MODAL CONTROLLER DESIGN IN THE PRESENCE OF OVERSHOOT MAGNITUDE RESTRICTIONS

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MODAL CONTROLLER DESIGN IN THE PRESENCE OF OVERSHOOT MAGNITUDE RESTRICTIONS

Insur Zavdyatovich Ahmetzyanov¹, Dmitry Nikolayevich Dem’yanov¹

1Kazan Federal University

e-mail (for correspondence): demyanovdn@mail.ru

ABSTRACT

In this paper, they considered the effect of zeros and poles location of the automatic control system with the modal controller on one of the transient process quality indicators - the overshoot value. The calculations were performed on the model of an asymptotically stable automatic control system with a positive static gain ratio, the binomial law of pole distribution, and a single transmission zero described by a linear differential equation of the order higher than the first one. For an indicated case, they developed analytical relations that allow one to determine its largest and smallest value, and also the magnitude of a positive or a negative overshooting, without the development of a transition characteristic curve. The nature of the overshoot value dependence on the parameters of the transfer function is analyzed and the conditions are determined under which this value will not be greater than the predetermined value. The examples of obtained formula practical use for the research and design of control systems with given quality indicators are presented. The correctness of the results is confirmed by the results of computer simulation in MATLAB system. The results of the work can be easily implemented on a computer and included as the additional modules in the composition of specialized software products for the analysis and the synthesis of automatic control systems.

Keywords: modal control, transfer function, zero, poles, transient response, overshoot.

INTRODUCTION

During the design of regulators by modal control methods, the roots of the characteristic polynomial of the closed system move to a certain region of a complex plane [1]. It is assumed that the position of the transfer function poles has the most significant effect on the quality of control processes. In this case, the position of the transfer function zeros with this approach is not taken into account and is not corrected most often.

However, the practical experience of automatic control system design and the results of a number of theoretical studies allow us to state that the quality of the transient process is determined by the location of not only the poles, but also by the zeros of the transfer function [2-4]. This effect is particularly evident in the case of non-minimal-phase objects having one or several zeros with a positive real part [5].

The establishment of a connection between the quality of control processes and localization areas on a complex plane of zeros and poles of a closed system is a fairly well-known scientific problem, the solution of which has been dealt with by a number of authors [6-8]. However, the results obtained so far have, as a rule, only qualitative character. The most general criteria are formulated, which allow to determine the form of time characteristics and to give only rough quantitative estimates of quality indicators for control system functioning [9].

As an example, you can specify an overshoot, the value of which is defined as the percentage of the maximum deviation ratio from the steady-state value to the most stable value of the transition

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determine the fact of a nonzero overshoot existence, as well as its type (a positive or a negative one) reliably only in some cases. In order to obtain the same quantitative estimates of this index, it is necessary to integrate the dynamics equation, which significantly complicates the analysis and synthesis procedures for control systems with a modal regulator.

Thus, the problem of the overshoot value dependence establishment on the parameters of the transfer function of the control system can be considered as a highly relevant one.

METHODS

As a rule, when modal control methods are used, the desired location area of closed system poles is given in accordance with one of the standard distributions [1]. In the framework of this paper, we will consider the binomial distribution, which is one of the most frequently used in practice.

The transfer function of the control system under the assumed assumptions has the following form:

( ^s )

ⁿ

^.

k s

W α

β +

= +

(1 )

Without the loss of generality, we assume that

α > 0

(the system is asymptotically stable),

^k β > 0

^(the

static gain is positive),

n ≥ 2

. Based on the physical meaning of the parameters under consideration, it can be argued that such assumptions will be valid for any real control system.

The control system described by model (1) can have both positive and negative overshoot [2].

The value of positive overshoot is given by the following formula:

.

%

max

− ⋅ 100

=

∞ + ∞

h h σ h

(2 )

Here

h

_max – the largest value of the transient response,

h

_∞ – the steady-state value of the transient response (its limit at

t → ∞

).

The value of the negative overshoot is given by the following formula:

.

%

min

⋅ 100

−

=

∞

−

h σ h

(3 )

Here

h

_min is the smallest value of the transition characteristic.

It is required to determine the dependence of the values

σ

⁺^and

σ

⁻ on the parameters

k , α , β , n

^.

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RESULTS AND DISCUSSION

It was shown in [10] that three different situations can arise for a system with a single transfer zero and a binomial distribution of poles depending on the parameter

β

^value.

1. If

β > α

, then the transient response has no extremum points. In this case, the transient process is monotonic and the overshoot value (both positive and negative one) makes zero.

2. If

β = α

, then there is the reduction of zero and one of the poles of the transfer function takes place.

In this case, the expression (1) takes the following form:

( ) ^.

1

−1

= +

_n

k s

W α

As is known, in this case the transition characteristic also has no extremum points [11], the transient process is monotonic one, and the overshoot value is zero.

3. If

β < α

, then the transient response has one extremum point. In this case, there may be a positive or a negative overshoot, the type of which is determined by the nature of an extremum point (the maximum point or the minimum point).

In order to calculate an overshoot value, we determine a set and an extreme value of the transition characteristic.

The image of the Laplace transition characteristic is determined by the known relation [12]:

( ) ( )

( ^s )

ⁿ

s s k s s W

h α

β +

= +

=

^.

Using the limiting properties of images, we calculate the quantity of

h

_∞:

[ ( ) ] ⁽ ⁾

( )

ⁿ ⁿ

s s

k s

s s k

h s

h α

β α

β = +

= +

⋅

=

→ →

∞

lim

0

lim

0 ^.

(4 )

The value of the transition characteristic at the extremum point t^* can be found from the well-known formula [12]:

( ) ^t

^∗

⁼

^t

_∫

^∗

^w ^{( )} ^d

h

0

τ

^.

(5 )

At that it was shown in [10] that the expression for the weight characteristic in the case under consideration has the following form:

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( ) ( )

( ) ( ) ^⎟ ⎠

⎜ ⎞

⎝

⎛

− + −

= −

⎭ ⎬

⎫

⎩ ⎨

⎧ +

= +

−

− −

t

n n

e k t s

s L k

t w

t n

n

1 1

! 2

1 2

β α

α

β

^α _.

(6 )

Combining formulae (5) - (6), we obtain an analytical expression for the value of the transition characteristic at the extremum point:

( ) _∫

^∗

₍ ₎ ^⎟

⎠

⎜ ⎞

⎝

⎛

− + −

= −

−

∗ −

t n

n d n

k e t

h

0

2

1 1

!

2 β α τ τ

τ

^α^τ

.

(7 )

Let's integrate by the parts of the right-hand side of the expression (7) the required number of times and substitute the true value of the coordinate of the extremum point in the resulting formula [10]:

β α −

= −

∗

n 1

t

^.

(8 )

If

n = 2

, then the expression (7) taking into account the formula (8) will have the following form:

( ) ⁽ ⁽ ⁾ ⁾ _⎟⎟

⎠

⎞

⎜⎜ ⎝

⎛ ⎟

⎠

⎜ ⎞

⎝

⎛

− + −

=

− +

= ∫

∗

−

∗

α β

α α

β α α τ β

τ α

τ

β

α

1

₂ ₂

exp

0

k d ke

t h

t

.

(9 )

If

n > 2

, then the expression (7) taking into account the formula (8) will have the following form:

( ) ₍ ₎ ⁽ ⁾ ^⎟

⎠

⎜ ⎞

⎝

⎛

−

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛ ⎟

⎠

⎜ ⎞

⎝

⎛

−

− −

⎟

⎠

⎜ ⎞

⎝

⎛

−

− + −

= ∑

⁻

=

− −

∗

α β α β

α α β

α β

α α

β α

β 1

1 exp

! 1

! 2

2

0

1 2

n n

m n

n k

t k

h

ⁿ

m m m n n

n

n . (10)

In order to determine the type of extremum, we find the value of the second derivative of the transition characteristic at the moment t^*.

Having differentiated the expression (6) with respect to time, we obtain [11]:

( ) ^⎟⎟ _⎠

⎞

⎜⎜ ⎝

⎛ ⎟

⎠

⎜ ⎞

⎝

⎛

− + −

= −

=

−

t

n n

e t dt k d dt dw dt

h

d

ⁿ ^t

1 1

! 2

2 2

2 ^α

β α

_. ₍₁₁₎

Combining formula (8) and (11), we obtain the expressions for the value of the second derivative at the extremum point.

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If

n = 2

^{, then:}

( ) ^⎟

⎠

⎜ ⎞

⎝

⎛

− −

−

=

= ∗

β α

α exp

2 2

dt k h d

t t

. (12)

If

n > 2

^{, then:}

( ) ^⎟ _⎠

⎜ ⎞

⎝

⎛

−

− −

⎟ ⎠

⎜ ⎞

⎝

⎛

−

− −

=

−

=^∗

α α β

β α

exp 1 1

! 2

3 2

2

n n

n k dt

h

d

ⁿ

t t

. (13)

Since

β < α

, then the signs of the right-hand sides in the expression (12) and (13) will be determined only by the value of the parameter k. If

k > 0

^{, then}

^h ^ʹʹ ( ) ^t

^∗

^< ⁰

and at the considered point the transition characteristic reaches its maximum. If

k < 0

^{, then}

^h ^ʹʹ ( ) ^t

^∗

^> ⁰

and at the point under consideration, the transition characteristic reaches its minimum [13].

According to accepted assumptions

^k β > 0

^{, thus at}

β > ⁰

the transient response has a maximum point and a positive overshoot, and at

β < 0

the transient response has a minimum point and a negative overshoot. The correctness of the obtained result is confirmed by known data on the transient characteristics of minimum-phase and non-minimal-phase dynamic objects [9].

Let us determine an overshoot amount by substituting the expressions (4), (9), (10) into the formula (2) and (3).

At

α > β > 0

an overshoot is a positive one.

If

n = 2

^{, then:}

% 100

exp ⎟ ⋅

⎠

⎜ ⎞

⎝

⎛

−

= −

+

α β

β

σ α

^. ⁽¹⁴⁾

If

n > 2

^{, then:}

( )

( ¹ ) ₁₀₀ _%

1 exp

! 1

1

²

1

1 2

⎟ ⋅

⎠

⎜ ⎞

⎝

⎛

−

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛

⎟ ⎠

⎜ ⎞

⎝

⎛

−

− + −

−

= ∑

⁻

=

−

− − + −

α β α β

α β α

β α

σ ⁿ ⁿ

k n

k

n

k

k k n

n

. (15)

At

β < 0

an overshoot is negative one.

If

n = 2

^{, then:}

% 100 1

exp ⎟⎟ ⋅

⎠

⎞

⎜⎜ ⎝

⎛ ⎟ −

⎠

⎜ ⎞

⎝

⎛

−

= −

−

α β

α

σ β

^. ⁽¹⁶⁾

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( )

( ¹ ) ₁ ₁₀₀ _%

1 exp

! 1

1

²

1

1 2

⎟ ⋅

⎟

⎠

⎞

⎜ ⎜

⎝

⎛

⎟ −

⎠

⎜ ⎞

⎝

⎛

−

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛

⎟ ⎠

⎜ ⎞

⎝

⎛

−

− −

−

= ∑

⁻

=

−

− −

α β α β

α α α

β β

σ ⁿ ⁿ

k n

k

n

k

k k n

n

. (17)

The calculated formulas (14) - (17) allow to solve the problem of analysis: to determine the amount of an overshoot by the given parameters of the control system transfer function. However, in practice, the solution of the synthesis problem is also of great interest: the determination of the regulator parameters according to the specified requirements for the quality of the transient process (in this case, according to a known overshoot value).

Let us analyze the nature of an overshoot value change from the value

α

at the fixed parameter

β

^.

If

α > β > 0

, then the transient response has a positive overshoot, the value of which is given by expression (14) and (15). Having differentiated them according to

α

, we get the following dependence:

( )

( ¹ ) ₁₀₀ _%

1 exp

! 1

1 1

⋅

⎟ ⎠

⎜ ⎞

⎝

⎛

−

⎟ −

⎠

⎜ ⎞

⎝

⎛

−

= −

∂

⁺ ⁻ ⁻

α β α β

α β

α α

σ n n

n

n n

. (18)

Taking into account the assumptions made earlier, it can be asserted that the right-hand side of expression (18) will always be positive. Consequently, the dependence

σ

⁺

( ) α

is an increasing function (the increase of the pole leads to the increase of a positive overshoot).

If

α ^{> 0} ^> β

, then the transient response has a negative overshoot, the value of which is given by expressions (16) and (17).

Having differentiated them with respect to

α

, we get the following relationship:

( )

( ¹ ) ₁₀₀ _%

1 exp

! 1

1 1

⋅

⎟ ⎠

⎜ ⎞

⎝

⎛

−

⎟ −

⎠

⎜ ⎞

⎝

⎛

−

− −

∂ =

∂

⁻ ⁻ ⁻

α β α β

α β

α α

σ n n

n

n n

. (19)

Taking into account the assumptions made earlier, it can be asserted that the right-hand side of expression (19) will always be a positive one. Consequently, the dependence

σ

⁻

( ) α

is also an increasing function (the pole increase leads to the increase of the negative overshoot).

Thus, at a fixed value of the zero transfer, the removal of the pole location point of the closed system from the coordinate origin leads to the increase of positive and negative overshoot. Therefore, if the limiting value of the overshoot

σ

⁺^or

σ

⁻ is specified in the requirements for the designed control system with zero transmission, then the poles of this system must be removed from the coordinate origin no more than by the value

α

given by conditions (14) - (17). In fact, this corresponds to the limitation on the admissible speed of the control system.

All formulated analytical relationships can be used in practice during the solution of analysis and synthesis issues of automatic control systems. We show this from two methodological examples.

Analysis problem solution. Let us consider an automatic control system whose model is given by a

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( ) ( ³ )

⁴

1 81

+

= + s

W s

^.

Let us determine the nature of the transient process and the amount of overshoot.

The control system has one zero,

β = 1

^,

α = ³

^,

n = ⁴

^.

Since

α ^> β

^and

β ^> ⁰

, then the transient process has a positive overshoot.

Let us calculate the overshoot by the following formula (15):

( ) ² ¹⁰⁰ ^% ¹⁶ ^, ³⁹ ^%

exp 9 2

3 ! 3 2 3

1

²

1 3 2

≈

⎟ ⋅

⎠

⎜ ⎞

⎝

⎛ −

⎟ ⋅

⎟

⎠

⎞

⎜ ⎜

⎝

⎛ ⎟

⎠

⎜ ⎞

⎝

⎛

− + ⋅

= ∑

=

− − +

k

k k

k

σ k

^.

The obtained results are confirmed by the computer simulation results using MATLAB system. Figure 1 shows the graph of the transient process for a control system with a given transfer function, illustrating the validity of the conclusion.

0 2 4 6

0 0.5 1

1.5 Step Response

Time (sec)

Amplitude

Fig. (1). – Transient indicator curve for the system from the first example

Synthesis problem solution. Suppose we are given a control object whose dynamics is described by a linear second-order differential equation. The transfer function of the object has a single zero

β = 2

^.

Let us determine the value of the parameter

α

, at which the modal regulator by state with the binomial

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Since at

β > 0

, then two options are possible: at

α > 2

the transient characteristic of a closed system will have a positive overshoot, there will be no overshoot at

α ≤ 2

^.

The limiting value of the parameter

α

is determined by the formula (14):

2 100 2 exp

5 2 ⎟ ⋅

⎠

⎜ ⎞

⎝

⎛

⋅ −

= −

α α

α

_.

The root of the resulting equation can be found using standard numerical methods:

α

₀

≈ 3 , 29

. Thus, the projected control system will have a permissible overshoot value if

α ≤ 3 , 29

^.

Figure 2 shows the graphs of the transient characteristics at different values of the parameter

α

^{. The}

analysis of the results confirms the correctness of the drawn conclusions.

0 0.5 1 1.5 2 2.5

0 0.2 0.4 0.6 0.8 1

1.2 Step Response

Time (sec)

Amplitude

α=3 α=3.29 α=4

Fig. (2). – The graphs of the transient characteristics for the system from the second example

It should be noted that the obtained limit value

α

actually limits the speed of the control system at the accepted requirements to an overshoot value.

CONCLUSIONS

The obtained relationships allow us to conclude that an overshoot value in the control systems with a modal regulator and a binomial distribution of poles depends essentially on the location of a complex zero plane of the transmission zero. If zero is located to the left of the poles, then the transient process in such a system is characterized by the absence of an overshoot. If zero is located to the right of the poles, then the

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left half-plane) or a negative (if zero is located in the right half-plane) overshoot. In this case, an overshoot value increases with the increasing distance between zero and poles.

SUMMARY

In this paper they formulated analytical relations that describe the dependence of an overshoot value on the parameters of the control system with the modal regulator under the binomial law of pole distribution and a single transmission zero. The obtained results can be used in practice to solve the problems of analysis and synthesis of automatic control systems with specified transient process quality parameters.

Then, we propose the development of the obtained results for the systems with a large number of transmission zeros, as well as for the systems with a different type of the modal regulator pole distribution.

ACKNOWLEDGEMENTS

The work is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University.

This work was supported by the Russian Foundation for Basic Research (Grant No. 16-38-00042).

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MODAL CONTROLLER DESIGN IN THE PRESENCE OF OVERSHOOT MAGNITUDE RESTRICTIONS