture: Pole placement by dynamic output feedback Dynamic compensator Example: Control of a DC Motor 10 15 20 −0.5 0.5 1.5 time (s) )= )= 10 15 20 0.5 1.5 2.5 time (s) )= )= Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 17 18

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Lecture: Pole placement by dynamic output feedback Dynamic compensator English-Italian Vocabulary root locus luogo delle radici separation principle principio di separazione dynamic compensator compensatore dinamico Translation is obvious otherwise. Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 18 18

Alberto Bemporad University of Trento Academic year 20102011 Prof Alberto Bemporad University of Trento Automatic Control 1 Academic year 20102011 1 18 brPage 2br Lecture Pole placement by dynamic output feedback Introduction Output feedback control ID: 23946

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Page 1

Lecture: Pole placement by dynamic output feedback Automatic Control 1 Pole placement by dynamic output feedback Prof. Alberto Bemporad University of Trento Academic year 2010-2011 Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 1 18

Page 2

Lecture: Pole placement by dynamic output feedback Introduction Output feedback control We know how to arbitrarily place the closed-loop poles by state feedback However, we may not want to directly measure the entire state vector Can we still place the closed-loop poles arbitrarily even if we only measure the output y Open-loop model: ) = Ax )+ Bu ) = Cx {z state-spacemodel )= {z transferfunction Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 2 18

Page 3

Lecture: Pole placement by dynamic output feedback Root locus Static output feedback (and “root locus”) Simple static feedback law: )= Ky Closed-loop poles can be only placed on the root locus by changing the gain Examples: MATLAB rlocus(sys) Root locus of a system with two asymptotically stable open-loop poles. The system is closed-loop asymptotically stable Root locus of a system with two asymptotically stable poles and an unstable open-loop pole. The system is closed-loop unstable (Walter R. Evans, “Graphical analysis of control systems”, 1948) (1920-1999) Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 3 18

Page 4

Lecture: Pole placement by dynamic output feedback State feedback control State feedback control (review) Assume the system is completely reachable State feedback control law )= Kx )+ Closed-loop system ) = ( BK )+ Bv ) = Cx )= where zI BK Adj zI BK det zI BK We can assign the roots of arbitrarily in the complex plane by properly choosing the state gain (complex poles must have their conjugate) Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 4 18

Page 5

Lecture: Pole placement by dynamic output feedback State feedback control State-feedback control (review) Assume in canonical reachability form . . . Let . . . The closed-loop matrix BK . . . is also in canonical form, so by choosing we can decide its eigenvalues arbitrarily Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 5 18

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Lecture: Pole placement by dynamic output feedback State feedback control Zeros of closed-loop system Fact Linear state feedback does not change the zeros of the system: )= Example for Change the coordinates to canonical reachability form 0 1 0 0 0 1 Compute Adj zI Adj zI does not depend on the coefﬁcients Then Adj zI BK also does not depends on Hence )= Adj zI Adj zI BK Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 6 18

Page 7

Lecture: Pole placement by dynamic output feedback State feedback control Potential issues in state feedback control Measuring the entire state vector may be too expensive (many sensors) even impossible (high temperature, high pressure, inaccessible environment) Can we use the estimate instead of to close the loop ? Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 7 18

Page 8

Lecture: Pole placement by dynamic output feedback Dynamic compensator Dynamic compensator Assume the open-loop system is completely observable (besides being reachable) Construct the linear state observer )= )+ Bu )+ )) Set )= )+ The dynamics of the error estimate )= is )= Ax )+ Bu Bu )+ Cx ))=( LC The error estimate does not depend on the feedback gain Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 8 18

Page 9

Lecture: Pole placement by dynamic output feedback Dynamic compensator Closed-loop dynamics Let’s combine the dynamics of the system, observer, and feedback gain ) = Ax )+ Bu ) = )+ Bu )+ )) ) = )+ ) = Cx Take as state components of the closed-loop system (itisindeedachangeofcoordinates) The closed-loop dynamics is BK BK LC ) = Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 9 18

Page 10

Lecture: Pole placement by dynamic output feedback Dynamic compensator Closed-loop dynamics The transfer function from to is ) = zI BK BK zI LC zI BK zI LC zI BK Even if we substituted with , the input-output behavior of the closed-loop system didn’t change ! The closed-loop poles can be assigned arbitrarily using dynamic output feedback, as in the state feedback case The closed-loop transfer function does not depend on the observer gain Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 10 18

Page 11

Lecture: Pole placement by dynamic output feedback Dynamic compensator Separation principle Separation principle The design of the control gain and of the observer gain can be done independently Watch out ! )= zI BK only represents the I input output) behavior of the closed-loop system The complete set of poles of the closed-loop system are given by det zI BK BK LC )= det zI BK det zI LC )= A zero pole cancellation of the observer poles has occurred: )= zI BK BK LC Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 11 18

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Lecture: Pole placement by dynamic output feedback Dynamic compensator Transient effects of the estimator gain The estimator gain seems irrelevant ... However, consider the effect of a nonzero initial condition for ) = Cx ) = —” BK BK LC —” —” BK BK LC BK CBK ) = —” BK BK LC —” BK CBK BK BK BK CBK The observer gain has an effect on the transient ! Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 12 18

Page 13

Lecture: Pole placement by dynamic output feedback Dynamic compensator Choosing the estimator gain Intuitively, if is a poor estimate of then the control action will also be poor Rule of thumb : place the observer poles 10 times faster than the controller poles Optimal methods exist to choose the observer poles (Kalman ﬁlter) Fact: The choice of is very important for determining the sensitivity of the closed-loop system with respect to input and output noise Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 13 18

Page 14

Lecture: Pole placement by dynamic output feedback Dynamic compensator Zero pole cancellations We have zero pole cancellations, the system has uncontrollable and or unobservable modes Intuitively: does not depend on is not controllable depends on during transient observable The reachability matrix is BK BK LC ··· BK BK LC BK ··· BK 0 0 ··· Since is reachable, rank )= uncontrollable modes The observability matrix doesn’t have a similar structure Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 14 18

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Lecture: Pole placement by dynamic output feedback Dynamic compensator Dynamic compensator The state-space equations of the dynamic compensator are ) = ( BK LC )+ Bv )+ Ly ) = )+ Equivalently, its transfer function is given by (superposition of effects) )=( zI BK LC )+ zI BK LC {z dynamicoutputfeedback MATLAB con=-reg(sys,K,L) Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 15 18

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Lecture: Pole placement by dynamic output feedback Dynamic compensator Example: Control of a DC Motor )= MATLAB K=1; beta=.3; alpha=1; G=tf(K,[1 beta alpha 0]); ts=0.5; % sampling time Gd=c2d(G,ts); sysd=ss(Gd); [A,B,C,D]=ssdata(sysd); % Controller polesK=[-1,-0.5+0.6 j,-0.5-0.6 j]; polesKd=exp(ts polesK); K=-place(A,B,polesKd); % Observer polesL=[-10, -9, -8]; polesLd=exp(ts polesL); L=place(A’,C’,polesLd)’; MATLAB % Closed-loop system, state=[x;xhat] bigA=[A,B K;L C,A+B K-L C]; bigB=[B;B]; bigC=[C,zeros(1,3)]; bigD=0; clsys=ss(bigA,bigB,bigC,bigD,ts); x0=[1 1 1]’; % Initial state xhat0=[0 0 0]’; % Initial estimate T=20; initial(clsys, [x0;xhat0],T); pause t=(0:ts:T)’; v=ones(size(t)); lsim(clsys,v); Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 16 18

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Lecture: Pole placement by dynamic output feedback Dynamic compensator Example: Control of a DC Motor 10 15 20 −0.5 0.5 1.5 time (s) )= )= 10 15 20 0.5 1.5 2.5 time (s) )= )= Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 17 18

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Lecture: Pole placement by dynamic output feedback Dynamic compensator English-Italian Vocabulary root locus luogo delle radici separation principle principio di separazione dynamic compensator compensatore dinamico Translation is obvious otherwise. Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 2010-2011 18 18

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