Optimal Controller Synthesis for Nonlinear Systems

Abstract

Optimal controller synthesis is a challenging problem to solve. However, in many applications such as robotics, nonlinearity is unavoidable. Apart from optimality, correctness of the system behaviors with respect to system specifications such as stability and obstacle avoidance is vital for engineering applications. Many existing techniques consider either the optimality or the correctness of system behavior. Rarely, a tool exists that considers both. Furthermore, most existing optimal controller synthesis techniques are not scalable because they either require ad-hoc design or they suffer from the curse of dimensionality. This thesis aims to close these gaps by proposing optimal controller synthesis techniques for two classes of nonlinear systems: linearly solvable nonlinear systems and hybrid nonlinear systems. Linearly solvable systems have associated Hamilton- Jacobi-Bellman (HJB) equations that can be transformed from the original nonlinear partial differential equation (PDE) into a linear PDE through a logarithmic transformation. The first part of this thesis presets two methods to synthesize optimal controller for linearly solvable nonlinear systems. The first technique uses a hierarchy of sums-of-square programs to compute a sequence of suboptimal controllers that have non-increasing suboptimality for first exit and finite horizon problems. This technique is the first systematic approach to provide stability and suboptimal performance guarantees for stochastic nonlinear systems in one framework. The second technique uses the low rank tensor decomposition framework to solve the linear HJB equation for first exit, finite horizon, and infinite horizon problems. This technique scale linearly with dimensions, alleviating the curse of dimensionality and enabling us to solve the linear HJB equation for a quadcopter model that is a twelve-dimensional system on a personal laptop. A new algorithm is proposed for a key step in the controller synthesis algorithm to solve the ill-conditioning issue that arises in the original algorithm. A MATLAB toolbox that implements the algorithms is developed, and the performance of these algorithms is illustrated by a few engineering examples. Apart from stability, in many applications, more complex specifications such as obstacle avoidance, reachability, and surveillance are required. The second part of the thesis describes methods to synthesize optimal controllers for hybrid nonlinear systems with quantitative objectives (i.e., minimizing cost) and qualitative objectives (i.e., satisfying specifications). This thesis focuses on two types of qualitative objectives, regular objectives, and ω-regular objectives. Regular objectives capture bounded time behavior such as reachability, and ω-regular objectives capture long term behavior such as surveillance. For both types of objectives, an abstraction-refinement procedure that preserves the cost is developed. A two-player game is solved on the product of the abstract system and the given objectives to synthesize the suboptimal controller for the hybrid nonlinear system. By refining the abstract system, the algorithms are guaranteed to converge to the optimal cost and return the optimal controller if the original systems are robust with respect to the initial states and the optimal controller inputs. The proposed technique is the first abstraction-refinement based technique to combine both quantitative and qualitative objectives into one framework. A Python implementation of the algorithms are developed, and a few engineering examples are presented to illustrate the performance of these algorithms.