Linear System Theory and Design Chapter 4

Chapter 4 dot x = [[0, i], [- i, 0]] * x = Ax A(i) = i * d*epsilon*t * [[A, - i], [i, lambda]] = lambda ^ 2 + i, lambda = epsilon*j f(lambda) = e ^ (lambda*t) ۲۰ دامدا ) (۲۱۶در 1=j e^ alpha*beta_{0} +j emptyset, j_{x} = (e ^ (jt) - e ^ (- j * t))/(lambda_{j} ^ j) = sin t beta 1 = phi^ it - 1 beta_{i} = se * s ^ 2 * t AEC Ient A Thus x(t)=[ matrix cos t&sin t\\ -sin t& cint matrix ]x( theta) 4.2 Fiind unit step response of x' = [[0, t], [- 2, - 2]] * x + [[1], [1]] * u y = r 23] 2 Method Laplace trous foun hat f(s) = [[2, 3]] * [[5, - i], [2, 5 + 2]] ^ - 1 * [[1], [1]] 231 1/(sigma ^ 2 + 2s + 2) * [[s + 2, 1], [- 2, 5]] * [[1], [1]] = (5s)/(s ^ 2 + 2s + 2) hat J(s) = j - 5 hat g (s)= hat g ( omega) hat u(s) = (zeta*s)/((s + 1) ^ 2 + 1) * 1/s y(t) = s ^ (- e ^ (- tau)) * sin t Method 2 doing (4.7) 2 1- lambda = - 1 plus/minus j f(lambda) = e ^ (lambda*tau) K(lambda) = beta_{c} + beta_{1}*lambda Aj etrist Be + P.(-1-j) A=-rj e ^ ((- 1 + j) * Gamma) = beta_{c} + beta_{1}(- 1 + j) Rightarrow beta_{0} * e ^ (- t) * sin t beta 1 = e ^ (- tau) ( mu dot r t+cot) rho*AF = rho_{rho} plus/minus tau * beta_{rho} * [[0, i], [- 2, - 2]] =[ matrix e^ - epsilon (sin epsilon+ varphi i epsilon)&e^ - epsilon sin epsilon \\ -2e^ - epsilon sin epsilon& e ^ (- epsilon) * (cos epsilon - gin*epsilon) matrix ] u( phi)=1 p forall 0 >= 0 g(t)=[ matrix 2& 37 * integrate e ^ (a(t - tau)) * [[i], [i]] * 9 dtau from 0 to t = int 0 ^ z (se^ -it-z) cos(t - z) + - s * e ^ (- (t - z)) * dim(t - c) )dc Consider S * integrate e ^ (- (t - tau)) * cos(t - tau) dtau from 0 to pi =-s int 0 ^ pi e^ -(t- tau) d id tau dos(t- tau)d tau =x[e^ -(t- tau) d omega( epsilon- tau) int t= epsilon ^ tau - integrate e ^ (- (tau - tau)) * sin(t - tau) dtau from 0 to tau ] Thus we have y(epsilon) = - S * [e ^ (- theta) * sin D - e ^ (- t) * sin t] = 5e ^ (- t) * sin Omega 4.3 claing computed in Prob. 4. For Tol d d =e^ aT =[ matrix e^ -1 (4 omega t+cos t)&e^ -1 4 omega t\\ -2e^ -1 +cos t& e ^ - 1 * (cos t - 4omega*t) matrix ] 0.5083 0.3046 -0,6191-0.1108 Becauss A is nenimgulas (el has no zero sipenvalue), we may use (4.18) l compute b_{d} = A ^ - 1 * (A_{d} - I) * b =[ matrix 1,00471\\ -0,1821 matrix ] 114 Thus the discretized equation with T 05083 0.3096 *[61] -0,6191 -0,4191-01108 y[k] /[2 3]x[k] For T = pi we have [+1]=0,0432 0 -00432 y( ff ,1=r2 ]x[] 1,0471 4) 1,5648 1611.0432 4.5 dot x = [[- 2, 0, 0], [i, 0, i], [0, - 2, - 2]] * chi + [[i], [0], [i]] * 4 y = [[1 - 1, 0]] * x Companion form Q=[ matrix b&A dot b &A^ 2 dot b matrix ]= [[i, - z, 4], [0, 2, - 4], [i, - 2, 0]] P = Q ^ - 1 =[ matrix 1&i&0\\ 0,5&0,5&-0,5\\ 8,aS&0&- rho,i5 matrix ] tilde overline x =Q^ -1 A Q overline x + a ^ - 1 * bu = [[0, 0, - 4], [1, 0, - 6], [0, 1, - 4]] overline pi +[ matrix i\\ 0\\ 0. matrix ]u q= angle Q overline x = [1 - 48] overline x Modal form Eigenvatices - 1 + j, - 1 - j, - 2 K 1 =[ matrix 0.57744 j \\ 2.5774 - 6.5794j \\ 577794-6.5794j\\ 57794+4.5774+4.5779+j matrix ] Enginvectors F x =[ matrix 0.7e&71\\ 0\\ - g * 0.98871 matrix ] Q = [[theta_{e}(v_{i}), Ion(v_{i} ^ gamma), v_{3}]] =[ matrix 0&0&e.70 gamma1\\ 0&0.577w&0\\ -0.577w-e.59 eta w-e.70 gamma i matrix ] PQ= -1.7319 -1.7319 -1.7319 C 11.4142 1.7319 0 dot overline x =Q^ -1 AQ overline x + Q ^ - 1 * bu + 00-2 0 0 y=c R overline x = [0 - e * 0.5774 * 0.7671] vec x +2 01 0-22 y = [[1, - 1, 0]] * x for unit step imput ( u(t) = 1 , fortzo), we the MATLAB 20 |y| max = 0.55 . |x_{2}| stex =1.05 ol-lain |x r | max = c ,s |x J | max = 0.52 der x=x tilde x_{2} = 0.5x_{2} overline x =[ matrix t&0&e\\ e&e.5&0\\ e&e&t matrix ] x = Px P ^ - 1 = [[t, 0, 0], [0, 2, 0], [0, 0, t]] dot overline x =PAP^ - prime overline x + Pbw ,y=eP^ -1 overline x a overline overline x =[ matrix -2&0&0\\ 0,5&0&e,5\\ 0&-q&-2 matrix ] overline x + [[9], [0], [i]] * 4 y= [[1, - 2, 0, 0]] tilde x For this equation, we have |g| max = 0.55 ,| dot i r | max =0.5 | tilde x 2 | avx = 0.525 171mx 0.52 The step imput must have magnitude less than (10 / e * 0.55) = 18.256 avoid asturation. overline x_{j} = x_{j} -15-