arXiv subjects

Behcet Acikmese

Publications and source records attributed to Behcet Acikmese.

At least 37 records · Page 2Linked to original sources

arXiv2020

Synthesis of Interval Observers for Nonlinear Discrete-Time Systems

A systematic procedure to synthesize interval observers for nonlinear discrete-time systems is proposed. The feedback gains and other matrices are found from the solutions to semidefinite feasibility programs. Two cases are considered: (1) the interval observer is in the same coordinate frame as the given system, and (2) the interval observer uses a coordinate transformation. The conditions where coordinate transformations are necessary are detailed. Numerical examples are provided to showcase the effectiveness of the interval observers and demonstrate their application to sampled-data systems.

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arXiv2020

Approximation of The Constrained Joint Spectral Radius via Algebraic Lifting

This paper studies the constrained switching (linear) system which is a discrete-time switched linear system whose switching sequences are constrained by a deterministic finite automaton. The stability of a constrained switching system is characterized by its constrained joint spectral radius that is known to be difficult to compute or approximate. Using the semi-tensor product of matrices, the matrix-form expression of a constrained switching system is shown to be equivalent to that of a lifted arbitrary switching system. Then the constrained joint/generalized spectral radius of a constrained switching system is proved to be equal to the joint/generalized spectral radius of its lifted arbitrary switching system which can be approximated by off-the-shelf algorithms.

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arXiv2020

Periodic Event-triggered Control for Incrementally Quadratic Nonlinear Systems

Periodic event-triggered control (PETC) evaluates the triggering rule periodically and is well-suited for implementation on digital platforms. This paper investigates PETC design for nonlinear systems affected by external disturbances under the impulsive system formulation. Sufficient conditions are provided to ensure the input-to-state stability of the resulting closed-loop system for the state feedback and the observer-based output feedback configurations separately. For each configuration, the sampling period and the triggering functions are provided explicitly. Sufficient conditions in the form of linear matrix inequalities are provided for the PETC design of incrementally quadratic nonlinear systems. Two examples are given to illustrate the effectiveness of the proposed method.

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arXiv2020

Observer-Based Controllers for Incrementally Quadratic Nonlinear Systems with Disturbances

Robust global stabilization of nonlinear systems by observer-based feedback controllers is a challenging task. This article investigates the problem of designing observer-based stabilizing controllers for incrementally quadratic nonlinear systems with external disturbances. The nonlinearities considered in the system model satisfy the incremental quadratic constraints, which are characterized by incremental multiplier matrices and encompass many common nonlinearities. The simultaneous search for the observer and the controller gain matrices is formulated as a feasibility problem of linear matrix inequalities, for two parameterizations (i.e., the block diagonal parameterization and the block anti-triangular parameterization) of the incremental multiplier matrices, respectively. The closed-loop system implementing the observer-based feedback controller is proven to be input-to-state stable with respect to external disturbances. Using the proposed continuous-time observer-based controllers, event-triggered controllers with time regularization are constructed for globally Lipschitz systems, such that the closed-loop system is Zeno-free and input-to-state practically stable.

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arXiv2019

Lossless convexification of non-convex optimal control problems with disjoint semi-continuous inputs

This paper presents a convex optimization-based method for finding the globally optimal solutions of a class of mixed-integer non-convex optimal control problems. We consider problems that are non-convex in the input norm, which is a semi-continuous variable that can be zero or lower- and upper-bounded. Using lossless convexification, the non-convex problem is relaxed to a convex problem whose optimal solution is proved to be optimal almost everywhere for the original problem. The relaxed problem can be solved using second-order cone programming, which is a subclass of convex optimization for which there exist numerically reliable solvers with convergence guarantees and polynomial time complexity. This is the first lossless convexification result for mixed-integer optimization problems. An example of spacecraft docking with a rotating space station corroborates the effectiveness of the approach and features a computation time almost three orders of magnitude shorter than a mixed-integer programming formulation.

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arXiv2019

Lossless Convexification of Optimal Control Problems with Semi-continuous Inputs

This paper presents a novel convex optimization-based method for finding the globally optimal solutions of a class of mixed-integer non-convex optimal control problems. We consider problems with non-convex constraints that restrict the input norms to be either zero or lower- and upper-bounded. The non-convex problem is relaxed to a convex one whose optimal solution is proved to be optimal almost everywhere for the original problem, a procedure known as lossless convexification. This paper is the first to allow individual input sets to overlap and to have different norm bounds, integral input and state costs, and convex state constraints that can be activated at discrete time instances. The solution relies on second-order cone programming and demonstrates that a meaningful class of optimal control problems with binary variables can be solved reliably and in polynomial time. A rocket landing example with a coupled thrust-gimbal constraint corroborates the effectiveness of the approach.

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arXiv2019

Sensitivity Analysis for Markov Decision Process Congestion Games

We consider a non-atomic congestion game where each decision maker performs selfish optimization over states of a common MDP. The decision makers optimize for their own expected costs, and influence each other through congestion effects on the state-action costs. We analyze on the sensitivity of MDP congestion game equilibria to uncertainty and perturbations in the state-action costs by applying an implicit function type analysis. The occurrence of a stochastic Braess paradox is defined, analyzed based on sensitivity of game equilibria and demonstrated in simulation. We further analyze how the introduction of stochastic dynamics affects the magnitude of Braess paradox in comparison to deterministic dynamics.

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arXiv2019

Robust Model Predictive Control for Linear Systems with State and Input Dependent Uncertainties

This paper presents a computationally efficient robust model predictive control law for discrete linear time invariant systems subject to additive disturbances that may depend on the state and/or input norms. Despite the dependency being non-convex, we are able to capture it exactly for input dependency and approximately for state dependency in at most a second order cone programming problem. The formulation has linear complexity in the planning horizon length. The approach is thus amenable to efficient real-time implementation with a guarantee on recursive feasibility and global optimality. Robust position control of a satellite is considered as an illustrative example.

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arXiv2019

Fast Trajectory Optimization via Successive Convexification for Spacecraft Rendezvous with Integer Constraints

In this paper we present a fast method based on successive convexification for generating fuel-optimized spacecraft rendezvous trajectories in the presence of mixed-integer constraints. A recently developed paradigm of state-triggered constraints allows to efficiently embed a subset of discrete decision constraints into the continuous optimization framework of successive convexification. As a result, we are able to solve difficult trajectory optimization problems at interactive speeds, as opposed to a mixed-integer programming approach that would require significantly more solution time and computing power. Our method is applied to the real problem of transposition and docking of the Apollo command and service module with the lunar module. We demonstrate that, within seconds, we are able to obtain trajectories that are up to 90 percent more fuel efficient (saving up to 45 kg of fuel) than non-optimization based Apollo-era design targets. Our trajectories take explicit account of minimum thrust pulse width and plume impingement constraints. Both of these constraints are naturally mixed-integer, but we handle them as state-triggered constraints. In its current state, our algorithm will serve as a useful off-line design tool for rapid trajectory trade studies.

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arXiv2019

Approximate Multiparametric Mixed-integer Convex Programming

We propose an algorithm for generating explicit solutions of multiparametric mixed-integer convex programs to within a given suboptimality tolerance. The algorithm is applicable to a very general class of optimization problems, but is most useful for hybrid model predictive control, where on-line implementation is hampered by the worst-case exponential complexity of mixed-integer solvers. The output is a simplicial partition which defines a static map from the current state to a suboptimal solution. The primary theoretical contribution of this paper is to introduce a non-zero optimal cost overlap metric which is necessary and sufficient for convergence. The overlap size is also linked to partition complexity. The algorithm is massively parallelizable and our implementation, which is publicly available, is run on a cluster of several hundred processors. Not only does our solution have a deterministic runtime, simulations show that our approach is faster than on-line optimization by up to three orders of magnitude.

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arXiv2019

Dual Quaternion Based Powered Descent Guidance with State-Triggered Constraints

This paper presents a numerical algorithm for computing 6-degree-of-freedom free-final-time powered descent guidance trajectories. The trajectory generation problem is formulated using a unit dual quaternion representation of the rigid body dynamics, and several standard path constraints. Our formulation also includes a special line of sight constraints that is enforced only within a specified band of slant ranges relative to the landing site, a novel feature that is especially relevant to Terrain and Hazard Relative Navigation. We use the newly introduced state-triggered constraints to formulate these range constraints in a manner that is amenable to real-time implementations. The resulting non-convex optimal control problem is solved iteratively as a sequence of convex second-order cone programs that locally approximate the non-convex problem. Each second-order cone program is solved using a customizable interior point method solver. Also introduced are a scaling method and a new heuristic technique that guide the convergence process towards dynamic feasibility. To demonstrate the capabilities of our algorithm, two numerical case studies are presented. The first studies the effect of including a slant-range-triggered line of sight constraint on the resulting trajectories. The second study performs a Monte Carlo analysis to assess the algorithm's robustness to initial conditions and real-time performance.

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arXiv2019

Tolling for Constraint Satisfaction in Markov Decision Process Congestion Games

Markov decision process (MDP) congestion game is an extension of classic congestion games, where a continuous population of selfish agents solves Markov decision processes with congestion: the payoff of a strategy decreases as more population uses it. We draw parallels between key concepts from capacitated congestion games and MDP. In particular, we show that population mass constraints in MDP congestion games are equivalent to imposing tolls/incentives on the reward function, which can be utilized by social planners to achieve auxiliary objectives. We demonstrate such methods in a simulated Seattle ride-share model, where tolls and incentives are enforced for two separate objectives: to guarantee minimum driver density in downtown Seattle, and to shift the game equilibrium towards a maximum social output.

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arXiv2019

Partition-based Feasible Integer Solution Pre-computation for Hybrid Model Predictive Control

For multiparametric mixed-integer convex programming problems such as those encountered in hybrid model predictive control, we propose an algorithm for generating a feasible partition of a subset of the parameter space. The result is a static map from the current parameter to a suboptimal integer solution such that the remaining convex program is feasible. Convergence is proven with a new insight that the overlap among the feasible parameter sets of each integer solution governs the partition complexity. The partition is stored as a tree which makes querying the feasible solution efficient. The algorithm can be used to warm start a mixed integer solver with a real-time guarantee or to provide a reference integer solution in several suboptimal MPC schemes. The algorithm is tested on randomly generated systems with up to six states, demonstrating the effectiveness of the approach.

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arXiv2019

Successive Convexification: A Superlinearly Convergent Algorithm for Non-convex Optimal Control Problems

This paper presents the SCvx algorithm, a successive convexification algorithm designed to solve non-convex constrained optimal control problems with global convergence and superlinear convergence-rate guarantees. The proposed algorithm can handle nonlinear dynamics and non-convex state and control constraints. It solves the original problem to optimality by successively linearizing non-convex dynamics and constraints about the solution of the previous iteration. The resulting convex subproblems are numerically tractable, and can be computed quickly and reliably using convex optimization solvers, making the SCvx algorithm well suited for real-time applications. Analysis is presented to show that the algorithm converges both globally and superlinearly, guaranteeing i) local optimality recovery: if the converged solution is feasible with respect to the original problem, then it is also a local optimum; ii) strong convergence: if the Kurdyka-Lojasiewicz (KL) inequality holds at the converged solution, then the solution is unique. The superlinear rate of convergence is obtained by exploiting the structure of optimal control problems, showcasing that faster rate of convergence can be achieved by leveraging specific problem properties when compared to generic nonlinear programming methods. Numerical simulations are performed for a non-convex quad-rotor motion planning problem, and corresponding results obtained using Sequential Quadratic Programming (SQP) and general purpose Interior Point Method (IPM) solvers are provided for comparison. The results show that the convergence rate of the SCvx algorithm is indeed superlinear, and that SCvx outperforms the other two methods by converging in less number of iterations.

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arXiv2019

Real-Time Quad-Rotor Path Planning Using Convex Optimization and Compound State-Triggered Constraints

The contribution of this paper is the application of compound state-triggered constraints (STCs) to real-time quad-rotor path planning. Originally developed for rocket landing applications, STCs are made up of a trigger condition and a constraint condition that are arranged such that satisfaction of the former implies satisfaction of the latter. Compound STCs go a step further by allowing multiple trigger and constraint conditions to be combined via Boolean "and" or "or" operations. The logical implications embodied by STCs can be formulated using continuous variables, and thus enable the incorporation of discrete decision making into a continuous optimization framework. In this paper, compound STCs are used to solve quad-rotor path planning problems that would typically require the use of computationally expensive mixed-integer programming techniques. Two scenarios are considered: (1) a quad-rotor flying through a hoop, and (2) a pair of quad-rotors carrying a beam-like payload through an obstacle course. Successive convexification is used to solve the resulting non-convex optimization problem. Monte-Carlo simulation results show that our approach can reliably generate trajectories at rates upwards of 3 and 1.5 Hz for the first and second scenarios, respectively.

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arXiv2019

Successive Convexification for 6-DoF Powered Descent Guidance with Compound State-Triggered Constraints

This paper introduces a continuous formulation for compound state-triggered constraints, which are generalizations of the recently introduced state-triggered constraints. State-triggered constraints are different from ordinary constraints found in optimal control in that they use a state-dependent trigger condition to enable or disable a constraint condition, and can be expressed as continuous functions that are readily handled by successive convexification. Compound state-triggered constraints go a step further, giving designers the ability to compose trigger and constraint conditions using Boolean and and or operations. Simulations of the 6-degree-of-freedom (DoF) powered descent guidance problem obtained using successive convexification are presented to illustrate the utility of state-triggered and compound state-triggered constraints. The examples employ a velocity-triggered angle of attack constraint to alleviate aerodynamic loads, and a collision avoidance constraint to avoid large geological formations. In particular, the velocity-triggered angle of attack constraint demonstrates the ability of state-triggered constraints to introduce new constraint phases to the solution without resorting to combinatorial techniques.

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arXiv2018

Successive Convexification for Real-Time 6-DoF Powered Descent Guidance with State-Triggered Constraints

In this paper, we present a real-time successive convexification algorithm for a generalized free-final-time 6-degree-of-freedom powered descent guidance problem. We build on our previous work by introducing the following contributions: (i) a free-ignition-time modification that allows the algorithm to determine the optimal engine ignition time, (ii) a tractable aerodynamics formulation that models both lift and drag, and (iii) a continuous state-triggered constraint formulation that emulates conditionally enforced constraints. In particular, contribution (iii) effectively allows constraints to be enabled or disabled by if-statements conditioned on the solution variables of the parent continuous optimization problem. To the best of our knowledge, this represents a novel formulation in the optimal control literature, and enables a number of interesting applications, including velocity-triggered angle of attack constraints and range-triggered line of sight constraints. Our algorithm converts the resulting generalized powered descent guidance problem from a non-convex free-final-time optimal control problem into a sequence of tractable convex second-order cone programming subproblems. With the aid of virtual control and trust region modifications, these subproblems are solved in succession until convergence is attained. Simulations using a third-party solver demonstrate the real-time capabilities of the proposed algorithm, with a maximum execution time of less than 0.7 seconds over a multitude of problem feature combinations.

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arXiv2018

Voronoi Partition-based Scenario Reduction for Fast Sampling-based Stochastic Reachability Computation of LTI Systems

In this paper, we address the stochastic reach-avoid problem for linear systems with additive stochastic uncertainty. We seek to compute the maximum probability that the states remain in a safe set over a finite time horizon and reach a target set at the final time. We employ sampling-based methods and provide a lower bound on the number of scenarios required to guarantee that our estimate provides an underapproximation. Due to the probabilistic nature of the sampling-based methods, our underapproximation guarantee is probabilistic, and the proposed lower bound can be used to satisfy a prescribed probabilistic confidence level. To decrease the computational complexity, we propose a Voronoi partition-based to check the reach-avoid constraints at representative partitions (cells), instead of the original scenarios. The state constraints arising from the safe and target sets are tightened appropriately so that the solution provides an underapproximation for the original sampling-based method. We propose a systematic approach for selecting these representative cells and provide the flexibility to trade-off the number of cells needed for accuracy with the computational cost.

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