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Robust Discretization of LTI Systems with Polytopic Uncertainties and Aperiodic Sampling
Robust Discretization of LTI Systems with Polytopic Uncertainties and Aperiodic Sampling
Journal of Electrical Engineering and Technology. 2015. May, 10(3): 1255-1263
Copyright © 2015, The Korean Institute of Electrical Engineers
This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/)which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
  • Received : October 04, 2014
  • Accepted : December 29, 2014
  • Published : May 01, 2015
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About the Authors
Dong Hwan Lee
Dept. of Electrical and Computer Engineering, Purdue University, USA. (lee1923@purdue.edu)
Jin Bae Park
Corresponding Author: Dept. of Electrical and Electronic Engineering, Yonsei University, Korea (jbpark@yonsei.ac.kr).
Young Hoon Joo
Dept. of Control and Robotics Engineering, Kunsan National University, Korea. (yhjoo@kunsan.ac.kr)

Abstract
In the previous work, the authors studied the problem of robust discretization of linear time-invariant systems with polytopic uncertainties, where linear matrix inequality (LMI) conditions were developed to find an approximate discrete-time (DT) model of a continuous-time (CT) system with uncertainties in polytopic domain. The system matrices of obtained DT model preserved the polytopic structures of the original CT system. In this paper, we extend the previous approach to solve the problem of robust discretization of polytopic uncertain systems with aperiodic sampling. In contrast with the previous work, the sampling period is assumed to be unknown, time-varying, but contained within a known interval. The solution procedures are presented in terms of unidimensional optimizations subject to LMI constraints which are numerically tractable via LMI solvers. Finally, an example is given to show the validity of the proposed techniques.
Keywords
1. Introduction
Continuous-time (CT) systems controlled by digital controllers are referred to as sampled-data (SD) control systems, which are composed of CT systems to be controlled, discrete-time (DT) controllers controlling them, and the ideal sampler and zero-order holder to convert the CT signals into DT ones and vice versa [4] . When the digital controller is implemented on an actual CT plant, the control action through the zero-order holder appears as a piecewise constant signal in time, which is termed a SD controller. Significant research efforts on the SD control design have been made in the literature, and they can be divided into several categories. For instance, the so-called the direct DT design [17] is a design method based on the discretization of the CT system, where a DT controller is designed in DT domain directly. In the so-called lifting techniques [1 , 4 , 27] , the SD controller design problem is transformed into an equivalent finite-dimensional discrete problem. The so-called jump system-based method [14 , 25] is based on the representation of the system in the form of hybrid discrete/continuous model. The input delay approach [10 , 11 , 20] treats the SD systems as a CT system with uncertain but bounded time-varying delay in the control input.
Among the promising results, this paper focuses on the direct DT design method, in which the computation of an exact DT model of the original CT plant is required. While for LTI systems, the exact DT model is available in principle, this is not the case for nonlinear systems [15 , 16] or uncertain LTI systems [3 , 17] . Rather, an approximate DT model can be used in replacement of the exact DT model for the SD control design. A major drawback of the approximation technique is that they can suffer from degradation in performance or even lead to instability of the resulting SD control system when the approximation error is relatively large [18] .
Especially for DT LTI systems with poyltopic uncertainties, substantial LMI-based results on robust control problems have been made up to date (e.g., [5 - 9 , 13 , 19 , 28 - 38] ), and most of them implicitly assumed that either exact or approximate polytopic DT model of the original CT plant is available. In order to apply the linear matrix inequality (LMI) methods for control design of DT systems, it is essential for the obtained approximate DT model to preserve the polytopic structure of the original CT system. A widely used simplest method is to take an approximation via the first-order Taylor series of the exact DT model under the assumption of fast sampling/fast hold [18] . This strategy usually works well under fast sampling, but the approximation error may become prohibitively large if the sampling period is relatively long. To alleviate this problem, in the previous work [18] , we developed new LMI-based techniques to search for more exact approximation of the exact DT models of the original CT polytopic uncertain LTI systems, in which the discrepancy between the exact and the approximate DT models was minimized. To this end, we exploited higher-order truncated Taylor series of the exact DT model so that the truncation error of the approximate DT model can be reduced.
Although the proposed method was successful in reducing the approximation error, there was still an unsolved problem: it can be applied only to the case that the sampling period is constant in time. To resolve this problem, in this paper, we investigate the robust discretization problem under aperiodic sampling. Specifically, it is assumed that the sampling period is time-varying and unknown, but lies within a known interval. Similarly to [18] , this problem is tackled by minimizing the norm distances between the system matrices of the approximate and exact DT models. To obtain numerically tractable method to compute the approximation, the truncated Taylor series of the exact DT model is used similarly to [18] . The solution procedures are given in terms of unidimensional optimizations subject to LMIs, which can be readily tractable via convex optimizations [2] . To derive the LMI constraints, the socalled matrix-dilation technique [9] , [22 - 24] is applied. A sufficient LMI condition to design a state feedback SD controller for the computed DT models is also studied briefly as one of applications of the proposed robust discretization strategy. Finally, an illustrative example is given to demonstrate the potential of the developed method.
2. Preliminaries
- 2.1 Notations
The adopted notation is as follows:
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and
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: sets of nonnegative real numbers and nonnegative integer, respectively;
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and
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: the n -dimensional Euclidean space and the set of all m × n real matrices, respectively; AT : transpose of matrix A ; A >0 ( A <0, A ≥ 0 , and A ≤ 0 , respectively): symmetric positive definite (negative definite, positive semi-definite, and negative semi-definite, respectively) matrix A ; A B : Kronecker’s product of matrices A and B ; He{ A } : a shorthand notion for A + AT ; In : n × n identity matrix; 0 n and 0 m×n : n × 1 zero vector and m × n zero matrix, respectively
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:
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; * inside a matrix: transpose of its symmetric term; || ⋅ || : Euclidean vector norm for vectors or the matrix two-norm for matrices; A : any matrices whose columns form bases of the right nullspace of matrix
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: unit vector of dimension j with a 1 in the i -th component and 0 ’s elsewhere.
- 2.2 Problem formulation
Consider the CT LTI polytopic uncertain system
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where
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,
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is the state,
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is the control input, and matrices
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and
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are not precisely known but assumed to belong to the convex set
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where Δ N is the unit simplex given by
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It is assumed that the system is controlled by the SD controller
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where 0 1 { t 0 , t 1 ,…} represents an unbounded monotonously increasing sequence of sampling instants with elements in
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, i.e.,
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. We assume that the sampling interval, denoted by θk = t k+1 tk , is time-varying and unknown but lies in a known compact set, θk ∈ [ θmin , θmax ] where 0 < θ min < θ max < ∞ . The closed-loop SD control system composed of (1) and (2) is given by
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The state at time t k+1 is
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Introducing notation Ad ( α , θk ) := eAc(α)θk
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ud ( k ) := uc ( tk ) , system (3) can be converted to the uncertain DT linear time-varying (LTV) system
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where
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. DT LTV system (4) can be viewed as the exact discretization of (3) in the sense that || xc ( tk ) − xd ( k ) || = 0,
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, α ∈ Δ N is satisfied with xc (0) = xd (0) and any control input sequence { ud (0), ud (1),…}. Note that (4) is the exact DT model of the SD control system (3) (or CT system (1)). As indicated in [18] , due to the nonlinear and infinite dimensional nature of Ad ( α , θk ) Bd ( α , θk )with respect to the uncertain parameters and sampling period θk , it may be difficult to find their exact representations that preserve the polytopic structures of Ac ( α ) and Bc ( α ) . To simplify the problem, let us consider uniform sampling period θ = θ 0 = θ 1 =⋯. In this case, most researches addressing the robust control of DT polytopic uncertain LTI systems approximate Ad ( α , θk ) and Bd ( α , θk ) to their first-order power series with the assumption that the sampling period θ is sufficiently small. However, when θ is relatively large, the approximations become inaccurate. To alleviate this problem, the concept of the robust discretization was suggested in the previous work [18] . Roughly speaking, the robust discretization problem is finding approximations G ( α ) and H ( α ) of matrices Ad ( α , θk ) and Bd ( α , θk ) , respectively, such that both G ( α ) and H ( α ) preserve the polytopic structures of Ac ( α ) and Bc ( α ) . In other words, it is required that the approximations can be expressed as convex combinations of given vertices. Specifically, a simplified robust discretization problem addressed in [18] can be expressed as finding matrices Gi , Hi , i ∈ {1,2, … , N } that solve the optimizations
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where
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and
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are the h -order Taylor series approximations of matrices
Ad ( α , θk )and Bd ( α , θk ) , respectively. As mentioned in the introduction, the research in [18] only considered the case of the uniform sampling period. If the sampling period is time-varying within a known bound, the problem becomes more complicated. In this paper, we cope with the robust discretization problem under aperiodic sampling. The robust discretization problem considered in [18] is modified as follows.
Problem (Robust discretization under aperiodic sampling) . Let integer h ≥ 1 be given. Compute matrices Gij , Hij , ( i , j )) ∈ {1, 2, …, N } × {1, 2}that solve the following optimizations:
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where
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Note that G ( α , θk ) and H ( α , θk ) depend on θk and have poyltopic structures with respect to θk .
3. Main Result
In this section, LMI solutions to the robust discretization with aperiodic sampling are presented. As in [18] , optimizations (5) and (6) can be rewritten by
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Alternative expressions are
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which are equivalent to (9) and (10), respectively. We will use expressions (11) and (12) rather than (9) and (10) since (11) and (12) are more suitable to be converted into LMI conditions. The following results can be viewed as the main results of this paper. They establish sufficient LMI conditions that ensure constraints (11) and (12).
Theorem 1 : Let h ≥ 1 be given. If there exist matrices
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and a scalar γA ≥ 0 such that
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where ( θ 1 , θ 2 ) = ( θ min , θ max ) , then constraint in (11) is satisfied.
Proof . First of all, multiplying (13) by αiβj ( θk ) and summing for ( i , j )∈{1, 2,…, N }×{1,2} , we obtain
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where G ( α , θk ) and H ( α , θk ) are defined in (8). Applying the Schur complement to the above inequalities yields
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Pre- and post-multiplying the last inequality by Π [h]T and its transpose, where
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and using relation ( Lh θAc ( α ) T Rh In [h] = 0 nh×n we can obtain the constraint in (11). This completes the proof.
Similarly to Theorem 1, an LMI condition that ensures constraint (12) can be obtained.
Theorem 2 : Let h ≥ 1 be given. If there exist matrices
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and a scalar γ B ≥ 0 such that
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where ( θ 1 , θ 2 ) = ( θ min , θ max ) , then constraint (12) is satisfied.
Proof . The proof is straightforwardly extended from the proof of Theorem 1 so omitted for brevity.
In this regard, the optimizations in (11) and (12) can be replaced by the following optimizations subject to LMI constraints:
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Remark. Optimizations (16) and (17) are singleparameter minimization problems subject to LMI constraints, and hence, can be solved by means of a sequence of LMI optimizations, i.e. a line search or a bisection process over γA and γB , respectively, or solved by the eigenvalue problem (EVP) [2] , which is convex optimization, and hence, can be directly treated with LMI solvers [12 , 21 , 26] .
4. Application
Although the proposed strategy provides only approximate solutions to the robust discretization problem with aperiodic sampling, it may be at least more precise than the firstorder Taylor series approximation. Moreover, the proposed technique would be effective from the practical point of view since as stated in [18] , once a discretized model of a CT system is obtained, then it can be stored in database and used repeatedly for various SD control design purposes through existing LMI-based DT control design techniques (e.g., [5 - 7 , 9] to name a few) in the literature. For instance, let us assume that matrices
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,
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( i , j )∈{1, 2,…, N }×{1, 2} are solutions to optimizations (16) and (17), respectively. Instead of considering exact discretization (4) of the original CT system (3), consider the following DT system which is an approximate discretization of (3) under aperiodic sampling:
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where
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is the state and
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is the control input. Note that DT system (18) can be viewed as an approximate DT model of the exact DT model (4). In addition, let us consider the following state-feedback control law:
π (k) = (k).
The closed-loop system is
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Based on the LMI design approach developed in [6] , we can readily establish the following LMI-based state-feedback design condition.
Proposition 1. If there exist matrices
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,
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and
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such that LMIs
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hold, then state-feedback gain given by F = KS −1 stabilizes closed-loop system (19) for all α ∈ Δ N and for all time-varying sampling period θk ∈ [ θ min , θ max ]
Proof. Multiplying (20) by αi βj ( θk ) βl ( θ k+1 ) and summing for ( i, j,l )∈{1, 2,…, N }×{1, 2} 2 , we obtain
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where
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and G ( α , θk ) and H ( α , θk ) are defined in (8). Next, by pre- and post-multiplying the last inequality by
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and its transpose, and by applying the extended Schur complement in [6] , it follows that
(G( α, θk) +H( α, θk)F)T X( α, θk+1) ×(G( α, θk) +H( α, θk)F) − X( α, θk) < 0, ∀(α, θk, θk+1 ∈ ΔN ×[θmin ,θmax]2
where F = KS −1 and X ( α , θk ) = S T P ( α , θk ) S −1 . By means of the Lyapunov theory, one concludes that (19) is asymptotically stable for all α ∈ Δ N and for all timevarying sampling period θk ∈ [ θ min , θ max ]. This completes the proof.
On the other hand, let us consider the SD state-feedback controller
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for system (3). The closed-loop SD control system is
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If G ( α , θk ) = Ad ( α , θk ) and H ( α , θk ) = Bd ( α , θk ) for all α ∈ Δ N and θk ∈ [ θ min , θ max ] , then one can expect that || xc ( tk ) − ξ ( k ) || = 0,
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α ∈Δ N is satisfied with xc (0) = ξ (0) and any control input sequence { π (0), π (1),…} . Although the idealistic case may not occur in reality, we can still expect that if G ( α , θk ) ≅ Ad ( α , θk ) and H ( α , θk ) ≅ Bd ( α , θk ), then the solution xc ( t ) to (22) closely matches the solution ξ ( k ) to (19) at every sampling instants { t 0 , t 1 ,…} . In this respect, the proposed robust discretization under aperiodic sampling can be viewed as a practically useful and simple approach to deal with various SD control problems.
All numerical examples in the sequel were treated with the help of MATLAB R2012b running on a PC with Intel Core i7-3770 3.4GHz CPU, 32GB RAM. The LMI problems were solved with SeDuMi 1.3 [26] and Yalmip [21] .
Example 1 . Let us consider the linearized model of the inverted pendulum system taken from [3] . Its state-space realization is given by
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xc,1 ( t ) is the position of the cart,
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, x c,3 ( t ) is the angle of the pendulum from the vertical,
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,
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is the mass of the pendulum,
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is the mass of the cart,
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is the length of the pendulum, and uc ( t ) is the horizontal force applied to the cart. We assume
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and
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. Then, the system can be described by (1) with two vertices. By applying Theorems 1 and 2 with min max ( θ min , θ max , h ) = (0.01s,0.1s,7) , we obtain the approximate DT system (18) with ( γA , γB ) = (0.1112,4.8251×10 −8 ) and
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By using Proposition 1, the state-feedback gain is calculated as follows:
F = [37.8 79.5 1148.3 210.5].
The simulation results with x 0 = [5 −3 2 −3] T and α = [0.5,0.5] T are depicted in Figs. 1(a)-(d) , where xc ( t ) (solid line) is the solution to the SD closed-loop system (22) and ξ ( k ) at each sampling instant (dot) is the solution to the DT closed-loop system (19). In other words, the dotted lines in Figs. 1(a)-(d) can be viewed as the state trajectories ξ ( k ) of the approximately discretized model of the original CT system (1) and the solid lines indicate the state trajectories xc ( t ) of the CT plant (1). The closeness of the two trajectories implies that the robust discretization approach proposed in this paper is an exact approximation of the exact discritization of the CT plant (1). From the figure, we confirm that the trajectory of ξ ( k ) closely matches the trajectory of xc ( t ) at sampling instants { t 0 , t 1 ,…} .
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The solid line is the solution to the SD closed-loop system (22) xc,1(t) and the dotted line is the solution to the DT closed-loop system (19) ξ1 (k) at each sampling instant.
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The solid line is the solution to the SD closed-loop system (22) xc,2(t) and the dotted line is the solution to the DT closed-loop system (19) ξ2 (k) at each sampling instant.
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The solid line is the solution to the SD closed-loop system (22) xc,3(t) and the dotted line is the solution to the DT closed-loop system (19) ξ3 (k) at each sampling instant.
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The solid line is the solution to the SD closed-loop system (22) xc,4(t) and the dotted line is the solution to the DT closed-loop system (19) ξ4 (k) at each sampling instant.
5. Conclusion
In this paper, our previous work on the robust discretization problem has been extended to deal with the same problem with aperiodic sampling. LMI conditions to compute approximate DT models of the original CT plants have been developed. Finally, an example has been given to illustrate the developed method.
Acknowledgements
This work was supported by National Research Foundation of Korea(NRF) grant by the Korea government (MEST) (NRF-2012R1A2A2A01014088).
BIO
Dong Hwan Lee received the B.S. degree in Electronic Engineering from Konkuk University, Seoul, Korea, in 2008 and M.S. degree in Electrical and Electronic Engineering, Yonsei University, Seoul, Korea, in 2010. Since 2014, he is working toward a Ph.D. degree in the Department of Electrical and Computer Engineering, Purdue University, USA. His current research interests include stability analysis in fuzzy systems, fuzzy-model-based control, and robust control of uncertain linear systems.
Jin Bae Park received his B.S. degree in Electrical Engineering from Yonsei University, Seoul, Korea, and his M.S. and Ph.D. degrees in Electrical Engineering from Kansas State University, Manhattan, in 1977, 1985, and 1990, respectively. Since 1992, he has been with the Department of Electrical and Electronic Engineering, Yonsei University, Seoul, Korea, where he is currently a professor. His major interest is mainly in the field of robust control and filtering, nonlinear control, intelligent mobile robot, fuzzy logic control, neural networks, Hadamard transform, chaos theory, and genetic algorithms. He served as Editor-in-Chief for the International Journal of Control, Automation, and Systems (IJCAS) (2006-2010) and President for the Institute of Control, Robot, and Systems Engineers (ICROS) (2013), and is serving as the Vice-President for Yonsei University (2014-present)
Young Hoon Joo received the B.S., M.S., and Ph.D. degrees in Electrical Engineering from Yonsei University, Seoul, Korea, in 1982, 1984, and 1995, respectively. He worked with Samsung Electronics Company, Seoul, Korea, from 1986 to 1995, as a project manager. He was with the University of Houston, Houston, TX, from 1998 to 1999, as a visiting professor in the Department of Electrical and Computer Engineering. He is currently a professor in the Department of Control and Robotics Engineering, Kunsan National University, Korea. His major interest is mainly in the field of intelligent robot, intelligent control, human-robot interaction, wind farm control, and intelligent surveillance systems. He served as President for Korea Institute of Intelligent Systems (KIIS) (2008-2009) and is serving as Editor-in-Chief for the Intelligent Journal of Control, Automation, and Systems (IJCAS) (2014-present) and is serving as the Vice- President for the Korean Institute of Electrical Engineers (KIEE) (2013-present).
References
Bamieh B. , Pearson J. , Francis B. , Tannenbaum A. 1991 “A lifting technique for linear periodic systems,” Systems and Control Letters 17 (2) 79 - 88    DOI : 10.1016/0167-6911(91)90033-B
L¨ofberg J. “YALMIP: a toolbox for modeling and optimization in MATLAB,” IEEE International Symposium on Computer Aided Control Systems Design Taipei, Taiwan 2-4 September 2004 284 - 289
Chen C. -T. 1995 Linear System Theory and Design Oxford University Press New York
Chen T. , Francis B. A. 1995 Optimal Sampled-Data Control Systems Springer-Verlag London
Cuzzola F. A. , Geromel J. C. , Morari M. 2002 “An improved approach for constrained robust model predictive control,” Automatica 38 (7) 1183 - 1189    DOI : 10.1016/S0005-1098(02)00012-2
de Oliveira M. C. , Bernussou J. , Geromel J. C. 1999 “A new discrete-time robust stability condition,” System and Control Letters 37 (4) 261 - 265    DOI : 10.1016/S0167-6911(99)00035-3
de Oliveira M. C. , Geromel J. C. , Bernussou J. 2002 “Extended 2 H and H∞ norm characterizations and controller parameterizations for discrete-time systems,” International Journal of Control 75 (9) 666 - 679    DOI : 10.1080/00207170210140212
Ding B. , Xi Y. , Li S. 2004 “A synthesis approach of on-line constrained robust model predictive control,” Automatica 40 (1) 163 - 167    DOI : 10.1016/j.automatica.2003.07.007
Ebihara Y. , Peaucelle D. , Arzelier D. 2011 “Periodically timevarying memory state-feedback controller synthesis for discrete-time linear systems,” Auto-matica 47 (1) 14 - 25
Fridman E. , Seuret A. , Richard J. -P. 2004 “Robust sampled data stabilization of linear systems: an input delay approach,” Automatica 40 (8) 1441 - 1446    DOI : 10.1016/j.automatica.2004.03.003
Fridman E. 2010 “A refined input delay approach to sampled-data control,” Automatica 46 (2) 421 - 427    DOI : 10.1016/j.automatica.2009.11.017
Gahinet P. , Nemirovski A. , Laub A. J. , Chilali M. 1999 LMI Control Toolbox MathWorks Natick
Goncalves E. N. , Palhares R. M. , Takahashi R. H. C. , Mesquita R. C. 2007 “New strategy for robust stability analysis of discrete-time uncertain systems,” Systems and Control Letters 56 (7-8) 516 - 524    DOI : 10.1016/j.sysconle.2007.02.004
Hu T. , Lam J. , Cao Y. , Shao H. 2003 “A LMI approach to robust H2 sampled-data control for linear uncertain systems,” IEEE Transaction Systems Man and Cybernetics 33 (1) 149 - 155
Lee D. H. , Joo Y. H. , Tak M. H. 2014 “Local stability analysis of continuous-time Takagi-Sugeno fuzzy systems: A fuzzy Lyapunov function approach,” Inf. Sci. 257 (1) 163 - 175    DOI : 10.1016/j.ins.2013.08.036
Kim D. W. , Park J. B. , Joo Y. H. 2007 “Effective digital implementation of fuzzy control systems based on approximate discrete-time models,” Automatica 43 (10) 1671 - 1683    DOI : 10.1016/j.automatica.2007.01.025
Laila D. S. 2003 “Design and analysis of nonlinear sampleddata control systems,” Ph.D. dissertation University of Melbourne, Dept. of Electrical and Electronic Engineering
Lee D.H. , Tak M.H. , Joo Y.H. 2013 “An asymptotically exact LMI solution to the robust discretisation of LTI systems with polytopic uncertainties and its application to sampled-data control,” International Journal of Systems Science
Li D. , Xi Y. , Zheng P. 2009 “Constrained robust feedback model predictive control for uncertain systems with polytopic description,” International Journal of Control 82 (7) 1267 - 1274    DOI : 10.1080/00207170802530883
Liu Y. -Y. , Yang G. -H. 2012 “Sampled-data H∞ control for networked control systems with digital control inputs,” International Journal of Systems Science 43 (9) 1728 - 1740    DOI : 10.1080/00207721.2010.550401
L¨ofberg J. “YALMIP: a toolbox for modeling and optimization in MATLAB,” IEEE International Symposium on Computer Aided Control Systems Design 284 - 289
Oishi Y. 2009 “An asymptotically exact approach to robust semidefinite programming problems with function variables,” IEEE Transactions on Automatic Control 54 (5) 1000 - 1006    DOI : 10.1109/TAC.2009.2017141
Oliveira R. C. L. F. , de Oliveira M. C. , Peres P. L. D. 2008 “Convergent LMI relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter-dependent Lyapunov functions,” Systems and Control Letters 57 (8) 680 - 689    DOI : 10.1016/j.sysconle.2008.01.006
Peaucelle D. , Sato M. 2009 “LMI tests for positive definite polynomials: slack variable approach,” IEEE Transactions on Automatic Control 54 (4) 886 - 891    DOI : 10.1109/TAC.2008.2010971
Sivashankar N. , Khargonekar P. P. 1994 “Characterization of the L2-induced norm for linear systems with jumps with application to sampled-data systems,” SIAM Journal on Control and Optimization 32 (4) 1128 - 1150    DOI : 10.1137/S0363012991223121
Strum J. F. 1999 “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,” Optimization Methods and Software http://sedumi.mcmaster.ca/ 11-12 625 - 653
Yamamoto Y. “New approach to sampled-data control systems-a function space method,” Proceedings of the 29th Conference on Decision and Control Honolulu, Hawaii 5-7 December 1990 1882 - 1887
Lee D. H. , Tak M. H. , Joo Y. H. “A Lyapunov functional approach to robust stability analysis of continuous-time uncertain linear systems in polytopic domains” International Journal of Control, Automation, and Systems 11 (3) 460 - 469    DOI : 10.1007/s12555-012-0426-1
Sung H. C. , Park J. B. , Joo Y. H. “Observerbased sampled-data control for uncertain nonlinear systems: intelligent digital redesign approach,” International Journal of Control, Automation, and Systems 12 (3) 486 - 496    DOI : 10.1007/s12555-013-0129-2
Lee D. H. , Joo Y. H. 2015 “LMI-based robust sampleddata stabilization of polytopic LTI systems: a truncated power series expansion approach” International Journal of Control, Automation, and Systems
Lee D. H. , Joo Y. H. , Tak M. H. 2014 “Periodically timevarying H∞ memory filter design for discrete-time LTI systems with polytopic uncertainty,” IEEE Transactions on Automatic Control 59 (5) 1380 - 1385    DOI : 10.1109/TAC.2013.2289705
Song M. K. , Park J. B. , Joo Y. H. 2013 “Stability and stabilization for discrete-time Markovian jump fuzzy systems with time-varying delays; Partially known transition probabilities case” International Journal of Control, Automation, and Systems 11 (1) 136 - 146    DOI : 10.1007/s12555-011-9112-y
Lee D. H. , Joo Y. H. “On the generalized local stability and local stabilization conditions for discretetime Takagi-Sugeno fuzzy systems” IEEE Transactions on Fuzzy Systems 22 (6) 1654 - 1668    DOI : 10.1109/TFUZZ.2014.2302493
Lee D. H. , Joo Y. H. “Extended robust H2 and H∞ filter design for discrete time-invariant linear systems with Polytopic Uncertainty” Circuits Syst Signal Process 33 (2) 393 - 419    DOI : 10.1007/s00034-013-9658-4
Lee D. H. , Joo Y. H. , Tak M. H. “Periodically timevarying memory static output feedback control design for discrete-time LTI systems” Automatica 52 (1) 47 - 54    DOI : 10.1016/j.automatica.2014.10.119
Lee D. H. , Joo Y. H. , Tak M. H. “A convex optimization approach to adaptive stabilization of discrete-time LTI systems with polytopic uncertainties” International Journal of Adaptive Control and Signal Processing
Koo G. B. , Park J. B. , Joo Y. H. “Intelligent digital redesign for nonlinear systems using a guaranteed cost control method” International Journal of Control, Automation, and Systems 11 (6) 1075 - 1083    DOI : 10.1007/s12555-013-0093-x
Lee D. H. , Joo Y. H. , Tak M. H. 2015 “FIR-type robust H2 and H∞ control of discrete linear timeinvariant polytopic systems via memory state-feedback control laws” International Journal of Control, Automation, and Systems