Optimal control of nonlinear systems with input constraints using linear time varying approximations

We propose a new method to solve input constrained optimal control problems for autonomous nonlinear systems affine in control. We then extend the method to compute the bang-bang control solutions under the symmetric control constraints. The most attractive aspect of the proposed technique is that it enables the use of linear quadratic control theory on the input constrained linear and nonlinear systems. We illustrate the effectiveness of our technique both on linear and nonlinear examples and compare our results with those of the literature

Consistent Approximations for the Optimal Control of Constrained Switched Systems

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 continuous-valued input and discrete-valued 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 discrete-valued input, then 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 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

Constrained nonlinear optimal control: a converse HJB approach

Extending the concept of solving the Hamilton-Jacobi-Bellman (HJB) optimization equation backwards [2], the so called converse constrained optimal control problem is introduced, and used to create various classes of nonlinear systems for which the optimal controller subject to constraints is known. In this way a systematic method for the testing, validation and comparison of different control techniques with the optimal is established. Because it naturally and explicitly handles constraints, particularly control input saturation, model predictive control (MPC) is a potentially powerful approach for nonlinear control design. However, nonconvexity of the nonlinear programs (NLP) involved in the MPC optimization makes the solution problematic. In order to explore properties of MPC-based constrained control schemes, and to point out the potential issues in implementing MPC, challenging benchmark examples are generated and analyzed. Properties of MPC-based constrained techniques are then evaluated and implementation issues are explored by applying both nonlinear MPC and MPC with feedback linearization

Nonlinear Model Predictive Control for Constrained Output Path Following

We consider the tracking of geometric paths in output spaces of nonlinear systems subject to input and state constraints without pre-specified timing requirements. Such problems are commonly referred to as constrained output path-following problems. Specifically, we propose a predictive control approach to constrained path-following problems with and without velocity assignments and provide sufficient convergence conditions based on terminal regions and end penalties. Furthermore, we analyze the geometric nature of constrained output path-following problems and thereby provide insight into the computation of suitable terminal control laws and terminal regions. We draw upon an example from robotics to illustrate our findings.Comment: 12 pages, 4 figure

Model predictive control of markovian jump nonlinear stochastic systems

In this paper we consider model predictive control for a class of constrained discrete-time Markovian switching systems consisting of a family of nonlinear stochastic subsystems whose nonlinear stochastic term for a particular mode is described by its statistical properties. The additive nonlinearity of the subsystems is allowed to contain state, input, and independent noise vectors. It is allowed also that hard constraints are imposed on the input manipulated variables

Model Predictive Control: Multivariable Control Technique of Choice in the 1990s?

The state space and input/output formulations of model predictive control are compared and preference is given to the former because of the industrial interest in multivariable constrained problems. Recently, by abandoning the assumption of a finite output horizon several researchers have derived powerful stability results for linear and nonlinear systems with and without constraints, for the nominal case and in the presence of model uncertainty. Some of these results are reviewed. Optimistic speculations about the future of MPC conclude the paper

Event-triggered robust control for multi-player nonzero-sum games with input constraints and mismatched uncertainties

In this article, an event-triggered robust control (ETRC) method is investigated for multi-player nonzero-sum games of continuous-time input constrained nonlinear systems with mismatched uncertainties. By constructing an auxiliary system and designing an appropriate value function, the robust control problem of input constrained nonlinear systems is transformed into an optimal regulation problem. Then, a critic neural network (NN) is adopted to approximate the value function of each player for solving the event-triggered coupled Hamilton-Jacobi equation and obtaining control laws. Based on a designed event-triggering condition, control laws are updated when events occur only. Thus, both computational burden and communication bandwidth are reduced. We prove that the weight approximation errors of critic NNs and the closed-loop uncertain multi-player system states are all uniformly ultimately bounded thanks to the Lyapunov's direct method. Finally, two examples are provided to demonstrate the effectiveness of the developed ETRC method

Data-Driven Nonlinear Control Designs for Constrained Systems

Systems with nonlinear dynamics are theoretically constrained to the realm of nonlinear analysis and design, while explicit constraints are expressed as equalities or inequalities of state, input, and output vectors of differential equations. Few control designs exist for systems with such explicit constraints, and no generalized solution has been provided. This dissertation presents general techniques to design stabilizing controls for a specific class of nonlinear systems with constraints on input and output, and verifies that such designs are straightforward to implement in selected applications. Additionally, a closed-form technique for an open-loop problem with unsolvable dynamic equations is developed. Typical optimal control methods cannot be readily applied to nonlinear systems without heavy modification. However, by embedding a novel control framework based on barrier functions and feedback linearization, well-established optimal control techniques become applicable when constraints are imposed by the design in real-time. Applications in power systems and aircraft control often have safety, performance, and hardware restrictions that are combinations of input and output constraints, while cryogenic memory applications have design restrictions and unknown analytic solutions. Most applications fall into a broad class of systems known as passivity-short, in which certain properties are utilized to form a structural framework for system interconnection with existing general stabilizing control techniques. Previous theoretical contributions are extended to include constraints, which can be readily applied to the development of scalable system networks in practical systems, even in the presence of unknown dynamics. In cases such as these, model identification techniques are used to obtain estimated system models which are guaranteed to be at least passivity-short. With numerous analytic tools accessible, a data-driven nonlinear control design framework is developed using model identification resulting in passivity-short systems which handles input and output saturations. Simulations are presented that prove to effectively control and stabilize example practical systems

A metaheuristic particle swarm optimization approach to nonlinear model predictive control

This paper commences with a short review on optimal control for nonlinear systems, emphasizing the Model Predictive approach for this purpose. It then describes the Particle Swarm Optimization algorithm and how it could be applied to nonlinear Model Predictive Control. On the basis of these principles, two novel control approaches are proposed and anal- ysed. One is based on optimization of a numerically linearized perturbation model, whilst the other avoids the linearization step altogether. The controllers are evaluated by simulation of an inverted pendulum on a cart system. The results are compared with a numerical linearization technique exploiting conventional convex optimization methods instead of Particle Swarm Opti- mization. In both approaches, the proposed Swarm Optimization controllers exhibit superior performance. The methodology is then extended to input constrained nonlinear systems, offering a promising new paradigm for nonlinear optimal control design.peer-reviewe