Robust decentralized controller design based on equivalent subsystems

Automatica 107 (2019) 29–35

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Brief paper

Robust decentralized controller design based on equivalent subsystems✩ ∗

Alena Kozáková a , , Vojtech Veselý a , Vladimír Kučera b a b

Faculty of Electrical Engineering and Information Technology, Slovak University of Technology in Bratislava, Slovakia Czech Technical University in Prague, Czech Institute of Informatics, Robotics, and Cybernetics, Prague, Czech Republic

article

info

Article history: Received 18 July 2018 Received in revised form 8 March 2019 Accepted 26 April 2019 Available online xxxx Keywords: Decentralized controller Equivalent subsystems method Frequency domain Robust controller Uncertainty

a b s t r a c t A method for robust decentralized controller design of continuous and discrete-time controllers for SISO and MIMO systems is proposed. The uncertain MIMO system is given by a set of square transfer matrices of any dimension. The design procedure relies on the Equivalent Subsystems Method (ESM) – an independent design of local controllers for stability and required performance of the overall system. The ESM has been extended by integrating robust stability conditions directly in the independent design of local controllers to obtain a unified robust decentralized controller design method. To demonstrate the design procedure, a discrete-time robust decentralized PID controller for a minimum-phase system was designed. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Industrial processes are naturally multivariable (MIMO) systems modelled as interconnection of a finite number of subsystems. Rather than using complicated MIMO controllers, decentralized controllers made up of local controllers of individual subsystems are preferred due to simple tuning, easy implementation, loop failure tolerance and reasonable performance (Chiu & Arkun, 1992). In general, closing individual loops affect performance of other loops if interactions are not sufficiently weak. Depending on the method of considering interactions in the design of local controllers, decentralized controller design can be classified as a sequential design (Hovd & Skogestad, 1994; Perez, Romero, Herrero, & Balaguer, 2012) or independent design (Hovd & Skogestad, 1993; Husek & Kucera, 2014; Kozáková & Veselý, 2009). While in the sequential design closing subsequent loops changes responses of previously designed loops, in the independent design, local controllers are designed independently with regard to bounds on interactions to guarantee stability and performance of the overall system. As information from other ✩ The work has been partially supported by the Slovak Scientific Grant Agency, Grant Nos. 1/0475/16 and 1/0733/16, and by the Research and Development Fund of the Czech Technical University in Prague. The material in this paper was partially presented at the 13th IFAC Symposium on Large Scale Complex Systems: Theory and Applications. July 7–10, 2013, Shanghai, China. This paper was recommended for publication in revised form by Associate Editor Linda Bushnell under the direction of Editor Richard Middleton. ∗ Corresponding author. E-mail addresses: [email protected] (A. Kozáková), [email protected] (V. Veselý), [email protected] (V. Kučera). https://doi.org/10.1016/j.automatica.2019.05.031 0005-1098/© 2019 Elsevier Ltd. All rights reserved.

loops is not exploited, resulting stability conditions are sufficient and hence conservative. The Equivalent Subsystems Method (Kozáková & Veselý, 2009; Kozáková, Vesely, & Osusky, 2009) eliminates the above shortcomings. Local controllers are designed independently for equivalent subsystems generated from frequency responses of decoupled subsystems using one selected characteristic locus of the matrix of interactions. Stability of equivalent subsystems closed-loops is necessary and sufficient condition for the overall system stability. Similarly, performance achieved in equivalent closed-loops is maintained in the overall system. For practical implementation it is necessary to consider model uncertainty in the controller design. Many frequency-domain engineering methods were developed to design robust multi-loop controllers for mainly for two-input two-output (TITO) systems, e.g. the individual channel design (O’Reilly & Leithead, 1991), effective transfer function methods (Balaguer & Romero, 2012; Jin, Wang, & Liu, 2016), etc. Decentralized controllers were designed and successfully applied in robotic systems (Li, Melek, & Clark, 2009), boilers (Balko & Rosinova, 2015; Labibi, Marquez, & Chen, 2009), power systems (Veselý, 2018), distillation columns (Hovd & Skogestad, 1993), etc. The presented unified robust decentralized controller design methodology is an enhanced version of ESM with robust stability conditions integrated in the design of local controllers; the resulting robust decentralized controller guarantees closed loop stability and performance of the uncertain system described by a nominal transfer matrix and unstructured uncertainty. The theoretical development as presented in Kozáková and Veselý

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A. Kozáková, V. Veselý and V. Kučera / Automatica 107 (2019) 29–35

Fig. 2. M -∆ structure.

Fig. 1. Standard feedback loop.

(2013) has been revised and modified to obtain a consistent unified methodology applicable for MIMO systems described by a set of square transfer function matrices of any dimension with stable interactions. Based on main results on stability, robust stability and performance of MIMO systems in the frequency domain presented in Section 2, the Equivalent Subsystems Method is revisited in Section 3. The unified robust decentralized controller design procedure is developed in Section 4 and demonstrated on a case study in Section 5. Conclusions are drawn at the end of the paper.

Individual uncertainty forms generate related sets Πk and norms ℓk (ω), k ∈ {a, o}; G denotes any member of Π and G0 is the nominal model: Additive uncertainty:

Πa : = {G(jω): G(jω) = G0 (jω) + Ea (jω), Ea (jω) = wa (jω)∆(jω)}, ℓa (ω) = max σmax [G p (jω) − G0 (jω)] p

Multiplicative output uncertainty:

Πo : = {G(jω): G(jω) = [I + Eo (jω)]G0 (jω), Eo (jω) = wo (jω)∆(jω)}

2. Preliminaries and problem formulation

(5)

(6)

ℓo (ω) = max σmax {[G p (jω) − G0 (jω)]G0−1 (jω)}. Consider the feedback system in Fig. 1 consisting of an interconnected system G(s) ∈ Rm×m , m ≥ 1, and a decentralized controller R(s) ∈ Rm×m . Theorem 1 (Generalized Nyquist Stability Theorem). Suppose that the system and the controller are minimal realizations of G(s) and R(s), respectively. Then the closed-loop in Fig. 1 is internally stable if and only if

m ∑

N {0, [1 + qi (s)]} = nq

(1)

where Q (s) = G(s)R(s) is the open-loop transfer function with no internal right half-plane pole-zero cancellations between G(s) and R(s); nq is number of unstable poles of Q (s); N {0, det [I + Q (s)]} is number of anticlockwise encirclements of (0, 0j) by det [I + Q (s)], D is the Nyquist contour, and qi (s), i = 1, . . ., m are characteristic functions of Q (s) defined by (2)

Characteristic loci (CL) are traced out in the complex plane by characteristic functions of Q (s) as s traverses the Nyquist contour. Let the uncertain plant G(s) be specified by a set Π of N transfer function matrices

{ p } Π : = {G p (s)}, p = 1, 2, . . . , N where G p (s) = Gij (s) m×m .

To assess performance of MIMO system in the frequency domain, sensitivity and complementary sensitivity functions T (s) = Q (s)[I + Q (s)]−1

The uncertain system (3) can be modelled using a nominal model G0 (s) and unstructured uncertainty DU defined as (4)

where w (jω) is a weight on the norm-bounded stable uncertainty ∆ (s) ∈ Rm×m , σmax [∆(jω)] ≤ 1. The set Π can be generated by six types of unstructured uncertainty models (Skogestad & Postlethwaite, 2009). In the sequel, the additive (Ea ) and the multiplicative output (Eo ) uncertainties will be considered, relations for other uncertainty types are obtained by analogy.

(8)

are the common measures. Their maximum peaks

∥S ∥∞ = max σmax [S(jω)] ≤ MS ∀ω ω

∥T ∥∞ = max σmax [T (jω)] ≤ MT ω

∀ω

(9)

are robustness measures with recommended values MS < 2 and MT < 1.3 (Skogestad & Postlethwaite, 2009). In SISO systems, generally used robustness measures are stability margins. Problem formulation Consider an uncertain system G(s) consisting of m subsystems (m ≥ 1) given as a set of N transfer function matrices, described by a nominal model G0 (s) and unstructured uncertainty. Let the nominal model G0 (s) be split as follows: G0 (s) = Gd (s) + Gm (s),

(3)

DU : = {E(jω): E(jω) = w (jω)∆(jω)}, |w (jω)| ≥ ℓ(jω),

(7)

∀ω.

S(s) = [I + Q (s)]−1

i=1

det[qi (s)I − Q (s)] = 0 i = 1, . . . , m ∀s ∈ D .

The feedback loop in Fig. 1 with the uncertain plant G(s) described by unstructured uncertainty can be recast into the M − ∆ structure (Fig. 2) where the particular form of M (jω) depends on the type of uncertainty description. According to the general robust stability condition, if M (jω) (nominal closed-loop) is stable, then the M − ∆ system is stable for all ∆(jω): σmax (∆) ≤ 1 if and only if

σmax [M (jω)] < 1,

1. det [I + Q (s)] ̸ = 0 ∀s ∈ D 2. N {0, det [I + Q (s)]} =

p

(10)

where Gd (s) is the diagonal part of G0 (s) modelling the individual subsystems: Gd (s) = diag{Gi (s)}m×m ,

det Gd (s) ̸ = 0 ∀s

(11)

and Gm (s) = G0 (s) − Gd (s) is a stable transfer function matrix of interactions. A decentralized controller R(s) = diag{Ri (s)}m×m

det R(s) ̸ = 0 ∀s

(12)

is to be designed to guarantee closed-loop stability over the whole operating range of the uncertain plant described by the nominal model and the unstructured uncertainty (robust stability), and required performance of the overall plant.

A. Kozáková, V. Veselý and V. Kučera / Automatica 107 (2019) 29–35

3. Equivalent subsystems method

Then the robust stability condition (7) modifies to:

Consider the feedback loop in Fig. 1 with the nominal MIMO system split according to (10) and the decentralized controller R(s) as in (12). Consider the following factorization ∀s ∈ D: det[I + G0 (s)R(s)] =

(13)

= det[R −1 (s) + Gd (s) + Gm (s)] det R(s).

Characteristic functions of Gm (s) denoted gi (s), i = 1, 2, . . . , m are defined as follows: det [gi (s)I − Gm (s)] = 0 i = 1, . . . , m.

(14)

With regard to stability, Gm (s) in (13) can be replaced by its characteristic function matrix gk (s)I , k ∈ {1, . . . , m} in the sense of the Cayley–Hamilton Theorem to obtain (15)

= N 0, det R −1 (s) + Gd (s) + gk (s)I det R(s) .

[

]

}

Let the m × m diagonal matrix eq

eq

Gk (s): = Gd (s) + gk (s)I = diag{Gik (s)}m×m

(16)

eq Gik (s)

define the m equivalent subsystems (the subscript k indicates the k-th characteristic function of Gm (s) applied to generate eq Gik (s) where eq

Gik (s) = Gi (s) + gk (s) ∀s ∈ D, i = 1, 2, . . . , m,

(17)

k ∈ {1, 2, . . . , m} .

N {0, det[I + G0 (s)R(s)]} = N {0, det[I +

eq Gk (s)R(s)

]} =

m

=

eq

(18)

N −1, Gik (s)Ri (s) = nq .

{

}

i=1

From (18) results that closed-loop stability of the overall system under a decentralized controller is guaranteed if and only if each closed-loop equivalent subsystem under its related local controller is stable. The above development is summarized in Theorem 2. Theorem 2 (Nominal stability under a DC). The closed-loop in Fig. 1 comprising a MIMO system (10) and a decentralized controller (12) is stable if and only if there exists gk (jω) such that ∀ω 1. det[gk (jω)I − Gm (jω)] = 0 k ∈ {1, . . . , m} 2.

m ∑

N {−1, [Gi (jω) + gk (jω)] Ri (jω)} = nq .

□

i=1

Simply put, a decentralized controller is obtained by independently designing local controllers for individual equivalent subsystems using any SISO design, preferred are frequency-domain approaches, e.g. the Neymark D-partition method (Kozáková & Veselý, 2009; Kozáková et al., 2009) or standard Bode design (Kozáková & Veselý, 2013), etc. ESM can effectively be extended to design robust decentralized controllers by integrating the robust stability condition (7) directly into the local controller design. 4. Unified robust decentralized controller design method Let the uncertain system in Fig. 1 be described by a full nominal model (10) and additive unstructured uncertainty (5). Consider the general robust stability condition (7) with the decentralized controller (12) and the characteristic function matrix gk (s)I , k ∈ {1, . . . , m} substituted for Gm (s): eq

Ma (jω) = ℓa (ω)R(jω) I + Gk (jω)R(jω)

[

(20)

Applying singular value properties in (20) we obtain

σmax [Tkeq (jω)] <

σmin [Gkeq (jω)] = LA (ω) ∀ω, ℓ a (ω )

eq

eq

(21)

eq

where Tk (jω) = Gk (jω)R(jω)[I + Gk (jω)R(jω)]−1 is the equivalent complementary sensitivity. For uncertain systems described by multiplicative uncertainty the robust stability condition (7) modifies to

σmax [Tkeq (jω)] <

1

|ℓo (ω)|

= LO (ω) ∀ω.

(22)

]−1

.

Due to diagonal a structure of Tk (jω) inequalities (21) and (22) have to be satisfied by each individual equivalent subsystem. In case of inverse uncertainty, corresponding bounds are obtained in terms of equivalent sensitivity norm. Overall system performance in terms of maximum overshoot and settling time can be translated into upper bound for phase margin Φ for the equivalent subsystems using the known relationships between maximum overshoot, maximum peak of equiveq alent complementary sensitivity MT (or equivalent sensitivity eq MS ) and equivalent subsystems phase margin:

Φ ≥ 2 arcsin

Substituting (17) into (15) implies the key relation:

∑

} { σmax ℓa (ω)R(jω)[I + Gkeq (jω)R(jω)]−1 < 1.

eq

N {0, det[I + G0 (s)R(s)]} =

{

31

(19)

(

1

)

eq 2MT

≥

1 180◦ eq

MT

π

[deg]

(23)

Maximum settling time is interpreted in terms of a required bandwidth for equivalent subsystems. The unified robust decentralized controller design procedure based on (21)–(23) integrated into the ESM is described next.

4.1. Design procedure The proposed procedure can be used to design robust continuous-time or digital controllers for MIMO systems of any dimension and order including non-minimum phase ones. The main limitation is stability of interactions. The design procedure has the following steps: 1. Calculation of the nominal model and ℓk (ω) depending on the chosen uncertainty type. eq 2. Calculation of upper bound LK (ω) on σmax [Tk (jω)] (21), (22) specification of the minimum phase margin Φmin for equivalent subsystems according to (23) where eq

MT = min(LK (ω)).

(24)

ω

3. Generating m equivalent subsystems (17) using a selected gk (jω) of Gm (s) (it is suggested to select gk (jω) providing maximum possible phase margin of all subsystems). 4. Independent design of local controllers for equivalent subsystems to guarantee required phase margin Φreq ≥ Φmin and a bandwidth ω0 related with settling time ts as follows: ts ≈

3

ω0

for MT ∈ (1.3; 1.5)

or

π ts

< ω0 <

4π ts

.

(25)

5. Evaluation of stability and achieved performance of equivalent subsystems and overall system. 6. Verification of the robust stability condition (7). To design a digital robust decentralized controller the design procedure is applied for the digitized transfer function matrix

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A. Kozáková, V. Veselý and V. Kučera / Automatica 107 (2019) 29–35

eq

Fig. 4. LA (ω) and LO (ω)– upper bounds on σmax [Tk (iω)].

Fig. 3. Quadruple tank system.

G(z), z = ejωT . Finally, the designed local controllers are digitized with the sampling period T. In this way, feasibility of the digital controller is guaranteed. 5. Example Consider the quadruple tank system in Fig. 3 (Johansson, 2000). Outputs y1 and y2 are levels in lower tanks, inputs v1 and v2 are flows delivered by the related pumps. The plant can operate as a minimum- or nonminimum phase system by changing flow ratios γ1 , γ2 . Here, the minimum-phase configuration is considered for which 1 < γ1 + γ2 < 2. Robust controller design for the nonminimum phase configuration specified by 0 < γ1 + γ2 < 1 is presented in (Kozáková & Veselý, 2013). In the linearized 2 × 2 transfer function matrix

⎡

c1 γ1

⎢

T1 s + 1 c2 (1 − γ1 )

G(s) = ⎢ ⎣

(T4 s + 1)(T2 s + 1)

c1 (1 − γ2 )

⎤

(T3 s + 1)(T1 s + 1) ⎥ ⎥, ⎦ c2 γ2

(26)

T2 s + 1

different working points can be adjusted by changing γ1 , γ2 . The uncertainty domain is specified by transfer function matrices in 3 working points GWPi (s), i = 1, 2, 3: GWP1 (s): γ1 = 0.4, γ2 = 0.8; GWP2 (s): γ1 = 0.8, γ2 = 0.4; GWP3 (s): γ1 = 0.8, γ2 = 0.8. The above procedure has been used to design a digital decentralized controller to keep required levels y1 and y2 in the lower tanks for γ1 , γ2 varying over the specified uncertainty domain. 1. The nominal model was obtained as a mean value parameter model from transfer function matrices GWPi (s), i = 1, 2, 3 according to (26):

⎡

2.4667

⎢

62s + 1 1.5667

G0 (s) = ⎢ ⎣

(30s + 1)(90s + 1)

1.2333

Fig. 5. Discrete Bode plots of equivalent subsystems generated by g2 (z): eq eq G12 (z)(upper plot), G22 (z) (lower plot).

⎤

(23s + 1)(62s + 1) ⎥ ⎥, ⎦ 3.1333

(27)

90s + 1

ℓa (ω) and ℓo (ω) were calculated using (5) and (6). eq 2. Upper bounds LA (ω) and LO (ω) for σmax [Tk (jω)] calculated using (21) and (22) are plotted in Fig. 4. The least upper-bound on the closed-loop magnitude MTO = minω LO = 1.22 was used in the next steps because for MTA = minω LA = 0.77 < 1 set-point tracking is not feasible.

3. To design a digital controller, the nominal model (27) was digitized with a sampling period TS = 30 s chosen with respect to plant dynamics. Characteristic loci g1 (z), g2 (z) of Gm (z), z = ejωT were calculated; g2 (z) was selected to generate equivalent eq eq subsystems G12 (z), G22 (z) using (17). Bode plots of discrete equivalent subsystems are in Fig. 5. 4. Required performance of the overall system was specified in terms of settling time ts = 600 s and maximum overshoot ηmax =

A. Kozáková, V. Veselý and V. Kučera / Automatica 107 (2019) 29–35

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Fig. 7. Nominal closed-loop stability test using the Nyquist plot of det[I + G(z)R(z)].

Fig. 6. Discrete Bode plots of equivalent open-loop systems: eq eq eq eq Q12 (z) = G12 (z)R1 (z), Q12 (z) = G22 (z)R2 (z).

5% corresponding to MT = 1.05 < MTO . Related frequencyeq domain design parameters for both equivalent subsystems G12 (z) eq and G22 (z) were obtained using (23): the minimum phase margin Φmin = 56.87◦ and the required bandwidth given by the crossover frequency ω0 = 0.0131 s−1 . Standard Bode design was completed for a required phase margin Φreq = 65◦ > Φmin . The continuous-time controllers R1 (s), R2 (s) were digitized to obtain R1 (z), R2 (z) as follows: eq

eq

G12 (z) R1 (s) = 0.1988 + R1 (z) =

0.0039 s

0.199 − 0.082z −1 1 − z −1

G22 (z) R2 (s) = 0.2212 + R2 (z) =

0.0034 s

0.221 − 0.119z −1 1 − z −1

Φachieved = 58.35

Φachieved = 65.70◦

ω0_achieved = 0.0122 s−1

ω0_achieved = 0.0121 s−1 .

◦

Bode plots of equivalent subsystems under related local controllers shown in Fig. 6 prove stability and achieved required performance and thus also closed-loop stability of the overall system (according to Theorem 2). Nominal closed-loop stability was verified using the Generalized Nyquist criterion (Fig. 7). The discrete-time sensitivity magnitude-versus-frequency plot with a peak around the crossover frequency proves good closed-loop nominal performance (Fig. 8). Nominal closed-loop responses are presented in Fig. 9.

Fig. 8. Magnitude-versus-frequency plot of σmax [S(z)]z =ejωT .

5. Fulfilment of the robust stability condition (22) examined in Fig. 10 proves closed-loop stability over the specified uncertainty region. 6. Conclusions A unified frequency-domain methodology for designing robust decentralized continuous-time and digital controllers for both MIMO and SISO uncertain systems described by a set of transfer function matrices was presented. The design procedure is based on the nominal model of the controlled system transformed into a set of equivalent subsystems using a selected characteristic locus of the matrix of interactions (for SISO plants the equivalent subsystem is directly the plant model). Local controllers that constitute the resulting decentralized controller are designed independently for each equivalent subsystem. Closedloop performance of the overall system is equivalent to the performance achieved in equivalent subsystems. Unlike standard robust approaches to decentralized controller design that use diagonal nominal model, the proposed methodology is based on the full nominal model which reduces conservatism of the robust stability conditions. The decentralized controller design relies on frequency-domain plots (Nyquist plots, Bode plots, CL) and is therefore applicable to design controllers of any structure for plants of any order including those with time delays and non-minimum phase dynamics.

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Fig. 9. Nominal closed-loop responses to – reference steps 0.1 m at the inputs of the 1st subsystem (in t1 = 0 s) and the 2nd subsystem (in t2 = 300 s) (l.h.s. plot); – reference steps 0.1 m at the inputs of the 1st subsystem (in t1 = 0 s) and the 2nd subsystem (in t3 = 10 s) (r.h.s. plot). O’Reilly, J., & Leithead, W. E. (1991). Multivariable control by ‘individual channel design’. International Journal of Control, 54(1), 1–46. Perez, J., Romero, A., Herrero, P., & Balaguer, P. (2012). Extending the AMIGO PID tuning method to MIMO systems. IFAC Proceedings Volumes, 45(3), 211–216. Skogestad, S., & Postlethwaite, I. (2009). Multivariable feedback control: Analysis and design. Chichester: Wiley. Veselý, V. (2018). On the subsystem level gain scheduled controller design for MIMO systems. International Journal of Control, Automation and Systems, 16(X), 1–10.

Fig. 10. Verification of the robust stability condition (22).

References Balaguer, P., & Romero, J. A. (2012). Model order reduction for decentralized PID control design on TITO processes. IFAC Proceedings Volumes, 45(3), 211–216. Balko, P., & Rosinova, D. (2015). Robust decentralized control of nonlinear drum boiler. IFAC-PapersOnLine, 48(14), 432–437. Chiu, M. S., & Arkun, Y. (1992). A methodology for sequential design of robust decentralized control systems. Automatica, 28(5), 997–1001. Hovd, M., & Skogestad, S. (1993). Improved independent design of robust decentralized controllers. Journal of Process Control, 3(1), 43–51. Hovd, M., & Skogestad, S. (1994). Sequential design of decentralized controllers. Automatica, 30(10), 1601–1607. Husek, P., & Kucera, V. (2014). Robust decentralized PI control design. IFAC Proceedings Volumes, 47(3), 4699–4703. Jin, Q., Wang, Q., & Liu, L. (2016). Design of decentralized proportional–integral– derivative controller based on decoupler matrix for TITO process with active disturbance rejection structure. Advances in Mechanical Engineering, 8(6), 1–18. Johansson, K. H. (2000). The quadruple-tank process: a multivariable laboratory process with an adjustable zero. IEEE Transactions on Control Systems Technology, 8(3), 456–465. Kozáková, A., & Veselý, V. (2009). Design of robust decentralized controllers using the M-∆ structure robust stability conditions. International Journal of Systems Science, 40(5), 497–505. Kozáková, A., & Veselý, V. (2013). A unified frequency domain method for designing robust decentralized controllers. IFAC Proceedings Volumes, 46(13), 266–271. Kozáková, A., Vesely, V., & Osusky, J. (2009). A new nyquist-based technique for tuning robust decentralized controllers. Kybernetika, 45(1), 63–83. Labibi, B., Marquez, H. J., & Chen, T. (2009). Decentralized robust PI controller design for an industrial boiler. Journal of Process Control, 19, 216–230. Li, Z., Melek, W., & Clark, C. (2009). Decentralized robust control of robot manipulators with harmonic drive transmission and application to modular and reconfigurable serial arms. Journal Robotica, 27(2), 291–302.

Alena Kozáková was born in Bratislava, Slovakia in 1960. She graduated in electrical engineering from the Slovak University of Technology in Bratislava, Slovakia in 1985 and received the Ph.D. in Technical Cybernetics in 1998. From 1985 to 2013 she was with the Department of Automatic Control Systems (later Institute of Control and Industrial Informatics). From 2018 she is a full professor at the Institute of Automotive Mechatronics, Faculty of Electrical Engineering and Information Technology, STU in Bratislava. Her research interests include robust, optimal and decentralized control.

Vojtech Veselý was born in Vel’ké Kapušany, Slovakia in 1940. He received the M.Sc. in electrical engineering from the Leningrad Electrical Engineering Institute, St. Peterburg, Russia in 1964, and the Ph.D. and the D.Sc. degrees from the Slovak University of Technology in Bratislava, Slovakia in 1971 and 1985, respectively. Since 1986 he has been a full professor. Since 1964 he has been with the Department of Automatic Control Systems, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology in Bratislava. His special fields of interest include robust control, decentralized control, H2 /Hinf optimization, LMI control, power systems control and process control. He supervised 24 Ph.D. students, and is author or coauthor of more than 450 scientific papers.

Vladimír Kučera was born in Prague, Czech Republic in 1943. He graduated in electrical engineering from Czech Technical University, Prague, in 1966 and obtained his research degrees in control engineering from the Czechoslovak Academy of Sciences, in 1970 and 1979. During 1967–2017 he was a Researcher, and from 1990 to 1998 Director of the Institute of Information Theory and Automation, one of the research institutes of the Academy of Sciences of the Czech Republic. Currently, he is an Emeritus Scientist of the Academy of Sciences. Since 1996 he has been a Professor of Control Engineering at the Czech Technical University, Prague. He served the university as Head of Control Engineering Department, Dean of Electrical Engineering, Director of the Masaryk

A. Kozáková, V. Veselý and V. Kučera / Automatica 107 (2019) 29–35 Institute of Advanced Studies, and currently as Distinguished Researcher and Vice Director of the Czech Institute of Informatics, Robotics, and Cybernetics. He held many visiting positions at prestigious European, American, Asian, and Australian universities. His research interests include linear systems and control theory. His well-known result is the parameterization of all controllers that stabilize a given plant, known as the Youla–Kučera parameterization, which has become a new paradigm in robust and optimal control. Recently, he has resolved a long-standing open problem of linear systems theory, the decoupling by static-state feedback.

35

He is an Advisor, Fellow, and a former President of IFAC. He is a Life Fellow of IEEE. His awards include the Prize of the Czechoslovak Academy of Sciences in 1972, the National Prize of the Czech Republic in 1989, and the Automatica Prize Paper Award in 1990. He is an Honorary Professor of the Northeastern University, Shenyang, China, and received honorary doctorates from Université Paul Sabatier, Toulouse, France and Université Henri Poincaré, Nancy, France. He is a Chevalier dans l’ordre des palmes académiques, a national order of France for distinguished academics.