H∞ LOOPSHAPING USING GRAPHICAL LOOPSHAPING IDEAS

Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain

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H∞ LOOPSHAPING USING GRAPHICAL LOOPSHAPING IDEAS Fernando Tadeo*, Francisco del Valle Departamento de Ingeniería de Sistemas y Automática Facultad de Ciencias, Universidad de Valladolid, 47011 Valladolid, Spain [email protected] Omar Pérez, William Colmenares Universidad Simón Bolívar, Centro de Automatización Industrial Apartado 89000, Caracas 1080, Venezuela Abstract: This paper studies the design of robust controllers using a robust loopshaping approach, by selection of the weighted plant using graphical loopshaping ideas. The H∞ Loop Shaping method is applied to calculate an optimal controller. The problem of choosing the desired shape of the open-loop transfer function needed by this method is addressed by considering the available uncertainty information and applying Graphical Loopshaping ideas. Thus, this methodology considers not only the robustness properties of the shaped plant, but also those of the real plant. Several examples show the advantages of using this approach. Copyright © 2002 IFAC Keywords: Robust Control, Hinfinity Control, Distillation columns

1.

INTRODUCTION

Among all the available Robust Control techniques the H∞ Loop Shaping (H∞LS) procedure has been chosen, because it has been proved to be efficient in process control applications (Green and Limebeer, 1995). The approach involves the robust stabilization to additive perturbations of normalized coprime factors of a shaped plant. Prior to robust stabilization, the open-loop singular values are shaped using preand post- compensators. Then, the resulting shaped plant is robustly stabilized with respect to coprime factor uncertainty using H∞ optimization.

Obtaining a precise non-linear model that accurately matches the plant at all working points is a difficult problem. So, most of the advanced controllers are usually designed using a linear model of the process based on fixed information of the plant that is imperfect and incomplete. Thus, control quality may deteriorate when working conditions change. In view of these difficulties, robust control design methods seem especially appropriate, since they give linear controllers with good stability margins (Zhou et al., 1996).

As pointed out by Pantas and Walsh (1996), one difficulty of the H∞LS design method is that it does not directly address the robustness properties of the real plant, but rather it is concerned with the shaped plant. Unfortunately, there is no direct connection between the uncertainties of the shaped and unshaped plant, as they are modified by the weighting functions considered.

The essence of robust control is to model the uncertainties themselves and to incorporate them in the design procedure of the control system, with the aim of ensuring stability and performance at all working points. Usually it is possible to identify multiple local linear models at different operating regions, which can be used to evaluate the expected uncertainty of the nominal model. Then this uncertainty information is used to design a controller that ensures robust stability and performance.

*

This paper shows a methodology that solves this problem by considering the robustness properties of the real plant in the selection of the weights of the shaped plant. Constraints are added to the selection of the shaped plant by considering the uncertainty in the

Author to whom correspondence should be addressed

301

real plant, as in Doyle et al. (1992). Once selected a robust shaped plant the controller is designed by application of the H∞LS design method. This method is simpler and more intuitive than other robust control techniques.

u

The paper is organized as follows. First, the H∞ Loopshaping is presented. Then the use of Graphical Loopshaping Methods to select the “shaped” plant is discussed. Finally, results of the application of this technique to several examples are given and discussed. 2.

NX + DY = I

-

In practical designs, the Loop Shaping Design Procedure (LSDP) can be applied (McFarlane and Glover, 1990). The complete design procedure is the following: 1. Using pre- and post-compensators (W1 and W2) ~ the singular values of the nominal plant G are modified to give a desired loop shape: ~ GS = W1GW2 , which should not contain unstable hidden modes. 2. GS is considered to be perturbed by normalized coprime uncertainties, and an optimal feedback controller KS is then synthesized using the H∞LS approach. 3. The combination of the H∞LS controller and the compensators gives the final controller: (4) K = W2 K SW1

G = ( D + ∆D )−1( N + ∆N ) ,

where ∆D ,∆N ∈ RH ∞ . The objective of robust stabilization is to stabilize the ~ nominal plant G and the family of perturbed plants defined by (1) G = ( D + ∆ )−1( N + ∆ ) / [∆ ∆ ] <ε

{

D

G

Figure 2: Augmentation of the GLS controller with the H∞LS compensator

In this technique two uncertainty blocks are used, as depicted in Figure 1, one on each of the factors in the factorization:

KG

KH ∞

(Where H *

denotes H T ( − s ) ).

coprime

y

-K s

which fulfils the identities

and NN + DD* = I

D-1

LG

transfer function matrices ( N , D ∈ RH ∞ ) such that *

N

Figure 1: Coprime perturbed plant

H∞LS, as introduced and solved by Glover and McFarlane (1989), considers the stabilization of a plant, which has a normalized left coprime ~ factorization: G = D −1N . That is, N and D are stable there exists X ,Y ∈ RH

∆D

K

H∞ LOOPSHAPING

∞

∆N

N

D

N ∞

}

where ε is the Stability Margin. The objective of H∞LS is the maximization of this Stability Margin. It can be shown that this is equivalent to find a stabilizing K that minimizes (2) éK ù ~ ~ γ = ê ú( I + GK )−1 D −1 ëI û ∞

This minimization can be calculated by solving an Algebraic Riccatti Equation, as shown by McFarlane and Glover (1990).

Different methods to select the compensators have been studied: Whidborne et al. (1994) propose the use of the Inequalities Method, Pantas and Walsh (1996) the use of the Phase Crossover Frequency, and Tang et al. (1996) the use of Genetic Algorithms. In order to consider the robustness properties of the real plant in the design, this paper shows a methodology that solves this problem by considering the robustness properties of the real plant in the selection of the weights of the shaped plant. This technique is based on a graphical approach to Loop Shaping proposed by Doyle et al. (1992).

Compared with other H∞ design methods, the main advantage of the H∞LS method is that it does not require the so-called γ-iteration to calculate the optimal controller. Also there are available relatively simple formulas to calculate the controller. On the other hand it does not (directly) include any closedloop specification, which must be included by considering, instead of the nominal plant, a shaped plant.

302

3.

MULTIVARIABLE GRAPHICAL LOOP SHAPING

σ (L ) ≤

Prior to coprime robust stabilization, the open-loop singular values must be shaped in order to obtain an adequate (and allowable) open-loop transfer function. This paper proposes to select this weights using a Multivariable Graphical Loop Shaping (MGLS) method, based on the ideas of Doyle et al. (1992). In (Tadeo et al. 2000) the technique was presented for the SISO case, deriving necessary conditions. This paper presents the technique for the MIMO case, deriving sufficient conditions.

σ (WS S )σ (I + ∆WT T )−1 < 1 Using the facts that σ (∆ ) < 1 σ (I + A) ≥ 1 − σ ( A) , that condition (sufficient condition) σ (W S S ) <1 1 − σ (WT T ) Or equivalently: σ (W S S ) + σ (WT T ) < 1

(8) and holds

if:

(9)

W S = wS I and WT = wT I , then, as σ (wA) = wσ ( A) (where w is any transfer function), eq. 7 is equivalent to say that wS σ (S ) + wT σ (T ) < 1 From the definition of S and T, and the singular value properties: 1 σ (S ) = σ (I + L )−1 = σ (I + L )

If

Let P contains plants with the same number of unstable poles. Consider perturbed plant transfer functions of ~ the form G = ( I + ∆WT )G , where ∆ is a variable

multiplicative direct uncertainty).

(7)

+1

wT − 1

Proof: The robustness condition given in eq 5 is fulfilled if the following (sufficient) condition holds:

Suppose that the plant can be represented by a transfer function G, which belongs to a set of possible plants P. A controller K provides Robust Stability if the feedback system is internally stable for every plant in P (that is, it gives feedback stability to every plant in the set P without cancelling any unstable pole ore zero of the plant).

stable transfer function such that

wS

(

∆ ∞ ≤ 1 (output

(

)

) (

σ (T ) = σ L(I + L )−1 = σ L−1 + I

)

−1

=

(

1

σ I + L−1

)

It is well known (Doyle et al., 1992) that K provides robust stability if and only if WT T ∞ < 1 , where ~ ~ T = GK ( I + GK ) −1 is the complementary sensitivity.

Then that condition is equivalent to 1 1 + wT <1 wS σ (I + L ) σ I + L−1

Similarly, if uncertainty

Using the fact that σ (I + A) ≤ 1 − σ ( A) , this condition holds if: (sufficient condition) (10) 1 1 + wT <1 wS −1 1 − σ (L ) 1−σ L

(

an

output inverse multiplicative description is used −1 ~ ( P = { G / G = ( I + ∆WS ) G } ) the controller K

provides robust stability if and only if WS S ~ where S = ( I + GK )−1 is the sensitivity.

∞

)

( )

<1,

From singular values properties, this holds if:

The Robust Performance problem can be stated as calculating a controller K that provide robust stability and certain performance specifications are met for all the plants in the uncertainty set. The following Robust Performance problem will be considered (SanchezPeña and Sznaier, 1998): design a controller K such that the feedback system for the nominal plant is internally stable and (5) WS S (I + ∆WT T )−1 < 1

σ (L ) ≥

wS − 1

σ (L ) ≤

wS

wT

−1

wT − 1

+1

It is convenient to study the shape of L in the regions where ı (L ) >> 1 (usually at low frequencies) or

( )

ı L−1 >> 1 (at high frequencies).

∞

The main result of the paper is now presented:

The design can be carried out graphically: wS − 1 − 1 at frequencies where wS > 1 1. Plot wT

Lema 1: if diagonal weights are selected ( W S = wS I and WT = wT I ), sufficient conditions to ensure internal stability and fulfill the robust performance condition (5) are: wS − 1 σ (L ) ≥ −1 (6) wT

2. Plot

wS wT − 1

+ 1 at frequencies where

wT > 1

3. Find an open-loop transfer function L(s) such that ı (L ) lies above the first curve (eq 9) and σ (L ) 303

responses of the system are compared with the graphical loopshaping controller in Figure 4 with the nominal model. It can be seen that the controller designed using the technique presented in this paper improves the nominal response. Figure 5 compares the responses when θ=1, with the robust controller and the controller proposed by (Doyle et al., 1992). Although for this parameter the original controller gives marginal stability, the robust controller maintains the nominal performance.

below the second (eq 10). To make the controller proper, roll-off at high frequencies at least as fast as ı (G ) >> 1 . To obtain internal stability the slope of ı (L ) should be small near the desired crossover frequency (about 20 dB/dec for step changes in references and disturbances (Skogestad and Postlethwaite (1998)).

It must be pointed out that the design procedure is valid also for plants with poles or zeros in the Right Half Plane (RHP), as long as they are included in the selected open-loop transfer function, and the crossover frequency is selected to fulfill the achievable bandwidth limitations. A complete study of these limitations (for the SISO case) can be found in Skogestad and Postlethwaite (1998). By selecting adequate weights it is possible to find an appropriate L(s), although it is not guaranteed that an appropriate L(s) exists. The MGLS method can be applied to obtain an open-loop transfer function LG, which then is robustly stabilized by application of the H∞LS approach, obtaining a robust open-loop transfer function LH∞. It is important to notice that the available information about uncertainties in the model and performance specifications is considered when applying the GLS method to design L(s). When applying the H∞LS method the uncertainty is considered unknown and coprime (eq 1). 4.

2

10

1

10

0

10

-1

10

-2

10

-3

10

-4

10

-2

10

-1

10

0

10

1

10

Figure 3: Loopshaping for first example (dashed: Graphical Loops.; continuous: Robust Loops.) Control signal: Step Input 1.5

Output: Step Input

1.5

EXAMPLES

Several numerical examples that illustrate the robust loopshaping technique discussed in this paper are now presented: first a SISO numerical example is presented. Then the application of this technique to a distillation column is discussed. An application to a real system using necessary conditions can be found in (Tadeo et al., 1999). 4.1.

R o b u s t L o o p s h a p in g

10

1

1

0.5

0.5

0

0

-0.5 0

5 Time (secs)

10

-0.5 0

5 Time (secs)

10

Figure 4: Step responses for first example with Nominal Conditions (dashed: Graphical Loops.; continuous: Robust Loops.)

3

Control signal: Step Input 2

2

1

1

0

0

-1

Output: Step Input

SISO example

In the first example a controller is designed for the θ −s plant G( s ) = , where θ has a nominal value s( θ + s )

-1 0

of θ=2. In (Doyle et al, 1992) it is shown that an initial controller for this plant is the trivial controller K=1, which fulfils the graphical loopshaping conditions given by Lema 1, as shown in Figure 3. Application of the Robust Loopshaping approach (with γ=1.1) gives a compensator 6.24s + 12.5 KS ( s ) = 2 s + 7.06s + 24.8

5 Time (secs)

10

-2 0

5 Time (secs)

10

Figure 5: Step responses for first example with Parameter variation: θ=1 (dashed: Graphical Loops.; continuous: Robust Loops.)

4.2. MIMO example The application of the technique presented in this paper to a distillation column model, similar to the one in (Skogestad and Postlethwaite, 1998) is now presented. The plant model is − 86.4 ù é 87.8 êτ s + 1 τ s + 1 ú 2 ú G( s ) = ê 1 ê 108.2 − 109.6 ú êëτ 1 s + 1 τ 2 s + 1 úû

The complete controller fulfils the conditions in Lema 1, improving the Phase Margin from 37º to 67º and the Gain Margin from 6dB to 9.8dB. It must be pointed out that this improvement is obtained without being necessary to consider any tuning parameter (for simple problems the effect of γ is negligible). The step 304

2

where in the nominal case τ1 = 75, τ 2 = 75 . From the analysis of the uncertainty in the plant poles as an input multiplicative uncertainty the following weight on the sensitivity can be estimated when 50 ≤ τ1 ≤ 100, 50 ≤ τ 2 ≤ 100 :

controller might not be so robust for other kind of problems, like multiplicative input uncertainty. 10

s + 0.1 s + 0.0001 The weight on the complementary sensitivity is s / 0.01 + 1 . The frequency selected to be WT = 0.5 s / 0.1 + 1 response of this weights is shown in Figure 6. WS = 0.1

10

10

10

Selection of the initial open-loop transfer function is now presented: to avoid interaction between loops, the open-loop transfer function is selected to be diagonal. Also, to simplify the design, the non-null transfer functions are selected to be the same: an integrator to eliminate steady-state error, plus an additional pole at high frequencies to increase roll-off. The gain is then selected to fulfill the conditions in Lema 1, and have an adequate bandwidth. That is, the selected open-loop transfer function is: 0.1 é1 0ù L( s ) = ê ú s (10 s + 1) ë0 1û It can be seen in figure 7 that the selected loop shape fulfills the robustness conditions in Lema 1. The controller designed using the graphical loopshaping approach is then: 0.1(75s + 1) é 0.39942 -0.31487 ù KG = G −1L = ê ú s (10 s + 1) ë 0.39431 -0.31997 û

2

WS 1

WT

0

-1

10

-5

10

-4

10

-3

-2

-1

10 10 10 f re q u e n c y (r a d / s e c )

0

10

Figure 6: Graphical Loopshaping for distillation column 4

10

Graphical Loopshaping

wS −1 wT

2

10

−1

0

10

wS

-2

10

wT −1

+1 Robust Loopshaping

-4

10

-6

10

Observe that using this simple selection of L(s) this controller inverts the plant, which is not always a good idea from the robustness point of view. However, with the approach presented in this paper the robustness is increased automatically to a selected value in the second step, by adding a compensator to maximize the accepted coprime uncertainty.

-5

-4

10

10

-3

10

-2

-1

10 10 frequency (rad/sec)

0

10

1

10

Figure 7: Graphical Loopshaping for distillation column (dashed: Graphical Loops.; continuous: Robust Loops.) Characteristic Transfer Functions

1

10

M

A compensator was then calculated that maximizes the 0 coprime uncertainty in the generalized plant (with 10 γ=1.2). The final controller is then: 0.1(75s + 1)(0.3068s+0.03373) é 0.39942 -0.31487 ù K = K S KG = ê ú s(10 s + 1)(s+0.3965s+0.06008) ë 0.39431 -0.31997 û10-1

T S K

It can be seen in figure 7 how the loopshape after adding the robust compensator fulfills the robustness conditions (observe that σ ( L ) = σ ( L ) ) The characteristics transfer functions of the feedback system with the controller KG is compared with the one obtained with the final controller K in figure 8. The disturbance rejection characteristics with the nominal model are shown in figure 9 and the closeloop step responses in figure 10. It can be seen that the controller designed using the technique presented in this paper improves the nominal response. However, this is a extreme situation, because we have considered coprime factor uncertainty. As decoupling is used, the

-2

10

-3

10 -3 10

-2 -1 10 Frequency (rad/sec) 10

0

10

Figure 8: Characteristic Transfer Functions for distillation column (dashed: Graphical Loops.; continuous: Robust Loops.)

305

1

Sensitivity

1

promising. It has been shown by several examples that the theory presented in this paper can be applied to different process, it being only necessary to consider the possible uncertainty in the nominal model and using available software to design the controller. Compared with other robust control approaches this technique is more intuitive to the control engineer, thanks to the fact that the design parameter is the open-loop transfer function itself.

0.5

0

-0.5 1

0.5

Further work must be done in the selection the graphical loopshaping, and the calculation of a final robust loopshaping controller that ensures that the the graphical loopshaping conditions are also fulfilled.

0

-0.5

0

20

40

60

80

1000

20

40

60

80

100

Figure 9: Disturbance Rejection for distillation column (dashed: Graphical Loops.; continuous: Robust Loops.)

ACKNOWLEGEMENTS

Complementary Sensitivity

This work was supported by the CYTED (Proyecto Precompetitivo VII-5) and Junta de Castilla y León (VA055/02)

1.5

1

REFERENCES

0.5

Chiang, R.Y.; Safonov, M.G., (1992). Robust Control Toolbox, The Mathworks Inc. Doyle, J.C.; Francis, B.A.; Tannenbaum, A.R. (1992) Feedback Control Theory, Macmillan Publishing Company Glover, K.; McFarlane, D. (1989). Robust stabilization of normalized coprime factor plant descriptions with H∞ bounded uncertainty, IEEE Trans. Automat. Contr., 34, 821-830 Green, M.; Limebeer, D. J. N. (1995) Linear Robust Control, Prentice Hall: New Jersey, McFarlane, D.C.; Glover, K., (1990) Robust Controller Design Using Normalized Coprime Factor Plant Descriptions, Lecture Notes Control & Information Sciences. Springer Verlag Pantas, A.; Walsh. S., (1996) Evaluation of H∞ Loop Shaping Controller Design on a Process Control Problem, UKACC Control’96, Exeter, UK Sanchez-Peña, R.S; Sznaier, M., (1998). Robust Systems: Theory and Applications, John Wiley & Sons, New York Skogestad, S; Morari, M., (1992). Some New Properties of the Structured Singular Value, IEEE Trans. Autom. Contr., 33, 1151-1154 Skogestad, S.; Postlethwaite, I., (1996). Multivariable Feedback Control, Analysis and Design, John Wiley & Sons Tadeo, F.; Perez, O.; Alvarez, T., (2000). Control of Neutralization Processes by Robust Loopshaping, IEEE Trans. On Control System Tech., 8, 236-246 Tang, K.S.; Man, K.F.; Gu, D.W., (1996). Structured Genetic Algorithm for Robust H∞ Control Systems Design, IEEE Trans. on Ind. Electr., 43, 575-582 Whidborne, J.F.; Postlethwaite, I.; Gu, D.W., (1994). Robust Controller Design using H∞ loop-shaping and the method of inequalities, IEEE Trans. Contr. Syst. Technol. 2, 455-461 Zhou, K.; Doyle, J.C.; Glover, K., (1996). Robust and Optimal Control, Prentice Hall: New Jersey

0 1.5

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Figure 10: Step Response for distillation column (dashed: Graphical Loops.; continuous: Robust Loops.)

5.

CONCLUSIONS

This paper has presented the improvement of the robustness of processes by using a robust loopshaping approach, considering coprime uncertainties. H∞ loopshaping is an appealing approach for controller design, as it addresses explicitly the problem of model uncertainty. However this design method does not directly address the robustness properties of the real plant, but rather it is concerned with the shaped plant. This paper has discussed a methodology that solves this problem by considering the robustness properties of the real plant in the selection of the weights of the shaped plant. Then a shaped plant is selected following the Graphical Loopshaping ideas, where instead of using sufficient conditions (which are only valid for linear uncertainty), necessary conditions for robust performance have been applied. Once selected a robust shaped plant the controller is designed by application of the H∞ loopshaping design method, following McFarlane/Glover ideas. This two-step design methodology makes possible to take advantage of the positive properties of the H∞ loopshaping design method, but considering at the same time the robustness properties of the real plant. The idea shown in this paper of combining graphical and robust loopshaping has been shown to be 306