ON THE SETTLING AND RESPONSE TIMES OF UNDERDAMPED SECOND-ORDER SYSTEMS

Chinese hamster ovary (CHO) cells is one of the most widely used production host for the commercial production of biopharmaceuticals product. They have been extensively studied and developed, and today provide a stable platform for producing monoclonal antibodies and recombinant proteins. This study focused on the production of recombinant protein in suspension culture of CHO cells in spinner flask and shake flask.The CHO cells were transfected with DNA plasmid containing lac Z gene which codes for β-galactosidase. The β-galactosidase-expressing CHO cells were adapted to suspension culture. The agitation speed for both spinner and shake flask were adjusted accordingly. The experiments were carried out in duplicate and samples were taken for cell count, determination of glucose consumption, lactate production and protein level by using biochemical assay. Results showed that cell growth in spinner flask is more favorable than in shake flask. The cell concentration in spinner flask is 58% higher than in shake flask. On the other hand, specific activity of β-galactosidase is 25% higher in spinner flask compared to shake flask, at the same agitation speed. ABSTRAK: Masa enapan ternormal (ts/τ) nilai ayunan system terbit kedua, apabila fungsi memaksa ubah berperingkat (step-change forcing function (SCFF)) dijalankan ke atasnya, bergantung kepada kepekaan alat pengukur yang digunakan untuk mengukur respons (± x%). Satu percubaan dijalankan secara matematik untuk mengaitkan ts/τ to ± x% dengan mempergunakan ekspresi yang tepat dan mudah, pada sempadan bawah sampul reputan (lower boundary of the decay envelope (LBDE)). Dua hubungan yang diperolehi dikaji terhadap nilai ts/τ sebenar untuk julat jalur enapan ±1% ≤ ±x% ≤ ±6%, melingkungi julat pekali redaman 0.1 ≤ ζ ≤ 0.65. Walaupun hubungannya tidak tepat, trend umum merupakan penganggaran marginal ts/τ. Hubungan berdasarkan LBDE adalah berdasarkan LBDE yang telah dipermudahkan, ia dipilih kerana ianya senang dan agak tepat antara keduanya. Ini mendorong kepada perbezaan yang disarankan antara ts/τ dan waktu respons ternormal (tR/τ), dengan nilai 5/ ζ yang ditetapkan kemudiannya.


INTRODUCTION
An underdamped second order system with 0 < ζ ≤ 0.65, when subjected to a SCFF undergoes a response which is significantly oscillatory.Under such condition, the settling time (also called the response or recovery time) is defined as the time required for the normalized response to enter the ±5% band of the step change magnitude.Other definitions for t s exist; notably, that related to the ±2% band or as determined by the sensitivity of the measuring instrument [1][2][3][4][5].

ESTIMATION OF SETTLING TIME
Pollard [2] pointed out that owing to the arbitrary nature of the settling band limits (due to the specific sensitivity of the measuring instrument employed), a mathematical definition for t s is not possible.He concluded that it can be easily measured from the response curve of a recording instrument (i.e. a posteriori).In spite of the aforementioned viewpoint, an estimate of t s value a priori is an advantage in many instances, e.g. in the design and analysis of control loops.The normalized response of an underdamped 2 ndorder system to a SCFF of magnitude A is, For a ± x% settling band, its limits would correspond to Y(t)/AK values of (1 + 0.01 x) and (1 -0.01 x) respectively.This renders the value of the second term of Eq. ( 1) equal to 0.01x in absolute value.Therefore, t s /τ is the shortest normalized time which satisfies this condition; provided that the second term will not exceed |0.01 x| for t/τ > t s /τ.
The exact expression for the lower boundary of the decay envelope (LBDE) related to the normalized response represented by Eq.( 1) is (1 Pollard [2], however, gave it as (1 െ e ି ಎ ಜ ୲ ሻ ; neglecting ߞ ଶ which is justifiable for small values of ߞ.Pollard further pointed out that (1 െ e ି ಎ ಜ ୲ ሻ is the normalized response of a 1 st -order system, whose time constant is τ/ζ, to a SCFF.These two expressions for the LBDE were utilized to obtain the following equations for the estimation of t s /τ as related to a -x% settling band limit.Hence, Corresponding to (1 െ e ି ಎ ಜ ୲ ) LBDE , and Eq.'s (2) and (3) were tested against the actual t s /τ values for settling bands ranging from ±1% to ±6% over the range 0.1≤ ζ ≤ 0.65.The results are shown in Fig. 's (1) to (6).The two equations generally overestimate t s /τ but give reasonably close values to the actual ones.Their respective values of t s /τ were too close to be distinguishable on the same graph; which necessitated the use of separate plots for each ±x% value.

SETTLING TIME VS. RESPONSE TIME
As a consequence of the adoption of Eq.( 1), the following argument is presented as a basis for suggesting that a distinction should be made between t s and the response time (t R ).
For a 1 st -order system subjected to a SCFF, the response time may be defined as that at which the normalized response exceeds 99% of the step change magnitude.Hence, a time interval equal to five times the system's time constant, corresponding to 99.3% of the step change magnitude, would serve this purpose.If this criterion is adopted, then by referring back to Pollard's [2] expression for the LBDE, i.e. ൬1 െ e ି ಎ ಜ ୲ ൰, the normalized response time for an underdamped 2 nd -order system would accordingly be 5/ζ .In other words, t R /τ would be the normalized settling time for a ± 0.7% settling band according to Eq.( 2 2), and t s /t R , based on Eq.( 6), for the range 0.7%≤ ±x ≤6% for comparison purposes.

CONCLUSION
A simple mathematical formula is presented to estimate a priori the normalized settling time values of oscillatory 2 nd -order systems, when subjected to a SCFF, for any value of the measuring instrument sensitivity.A distinction is made between normalized settling and response times of such systems, with the latter assigned the value of 5/ζ .Accordingly, ratios of settling to response times can readily be established.

Table ( 1
) gives rounded-off values of t s /τ, based on Eq.(