## Stabilizing Model Predictive Control of Hybrid Systems

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### BibTeX

@MISC{Lazar_stabilizingmodel,

author = {M. Lazar and W. P. M. H. Heemels and S. Weil and A. Bemporad},

title = {Stabilizing Model Predictive Control of Hybrid Systems},

year = {}

}

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### Abstract

Abstract—In this note, we investigate the stability of hybrid systems in closed-loop with model predictive controllers (MPC). A priori sufficient conditions for Lyapunov asymptotic stability and exponential stability are derived in the terminal cost and constraint set fashion, while allowing for discontinuous system dynamics and discontinuous MPC value functions. For constrained piecewise affine (PWA) systems as prediction models, we present novel techniques for computing a terminal cost and a terminal constraint set that satisfy the developed stabilization conditions. For quadratic MPC costs, these conditions translate into a linear matrix inequality while, for MPC costs based on 1,-norms, they are obtained as norm inequalities. New ways for calculating low complexity piecewise polyhedral positively invariant sets for PWA systems are also presented. An example illustrates the developed theory. Index Terms—Hybrid systems, Lyapunov stability, model predictive control (MPC), piecewise affine systems. I.

### Citations

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Constrained model predictive control: Stability and optimality
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(Show Context)
Citation Context ...apunov stability theory [1] can be used to prove that the MPC control law is stabilizing. For a comprehensive overview on stability of receding horizon control in discrete-time we refer the reader to =-=[2]-=- and the references therein. The recent development of MPC for hybrid systems, which are inherently discontinuous and nonlinear, requires a reconsideration of the stability results, as it was also poi... |

379 | D.; Smooth stabilization implies coprime factorization
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Citation Context ...yapunov stability in discrete-time. For continuous-time nonlinear systems affected by external (disturbance) inputs, the input-to-state stability (ISS) framework, introduced in the fundamental works (=-=Sontag, 1989-=-, 1990; Sontag and Wang, 1995), has proven to be very successful, as it generalizes the Lyapunov stability concept to systems affected by perturbations. Extensions to the discrete-time case have recen... |

307 | Control of systems integrating logic, dynamics, and constraints
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(Show Context)
Citation Context ...ires a reconsideration of the stability results, as it was also pointed out in the excellent survey [2]. Attractivity was proven for hybrid systems in closed-loop with model predictive controllers in =-=[3]-=- and [4]. However, proofs of Lyapunov stability only appeared in the literature recently; for particular classes of hybrid systems and MPC cost functions, see, for example, [5]–[7]. In these works, ei... |

230 | A uni framework for hybrid control: Model and optimal control theory
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Citation Context ... Input-to-state stability. 1. INTRODUCTION Discrete-time discontinuous nonlinear and hybrid systems form a powerful and general modeling class for the approximation of hybrid and nonlinear phenomena [=-=Branicky et al., 1998-=-], which also includes the class of piecewise affine (PWA) systems [Heemels et al., 2001]. The modeling capability of the latter class of systems has already been shown in several applications, includ... |

165 | On characterizations of the input-to-state stability property,” Syst
- Sontag, Wang
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(Show Context)
Citation Context ... discrete-time. For continuous-time nonlinear systems affected by external (disturbance) inputs, the input-to-state stability (ISS) framework, introduced in the fundamental works (Sontag, 1989, 1990; =-=Sontag and Wang, 1995-=-), has proven to be very successful, as it generalizes the Lyapunov stability concept to systems affected by perturbations. Extensions to the discrete-time case have recently been developed in (Jiang ... |

163 |
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Citation Context ...resentation and more structure, but still include a wide range of industrially relevant processes. The following classes of hybrid systems are part of this category: • Piecewise affine (PWA) systems (=-=Sontag, 1981-=-, 1996; Johansson and Rantzer, 1998; Ferrari-Trecate et al., 2002); • Switched systems (Liberzon, 2003); • Mixed logical dynamical (MLD) systems (Bemporad and Morari, 1999; Bemporad, 2004); • Linear c... |

150 | An introduction to hybrid dynamical systems - Schaft, Schumacher - 2000 |

99 | Input-to-state stability for discrete-time nonlinear systems
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(Show Context)
Citation Context ...r this task, see, for example, [Bemporad and Morari, 1999, Kerrigan and Mayne, 2002, Lazar et al., 2006] and the references therein. In this paper we are interested in input-to-state stability (ISS) [=-=Jiang and Wang, 2001-=-] as a property to characterize robust stability of hybrid systems in closed-loop with MPC. More precisely, we consider systems that are piecewise continuous, affected by additive disturbances. It is ... |

98 | Equivalence of hybrid dynamical models
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(Show Context)
Citation Context ...s for PWA prediction models and the result of Theorem III.2 applies. Although we focus on PWA systems of the form (5), the results presented in the sequel have a wider applicability since it is known =-=[12]-=- that PWA systems are equivalent under certain mild assumptions with other relevant classes of hybrid systems, such as mixed logical dynamical systems and linear complementarity systems, and they can ... |

96 |
Model Predictive Heuristic Control: Applications to Industrial Processes
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Citation Context .... In this framework, several solutions have been proposed both in the industrial world and in the academic world: IDCOM - Identification and command (later MAC - Model algorithmic control) at ADERSA (=-=Richalet et al., 1978-=-) and DMC - Dynamic matrix control at Shell (Cutler and Ramaker, 1980), which use step and impulse response models, (the adaptive control branch) MUSMAR - Multistep multivariable adaptive regulator (M... |

81 |
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Citation Context ...econsideration of the stability results, as it was also pointed out in the excellent survey [2]. Attractivity was proven for hybrid systems in closed-loop with model predictive controllers in [3] and =-=[4]-=-. However, proofs of Lyapunov stability only appeared in the literature recently; for particular classes of hybrid systems and MPC cost functions, see, for example, [5]–[7]. In these works, either con... |

78 | Complementarity modeling of hybrid systems - Schaft, Schumacher - 1998 |

66 | A Lyapunov Formulation of Nonlinear Small Gain Theorem for
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(Show Context)
Citation Context ...ptimization problems) that are not optimal, but within a margin δ ≥ 0 from the optimum, we introduce the notion of ε-ISS as a particular case of the input-tostate practical stability (ISpS) property [=-=Jiang et al., 1996-=-]. Furthermore, we establish an analytic relation between the optimality margin δ of the solution of the MPC optimization problem and the ISS margin ε(δ). Revealing this explicit relation is an import... |

61 | Interconnected automata and linear systems: A theoretical framework in discrete time
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(Show Context)
Citation Context ...and MMPS systems. Also, it is well known that PWA systems can approximate nonlinear systems arbitrarily well (Sontag, 1981) and they can arise from the interconnection of linear systems and automata (=-=Sontag, 1996-=-). The modeling power of PWA systems has already been shown in several applications, such as switched power converters (Leenaerts, 1996), optimal control of DC-DC converters and direct torque control ... |

60 | Further facts about input to state stabilization - Sontag - 1990 |

59 |
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(Show Context)
Citation Context ...nteeing stability of the controlled system, on the other hand. In this chapter we are interested in stability issues and therefore, we focus our attention on results developed in this direction. See (=-=Scokaert and Mayne, 1998-=-; Mayne et al., 2000; Bemporad et al., 2003b; Kerrigan and Maciejowski, 2004; Pluymers et al., 2005; Mayne et al., 2004; Alamo et al., 2005; Muñoz de la Peña et al., 2005) and the references therein f... |

59 | Suboptimal model predictive control (Feasibility implies stability
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(Show Context)
Citation Context ...on simpler optimization problems, which can be solved faster, and that still have an a priori stability guarantee. An important result regarding sub-optimal MPC of nonlinear systems was presented in (=-=Scokaert et al., 1999-=-), where it is shown that feasibility of the MPC optimization problem rather than optimality is sufficient for stability. In (Scokaert et al., 1999), stability is achieved without optimality, by forci... |

57 |
Stability theory of dynamical systems
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Citation Context ...ral works that also cover the discrete-time case have 1 Copies of these two articles can be supplied upon request.s38 Classical stability results revisited appeared, see, for example, (Freeman, 1965; =-=Willems, 1970-=-; LaSalle, 1976) and the more recent books (Vidyasagar, 1993; Khalil, 2002). In discrete-time the candidate Lyapunov function and the system dynamics do not necessarily have to be continuous, as it wi... |

50 | Optimal control of constrained piecewise affine discretetime systems
- Mayne, Raković
- 2003
(Show Context)
Citation Context ...hybrid systems. The model predictive control (MPC) methodology [Mayne et al., 2000] has proven to be one of the most successful frameworks for this task, see, for example, [Bemporad and Morari, 1999, =-=Kerrigan and Mayne, 2002-=-, Lazar et al., 2006] and the references therein. In this paper we are interested in input-to-state stability (ISS) [Jiang and Wang, 2001] as a property to characterize robust stability of hybrid syst... |

50 |
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Citation Context ...y of discrete-time nonlinear MPC. Assumption III.1: There exist I@1A, P@1A Pu, a neighborhood of the origin x � @x A and a function �@1A such that „ , with H P ���@ „ A, is a positively invariant set =-=[8]-=- for system (1) in closed-loop with �� a �@��A, � P C v@�Y �A ! I @���A for all � P � @x A and all � P (4a) p @�A P @���A for all � Px and (4b) p @� @�Y �@�AAA 0 p @�A C v @�Y �@�AA H for all � P „ X ... |

46 |
Control System Analysis and Design Via the "Second Method" of Lyapunov
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Citation Context ...oach to ensure stability in MPC is to consider the value function of the MPC cost as a candidate Lyapunov function. Then, if the system dynamics is continuous, the classical Lyapunov stability theory =-=[1]-=- can be used to prove that the MPC control law is stabilizing. For a comprehensive overview on stability of receding horizon control in discrete-time we refer the reader to [2] and the references ther... |

44 |
Predictive Control - A Unified Approach
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Citation Context ...t/output models were also developed later on, from which thes20 Introduction most significant ones are GPC - Generalized predictive control (Clarke et al., 1987) and UPC - Unified predictive control (=-=Soeterboek, 1992-=-). The next step of the academic community was to extend the MPC algorithms based on state-space models to continuous (smooth) nonlinear systems, which includes the following approaches: nonlinear MPC... |

33 |
Discretetime stability with perturbations: application to model predictive control
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(Show Context)
Citation Context ...ions above. Several solutions that rely on continuous (or even Lipschitz continuous) system dynamics and/or MPC value functions were proposed for smooth nonlinear systems in the literature, e.g. see (=-=Scokaert et al., 1997-=-; Limon et al., 2002a; Grimm et al., 2003b, 2004), regarding the second and the third question posed here. Since hybrid systems are inherently nonlinear and discontinuous, and hybrid MPC value functio... |

32 | Multi-Parametric Toolbox (MPT
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- 2004
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Citation Context ... 1, I-norm MPC costs. These problems can be solved using thesIEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 11, NOVEMBER 2006 1815 hybrid toolbox (HT) [10] or the multi parametric toolbox (MPT) =-=[11]-=-. Note that if a MIQP (MILP) problem is feasible, the global optimum is attained because, in principle, a MIQP (MILP) consists of a finite number of QP (LP) problems (see, for example, [4]). Then, due... |

28 |
A class of hybrid-state continuous-time dynamic systems
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Citation Context ...ge class of hybrid systems as they con1 The term “hybrid systems” was originally introduced by Witsenhausen in 1966, for describing the combination of continuous dynamical and discrete event systems (=-=Witsenhausen, 1966-=-).s16 Introduction mode 5 x � ( t) = f5( x( t), u( t), t) y ( t) = g5( x( t), u( t), t) mode 1 x � ( t) = f1( x( t), u( t), t) y ( t) = g1 ( x( t), u( t), t) mode 4 x �( t) = f4( x( t), u( t), t) y ( ... |

26 |
The TOMLAB Optimization Environment
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Citation Context ...linear programming (MILP) solver proposed in [Spjøtvold et al., 2007]. Furthermore, quadratic programming (QP) and linear programming (LP) solvers that are employed by existing MIQP and MILP solvers [=-=Holmström, 1999-=-] usually allow the specification of a tolerance with respect to achieving the optimum. Therefore, the tolerance parameter can be used to set a desired optimality margin. In the next section we will e... |

21 |
Analysis of discretetime piecewise affine and hybrid systems. Automatica
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Citation Context ...s only been solved partially, in [7], i.e., by employing a common quadratic Lyapunov function for system (5) in closed-loop with �� a �@��A, � P C. This is known to be conservative (see, for example, =-=[13]-=-) because there are PWA systems which only admit a piecewise quadratic (PWQ) Lyapunov function (see also [9, Sec. 3.7] for an example where the method of [7] fails). In the case of MPC costs based on ... |

20 |
Input-to-state stable MPC for constrained discrete-time nonlinear systems with bounded additive uncertainties
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Citation Context ...for example, see [Lazar, 2006] and the references therein) that for such discontinuous systems most of the results obtained for predictive control of continuous nonlinear systems [Mayne et al., 2000, =-=Limon et al., 2002-=-, Grimm et al., 2007] do not necessarily apply. Only the min-max MPC methodology [Magni et al., 2006, Lazar et al., 2008] could be an alternative, but the prohibitive computational complexity of min-m... |

17 |
An algorithm to find maximal state constraint sets for discrete-time linear dynamical systems with bounded control and states
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Citation Context ... that € is the maximal positively invariant set for system (20) with arbitrary switching. Then, we have that € is a positively invariant set for any linear subsystem in (20) and thus, it follows from =-=[16]-=- that the convex hull of € is also a positively invariant set for any linear system in (20). Hence, the convex hull of € is a positively invariant set for system (20) under arbitrary switching. Since ... |

16 | Regional input-to-state stability for nonlinear model predictive control - Magni, Raimondo, et al. - 2006 |

15 | On the stability of min-max nonlinear model predictive control - Lazar, Pena, et al. |

14 | Computation of invariant sets for piecewise affine discrete time systems subject to bounded disturbances
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- 2004
(Show Context)
Citation Context ...ng positively invariant sets for PWA systems, which is a notoriously difficult problem. An algorithm for calculating the maximal positively invariant set [8] for PWA systems was recently presented in =-=[14]-=-. However, it is known that the maximal positively invariant set inside a given compact set is a piecewise polyhedral set for PWA systems, which can be very complex (i.e., it may consist of the union ... |

14 |
Nonlinear Systems Analysis, second edition. Prentice-Hall, Englewood Cliffs
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Citation Context ...opies of these two articles can be supplied upon request.s38 Classical stability results revisited appeared, see, for example, (Freeman, 1965; Willems, 1970; LaSalle, 1976) and the more recent books (=-=Vidyasagar, 1993-=-; Khalil, 2002). In discrete-time the candidate Lyapunov function and the system dynamics do not necessarily have to be continuous, as it will be shown in this chapter. This is extremely interesting f... |

11 |
Nominally robust model predictive control with state constraints
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(Show Context)
Citation Context ...zar, 2006] and the references therein) that for such discontinuous systems most of the results obtained for predictive control of continuous nonlinear systems [Mayne et al., 2000, Limon et al., 2002, =-=Grimm et al., 2007-=-] do not necessarily apply. Only the min-max MPC methodology [Magni et al., 2006, Lazar et al., 2008] could be an alternative, but the prohibitive computational complexity of min-max MPC schemes preve... |

10 |
ISS of switched systems and applications to switching adaptive control
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(Show Context)
Citation Context ...ally stable PWA system, then local robust convergence is ensured. For continuous-time input-to-state stability results for switched systems and hybrid systems we refer the reader to the recent works (=-=Vu et al., 2005-=-; Hespanha et al., 2005; Cai and Teel, 2005). However, to the best of the author’s knowledge, a global robust stability analysis methodology for discrete-time PWA systems that can be used for both ana... |

8 | On the stability and robustness of non-smooth nonlinear model predictive control
- Lazar, Heemels, et al.
- 2005
(Show Context)
Citation Context ...umerical solver by choosing a desirable ISS margin ε(δ) and finding the corresponding value of δ. Also, Theorem 9 recovers as a particular case the following result for the optimal case published in [=-=Lazar et al., 2005-=-], where only PWA systems were considered. Corollary 12. Suppose that the hypothesis of Theorem 9 is satisfied and the global optimum is attained in Problem 6 for all k ∈ Z+. Then, the closed-loop sys... |

7 |
SeDuMi: Matlab toolbox for solving optimization problems over symmetric cones. Web: http://fewcal. kub. nl/sturm/software/sedumi. html, 2001. 47th
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(Show Context)
Citation Context ...e (4.8) is a bilinear matrix inequality (i.e. due to γ1Pi, γ2Pi), since the unknowns γ1, γ2 are scalars, this problem can be solved efficiently via semi-definite programming solvers (software), e.g. (=-=Sturm, 2001-=-), (Löfberg, 2002), by setting lower and upper bounds for γ1, γ2 and doing bisections. Note that, since we take Mji = 0 for all (j, i) ∈ Slin × S, the LMI-based sufficient conditions for ISS (4.7) rec... |

6 |
Model predictive control of constrained piecewise affine discrete-time systems
- Mayne, Rakovic
(Show Context)
Citation Context ... literature recently; for particular classes of hybrid systems and MPC cost functions, see, for example, [5]–[7]. In these works, either continuous piecewise affine (PWA) systems are considered, [5], =-=[6]-=- which are in fact Lipschitz continuous systems or, in [7], asymptotic stability is established via the results of [2], where continuity of the MPC value function is assumed. Note that this property d... |

6 |
Geometric Bounding Toolbox (GBT) for Matlab. Official website: http://www. sysbrain.com/gbt
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Citation Context ...E, it holds that P � Co(P) ⊆ Co(E) = E. Note that the computation of the vertices of P and of their convex hull can be performed efficiently using, for instance, the Geometric Bounding Toolbox (GBT) (=-=Veres, 1995-=-). Also, an ellipsoidal β-contractive set with β ∈ (0, 1) as small as possible is desirable, as this will result in a polyhedral positively invariant (λ-contractive) set of lower complexity. Remark 3.... |

5 |
Stabilizing low complexity feedback control of constrained piecewise affine systems
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- 2005
(Show Context)
Citation Context ...tive controllers in [3] and [4]. However, proofs of Lyapunov stability only appeared in the literature recently; for particular classes of hybrid systems and MPC cost functions, see, for example, [5]–=-=[7]-=-. In these works, either continuous piecewise affine (PWA) systems are considered, [5], [6] which are in fact Lipschitz continuous systems or, in [7], asymptotic stability is established via the resul... |

3 |
Constrained optimal control of an electronic throttle
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- 2006
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Citation Context ... when the state crosses the switching boundaries (denoted by dotted diagonal lines). induction motors (Geyer et al., 2005), applications to automotive systems (Bemporad et al., 2003a; Borrelli, 2003; =-=Vasak et al., 2006-=-) and systems biology (Musters and van Riel, 2004; Drulhe et al., 2006), to mention just a few. A significant part of this thesis focuses on discrete-time 2 PWA systems of the form: xk+1 = Ajxk + Bjuk... |

2 |
D.: Robust model predictive control of contrained piecewise affine discrete time systems
- Rakovic, Mayne
- 2004
(Show Context)
Citation Context ...e available for re-tuning the parameters of the MPC algorithm such that stability is achieved. Recently, results on robust MPC of hybrid systems have been presented in (Kerrigan and Mayne, 2002) and (=-=Rakovic and Mayne, 2004-=-), which deal with dynamic programming and tube based approaches, respectively, for solving feedback min-max MPC problems (Mayne, 2001) for continuous PWA systems, and also provide a robust stability ... |

2 | Stabilizing output feedback nonlinear model predictive control: An extended observer approach
- Roset, Lazar, et al.
(Show Context)
Citation Context ...e. Some other results on similar topics, which are not included in this thesis, can be found in (Lazar and Heemels, 2003), (Lazar and De Keyser, 2004), (Roset, Lazar, Nijmeijer, and Heemels, 2006a), (=-=Roset, Lazar, Nijmeijer, and Heemels, 2006b-=-) and (Corona, Lazar, De Schutter, and Heemels, 2006). 1.6 Basic mathematical notation and definitions In this section, some basic mathematical notation and standard definitions are recalled to make t... |

2 |
Delaunay partitions in R n applied to nonconvex programs and vertex/facet enumeration problems
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- 2005
(Show Context)
Citation Context ...P0 ⊂ Rn that contains E. Let ( ˜ θ0, .., ˜ θm), with m ≥ n, be the vertices of P0. An initial set of simpleces S0 1 , . . . , S0 l0 that contains these points is determined by Delaunay triangulation (=-=Yepremyan and Falk, 2005-=-). Then, for every simplex S0 i � Co(θ0 0i , . . . , θ0 ni ), i = 1, . . . , l0, the following operations are performed. Algorithm 3.5.10 1. Let k = 0. j∈S 2. For every simplex S k i , i = 1, . . . , ... |

1 |
Hybrid Toolbox—User’s Guide [Online]. Available: http://www.dii.unisi.it/hybrid/toolbox 2003
- Bemporad
(Show Context)
Citation Context ...r programming (MILP) problem in the case of 1, I-norm MPC costs. These problems can be solved using thesIEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 11, NOVEMBER 2006 1815 hybrid toolbox (HT) =-=[10]-=- or the multi parametric toolbox (MPT) [11]. Note that if a MIQP (MILP) problem is feasible, the global optimum is attained because, in principle, a MIQP (MILP) consists of a finite number of QP (LP) ... |

1 |
Squaring the circle: An algorithm for obtaining polyhedral invariant sets from ellipsoidal ones
- Lazar, Alessio, et al.
- 2006
(Show Context)
Citation Context ...ƒ, then the set € defined in (23) is a polyhedral set. Another method for computing low complexity positively invariant sets for PWA systems that admit a (local) PWQ Lyapunov function is presented in =-=[15]-=-. VI. ILLUSTRATIVE EXAMPLE Consider the following open-loop unstable discontinuous 3-D PWA system with four linear sub-systems: ��CI a e� �� C f���Y if �� P �Y � aIY PY QY R (24) subject to the constr... |

1 | A stabilizing output based nonlinear model predictive control scheme, submitted to a conference
- Roset, Lazar, et al.
(Show Context)
Citation Context ...ctive Control for Fast Systems, Grenoble, France. Some other results on similar topics, which are not included in this thesis, can be found in (Lazar and Heemels, 2003), (Lazar and De Keyser, 2004), (=-=Roset, Lazar, Nijmeijer, and Heemels, 2006a-=-), (Roset, Lazar, Nijmeijer, and Heemels, 2006b) and (Corona, Lazar, De Schutter, and Heemels, 2006). 1.6 Basic mathematical notation and definitions In this section, some basic mathematical notation ... |

1 | received the Dipl.Ing. degree in Control Engineering and Industrial Informatics from the Faculty of Automatic Control and - He |

1 | received the Master of Science degree in Control Engineering from the same faculty in 2002 for the thesis “Nonlinear model predictive control - He |

1 | 2002 to September 2006 he pursued his Ph.D. degree with the Control Systems Group at the Faculty of Electrical Engineering, Eindhoven University of Technology, The Netherlands, under the supervision of Prof. Paul van den Bosch. The research conducted in t - From |