Matlab信号与系统应用

拉氏变换： help laplace --- help for sym/laplace.m --- LAPLACE Laplace transform. L = LAPLACE(F) is the Laplace transform of the scalar sym F with default independent variable t. The default return is a function of s. If F = F(s), then LAPLACE returns a function of t: L = L(t). By definition L(s) = int(F(t)*exp(-s*t),0,inf), where integration occurs with respect to t. L = LAPLACE(F,t) makes L a function of t instead of the default s: LAPLACE(F,t) <=> L(t) = int(F(x)*exp(-t*x),0,inf). L = LAPLACE(F,w,z) makes L a function of z instead of the default s (integration with respect to w). LAPLACE(F,w,z) <=> L(z) = int(F(w)*exp(-z*w),0,inf). Examples: syms a s t w x laplace(t^5) returns 120/s^6 laplace(exp(a*s)) returns 1/(t-a) laplace(sin(w*x),t) returns w/(t^2+w^2) laplace(cos(x*w),w,t) returns t/(t^2+x^2) laplace(x^sym(3/2),t) returns 3/4*pi^(1/2)/t^(5/2) laplace(diff(sym('F(t)'))) returns laplace(F(t),t,s)*s-F(0) See also ILAPLACE, FOURIER, ZTRANS. 拉氏逆变换： >> help ilaplace --- help for sym/ilaplace.m --- ILAPLACE Inverse Laplace transform. F = ILAPLACE(L) is the inverse Laplace transform of the scalar sym L with default independent variable s. The default return is a function of t. If L = L(t), then ILAPLACE returns a function of x: F = F(x). By definition, F(t) = int(L(s)*exp(s*t),s,c-i*inf,c+i*inf) where c is a real number selected so that all singularities of L(s) are to the left of the line s = c, i = sqrt(-1), and the integration is taken with respect to s. F = ILAPLACE(L,y) makes F a function of y instead of the default t: ILAPLACE(L,y) <=> F(y) = int(L(y)*exp(s*y),s,c-i*inf,c+i*inf). Here y is a scalar sym. F = ILAPLACE(L,y,x) makes F a function of x instead of the default t: ILAPLACE(L,y,x) <=> F(y) = int(L(y)*exp(x*y),y,c-i*inf,c+i*inf), integration is taken with respect to y. Examples: syms s t w x y ilaplace(1/(s-1)) returns exp(t) ilaplace(1/(t^2+1)) returns sin(x) ilaplace(t^(-sym(5/2)),x) returns 4/3/pi^(1/2)*x^(3/2) ilaplace(y/(y^2 + w^2),y,x) returns cos(w*x) ilaplace(sym('laplace(F(x),x,s)'),s,x) returns F(x) See also LAPLACE, IFOURIER, IZTRANS. 实例1： 传递函数：G(s)=(2*s+1)/(3*s^2+4*s+1)求拉氏逆变换。 解： syms s; G=(2*s+1)/(3*s^2+4*s+1); Gt=ilaplace(G) 结果：Gt=1/2*exp(-t)+1/6*exp(-1/3*t) 实例2：求传递函数: s^3 + 5 s^2 + 9 s + 7 ------------------------- s^2 + 3 s + 2 的拉氏逆变换。 解： >> Gt=(s^3+5*s^2+9*s+7)/(s^2+3*s+2); >> ilaplace(Gt) ans =dirac(1,t)+2*dirac(t)-exp(-2*t)+2*exp(-t) 即: δ(t)+2*δ(t) -exp(-2*t)+2*exp(-t) 冲激函数： >> help dirac DIRAC Delta function. DIRAC(X) is zero for all X, except X == 0 where it is infinite. DIRAC(X) is not a function in the strict sense, but rather a distribution with int(dirac(x-a)*f(x),-inf,inf) = f(a) and diff(heaviside(x),x) = dirac(x). See also heaviside. 阶跃函数： >> help heaviside HEAVISIDE Step function. HEAVISIDE(X) is 0 for X < 0, 1 for X > 0, and NaN for X == 0. HEAVISIDE(X) is not a function in the strict sense. See also dirac. 一些常用函数的拉氏变换Matlab实现： 参见《信号与系统》第二版（上册） 郑君里，P181 表 关于同志近三年现实表现材料材料类招标技术评分表图表与交易pdf视力表打印pdf用图表说话 pdf 4-1 1．冲激函数：δ(t) >> laplace(dirac(t)) ans =1 2．阶跃函数：u(t) >> laplace(heaviside(t)) ans =1/s 3．指数函数：exp(-a*t) >> laplace(exp(-a*t)) ans =1/(s+a) 4．幂函数：t^n（n是正整数） 如:t^5 >> laplace(t^5) ans =120/s^6 注:不能直接用laplace(t^n),此时n应该是一个具体的数值. 5．正弦函数：sin(w*t) >> laplace(sin(w*t)) ans =w/(s^2+w^2) 6．余弦函数：cos(w*t) >> laplace(cos(w*t)) ans =s/(s^2+w^2) 7．指数函数乘正弦函数：exp(-a*t)*sin(w*t) >> laplace(exp(-a*t)*sin(w*t)) ans =w/((s+a)^2+w^2) 8．指数函数乘余弦函数：exp(-a*t)*cos(w*t) >> laplace(exp(-a*t)*cos(w*t)) ans = (s+a)/((s+a)^2+w^2) 9．斜坡函数乘指数函数：t*exp(-a*t) >> laplace(t*exp(-a*t)) ans =1/(s+a)^2 10．幂函数乘指数函数：t^n*exp(-a*t) >> laplace(t^5*exp(-a*t)) ans =120/(s+a)^6 注:不能直接用laplace(t^n),此时n应该是一个具体的数值,此例中n=5。 11．斜坡函数乘正弦函数：t*sin(w*t) >> laplace(t*sin(w*t)) ans =2/(s^2+w^2)^2*s*w 12．斜坡函数乘余弦函数：t*cos(w*t) >> laplace(t*cos(w*t)) ans =1/(s^2+w^2)^2*(s^2-w^2) 13．双曲正弦函数：sinh(a*t) >> laplace(sinh(a*t)) ans =a/(s^2-a^2) 14．双曲余弦函数：cosh(a*t) >> laplace(cosh(a*t)) ans =s/(s^2-a^2) 解微分方程： 求以下微分方程的解：dy/dt+a*y=0，y(0)=1。 解法一（常用方法）： >> syms a y t >> y=dsolve('Dy+a*y=0','y(0)=1','t') y =exp(-a*t) 解法二（拉氏变换法）： MATLAB实现： >> syms a y t >> z0=diff(sym('y(t)'))+a*sym('y(t)'); >> z=laplace(z0) >> F1=subs(z,'laplace(y(t),t,s)',sym('Y')) >> Y=solve(F2,sym('Y')) >> ilaplace(Y) ans =exp(-a*t) PAGE 4