Openloop and Closedloop Characteristic Equations

The characteristic equation of any system, closedloop or openloop, is the equation that you get when you take the denominator of the transfer function describing the system and set it equal to zero. The resulting Nth-order polynomial 

For openloop systems, the denominator of the transfer functions in the matrix gives the openloop characteristic equation. In Example 15.14 the denominator of the elements in was (s + 2X + 4). Therefore the openloop characteristic equation was 

The system was openloop stable since both roots were in the left half of the s plane. Putting it another way, both openloop eigenvalues were in the left half of the 5 plane. 

For closedloop systems, the denominator of the transfer functions in the closedloop servo and load transfer function matrices gives the closed-loop characteristic equation. This denominator was shown in Chap. 15 to be [I + which is a scalar Nth-order polynomial in s. Therefore, the 

This is the most important equation in multivariable control. It applies for any type of controller, diagonal (multiloop SISO) or full multivariable controller. If any of the roots of this equation are in the right half of the s plane, the system is closedloop unstable. 

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