Constrained Finite-time Optimal Control Research Articles

Data-Driven Optimization for Deposition with Degradable Tools

We present a data-driven optimization approach for robotic controlled deposition with a degradable tool. Existing methods make the assumption that the tool tip is not changing or is replaced frequently. Errors can accumulate over time as the tool wears away and this leads to poor performance in the case where the tool degradation is unaccounted for during deposition. In the proposed approach, we utilize visual and force feedback to update the unknown model parameters of our tool-tip. Subsequently, we solve a constrained finite time optimal control problem for tracking a reference deposition profile, where our robot plans with the learned tool degradation dynamics. We focus on a robotic drawing problem as an illustrative example. Using real-world experiments, we show that the error in target vs actual deposition decreases when learned degradation models are used in the control design.

Cooperative signal-free intersection control using virtual platooning and traffic flow regulation

The emerging technologies of connectivity and automation enable the potential for signal-free intersection control. In this context, virtual platooning is posited to be an innovative, decentralized control strategy that maps two-dimensional vehicle movements onto a one-dimension virtual platoon to enable intersection operations. However, the effectiveness of virtual platooning-based control can be limited or degraded by parametric inaccuracies and unparameterized disturbances in vehicle dynamics, heavy traffic congestion, and/or uncoordinated platoons in multi-lane intersections. To explicitly address these limitations, this study proposes a hybrid cooperative intersection control framework consisting of microscopic-level virtual platooning control and macroscopic-level traffic flow regulation for traffic environments with connected autonomous vehicles. In virtual platooning control, vehicles approaching an intersection are organized into coordinated independent virtual platoons to avoid potential conflicts triggered by platoon formation changes. Through coordination, vehicles in a platoon are grouped into compatible passing sets to maintain desired safe spacing when proceeding through the intersection. We propose a distributed adaptive sliding mode controller (DASMC) which uses the backstepping control method and model reference adaptive control method to address parametric inaccuracies, and the sliding mode control method to consistently suppress the negative effects of the unparameterized disturbances. Each vehicle approaching the intersection utilizes the kinematic information from neighboring vehicles to implement the DASMC in a distributed manner such that vehicles within the same virtual platoon can achieve consensus safely. However, virtual platooning control cannot preclude excessive traffic from approaching the intersection, which can cause undesired spillbacks and degrade intersection control performance. To address this issue, traffic flow regulation is integrated with the virtual platooning control using an iterative feedback loop mechanism. In each iteration of the iterative feedback loop, a constrained finite-time optimal control (CFTOC) problem is solved to determine the optimal input flow permitted to proceed through the intersection, and the virtual platooning control provides feedback on the queue status to the CFTOC to initiate the next iteration. The effectiveness of the proposed intersection control framework is evaluated through numerical experiments. The results indicate that the proposed virtual platooning DASMC controller can mitigate the effects of parametric inaccuracies and unparameterized disturbances to achieve consensus for approaching vehicles, as well as guarantee string stability. Further, the proposed framework can alleviate traffic spillbacks and travel delays effectively through traffic flow regulation.

Direct Parallel Computations of Second-Order Search Directions for Model Predictive Control

Model predictive control ( MPC ) is one of the most widely spread advanced control schemes in industry today. In MPC , a constrained finite-time optimal control ( CFTOC ) problem is solved at each iteration in the control loop. The CFTOC problem can be solved using, for example, second-order methods, such as interior-point or active-set methods, where the computationally most demanding part often consists of computing the sequence of second-order search directions. Each search direction can be computed by solving a set of linear equations that corresponds to solving an unconstrained finite-time optimal control ( UFTOC ) problem. In this paper, different direct (noniterative) parallel algorithms for solving UFTOC problems are presented. The parallel algorithms are all based on a recursive variable elimination and solution propagation technique. Numerical evaluations of one of the parallel algorithms indicate that a significant boost in performance can be obtained, which can facilitate high-performance second-order MPC solvers.

Reconfigurable Model Predictive Control for Multievaporator Vapor Compression Systems

This paper considers the control of a multievaporator vapor compression system (ME-VCS) where individual evaporators are permitted to turn ON or OFF. We present a model predictive controller (MPC) that can be easily reconfigured for different ON/OFF configurations of the system. In this approach, only the cost function of the constrained finite-time optimal control problem is updated depending on the system configuration. Exploiting the structure of the system dynamics, the cost function is modified by zeroing elements of the state, input, and terminal cost matrices. The advantage of this approach is that cost matrices for each configuration of the ME-VCS do not need to be stored or computed online. This reduces the effort required to tune and calibrate the controller and the amount of memory required to store the controller parameters in a microprocessor. The reconfigurable MPC is compared with a conventional approach in which individual model predictive controllers are independently designed for each ON/OFF configuration. The simulations show that the reconfigurable MPC method provides a similar closed-loop performance in terms of reference tracking and constraint satisfaction to the set of individual model predictive controllers. Further, we show that our controller requires substantially less memory than the alternative approaches. Experiments on a residential two-zone vapor compression system further validate the reconfigurable MPC method.

Low-Rank Modifications of Riccati Factorizations for Model Predictive Control

In model predictive control ( MPC ), the control input is computed by solving a constrained finite-time optimal control ( CFTOC ) problem at each sample in the control loop. The main computational effort when solving the CFTOC problem using an active-set ( AS ) method is often spent on computing the search directions, which in MPC corresponds to solving unconstrained finite-time optimal control ( UFTOC ) problems. This is commonly performed using Riccati recursions or generic sparsity exploiting algorithms. In this paper, the focus is efficient search direction computations for AS type methods. The system of equations to be solved at each AS iteration is changed only by a low-rank modification of the previous one, and exploiting this structured change is important for the performance of AS -type solvers. In this paper, theory for how to exploit these low-rank changes by modifying the Riccati factorization between AS iterations in a structured way is presented. A numerical evaluation of the proposed algorithm shows that the computation time can be significantly reduced by modifying, instead of re-computing, the Riccati factorization. This speedup can be important for AS -type solvers used for linear, nonlinear, and hybrid MPC .

Stabilisation of constrained uncertain systems by multi-parametric optimisation

This paper aims to construct a controlled invariant set while determining a stabilising piecewise linear control law for constrained uncertain systems. The proposed approach is based on the equivalence between the considered system and its Euler auxiliary system (EAS). The control law is obtained by the resolution of a constrained finite time optimal control (CFTOC) problem. Due to the exhaustive computing demands, a multi-parametric formulation and an offline resolution of the problem are considered to extend the set parameters variation.

Experimental constrained optimal attitude control of a quadrotor subject to wind disturbances

The design and experimental verification of a Constrained Finite Time Optimal Controller (CFTOC) for attitude maneuvers of an Unmanned Quadrotor operating under severe wind conditions is the subjec ...

Piecewise Affine Modeling and Constrained Optimal Control for a Pneumatic Artificial Muscle

In this paper, the modeling and control problem of a pneumatic artificial muscle (PAM) is being considered. The PAM is an actuator characterized by a decrease in the actuating length when pressurized. Its nonlinear nature and time-varying parameters cause difficulties in modeling their characteristics, as well as in designing controllers for high-performance positioning systems. A constrained linear and piecewise affine system model approximation is formulated, and a control scheme composed of the following is being synthesized: 1) a feedforward term regulating the control input at specific set points and 2) a constrained finite-time optimal controller handling any deviations from the system's equilibrium points. Extended experimental studies are utilized to prove the efficacy of the suggested controller.

Explicit MPC for LPV Systems: Stability and Optimality

This paper considers high-speed control of constrained linear parameter-varying systems using model predictive control. Existing model predictive control schemes for control of constrained linear parameter-varying systems typically require the solution of a semi-definite program at each sampling instance. Recently, variants of explicit model predictive control were proposed for linear parameter-varying systems with polytopic representation, decreasing the online computational effort by orders of magnitude. Depending on the mathematical structure of the underlying system, the constrained finite-time optimal control problem can be solved optimally, or close-to-optimal solutions can be computed. Constraint satisfaction, recursive feasibility and asymptotic stability can be guaranteed a priori by an appropriate selection of the terminal state constraints and terminal cost. The paper at hand gathers previous developments and provides new material such as a proof for the optimality of the solution, or, in the case of close-to-optimal solutions, a procedure to determine a bound on the suboptimality of the solution.

On Predictive Control for Systems with Information Structured Constraints

In this paper, we investigate a predictive control problem with information structured constraints motivated by control of micro grid. A system with information structures is defined as a system in which each subsystem collects spatio-temporally different information. For the system, we consider a predictive control law and the finite time con- strained optimization problem to be solved online is reduced to a deterministic convex programming problem. Then, we reduce a control problem for a simple micro grid system to the framework and the effectiveness of the proposed control and estimation law is demonstrated through a numerical simulation. The main objective of this paper is thus to present a pre- dictive control scheme, which is known to be a useful control methodology (22),(23) in order to meet the above requirements for systems with information structures. We first propose a pre- dictive control scheme for a finite-time optimal control prob- lem. Then, we prove that the constrained finite-time optimal control problem is reduced to a deterministic convex program- ming problem. Finally, a control problem for a simple micro grid system is shown to be reduced to the present framework and the effectiveness of the proposed control scheme is shown via a numerical simulation. The following notation is used in this paper. Z+ is the non-negative integer set, R n is the n-dimensional real space, R n is the n-dimensional non-negative real space, R m×n is the m × n-dimensional real space, E is an expectation operator, Q > 0( Q ≥ 0) means that Q is a positive (non-negative) defi- nite matrix, and Tr is a trace operator.

Explicit nonlinear predictive control for a magnetic levitation system

AbstractThe paper presents a methodology for the construction of an explicit nonlinear control law via approximation of the nonlinear constrained finite‐time optimal control (CFTOC). This is achieved through an approximate mapping of a general nonlinear system in a set of linear piecewise affine (PWA) systems. The key advantages of this methodology are two‐fold. First, the construction of an analytic solution of the CFTOC problem leads to an efficient explicit implementation. Second, by taking advantage of model predictive control's systematic fashion to handle constraints, an improved performance can be obtained for the closed‐loop system. The proposed theory is applied in real‐time for a system with fast dynamics: a magnetic levitation benchmark. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Efficient On-Line Computation of Constrained Optimal Control

We consider constrained finite-time optimal control problems for discrete-time linear time-invariant systems with constraints on inputs and outputs based on linear and quadratic performance indices. The solution to such problems is a time-varying piecewise affine (PWA) state-feedback law and can be computed by means of multiparametric programming. By exploiting the properties of the value function and the piecewise affine optimal control law of the constrained finite-time optimal control (CFTOC), we propose two new algorithms that avoid storing the polyhedral regions. The new algorithms significantly reduce the on-line storage demands and computational complexity during evaluation of the PWA feedback control law resulting from the CFTOC.

Parametric optimization and optimal control using algebraic geometry methods

We present two algebraic methods to solve the parametric optimization problem that arises in non-linear model predictive control. We consider constrained discrete-time polynomial systems and the corresponding constrained finite-time optimal control problem. The first method is based on cylindrical algebraic decomposition. The second uses Gröbner bases and the eigenvalue method for solving systems of polynomial equations. Both methods aim at moving most of the computational burden associated with the optimization problem off-line, by pre-computing certain algebraic objects. Then, an on-line algorithm uses this pre-computed information to obtain the solution of the original optimization problem in real time fast and efficiently. Introductory material is provided as appropriate and the algorithms are accompanied by illustrative examples.

Constrained optimal control of an electronic throttle

The overall vehicle performance is strongly influenced by the quality of the control of the electronic throttle – a DC motor driven valve that regulates the inflow of air to the vehicle's engine. Designing a controller for the throttle system is a challenging task since one has to cope with two strong non-linearities: the gearbox friction and the so-called “limp-home” non-linearity. In this paper we address these issues by solving a constrained optimal control problem formulated for the discrete-time piecewise affine (PWA) model of the throttle. In an off-line, dynamic programming procedure we obtain the look-up table like solution to the optimal control problem. Such a solution allows the real-time controller implementation that would otherwise be impossible to achieve due to the small sampling time needed for the application at hand. We address the issue of the PWA friction modelling in more detail by considering both static and dynamic friction models. Two different control strategies are studied: constrained finite time optimal control (CFTOC), used in the regulator case, and constrained time-optimal control (CTOC), used in the reference tracking case. We report experimental results with both control strategies. The reference tracking controller significantly outperformed a tuned PID controller with a feedforward compensation of non-linearities in terms of the response speed while preserving the response quality regarding the absence of an overshoot and the static accuracy within the measurement resolution.

MULTI–PARAMETRIC H∞ CONTROL OF A MICRO ACTUATOR

In this article the control problem of a micro-actuator (μ – A is considered. The μ – A is composed of a micro-capacitor, whose one plate is clamped while its other flexible plate's motion is constrained by hinges acting as a combination of springs and dashpots. The distance of the plates is varied by the applied voltage between them. The flexibility of the moving plate coupled to the dynamics of the plate's rigid-body motion results in an unstable, nonlinear system of distributed nature. Utilization of FEM can approximate the μ – A dynamics nonlinear-PDE to a finite nonlinear-ODE. The nonlinearity stems from the plate's rigid-body motion, while all flexibility effects are considered as additive linear-terms. A controller composed of: a) a feedforward term for regulation at selected setpoints, and b) a constrained finite time optimal controller to handle any deviations from the equilibrium is synthesized. Simulation studies are used to investigate the efficacy of the suggested controller.