ROBUST DECENTRALIZED CONTROLLER DESIGN FOR STABLE PLANTS

Robust decentralized controller design for stable plants

20. November, 2009, Autor článku: Osuský Jakub, Elektrotechnika, Študentské práce
Ročník 2, číslo 11 Pridať príspevok

control The paper deals with the robust decentralized controller design in the frequency domain for stable plants. Nominal and robust stability conditions under a decentralized IMC controller are developed. An illustrative example is included.

1 INTRODUCTION

PID controllers are standard and well-proven solution for the majority of industrial applications. Over the years, a plenty of PID tuning rules were developed see e.g. (Šulc and Vítečková, 2005).

In this paper decentralized PID controller design approaches are developed for stable systems and further extended to satisfy robust stability conditions in terms of unstructured uncertainty developed in (Kozáková A., Veselý, 2005; Kozáková and Veselý, 2009). The approach is IMC structure-based and applicable just for stable plants; an innovation brought about by its development is derivation of robust stability conditions in terms of a decentralized IMC controller.

The paper is organized as follows: preliminaries and problem formulation are given in Section 2, robust decentralized IMC controller for stable plants is addressed in Section 3. In Section 4, a detailed robust decentralized PID controller design procedure is illustrated on an example in Section 5. Conclusions are drawn at the end of the paper.

2 PRELIMINARIES AND PROBLEM FORMULATION

The Internal Model Control (IMC) structure was introduced as an alternative to the classic feedback structure. Its main advantage is that closed-loop stability is guaranteed simply by choosing a stable IMC controller.

Consider the simplified block diagram of the IMC structure (Morari and Zafiriou, 1989) depicted in Fig. 1

Fig. 1 – IMC structure

where $C (s) \in R^{m \times m}$ denotes the controller, $G (s) \in R^{m \times m}$ is the plant and $G_M (s) \in R^{m \times m}$ is the plant model. Exact knowledge of the output y is assumed. If the model is exact, i.e. if $G (s) = G_M (s)$ then

$y (s) = G (s) C (s) w (s)$ ,

(1)

Relationship with the standard feedback structure in Fig. 2 is obtained simply by comparing outputs from both structures

$[G (s) R (s) + I]^{-1} G (s) R (s) = G (s) R (s)$ ,

(2)

where $R (s) \in R^{m \times m}$ is the controller

Fig. 2 – Standard feedback configuration

The standard controller $R (s)$ and the IMC controller $C (s)$ are related through

$R (s) = [I - C (s) G (s)]^{-1} C (s)$ ,

(3)

If $G (s)$ is stable and $G (s) = G_M (s)$ , the standard feedback system with controller (3) is internally stable if and only if $G (s)$ is stable. In such a case (3) is a simple parameterization of all stabilizing controllers $R (s)$ in terms of stable $C (s)$ . Thus, instead of searching for $R (s)$ it is possible and simpler to search for $C (s)$ (Morari and Zafiriou, 1989).

When designing a controller a major source of difficulty is plant model inaccuracy; hence uncertainty models are to be used which means that instead of a single model a class $\Pi$ of perturbed models is to be considered. Denote $\tilde{G} (s) \in \Pi$ any perturbed plant model and $G (s) \in \Pi$ the nominal plant model. A simple uncertainty model is obtained using unstructured uncertainty $\Delta (s)$ . Three uncertainty forms are commonly used: additive $(a)$ , multiplicative input $(i)$ and multiplicative output $(o)$ uncertainties.

Standard feedback configuration with unstructured uncertainty of any type can be rearranged to obtain the general $M - \Delta$ structure in Fig. 3 where $M (s)$ represents the nominal model and $\Delta (s) : \sigma_{max} [\Delta (j \omega)] \leq 1$ the normalized perturbation.

Fig. 3 – $M - \Delta$ structure

Robust stability condition for unstructured perturbations is formulated in terms of stability of the $M - \Delta$ system: if both the nominal system $M (s)$ is stable (nominal stability) and the normalized perturbation $\Delta (s)$ is stable, closed-loop stability is guaranteed for

$\sigma_{max} [M (j \omega)] < 1, \quad \forall \, \omega [/latex] ,</td> <td style="width:20px; text-align:center;">(4)</td> </tr> </table> <p align="justify">For individual uncertainty types <img src='http://s0.wp.com/latex.php?latex&bg=ffffff&fg=000000&s=0' alt='' title='' class='latex' />M = l_k M_k$ , $k = a, i, o$ in particular

for additive uncertainty

$M_a (s) = - [I + R (s) G (s)]^{-1} R (s)$
$l_a (\omega) =$ $\begin{smallmatrix} {max} \\ _{\tilde{G} \in \Pi} \end{smallmatrix}$ $\sigma_{max} [\tilde{G} (s) - G (s) ]$	(5)

for multiplicative input uncertainty

$M_i (s) = - [I + R (s) G (s)]^{-1} R (s) G (s)$
$l_i (\omega) =$ $\begin{smallmatrix} {max} \\ _{\tilde{G} \in \Pi} \end{smallmatrix}$ $\sigma_{max} \{ G^{-1} (s) [ \tilde{G} (s) - G (s) ] \}$	(6)

for multiplicative output uncertainty

$M_o (s) = - G (s) R (s) [I + R (s) G (s)]^{-1}$
$l_o (\omega) =$ $\begin{smallmatrix} {max} \\ _{\tilde{G} \in \Pi} \end{smallmatrix}$ $\sigma_{max} \{ [ \tilde{G} (s) - G (s) ] G^{-1} (s) \}$	(7)

In view of (5), (6, (7) condition (4) reads as follows

$\sigma_{max} (M_k (s)) < \frac{1}{\mid l_k (s) \mid}, \quad k = a, i, o [/latex] </td> <td style="width:20px; text-align:center;">(8)</td> </tr> </table> <p align="justify"><strong>Problem Formulation</strong>. For the uncertain plant <img src='http://s0.wp.com/latex.php?latex&bg=ffffff&fg=000000&s=0' alt='' title='' class='latex' />\tilde{G} (s)$ a decentralized controller is to be designed

$R (s) = diag\{ R_i (s) \} _{m \times m}$

(9)

where $R_i (s)$ is the transfer function of the i – th local controller. The designed controller has to guarantee stability of the controlled plant over the entire operating range specified by (5), (6), (7), i.e. robust stability (RS).

3 MAIN RESULTS

With a view of designing a decentralized (diagonal) controller, the nominal plant $G (s)$ is split into the diagonal part $G_d (s)$ and the off-diagonal part $G_m (s)$ , i.e.

$G (s) = G_d (s) + G_m (s)$

(10)

Then, the controller (3) with a decentralized structure is

$R (s) = [I - C (s) G_d (s)]^{-1} C (s)$

(11)

where $C (s) \in R^{m \times m}$ is a suitably chosen stable diagonal matrix.

3.1 Guaranteeing robust stability

To guarantee robust stability using the IMC structure it is necessary to suitably recast the $M - \Delta$ robust stability condition (8). Uncertainties in the three standard forms (5), (6), (7) will be considered.

For the centralized IMC controller and different uncertainty types, substituting (3) in (8) yields

for the additive uncertainty

$\sigma_{max} [C (s)] < \frac{1}{\mid l_a (s) \mid} [/latex] </td> <td style="width:20px; text-align:center;">(12)</td> </tr> </table> <li>for the multiplicative input uncertainty </li> <table style="width:100%;"> <tr> <td style="text-align:center;"><img src='http://s0.wp.com/latex.php?latex&bg=ffffff&fg=000000&s=0' alt='' title='' class='latex' />\sigma_{max} [C (s) G (s)] < \frac{1}{\mid l_i (s) \mid} [/latex] </td> <td style="width:20px; text-align:center;">(13)</td> </tr> </table> <li>for the multiplicative output uncertainty </li> <table style="width:100%;"> <tr> <td style="text-align:center;"><img src='http://s0.wp.com/latex.php?latex&bg=ffffff&fg=000000&s=0' alt='' title='' class='latex' />\sigma_{max} [G (s) C (s)] < \frac{1}{\mid l_o (s) \mid} [/latex] </td> <td style="width:20px; text-align:center;">(14)</td> </tr> </table> </ul> <p align="justify">For the <em>decentralized </em>controller (DC) (11) the above robust stability conditions develop into <ul> <li>for the additive uncertainty</li> <table style="width:100%;"> <tr> <td style="text-align:center;"><img src='http://s0.wp.com/latex.php?latex&bg=ffffff&fg=000000&s=0' alt='' title='' class='latex' />\sigma_{max} \{ [C^{-1} (s) + G_m (s)]^{-1} \} < \frac{1}{\mid l_a (s) \mid} [/latex] </td> <td style="width:20px; text-align:center;">(15)</td> </tr> </table> <li>for the multiplicative input uncertainty</li> <table style="width:100%;"> <tr> <td style="text-align:center;"><img src='http://s0.wp.com/latex.php?latex&bg=ffffff&fg=000000&s=0' alt='' title='' class='latex' />\sigma_{max} \{ [C^{-1} (s) + G_m (s)]^{-1} G (s) \} < \frac{1}{\mid l_i (s) \mid} [/latex] </td> <td style="width:20px; text-align:center;">(16)</td> </tr> </table> <li>for the multiplicative output uncertainty</li> <table style="width:100%;"> <tr> <td style="text-align:center;"><img src='http://s0.wp.com/latex.php?latex&bg=ffffff&fg=000000&s=0' alt='' title='' class='latex' />\sigma_{max} \{ G (s) [C^{-1} (s) + G_m (s)] \} < \frac{1}{\mid l_o (s) \mid} [/latex] </td> <td style="width:20px; text-align:center;">(17)</td> </tr> </table> </ul> <p align="justify"><strong>3. 2 Guaranteeing nominal stability </strong> <p align="justify">According to the General robust stability theorem formulated for the <img src='http://s0.wp.com/latex.php?latex&bg=ffffff&fg=000000&s=0' alt='' title='' class='latex' />M - \Delta$ structure (Morari and Zafiriou, 1989), stability of $M (s)$ , i.e. nominal stability is the prerequisite for robust stability.

Next, the additive type uncertainty is considered, development for other uncertainty types is similar.

Consider nominal stability of the uncertain system described with additive uncertainty in the IMC structure, i.e. stability of (5) transformed to (15).

$M_a (s) = [C ^{-1} (s) + G_m (s)]^{-1}$

(18)

Further manipulations yield

$M_a (s) =$ $- \frac{adj [C^{-1} (s) + G_m (s)]}{\det [C^{-1} (s)] \, \det [I + C (s) G_m (s)]}$

(19)

Because $C (s)$ is supposed to be a stable diagonal matrix, the polynomial $\det [C^{-1} (s)]$ is stable and stability of $M_a (s)$ depends on the stability of the factor $\det [I + C (s) G_m (s)].$ Applying to it the Small gain theorem (Morari and Zafiriou; Skogestad and Postlethwaite, 1997) the sufficient stability condition reads as

$\sigma _{max} [C (s) G_m (s)] < 1 [/latex] </td> <td style="width:20px; text-align:center;">(20)</td> </tr> </table> <p align="justify">or using the singular value properties. <table style="width:100%;"> <tr> <td style="text-align:center;"><img src='http://s0.wp.com/latex.php?latex&bg=ffffff&fg=000000&s=0' alt='' title='' class='latex' />\sigma _{max} [C (s)] < [/latex] <img src='http://s0.wp.com/latex.php?latex=++%5Cfrac%7B1%7D%7B%5Csigma_%7Bmax%7D+%5BG_m+%28s%29%5D%7D+++&bg=ffffff&fg=000000&s=2' alt=' \frac{1}{\sigma_{max} [G_m (s)]} ' title=' \frac{1}{\sigma_{max} [G_m (s)]} ' class='latex' /> </td> <td style="width:20px; text-align:center;">(21)</td> </tr> </table> <p align="justify">Considering <img src='http://s0.wp.com/latex.php?latex&bg=ffffff&fg=000000&s=0' alt='' title='' class='latex' />C (s)$ a diagonal matrix $C (s) = \{ C_i (s) \}_{m \times m}$ with $C_i (s)$ being stable transfer functions, condition (21) can be rewritten as

$\begin{smallmatrix} max \\ {i} \end{smallmatrix}$ $\mid C_i (s) \mid < \frac{1}{\sigma _{max} [G_m (s)]}, \quad i = 1, ... , m [/latex] </td> <td style="width:20px; text-align:center;">(22)</td> </tr> </table> <p align="justify">Consider a decentralized PID controller with local controllers in the form <table style="width:100%;"> <tr> <td style="text-align:center;"><img src='http://s0.wp.com/latex.php?latex&bg=ffffff&fg=000000&s=0' alt='' title='' class='latex' /> R_i = \frac{r_2 s^2 + r_1 s + r_0}{k_r s}$

(23)

Tuning parameter $k_r$ will used to stabilize the full closed-loop system in the sense of (22) and ensuring robust stability (15), (16), (17).

3.3 Robust decentralized controller design procedure for stable systems

Nominal model calculating;
Controller design for each subsystems of nominal model using any SISO approach;
Once subsystems controllers are known, tuning parameters $k_r$ are adjusted, so that the nominal stability condition (22) and any of the robust stability conditions (15), (16), (17) are met simultaneously;

4 EXAMPLE

Consider plant with two inputs and two outputs represented engine-generator identified in three working points

$\tilde{G} _1 (s) =$ $\Bigg [ \begin{smallmatrix} {\frac{0.12 s + 4.76}{s^2 + 12.61 s + 9.689}} & {\frac{- 0.02 s - 1.52}{s^2 + 16.87 s + 11.55}} \\ {\frac{0.12 s + 1.21}{s^2 + 0.98 s + 6.26} } & {\frac{1.3 s + 30.09}{s^2 + 13.75 s + 52.63}} \end{smallmatrix} \Bigg ]$

$\tilde{G} _2 (s) =$ $\Bigg [ \begin{smallmatrix} {\frac{0.12 s + 4.76}{1.6 s^2 + 20.18 s + 15.5}} & {\frac{- 0.02 s - 1.52}{1.64 s^2 + 27.67 s + 18.94}} \\ {\frac{0.12 s + 1.22}{1.58 s^2 + 1.54 s + 9.89} } & {\frac{1.3 s + 30.1}{1.62 s^2 + 22.28 s + 85.26}} \end{smallmatrix} \Bigg ]$

$\tilde{G} _3 (s) =$ $\Bigg [ \begin{smallmatrix} {\frac{0.11 s + 4.76}{0.4 s^2 + 5.04 s + 3.88}} & {\frac{- 0.02 s - 1.52}{0.36 s^2 + 6.07 s + 4.16}} \\ {\frac{0.12 s + 1.22}{0.42 s^2 + 0.41 s + 2.63} } & {\frac{1.3 s + 30.1}{0.38 s^2 + 5.23 s + 20}} \end{smallmatrix} \Bigg ]$

Nominal model of this plant was calculated:

$G (s) =$ $\Bigg [ \begin{smallmatrix} {\frac{0.12 s + 4.76}{s^2 + 12.61 s + 9.69}} & {\frac{- 0.02 s - 1.52}{ s^2 + 16.87 s + 11.55}} \\ {\frac{0.12 s + 1.21}{s^2 + 0.98 s + 6.26} } & {\frac{1.3 s + 30.1}{ s^2 + 13.75 s + 52.63}} \end{smallmatrix} \Bigg ]$

Assume that input-output pairing was done and diagonal pairing is recommended. For diagonal elements PI controllers were designed using D-curves method.

$R_1 (s) = \frac{1.2 s + 1.5}{s} \quad and \quad R_2 (s) = \frac{0.7 s + 6}{s}$

Tuning parameters $k_{r1,2} = 1$

Decentralized feedback controller $R (s) = \big [ \begin{smallmatrix} {R_1} & {0} \\ {0} & {R_2} \end{smallmatrix} \big ]$ was calculated into IMC form $C (s) = R (s) [I + G_d R]^{-1}$ and nominal stability condition (22) was tested (fig. 4).

Fig. 4 Nominal stability condition (22)

Nominal stability condition is passed. Robust stability condition (12) is depicted in Fig. 5.

Fig. 5 Robust stability condition (12)

From fig. 5 it is evident that robust stability condition is not passed so tuning parameters $k_{r-i}$ has to be changed.

Increasing of first tuning parameter on value $k_{r-i} = 1.2$ robust stability was satisfied (Fig. 6)

Fig. 6 Robust stability condition with $k_{r-i} = 1.2$

Resulting sets $\Lambda _i$ of closed-loop poles proves the stability in three working points.

$\Lambda _1 = \{ - 11.44 , - 6.23 , 4.22 \pm 3.35 i , - 0.47 \pm 2.48 i ,$
$- 0.67 \pm 0.47 i , -0.73 \}$

$\Lambda _2 = \{ - 11.57 , 5.53 \pm 2.01 i , - 3.25 , -0.48 \pm 2.45 i , - 0.73$
$- 0.57 \pm 0.32 i \}$

$\Lambda _3 = \{ - 10.94 , - 4.3 \pm 6.41 i , - 7.38 , - 0.44 \pm 2.48 i ,$
$- 1.04 \pm 0.73 i , - 0.72 \}$

5 CONCLUSION

The paper deals with the robust decentralized controller design in the frequency domain for stable plants. Nominal and robust stability conditions under a decentralized IMC controller are developed. An illustrative example is included.

ACKNOWLEDGEMENT

This work was supported by the Scientific Grant Agency of the Ministry of Education of the Slovak Republic and the Slovak Academy of Sciences under Grants No. 1/0544/09 and 1/0822/08/4.

REFERENCES

Morari M., Zafiriou E., (1989) Robust process control. Prentice Hall.
Skogestad S, Postlethwaite I., (1997) Multivariable feedback control: analysis and design. John Wiley
Kozáková A., Veselý V. (2005) A frequency domain design technique for robust decentralized controllers. Preprints of the 16^th IFAC World Congress. Prague, Czech Republic July 3-8, p. Mo-E21-TO/6.
Kozáková A., Veselý V., (2009) Design of robust decentralized controllers using the M- structure robust stability conditions, Int. Journal of Systems Science.
Šulc B., Vítečková, M., (2004) Theory and practice of control loops design. ČVUT, Praha (in Czech).

Co-authors of this paper are Vojtech Veselý and Alena Kozáková, Slovak University of Technology, Faculty of Electrical Engineering and Information Technology, Ilkovičova 3, 812 19 Bratislava.

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