Control Systems

Transcription

1 Conider the ytem: x Ax Bu; y Cx Du Given x() and u(t) for t. The olution i x(t) e At y(t) Ce 6.5 Control Sytem Summary of Reult From Lat Lecture: x() At x() t e t A(t τ) Ce Bu ( )d ; A(t τ) Bu ( )d Du(t) The main problem involved i to compute e At. The ytem i internally table iff all the eigenvalue of A have negative real part: Re i (A) < for all i x(t) if u(t)= Solution of Dicrete-time Equation The DT ytem: x[k ] Ax[k] Bu[k] y[k] Cx[k] Du[k] Given x[] and u(k), k, the olution i: x[k] A x[] k k k y[k] CA x[] A m k m k m CA Bu[m] k m Bu[m] Du[k] The main problem involved i to compute A k. The ytem i internally table iff all the eigenvalue of A are within the unit dik: i (A) < for all i x[k] if u[k]=

2 Today: we will addre ome micellaneou problem about LTI ytem How to deal with complex eigenvalue Realization of a tranfer function Simulation of ytem by uing Simulink Coure project To prepare for new topic in thi coure, we will alo tudy Quadratic function and poitive-definitene Next Time: We tart another topic (Chapter 6) Controllability and obervability Deal with complex eigenvalue Typically, we would like to tranform a matrix AR n n into a diagonal form through equivalent tranformation λ - A Q AQ A λ λ Suppoe that A ha complex eigenvalue. Some i will be complex The tranformation matrix Q - will have complex entrie What doe the new tate z = Q - x tand for? Phyically, it i meaningle. Numerically, it may render analyi or deign reult invalid, uch a a feedback law with complex number 4

3 Since A i a real matrix, if A ha complex eigenvalue, The complex eigenvalue appear a conjugate pair: i j i, i j i. The eigenvector alo appear a conjugate pair v i +jw i, v i -jw i. There i a way to avoid complex number. Aume A(v jw) (α j )(v jw) v - w j( v w); A(v jw) (α j )(v jw) v - w j( v w);; Add the two: Subtract: Av αv -βw Aw βv αw A[v w] [v? w]??? A[v w] [v w] - 5 Suppoe we have two real eigenvalue and two pair of complex eigenvalue, all ditinct, λ, λ, α jβ, α jβ, α jβ, α jβ, with correponding eigenvector;, q, v jw, v jw, v jw, v Aq q, A[v w ] [v w ] α q jw, β i i i i i i i i i -βi α i In matrix form; Q Q A Q AQ A[q q v w v w] [q q v w v w] Not trictly a diagonal form but real: a block diagonal 6form A

4 Example: A Ue matlab [V,D]=eig(A), you get: V = complex eigenvector.6 -.6i.6 +.6i D = diagonal form. +.i. -.i. Where inv(v)av=d Let q=real(v(:,)); q=imag(v(:,)); q=v(:,) Form Q=[q q q], then Find the block diagonal form and the tranformation matrix Q = >> inv(q)aq an = What i e At with α β A? -β α Ue the firt definition for a matrix function. Let f()=e t. Find g()=k + k. The eigenvalue of A: λ α jβ, λ α jβ, g(λ ) f(λ ) g(λ ) f(λ ) k k k (α jβ) e k (α jβ) e (α jβ)t (α jβ)t α t e in βt αt α k, k e (co βt in βt) β β e g(a) e (co β t in ) in β βt β βt -β α αt At αt α e α β co co in αt co t in t e in t co t 8 4

5 In ummary: α β With A, -β α co βt in βt At e e αt, -in βt co βt For the ytem: x Ax Bu The zero-input olution i αt co βt in βt At x(t) e x() e x() -in βt co βt The real part of the eigenvalue) determine the tability of the ytem; The imaginary part determine the frequency of the ocillation. 9 e For a matrix A λt e λt e e coβ t e inβ t -e inβ t e coβ t αt αt e coβt e inβt αt αt -e inβt e coβt αt αt At αt αt 5

6 Realization of a periodic ignal, e.g., with two harmonic: in in Can be realized a the output of a 5 th -order ytem: ; Γ, A, Γ a cot int b vt () int cot c, cot int d int cot e Will learn how to build an oberver to recontruct a periodic ignal in co in co Today: ome micellaneou problem about LTI ytem How to deal with complex eigenvalue Realization of a tranfer function Simulation of ytem by uing Simulink Coure Project And more from linear algebra Quadratic function and poitive-definitene 6

7 State-pace realization of tranfer function Given tate equation x Ax Bu; y Cx Du The tranfer function i: G() C(I A) Now, given G(), how to find (A,B,C,D)? () B D Background: Sometime it i hard to obtain a tate-pace decription. But you can identify the tranfer function uing frequency repone. We have more advanced deign method for tate-pace model Example: G( ) It can be verified that G() = C(I-A) - B + D with A, B, C 4 5 6, D Firt, I A Then, (I - A) The adjunct matrix 4 7

8 8 5 (I - A) D B A) C(I G() We ay that (A,B,C,D) i a realization of G() If there exit (A,B,C,D) uch that G()=C(I-A) - B+D then we ay that G() i realizable. D, C, B, A 6 Theorem: A tranfer matrix G() i realizable with LTI tate equation if and only if it i a proper rational matrix. A proper rational matrix can be decompoed a the um of a contant matrix and a trictly proper rational matrix: G()=G p ()+D, D= G() Let d()= r + a r- + a r- +.+ a r- +a r be the leat common denominator of all entrie of G p () Then G p () can be expreed a (aume G i qp) Firt oberve that C(I-A) - B+D i alway proper and rational (neceity proved). p q i r r r r p R N, N N N N d() () G

9 With G () N N N N, N R d() r r mp p r r i d() a a a r r r r The realization of G p () i given a: ai Ip A C N p a I Ip p a ri I N Nr Nr p p a ri p, I p B Another form of realization: Problem Example: 4 G( ) ( )( ) ( ) Step : break it into a contant part and a trictly proper part 6.5 G( ).5 (.5)( ), ( ) G p () Step : the monic leat common denominator d()=(+.5)(+)(+)= a a a Step : 6( ) ( )(.5) Gp() (.5)( ).5( ) ( )(.5) d()

10 d()= =a + a + a From tep : G (S) p d() Step 4: 6 d() N a I p a I p a I p A, N B, N.5 C ,.5 D N N N 9 Another realization of the ame ytem i given in p.5 Example 4.7, where the dimenion i only 4. Dicuion: The realization (A,B,C,D) for a particular G() i not unique; All the equivalence tranformation are alo valid realization; With different method, the dimenion of the reulting ytem, i.e., the number of tate variable, may be different. There exit a minimal-order realization We will learn later how to reduce the order of a realization to the minimal number.

11 Today: ome micellaneou problem about LTI ytem How to deal with complex eigenvalue Realization of a tranfer function Simulation of ytem by uing Simulink Coure project And more from linear algebra Quadratic function and poitive-definitene A Tool for Sytem Simulation: SIMULINK Can be ued for imulation of variou ytem: Linear, CT or DT, Nonlinear; Switched; Hybrid: CT + DT component, ignal; Input ignal can be arbitrarily generated: Standard: inuoidal, polynomial, quare, impule Cutomized: from a function, look-up table Output ignal can be tored or demontrated in different way.

12 y Example: Input u y y y y y u y y y y y y y u Click imulation and ue plot(t,y), you will get a time repone of y The parameter can be eaily changed; The initial condition can be eaily changed t (ec) Simulink for linear ytem Main component with dynamic: integrator, tate-pace decription (A,B,C,D) tranfer function derivative (rarely ued) The firt two component need initial condition Math component: addition (a+b+c); product (ab); dot (inner) product <x,y>; gain (amplifier) kx : x a calar matrix gain Kx: x a vector 4

13 Source: input ignal contant, tep, ramp pule, ine wave, quare wave from data file ignal generator Sink: for output demontration or torage digital diplay cope ave to file export to workpace XY graph Nonlinear: function and operation aturation, deadzone, witch 5 Signal and ytem: Demux: input a vector ignal and output all the component Mux: input a bunch of calar ignal and output a vector ignal Function and table: input u output y: y=f(u); f compoed from available function or operation; e.g, y=in(u)+uu matlab function: y=f(u); f written by a matlab file look-up table. 6

14 Example: Find the olution to the LTI ytem x y x u x where x()=; u(t) i a quare wave. Step:. Open matlab workpace. type imulink and return - imulink library brower window i open. Click file and chooe new then chooe model - a blank window i open 4. Open one of the commonly ued block and drag and drop whatever you need to the blank window. 5. Connect the component by arrow. 7 Firt approach: ue tate-pace decription: Click each component to etup the parameter properly ink labeled t, u, y : chooe array for ave format ampling time can be a parameter inputted from workpace When ready, click imulation and chooe configuration parameter to etup imulation time. Finally, click imulation and chooe tart When finihed, type plot(t,y,t,u) to plot the input and output 8 4

15 Second approach: ue integrator and amplifier: 9 You can make any kind of change to the model: Change the parameter, the ampling time, add ome nonlinear component uch a a aturation: 5

16 Simulation for nonlinear ytem: u u u x x x x f ( x, u) v Matlab function x x x, x u u u n u m dx x x x Click on matlab function to chooe fun function dx=fun(v) x=v(); x=v(); xn=v(n) u=v(n+); u=v(n+); um=v(n+m); dx()=f(x,...,u, ) dx()=f(x,,u, ) dx(n)=f(x,,u, ) Simulation for a two-link pendulum l T l m g T 4 4 l u x x ; g mg x in x co x in( x l m l 4 g x in x x x ;.x x ).x m g function dx=ff(x) g=9.8; m=;m=;a=;a=; x=x();x=x(); x=x();x4=x(4); dx=-(g/a)in(x)+(mg/(ma))co(x)in(x-x)-.x; dx4=-(g/a)in(x)-.x4; dx=[dx;dx4] 6

17 At thi point, it i time to give a ummary on what we have achieved and what will be tudied Main Problem of the Coure Analyi: Solution to LTI ytem, tability etc. Controllability and obervability; Feedback deign and contruction of oberver Optimal control Lyapunov tability Coure project will involve feedback deign of an inverted pendulum ytem. Deign a feedback law through the linearized ytem Apply the feedback law to the nonlinear ytem Ue imulink to check if deirable performance requirement are atified. 4 7

18 Coure Project A cart with an inverted pendulum (page, Chen book) u M m u: control input, external force (Newton) y: diplacement of the cart (meter) angle of the pendulum (radian) State: y y x θ θ y The control problem are : Stabilization: Deign a feedback law u=fx uch that x(t) for x() cloe to the origin. : For x()=(,,), apply an impule force (u(t)=u max for t[, ]) to bring to a certain range and then witch to the linear controller o that x(t). Aume that there i no friction or damping. The nonlinear model i a follow. M m coθ ml coθ y u mlθ in θ l θ g in θ m kg : ma of the pendulum l.m : length of the pendulum 5 M 5kg : ma of the cart, g 9.8 Linearize the ytem at x=; M m ml y u l θ gθ y y x θ θ The tate pace decription for the linearized ytem x Ax Bu Problem:. Find matrice A, B for the tate pace equation.. Deign a feedback law u=f x o that A+BF ha eigenvalue at -±j;-.5 and -5. Build a imulink model for the cloed-loop linear ytem. Plot the repone under initial condition x()=[.5,,,-].. Build a imulink model for the original nonlinear ytem, verify that tabilization i achieved by u=f x when x() i cloe to the origin. Find the maximal o that the nonlinear ytem can be tabilized from x()=(,,,). 4. For x()=(,,compare the repone y(t) and (t) for the linearized ytem and the nonlinear ytem under the ame feedback u=f x. 6 8

19 5. Aume that the initial condition i x()=(,,,). For the nonlinear ytem, contruct a witching law to bring the pendulum upward and tabilized at x=. (cart till at y=, pendulum inverted, =). An initial impule control i applied with u(t)= u max for t(,t ] and u(t)= for t t. After the angle i within a mall range, i.e., d, witch to a linear controller u=f x. Find u max,t, d, and F o that the following requirement are atified: ) y(t) for all t> or keep the maximal y a mall a poible. ) y(t). for t >.5. ) u 5 for all t>. Note: In all the imulation, pleae chooe a fixed ampling period:.econd 7 Some guideline: The imulink model Ue a matlab function to realize the nonlinear/linear model Ue a matlab function to realize the witching control law You may ue a not o good control law to check if your imulink model i built correctly. u x.77y.8y

20 v Sample deign reult: y (meter) By a good controller time (econd) y (meter) Controller not o good time (econd) 5 u (newton) 5-5 An animation code will be provided time (econd) 9 Continuou pow ergui + - Coure project: a new option-- Ripple reduction in power converter Clock t To Workpace vc To W9 i To W5 Vment Mofet6 R RL To W v + v - Vment To W7 v6 v5 To W6 R R L R V d g m D Mofet S To W4 v T m Mofet5 g m g m D Mofet4 k Gain L C g m g D S D S L C Contant.5 Mofet g m D S Mofet m S D S v7 To W8 Add4 To W4 -u Pule Signal PWM Fcn Sine Wave Signal() Pule PWM Saturation Add k Gain k Integrator Gain Add Contant 4, vce To W Ku Ku Gain4 Add n)=cx(n)+du( +)=Ax(n)+Bu Dicrete State-Space xe 4 Gain To W

21 Everything outide the red box i provided: V=4;hh=.;%(hh i ampling time for imulation) %boot converter parameter R=.;RL=.67;L=.;C=.; %inverter parameter; R=.56;L=.;R=.5;L=.68; C=.;R=5; %Mofet parameter Ron=.7;Rd=e-4;Vf=.;Rn=e6;(nubber reitance);c=e-6; k=-.4;k=-.6;k=-; PWM frequency=hz, inuoidal function frequency = 6Hz Duty cycle for the boot converter < d <.75 4 The boot converter i deigned o that the dc-link voltage vc will track a reference voltage,,. In the figure,, 4. When an inverter i connected a a load, there will be ripple in vc, with frequency twice that of the inverter output voltage. Objective:. Deign a 5 th -order dicrete-time oberver, to etimate the firt order and econd order harmonic of vc. (ee lide ). For computer implementation, the oberver i dicretized with ampling frequency Hz (ame a PWM frequency). Denote the etimated tate a vce.. Ue the firt order harmonic a feedback to reduce the ripple of vc The gain Kh will be Kh=[ k4 k5 ]. Chooe k4, k5 by trial and error to achieve minimal ripple. 4

22 Reult: Blue curve: dc link voltage: vc Red curve: etimated dc link voltage, vce Kh 5 Kh= We are going to learn how to deign a good control law. Before that, we need to tudy Controllability and obervability; We need ome background on linear algebra: poitive-definitene of a quare matrix. They are alo eential to Lyapunov tability and optimal control. 44

23 Quadratic function and poitive-definitene (.9 ) Given a ymmetric matrix P=P (p ij =p ji ). A quadratic function can be defined a V(x) = x Px Example: a b x V (x) x x ax bx x cx b c x x P x a d ex V (x) x x xd b f x e f c x ax bx cx dx x ex x For higher order vector pace, fx x V(x) x'px n i n j p x x ij i j 45 Definition: A ymmetric matrix P i aid to be poitive definite, denoted by P >, if x Px > for all x. It i aid to be poitive emidefinite, denoted by P, if x Px for all x. Under what condition i V(x)=x Px poitive definite? Thi depend on the eigenvalue of P. Compare the eigenvalue of a b b a and P b a P {a+jb, a-jb} for P, a c b c (a c) 4b for P 46

24 Theorem: A real ymmetric matrix ha real eigenvalue. Proof: Suppoe that i an eigenvalue, poibly complex, v i the eigenvector uch that Pv = v. The complex conjugate tranpoe of v i v, the complex conjugate tranpoe of P i P. We have (vpv)=vpv=vp v=vpv vpv mut be a real number. Alo recall that vv i a real number. From Pv=v, we have vpv=vv mut be a real number Theorem: A real ymmetric matrix P i alway diagonalizable. 47 Theorem: A ymmetric matrix P i poitive definite (P > ) if and only if it eigenvalue are all poitive. Proof: There exit diagonal matrix D and orthogonal matrix U uch that P=UDU. Conider the quadratic form z Dz. Have z Dz = d z + d z +. +d nn z n > for all z Let z=u x. z= iff x=. Hence x Px=x UDU x=z D z > for all x Theorem: A ymmetric matrix P i poitive definite iff there exit a noningular matrix N uch that P=NN. Proof:. 48 4

25 In ummary: Given a ymmetric matrix P. All the eigenvalue and eigenvector are real. Exit a matrix U, UU =U U=I, and a diagonal D, uch that P=UDU. P i poitive definite iff all eigenvalue are poitive; exit noningular N uch that P=NN ; P poitive emi-definite iff all eigenvalue are non-negative; exit N uch that P=NN ; P negative definite iff all eigenvalue are negative; exit noningular N uch that P= NN 49 Today: ome micellaneou problem about LTI ytem How to deal with complex eigenvalue Realization of a tranfer function Simulation of ytem by uing Simulink Coure project Quadratic function and poitive-definitene Next Time: Chapter 6. Controllability and Obervability 5 5

26 Problem et #8. Ue the firt definition of a matrix function to compute e At for A. Find a tate pace realization for G() Ue integrator and amplifier to contruct a Simulink model for it. Let the input be a tep ignal: u(t)= for t< and u(t)= for t >. Chooe the ampling time to be T=.. Simulate the output under initial condition and plot the output repone for t= to t=5. (print the model and the output repone). You can try different input ignal. 5. Contruct the imulink model on page (two link pendulum) and run imulation from t= to t=, with initial condition x()=(.5,,-,). Chooe ampling time=.econd. Plot the two output theta and theta. 5 6