多项式的增广图示及其在工程控制论中的应用

本文图示复变数s(=x+iy)的实系数多项式K=f(s)≡a₀sn+a₁sn-1+……+a_n-1s+a_nK是实参数,因此上式的图示表为(x,iy,K)中的一集空间曲线.这曲线在三个坐标平面上的投影就是本文图示的内客.在(x,iy)上的投影就是根轨迹.不论n=2m+1或n=2m+2,根轨迹方程都是y2的m次方程.(K,x)图线除了包含实曲线K_r=f(x)以外,尚包含复根的实数部随K变化的曲线,这是新增的曲线.(K,x)曲线对判别系统的绝对和相对稳定性是很有用的.(K,iy)曲线对控制系统来说,表示放大K和自然频率ω(≡y)的关系曲线.这三幅图线可应用于方程式论和工程控制论.

Extended Graphical Representation of Polynomials with Applications to Cybernetics

In this paper,the polynomial of a complex variable s(≡x+iy) with real coefficients K=f(s)≡a₀sn+a₁sn-1+……+a_n-1s+a_n is graphically represented by three plane curves which are the projections of a space curve on three coordinate planes of the coordinate system(x,iy,K)in which K is confined to be real,The projection on(x,iy)plane is just the root locus of polynomial with K as a real parameter,It is remarkable that the equation of the root locus is mth degree of y2,whether。n=2m+1 or n=2m+2.In addition to the real curve K_r=f(x) in the figure(K,x),there exists another curve K_c which is plotted by the real parts of all complex roots against those of K.The(K,iy)curve is particularly important to determine the absolute as well as the relative stability interval of K for linear systems,For cybernetics,the(K,iy)curve can be used to show the relation between the natural frequency ω and the gain K.Such three figures are useful for studying the theory of equation and cybernetics.