Linear Regulator Problem Research Articles

Direct integral pseudospectral and integral spectral methods for solving a class of infinite horizon optimal output feedback control problems using rational and exponential Gegenbauer polynomials

This study is concerned with the numerical solution of a class of infinite-horizon linear regulation problems with state equality constraints and output feedback control. We propose two numerical methods to convert the optimal control problem into nonlinear programming problems (NLPs) using collocations in a semi-infinite domain based on rational Gegenbauer (RG) and exponential Gegenbauer (EG) basis functions. We introduce new properties of these basis functions and derive their quadratures and associated truncation errors. A rigorous stability analysis of the RG and EG interpolations is also presented. The effects of various parameters on the accuracy and efficiency of the proposed methods are investigated. The performance of the developed integral spectral method is demonstrated using two benchmark test problems related to a simple model of a divert control system and the lateral dynamics of an F-16 aircraft. Comparisons of the results of the current study with available numerical solutions show that the developed numerical scheme is efficient and exhibits faster convergence rates and higher accuracy.

Event-triggered output-based adaptive distributed observer over jointly connected networks and its application

The adaptive distributed observer for a leader system is a fully distributed dynamic compensator for estimating the state variables and the system matrix of a linear leader system over a communication network. In this paper, we first propose an event-triggered implementation of the output-based adaptive distributed observer for a linear leader system over jointly connected communication networks. Then we show that the proposed triggering mechanism not only retains the exponential convergence property of the analog adaptive distributed observer but also prevents the Zeno phenomenon from happening. Our result includes several existing results as special cases and is applicable when the state of the leader system is unavailable. Finally, we apply the proposed event-triggered output-based adaptive distributed observer to solve the cooperative linear output regulation problem. A numerical example is used to validate our design.

A Game Representation for a Finite Horizon State Constrained Continuous Time Linear Regulator Problem

A supremum-of-quadratics representation for a class of extended real valued barrier functions is developed and applied in the context of solving a continuous time linear regulator problem subject to a single state constraint of bounded norm. It is shown that this very simple state constrained regulator problem can be equivalently formulated as an unconstrained two-player game. By demonstrating equivalence of the upper and lower values, and exploiting existence and uniqueness of the optimal actions for both players, state feedback characterizations for the corresponding optimal policies for both players are developed. These characterizations are illustrated by a simple example.

Constrained linear output regulation via measurement output feedback model predictive control

Recently, the output regulation problem with input constraints for incrementally stable plants was solved by an error feedback model predictive control (MPC) algorithm. This paper further considers the case with both state and input constraints for linear systems and uses a measurement output feedback MPC algorithm to solve the problem. By relaxing two restrictions, the result of this paper can handle a larger class of linear systems than the existing result. Due to the uncontrollability of the augmented system composed of the plant and the exosystem, a key assumption in the literature fails, we manage to overcome this difficulty by introducing novel tightened constraints and the terminal constraint. As a result, the proposed algorithm solves the linear output regulation problem in the presence of both state and input constraints. The effectiveness of our design is illustrated by a numerical example.

Optimal Output Regulation for General Linear Systems via Adaptive Dynamic Programming.

In this article, we consider an adaptive optimal output regulation problem for general linear systems. The purpose of the optimal output regulation problem is to guarantee the stability of the closed-loop system and disturbance rejection, as well by minimizing some predefined performance indices. It can be realized by using an optimal controller, in which both optimal feedback control gain and optimal feedforward control gain are included. First, an adaptive dynamic programming (ADP) technique is used to solve the optimal feedback control gain. Next, the unknown system matrices of the plant are explicitly computed. In addition, based on the property of the minimal polynomial, the coefficient of the exogenous disturbance in the expression of the regulated output can also be calculated. Finally, according to the regulator equation, an extra cost function is given, which aims to obtain the optimal feedforward control gain. The linear vector space optimization methods are used to solve the optimal problem. As a result, the linear optimal output regulation problem can be solved by the approximately optimal feedback and feedforward control gains.

Parametric variational procedure to solve index-3 semi explicit linear-regulator control problem with constrained robot illustration

The aim of this work is to solve 2-dimensional, 3-links linearized constrained robot problem. This problem is of index-three linear descriptor control problem. Since the aim is to regulate its output, a quadratic performance index is proposed to form an index-3 linear regulator optimal control problem. Firstly, the problem has been transformed into standard constrained optimal control problem over class of consistent initial condition with solvability assumptions. Secondly, the solution of differential-algebraic equation has been shown as equivalent to find the critical points of some variational formulation and vise- versa. Then aggregating all functional (performance one) with consistent initial conditions in one performance (weighted) index. Under standard topology of the solvability, a parametric approach has been proposed to approximate the state-space solutions with control over the class of constraints. Hence, an infinite dimensional problem is transformed into finite one, and then an approximate solutions is obtained by solving the class of linear algebraic equations. The proposed approach is easily implemented and accurate up to decision maker selections of the parameterization basis functions and their numbers.

Scalable clock synchronization analysis: A symmetric noncooperative output feedback tubes-MPC approach

In the cyber-physical environment, the clock synchronization algorithm is required to have better expansion for network scale. In this paper, a new measurement model of observability under the equivalent transformation of minimum mean square error ( MMSE ) is constructed based on basic measurement unit ( BMU ) , which can realize the scaled expansion of MMSE measurement. Based on the state updating equation of absolute clock and the decoupled measurement model of MMSE-like equivalence, which is proposed to calculate the positive definite invariant set by using the theoretical-practical Luenberger observer as the synthetical observer, the local noncooperative optimal control problem is built, and the clock synchronization system driven by the ideal state of local clock can reach the exponential convergence for synchronization performance. Different from the problem of general linear system regulators, the state estimation error and state control error are analyzed in the established affine system based on the set-theory-in-control to achieve the quantification of state deviation caused by noise interference. Based on the BMU for isomorphic state map, the synchronization performance of clock states between multiple sets of representative nodes is evaluated, and the scale of evaluated system can be still expanded. After the synchronization is completed, the state of perturbation system remains in the maximum range of measurement accuracy, and the state of nominal system can be stabilized at the ideal state for local clock and realizes the exponential convergence of the clock synchronization system.

Gain Selection for Attitude Stabilization of Earth-Pointing Spacecraft Using Magnetorquers

This paper considers a feedback control law that achieves attitude stabilization for Earth-pointing spacecraft using only magnetorquers as torque actuators. The control law is proportional derivative (PD)-like with matrix gains, and it guarantees asymptotic stability. The PD matrix gains are determined through the numerical solution of a periodic linear quadric regulator problem. A case study shows the effectiveness of the considered control law, and specifically of the gain selection method, in a simplified simulation scenario.

Cooperative Output Regulation of Heterogeneous Multiagent Systems: A Global Distributed Control Synthesis Approach

Agent-wise local design methods to synthesize distributed control gains focus on the individual dynamics of each agent to guarantee the overall stability of the system. They are powerful tools due to their scalability. However, the agent-wise local design methods are incapable of maximizing the overall system performance through, for example, decay rate assignment. On the other hand, design methods, which are predicated on a global condition, leads to nonconvex optimization problems. This article considers a global design of an internal model-based distributed dynamic state feedback control law for the linear cooperative output regulation problem. We present a convex formulation of this global design problem based on a structured Lyapunov inequality. Then, the existence of solutions to the structured Lyapunov inequality is investigated. Specifically, we analytically show that the solutions exist for the systems satisfying the agent-wise local sufficient condition. Finally, we compare the proposed method with the agent-wise local design method through numerical examples in terms of conservatism, performance maximization, graph dependency, and scalability.

Robust Output Regulation of Linear Systems by Event-Triggered Dynamic Output Feedback Control

In this article, we study the robust output regulation problem of linear systems by an event-triggered output feedback control law. We first summarize the solvability of the linear robust output regulation problem by the internal model design. Then, we implement this continuous-time control law by an event-triggered output feedback control law together with an output-based triggering mechanism. We show that, under almost the same solvability conditions as those for the linear robust output regulation problem, the robust output regulation problem can also be solved by an event-triggered output feedback control law without incurring Zeno phenomenon. Furthermore, we show that, it is possible to solve the robust output regulation problem practically while preventing the interevent time from approaching zero as time tends to infinity.

Supervised Output Regulation via Iterative Learning Control for Rejecting Unknown Periodic Disturbances

The internal model principle (IMC) in linear robust output regulation theory states that a dynamical controller needs to incorporate a copy of the model generating the periodic signals in order to achieve perfect rejection/tracking, robustly with respect to plant’s parameters. On the other hand Iterative Learning Control (ILC) is a data-based approach which not requires any a priori knowledge, and can be used to find the required control action for attenuating periodic disturbances or tracking periodic references. The control signal generated by ILC includes the frequency and amplitude information of the disturbance and can be used to build the internal model needed for a linear output regulator problem. The objective of this work is therefore that of trying to combine the two approaches, that is IMC and ILC, in order to retain the advantages of each methodology. The proposed methodology, denoted as Supervised Output Regulation via Iterative Learning Control (SOR-ILC), allows to address the problem of output regulation in presence of unknown frequencies. The performances of SOR-ILC are validated through numerical simulations in case of complex periodic disturbances and parameter uncertainties.

A distributed control approach for heterogeneous linear multiagent systems

This paper considers an internal model-based distributed control approach to the cooperative output regulation problem of heterogeneous linear time-invariant multiagent systems over fixed directed communication graph topologies. First, a new definition of the linear cooperative output regulation problem is introduced in order to allow a broad class of functions to be tracked and rejected by a network of agents. Second, the solvability of this problem with three distributed control laws, namely dynamic state feedback, dynamic output feedback with local measurement and dynamic output feedback, is investigated by first considering a global condition and then providing an agent-wise local sufficient condition under standard assumptions. Finally, two numerical examples are provided to illustrate the selected contributions of this paper.

Infinite Horizon Optimal Output Feedback Control for Linear Systems with State Equality Constraints

This note deals with the infinite horizon linear regulation problem using output feedback with state equality constraints. Similar to the corresponding state feedback problem of a previous work, an existence condition for the output feedback gain and, if it exists, all constrainable output feedback gains are determined. However, different from the fore-mentioned state feedback case, only the necessary conditions for the optimal output feedback gain which minimizes the given standard cost function are determined. The performance of the developed algorithm is demonstrated using numerical simulations for a simple model of divert control system and the lateral dynamics of an F-16 aircraft.

An output-based distributed observer and its application to the cooperative linear output regulation problem

In this paper, we first extend an existing stability result for a class of linear switched systems. This extended result will relax the existence conditions of the output-based distributed observer for a leader system subject to jointly connected switching communication networks in the literature. As an application of this output-based distributed observer, we solve the cooperative output regulation problem of a linear multi-agent system subject to jointly connected switching communication networks by composing a purely decentralized control law and the output-based distributed observer based on the certainty equivalence principle.

Improving power system damping using a combination of optimal control theory and differential evolution algorithm

This paper presents a novel control strategy to improve the damping capability of sub-synchronous oscillations by tuning of Linear Quadratic Regulator (LQR) optimally in order to reduce the fluctuations in the power system. The proposed model includes the coordination of Power System Stabilizer (PSS) and Thyristor-Controlled Series Capacitors (TCSC) in combination with LQR controllers which is formulated as an optimal control problem. The problem is formulated as a linear regulator problem and then the Differential Evolution (DE) algorithm is utilized to optimize the proposed controlling parameters. Several cases have been studied to show the efficiency of the proposed strategy. Obtained results from a case study on a typical generator demonstrated that the proposed method has the best response and quickest function among conventional controller systems. Moreover, the coordination of the LQR with the other control systems, as an optimal linear regulator problem in order to damp system oscillations provides robust stability for optimizing system performance index. Besides, the flexibility and usability of the LQR controller guarantee the stability of the system to cope with the oscillations.

Linear output regulation with dynamic optimization for uncertain linear over-actuated systems

This paper considers the linear output regulation problem for uncertain over-actuated plants. The general form of input redundancy considered in this work implies the existence of multiple control inputs and state trajectories compatible with a prescribed reference for the output. On-line selection, according to certain performance criteria, of the most suitable of these inputs-state trajectories leads to a linear output regulation problem with dynamic redundancy allocation. We present a solution that augments the well known internal model control scheme with two additional dynamical systems. The first one, named annihilator, parametrizes the inputs and the corresponding state trajectories that are invisible from the output. The second one, named redundancy allocator, dynamically selects the best solution according to a predefined performance criterion. We derive explicit solutions for the performance criterion equal to relaxed 1, 2, and ∞- norms of the plant input. This set-up is a particular case of the dynamic redundancy allocation problem named dynamic input allocation. The proposed solutions can be implemented in an error feedback form and are especially suitable for optimizing sparsity, power and amplitude of the control input. Finally, structural stability, robustness and existence of a unique steady-state are proven.

Pitch and flight path controller design for F-16 aircraft using combination of LQR and EA techniques

In a linear regulator problem with a quadratic cost function, there are two weighting matrices which determine the feedback gain matrix, and thereby the closed loop eigenvalues and eigenvectors of the system. To find the weighting matrices leading to a desired eigenstructure, trial and error is a common approach. In spite of its simplicity, trial and error is too time-consuming and does not guarantee achieving expected results. In this regard, this paper presents a step-by-step algorithm for calculation of state and control weighting matrices in such a way that the pre-specified eigenstructure is generated with acceptable accuracy. Accordingly, once the designer chooses preferred eigenvalues and eigenvectors, the feedback gain matrix will be available uniquely. In design procedure, tracking of flight path and pitch attitude commands for an F-16 aircraft will be provided. To achieve this aim completely, the feedforward gain matrix necessary for getting a zero steady state error is calculated based on the mathematical model of the aircraft as well as the pseudoinverse matrix concept. Various simulations demonstrate that the resulted controller has good performance, gives the desired eigenstructure satisfactorily and also shows the expected robustness of a linear quadratic regulator.

A comparison of optimal policy rules prior to and during inflation targeting: empirical evidence from Bank of Canada

ABSTRACTWe examine policy rules that are consistent with inflation targeting (IT) framework in a small macroeconomic model of the Canadian economy. We set up an optimal linear regulator problem and derive policy rules to compare the dynamics of pre-IT and IT eras. We find that while the optimal monetary policy rule in the pre-IT period is best described with a loss function that attaches equal weight to price stability, financial stability and output stability; the IT era is dominated by the price stability objective followed by the financial stability and output stability, consecutively. Moreover, we do not find an explicit role for exchange rate stability in the objective function of the Bank of Canada for both monetary policy eras. We, then, compare the properties of the derived optimal rules with those of an ad hoc Taylor rule for the IT period. In response to inflationary shocks, Taylor rule brings down inflation rates more quickly compared to the derived policy rules, but at the cost of a higher sacrifice ratio and more volatile interest rates.

Application of Scaled Extended Conjugate Gradient Method in Solving Continuous-Time Linear Regulator Problems

Application of Scaled Extended Conjugate Gradient Method in Solving Continuous-Time Linear Regulator Problems

A Computational Method for Stochastic Optimal Control Problems in Financial Mathematics

AbstractPrinciple of optimality or dynamic programming leads to derivation of a partial differential equation (PDE) for solving optimal control problems, namely the Hamilton‐Jacobi‐Bellman (HJB) equation. In general, this equation cannot be solved analytically; thus many computing strategies have been developed for optimal control problems. Many problems in financial mathematics involve the solution of stochastic optimal control (SOC) problems. In this work, the variational iteration method (VIM) is applied for solving SOC problems. In fact, solutions for the value function and the corresponding optimal strategies are obtained numerically. We solve a stochastic linear regulator problem to investigate the applicability and simplicity of the presented method and prove its convergence. In particular, for Merton's portfolio selection model as a problem of portfolio optimization, the proposed numerical method is applied for the first time and its usefulness is demonstrated. For the nonlinear case, we investigate its convergence using Banach's fixed point theorem. The numerical results confirm the simplicity and efficiency of our method.