Root hodograph

The root hodograph is the trajectory in control theory , described on the complex plane by the poles of the transfer function of a dynamic system when one of its parameters changes. The usually modifiable parameter is the gain of the system. Root hodographs are widely used in the analysis and synthesis of linear SISO systems.

Usually root hodographs are used in the analysis of system stability .

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Root Hodograph Method

An example of a root hodograph of a system

W(s)={\frac {(s+1)(s+2)(s+3)}{s^{2}(s+0,4)(s+0,5)(s+0,6)}}

{\ displaystyle W (s) = {\ frac {(s + 1) (s + 2) (s + 3)} {s ^ {2} (s + 0.4) (s + 0.5) (s +0.6)}}}

.

Let the transfer function of a closed system

W(s)={\frac {A(s)}{B(s)}}

{\ displaystyle W (s) = {\ frac {A (s)} {B (s)}}}

,

and the order of the polynomial of the numerator is $m$ ${\ displaystyle m}$ , the order of the polynomial of the denominator is $n\!,m\leq n$ ${\ displaystyle n \ !, m \ leq n}$ for physically feasible systems .

The root hodograph method associates the dynamic characteristics of a system with the behavior of the zeros and poles of its transfer function, which are found from the zeros and poles of an open system when a parameter (usually the gain of an open system) changes. A closed system is connected to an open system using the following relation:

W(s)={\frac {W_{\Pi }}{1+W_{p}}}

{\ displaystyle W (s) = {\ frac {W _ {\ Pi}} {1 + W_ {p}}}}

Where $W_{\Pi }$ ${\ displaystyle W _ {\ Pi}}$ - transfer function of the direct system, $W_{p}$ ${\ displaystyle W_ {p}}$ - transfer function of an open system. This formula is valid only for negative feedback, otherwise the sign after the unit will be negative. Let the point $s$ ${\ displaystyle s}$ is a closed system pole. Draw a vector of all zeros to this point $W_{p}$ ${\ displaystyle W_ {p}}$ open system (denote the arguments of these vectors $\theta _{j}^{0}$ ${\ displaystyle \ theta _ {j} ^ {0}}$ ) and all poles $W_{p}$ ${\ displaystyle W_ {p}}$ (the arguments of these vectors are denoted by $\theta _{j}^{P}$ ${\ displaystyle \ theta _ {j} ^ {P}}$ ) Then the root locus will be the locus of the points satisfying the following equation:

\sum _{j=1}^{n}\theta _{j}^{0}-\sum _{j=1}^{n}\theta _{j}^{P}=\pm (2u+1)\pi ,u=0,1,2,\dots

{\ displaystyle \ sum _ {j = 1} ^ {n} \ theta _ {j} ^ {0} - \ sum _ {j = 1} ^ {n} \ theta _ {j} ^ {P} = \ pm (2u + 1) \ pi, u = 0,1,2, \ dots}

The method of the root hodograph allows you to choose the gain of the control system, evaluate the oscillation of movement, choose the location of the zeros and poles of the correcting links of the control system .

Root Hodograph Properties

Consider the properties of the root hodograph when changing the gain:

The branches of the root hodograph are continuous and symmetrical about the real axis of the complex plane.
The number of branches of the root hodograph is equal to the order of the system $n$ ${\ displaystyle n}$ .
Branches begin at the poles of an open system (since at zero gain $K$ ${\ displaystyle K}$ the poles of open and closed systems coincide). With increasing $K$ ${\ displaystyle K}$ from 0 to infinity, the poles of a closed system move along the branches of the root hodograph.
Since when $K=\infty$ ${\ displaystyle K = \ infty}$ the poles of a closed system become equal to the zeros of the open system, then exactly $m$ ${\ displaystyle m}$ branches of the root hodograph ends at the zeros of the closed system, and the remaining branches go to infinity.
A closed system is stable if its poles lie in the left half-plane of the root plane. Accordingly, when the branches of the hodograph cross the imaginary axis from left to right, the system from stable becomes unstable. The gain corresponding to this transition is called critical . This property is useful in assessing the stability of a system.

External links

E.V. Nikulchev . Control Systems Toolbox Tutorial

Root hodograph

Content

Root Hodograph Method

Root Hodograph Properties

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