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32
Optimal control of switching times in switched dynamical systems
 Proc. 42nd Conference on Decision and Control, Maui, HI
, 2003
"... Abstract — This paper considers an optimal control problem for switched dynamical systems, where the objective is to minimize a cost functional de£ned on the state, and where the control variable consists of the switching times. The gradient of the cost functional is derived on an especially simple ..."
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Cited by 36 (6 self)
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Abstract — This paper considers an optimal control problem for switched dynamical systems, where the objective is to minimize a cost functional de£ned on the state, and where the control variable consists of the switching times. The gradient of the cost functional is derived on an especially simple form, which lends itself to be directly used in gradientdescent algorithms. This special structure of the gradient furthermore allows for the number of switching points to become part of the control variable, instead of being a given constant. Numerical examples testify to the viability of the proposed approach. I.
Optimal control of continuoustime switched affine systems
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 2006
"... This paper deals with optimal control of switched piecewise affine autonomous systems, where the objective is to minimize a performance index over an infinite time horizon. We assume that the switching sequence has a finite length, and that the decision variables are the switching instants and the s ..."
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Cited by 26 (5 self)
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This paper deals with optimal control of switched piecewise affine autonomous systems, where the objective is to minimize a performance index over an infinite time horizon. We assume that the switching sequence has a finite length, and that the decision variables are the switching instants and the sequence of operating modes. We present two different approaches for solving such an optimal control problem. The first approach iterates between a procedure that finds an optimal switching sequence of modes, and a procedure that finds the optimal switching instants. The second approach is inspired by dynamic programming and identifies the regions of the state space where an optimal mode switch should occur, therefore providing a state feedback control law.
Stabilization of ContinuousTime Switched Linear Positive Systems
, 2009
"... In this paper we tackle a few problems related to linear positive switched systems. First, we provide a result on statefeedback stabilization of autonomous linear positive switched systems through piecewise linear copositive Lyapunov functions. This is accompanied by a side result on the existence ..."
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Cited by 25 (5 self)
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In this paper we tackle a few problems related to linear positive switched systems. First, we provide a result on statefeedback stabilization of autonomous linear positive switched systems through piecewise linear copositive Lyapunov functions. This is accompanied by a side result on the existence of a switching law guaranteeing an upper bound to the optimal L1 cost. Then, the induced L1 guaranteed cost cost is tackled, through constrained piecewise linear copositive Lyapunov functions. The optimal L1 cost control is finally studied via Hamiltonian function analysis.
Switching mode generation and optimal estimation with application to skidsteering
 Automatica
"... Skidsteered vehicles, by design, must skid in order to maneuver. The skidding causes the vehicle to behave discontinuously during a maneuver as well as introduces complications to the observation of the vehicle’s state, both of which affect a controller’s performance. This paper addresses estimatio ..."
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Cited by 15 (13 self)
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Skidsteered vehicles, by design, must skid in order to maneuver. The skidding causes the vehicle to behave discontinuously during a maneuver as well as introduces complications to the observation of the vehicle’s state, both of which affect a controller’s performance. This paper addresses estimation of contact state by applying switched system optimization to estimate skidding properties of the skidsteered vehicle. In order to treat the skidsteered vehicle as a switched system, the vehicle’s ground interaction is modeled using Coulomb friction, thereby partitioning the system dynamics into four distinct modes, one for each combination of the forward and back wheel pairs sticking or skidding. Thus, as the vehicle maneuvers, the system propagates over some mode sequence, transitioning between modes over some set of switching times. This paper presents secondorder optimization algorithms for estimating these switching times. We emphasize the importance of the secondorder algorithm because it exhibits quadratic convergence and because even for relatively simple examples, firstorder methods fail to converge on time scales compatible with realtime operation. Furthermore, the paper presents a technique for estimating the mode sequence by optimizing a relaxation of the switched system.
Optimal control of switching times in hybrid systems
 MMAR'2003, Miedzyzdroje
, 2003
"... Abstract. This paper considers the problem of determining optimal switching times at which mode transitions should occur in multimodal, hybrid systems. An expression for the gradient of the cost functional, defined with respect to the switching times, is presented in such a way that not only the sw ..."
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Cited by 9 (0 self)
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Abstract. This paper considers the problem of determining optimal switching times at which mode transitions should occur in multimodal, hybrid systems. An expression for the gradient of the cost functional, defined with respect to the switching times, is presented in such a way that not only the switching times, but also the number of switches can be incorporated in the problem formulation. Numerical examples testify to the viability of the proposed approach. Key Words. Switched dynamical systems, switchingtime control, optimal control, hybrid systems, gradientdescent algorithms. 1.
Optimal control of hybrid autonomous systems with state jumps
 Proceedings of the American Control Conference 2003, Volume: 6, Pages: 5191  5196
, 2003
"... In this paper, optimal control problems for hybrid autonomous systems with state jumps are studied. In particular, we focus on problems in which a prespecified sequence of active subsystems is given and propose an approach to finding the optimal switching instants. Specifically, the derivatives of t ..."
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Cited by 6 (1 self)
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In this paper, optimal control problems for hybrid autonomous systems with state jumps are studied. In particular, we focus on problems in which a prespecified sequence of active subsystems is given and propose an approach to finding the optimal switching instants. Specifically, the derivatives of the cost with respect to the switching instants are derived and nonlinear optimization techniques are used to locate the optimal switching instants. Using the approach, accurate numerical values of local optimal solutions can be obtained. An example illustrates the approach. 1
Optimal Impulsive Control for Point Delay Systems with Refractory Period
 IFAC Workshop on TimeDelay Systems
, 2004
"... Abstract: The optimal impulsive control problem for a system with a single discrete delay is studied. In such systems the control consists only of a sequence of modulated impulses, the control variables being the impulse times and their magnitudes. It is assumed that the systems considered all have ..."
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Cited by 5 (1 self)
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Abstract: The optimal impulsive control problem for a system with a single discrete delay is studied. In such systems the control consists only of a sequence of modulated impulses, the control variables being the impulse times and their magnitudes. It is assumed that the systems considered all have a refractory period, in the sense that once an action is taken, it takes a noninfinitesimal amount of time before a subsequent action can be taken. Necessary conditions for a stationary solution are derived and shown to extend those of the delay free case.
Consistent approximations for the optimal control of constrained switched systems—part 1: A conceptual algorithm
 SIAM Journal on Control and Optimization
, 2013
"... Though switched dynamical systems have shown great utility in modeling a variety of physical phenomena, the construction of an optimal control of such systems has proven difficult since it demands some type of optimal mode scheduling. In this paper, we devise an algorithm for the computation of an ..."
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Cited by 4 (1 self)
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Though switched dynamical systems have shown great utility in modeling a variety of physical phenomena, the construction of an optimal control of such systems has proven difficult since it demands some type of optimal mode scheduling. In this paper, we devise an algorithm for the computation of an optimal control of constrained nonlinear switched dynamical systems. The control parameter for such systems include a continuousvalued input and discretevalued input, where the latter corresponds to the mode of the switched system that is active at a particular instance in time. Our approach, which we prove converges to local minimizers of the constrained optimal control problem, first relaxes the discretevalued input, then performs traditional optimal control, and then projects the constructed relaxed discretevalued input back to a pure discretevalued input by employing an extension to the classical Chattering Lemma that we prove. We extend this algorithm by formulating a computationally implementable algorithm which works by discretizing the time interval over which the switched dynamical system is defined. Importantly, we prove that this implementable algorithm constructs a sequence of points by recursive application that converge to the local minimizers of the original constrained optimal control problem. Four simulation experiments are included to validate the theoretical developments. 1
A Provably Convergent Algorithm for TransitionTime Optimization in Switched Systems
"... Abstract — This paper concerns a modesequencing and switchingtime optimization problem defined on autonomous switchedmode hybrid dynamical systems. The design parameter consists of two elements: (i) the sequence of dynamicresponse functions associated with the modes, and (ii) the duration of each ..."
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Cited by 4 (1 self)
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Abstract — This paper concerns a modesequencing and switchingtime optimization problem defined on autonomous switchedmode hybrid dynamical systems. The design parameter consists of two elements: (i) the sequence of dynamicresponse functions associated with the modes, and (ii) the duration of each mode. The sequencing element is a discrete parameter which may render the problem of computing the optimal schedule exponentially complex. Therefore we are not seeking a global minimum, but rather a local solution in a suitable sense. To this end we endow the parameter space with a local continuous structure which allows us to apply gradientdescent techniques. With this structure, the problem is cast in the form of a nonlinearprogramming problem defined on a sequence of nested Euclidean spaces with increasing dimensions. We characterize suboptimality in an appropriate sense, define a corresponding convergence criterion, and devise a provablyconvergent optimization algorithm.
A hybrid bellman equation for systems with regional dynamics
 In 46th IEEE Conf. on Decision and Control
, 2007
"... AbstractIn this paper, we study hybrid systems with regional dynamics, i.e., systems where transitions between different dynamical regimes occur as the continuous state of the system reaches given switching surfaces. In particular, we focus our attention on the optimal control problem associated w ..."
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Cited by 3 (1 self)
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AbstractIn this paper, we study hybrid systems with regional dynamics, i.e., systems where transitions between different dynamical regimes occur as the continuous state of the system reaches given switching surfaces. In particular, we focus our attention on the optimal control problem associated with such systems, and we present a Hybrid Bellman Equation for such systems that provide a characterization of global optimality, given an upper bound on the number of switches. Not surprisingly, the solution will be hyrbid in nature in that it will depend on not only the continuous control signals, but also on discrete decisions as to what domains the system should go through in the first place. A number of examples are presented to highlight the operation of the proposed approach.