Horizon Linear Quadratic Optimal Control Research Articles

Autonomous Vehicle Convoy Formation Control with Size/Shape Switching for Automated Highways

Today’s semi-autonomous vehicles are gradually moving towards full autonomy. This transition requires developing effective control algorithms for handing complex autonomous tasks. Driving as a group of vehicles, referred to as a convoy, on automated highways is a highly important and challenging task that autonomous driving systems must deal with. This paper considers the control problem of a vehicle convoy modeled with linear dynamics. The convoy formation requirement is presented in terms of a quadratic performance index to minimize. The convoy formation control is formulated as a receding horizon linear-quadratic (LQ) optimal control problem. The receding horizon control law is innovatively defined via the solution to the algebraic Riccati equation. The solution matrix and therefore the receding horizon control law are obtained in the closed-form. A control architecture consisting of four algorithms is proposed to handle formation size/shape switching. The closed-form control law is at the core of these algorithms. Simulation results are provided to justify the models, solutions, and proposed algorithms.

Linear quadratic optimal control for a class of continuous-time nonhomogeneous Markovian jump linear systems in infinite time horizon

In this paper, the infinite time horizon linear quadratic optimal control problem is investigated for the continuous-time Itô stochastic Markovian jump linear systems (MJLSs) with time-varying transition rates. It is assumed that the time-varying transition rates of the MJLSs possess piecewise homogeneous time-varying property, which implies that the transition rates are time-varying in the whole time domain but they are time-invariant in some small time intervals. The variations of the transition rates in these small time intervals are considered to be in two cases: arbitrary variation and stochastic variation. With this, the considered system becomes a piecewise homogeneous Itô stochastic MJLS. The main contribution of this paper is that two linear quadratic optimal controllers in infinite time horizon are proposed for the above modeled continuous-time piecewise homogeneous Itô stochastic MJLS in the sense of arbitrary variation and stochastic variation, respectively. Moreover, the sufficient and necessary conditions for the existence of the designed controllers are established based on the existence of the unique positive definite solution of two coupled algebraic Riccati matrix equations. Finally, a simulation example is provided to illustrate the effectiveness of the proposed linear quadratic optimal controllers.

On linear-quadratic optimal control of implicit difference equations

Abstract In this work we investigate explicit and implicit difference equations and the corresponding infinite time horizon linear-quadratic optimal control problem. We derive conditions for feasibility of the optimal control problem as well as existence and uniqueness of optimal controls under certain weaker assumptions compared to the standard approaches in the literature which are using algebraic Riccati equations. To this end, we introduce and analyse a discrete-time Lur’e equation and a corresponding Kalman–Yakubovich–Popov (KYP) inequality. We show that solvability of the KYP inequality can be characterized via the spectral structure of a certain palindromic matrix pencil. The deflating subspaces of this pencil are finally used to construct solutions of the Lur’e equation. The results of this work are transferred from the continuous-time case. However, many additional technical difficulties arise in this context.

Optimal control of discrete-time linear fractional-order systems with multiplicative noise

ABSTRACTA finite horizon linear quadratic (LQ) optimal control problem is studied for a class of discrete-time linear fractional systems (LFSs) affected by multiplicative, independent random perturbations. Based on the dynamic programming technique, two methods are proposed for solving this problem. The first one seems to be new and uses a linear, expanded-state model of the LFS. The LQ optimal control problem reduces to a similar one for stochastic linear systems and the solution is obtained by solving Riccati equations. The second method appeals to the principle of optimality and provides an algorithm for the computation of the optimal control and cost by using directly the fractional system. As expected, in both cases, the optimal control is a linear function in the state and can be computed by a computer program. A numerical example and comparative simulations of the optimal trajectory prove the effectiveness of the two methods. Some other simulations are obtained for different values of the fractional order.

Linear quadratic Pareto optimal control problem of stochastic singular systems

This paper is concerned with the finite horizon linear quadratic (LQ) Pareto optimal control problem of stochastic singular systems. By means of the square completion technique, for the finite horizon LQ optimal control of stochastic singular systems, we establish a new kind of generalized differential Riccati equations (GDREs) and present the existence condition of the solution of the GDREs. Then, for the finite horizon LQ Pareto optimal control, it is shown that under the solvability of the corresponding GDREs, all Pareto candidates can be obtained by solving a weighting sum optimal control. Finally, an example is provided to show the effectiveness of our main results.

Optimal State Estimation for Non-Time Invertible Evolutionary Systems

In this paper, we formulate state estimation problem for usually non-time invertible evolutionary systems into a nonstandard finite horizon linear quadratic output tracking problem. This problem directly leads to the classical filtering problem with disturbance constraint. For a given reference trajectory, we give the explicit expressions of optimal initial state and optimal control. In addition, based on some well-known results for classical finite horizon linear quadratic optimal control problems, we obtain, in terms of measured output, the pointwise linear state feedbacks to optimal initial state and optimal control, and the representation of optimal cost is also given. As an important consequence, the differential equation satisfied by the optimal state estimate for filtering problem is developed. Some examples are given as applications.

LQ Nash Games With Random Entrance: An Infinite Horizon Major Player and Minor Players of Finite Horizons

We study Dynamic Games with randomly entering players, staying in the game for different lengths of time. Particularly, a class of Discrete Time Linear Quadratic (LQ) Games, involving a major player who has an infinite time horizon and a random number of minor players is considered. The number of the new minor players, entering at some instant of time, is random and it is described by a Markov chain. The problem of the characterization of a Nash equilibrium, consisting of Linear Feedback Strategies, is reformulated as a set of coupled finite and infinite horizon LQ optimal control problems for Markov Jump Linear Systems (MJLS). Sufficient conditions characterizing Nash equilibrium are then derived. The problem of Games involving a large number of minor players is then addressed using a Mean Field (MF) approach and asymptotic ε-Nash equilibrium results are derived. Sufficient conditions for the existence of a MF Nash equilibrium are finally stated.

Infinite horizon linear quadratic optimal control for stochastic difference time-delay systems

The aim of this paper is to investigate the infinite horizon linear quadratic (LQ) optimal control for stochastic time-delay difference systems with both state and control dependent noise. To do this, the notion of exact observability of a stochastic time-delay deference system is introduced and its PBH criterion is presented by the spectrum of an operator related with stochastic time-delay deference systems. Under the assumptions of stabilization and exact observability, it is shown that the optimal control law and optimal value exist, and also the properties of the associated general algebraic Ricatti equation (GARE) are discussed.

Optimal switching policy for performance enhancement of distributed parameter systems based on event-driven control

This paper aims to improve the performance of a class of distributed parameter systems for the optimal switching of actuators and controllers based on event-driven control. It is assumed that in the available multiple actuators, only one actuator can receive the control signal and be activated over an unfixed time interval, and the other actuators keep dormant. After incorporating a state observer into the event generator, the event-driven control loop and the minimum inter-event time are ultimately bounded. Based on the event-driven state feedback control, the time intervals of unfixed length can be obtained. The optimal switching policy is based on finite horizon linear quadratic optimal control at the beginning of each time subinterval. A simulation example demonstrate the effectiveness of the proposed policy.

A Risk-Based System of Systems Approach to Control the Transport Flows of Dangerous Goods by Road

A risk-based approach to control dangerous goods (DG) transport flows by roads is proposed, solving a real-time flow assignment problem. The model takes into account the planned scheduling of the DG fleets. The objective is to readapt the schedule in real time, controlling the DG flow to minimize both the risk on the network and the gap between the proposed modified delivery and the planned one. The innovative aspect of the proposed approach is to balance the social objective of a national authority, thus minimizing the risk on the road infrastructures, with the economical objective of the DG distribution companies that have to minimize the actual time, as defined by the planned deliveries. The proposed DG transport model is defined according to a system of systems view. Each subsystem (SS) represents either a regional area or, more commonly, a segment of a road. The proposed approach provides a useful tool for evaluating the optimal speed for DG vehicles in each SS and the optimal amount of DG flow that should transit from one SS to another, following the planned delivery schedule. The problem has been tackled in two different formulations. First, a nonlinear mathematical programming formulation is defined. Then, according to simplifying assumptions, the problem is solved as a discrete-time finite horizon linear quadratic optimal control problem with a state feedback control. An exemplificative case study is used to show a comparison between the two formulations, as well as the effects of a risk sudden change in the overall DG routing.

Infinite horizon linear quadratic optimal control for discrete‐time stochastic systems

AbstractThis paper is concerned with the infinite horizon linear quadratic optimal control for discrete‐time stochastic systems with both state and control‐dependent noise. Under assumptions of stabilization and exact observability, it is shown that the optimal control law and optimal value exist, and the properties of the associated discrete generalized algebraic Riccati equation (GARE) are also discussed. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Infinite horizon quadratic control of linear singularly perturbed systems with small state delays: an asymptotic solution of Riccati-type equations

An infinite horizon linear-quadratic optimal control problem for a singularly perturbed system with multiple point-wise and distributed small delays in the state variable is considered. The set of Riccati-type equations, associated with this problem by the control optimality conditions, is studied. Since the system in the control problem is singularly perturbed, the equations of this set are also perturbed by a small parameter of the singular perturbations. The zero-order asymptotic solution to this set of equations is constructed and justified. Based on this asymptotic solution, parameter-free sufficient conditions for the existence and uniqueness of solution to the original optimal control problem are established.

Optimal design of networked control systems: computer control via asynchronous communication channels

We study a linear discrete-time partially observed system perturbed by white noises. The observations are transmitted to the controller, and its outputs are sent to the actuators via communication channels that provide random transmission times and may lose messages. Various signals may incur independent delays and arrive at the destination point out of order. Under certain assumptions, a recursive minimum variance state estimator is proposed. A finite horizon linear-quadratic optimal control problem is also solved.

Mixed Constrained Infinite Horizon Linear Quadratic Optimal Control

ABSTRACTFor a given initial state, a constrained infinite horizon linear quadratic optimal control problem can be reduced to a finite dimensional problem [12]. To find a conservative estimate of the size of the reduced problem, the existing algorithms require the on‐line solutions of quadratic programs [10] or a linear program [2]. In this paper, we first show based on the Lyapunov theorem that the closed‐loop system with a mixed constrained infinite horizon linear quadratic optimal control is exponentially stable on proper sets. Then the exponentially converging envelop of the closed‐loop trajectory that can be computed off‐line is employed to obtain a finite dimensional quadratic program equivalent to the mixed constrained infinite horizon linear quadratic optimal control problem without any on‐line optimization. The example considered in [2] showed that the proposed algorithm identifies less conservative size estimate of the reduced problem with much less computation.

Optimal computer control via communication channels with irregular transmission times

We study a linear discrete-time partially observed system perturbed by white noises. The observations are transmitted to the controller via communication channels with random transmission times. Various measurement signals may incur independent delays, be lost or corrupted. Under certain assumptions, a recursive minimum variance state estimator is proposed. A finite horizon linear-quadratic optimal control problem is solved. We also find a solution for the problem of minimizing the average cost per unit time over the infinite planning horizon.

An Infinite Horizon Linear Quadratic Problem with Unbounded Controls in Hilbert Space

An infinite horizon linear quadratic optimal control problem for analytic semigroup with unbounded control in Hilbert space is considered. The state weight operator is allowed to be indefinite while the control weight operator is coercive. Under the exponential stabilization condition, it is proved that any optimal control and its optimal trajectory are continuous. The positive real lemma as a necessary and sufficient condition for the unique solvability of this problem is established. The closed-loop synthesis of optimal control is given via the solution to the algebraic Riccati equation.

Constrained linear quadratic optimal control of chemical processes

In this paper, we address the constrained infinite horizon linear quadratic optimal control (CIHLQOC) of chemical processes. Employing the exponential bound of the closed loop system trajectory that can be computer off-line from the exponential stability properties, it was shown that a finite dimensional quadratic program equivalent to the CIHLQOC problem without any on-line optimization. This technique is illustrated with a chemical reactor example.

Fast algorithm for a constrained infinite horizon LQ problem

This paper presents an efficient algorithmic solution to the infinite horizon linear quadratic optimal control problem for a discrete-time SISO plant subject to bound constraints on a scalar variable. The solution to the corresponding quadratic programming problem is based on the active set method and on dynamic programming. It is shown that the optimal solution can be updated after inclusion or removal of an active constraint by a simple procedure requiring in the order of kn operations, n being the system order and k the time at which the constraint is included or removed.

Receding Time Horizon Linear Quadratic Optimal Control for Multi-Axis Contour Tracking Motion Control1

A receding time horizon linear quadratic optimal control approach is formulated for multi-axis contour tracking problem. The approach employs a performance index with fixed weights on quadratic contouring error, tracking error, and control input over a future finite horizon. The problem is then cast into a standard receding horizon LQ problem with time varying weighting matrices, which are functions of the future contour trajectory within the horizon. The formulation thus leads to a solution of time varying state feedback and finite preview gains. Stability is proven for the linear trajectory case. Experimental and simulated results for an X-Y motion control problem are presented, which demonstrate the effectiveness of the control scheme and the effects of the key controller design parameters. [S0022-0434(00)01202-8]

Linear quadratic problem with unbounded control in Hilbert spaces

A finite horizon linear quadratic optimal control problem in Hilbert space is considered, in which the controls are unbounded and the state weight operators are allowed to be indefinite and not pecessarily smoothing. It is proved that any optimal control and optimal trajectory are continuous. The closed-loop synthesis for optimal control is given via the solution to Fredholm integral equation and also the solution to Riccati integral equation, which exists under some mild conditions.