Static Output Feedback Problem Research Articles

Optimization Landscape of Policy Gradient Methods for Discrete-Time Static Output Feedback.

In recent times, significant advancements have been made in delving into the optimization landscape of policy gradient methods for achieving optimal control in linear time-invariant (LTI) systems. Compared with state-feedback control, output-feedback control is more prevalent since the underlying state of the system may not be fully observed in many practical settings. This article analyzes the optimization landscape inherent to policy gradient methods when applied to static output feedback (SOF) control in discrete-time LTI systems subject to quadratic cost. We begin by establishing crucial properties of the SOF cost, encompassing coercivity, L -smoothness, and M -Lipschitz continuous Hessian. Despite the absence of convexity, we leverage these properties to derive novel findings regarding convergence (and nearly dimension-free rate) to stationary points for three policy gradient methods, including the vanilla policy gradient method, the natural policy gradient method, and the Gauss-Newton method. Moreover, we provide proof that the vanilla policy gradient method exhibits linear convergence toward local minima when initialized near such minima. This article concludes by presenting numerical examples that validate our theoretical findings. These results not only characterize the performance of gradient descent for optimizing the SOF problem but also provide insights into the effectiveness of general policy gradient methods within the realm of reinforcement learning.

A Novel relaxation of the static output feedback problem for a class of plants

The static output feedback control problem is important, as it is concerned with the case when one cannot measure all state variables. It seeks to obtain a stabilising control under these conditions and is, often, difficult to solve. For a certain class of linear dynamical systems, we provide a novel approach to obtain a stabilising control gain matrix. The class of plants considered is of those, for which we can measure at least half of all state variables and where those measurements affect at least half of all state variables. Moreover, when we write the (linearly transformed) system matrix in block form, we require that either the two lower block matrices are nonsingular, or, if the inputs affect the measured state variables directly, at least the lower left one is nonsingular. We then show that we can determine the stabilising control gain matrix from the solution of a linear matrix inequality. Finally, we apply the approach to different benchmark problems, where it performs well, and confirm its good performance through additional numerical experiments.

A non-iterative approach to Linear Quadratic Static Output Feedback

This paper considers the problem of static output feedback (SOF) synthesis for linear time-invariant (LTI) systems. Static output feedback, and more generally structured controller synthesis, is of special interest to any industrial application where a reduced-order controller is desired, e.g., high-order systems, or a specific structure is to be imposed, i.e., distributed/decentralised control. A simple two-step process is proposed, solving one Riccati equation and one optimisation problem with linear matrix inequality (LMI) constraints, enabled via a dilation using the distance to the full-state feedback optimum gain. Numerical analysis shows considerable computational savings, with negligible differences in the performance, when compared to current iterative methods.

Finite-Time Static Output-Feedback H∞ Control for Discrete-Time Singular Markov Jump Systems Based on Event-Triggered Scheme

The problem of finite-time static output feedback H∞ control for a class of discrete-time singular Markov jump systems is studied in this paper. With the consideration of network transmission delay and event-triggered schemes, a closed-loop model of a discrete-time singular Markov jump system is established under the static output feedback control law, and the corresponding sufficient condition is given to guarantee this system will be regular, causal, finite-time bounded and satisfy the given H∞ performance. Based on the matrix decomposition algorithm, the output feedback controller can be reduced to a feasible solution of a set of strict matrix inequalities. A numerical example is presented to show the effectiveness of the presented method.

Second-Order NonConvex Optimization for Constrained Fixed-Structure Static Output Feedback Controller Synthesis

For linear time-invariant systems, the design of an optimal controller is a commonly encountered problem in many applications. Among all the optimization approaches available, the linear quadratic regulator (LQR) methodology certainly garners much attention and interest. As is well known, standard numerical tools in linear algebra are readily available to determine the optimal static LQR feedback gain matrix when all the system state variables are measurable. However, in various certain scenarios where some of the system state variables are not measurable, and consequent prescribed structural constraints on the controller structure arise, the optimization problem can become intractable due to the nonconvexity characteristics that can then be present. In such cases, some first-order methods have been proposed to cater to these problems, but all of these methods, if at all successful, are limited to linear convergence. To speed up the convergence, a second-order approach in the matrix space is essential, with appropriate methodology to solve the linear equality constrained static output feedback (SOF) problem with a suitably defined linear quadratic cost function. Thus, in this article, an efficient method is proposed in the matrix space to calculate the Hessian matrix by solving several Lyapunov equations. Then, a new optimization technique is applied to deal with the indefiniteness of the Hessian matrix. Subsequently, through Newton’s method with linear equality constraints, a second-order optimization algorithm is developed to effectively solve the constrained SOF LQR problem. Finally, two numerical examples are described, which demonstrate the applicability and effectiveness of the proposed method.

Cluster consensus for leader-following high-order linear multiagent systems

<p indent="0mm">Cluster consensus for leader-following high-order linear multiagent systems (MASs) is investigated. Based on the information of an agent and its neighbors, state feedback and output feedback control protocols are designed for each follower, respectively. Sufficient conditions for leader-following cluster consensus are given using tools from the algebraic Riccati equation and matrix analysis. It is shown that if the communication topology of an MAS composed of the leader and followers is interactively balanced and contains a spanning tree, the MAS can achieve leader-following cluster consensus under the proposed control protocols by properly selecting control gains. Particularly, it is found that the leader-following cluster consensus problem based on the output feedback protocol can be transformed into a static output feedback problem using an error system. If the input and output matrices of a follower satisfy a certain rank condition, the MAS can achieve leader-following cluster consensus under the designed output feedback control protocol.

Static output feedback negative imaginary controller synthesis with an [formula omitted] norm bound

A large number of practical systems can be modeled as negative imaginary (NI) systems. In this paper, we investigate the problem of static output feedback (SOF) NI controller design with H∞ performance. Motivated by the two-stage framework for SOF stabilization, a sufficient condition is derived for the SOF controller synthesis. An iterative LMI-based algorithm is suggested to find an SOF controller. The algorithm consists of two stages: (1) finding an initial output injection controller; (2) computing a desired SOF controller. In addition, a numerical example is provided to verify the proposed method.

Optimizing Static Linear Feedback: Gradient Method

The linear quadratic regulator is the fundamental problem of optimal control. Its state feedback version was set and solved in the early 1960s. However, the static output feedback problem has no ex...

Designing a Generalized PIO for Polytropic Discrete Time Systems

Considering the Linear Parameter Varying (LPV) systems of discrete time, in this paper an approach for the synthesis of robust Proporcional+Integral Observers (PIO) is presented. From LPV systems characterized with polytopical uncertainties, the method of design is based on considering a dynamics extended of the typical PIO, in order to transform the design of the matrices of the dynamics of the observer, as a design of the gain of Static Output Feedback (SOF) of a problem of robust control. Under these conditions and from the norms H2=H1 described as Linear Matrix Inequalities (LMI), the criteria to obtain the gain in the SOF problem are established; taking into account performance indices in H2 and H1, under the presence of uncertainties and disturbances.

Static output feedback control of uncertain systems over uncertain communication links with guaranteed H∞ performance

AbstractThis article is concerned with the problem of robust static output feedback H∞ control for networked control systems (NCSs) by considering uncertainties in both controlled‐plant and communication subsystems. Markov chains are employed to model the bounded random network‐induced delays occurring in the communication channels. This modeling framework can be used to describe the packet dropout phenomena over a lossy communication channel as well. It is assumed that the transition probabilities of the Markov chains are uncertain but lying within given intervals. By utilizing the augmentation technique, the NCS model is transformed into a delay‐free Markovian jump linear system. A new sufficient condition is derived for the problem of robust H∞ control using static output feedback in NCSs with network‐induced delays. This design condition is expressed in terms of linear matrix inequalities without requiring any constraints on the matrix variables. Furthermore, the proposed method is extended to the case of NCSs with bounded Markovian packet dropouts. Numerical examples are provided to make comparisons with other works and demonstrate the advantages of the proposed method.

On LMI conditions to design robust static output feedback controller for continuous-time linear systems subject to norm-bounded uncertainties

This paper addresses the problem of Static Output Feedback (SOF) stabilisation for continuous-time linear systems subject to norm-bounded parameter uncertainties using the Linear Matrix Inequality (LMI) approach. Usually, this issue leads to the feasibility of a Bilinear Matrix Inequality (BMI), which is difficult to linearise to get non-conservative LMI conditions. We present first, in this paper, some background results on the SOF controller design and that are found to be extended to the case of norm-bounded uncertainties. We show that some restrictions on the feasibility of these results should be guaranteed. Moreover, by means of some technical Lemmas, we transform the BMI into a new LMI with a line search over a scalar variable. An enhanced and less conservative LMI condition with a line search over two scalar variables is also developed. Furthermore, a simplified version of each LMI condition without a priori fixed parameters is also presented. An extensive portfolio of numerical examples is presented in order to evaluate the conservativeness and to show the superiority of the proposed design method to the background results.

Robust Static Output Feedback H2/H∞ Control Synthesis with Pole Placement Constraints: An LMI Approach

This paper studies the robust static output feedback (SOF) problem considering pole placement constraints for linear systems with polytopic uncertainty as well as linear parameter varying (LPV) systems. New linear matrix inequality (LMI) approaches are proposed for the SOF controller design while the pole placement, H2, and H∞ constraints are guaranteed. In addition, the gain-scheduled SOF controller will be designed for LPV systems if system parameters are measured. The proposed methods can be applied to general linear systems without imposing any constraints on system matrices. The performance and effectiveness of the proposed methods are shown using two examples.

Eigenvalue Placement by Quantifier Elimination - the Static Output Feedback Problem

This contribution deals with the static output feedback problem of linear time-invariant systems. This is still an area of active research, in contrast to the observer-based state feedback problem, which has been solved decades ago. We consider the formulation and solution of static output feedback design problems using quantifier elimination techniques. Stabilization as well as more specified eigenvalue placement scenarios are the focus of the paper.

Robust Gain-Scheduled PID Control: A Parameter Dependent BMI Solution

Abstract In control practices, problems of parametric or time-varying uncertainties must be dealt with. Robust control based on norm theory and convex and non-convex optimization algorithms is a powerful tool to solve these problems in theory, but it is employed rarely in applications. In most engineering cases, Proportional-Integration-Derivative (PID) control is still the most popular method for its easy-to-tune and controllable properties. The control method proposed in this paper integrates the PID control into robust control formulation as a robust Structured Static Output Feedback (SSOF) problem of Linear-Parameter-Varying (LPV) systems, which can be converted into a Parameter Dependent Bilinear-Matrix-Inequality (PDBMI) optimization problem. A convex-concave decomposition based method is given to solve the proposed PDBMI problem. The proposed solution has a simple structure in PID form and can guarantee stability and robustness of the system being controlled in the whole operation range with less conservativeness than existing solution.

Minimal-norm static feedbacks using dissipative Hamiltonian matrices

In this paper, we characterize the set of static-state feedbacks that stabilize a given continuous linear-time invariant system pair using dissipative Hamiltonian matrices. This characterization results in a parametrization of feedbacks in terms of skew-symmetric and symmetric positive semidefinite matrices, and leads to a semidefinite program that computes a static-state stabilizing feedback. This characterization also allows us to propose an algorithm that computes minimal-norm static feedbacks. The theoretical results extend to the static-output feedback (SOF) problem, and we also propose an algorithm to compute the minimal-norm SOF. We illustrate the effectiveness of our algorithm compared to state-of-the-art methods for the SOF problem on numerous numerical examples from the COMPLeIB library.

Finite-Frequency Fuzzy Output Feedback Controller Design for Roesser-Type Two-Dimensional Nonlinear Systems

This article studies the problem of finite-frequency static output feedback (SOF) \mathscr H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> controller design for discrete-time Roesser-type two-dimensional (2-D) nonlinear systems based on Takagi-Sugeno (T-S) fuzzy models. The 2-D Roesser nonlinear systems are described by T-S fuzzy models with parameter uncertainties. The objective is to design a SOF controller guaranteeing the asymptotic stability of the resulting closed-loop system with finite frequency \mathscr H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance. Via a system state-input augmentation technique, the closed-loop system is formulated in a descriptor form. Then, based on fuzzy Lyapunov functions and some elegant convexification procedures, the SOF controller design approach is proposed. It is shown that the controller gains can be obtained by solving a set of linear matrix inequalities. Finally, simulation studies are given to demonstrate the effectiveness of the proposed method.

Robust Static Output Feedback Stabilization of Continuous-Time Linear Systems via Enhanced LMI Conditions

This paper addresses the problem of Static Output Feedback (SOF) stabilization for continuous-time linear systems subject to norm-bounded parameter uncertainties. Usually this issue leads to the feasibility of a Bilinear Matrix Inequality (BMI), which is difficult to linearize to get non conservative Linear matrix inequality (LMI) conditions. In this paper, by means of some technical lemmas, we transform the BMI into a new LMI with a line search over two scalar variables. The obtained LMI conditions are less conservative than those existing in the literature. Numerical evaluations are presented to show the superiority of the proposed method.

Design of Robust PID Controller using Static Output Feedback framework

This paper considers the design of a robust PID controller for higher order MIMO plants. The design problem is first recast into a Static Output Feedback (SOF) controller design problem and then the transformed SOF problem is solved within the framework of Linear Matrix Inequalities (LMIs) through a decomposition of the Lyapunov matrix variable. Sufficient LMI criteria are derived that ensure H∞ performance of the underlying system. By means of a numerical example, it is shown that the designed controller yields less conservative results. Also, a comparative study is done with the existing techniques to demonstrate the efficacy of the proposed method.

Asynchronous Resilient Output Consensus of Switched Heterogeneous Linear Multivehicle Systems With Communication Delay

The resilient output consensus problem for a network of switched heterogeneous linear vehicle systems with asynchronous switching, controller gain variation, unknown disturbance, and communication delay is investigated. Based on the topology decoupling technique, the investigated problem is transformed into two subproblems: one is a robust asynchronous static output feedback (SOF) control problem and the other is an asynchronous SOF problem. Based on the piecewise Lyapunov functional method and the average dwell time switching scheme, some new sufficient conditions are proposed for the solutions to those subproblems. Moreover, some novel linear-matrix-inequality-based conditions are developed to calculate the SOF controller gains. A network of six vehicles is finally presented to show the effectiveness of the proposed resilient consensus controller.

Static output feedback control of positive linear systems with output time delays

ABSTRACTIn this paper, we discuss a new problem of static output feedback controllers design for positive systems with delayed output measurements. When delayed output measurements are exclusively used as feedback control signals, previous control design methods for positive systems relying on the so-called delay-independent positivity and stability conditions are unable to synthesise a stable static output feedback controller for positive systems. A new method based on delay-dependent positivity and stability conditions is proposed in this paper to tackle this issue. We show that the synthesis of static output feedback controllers for positive systems under the effect of delayed output measurements is feasible under the newly proposed design method. Numerical examples are given to demonstrate the effectiveness of the new result.