On the Robustness of Integral Time Delay Systems with PD Controllers

Eduardo Zuñiga ¹ , Omar Santos ² and M.A. Paz Ramos ¹

1 Instituto Tecnológico y de Estudios Superiores de Monterrey, Campus Ciudad de México, Calle del Puente 222, Col. Ejidos de Huipulco, Tlalpan, 14380, México D.F.

2 CITIS, Universidad Autónoma del Estado de Hidalgo, Carretera a Tulancingo, Km 2.5, Pachuca, Hgo., México.

a00970554@itesm.mx, omarj@uaeh.reduaeh.mx, marco.paz@itesm.mx Abstract-We studied the robust stability of first-order integral

systems in closed loop with a PD controller when general tables for tuning the controller are used. The frequency approach is used to obtain necessary and sufficient conditions for the robust stability of the characteristic equation in closed loop. The Linear Matrix Inequalities (LMI) approach is employed to obtain sufficient stability conditions when model non-linearities are considered.

I. INTRODUCTION

The robust stability analysis for time delayed systems has been widely studied in the last decades [8].The time delay effect on the performance on the closed loop characteristic of the system when a control law is introduced may induce complex behaviors (instability, oscillations, undesired performance), therefore the study of the stability regions is a problem of great interest. Note that complete characterization of the corresponding stability regions is a very complex problem and still open in the general case [2]. Furthermore, a chaotic behavior may occur if the system is perturbed with a non-linear function that depends on the delayed state [5]. The Proportional Integral Derivative (PID) controller is one of the most popular strategies for control in industrial plants. The relatively easy implementation, robustness and the availability of an enormous set of rules and tables [15], [11], [13], [12] for tuning this type of controller makes the PID control the most adopted in a variety of applications. The robust stability of time delayed systems in closed loop with PID controllers has been studied in [6], [8] and [10]; however the papers that propose tables and rules for the PID tuning, for specific systems, do not present any robust stability analysis.

In this contribution, we present a robust stability analysis via the frequency approach for integral time delayed systems in closed loop with a PD controller. In addition, robust stability conditions are obtained when non-linear disturbances are considered in the model. The parameters of the controller for integral time delayed systems were obtained from two tables presented in [12] and [9]. The analysis can be extended to include other types of systems, for example, unstable first-order processes. Our contribution is organized as follows: in the first section we introduced our contribution, in section 2 we established the problem analyzed; the frequency domain analysis with illustrative examples is included in section 3,

section 4 deals with time domain analysis with an illustrative example, section 5 provide an analysis of the roots' behavior, and section 6 is dedicated to final comments.

II. PROBLEM FORMULATION

Consider the following integral time delayed system

( ) ,

( ) Y s Kesh

U s s

= − (1)

when the input signal

U s ( )

is a PD controller, two constants must be determined: K_p and T_d . These constants can be obtained from the tables or rules when an optimization problem is solved considering a performance index. For example in [12] the ISE (Integral Square Error) and ITSE (Integral time square error) criteria are considered. The rules given in [12] are improved in [9], but neither [9] nor [12]

present a robust stability analysis. So if there are variations in the system (1) parameters, it is interesting to estimate these variations on the parameters, which is shown in section 3.

When the system presents linear perturbances or non-modeled dynamics, using the results given in [4], a time domain analysis is presented in section 4.

III. FRECUENCY DOMAIN ANALYSIS

In this section we analyze the robust stability of the system (1) in closed loop with a PD controller. Parameters space hyper-surfaces are obtained using the D− partitions method [7].

We are considering the following system

( ) ,

( ) Y s Kesh

U s s

= − (2)

where Y s( ) and U s( ) are the Laplace transformed of output

( )

y t

and input

u t ( )

respectively. Consider that input

U s ( )

is a PD controller, so we have that

T. Sobh et al. (eds.), Innovative Algorithms and Techniques in Automation, Industrial Electronics and Telecommunications, 119–124.

© 2007 Springer.

119

( ) ( ) _d ( ), U s =KpE s +K sE s

where

E s ( )

is the error signal given by

( ) ( ) ( ).

E s = R s − Y s

So, we have that the characteristic equation (considering the transfer function of output

Y s ( )

and reference

R s ( )

) is given by

(

¹ d ^sh

)

p ^{sh 0.}

s +KK e⁻ +KK e⁻ = (3)

Observe that the cuasipolinomyal (3) is a neutral cuasipolinomyal [1], [3]. As it is well know, [3], [8], a necessary condition for the stability of a neutral cuasipolinomyal is that the atomic part has to be stable [3]; for equation (3) this necessary condition implies that

d 1.

KK < (4)

Now, if we consider that the system (2) has K and h uncertain parameters

,

we want to obtain robust stability conditions for the parameters

K

and h if

K

_p and K_d are given.

Observe that when h=0, according to the equations (3) and (4) the condition to conclude stability for the polynomial is that KK_p >0, so, if we employ the roots' continuity principle with respect to the parameters [1], this implies that there exists an

h

^∗

> 0

such that the system (3) remains stable. We can use the D−partitions method to obtain the stability regions for the equation (3).

Now, according to the D−partitions method, the first boundary of the hyper-surfaces is when s=0 in (3), if we

direct calculations tell us that

2 2.

The set of equations (set1) implies that cos (1 KK^d),

h

ω

− −

= (7)

combining the equations (w) and (h) we find that

1 2 2

Now we are able to establish the following proposition.

Lemma 1. Assuming that KK_d <1, the characteristic

We illustrate the use of Lemma 1 by analyzing the robust stability of the rules given in [12] and [9].

The rules for integral processes given in [12] when a ISE criteria is minimized are

PID parameter ISE

Using the conditions given in Lemma 1 we obtain that the regions for parameters

K

and h of the model (2) are defined as follows

0, ,

Kn and h_n are the nominal values for the parameters in model (2). These nominal values are used for tuning the PD controller. Observe that the boundaries of the robust stability regions are defined by these nominal values. The analysis can be expanded to include the ITSE and ISTE criteria given in [12].

Now we analyze the robust stability of the model (2) parameters

K

and h when the following tables given in [9] are used

PID parameter ISE

0.0747/

-0.015

p i d

K M

T

T h

(11)

here, M is the slope for the open loop response and h is the delay. Using Lemma 1, we find the following conditions

0,0.0011 _n K M

h

⎛ ⎞

∈⎜ ⎟

⎝ ⎠ ⁽¹²⁾

and

)

0,

h∈ ⎣⎡ h^{∗ (13)}

Where

( ) ( )

1 0.0011 2 2 6

cos 1.2 10

0.00747 .

K n

M h M K

h K

− −

∗ − − ×

=

In the next section, we illustrate the robust stability conditions obtained for some plants.

ILUSTRATIVE EXAMPLES

In this subsection, we illustrate the use of Lemma 1 for two given plants; the controllers were tuned with tables (8) and (11).

Example 1. Consider the following plant referred to [9]

0.654

10

( )

^s

,

G s e

s

=

− (14)

considering the nominal parameters K_n =0.654 and

n 10,

h = we can use the conditions (9), (10) (12) and (13) in order to obtain the following hyper-surfaces:

Fig. 1. Stability zones for plant (14).

Zone I is the systems (2) stability zone with a PD controller tuned using table (8). Zone II (which also includes zone I ) is the stability region when table (11) is used. If we look closer, we can see that the pair (0.654,10) is on both stability zones:

Fig. 2. Nominal values for plant (14).

Based on this example, we can conclude that the controller tuned as per table (11) is more robust than the controller tuned as per table (8).

Example 2. Now consider the following plant 0.0506 6

( ) ^s.

G s e

s

= − (15)

The stability regions for closed loop systems are shown in the following figure

ROBUSTNESS OF INTEGRAL TIME DELAY SYSTEMS 121

Fig. 3. Stability regions for plant (15).

Zones I and II correspond to tables (8) and (11) respectively. Again, region II includes region I; this implies that for this example the PD controller tuned as per table (11) is more robust than the controller tuned as per table (8).

In next section, we considered non-linear perturbances for the model. Using the results given in [4] we obtained sufficient conditions for robust stability.

IV. TIME DOMAIN ANALYSIS

In this section, we assumed that the model used had the following form

( ) ( ) ( ( ), )

( ( ), ) ( ( ), ),

y t Ku t h f y t t g y t h t h y t h t

= − +

+ − + −

⁽¹⁶⁾

where input

u t ( )

is a PD controller

( )

_p

( )

_d

( ), u t = K e t + K e t ( )

e t

is the error signal,

e t ( ) = − r y t ( ).

To simplify the expressions we assume that

r = 0,

which means that we want to carry output y t( ) to the origin. We assumed that constants

K

_p and K_d are calculated with the tables (8) and (11) for the nominal system:

( ) ( ) ( ).

n p n d n

y t = − KK y t h − − KK y t h −

(17) Now, considering that the equation (17) is non-linearly perturbed, the equation (16) is

( ) ( ) ( )

( ( ), ) ( ( ), ) ( ( ), ),

p d

y t KK y t h KK y t h f y t t g y t h t h y t h t

= − − − −

+ + −

+ −

(18)

with initial function conditions

[ ]

( ) ( ), ( ) ( ), for all ,0 .

y t y t

h

θ ϕ θ θ ϕ θ

θ

+ = + =

∈ −

(19)

We denote with y_t the state of the system (18), defined as

[ ]

( ), ,0 , for all 0 yt =y t+

η η

∈ −h t≥

the uncertain system (non-linear) has non-linear time-varying perturbances

f y t t ( ( ), )

,

g y t h t ( ( − ), )

and

( ( ), )

h y t h t − which satisfies

( ( ), ) ( )

f y t t ≤

α

y t (20)

( ( ), ) ( )

g y t h t− ≤

β

y t h− (21)

( ( ), ) ( )

h y t h t − ≤

γ

y t h − (22) where

α

≥0,

β

≥0 and

γ

≥0 are given constants . For mechanical systems, we can interpret the non-linearities boundaries as the position

y t ( )

and velocity

y t ( ).

Observe that the system (18) is a neutral non-linear system.

Observe that the inequalities (20), (21) and (22) can be rewritten as

2( ( ), ) 2 2( )

f y t t ≤

α

y t (23)

2( ( ), ) 2 2( )

g y t h t− ≤

β

y t h− (24)

2( ( ), ) 2 2( )

h y t h t − ≤

γ

y t h − (25) We assumed that the nominal system (17) is stable, so we want to know the constants

α

,

β

and

γ

so that the perturbed system (non-linear) remains stable.

This type of system is considered in [4]. In fact, using the LMI approach, a Lyapunov-Krasovskii functional and the S-procedure [14] for the inequalities (23), (24) and (25), in [4], sufficient robust stability conditions are found.

Sufficient robust stability system (18) conditions are given for the next proposition. The proposition establishes delay dependent robust stability conditions.

Proposition 1. [4] The system (18), with initial function condition (19) and disturbances, satisfying (23), (24) and (25), is asymptotically stable if KK_d + <

γ

1, and there exists a ZUÑIGA ET AL.

122

real number

X ,

and positive numbers

P ,

R ,

S and

Y

so that the following LMI is satisfied

1 the neutral systems stability [8].

In the next subsection, we illustrated the use of Proposition (1) to obtain robust stability conditions for an example.

ILUSTRATIVE EXAMPLE

In this section, we obtain explicitly the robust stability conditions using Proposition (1). The LMI Toolbox of Matlab is used to solve the LMI given in Proposition 1.

Example Consider the plant (14) in the time domain as the nominal plant

( ) 0.654 ( 10).

y t = u t −

(26)

The controller tuned with the Table (8) is

( ) 0.1574 ( ) 0.7717 ( ).

u t = − y t − y t

The closed loop is given by

( ) 0.103 ( 10) 0.5047 ( 10).

y t = − y t − − y t −

(27)

Lemma 1 shows us that the system (27) is stable. Now we consider the non-linearities with boundaries given by (23), (24) and (25): using the LMI Toolbox we found that the LMI given in Proposition 1 was feasible, therefore the system (28) is asymptotically stable. Now consider the following constants:

0.03,

α =

β = 0.05,

γ = 0.1

, for these values, we found that the LMI is feasible, therefore the system (28) is asymptotically stable.

Consider the system (26), this time we tuned the PD controller with the Table (11), so we obtained the following controller

( ) 0.7515 ( ) 0.1127 ( ).

u t = − y t − y t

(29)

We know that the system (26) in closed loop along with the control law (29) is stable. If we consider the perturbed system we obtain given in Proposition (1) was feasible, therefore the perturbed system (30) is asymptotically stable. If we consider that

α = 0.3,

β = 0.2,

γ = 0.6

we found that the LMI is feasible, then the system (30) is asymptotically stable.

Clearly the PD controller tuned with the Table (11) is more robust under non-linear perturbances than the PD controller tuned with the Table (8).

V. ROOTS BEHAVIOR IN CLOSED LOOP ANALYSIS

Now, for the previously proposed plants we analyze the behavior of the roots, tuned according to the two tables shown.

This analysis was based on data obtained using the software MAPLE specifically the “fsolve” function. With this tool we

ROBUSTNESS OF INTEGRAL TIME DELAY SYSTEMS 123

swept the left semi-plane for roots, with this data we generated a graph that gave us the behavior of the roots for a given system.

It is well known that roots of the neutral cuasipolinomyal behave chaotically in a certain radius close to the origin [1], and then they behave exponentially, until the real coordinate presents very small changes and stay trapped in a continuous chain. At this point we can assume that real coordinate distance to the intersection with cero will not change.

It is also well know that roots behave continuously with their respective parameters [1], so we can find out how robust our system will be depending upon the horizontal distance from the real part of the root to cero. So when plant disturbances appear, roots will have a bigger zone to move in before the system goes unstable.

Figure 4 shows the graph obtained for plant (14) and the behavior of its roots when the controller is tuned with the two given tables.

Fig. 4. Roots Behavior for the plant (14).

In this graph we can observe that the roots of the system tuned with table (8) are closer to the origin, which makes this system less robust than the other tuned with table (11). This shows a more robust root behavior because the zone where the roots' real part stays practically constant is considerably less likely to make the system unstable.

VI. CONCLUSIONS

An analysis of robust stability for the integral time delayed systems is presented. For the examples that we considered, the PD controller obtained from the table given in [9] presents a greater stability region than that generated with the PD controller tuned with the rules given in [12]. The approach can be extended in order to analyze the robust stability regions in other type of systems using other tables.

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