Second-order systems

In this section we consider the classical equations of motion of particles in cases where the highest-frequency oscillations are nearly harmonic The positions y t) = j/i (t) evolve according to the second-order system of differential equations... 

Method of Variation of Parameters This technique is applicable to general linear difference equations. It is illustrated for the second-order system -2 + yx i + yx = ( )- Assume that the homogeneous solution has been found by some technique and write yY = -I- Assume that a particular solution yl = andD ... 

FIG. 8-16 Resp onse of general second-order system. 

The Van Krevelen-Hoftyzer relationship was tested experimentally for the second-order system in which CO9 reacts with either NaOH or KOH solutions by Nijsing et al. [Chem. Eng. ScL, 10, 88 (1959)]. Nijsing s results for the NaOH system are shown in Fig. 14-15 and are in excellent agreement with the second-order-reaction theory. Indeed, these experimental results can be described very well by Eqs. (14-80) and (14-81) when values of V = 2 and T)JT = 0.64 are employed in the equations. 

Time domain response of second-order systems 3.6.1 Standard form... 

Equations (3.42) and (3.43) are the standard forms of transfer functions for a second-order system, where K = steady-state gain constant, Wn = undamped natural frequency (rad/s) and ( = damping ratio. The meaning of the parameters Wn and ( are explained in sections 3.6.4 and 3.6.3. 

The transient response of a second-order system is given by the general solution... 

Fig. 3.16 Effect that roots of the characteristic equation have on the clamping of a second-order system.

Generalized second-order system response to a unit step input... 

Consider a second-order system whose steady-state gain is K, undamped natural frequency is Wn and whose damping ratio is (, where C < 1 For a unit step input, the block diagram is as shown in Figure 3.18. From Figure 3.18... 

Fig. 3.18 Step response of a generalized second-order system for C < 1-Expanding equation (3.52) using partial fractions...

The denominator is now in the standard second-order system form of equation (3.42). The steady-state response may be obtained using the final value theorem given in equation (3.10). 

The characteristic equation was defined in section 3.6.2 for a second-order system as... 

These roots determine the transient response of the system and for a second-order system can be written as... 

Fig. 5.9 Root locus diagram for a second-order system.

General case for an underdamped second-order system... 

Fig. 5.10 Roots of the characteristic equation fora second-order system shown in the s-plane.

Frequency response characteristics of second-order systems... 

Second-order system closed-loop frequency response... 

Since many closed-loop systems approximate to second-order systems, a few interesting observations can be made. For the case when the frequency domain specification has limited the value of Mp to 3 dB for a second-order system, then from equation (6.72)... 

In general, for a second-order system, when Q is a diagonal matrix and R is a scalar quantity, the elements of the Riccati matrix P are... 

Example 6.2 Bode Diagram %Second-order system clf... 

Lastly, we should see immediately that the system steady state gain in this case is the same as that in Example 5.1, meaning that this second order system will have the same steady state error. 

If we have a second order system, we can derive an analytical relation for the controller. If we have a proportional controller with a second order process as in Example 5.2, the solution is unique. However, if we have, for example, a PI controller (2 parameters) and a first order process, there are no unique answers since we only have one design equation. We must specify one more design constraint in order to have a well-posed problem. 

With the expectation that the second order system may exhibit underdamped behavior, we rewrite the closed-loop function as... 

Transient response criteria Analytical derivation Derive closed-loop damping ratio from a second order system characteristic polynomial. Relate the damping ratio to the proportional gain of the system. 

In the case of a second order system, the first column of the Routh array reduces to simply the coefficients of the polynomial. The coefficient test is sufficient in this case. Or we can say that both the coefficient test and the Routh array provide the same result. 

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