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Papers by Humberto Gonzalez

arXiv (Cornell University), Sep 8, 2014

After the three DARPA Grand Challenge contests many groups around the world have continued to act... more After the three DARPA Grand Challenge contests many groups around the world have continued to actively research and work toward an autonomous vehicle capable of accomplishing a mission in a given context (e.g. desert, city) while following a set of prescribed rules, but none has been completely successful in uncontrolled environments, a task that many people trivially fulfill every day. We believe that, together with improving the sensors used in cars and the artificial intelligence algorithms used to process the information, the community should focus on the systems engineering aspects of the problem, i.e. the limitations of the car (in terms of space, power, or heat dissipation) and the limitations of the software development cycle. This paper explores these issues and our experiences overcoming them.

Learning-based model predictive control (LBMPC) is a technique that provides deterministic guaran... more Learning-based model predictive control (LBMPC) is a technique that provides deterministic guarantees on robustness, while statistical identification tools are used to identify richer models of the system in order to improve performance. This technical note provides proofs that elucidate the reasons for our choice of measurement model, as well as giving proofs concerning the stochastic convergence of LBMPC. The first part of this note discusses simultaneous state estimation and statistical identification (or learning) of unmodeled dynamics, for dynamical systems that can be described by ordinary differential equations (ODE's). The second part provides proofs concerning the epi-convergence of different statistical estimators that can be used with the learningbased model predictive control (LBMPC) technique. In particular, we prove results on the statistical properties of a nonparametric estimator that we have designed to have the correct deterministic and stochastic properties for numerical implementation when used in conjunction with LBMPC.

IEEE Transactions on Automatic Control, 2015

The study of controlled hybrid systems requires practical tools for approximation and comparison ... more The study of controlled hybrid systems requires practical tools for approximation and comparison of system behaviors. Existing approaches to these problems impose undue restrictions on the system's continuous and discrete dynamics. Metrization and simulation of controlled hybrid systems is considered here in a unified framework by constructing a state space metric. The metric is applied to develop a numerical simulation algorithm that converges uniformly, with a known rate of convergence, to orbitally stable executions of controlled hybrid systems, up to and including Zeno events. Benchmark hybrid phenomena illustrate the utility of the proposed tools.

IEEE Conference on Decision and Control and European Control Conference, 2011

Though hybrid dynamical systems are a powerful modeling tool, it has proven difficult to accurate... more Though hybrid dynamical systems are a powerful modeling tool, it has proven difficult to accurately simulate their trajectories. In this paper, we develop a provably convergent numerical integration scheme for approximating trajectories of hybrid dynamical systems. This is accomplished by first relaxing hybrid systems whose continuous states reside on manifolds by attaching epsilon-sized strips to portions of the boundary and then extending the dynamic and distance metric onto these strips. On this space we develop a numerical integration scheme and prove that discrete approximations converge to trajectories of the hybrid system. An example is included to illustrate the approach.

49th IEEE Conference on Decision and Control (CDC), 2010

Switched dynamical systems have shown great utility in modeling a variety of systems. Unfortunate... more Switched dynamical systems have shown great utility in modeling a variety of systems. Unfortunately, the determination of a numerical solution for the optimal control of such systems has proven difficult, since it demands optimal mode scheduling. Recently, we constructed an optimization algorithm to calculate a numerical solution to the problem subject to a running and final cost. In this paper, we modify our original approach in three ways to make our algorithm's application more tenable. First, we transform our algorithm to allow it to begin at an infeasible point and still converge to a lower cost feasible point. Second, we incorporate multiple objectives into our cost function, which makes the development of an optimal control in the presence of multiple goals viable. Finally, we extend our approach to penalize the number of hybrid jumps. We also detail the utility of these extensions to our original approach by considering two examples.

SIAM Journal on Control and Optimization, 2013

Switched systems, or systems whose control parameters include a continuous-valued input and a dis... more Switched systems, or systems whose control parameters include a continuous-valued input and a discrete-valued input which corresponds to the mode of the system that is active at a particular instance in time, have shown to be highly effective in modeling a variety of physical phenomena. Unfortunately, the construction of an optimal control algorithm for such systems has proved difficult since it demands some form of optimal mode scheduling. In a pair of papers, we construct a first order optimization algorithm to address this problem. Our approach, which we prove in this paper converges to local minimizers of the constrained optimal control problem, first relaxes the discrete-valued input, performs traditional optimal control, and then projects the constructed relaxed discrete-valued input back to a pure discrete-valued input by employing an extension to the classical chattering lemma that we formalize. In the second part of this pair of papers, we describe how this conceptual algorithm can be recast in order to devise an implementable algorithm that constructs a sequence of points by recursive application that converge to local minimizers of the optimal control problem for switched systems.

arXiv (Cornell University), Sep 8, 2014

After the three DARPA Grand Challenge contests many groups around the world have continued to act... more After the three DARPA Grand Challenge contests many groups around the world have continued to actively research and work toward an autonomous vehicle capable of accomplishing a mission in a given context (e.g. desert, city) while following a set of prescribed rules, but none has been completely successful in uncontrolled environments, a task that many people trivially fulfill every day. We believe that, together with improving the sensors used in cars and the artificial intelligence algorithms used to process the information, the community should focus on the systems engineering aspects of the problem, i.e. the limitations of the car (in terms of space, power, or heat dissipation) and the limitations of the software development cycle. This paper explores these issues and our experiences overcoming them.

Learning-based model predictive control (LBMPC) is a technique that provides deterministic guaran... more Learning-based model predictive control (LBMPC) is a technique that provides deterministic guarantees on robustness, while statistical identification tools are used to identify richer models of the system in order to improve performance. This technical note provides proofs that elucidate the reasons for our choice of measurement model, as well as giving proofs concerning the stochastic convergence of LBMPC. The first part of this note discusses simultaneous state estimation and statistical identification (or learning) of unmodeled dynamics, for dynamical systems that can be described by ordinary differential equations (ODE's). The second part provides proofs concerning the epi-convergence of different statistical estimators that can be used with the learningbased model predictive control (LBMPC) technique. In particular, we prove results on the statistical properties of a nonparametric estimator that we have designed to have the correct deterministic and stochastic properties for numerical implementation when used in conjunction with LBMPC.

IEEE Transactions on Automatic Control, 2015

The study of controlled hybrid systems requires practical tools for approximation and comparison ... more The study of controlled hybrid systems requires practical tools for approximation and comparison of system behaviors. Existing approaches to these problems impose undue restrictions on the system's continuous and discrete dynamics. Metrization and simulation of controlled hybrid systems is considered here in a unified framework by constructing a state space metric. The metric is applied to develop a numerical simulation algorithm that converges uniformly, with a known rate of convergence, to orbitally stable executions of controlled hybrid systems, up to and including Zeno events. Benchmark hybrid phenomena illustrate the utility of the proposed tools.

IEEE Conference on Decision and Control and European Control Conference, 2011

Though hybrid dynamical systems are a powerful modeling tool, it has proven difficult to accurate... more Though hybrid dynamical systems are a powerful modeling tool, it has proven difficult to accurately simulate their trajectories. In this paper, we develop a provably convergent numerical integration scheme for approximating trajectories of hybrid dynamical systems. This is accomplished by first relaxing hybrid systems whose continuous states reside on manifolds by attaching epsilon-sized strips to portions of the boundary and then extending the dynamic and distance metric onto these strips. On this space we develop a numerical integration scheme and prove that discrete approximations converge to trajectories of the hybrid system. An example is included to illustrate the approach.

49th IEEE Conference on Decision and Control (CDC), 2010

Switched dynamical systems have shown great utility in modeling a variety of systems. Unfortunate... more Switched dynamical systems have shown great utility in modeling a variety of systems. Unfortunately, the determination of a numerical solution for the optimal control of such systems has proven difficult, since it demands optimal mode scheduling. Recently, we constructed an optimization algorithm to calculate a numerical solution to the problem subject to a running and final cost. In this paper, we modify our original approach in three ways to make our algorithm's application more tenable. First, we transform our algorithm to allow it to begin at an infeasible point and still converge to a lower cost feasible point. Second, we incorporate multiple objectives into our cost function, which makes the development of an optimal control in the presence of multiple goals viable. Finally, we extend our approach to penalize the number of hybrid jumps. We also detail the utility of these extensions to our original approach by considering two examples.

SIAM Journal on Control and Optimization, 2013

Switched systems, or systems whose control parameters include a continuous-valued input and a dis... more Switched systems, or systems whose control parameters include a continuous-valued input and a discrete-valued input which corresponds to the mode of the system that is active at a particular instance in time, have shown to be highly effective in modeling a variety of physical phenomena. Unfortunately, the construction of an optimal control algorithm for such systems has proved difficult since it demands some form of optimal mode scheduling. In a pair of papers, we construct a first order optimization algorithm to address this problem. Our approach, which we prove in this paper converges to local minimizers of the constrained optimal control problem, first relaxes the discrete-valued input, performs traditional optimal control, and then projects the constructed relaxed discrete-valued input back to a pure discrete-valued input by employing an extension to the classical chattering lemma that we formalize. In the second part of this pair of papers, we describe how this conceptual algorithm can be recast in order to devise an implementable algorithm that constructs a sequence of points by recursive application that converge to local minimizers of the optimal control problem for switched systems.