Jan M. Maciejowski

Optimization over state feedback policies for robust control with constraints

This paper is concerned with the optimal control of linear discrete-time systems subject to unknown but bounded state disturbances and mixed polytopic constraints on the state and input. It is shown that the class of admissible affine state feedback control policies with knowledge of prior states is equivalent to the class of admissible feedback policies that are affine functions of the past disturbance sequence. This implies that a broad class of constrained finite horizon robust and optimal control problems, where the optimization is over affine state feedback policies, can be solved in a computationally efficient fashion using convex optimization methods. This equivalence result is used to design a robust receding horizon control (RHC) state feedback policy such that the closed-loop system is input-to-state stable (ISS) and the constraints are satisfied for all time and all allowable disturbance sequences. The cost to be minimized in the associated finite horizon optimal control problem is quadratic in the disturbance-free state and input sequences. The value of the receding horizon control law can be calculated at each sample instant using a single, tractable and convex quadratic program (QP) if the disturbance set is polytopic, or a tractable second-order cone program (SOCP) if the disturbance set is given by a 2-norm bound.

An Overview of Sequential Monte Carlo Methods for Parameter Estimation in General State-Space Models

Abstract Nonlinear non-Gaussian state-space models arise in numerous applications in control and signal processing. Sequential Monte Carlo (SMC) methods, also known as Particle Filters, provide very good numerical approximations to the associated optimal state estimation problems. However, in many scenarios, the state-space model of interest also depends on unknown static parameters that need to be estimated from the data. In this context, standard SMC methods fail and it is necessary to rely on more sophisticated algorithms. The aim of this paper is to present a comprehensive overview of SMC methods that have been proposed to perform static parameter estimation in general state-space models. We discuss the advantages and limitations of these methods.

Asymptotic recovery for discrete-time systems

An asymptotic recovery design procedure is proposed for square, discrete-time, linear, time-invariant multivariable systems, which allows a state-feedback design to be approximately recovered by a dynamic output feedback scheme. Both the case of negligible processing time (compared to the sampling interval) and of significant processing time are discussed. In the former case, it is possible to obtain perfect recovery if the plant is minimum-phase and has the smallest possible number of zeros at infinity. In other cases good recovery is frequently possible. New conditions are found which ensure that the return-ratio being recovered exhibits good robustness properties.

MPC fault-tolerant flight control case study: Flight 1862

We demonstrate that the fatal crash of El Al Flight 1862 might have been avoided by using MPC-based fault-tolerant control. Simulation on a detailed nonlinear model shows that it is possible to reconfigure the controller so that the aircraft is flown successfully down to ground level, without entering the condition in which it was lost. We use a reference-model based approach, in which an MPC controller attempts to restore the original functionality of the pilot’s controls. For the purposes of simulation, we emulate the pilot by another MPC controller, running at a lower sampling rate. We assume in this paper that an FDI function delivers information about actuator damage, and about changes to aerodynamic coefficients in the failed condition.

Invariant sets for constrained nonlinear discrete-time systems with application to feasibility in model predictive control

An understanding of invariant set theory is essential in the design of controllers for constrained systems, since state and control constraints can be satisfied if and only if the initial state belongs to a positively invariant set for the closed-loop system. The paper briefly reviews some concepts in invariant set theory and shows that the various sets can be computed using a single recursive algorithm. The ideas presented in the first part of the paper are applied to the fundamental design goal of guaranteeing feasibility in predictive control. New necessary and sufficient conditions based on the control horizon, prediction horizon and terminal constraint set are given in order to guarantee that the predictive control problem will be feasible for all time, given any feasible initial state.

On Particle Methods for Parameter Estimation in State-Space Models

Nonlinear non-Gaussian state-space models are ubiquitous in statistics, econometrics, information engineering and signal processing. Particle methods, also known as Sequential Monte Carlo (SMC) methods, provide reliable numerical approximations to the associated state inference problems. However, in most applications, the state-space model of interest also depends on unknown static parameters that need to be estimated from the data. In this context, standard particle methods fail and it is necessary to rely on more sophisticated algorithms. The aim of this paper is to present a comprehensive review of particle methods that have been proposed to perform static parameter estimation in state-space models. We discuss the advantages and limitations of these methods and illustrate their performance on simple models.

Monte Carlo Optimization for Conflict Resolution in Air Traffic Control

The safety of flights, and, in particular, separation assurance, is one of the main tasks of air traffic control (ATC). Conflict resolution refers to the process used by ATCs to prevent loss of separation. Conflict resolution involves issuing instructions to aircraft to avoid loss of safe separation between them and, at the same time, direct them to their destinations. Conflict resolution requires decision making in the face of the considerable levels of uncertainty inherent in the motion of aircraft. In this paper, a framework for conflict resolution that allows one to take into account such levels of uncertainty using a stochastic simulator is presented. The conflict resolution task is posed as the problem of optimizing an expected value criterion. It is then shown how the cost criterion can be selected to ensure an upper bound on the probability of conflict for the optimal maneuver. Optimization of the expected value resolution criterion is carried out through an iterative procedure based on Markov chain Monte Carlo. Simulation examples inspired by current ATC practice in terminal maneuvering areas and approach sectors illustrate the proposed conflict resolution strategy

Embedded Model Predictive Control (MPC) Using a FPGA

Model Predictive Control (MPC) is increasingly being proposed for application to miniaturized devices, fast and/or embedded systems. A major obstacle to this is its computation time requirement. Continuing our previous studies of implementing constrained MPC on Field Programmable Gate Arrays (FPGA), this paper begins to exploit the possibilities of parallel computation, with the aim of speeding up the MPC implementation. Simulation studies on a realistic example show that it is possible to implement constrained MPC on an FPGA chip with a 25MHz clock and achieve MPC implementation rates comparable to those achievable on a Pentium 3.0 GHz PC.

Collective behavior coordination with predictive mechanisms

In natural flocks/swarms, it is very appealing that low-level individual intelligence and communication can yield advanced coordinated collective behaviors such as congregation, synchronization and migration. In the past few years, the discovery of collective flocking behaviors has stimulated much interest in the study of the underlying organizing principles of abundant natural groups, which has led to dramatic advances in this emerging and active research field. Inspired by previous investigations on the predictive intelligence of animals, insects and microorganisms, we seek in this article to understand the role of predictive mechanisms in the forming and evolving of flocks/swarms by using both numerical simulations and mathematical analyses. This article reviews some basic concepts, important progress, and significant results in the current studies of collective predictive mechanisms, with emphasis on their virtues concerning consensus improvement and communication cost reduction. Due to these advantages, such predictive mechanisms have great potential to find their way into industrial applications.

MULTIPLEXED MODEL PREDICTIVE CONTROL

Abstract Most academic control schemes for MIMO systems assume all the control variables are updated simultaneously. MPC outperforms other control strategies through its ability to deal with constraints. This requires on-line optimization, hence computational complexity can become an issue when applying MPC to complex systems with fast response times. The multiplexed MPC scheme described in this paper solves the MPC problem for each subsystem sequentially and updates subsystem controls as soon as the solution is available, thus distributing the control moves over a complete update cycle. The resulting computational speed-up allows faster response to disturbances, and hence improved performance, despite finding sub-optimal solutions to the original problem. The multiplexed MPC scheme is also closer to industrial practice in many cases. This paper presents initial stability results for two variants of multiplexed MPC, and illustrates the performance benefit by an example.