What is automatic control all about? Welcome to Automatic Control III!! "Automatic control is the art of getting things to behave as you want.

Transcription

1 What i automatic control all about? Welcome to Automatic Control III!! Lecture Introduction and linear ytem theory Thoma Schön Diviion of Sytem and Control Department of Information Technology Uppala Univerity. "Automatic control i the art of getting thing to behave a you want." thoma.chon@it.uu.e, www: uer.it.uu.e/~thoc2 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory 2 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory Example automotive Almot all abbreviation in the ale brochure hide control ytem. ABS (Anti-lock Braking Sytem, controlling the brake force ESC (Electronic Stability Control, controlling coure tability ACE (Active Cornering Enhancement, controlling hock aborber in curve TCS (Traction Control Sytem, controlling road grip ACC (Adaptive Cruie Control, controlling peed/ditance ANC (Active Noie Control, controlling ound Example indutrial robot A robot arm i relatively weak and ocillate trongly when moving. Automatic control i neceary to achieve peed and accuracy. 3 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory 4 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory

2 Example extremely large telecope Example mobile phone We have reached the limit of how large mirror can be made. Large telecope are built with lot of mall mirror which are then continuouly controlled o that the image i in focu (called adaptive optic. Controlling the ignal trength between the mobile phone and the bae tation Traffic control / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory 6 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory Example medical technology Example Aeropace pacemaker inuline pump anaetheia Stabilization Cruie control Altitude control Navigation Weapon ytem Quadrotor ( com/watch?v=3cry8qzfy 7 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory 8 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory

3 Ued almot everywhere Aim of the coure. Improve already exiting technical ytem 2. Central area for many high-tech companie 3. Thankful area filled with many fun application. 4. Very intereting mathematic To ummarize it i fair to ay that Automatic Control i ued almot everywhere, but it i often hidden. What i new in thi coure?. More on multivariable ytem 2. Sytematic deign method 3. Fundamental bound on control performance 4. Nonlinear dynamic. Optimal control Automatic control i ometime refereed to a the hidden cience. 9 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory Coure tructure and practicalitie (I/III Coure tructure and practicalitie (II/III People: Lecturer and examiner: Thoma Schön, uer.it.uu.e/~thoc2/ Problem olving eion: Andrea Svenon, lecture (theory and example 8 problem olving eion (olve problem, dicu and ak quetion 2 computer laboratorie (each 2h Complete coure information (including lecture lide i available from the coure home page: Feel free to ak quetion! Examination (paing the coure (grade 3: Three compulory homework aignment: Homework -2 i olved in group of up to four tudent. Each group hand in one olution in the form of a written report. Homework 3 i olved individually. Every tudent hand in hi/her olution a an individual report. Identical report/olution will be rejected. An oral exam that mut all be paed. The oral exam i baed on the home work aignment. Examination (for grade 4 and : Beide fulfilling the requirement for grade 3 you have to take an additional written exam (you can bring the textbook. / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory 2 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory

4 Coure tructure and practicalitie (III/III Outline entire coure Problem olving eion 8 eion Solve problem, dicu and ak quetion Exercie manual available a wiki at All regitered tudent (hould have acce to edit the wiki. Feel free to improve and clarify the olution, contruct and add new exercie, etc! Technical wiki-problem (no acce etc: talk to Andrea The exercie manual i alo available a pdf in the tudent portal Feedback i a central part of thi coure. So pleae help u by providing any feedback on the wiki ytem!. Lecture : Introduction and linear multivariable ytem 2. Lecture 2 : Multivariable linear control theory a ytem theory, cloed loop ytem b Baic limitation c Controller tructure and control deign d H 2 and H loop haping 3. Lecture 6 8: Nonlinear control theory a Linearization and phae portrait b Lyapunov theory and the circle criterion c Decribing function 4. Lecture 9: Optimal control. Lecture : Summary and repetition 3 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory 4 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory Outline lecture The control problem. Introduction 2. What i automatic control? 3. Coure adminitration 4. Sytem theory linear multivariable ytem a Signal ize b Gain d The frequency function and ingular value e Stability and the mall gain theorem z: Control entity r: Reference ignal w: Proce diturbance n: Meaurement error u: Control ignal y: Meaurement ignal We want r z to be mall (i.e. z hould follow r nicely depite w, n and model uncertaintie. At the ame time we want u to take on reaonable value. / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory 6 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory

5 Multivariable v. calar The following concept generalize traightforwardly from SISO to multivariable ytem: The weight function i now a matrix The tranfer function i now a matrix Stability State pace formulation and the olution of the tate equation Controllability and obervability The following require ome thought: Gain and plotting the gain Pole and zero Turning G( into a tate pace decription 7 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory Computing the gain Gain For a linear tranformation y = Ax, the operator norm ( the gain i defined a how much larger the norm of y can (maximally become compared to the norm of x: y A = up x x = up Ax x x You can think of A a the gain of the tranformation A. For a dynamical ytem S (y = S(u we can imilarly define the gain of the ytem a S = up u y 2 S(u 2 = up u 2 u u 2 Chooe u uch that the quotient between the ize of the output ignal and the input ignal i maximized. 8 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory How big i G(iω if G(iω i a matrix? A we have een it i fairly eay to compute the gain for:. Static nonlinearity The gain varie with the amplitude. The gain doe not vary with the frequency. 2. Linear dynamical ytem (calar The gain doe not vary with the amplitude. The gain varie with the frequency. Magnitude (db 2 3 K u f(u u When we have everal input and output ignal it get more complicated. G (iω G 2 (iω... G m (iω G(iω =... G p (iω G p2 (iω... G pm (iω Which Bode diagram are intereting? Frequency (rad/ec 9 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory 2 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory

6 An example with 2 input ignal (I/VII An example with 2 input ignal (II/VII Controlling the level x in a tank. The flow to the tank i controlled by the two flow u and u 2, which both can be either poitive or negative. u u 2 Conider a control law where we neglect u 2 (u 2 =. u u 2 ẋ = u + u 2, G( = ( ẋ = u + u 2 G( = ( Magnitude (ab Frequency (rad/ec What i the gain if we intead combine the ue of u and u 2? 2 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory 22 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory An example with 2 input ignal (III/VII An example with 2 input ignal (IV/VII Combine the ue of u and u 2, i.e., let the pump cooperate (u = u 2. The level adjut fater. ẋ = u + u 2, G( = ( Let the pump counter-act each other (u = u 2 : the level doe not change at all. ẋ = u + u 2, G( = ( u u 2 u u 2 Magnitude (ab Magnitude (ab Frequency (rad/ec Frequency (rad/ec 23 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory 24 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory

7 An example with 2 input ignal (V/VII An example with 2 input ignal (VI/VII Conider arbitrary combination (e.g., u 2 =.3u of the combined ue of the pump. ẋ = u + u 2, G( = ( Conider arbitrary combination of the combined ue of the pump. ẋ = u + u 2, G( = ( u u 2 Magnitude (ab Magnitude (ab Frequency (rad/ec Frequency (rad/ec 2 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory All combination lead to en himla många Bode diagram... Lead to a natural quetion: Can we omehow find the mot relevant? 26 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory An example with 2 input ignal (VII/VII ẋ = u + u 2, G( = ( Singular Value Singular Value Frequency (rad/ Ye we can! The mot relevant plot are given by the ingular value of the tranfer matrix. Now, what doe thi mean? 27 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory Computing ingular value SVD The ingular value are computed uing the ingular value decompoition (SVD, which tate that every matrix A R n m can be written a A = UΣV, where U R n n and V R m m are unitary (UU = I matrice and Σ R n m with the ingular value on the diagonal and zero elewhere, σ σ 2 Σ =.... σ n In MATLAB: [U,Sigma,V] = vd(a. 28 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory

8 Connection, ingular value and eigenvalue The SVD (applicable to any n m matrix i more general than the eigenvalue (applicable to certain quare matrice decompoition, but the two are connected. Let A = UΣV. Hence, σ 2 σ2 2 AA = UΣ V}{{} V Σ U = U.... U, = I σn 2 which i the eigenvalue decompoition of AA. Hence, σ 2 i are the eigenvalue of AA. How big i G(iω if G(iω i a matrix? (reviited We can now anwer thi quetion. G(iω = the larget ingular value of G(iω Y (iω G(iω U(iω G = the larget ingular value of G(iω for any ω. Y 2 G U 2 Plotting the ingular value of G(iω a a function of ω correpond to the plot of the amplitude curve for SISO ytem. For multivariable ytem the actual gain depend on the direction of the input vector U(iω. 29 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory 3 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory Example a heat exchanger (I/III Example a heat exchanger (II/III T Ci, fc T H, fh T C, fc f C, f H flow, V C, V H volume, T C, T H temperature. Aume (for implicity: f C = f H = f i contant. Input: u = T Ci and u 2 = T Hi. State: x = T C and x 2 = T H. Model T Hi, fh V C dt C dt V H dt H dt = f C (T Ci T C + β(t H T C, = f H (T Hi T H β(t H T C. Uing the following numerical value f =. (m 3 /min, β =.2 (m 3 /min and V H = V C = (m 3, reult in ( (.2.2. ẋ = x + u,.2.2. y = x. 3 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory 32 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory

9 Example a heat exchanger (III/III Stability G( = Plot the ingular value in MATLAB: A = [-.2.2;.2 -.2]; B =.*eye(2; C = eye(2; D=zero(2,2; igma(a,b,c,d ( ( +.( Singular Value (db Singular Value A ytem i input-output table if it gain i finite. A olution to a differential equation i table if a mall perturbation of the initial condition give mall, poibly a vanihing effect. For linear ytem, tability i a ytem property, all olution have the ame tability propertie. The tability theory for multivariable linear ytem i the ame a in the baic coure. In thi coure we will alo analyze tability for nonlinear ytem, which lead to Lyapunov theory (chapter Frequency (rad/ec 33 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory 34 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory Stability the mall gain theorem Two table ytem S and S 2 which are connected according to the figure below reult in a cloed loop ytem that i table if r + e + y 2 S S 2 <. S 2 S e2 Note that the mall gain theorem i valid both for linear and nonlinear ytem. For linear ytem the criterion i implified to S 2 S <. y + + r 2 Ueful MATLAB command 2zp, zp2, tf2zp, zp Tranformation between different repreentation (tate pace (, tranfer function (tf, zero and pole (zp tzero Calculation of zero (for multivariable ytem lim, tep, impule Simulation, tep and impule repone pole, eig, root Eigenvalue and pole bode Frequency function and Bode plot igma Compute the ingular value of the frequency function obv, ctrb Obervability and controllability 3 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory 36 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory

10 A few concept to ummarize lecture Automatic control: the art of getting thing to behave a you want. Singular value: The ingular value of a matrix A are given by σ i = λ i, where λ i denote the eigenvalue of A A. Singular value of the frequency function: Plotting the ingular value of G(iω (for a multivariable ytem a a function of ω correpond to the plot of the amplitude curve for SISO ytem. Input-output tability: A ytem i input-output table if it gain i finite. Small gain theorem: Let the two ytem S and S 2 be connected in a feedback loop, then the cloed loop ytem i input-output table if S S 2 <. 37 / 37 T. Schön, 2 Automatic Control III, Lecture Introduction and linear ytem theory