1

Interactive tool for analysis of time-delay systems with dead-time compensators

Jos´e Luis Guzm´an a , Pedro Garc´ıa b , Tore H¨agglund c , Sebasti´an Dormido d , Pedro Albertos b , Manuel Berenguel a a Dep. b Dep.

de Lenguajes y Computaci´ıon, Universidad de Almer´ıa, Spain.

Ingenier´ıa de Sistemas y Control, Univ. Polit´ecnica de Valencia, Spain. c Department

d Dep.

of Automatic Control, Lund University, Sweden.

Inform´ atica y Autom´ atica, ETSI Inform´ atica, UNED, Madrid, Spain.

Abstract The problem of controlling delayed systems has widely been studied in the control community. The Smith Predictor (SP) scheme was the most famous and efficient technique to solve these first problems. After that, robustness problems and those related to the control of integrating and unstable delayed processes were resolved. This work presents a summary of most relevant control structures which have been used in the solution of such problems. In this way an interactive tool have been developed which allows to study the problem on the control of systems with large delays. The developed tool allows to compare the performances of PID controllers and Smith Predictor approaches for processes with delay as well as other control structures to face the different problems that the original SP has with integrator and unstable models. Robustness problems due to presence of model uncertainties can also be studied. Furthermore, some real industrial processes are also described as motivation in the study of the problems presented in the paper. The tool incorporates a set of typical transfer functions based on these real industrial processes. Key words: time-delay systems, Smith predictor, delayed unstable processes, delayed integrating processes, interactive tools

1

Introduction

Most industrial processes are characterized by the presence of time delays. These time delays may be intrinsic to the process to be controlled, such as in 1 A preliminary shortened version of this work was presented in the 7th IFAC Symposium on Advances in Control Education, Madrid (Spain), June 2006.

Preprint submitted to Control Engineering Practice

26 March 2007

chemical and biological processes, distillation columns, mass or energy transport, etc. Delays can also be introduced in the controller design itself (computation time of the control algorithm, distributed systems, remote control, communication networks, sensors and/or actuators induced delays, etc.). From a control point of view, time delays produce an increase in the system phase lag, therefore decreasing the phase and gain margins and limiting the response speed of the system (system bandwidth). On the other hand, simple models are very important in the process industry in such a way that the dynamics of most industrial processes are usually approximated/modelled by either a first-order plus dead-time (FOPDT) model or an integrator plus dead-time (IPDT) model. Furthermore, there still exist many unstable processes available at industry mainly found in the chemical area. In general, there are modelled by unstable FOPDT and unstable secondorder plus dead-time (SOPDT) models [17]. For all these models the deadtime time part not only represents the current delay but also the high order dynamics of the real process [27]. As well-known, the control problem for delayed-systems have widely been studied [10, 25]. Conventional controllers, such as PID controllers, have been used when the dead-time is small [6] but they show poor performance when the process exhibits long dead-times since a significant amount of detuning is required to maintain closed-loop stability [11]. In these cases, it is convenient to use a dead-time compensating method [12, 27]. The Smith Predictor [26] and its many extensions, generically named DeadTime Compensator (DTC) [23], can be considered as the first control methods for single-input-single-output (SISO) linear processes showing a delay in their input or output [10, 25]. The main advantage of the Smith predictor method is that time delay is eliminated from the characteristic equation of the closed loop system. Thus, the design and analysis problem for processes with delay can be translated to the one without delay. On the other hand, it is well-known that the Smith Predictor presents several problems in presence of integrator or unstable processes and also due to modelling errors [23, 20]. During last years, the control community has performed a great effort against such drawbacks existing available several modifications in order to face them [30, 19, 21, 22, 29, 16, 5]. Therefore, considering the previous discussion, control engineers have available a great number of different control schemes for attempting the control of delayed systems. This availability is very rich for the control community but, for an engineer working in the industry, it can be sometimes difficult to choose between all possible solutions. Hence, in order to facilitate the study and tuning of all kind of DTC, this paper is focused on studying those DTC that after some basics manipulation it can be shown to be compatible with 2

the original Smith predictor structure. This choice is useful in order to help the understanding of the different problems (stable FOPDT, IPDT, unstable FOPDT, unstable SOPDT, and robustness) and also from an educational point of view. In this way, the selected DTC have been [19] for IPDT processes, [16] for systems described by unstable FOPDT and SOPDT, and [21] for solving robustness problems. A brief description of these control structures will be provided along the paper. So, taking the previous description as motivation, this work presented an interactive tool in order to facilitate the study of problems for systems with large delays. The developed tool allows working with the commented control algorithms, compare them, and study the different problems related with stability, performance, and robustness in an interactive way. The term interactivity is being more common each day in the field of teaching in Automatic Control. In general terms, interactivity could only be understood as the response to some action performed by the user. However, looking at the results obtained in the field of Automatic Control this term seems more ambitious [3, 4, 7, 9]. At the moment, interactivity is associated with a set of graphical representations, elements and parameters interconnecting each others, in such way that if some element is modified the rest are updated immediately and only spending few seconds. This powerful element has provided very important new possibilities in researching and teaching, being possible to study problems at different complexity levels and using a new interactive philosophy relatively difficult some years ago. Nowadays a new generation of software packages has appeared. Some of them are based on objects that admit a direct graphic manipulation and are automatically updated, so that the relationship among them is continuously maintained. Ictools and CCSdemo [14, 31], developed in the Department of Automatic Control at Lund Institute of Technology or SISO-GPCIT at the University of Almer´ıa and UNED [8] are good examples of this new educational philosophy applied to the field of automatic control. Therefore, this work deals with the application of all these interactive features to the control problem of delayed systems. In this way, an Interactive Tool is described using the control schemes commented above and studying the most common control problems. Several examples are presented in order to express better the interactive capabilities of the developed tool. The paper is organized as follows. The next sections briefly describes the Smith Predictor scheme and its variants to treat with robustness problems and also with integrate and unstable systems. Section 3 presents the developed interactive tool showing its main features. Some illustrative examples are shown in section 4 using the presented tool. Finally, conclusions are outlined in section 5. 3

2

Control schemes to control time delay systems

Many processes present delay in their input or output variables. In many cases we can represent their behavior by means of a transfer function described as P (s) = G(s)e−Ls

(1)

Considering this representation, this section deals with the main drawbacks and solutions of the Smith predictor scheme [26]. That is, robustness (plant and/or delay) and extensions of the Smith’s idea to the case of integrating or unstable delayed systems.

2.1 Smith Predictor for stable systems

As commented above, the Smith Predictor [26] is the best know and most widely used algorithm for dead-time compensation. The typical scheme for this approach is shown in Figure 1.

r (s) +

K (s)

d (s) + +

y(s)

Gr ( s )e - Lr s

G (s)

yˆ p (s)

e

- Ls

-

+

+ + Fig. 1. Smith Predictor scheme.

From this scheme, the following expressions can be obtained K(s)Gr (s)e−Lr s r(s) 1 + K(s) (G(s) − G(s)e−Ls + Gr (s)e−Lr s ) (1 + K(s)G( s) − K(s)G(s)e−Ls )Gr (s)e−Lr s + d(s) 1 + K(s) (G(s) − G(s)e−Ls + Gr (s)e−Lr s )

y(s) =

(2)

where Gr (s)e−Lr s represents the real process, K(s) any conventional controller, and G(s) and e−Ls the process model free of delay and the model delay, respectively. 4

If the process and delay models are considered to loyally represent the real system, G(s)e−Ls = Gr (s)e−Lr s , then the closed-loop response to set-point and disturbance inputs is given by: 



K(s)G(s)e−Ls K(s)G(s)e−Ls y(s) = r(s) + 1 − G(s)e−Ls d(s) 1 + K(s)G(s) 1 + K(s)G(s)

(3)

From (3) the well-known powerful of the Smith Predictor idea can be observed. The controller K(s) can be designed as if the process has no time delay, in such a way that the response of the closed-loop system will simply have an additional time delay. However, this original scheme presents some drawbacks against different situations such as described below. Remark 1 Note in (3) that, even in the ideal case if the system G(s) has any unstable pole, the closed-loop system will also be unstable. On the other hand, from some easy algebraic manipulation it is possible rewriting the transfer function (3) as

Gyd (s) =

y(s) G(s)e−Ls K(s)G(s)e−Ls = + (G(s) − G(s)e−Ls )d(s) d(s) 1 + K(s)G(s) 1 + K(s)G(s) (4)

Remark 2 Note that, the closed-loop system has   zero steady-state to step load −Ls = 0. However, when a system disturbances only if lims−→0 G(s) − G(s)e with an integrating mode G(s) = not zero:

Gs (s) s

is considered, the steady-state error is

lim (G(s) − G(s)e−Ls ) = lim Gs (s)L

s−→0

s−→0

(5)

Remark 3 It is well-known that the SP scheme presents performance and stability problems in presence of model uncertainties [23, 20]. A good understanding of this problem can be found in [27]. In the SP scheme the maximum value for maintaining closed-loop stability can be given by 1 |  P (iω)| < |P (iω)| |T (iω)| where T (iω) is the complementary sensitivity transfer function, and P = (Pr − P ) represents variations on the process transfer functions. 5

For controllers with integral action, T (0) = 1, and considering uncertainties in the time delay, variations in the phase are only presented in such a way that an estimate of permissible variations in the time delay are given by π 1 |  L| < ≈ L 3 ∗ ωb L ωb L where ωb is a frequency such that |T (iω)| is close to 1 for 0 ≤ ω ≤ ωb and L = (Lr − L) represents variations on the process delay. Form the previous analysis, it can be deduced that systems with large values of ωb L thus require that the time delay be known accurately. Otherwise stability problems due to uncertainties could raise [27].

2.2 Improving robustness in the Smith Predictor

As commented above, SP may be very sensitive to model uncertainties [20]. Therefore, in order to improve the robustness in the SP scheme, from [21] the structure shown in Figure 2 is proposed, where F (s) is defined as a n-order low-pass filter with unitary static gain F (0) = 1.

r (s) +

K (s)

d (s) + +

y(s)

Gr ( s )e - Lr s

G (s)

e

+ +

- Ls

-

+

F (s)

Fig. 2. Structure proposed by [21]

Let us consider multiplicative uncertainties in such a way that the process transfer function is given by Pr (s) = P (s)(1+Wm (s)), where Wm (s) defines the process multiplicative uncertainty bound. Hence, the robust stability condition is given by 1 :    



 KG F Wm  < 1 1 + KG

(6)

1

From now on, in order to simplify the notation, the argument (s) is omitted, if the meaning of the equations is clear.

6

Note that the filter F , does not affect the stability properties and it can be chosen to improve the robustness of the system at the desired frequency region. Let us assumed a FOPDT model:

P (s) =

Kp −Ls e Ts + 1

and a simple PI-controller

K(s) = K

(Ti s + 1) Ti s

If the tuning rules proposed by [11] are used in order to set PI parameters, that is Ti = T and K = k/Kp , and the filter F is defined as a first order low pass filter with filter time constant Tf , the maximum value for maintaining closed-loop stability is given by     1 Wn