1. Linearization of a nonlinear system given in the form of a system of ordinary differential equations

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1 . Liearizatio of a oliear system give i the form of a system of ordiary differetial equatios We ow show how to determie a liear model which approximates the behavior of a time-ivariat oliear system i a small regio aroud a equilibrium poit. This is importat sice there are umerous rigorous methods for developig cotrollers for liear systems ad these are widely accepted ad used i idustry. Eve if most systems are oliear it is usually required to cotrol such a system i a small regio of operatio where its behavior is approximately liear. (take for example a power system which eeds to operate at 5Hz. Let a dyamical system be described by the first order oliear differetial equatios = f( xu, y= hxu (, ( where x R is the iteral state vector, the measured outputs vector. m m p m u R is the cotrol iputs vector ad f : R R R, h: R R R deote geeral oliear fuctios. p y R is To determie a liear model of the oliear system we will use the first two terms of the Taylor series of the two oliear fuctios f( xu, ad hxu (,. Let s deote the operatig poit where we wat to obtai the liear model with The retaiig oly the first two terms of the Taylor series we get where f ( x, u ( x, u ( x, u ( x, u. f f f( xu, f( x, u + ( x x + ( u u u ( h h hxu (, hx (, u + ( x x + ( u u u (3 ( x, u ( x, u reads the derivative of f with respect to x evaluated at the operatig poit ( x, u. f Note that f( xu, is a vector valued fuctio R. The is i fact the Jacobia matrix (a matrix of fuctios which ca be explicitly writte as df df dx dx f =. df df dx dx

2 f h h Similarly oe ca write the Jacobia matrices,, u u : df df dh dh dh dh du du m dx dx du dum f h h = ; = ; = u x u df df dhp dh p dhp dh p du dum dx dx du du m These matrices are i fact matrices of fuctios depedig o the variables x ad u. Whe these matrices are evaluated at the omial poit ( x, u the oe obtais costat matrices ABCD.,,, For the case i which model of the form ( x, u = (, ad f(, ; h(, = = oe obtais the liear = Ax+ Bu. (4 y= Cx+ Du We have aalyzed the properties of systems described by (4 i Lectures, 3 ad 4. If a system is described by a set of oliear differetial equatio of order q>, the it ca be trasformed i a set of xq first order differetial equatios by defiig more states. Say for example that the system is described by x= f( xxx,, with x R, q = 3 the by deotig z= x; z= ; z3= x oe ow ca write z = = z z = x= z3 z 3= x= f( z, z, z3 which is the descriptio of the same system but ow usig a set of xq first order differetial equatios. A equilibrium poit of a system is defied by costat values of the iput ad the state (all derivatives are equal with zero. I the case of the above example the equilibrium poit is defied by z = z= z = z3= z 3= f( z, z, z3 = Thus is defied as ( z, z, z3 = ( z,, where z satisfies f( z,, =.

3 Exercise A secod-order differetial equatio of the sort occurrig i robotic systems is mq + mlq + mgl si q = τ where q(t is a agle ad τ(t is a iput torque. To solve the secod-order differetial equatio oe requires two iitial coditios q(, q (. These state compoets correspod to eergy storage variables. I this case oe could thik of potetial eergy mgh (the third term i the differetial equatio, which ivolves q (t, ad rotatioal kietic eergy m ω (the secod term, which ivolves ω = q (t. Defiig the state variables x = q, x = q, give a expressio of the system dyamics i the form of two first order ordiary differetial equatios. Calculate the equilibrium poit of the system. Obtai a liear model of this oliear system about the equilibrium poit. The fuctio f(x,u is give by the dyamics of the system. The output fuctio h(x,u is give by which measuremets the egieer decides to take. Suppose we decide to measure the robot joit agle x = q. The the output fuctio is give as y = h( x, u = [ ]x. Problem Obtai the liear model of the system described by y+ yy + y = u about the equilibrium poit ( y, y, u = (,,. x y Defie the system state as x= x = y. Problem Obtai the liear model of the system described by y+ α( y y + y= aroud the equilibrium poit. x y Defie the system state as x= x = y. 3

4 Problem 3 The system of equatios = ax bx x cx = dx ex x fx describes the growth of two competig species that prey o each other. The costats abcde,,,,, f are positive parameters ad it is assumed that x, x. Determie the liear model of the system aroud the equilibrium poit.. Solutio of a ordiary differetial equatio usig umerical itegratio The two methods outlied below provide umerical approximate solutios for first order ordiary differetial equatios such as = f( xu,, = Ax+ Bu or z = z z = z3 z 3= f( z, z, z3 give iitial coditios ad the iput sigal ut (. Notice that these equatios actually provide iformatio o the way i which the solutio, xt (, or zt ( = [ z( t z( t z3(] t T, varies with time. A. Euler s method Euler s method is of limited value for solvig iitial value problems umerically sice it gives arbitrarily close approximatios to the solutio by makig the itegratio step sufficietly small, which i tur ca lead to umerical istabilities. However, we choose to preset it sice it provides the most basic ituitio o how umerical itegratio methods work. For simplicity we preset the scalar versio. Give the scalar iitial value problem a approximatio to the solutio is give by y = f(, t y, yt ( = y (5 yk+ = yk + hf( tk, yk (6 where h= tk+ tk is a costat itegratio step. Notice that this meas that we replaced the ordiary differetial equatio (5 by a differece equatio (6. 4

5 B. Ruge-Kutta method The Ruge-Kutta itegratio methods are amog the most widely used techiques to solve iitial value problems. A simple versio is give by yi+ = yi + k where ad k= ( k+ k+ k3+ k4 (7 6 k = hf( t, y i k= hf( ti+ hy, i+ k k3= hf( ti+ hy, i+ k k = hf( t + hy, + k 4 i i 3 i with h= ti+ ti. The idea is to probe ahead i time by oe half or by a whole step h to determie the values of the derivative at several poits ad the form a weighted average. (8 The ext lecture is orgaized as a laboratory sessio as its goal is to give the opportuity of buildig some practical experiece with usig the Matlab software for simulatio of liear ad oliear systems. The Matlab fuctios that you will use are implemetig itegratio methods such as the oes that were preseted above. 5