Fundctrlsys chapter2

Lecture Notes Fundamentals of Control Systems Instructor: Assoc Prof Dr Huynh Thai Hoang Department of Automatic Control Faculty of Electrical & Electronics Engineering Ho Chi Minh City University of Technology Email: hthoang@hcmut.edu.vn huynhthaihoang@yahoo.com Homepage: www4.hcmut.edu.vn/~hthoang/ December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ Chapter Mathematical Models of Continuous Control Systems December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ Content       The concept of mathematical model Transfer function Block diagram algebra Signal flow diagram State space equation Linearized models of nonlinear systems  Nonlinear state equation  Linearized equation of state December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ The concept of mathematical models December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ Question  If you design a control system system, what you need to know about the plant or the process to be controlled?  What are the advantages of mathematical models? December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ Why mathematical model?    Practical control systems are diverse and different in nature nature It is necessary to have a common method for analysis and design of different type of control systems  Mathematics The relationship between input and output of a LTI system of can be described by linear constant coefficient equations: u(t) Linear TimeInvariant System y(t) dy (t ) d n 1 y (t ) d n y (t )  a1    an 1  an y (t )  a0 n n 1 dt dt dt d mu (t ) d m 1u (t ) du (t ) b0  b1    bm 1  bmu (t ) m m 1 dt dt dt n: system order, order for proper systems: nm m ai, bi: parameter of the system December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ Example: Car dynamics dv (t ) M  Bv (t )  f (t ) dt M: mass of the car, car B friction coefficient: system parameters f(t): engine driving force: input v(t): car speed: output December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ Example: Car suspension d y (t ) dy (t ) M B  Ky (t )  f (t ) dt dt M: equivalent mass B friction constant, K spring stiffness f(t): external force: input y(t): (t) travel t l off the th car body: b d output t t December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ Example: Elevator ML Cabin & load MB Counterbalance ML: mass of cabin and load, MB: counterbalance t b l B friction constant, gear box constant Kg (t): driving moment of the motor y(t): position of the cabin d y (t ) dy (t ) ML B  M T g  K (t )  M B g dt dt December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ Disadvantages of differential equation model  Difficult to solve differential equation order n (n>2) d n y (t ) d n 1 y (t ) dy (t ) a0  a1    an 1  an y (t )  n n 1 dt dt dt d mu (t ) d m 1u (t ) du (t ) b0 b b      bmu (t ) m 1 m m 1 dt dt dt  System analysis based on differential equation model is difficult  System design based on differential equations is almost impossible in general cases  It is necessary to have another mathematical model that makes the analysis and design of control systems easier:  transfer function  state space equation December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ 10 State space model of nonlinear system – Example State  l u  m  Differential equation: (t )   B (t )  (ml  MlC ) g cos   u (t ) 2 ( J  ml ) ( J  ml ) ( J  ml )  x1 (t )   (t )  Define the state variable:    x2 (t )   (t ) State equation: q where  x (t )  f ( x (t ), u (t ))  ) u (t ))  y (t )  h( x (t ),  x2 (t )   B f ( x , u )   ( ml  MlC ) g x2 (t )  u (t )  cos x1 (t )   2 ( J  ml ) ( J  ml )  ( J  ml )  h( x (t ), u (t ))  x1 (t ) December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ 106 Equilibrium points of a nonlinear system  Consider a nonlinear system described by the diff equation:  x (t )  f ( x (t ), u (t ))   y (t )  h( x (t ), u (t ))  The state x is called the equilibrium point of the nonlinear system if the system is at the state x and the control signal is fixed at u then the system will stay at state x forever  If ( x , u ) is equilibrium point of the nonlinear system then: f ( x (t ), u (t )) x  x ,u u   The equilibrium point is also called the stationary point of the nonlinear system December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ 107 Equilibrium point of nonlinear system – Example  Consider a nonlinear system described by the state equation:  x1 (t )   x1 (t ).x2 (t )  u   x (t )   x (t )  x (t )      Find the equilibrium point when u (t )  u   Solution: The equilibrium point(s) are the solution to the equation: f ( x (t ), u (t )) x  x ,u u    x1.x2     x1  x2    x1     x2   December 2013 or  x1      x2   © H T Hoang - www4.hcmut.edu.vn/~hthoang/ 108 Equilibrium point of nonlinear system – Example  C Consider id a nonlinear li system t d described ib d by b the th state t t equation: ti 2   x  x  x1  u   x    x3  sin( x1  x3 )    x   x3  u   y  x1 Find the equilibrium point when u (t )  u  December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ 109 Linearized model of a nonlinear system around an equilibrium point  Consider a nonlinear system described by the diff diff equation:  x (t )  f ( x (t ), u (t ))  ) u (t ))  y (t )  h( x (t ),  (1) Expanding g Taylor y series for f(x,u) ( ) and h(x,u) ( ) around the equilibrium point ( x , u ) , we can approximate the nonlinear system (1) by the following linearized state equation:  x~ (t )  Ax~(t )  Bu~(t ) ~ ~ ~  y (t )  Cx (t )  Du (t ) where: x~ (t )  x (t )  x u~ (t )  u (t )  u ~ y (t )  y (t )  y December 2013 (2) ( y  h( x , u )) © H T Hoang - www4.hcmut.edu.vn/~hthoang/ 110 Linearized model of a nonlinear system around an equilibrium point  The matrix Th t i off the th linearized li i d state t t equation ti are calculated l l t d as follow:  f1  x   f A   x1    f n   x1 f1 x2 f x2  f n x2  h C  x1 h x2 December 2013     f1  xn   f  xn    f n   xn  ( x,u ) h    xn  ( x,u )  f1   u   f   B  u        f n   u  ( x,u )  h  D   u  ( x,u ) © H T Hoang - www4.hcmut.edu.vn/~hthoang/ 111 Linearized statestate-space model – Example The parameter of the tank: u(t)  qin a  1cm , A  100cm y(t) qout Nonlinear state equation: where k  150cm3 / sec V , CD  0.8 g  981cm / sec2  x (t )  f ( x (t ), u(t ))  ) u(t ))  y (t )  h( x (t ), aCD g gx1 (t ) k f ( x, u )    u (t )  0.3544 x1 (t )  0.9465u (t ) A A h( x (t ), u (t ))  x1 (t ) December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ 112 Linearized statestate-space model – Example (cont’) Linearize the system around y = 20cm:  The equilibrium point: x1  20 f ( x , u )  0.3544 x1  1.5u  December 2013  u  0.9465 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ 113 Linearized statestate-space model – Example (cont’)  The matrix of the linearized state-space state space model: aCD g f1 A  x1 ( x,u ) A x1  0.0396 ( x,u ) f1 k B   1.5 u ( x,u ) A ( x,u )  h C 1 x1 ( x,u ) h D 0 u ( x,u ) The linearized state equation describing the system around the equilibrium point yy=20cm 20cm is: aCD gx1 (t ) k f (~x , u )   ~  u (t ) ~  x (t )  0.0396 x (t )  1.5u (t ) A ~ ~ y ( t )  x (t ) h( x (t ), u (t ))  x1 (t )  December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ A 114 Linearized statestate-space model – Example The parameters of the robot: l u l  0.5m, lC  0.2m, m  0.1kg m M  0.5kg , J  0.02kg.m  B  0.005, g  9.81m / sec2  Nonlinear state equation : where:  x (t )  f ( x (t ), u (t ))  ) u (t ))  y (t )  h( x (t ),  x2 (t )   B f ( x , u )   ( ml  MlC ) g cos x1 (t )  x2 (t )  u (t )  2 ( J  ml ) ( J  ml )  ( J  ml )  h( x (t ), u (t ))  x1 (t ) December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ 115 Linearized statestate-space model – Example (cont’) Linearize the system around the equilibrium point y = /6 (rad):  Calculating the equilibrium point: x1   /  x2  0 B f ( x , u )   (ml  MlC ) g cos x1  x2  u  2 ( J  ml ) ( J  ml )   ( J  ml )  x2    u  1.2744 Th the Then th equilibrium ilib i point i t is: i  x1   / 6 x   x   2  u  1.2744 December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ 116 Linearized statestate-space model – Example (cont’)  The system matrix around the equilibrium point: f1 0 a11  x1 ( x,u )  a11 a12  A  a a  21 22  f1 a12  x2 1 ( x,u ) (ml  MlC ) f a21   sin x1 (t ) x1 ( x,u ) ( J  ml ) ( x,u ) f a22  x2 ( x,u ) B  ( J  ml ) ( x,u )  x2 (t )   B f ( x , u )   ( ml  MlC ) g cos x1 (t )  x2 (t )  u (t )   2 ( J  ml ) ( J  ml )  ( J  ml )  December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ 117 Linearized statestate-space model – Example (cont’)  The input matrix around the equilibrium point:  b1  B  b2  ff1 b1  0 u ( x,u ) f b2  u ( x,u )  J  ml   x2 (t )  B f ( x , u )   ( ml  MlC ) g  cos x ( t )  x ( t )  u ( t )   2 2  J  ml ( J  ml ) ( ) ( ) J ml   December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ 118 Linearized statestate-space model – Example (cont’)  The output matrix around the equilibrium point: C  c1 c2  D  d1 h c1  1 x1 ( x,u ) d1  c2  h x2 0 ( x,u ) h 0 u ( x,u )  x~ (t )  Ax~ (t )  Bu~ (t )  Then the linearized state equation is:  ~ ~ ~ (t ) y ( t )  C x ( t )  D u   0 A  a a  21 22  0 B  b2  C  1 0 D0 h( x , u )  x1 (t ) December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ 119 Regulating nonlinear system around equilibrium point  Drive the nonlinear system to the neighbor of the equilibrium point (the simplest way is to use an ON-OFF controller)  Around the equilibrium point, use a linear controller to maintain the system around the equilibrium point r(t) + e(t) Linear control  u(t) Nonlinear system y(t) ON-OFF Mode select December 2013 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ 120