Lecture 9\10 Analysis in the time domain (III) North China Electric Power University Sun Hairong

Topics of this class  Second-order systems: (Reading Module 6) （ Disturbance rejection and Rate feedback ）  Higher-order system: (Reading Module 7)  Routh ’ s Method; (Reading Module 9)  System type and steady-state errors (Reading Module 8)

1. Second-order systems: Disturbance rejection and Rate feedback The benefits of utilizing feedback control system One, to improve the dynamic response beyond that available with an open loop system; Two, to ignore unwanted input, often called its disturbance rejection capability; Three, to improve the dynamic performance with adjusting some parameter.

2. Higher-order systems a. Evans, or root locus, form. Example: b. Bode form. Example: 2.1 Reduction to lower-order systems A higher-order system may be reduced to lower-order system through the identification of dominant poles. However, some qualifications need to be specified. Two forms of transfer function,

If the input is Then the steady-state output is If the reduced system is The Bode form must be used when reducing higher-order system in order to ensure the steady-state output the same. cf.

Given nth-order linear time-constant system The characteristic equation yields, The closed-loop system may have both real and complex poles. Assuming the system has q real poles, and r pairs of complex poles ( ), the output is given by 2.2 Effect of a closed-loop pole

2.4 Effect of a closed-loop zero (See page 114~117) Fig. 7.6 Effect of zero on overshoot 2.3 Third order systems (See page 112~114) Fig. 7.3 PO for general third-order system

Then the closed-loop transfer function is given by  Clearly, the poles of the closed-loop are decided by the characteristic equation.  And the zeros are the zeros of the forward path transfer function and the poles of the feedback transfer function. 2.5 Occurrence of closed-loop zeros and poles Taking a feedback system

Assuming the system has q real poles, and r pairs of complex poles ( ), the output is given Given nth-order linear time-constant system The characteristic equation yields, The closed-loop system may have both real and complex poles. 3. Routh’s Method 3.1 Review:

Routh’s criterion states that,  The number of closed-loop poles in the right-hand half complex plane is equal to the number of sign changes of the elements of the first column of Routh’s array.  The Routh’s criterion is both necessary and sufficient condition of a stable system. 3.2 Routh’s method

Example 1: The characteristic equation is given by D(s)=3s4+10s3+6s2+40s+9=0 Determine if the system is stable. Write Routh’s array,

Example 2: Consider the feedback system shown in following figure. Determine the range of the gain K on the condition that the system is stable.

Definition The feedback system is known as Error is defined as two forms (a). E(s) = R(s)-C(s)H(s) (b.) E(s) = R(s)-C(s) Calculate the steady-state error subjected to definition (a) The steady-state error is The steady-state error can also calculate by using the final-value theorem as (assuming the poles of sE(s) are all located on the left-hand complex plane) 4. System type and steady-state errors

Where Kk is called the open-loop gain. (Bode form). The system type is defined by the value of n. 4.1 Step input For the type zero system For the type one system For the type two system Firstly, A generalized open-loop transfer function will be defined as,the constant K p is called the position error constant.

4.2 Ramp input, the constant K v is called the velocity error constant. For the type one system For the type two system For the type zero system

4.3 Acceleration input, the constant K a is called the acceleration error constant. ， For the type one system For the type two system For the type zero system

So, if the input is The steady-state error will be Example: SP8.3(page 138)

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