ATOMATIC CONTROL LABS

	1 2
y	K	exp t sin				1 2	t
		0			0
K exp 0t cos 0			1 2 t K.						(22)

Expession (22) causes difficulties in its application, therefore it can be presented in simpler form. The first two terms of equation, describing the free response are solution of homogenic equation at zero initial conditions and can be rewritten in this way:

	y	K	exp t sin								1 2	t
		1 2
	h					0				0
	K exp 0t cos 0					1 2			t					(23)
								sin	0		1 2	t
				1			2
	K exp t													.
			0					1 2 t
			cos
						0

Demote a = tanφ and apply formula:

cos t asin t cos t sin sin t			cos t	.	(24)
		cos	cos
If	1 2 , a	1 2 and	arctan a ,		then
0

expression (24) can be applied for portion of Eq. (23):

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		1 2
									sin			1		2		t		cos						1 2 t
												0			.							0
			cos								1 2 t																			(25)
									0
										cos
In order to express cos φ by parameters of Eq. (3), the set
equations must be solved:
				2	cos						2	1;						sin	2	cos					2	1;
sin
tan													;			sin											cos ;			(26)
										1 2														1 2
			cos								1 2 .

Application of express (25) an (26) for (23) gives:

		1		2
y	h			K	exp t		cos				1 2	t	;	(28)
						0				0
where arctan								.
						1 2

Expressions (27) and (28) are more suitable for analysis. They show exponentially decaying oscillations. Frequency of oscillations:

	s		1 2	Im p ;	(29)
		0		1,2

is called frequency of selfoscillations.

It is evident from Eq. (28), that at ρ=0, systems output is described in this way:

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y K cos 0t K.

(30)

In the general case magnitude of oscillation is found from expression:

Y	K	exp t .	(31)
	1 2	0

Expression (31) shows that decaying speed is proportional to ρ, therefore ρ is called damping ratio, and if ρ = const, amplitude also remains constant.

Two step responses, shown in Fig. 1 correspond systems with ρ < 1. The overshoot appears and its magnitude and frequency increases with reducing ρ.

Generalizing analysis of the second order system in time do-

main, can be concluded:

at 1 the system has deadbed step response without

overshoot and oscillations;

at 0 1 the step response is oscillating with oscil-

lations frequency	1 2 ; the magnitude decays,
0

if ρ > 0 or remains constant, if ρ = 0.

In most cases for synthesis of controller there are considered (oscillating) systems with 0 1.

In Eq. 3 replacing dtd s , the transfer function of the system gets the form:

	G(s)	Y (s)					K		.	(32)
		U (s)		s	2		2	s 1
				02
							0

Substituting s = jω yields the frequency response:

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G( j )				K			.	(33)
			2
					2	j
				1
		0
					0

The polar plot of the first order system has two characteristic points. From Eq. (33) at ω = 0 the initial point is calculated as:

G( j0) K;								(34)
and final point at t → ∞:
G( j ) lim				K			0.	(35)
			2
					2	j
				1
		0
					0

Denoting real and imaginary part of Eq. (3) as:

0 2 2 j 0 1 1 0 2 ;

0 2 2 j 0 1 2 / 0;

(36)

the Eq. (33) can be rearranged to this form:

	K		K j
G( j )					.	(37)
	j		2 2

Real part of complex transfer function is:

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	G j		K	.	(38)
		2	2

Numerator of real parts in Eq. (38) will be equal to zero only if Kα(ω) = 0 and then:

	K 0;	0;				2
				1			0;	(39)
					0

0.

Imaginary part of Eq. (38) for the frequency ω0 becomes:

G( j )	K 0					2 K		K	.	(40)
						2
0								2
	0		0			4

Thus, the polar plot of the second order system intersects imaginary axis at the point x = (0,−K/2ρ) where the frequency is equal ω0. Polar plot of the system with ω0 = 5, K = 2,5 and ρ = [0,5; 0,75; 1; 1,25; 1,5] is presented in Fig. 2.

Fig. 2. Polar plot of the second order system with different ρ values

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It is evident, that all characteristics begins at the point x(0) = (K; 0) = (2,5; 0) and ends at the point x(∞) = (0; 0). In this figure the frequencies ω0 are marked by small squares.

Expression (37) is convenient for analysis of magnitude and phase Bode plots. Frequency response magnitude characteristic is:

								K					2				K			2
	G( j )
				2				2									2 2			(41)
				K					.
		2 2
Magnitude (41) is expressed in decibels:
																	K
	L 20lg						G( j )				20lg							.		(42)
															2 2

Expression (42) describes the actual characteristic, but it can be plotted just using computer. Asymptotic characteristics have more attractive form and advantages at design of regulators.

Asymptotic magnitude characteristic is constructed from straight lines, intersecting at corner frequency. In low frequency range, when ω<<ω0 , then α(ω)≈1, β(ω)≈0 and expression (42) gives magnitude:

	G( j )		T 1 K;		L		T 1 20lg		G j		T 1 (43)
20lg K.

Thus, in the low frequency range the magnitude is constant and represented by horizontal line parallel to frequency axis with ordinate K dB.

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In the high frequency range, at ω >> ω0, then α(ω) ≈ − (ω/ ω0)2, therefore magnitude is:

	G( j )		T 1					K								K 0				.	(44)
							4	4	2			2				2		4	2
						0					0
																0

Damping ratio ρ is not limited, but usually it reaches order of units, therefore, if ω >> ω0 , then inequality (ω/ω0)2 >> 4 ρ 2 and (44) is rewritten in this form:

				K						2
G( j )					0			K	0	.	(45)
	T 1						2
					0

Magnitude characteristic (45) is expressed in decibels:

L							2
			20lg	K		0		20lg K 40lg		.	(46)
		T 1
									0

Using semi-log scale for frequency, expression (46) is represented by straight line. If the frequency is increased 10 times, then:

L 10			20lg K 40lg 10	20lg K 40lg		40
		T 1	0		0		(47)

L T 1 40;

The magnitude, smaller by 40 decibels is obtained. Thus in the high frequency range the second order system Bode plot is expressed by straigh line with slope of -40 dB/dec. Bode plot in the low and

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high frequencies is constructed from two lines correspondingly according to (43) and (46) expressions intersecting at corner frequency

ωs:

		2		s 0.	(48)
K K	0	;
	s
Corner plot ωs is equal to system resonant frequency ω0.
Bode plot of the second order system with					ω0 = 5 , K = 2,5

and ρ = [0,25; 0,5; 0,75; 1,0; 1,25; 1,5] are presented in Fig. 3.

Fig. 3. Bode plot of the second order system with different ρ values

Fig. 3 shows that approaching the damping ratio to zero increases the magnitude and the frequency at which the maximum takes place, approaches to resonant frequency ω → ωs = ω0. Dependence the phase angle versus frequency is calculated as:

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						K
arctan			G( j )	arctan		2 2
			G( j )			K
						2 2
arctan		.

Substituting (36) to expression (49) yields:

	2
arctan
				0			.
						2
1
		0

(49)

(50)

In the low frequency range, i. e. at ω << ω0, and the damping ratio is units order, the phase angle:

							arctan 0 0 ;
			arctan	2				(51)
		T 1			0

thus, the curve is a bit below frequency axis. At a high frequency range, at ω >> ω0, expression (49) is rewritten in this way:

T 1 arctan	2	180 .
	0		(52)

Thus, the phase angle characteristic with increasing of frequency, approaches to -180º. The other characteristic point is

ω = ω0. At this point α(ω) has interrupt, therefore should be done limits analysis the left hand side and right hand side of this point:

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					lim	arctan		2			90 ;
				0
and
								2
					lim arctan					90 .
		0

(53)

(54)

Thus, is the frequency ω acquires value ω0, the phase angle at this frequency reaches -90º. Nevertheless, this result could be obtained from analysis of phase angle versus frequency, i. e. expression (39).

The phase angle dependence versus frequency in semilog scale for the second order system with parameters ω0 = 5, K = 2,5 and ρ = [0,25; 0,5; 0,75; 1,0; 1,25; 1,5] are presented in Fig. 4.

Fig. 4. Phase angle Bode plot of the second order system with different ρ values

The mechanical and electrical circuits, shown in Fig. 5 a and 5 b are typical examples of mechanical and electrical the second

order systems.

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