TIME DELAY ROBUST CONTROL – PROGRAM IMPLEMENTATION

TIME DELAY ROBUST CONTROL - PROGRAM IMPLEMENTATION Zdenka Prokopová, Roman Prokop Faculty of Applied Informatics, Tomas Bata University in Zlín, Nad Stráněmi 4511, 760 05 Zlín, Czech Republic e-mail: [email protected] and fax: +420 57 603 5255

Abstract: The contribution presents a control design method for robust tuning of continuous-time controllers for SISO with time delays. Controllers are obtained via general solutions of Diophantine equations in the ring of proper and stable rational functions. The methodology covers both stable and unstable systems. The control objectives are obtained through divisibility conditions in this ring. Robustness of proposed algorithms can be studied through the infinity norm and sensitivity function. Several approximations of time delay terms and two control structures of controlled loop were considered. A scalar parameter was proposed as a tuning knob for influencing of stability, minimization of the sensitivity function and uncertainty evaluation. A Matlab-Simulink package was developed for automatic design and simulation of proposed algorithms. Copyright © 2007 IFAC Key words: Rings, time delays, SISO systems, robust control, PID controllers.

1. INTRODUCTION Continuous-time controllers of the PID type have been widely used in many industrial applications for decades. There are several features to their success, e.g. structure simplicity, reliability, robustness in performance, see e.g. (Aström and Hägglund, 1995; Rad and Lo, 1992). However, the choice of the individual weighting of the three actions, i.e. proportional, integral and derivative has been a problem. Moreover, the number of tuning parameters in more sophisticated PID modifications proposed in (Aström, et al., 1992) is even higher. Robust controllers and plant uncertainty have become a requisite and popular notions in control theory during for many years. Robustness also influenced design and tuning of PID controllers (Morari and Zafiriou, 1989; Prokop and Corriou, 1997). The necessity of robust control was naturally developed by the situation when the nominal plant (used in control design) differs from the real (perturbed) one.

A suitable tool for parameter uncertainty is the infinity norm H∞. Hence, a polynomial description of transfer functions had to be replaced by another one. A convenient description adopted from (Vidyasagar, 1985; Kučera 1993; Doyle, et al., 1992) is a factorization approach where transfer functions are expressed as a ratio of two Hurwitz stable and proper rational functions. Then, the conditions of robust stability can be easily formulated in algebraic parlance and all controllers are obtained and parameterized via linear Diophantine equations in an appropriate ring. Moreover, further tasks as disturbance rejection can be easily solved through the divisibility conditions of all stabilizing controllers. A useful and interesting application of the mentioned robust philosophy can be seen in time delay systems. The dynamics of many technological plants can be adequately approximated by first order transfer functions plus dead time. This transfer function can be stable or unstable one. The situation where linear systems have unstable poles may occur e.g. in

a continuous-time stirred exothermic tank reactor, in polymerisation processes or in a class of biochemical processes where the processes must operate at an unstable steady state. Moreover, a time delay is an inherent part of many technological plants. For SISO systems of the first and second order this approach yields a class of PID like controllers. The methodology is proposed and analysed in (Prokop and Mészáros, 1996; Prokop and Corriou, 1997; Prokop and Prokopová, 1998; Prokop, et al., 2001). The algebraic approach gives for two degree of freedom structure a nontraditional PID controller proposed in e.g. (Aström, et al., 1992; Morari and Zafiriou, 1989) in a different way. The fractional approach for SISO controllers brings a scalar parameter m > 0 which strongly influences the dynamic of the feedback system as well as the robustness and sensitivity of proposed controllers. 2. THEORETICAL BACKGROUND Any transfer function G(s) of a (continuous-time) linear system has been traditionally expressed as a ratio of two polynomials in s. For the purposes of this contribution it is necessary to express the transfer functions as a ratio of two elements of RPS(s). It can be easily performed by dividing, both the polynomial denominator and numerator by the same stable polynomial of the order of the original denominator. Moreover, a scalar parameter m>0 seems to be a suitable „tuning parameter” influencing control behaviour as well as robustness of the closed loop system. Then all transfer functions could be described by

b( s ) b ( s ) ( s + m) n B ( s ) , = = G (s) = a( s) A( s ) a(s) ( s + m) n n = max(deg(a), deg(b)), m > 0

or

A − A% ;

B − B% ≤ ε

3. APPROXIMATION OF TIME DELAY TERMS Systems with time delay are obtained by an appropriate approximation of the term e-Θs see e.g. (Prokop, et al., 1997). The approximation for time delay processes can be shown by a first order model G1 ( s) =

Note that the divisibility in the ring RPS(s) is defined through all unstable zeros of elements in the ring. More precisely, element A divides element B in RPS(s) iff all zeros including infinite ones of A are also zeros of B. The H∞ norm in the ring RPS(s) is defined by

G = sup G ( s) = sup G ( jω ) ω ∈E

G1 2 = sup { G1 ( s) + G2 ( s ) } G2 Re s ≥ 0

1 2 2

(3)

Ke −τs s +α

(5)

The value of the parameter α > 0 represents stable systems, α < 0 represents unstable ones and α = 0 is an integrator. For linear control design, it is necessary to approximate the time delay term in (5). It can be done in several methods. The simplest case is to neglect the delay term e-τs. Then the time delay is considered as a perturbation of a nominal transfer function. So the first nominal approximation is K s +α

(6)

Next two approximations are based on the Taylor series approximation of e-τs in the numerator or in the denominator. Approximations e-τs ≈ (1-τs) ≈ (1+τs)-1 then give K (1 − τs ) b1 s + b0 = s +α s +α

(7)

K 1 b0 = 2 ( s + α ) (1 + τs ) s + a1s + a0

(8)

G3 ( s ) =

(2)

(4)

where ε1, ε2, ε are positive constants.

(1)

b( s ) −τ s e B(s) ( s + m)n , = G (s) = a( s) A( s) ( s + m) n τ > 0, m > 0

G1 ; G2 =

A − A% ≤ ε1 , B − B% ≤ ε 2

G 2 (s) =

Time delay systems with “pure dead time” will be considered as system (1) in the form

Re s ≥ 0

This (called infinity) norm is the radius of the smallest circle containing the Nyquist plot of the transfer function and it is a convenient tool for the evaluation of uncertainty. The distance of two elements in the ring can be easily expressed by this norm. Let G ( s ) = B( s ) / A( s ) be a nominal plant and consider a family of perturbed systems G% ( s ) = B% ( s ) / A% ( s ) where

G4 ( s ) =

The last model can be obtained by the traditional Padé approximation

K G5 ( s ) = s +α

1− 1+

τ 2

τ

2

s = s

b1 s + b0 2

s + a1 s + a 0

(9)

All approximated transfer function can be written in Rps in the form

b1 s + b0 G ( s) =

b1 s + b0 s 2 + a1 s + a 0

=

( s + m) 2 s 2 + a1 s + a 0

B(s) = (10) A( s )

( s + m) 2 Figs. 1, 2 show step responses and Nyquist plots of the stable plant (5) and its approximated transfer functions (6) - (9).

G v= v Fv

G n= n Fn

G w= w Fw

u

C(s)

G=

y

B A

Fig. 3. General feedback system. 2 G2

G3

G4

Disturbances v =

1

output and input, respectively and

G1

G5

Gv (s) G ( s) and n = n corrupt the Fv ( s ) Fn ( s ) w=

G w ( s) Fw ( s)

represents a reference. All transfer functions, B , Q ,

0

-1 0

5

Time (secs)

10

A P R Gv G n G w , , , are supposed to be coprime in P Fv Fn Fw

15

Fig. 1. Step responses (K=2,α=1,τ=5). 2 1.5

G1

1 0.5 0 -0.5

G2

G5

-1

G4

-1.5 -2

G3

-2.5 -3 -4

-3

-2

-1

0

1

2

Fig. 2. Nyquist plots (K=2, α=1,τ=5). The similar principle can be easily extended for higher order degrees (see Prokop, R. and A. Mészáros (1996)). 4. CONTROL DESIGN IN RPS represents for C ( s) = Q( s ) a classical feedback oneP( s)

degree-of freedom (1DOF) loop with the control law (11)

In a two-degree-of freedom (2DOF) control system, the controller C(s) consists of two transfer functions Q and R . The control law is then governed by P P P( s ) u = R( s) w − Q ( s) y

AP0 + BQ0 = 1

(13)

Q0 − AT P0 + BT

(14)

in a parametric form

A general feedback system is shown in Fig. 3. It

P( s ) u = Q( s ) [w − y ]

RPS(s). The disturbance v usually represents a harmonic signal while n can be modeled by a stepwise signal. The objective is to design controller transfer functions P(s), Q(s), R(s) such that the feedback system is internally BIBO stable, the reference error tends asymptotically to zero and the disturbances v and n are asymptotically eliminated from the plant output. Commonly, it is desirable that the feedback system be internally BIBO stable in the sense that any bounded input produces a bounded output (e.g. Kučera 1993, Vidyasagar 1985). All transfer functions of the feedback system (Fig. 3.) have common denominator AP+ BQ. One of the nice and convenient results of the algebraic philosophy is that this denominator should be a unit in the ring RPS(s). In other words, the term (AP+BQ)-1 resides in RPS(s) and the feedback system is BIBO stable. If the elements A and B are coprime in RPS(s) then all stabilizing controllers are given through a solution of Diophantine (Bézout) equation:

(12)

where T varies over RPS(s) while satisfying P0+BT≠0. From the practical point of view, it is often desirable to ensure more than stability. Probably the most frequent problem of importance is that of reference tracking. Then the tracking error e tends to zero iff a) Fw divides P for 1DOF b) Fw divides 1-BR for 2DOF Another control problem of practical importance is disturbance rejection and disturbance attenuation. In both cases, the effect of disturbances v and n should

be asymptotically eliminated from the plant output. Since the both disturbances are external inputs into the feedback part of the system, the effect must be processed by a feedback controller. The algebraic approach enables to express these conditions by the notion of divisibility. The details can be found e.g. (Kučera, 1993, Prokop and Corriou, 1997 or Prokop et al. 2001). More precisely, Fv, must divide the multiple AP and Fn the multiple BP. The final feedback controller is given by Cb =

Q Q = P P0 Fv 0 Fn 0

(15)

where P0, Q are the solution of the equation AFv 0 Fn 0 P0 + BQ = 1

controller for the nominal plant. The third strategy is based on robust conditions (18). The main menu window is shown in Fig. 4. The program system enables design and simulation of a wide spectrum of robust control problems. It is desirable for robust control that the nominal and perturbed (really controlled) plants are different. The program is user friendly as much as possible. User can set up two different plants with different degrees, time constants and delays. First, a nominal plant of a desired structure (first or second order) with its transfer function and dead/time has to be defined. Then, a transfer function of the given perturbed plant can be set up. The default choice is the identity of nominal and perturbed systems.

(16)

Moreover, for simultaneous reference tracking and disturbance rejection and attenuation Fw must divide P0 for 1DOF Fw Z + BR = 1

(17)

or for the 2DOF structure. For robust control, it is necessary to choose a part of all stabilizing controllers (13), (14) which stabilize also perturbed plants (4). The answer can be found in (Vidyasagar, 1985). For perturbed plants (4) choose such P, Q in (11), (12) which fulfill the conditions

ε1 P0 + BT + ε 2 Q0 − AT < 1 or

ε

P0 + AT Q0 − AT

< 1

(18)

Another insight into robustness the notion of the sensitivity function: ∈=

y = A( P0 + BT ) v

(19)

utilizes in the sense of (Doyle, et al., 1992). For the mentioned SISO systems, sensitivity function ∈ is a nonlinear function of m > 0 and it can be minimized by a simple scalar optimization method. In this way, the „most robust“ controller of given structure can be obtained.

Fig. 4. Main menu window. If the nominal plant contains transport delay the important step is to choose an approximation method of the time delay term. There are four possibilities of approximation as it is illustrated in Fig. 5. Program does not support Pade or Taylor approximation in denominator with 2nd order nominal plant since the resulting controller would be too high order. The program also covers both well-known control structures: feedback (1DOF) and feedbackfeedforward (2DOF) one. Further, there are two options for the control loop structure and three mentioned options for the control design. These choices are defined in the main menu according to Fig. 6 and Fig. 7.

5. PROGRAM IMPLEMENTATION A MATLAB-package with simulation support in SIMULINK was developed for nominal plants of the first and second orders with and without dead-time. The program for automatic control design was created in Matlab R12 and Simulink. It covers all three control design strategies and both structures of the closed loop. The first strategy generates the controller by a user defined value of the tuning parameter m >0. The second one minimizes the sensitivity function and without any knowledge of the perturbed plant tries to find the "most robust"

Fig. 5. Approximation of dead-time.

Fig. 6. Choices of control strategies.

Nyquist Diagram

0.8 0.4

Fig. 7. Structure of the control scheme.

0

The mentioned algebraic approach is able to design controllers which are prepared for simultaneous tracking and disturbance rejection. There are two possibilities which can be defined in the sub window according Fig. 8.

-0.4 0.8

-

-0.8

-0.4

0

0.4 Real Axis

0.8

Fig. 11. Nyquist plots of nominal and perturbed plants.

Fig. 8. Ability of the controller. Then the simulation of the designed controller and the perturbed plant can start. In some cases it is necessary to adjust simulation parameters: simulation horizon, initial and final reference values, time of the load disturbances and so on. It can be performed very simply in the part of the main window in Fig. 9.

For a deeper insight into control behaviour, stability and robustness the open loop Nyquist plot of the perturbed plant and final controller can be used for more experienced user. A typical example is shown in Fig. 11. Also the sensitivity function of the control system can be displayed in similar way. The simulation of the perturbed plant with the controller designed for the nominal transfer function is performed in the standard Simulink environment, an example is described in Fig. 12.

Fig. 9. Simulation parameters. Software tools also enable to corrupt the measured output of the controlled plant by harmonic or ramp signal. The user can defined the presence and type of the disturbance according to Fig. 4. A very important part of the simulation is the display of obtained results. The comparison between nominal and perturbed plant can be seen in two figures. First of them compares step responses and the second one Nyquist plots. A typical two step responses are shown in Fig. 10 while the Nyquist plots of nominal and perturbed plants are shown in Fig. 11.

Fig. 12. Simulation scheme. All simulation variables can be stored and transferred out of the Matlab workspace and ITAE, IAE or IE criteria can be calculated as a tool for comparison and quality evaluation of the control behavior. 6. ILLUSTRATIVE EXAMPLE

S tep R esp on se

Example 1: Let the nominal stable system has the transfer function 1

G0 ( s ) =

0

4

T im e (sec)

Fig. 10. Step responses.

8

12

e −s s2 + s +1

(20)

The aim of the controller is to stabilize the control loop, track a stepwise reference and to compensate a harmonic disturbance of a given frequency and arbitrary amplitude. The control law was constructed according to (11), (12) for both structures. All controller parameters are implicitly nonlinear functions of m >0. The control law is obtained for the nominal plant (8) with the Taylor approximation in the numerator of (20). The control response depicted in Fig. 13 shows the behavior m=1, ω=1.

7. CONCLUSIONS 2 1.5 1 0.5 0 -0.5 0

10

20 Time (second)

30

40

Fig. 13. Control responses of the second order system with Taylor numerator approximation.

A design method based on fractional representation was developed for SISO continuous-time systems generally with time delay. The time delay term can be approximated in various ways. Resulting control laws for first and second order systems give a class of generalized PI and PID structures. The algebraic methodology enables to derive controllers rejecting also disturbances. The robustness and control behavior can be tuned and influenced by a single scalar parameter m>0. The tuning parameter can be chosen arbitrarily or it is a result of the robust and sensitivity optimization. The proposed methodology is supported by a Matlab + Simulink program system for automatic design and simulation. ACKNOWLEDGEMENT

Example 2: Consider an unstable transfer function with time delay

GS ( s) =

e −0.5s s −1

(21) REFERENCES

The control responses for FB (m=1.5) and also FBFW (m=1.6) structures with plant (21) are shown in Fig. 14 and Fig. 15, respectively. Pade approximation was used.

4 3 2 1 0 0

10

20 30 Tim e (second)

40

Fig. 14. FB response of unstable plant.

2 1.5 1 0.5 0 -0.5 0

10

20 30 Time (second)

This work was supported in part by the grants of Ministry of Education of the Czech Republic No. MSM 708 835 2102 and by the FRVS 2194/2005.

40

Fig. 15. FBFW control response of unstable plant.

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